aa r X i v : . [ m a t h . G T ] J u l On Rho invariants of fiber bundles
Michael Bohn
Department of Mathematics, Bonn UniversityEndenicher Allee 60, D-53115 Bonn, Germany e-mail: [email protected] under http://hss.ulb.uni-bonn.de/diss_online/
Bonn, 2009 ummary
The content of this thesis is a detailed investigation of Rho invariants of the total spaces of fiberbundles. The main idea is to use adiabatic limits of Eta invariants to obtain a formula for Rhoinvariants that separates the contribution coming from the fiber and the one coming from thebase. An adiabatic metric on a fiber bundle rescales the metric of the base manifold in such away that the geometry of the fiber bundle approaches a product situation. Concerning the Etainvariant, this process has received a far-reaching treatment in the literature. For this reason,one concern of this thesis is to formulate the technical aspects of local index theory for familiesof Dirac operator in terms of the odd signature operator, and place known results in a contextwhich permits the treatment of Rho invariants.The resulting formula expresses the Rho invariant as a sum of three terms, each of which is ofa very different nature. First of all, a higher dimensional analog of the Rho invariants of the fiberhas to be integrated over the base. This term is of a local nature on the base, but contains globalspectral information about the fiber. The next term is essentially a Rho invariant of the base,where the underlying flat connection is defined on the bundle of cohomology groups of the fiber.Lastly, there is a purely topological term, which can be computed from the spectral sequence ofthe fiber bundle. Together, this formula casts the Rho invariant of the total space into a formwhich incorporates the structure of the fiber bundle in a satisfactory way.The main concern of this thesis is, however, to use this theoretical formula to compute Rhoinvariants for explicit classes of fibered 3-manifolds. More precisely, we consider principal S -bundles over closed, oriented surfaces as well as mapping tori with fiber a closed, oriented surface.For the first class of examples, one can compute U(1)-Rho invariants without using this generalformula. In particular, this yields the opportunity to compare the different approaches and testthe systematical advantage of the general formula.For 3-dimensional mapping tori, the presented theory can also be used for explicit compu-tations. We first consider the case that the monodromy map is of finite order. In this case,a general formula for Rho invariants can be derived. To investigate a further interesting classof mapping tori, we consider U(1)-Rho invariants in the case that the fiber is a 2-dimensionaltorus. Here, hyperbolic monodromy maps deserve particular attention. When discussing them,the logarithm of a generalized Dedekind Eta function naturally appears. A satisfactory formulafor U(1)-Rho invariants of hyperbolic mapping tori can then be deduced from a transformationformula for these Eta functions. i ontents Summary iIntroduction vAcknowledgements xii1 The Signature Operator and the Rho Invariant 1 S -Bundles over Surfaces . . . . . . . . . . . . . . . . 67ii ontents iii2.3.1 The U(1)-Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.3.2 The Odd Signature Operator. . . . . . . . . . . . . . . . . . . . . . . . . . 732.3.3 The Eta Invariant of the Truncated Odd Signature Operator. . . . . . . . 782.3.4 Adiabatic Metrics and the Spectral Flow . . . . . . . . . . . . . . . . . . . 81 p, q )-Connections . . . . . . . . . . . . . . . . . . . . 973.2 Elements of Bismut’s Local Index Theory for Families . . . . . . . . . . . . . . . 1003.2.1 The Index Theorem for Families . . . . . . . . . . . . . . . . . . . . . . . 1003.2.2 Superconnections and Associated Dirac Operators . . . . . . . . . . . . . 1013.2.3 The Families Index Theorem for the Signature Operator . . . . . . . . . . 1033.3 A General Formula for Rho Invariants . . . . . . . . . . . . . . . . . . . . . . . . 1083.3.1 Transgression and Adiabatic Limits . . . . . . . . . . . . . . . . . . . . . 1083.3.2 Dai’s Adiabatic Limit Formula . . . . . . . . . . . . . . . . . . . . . . . . 1123.3.3 Small Eigenvalues and the Leray Spectral Sequence . . . . . . . . . . . . . 1163.4 Circle Bundles Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 S , General Setup . . . . . . . . . . . . . . . . . . . . . . . . 1374.3.1 Geometry of Torus Bundles over S . . . . . . . . . . . . . . . . . . . . . 1374.3.2 The Bismut-Cheeger Eta Form . . . . . . . . . . . . . . . . . . . . . . . . 1444.4 Torus Bundles over S , Explicit Computations . . . . . . . . . . . . . . . . . . . 1524.4.1 The Elliptic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.4.2 The Parabolic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584.4.3 The Hyperbolic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 AppendixA Characteristic Classes and Chern-Simons Forms 177
A.1 Chern-Weil Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177A.1.1 Connections and Characteristic Forms . . . . . . . . . . . . . . . . . . . . 177A.1.2 Transgression and Characteristic Classes . . . . . . . . . . . . . . . . . . . 182A.2 Chern-Simons Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
B Remarks on Moduli Spaces 189
B.1 The Moduli Space of Flat Connections . . . . . . . . . . . . . . . . . . . . . . . . 189B.1.1 Flat Connections and Representations of the Fundamental Group . . . . . 189B.1.2 Flatness and Triviality of U( k )-Bundles . . . . . . . . . . . . . . . . . . . 196v Introduction
B.2 Flat Connections over Mapping Tori . . . . . . . . . . . . . . . . . . . . . . . . . 198B.2.1 Algebraic Description of the Moduli Space . . . . . . . . . . . . . . . . . . 200B.2.2 Geometric Description of the Moduli Space . . . . . . . . . . . . . . . . . 202B.3 Holomorphic Line Bundles over Riemann Surfaces. . . . . . . . . . . . . . . . . . 208
C Some Computations 215
C.1 Values of Zeta and Eta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 215C.1.1 The Gamma and the Hurwitz Zeta Function . . . . . . . . . . . . . . . . 215C.1.2 An Eta Function and a “Periodic” Zeta Function . . . . . . . . . . . . . . 217C.2 Generalized Dedekind Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218C.2.1 Some Finite Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 218C.2.2 Relation Among Some Dedekind Sums . . . . . . . . . . . . . . . . . . . . 222
D Local Variation of the Eta Invariant 227
D.1 More Results on the Heat Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 227D.1.1 Expression via the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . 227D.1.2 Perturbed Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231D.1.3 Variation of the Heat Operator . . . . . . . . . . . . . . . . . . . . . . . . 234D.2 Parameter Dependent Eta Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 237D.2.1 Large Time Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237D.2.2 Small Times and Meromorphic Extension . . . . . . . . . . . . . . . . . . 239D.2.3 The Case of Varying Kernel Dimension . . . . . . . . . . . . . . . . . . . 241
Bibliography 250 ntroduction
The Rho Invariant for Closed Manifolds.
In their famous series of articles [7, 8, 9], Atiyah,Patodi and Singer established an index theorem for manifolds with boundary. Part of theirmotivation was to find a generalization of Hirzebruch’s Signature Theorem to manifolds withboundary and give a differential geometric explanation for the signature defect.We recall this briefly. Let W be a closed, oriented 4-manifold, and let Sign( W ) be its signature.Then the Hirzebruch’s Signature Theorem states thatSign( W ) = 13 Z W p ( T W ) . (1)Here, p ( T W ) is the first Pontrjagin form, and since W is closed, it is a characteristic classindependent of the connection used to compute it. Let us now assume that W has a boundary M . If α : π ( W ) → U( k ) is a unitary representation of the fundamental group, one defines atwisted signature Sign α ( W ) using cohomology groups with local coefficients. Then an applicationof the signature formula (1) and its twisted version to the closed double W ∪ M − W shows thatthe difference Sign α ( W ) − k · Sign( W )depends only on the topology of the boundary ∂W = M as well as the restriction of α to π ( M ). This signature defect is in general non-trivial. Moreover, the signature itself fails to bemultiplicative under finite coverings of manifolds with boundary. Both observations show thatthe signature of a manifold with boundary is in general not expressible as in (1).The Atiyah-Patodi-Singer Index Theorem for manifolds with boundary identifies the correc-tion term in great generality. For a formally self-adjoint elliptic differential operator D of firstorder, acting on sections of a vector bundle over a closed manifold M , one defines the Eta function η ( D, s ) := X = λ ∈ spec( D ) sgn( λ ) | λ | s , Re( s ) large . (2)The function η ( D, s ) admits a meromorphic continuation to the whole s -plane, and it is a remark-able fact that s = 0 is not a pole. The Eta invariant η ( D ) is defined as this finite value. Thena special case of the Atiyah-Patodi-Singer Index Theorem for a compact, oriented 4-manifold W with boundary M is Sign α ( W ) = k Z W p ( T W, ∇ g ) − η ( B ev A ) . (3)Here, p ( T W, ∇ g ) is the first Pontrjagin form, computed with respect to a metric g in productform near the boundary, A is a flat U( k )-connection over M whose holonomy coincides withvi Introduction α | π ( M ) , and—most importantly— B ev A is the odd signature operator on M . It is defined ondifferential forms of even degree with values in the flat bundle E as B ev A ω = ( − p ( ∗ d A − d A ∗ ) ω, ω ∈ Ω p ( M, E ) . A simple consequence of (3) is thatSign α ( W ) − k · Sign( W ) = η ( B ev A ) − k · η ( B ev ) . (4)The right hand side of (4) is called the Rho invariant ρ A ( M ). It is defined for every odddimensional manifold M and flat unitary connection A over M , without any reference to abounding manifold. Moreover, ρ A ( M ) turns out to be independent of the choice of the metricused in defining the involved odd signature operators. Therefore, it is an intrinsically definedsmooth invariant of M , which extends the signature defect.Since the Eta invariants appearing in the definition of ρ A ( M ) are non-local spectral invari-ants, it is difficult to compute Rho invariants directly without using property (4). However,one cannot always find a bounding manifold in such a way that the flat connection extends.Therefore, intrinsic methods to compute Rho invariants are in demand. The concern of thisthesis is to investigate an intrinsic approach to this problem in the case that the manifold M isthe total space of an oriented fiber bundle of closed manifolds. Adiabatic Limits of Eta Invariants.
In a remarkable paper of Witten [97], Eta invariantsappeared in the interpretation of anomalies in physics. Associated to a family of Dirac operatorsis a determinant line bundle over the parameter space, first described by in [83] by Quillen. Itcomes equipped with a natural connection, defined in terms of the Ray-Singer analytic torsion[87]. The topology of this line bundle encodes the obstruction to defining in a consistent way aregularized determinant associated to the family of Dirac operators. In the physicists terminology,there is no “local anomaly” if Quillen’s connection on the determinant bundle is flat. However,the bundle might still not be trivial, and this “global anomaly” is encoded in the holonomy ofQuillen’s connection. Witten suggests an interpretation of this holonomy using the Eta invariant.For example, a family of spin Dirac operators is naturally associated to a fiber bundle over theparameter space B whose fiber F is a closed spin manifold. Here, we also restrict to the casethat F is even dimensional. Pulling back this structure using a closed loop c : S → B leads toa fiber bundle F ֒ → M → S . Now Witten considers an adiabatic metric , that is, a family ofsubmersion metrics of the form g ε = g S ε ⊕ g v , (5)where g v is a metric on the vertical tangent bundle of M . Then, if D ε denotes the Dirac operatorassociated to (5) on the total space M , Witten suggests that the holonomy of Quillen’s connectionaround the loop c : S → B is given bylim ε → exp (cid:0) πiη ( D ε ) (cid:1) . The mathematical treatment of this holonomy theorem is due to Bismut-Freed [18] and Cheeger[26].Motivated by this geometric interpretation, Bismut and Cheeger [16] gave a formula for the adiabatic limit of the Eta invariant, and generalized it to fiber bundles
F ֒ → M → B with ntroduction viihigher dimensional base spaces. Using ideas of Bismut’s local index theory for families [14],they construct a differential form b η on B , whose value at each point x ∈ B depends only onglobal information of the fiber over x . Under the assumption that the fiberwise Dirac operatoris invertible, they prove that lim ε → η ( D ε ) = Z B b A ( T B, ∇ g B ) b η, (6)where A ( T B, ∇ g B ) is the Hirzebruch b A -form of B . Very roughly, a consequence of (6) is thatthe adiabatic limit of the Eta invariant is a simpler object than the Eta invariant itself, since itis local on the base. The main matter of this thesis is to analyze in which way this effect can beused to simplify explicit computations of Rho invariants of fiber bundles. Dai’s Adiabatic Limit Formula.
The Rho invariants we are considering are associated tothe odd signature operator. Here, the kernel of the vertical operator forms a vector bundle H • v ( M ) → B whose fiber over each point is isomorphic to the cohomology of F . Therefore, theinvertibility hypothesis leading to (6) is too restrictive. Fortunately, the result of Bismut andCheeger has been generalized by Dai [30] to a setting which applies in particular to the case we areinterested in. The bundle H • v ( M ) → B of vertical cohomology groups is endowed with a naturalflat connection ∇ H v . Using this, one associates a twisted odd signature operator D B ⊗ ∇ H v overthe base. Then Dai proves the following, very remarkable adiabatic limit formula. Theorem 1 (Dai) . Let
F ֒ → M → B be an oriented fiber bundle of closed manifolds withodd dimensional total space, endowed with a submersion metric. Let B ev ε be the family of oddsignature operators on M associated to an adiabatic metric. Then lim ε → η ( B ev ε ) = 2 [ b +12 ] Z B b L ( T B, ∇ g B ) ∧ b η + η (cid:0) D B ⊗ ∇ H v (cid:1) + σ. Here, b is the dimension of B , and the differential forms b L ( T B, ∇ B ) and b η are the Hirzebruch b L -form and the Eta form of Bismut-Cheeger, respectively. Moreover, σ is a topological invariantcomputed from the Leray-Serre spectral sequence. We will give more details on the terms appearing here in the main body of this thesis.However, we already want to stress that η (cid:0) D B ⊗ ∇ H v (cid:1) and σ are of a very different nature thanthe integral of the Eta form. Where the latter is local on the base and contains spectral infor-mation about the fiber, the twisted Eta term is a spectral invariant of the base which containscohomological information of the fiber. Moreover, since σ arises from the Leray-Serre spectralsequence it is purely cohomological. In this respect, Theorem 1 is a very satisfactory decom-position of the adiabatic limit into contributions coming from the base and the fiber, respectively. Concern of this Thesis.
Since the treatment in [30] is more general than what we have stated inTheorem 1, Dai’s adiabatic limit formula continues to hold for the odd signature operator twistedby a flat connection A over M . We will see that there are natural analogs b η A , D B ⊗ ∇ H A,v and σ A of the quantities appearing in Theorem 1. As the Rho invariant ρ A ( M ) is independent of themetric, it is immediate, that with respect to every adiabatic metric, ρ A ( M ) = lim ε → η ( B ev A,ε ) − k · lim ε → η ( B ev ε ) . Then, Theorem 1 yieldsiii
Introduction
Theorem 2.
Let A be a flat U( k ) -connection over M . Then with respect to every submersionmetric ρ A ( M ) = 2 [ b +12 ] Z B b L ( T B, ∇ B ) ∧ (cid:0)b η A − k · b η (cid:1) + η (cid:0) D B ⊗ ∇ H A,v (cid:1) − k η (cid:0) D B ⊗ ∇ H v (cid:1) + σ A − k · σ. Now, the main matter of this thesis is to investigate how this rather straightforward con-sequence of Theorem 1 can be used for explicit computations of Rho invariants. Due to thetechnical nature of local families index theory, our first concern is to assemble the building blockswe need, and specialize many known results to the case of the odd signature operator. The moti-vation here is certainly not to exhibit new results, but to present the theory in such a way that itbecomes accessible for a treatment of Rho invariants of fiber bundles. Therefore, our perspectivewill usually be a geometric one rather than discussing analytical difficulties. For a discussion ofthese aspects, we usually refer to the wide variety of literature. Nevertheless, we include proofsof some folklore results, for instance a fibered version of the Hodge decomposition theorem, anda result about how to achieve that the mean curvature of a fiber bundle vanishes.Apart from the theoretical discussion, our true focus is on explicit examples. We will examinetwo important classes of fibered 3-manifolds in detail.
Circle Bundles over Surfaces.
The simplest class of fiber bundles for which a discussion of Rhoinvariants is meaningful, is given by principal S -bundles over Riemann surfaces. Nevertheless,this family of manifolds already deserves some attention as it is a model for two important classesof manifolds, namely 3-dimensional Seifert fibrations and higher dimensional principal bundles.In this spirit, Nicolaescu [78, 79] has analyzed the Seiberg-Witten equations of Seifert man-ifolds, and parts of our discussion are influenced by his work. Given a closed, oriented surfaceΣ, and an oriented principal bundle S ֒ → M → Σ, we will see that we can represent every flatU(1)-connection A over M by pulling back a line bundle of degree k over Σ. In terms of thisdata, the Rho invariant associated to A is given as follows, see Theorem 2.3.18. Theorem 3.
Assume that l = 0 , and that q ∈ [0 , is such that k/l ≡ q mod Z . Then ρ A ( M ) = 2 l ( q − q ) + sgn( l ) . If l = 0 , so that the fiber bundle is trivial, all Rho invariants vanish. We shall include two proofs of this result. The first one in Chapter 2 uses only basic con-siderations about the geometry of fiber bundles, and the second one in Chapter 3 shows howTheorem 2 can be used for this class of examples.
The second family of manifolds we will consider are fiberbundles Σ ֒ → M → S , where Σ is again a closed, oriented surface. A manifold of this type isdetermined by an element f of the mapping class group of Σ. Due to the rich algebraic structureencoded in the latter, we have not attempted to treat the class of 3-dimensional mapping tori infull generality.What we shall do instead, is to assume first that the monodromy f of the mapping torus M is of finite order. Under this assumption, the formula of Theorem 2 for the Rho invariant of aflat U( k )-connection A over M reduces to ρ A ( M ) = η (cid:0) D B ⊗ ∇ H A,v (cid:1) − k η (cid:0) D B ⊗ ∇ H v (cid:1) . ntroduction ixIn Theorem 4.2.4 we shall give an expression of the right hand side in terms of Hodge-de-Rhamcohomology of Σ, thus obtaining a cohomological formula for the Rho invariant. Since the precisestatement would need a longer preamble, we refer Chapter 4 for details.After this we will consider U(1)-Rho invariants of a mapping torus T M , where the fiber is the2-dimensional torus T , and M ∈ SL ( Z )—the mapping class group of T . One naturally has todistinguish between three cases, depending of whether M is elliptic, parabolic or hyperbolic. Thefirst two cases are rather special, and we shall not discuss them here. The explicit formulæ forthe corresponding Rho invariants can be found in Theorem 4.4.4 and Theorem 4.4.8, respectively.The case of a hyperbolic monodromy matrix requires more background material. Here wewill use ideas of Atiyah [3], who gives a far-reaching treatment of the untwisted Eta invariant formapping tori with fiber T . In particular, he uses the relation to Hirzebruch’s signature defect toshow that for a hyperbolic element M = (cid:0) a bc d (cid:1) ∈ SL ( Z ), the Eta invariant of the odd signatureoperator with respect to a natural metric on T M , is given by η ( B ev ) = a + d c − c ) s ( a, c ) − sgn (cid:0) c ( a + d ) (cid:1) . (7)Here, s ( a, c ) is the Dedekind sum s ( a, c ) = | c |− X k =1 P (cid:0) akc (cid:1) P (cid:0) kc (cid:1) , (8)where for x ∈ R , P ( x ) = ( , if x ∈ Z ,x − [ x ] − , if x / ∈ Z .Atiyah also relates the Eta invariant to the classical Dedekind Eta function , which is defined fora point σ in the upper half plane as η ( σ ) := q σ ∞ Y n =1 (1 − q nσ ) , q σ := e πiσ . One can define a natural logarithm of η , and the transformation property of log η under theaction of elements of SL ( Z ) has a long history, going back to Dedekind [34]. For an element M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) with c = 0, this transformation formula states thatlog η ( M σ ) − log η ( σ ) = 12 log (cid:16) cσ + d sgn( c ) i (cid:17) + πi (cid:16) a + d c − sgn( c ) s ( a, c ) (cid:17) . (9)Here, the logarithm on the right hand side is the standard branch on C \ R − , and s ( a, c ) is theDedekind sum (8). Atiyah’s explanation of the relation between (9) and the formula (7) makesessential use of the idea of taking the adiabatic limit.Motivated by this, we will study in detail the expression R S b η A appearing in Theorem 2 forthe case of a flat U(1)-connection over a hyperbolic mapping torus over S . Using ideas related toKronecker’s second limit formula, we will cast it into a form, where the logarithm of a generalizedDedekind Eta function naturally appears. We will then employ a transformation formula dueto Dieter [35], to obtain the analog of (9) for the logarithm of this generalized Dedekind Etafunction. From this we shall deduce the main result of Chapter 4, see Theorem 4.4.20. Introduction
Theorem 4.
Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) be hyperbolic, and let ( ν , ν ) ∈ R \ Z with ν ∈ [0 , satisfy (cid:18) m m (cid:19) = (Id − M t ) (cid:18) ν ν (cid:19) ∈ Z . This defines a flat connection A over T M , and ρ A ( T M ) = a + d ) − c ( ν − ν ) − | c |− r X k =1 P (cid:0) dkc (cid:1) + sgn (cid:0) c ( a + d ) (cid:1) − sgn( c ) δ ( ν ) (cid:0) − δ ( m c ) (cid:1) − P (cid:0) dm c (cid:1) − δ ( ν ) (cid:16) P (cid:0) m c (cid:1) − P (cid:0) dm c (cid:1)(cid:17) , where r ∈ { , . . . | c | − } is such that m ≡ r mod c , and δ is the characteristic function of R \ Z . Although this formula might appear to be somewhat involved, it is satisfactory in two ways.First of all, the involved terms are easy to compute for explicit choices of M = (cid:0) a bc d (cid:1) and ( ν , ν ).Secondly, we shall see that it contains previous computations of Chern-Simons invariants byFreed and Vafa [42] as a special case. In this respect, the author hopes that a possible general-ization to SU(2)-connections will reprove results of Kirk and Klassen [57] and might shed a newlight on Jeffrey’s conjecture [54] concerning the spectral flow associated to twisted odd signatureoperators on a mapping torus of the form considered here. Outline of this Thesis.
We end the introduction with a very brief outline of this thesis. Wewill keep this rather short since the beginning of each chapter contains a more detailed outlineof its contents. • Chapter 1 is a survey of results from index theory that we will need. In particular, we shallintroduce the signature of a manifold, discuss its relation to index theory, and recall theAtiyah-Singer Index Theorem in its cohomological version for geometric Dirac operators.Then we introduce the Eta and Rho invariant, and discuss how they appear in the indextheorem for manifolds with boundary. We also place some emphasis on variation formulæand sketch how they are related to local index theory. • In Chapter 2 we start with the discussion of fiber bundles. We will introduce the geometricsetup, paying close attention to the structure of the odd signature operator. Then, we shallencounter the basic idea of adiabatic limits and use this to give an elementary proof ofTheorem 3 above. • Chapter 3 contains the main theoretical part of this thesis. Here, the main objective is tointroduce all quantities appearing in Theorem 1. After discussing the bundle of verticalcohomology groups in some detail, we will give a heuristic derivation of Theorem 1. Forthis, we also include a short survey of local families index theory. All this discussion willlead to Theorem 2, which we will then use to reprove Theorem 3 in a more abstract way. • The content of Chapter 4 is the discussion of 3-dimensional mapping tori along the lineswe have already outlined above. • For the reader’s convenience, and to keep our discussion more self-contained, we have alsoincluded a couple of appendices, which contain material that we freely use, but that wouldlead to far afield if discussed in the main body of this thesis. ntroduction xi – Appendix A contains a discussion of Chern-Weil theory in a way which is particularlywell-suited for the applications we need. Moreover, we have included some aspectsconcerning Chern-Simons invariants, as they appear throughout our discussion. – Since the Rho invariants we are interested in depend only on the gauge equivalenceclass of the involved flat connection, we include some remarks concerning the modulispace of flat connections in Appendix B. We start giving some details on the relationto representations of the fundamental group. Moreover, the moduli space of flat con-nections over a mapping torus is discussed, since we need this in Chapter 4. We endthis appendix with a brief survey of the moduli space of holomorphic line bundles overa Riemann surface, which is an ingredient for discussing flat U(1)-connections overprincipal S -bundles over surfaces. – Appendix C contains some computations. On the one hand, we need explicit values ofbasic Eta and Zeta functions to which we reduce most computations in the main bodyof the thesis. On the other hand, we shall discuss the Dedekind sum in (8) and itsgeneralization, establishing a relation among them which we need to prove Theorem4. – Finally, Appendix D includes some more analytical details concerning the heat opera-tor. Specifically, we will give some remarks concerning families of heat operators, andderive the variation formula for the Eta invariant. cknowledgements
Foremost, I want to express my deep gratitude to Prof. Dr. Matthias Lesch, under whosesupervision I chose this topic and worked on this thesis. Without his interest and insight intovarious branches of mathematics, I would not have learnt to appreciate the beauty of indextheory. I am particularly thankful for his confidence in me with which he helped to overcomeany dry spell. I am also deeply indebted to Prof. Dr. Paul Kirk for hosting an inspiring one-year stay at the Indiana University in Bloomington. Many ideas in this thesis are influenced bynumerous discussions in which he explained to me the elegant way he thinks about mathematics.Furthermore, I would like to thank Prof. Dr. Matthias Kreck, Prof. Dr. Jens Frehse and Prof.Dr. Albrecht Klemm for joining my thesis committee. Special thanks are also due to ChristianFrey and Benjamin Himpel, who were my office mates in K¨oln and Bonn. Both were alwaysavailable for giving their mathematical advice as well as encouraging me with their friendship.Moreover, I would like to thank the SFB/TR12, the Department of Mathematics at IndianaUniversity, and the Bonn International Graduate School in Mathematics, for providing fundingduring the period when this thesis was written. Especially the financial support for attendingvarious conferences and workshops is gratefully acknowledged.I shall not end without expressing my deep gratitude to my family and friends for theirencouragement and love on which I could always rely. I cannot be thankful enough for all thesupport my parents, Lothar and Monika Bohn, offered me to pursue my studies.xii hapter 1
The Signature Operator and the RhoInvariant
In this chapter we will survey some results from index theory, which we need in our discussionlater on. The objective is not to give a systematic treatment but merely to introduce the objectswe are interested in and to fix notation. For this reason we shall include proofs only if they enrichthe discussion and do not lead too far afield.We start with a brief discussion of the signature of a closed manifold, placing emphasis onthe generalized version where the intersection form is associated to a flat unitary vector bundle.Introducing the signature operator relates the signature to the index of an elliptic operator, andthis leads us to a discussion of the main facts concerning the heat equation on closed manifolds.Although our focus is on the signature operator and its odd dimensional analog, we present theAtiyah-Singer Index Theorem in its version for geometric Dirac operators. This is because manyideas in the later chapters are influenced by local index theory which is more transparent whenformulated in terms of Clifford modules and Dirac operators.Then we will introduce the object which constitutes the main topic of this thesis—namely theEta invariant of an elliptic operator on a closed manifold. Variation formulæ for Eta invariantswill play a prominent role in the discussion in the next chapters. Therefore, we discuss this topicin some detail. Most notably, the behaviour under the variation of a flat twisting connection leadsnaturally to the first appearance of a Rho invariant as the difference of certain Eta invariants.After this we describe—briefly leaving the realm of closed manifolds—how the Eta invariantarises as a correction term in the index theorem for a manifold with boundary. From then onthe focus will be on the case that the elliptic operator in question is the odd signature operator.From the signature theorem for manifolds with boundary we derive some general and well-knownproperties of the Eta invariant. In particular, the relation of Rho invariants to Chern-Simonsinvariants will be exhibited.We close the general discussion of this chapter with a short outline of how local index the-ory methods for odd dimensional manifolds can be used to obtain important properties of Rhoinvariants without referring to the Atiyah-Patodi-Singer Index Theorem. Our interest in this isnot only of a purely academic nature, as similar ideas are the ones underlying local families indextheory, which we will need in the context of fiber bundles in Chapter 3.1
1. The Signature Operator and the Rho Invariant
Let M be a closed, oriented and connected manifold of dimension m . On the real cohomologygroups there exists the intersection pairing H p ( M, R ) × H m − p ( M, R ) → R , ( a, b ) (cid:10) a ∪ b , [ M ] (cid:11) , where h ., . i is the Kronecker pairing, ∪ is the cup product, and [ M ] denotes the fundamental classof M determined by the orientation. Expressing H p ( M, R ) in terms of de Rham cohomologygroups, the intersection pairing is induced byΩ p ( M ) × Ω m − p ( M ) R , ( α, β ) Z M α ∧ β. As a consequence of the Poincar´e duality theorem, the above pairing is non-degenerate. Inparticular, if dim M = m is even, there is a non-degenerate bilinear form Q : H m/ ( M, R ) × H m/ ( M, R ) → R , (1.1)which is called the intersection form of M . If ( m ≡ m ≡ Convention.
We also use the convention that the signature of a skew form is the number ofpositive imaginary eigenvalues minus the number of negative imaginary ones.
Definition 1.1.1.
Let M be a closed, oriented and connected manifold of even dimension m .Then the signature of M is defined asSign( M ) := Sign( Q ) . Remark.
In topology, the intersection form is usually considered as a form over Z . If oneuses cohomology groups with integer coefficients, one has to divide out the torsion subgroup of H m/ ( M, Z ) to get a non-degenerate form. Moreover, note that if we want to work with complexcoefficients, we have to extend Q anti-linearly in, say, the first variable to get a (skew) Hermitianform. Then the signature is also well-defined and agrees with the signature of the underlying realform. Note, however, that if a skew form comes from a form over R it has zero signature sinceits eigenvalues come in conjugate pairs. Cohomology with Local Coefficients.
We will also need a twisted version of the intersectionform. For this we briefly recall the construction of cohomology groups with local coefficients. Werefer to [33, Ch. 5] for more details and proofs.Let M be a connected manifold, not necessarily closed, and let f M be the universal cover of M . Let π = π ( M ) be the fundamental group of M , and let C [ π ] denote the group algebra of π . The fundamental group acts from the right on f M , so that the cellular chain groups C p ( f M )are C [ π ] right modules in a natural way. This is because a cell decomposition of M induces vialifting of cells a cell decomposition of f M which is compatible with the action of π on f M . .1. The Signature of a Manifold α : π → U( k ) be a unitary representation, and let C p ( M, E α ) := Hom C [ π ] (cid:0) C p ( f M ) , C k ) . Here, the action of C [ π ] on C k is by matrix multiplication x α ( x ) − . The differential oncochains on f M turns C • ( M, E α ) into a complex. As for untwisted cellular cohomology, thehomology of this complex does not depend on the particular cell decomposition and the chosenlifts. Definition 1.1.2.
Let M be a compact, connected manifold, and α : π ( M ) → U( k ) a represen-tation. Then the homology of C • ( M, E α ) is denoted by H • ( M, E α ) and is called the cohomologyof M with local coefficients given by α .If we now assume that M is also closed and oriented, then there exists a non-degeneratepairing H p ( M, E α ) × H m − p ( M, E α ) → C (1.2)induced by the cup product on the cohomology of f M and the scalar product on C k . If M is ofeven dimension m this yields a bilinear form Q α : H m/ ( M, E α ) × H m/ ( M, E α ) → C , which we call the twisted intersection form. Definition 1.1.3.
Let M be a closed, oriented and connected manifold of even dimension m ,and let α : π ( M ) → U( k ) be a unitary representation of the fundamental group of M . Then the twisted signature of M is defined as Sign α ( M ) := Sign( Q α ) , where we use again the convention that the signature of a skew form is the number of positiveimaginary eigenvalues minus the number of negative imaginary ones. Remark.
As we will see soon, the twisted signatures we have just defined give no new topologicalinformation for a closed manifold M . However, their version for manifolds with boundary are anon-trivial generalization of topological importance. Nevertheless, we have included the definitionhere to keep the discussion parallel. Local Coefficients and Flat Bundles.
Twisted cohomology groups can also be defined interms of differential forms. Let E → M be a Hermitian vector bundle of rank k , endowed witha unitary connection A . This gives rise to a twisted version of the exterior differential d A : Ω p ( M, E ) → Ω p +1 ( M, E ) , d A ( ω ⊗ e ) = dω ⊗ e + ω ∧ Ae.
The square of d A is given by exterior multiplication with the curvature F A ∈ Ω (cid:0) M, End( E ) (cid:1) .Therefore, we get a complex (cid:0) Ω • ( M, E ) , d A (cid:1) precisely if A is flat. Definition 1.1.4.
Let E → M be a Hermitian vector bundle of rank k , endowed with a unitaryflat connection A . Then we denote the homology of (cid:0) Ω • ( M, E ) , d A (cid:1) by H • ( M, E A ) and call itthe cohomology of M with values in the flat bundle ( E, A ).
1. The Signature Operator and the Rho Invariant
We briefly sketch the relation between cohomology with local coefficients and cohomology withvalues in a flat bundle. As a general reference, we refer to [86, Sec. 5.5]. In addition, we haveincluded a detailed discussion concerning the equivalence of flat connections and representationsof the fundamental group in Appendix B.1. The language there is in terms of principal bundles,but the translation to Hermitian vector bundles is done without effort.Let M be a connected manifold, not necessarily closed, and let E be a flat unitary bundleover M with connection A . As explained in Appendix B.1 we can lift every closed loop c in M horizontally to E with respect to A . The starting point and the end point of the lifted loop liein the same fiber of E and since A is unitary they differ by the action of an element in U( k ).This construction gives rise to the holonomy representation of the based loop group of M , see(B.3) on p. 192. Since A is flat, the holonomy representation depends only on homotopy classes.Thus, we obtain a representation hol A : π ( M ) → U( k ) , which is precisely the object we need to define cohomology with local coefficients.Conversely, let us start with a representation α : π ( M ) → U( k ) of the fundamental group.Via the action of π ( M ) as the group of deck transformations we may interpret the universalcover f M as a π ( M )-principal bundle over M . The representation α defines an associated vectorbundle E α = f M × α C k → M. Since α is unitary, one can define a natural Hermitian metric on E α . Moreover, the trivialconnection on f M × C k descends to a unitary, flat connection A α on E α .When taking suitable equivalence classes, the above constructions are inverses of each other,see Appendix B.1. Then there is the following twisted version of the de Rham Theorem, see forexample the discussion in [86, p. 154]. Proposition 1.1.5.
Let M be a connected manifold, and let E be a Hermitian vector bundlewith flat connection A . If α : π ( M ) → U( k ) is the holonomy representation of A , then there isa natural isomorphism H • ( M, E A ) ∼ = −→ H • ( M, E α ) . Moreover, if M is closed, the twisted intersection pairing of (1.2) corresponds under this isomor-phism to the bilinear form on H • ( M, E A ) induced by Ω p ( M, E ) × Ω m − p ( M, E ) C , ( ω, η ) Z M h ω ∧ η i , The notation h ω ∧ η i is shorthand for taking the exterior product in the differential form part andpairing with the Hermitian metric in the bundle part. Having the above canonical isomorphism in mind, we will henceforth not carefully distinguishbetween H • ( M, E A ) and H • ( M, E α ). Similarly, when concerned with the intersection form, wewill also write Q A and Sign A ( M ) if the focus is on a flat connection rather than a representationof the fundamental group. .1. The Signature of a Manifold Historically, one of the starting points of index theory is the observation that the signature canbe described as the index of an elliptic operator. Before we can define the signature operator,we need to fix some notation and conventions. Since they are of a purely linear algebraic nature,we formulate them for an oriented vector space V of dimension m over R which plays the role ofthe cotangent space T ∗ x M . Algebraic Preliminaries.
Let V be an Euclidean vector space with scalar product g , and letΛ • V C denote the complexified exterior algebra. We endow it with the Hermitian metric g h givenby extending g antilinearly in the first variable. Then g h ( α, β ) vol( g ) = α ∧ ∗ β, α, β ∈ Λ • V C , where vol( g ) is the volume element given by g and the orientation of V and ∗ is the complexlinear extension of the Hodge ∗ operator. V acts on Λ • V via exterior multiplicatione( v ) α = v ∧ α, α ∈ Λ • V. Using the metric g , one defines an interior multiplication by requiring thati( v ) w = g ( v, w ) and i( v )( α ∧ β ) = i( v )( α ) ∧ β + ( − | α | α ∧ i( v ) β, for every v, w ∈ V and α, β ∈ Λ • V with α homogeneous of degree | α | . For v ∈ V we extend i( v )and e( v ) complex linearly to Λ • V C , and define the Clifford multiplication c : V → End C (cid:0) Λ • V C (cid:1) , c ( v ) := e( v ) − i( v ) . (1.3)Then, for all v ∈ V , c ( v ) = − g ( v, v ) , c ( v ) ∗ = − c ( v ) . This means that c extends to a complex representation of the Clifford algebra c : Cl( V, g ) → End C (cid:0) Λ • V C (cid:1) . Equivalently, this is a representation of the complexified Clifford algebra Cl C ( V ). We define the symbol map σ : Cl C ( V ) → Λ • V C , a c ( a )1 . (1.4)The symbol map σ is an isomorphism of vector spaces, but certainly not of algebras. The inverse σ − is called the quantization map . Using this we define the chirality operator τ := i [ m +12 ] c ◦ σ − (cid:0) vol( g ) (cid:1) : Λ • V C → Λ • V C . (1.5)Here, [ m +12 ] denotes the integral part of m +12 . The following result is straightforward, see [13,Prop. 3.58]. Recall (e.g. from [13, Sec. 1.3]) that the
Clifford algebra
Cl(
V, g ) of an R vector space V with a metric g isthe R algebra generated by V and the relations c ( v ) c ( w ) + c ( w ) c ( v ) = − g ( v, w ) , v, w ∈ V.
1. The Signature Operator and the Rho Invariant
Lemma 1.1.6.
The chirality operator τ satisfies τ = Id and τ ∗ = τ = τ − . On Λ p V C it is explicitly given by τ = ( − p ( p − + mp i k ∗ p , (1.6) where k := [ m +12 ] and ∗ p is the complex linear Hodge ∗ operator on Λ p V C . Moreover, τ ◦ c ( v ) = ( − m +1 c ( v ) ◦ τ and τ ◦ i( v ) = ( − m e( v ) ◦ τ. Convention.
From now on we will always drop the subscripts C . Thus, for a real vector space V , we will use Λ • V to denote the complexified exterior algebra, and Cl( V ) will denote thecomplexified Clifford algebra. The Signature an the Index.
Now let M be an oriented manifold of dimension m , endowedwith a Riemannian metric g . We fix a Hermitian vector bundle E → M of rank k , endowed with aunitary connection A . Let Ω • ( M, E ) denote differential forms with values in E . The Riemannianmetric g and the bundle metric on E define an L scalar product on Ω • ( M, E ). With respect tothis, the formal adjoint of the twisted exterior differential d A : Ω • ( M, E ) → Ω • +1 ( M, E )is given in terms of the chirality operator τ = τ M from (1.5) by d tA = ( − m +1 τ ◦ d A ◦ τ, (1.7)see [13, Prop. 3.58]. Here, τ M acts only on the differential form part. Note that we are usingthat A is unitary. Now (1.7) implies that the twisted de Rham operator d A + d tA satisfies τ ( d A + d tA ) = ( − m +1 ( d A + d tA ) τ. (1.8)Let us assume from now on that m is even. Since τ is an involution, we may decompose Ω • ( M, E )into the ± τ , Ω • ( M, E ) = Ω + ( M, E ) ⊕ Ω − ( M, E ) . It follows from (1.8) that we can decompose d A + d tA = (cid:18) D − A D + A (cid:19) , where D + A : Ω + ( M, E ) → Ω − ( M, E ) , D − A = ( D + A ) t . Definition 1.1.7.
Let M be an even dimensional, oriented Riemannian manifold, and let E → M be a Hermitian vector bundle with a unitary connection A . Then D + A : Ω + ( M, E ) → Ω − ( M, E )is called the twisted signature operator of M twisted by A . .1. The Signature of a Manifold M is a closed manifold,we have a well-defined index problem. The name signature operator is only justified if A is a flatconnection, since then the index of D + A is indeed the twisted signature Sign A ( M ). Proposition 1.1.8.
Let M be a closed, oriented and connected Riemannian manifold of evendimension m . Let E be a Hermitian vector bundle, endowed with a unitary flat connection A .Then Sign A ( M ) = ind( D + A ) . Proof.
Although this result can be found in many textbooks, we include its proof as relatedarguments will appear again. First of all, since ( D + A ) t = D − A ,ind( D + A ) = dim (cid:0) ker D + A (cid:1) − dim (cid:0) ker D − A (cid:1) (1.9)To identify Ker D ± A in cohomological terms consider the twisted Laplacian∆ A = ( d A + d tA ) : Ω • ( M, E ) → Ω • ( M, E ) . (1.10)The twisted version of the Hodge isomorphism identifies H p ( M, E A ) with the space of harmonicforms H p ( M, E A ) := (cid:0) ker ∆ A (cid:1) ∩ Ω p ( M, E ) . It follows from (1.8) that the chirality operator τ commutes with ∆ A , hence it induces an invo-lution on H • ( M, E A ). On H m/ ( M, E A ) the chirality operator is τ = i k ∗ , where k := m / Q A (cid:0) α, β (cid:1) = Z M h α ∧ β i = (cid:0) α , i k τ β (cid:1) L , α, β ∈ H m/ ( M, E A ) . Therefore, Sign( Q A ) = Sign( i k τ | H m/ ) , where we use the same convention as before regarding the signature of a skew endomorphism.We deduce that Sign( Q A ) = dim H m/ ( M, E A ) + − dim H m/ ( M, E A ) − . If p = m/
2, there are isomorphismsΦ ± : H p ∼ = −→ (cid:0) H p ⊕ H m − p (cid:1) ± , Φ ± ( α ) := ( α ± τ α ) . From this and from the fact that Ker( d A + d tA ) = Ker ∆ A we finddim (cid:0) ker D ± A (cid:1) = dim (cid:0) H m/ ( M, E A ) ± (cid:1) + X p 1. The Signature Operator and the Rho Invariant Remark. We have already pointed out that for closed manifolds the twisted signatures do notcarry interesting topological information. If the flat twisting bundle is trivial, this can be deducedfrom the above result: Let A be a flat connection on the trivial vector bundle E = M × C k , andlet D + A be the associated signature operator. Furthermore, let D + ⊕ k denote the signature operatorassociated to the trivial connection on E . Clearly, D + A − D + ⊕ k is an operator of order 0. Ona closed manifold adding a 0-order perturbation to an elliptic operator of first order does notchange the index. This is because it is a compact perturbation of a Fredholm operator in theappropriate Hilbert space setting. Therefore,Sign A ( M ) = ind( D + A ) = ind( D + ⊕ k ) = k · ind( D + ) = k · Sign( M ) . This means that the only new information encoded in the twisted signature is the rank of thetwisting bundle. We will see in Corollary 1.2.10 below that this is also true for flat twistingbundles which are topologically non-trivial. The famous index theorem equates the index of an elliptic operator over a closed manifold M with the integral over certain characteristic classes over M . In this section we briefly recall thedefinitions occurring in the index theorem for Dirac type operators. We first recall some facts about the spectral theory of formally self-adjoint elliptic operators onclosed manifolds, see e.g. [49, Sec.’s. 1.3 & 1.6]. Definition 1.2.1. Let M be a Riemannian manifold, and let E → M be a Hermitian vectorbundle. We denote by P ds,e = P ds,e ( M, E )the space of formally self-adjoint elliptic differential operators of order d . Theorem 1.2.2. Let D ∈ P ds,e ( M, E ) , and assume that M closed. (i) The operator D extends to an unbounded self-adjoint operator in L ( M, E ) with domainthe Sobolev space L d ( M, E ) . It defines Fredholm operators D : L s + d ( M, E ) → L s ( M, E ) , s ∈ R , with Fredholm index independent of s . (ii) There exists a constant C such that for all ϕ ∈ C ∞ ( M, E ) k ϕ k L d ≤ C (cid:0) k ϕ k L + k Dϕ k L (cid:1) . (1.11)(iii) The spectrum spec( D ) is a discrete subset of R consisting of eigenvalues with finite multi-plicities. There is an orthonormal basis of L ( M, E ) consisting of smooth eigenvectors. .2. Dirac Operators and the Atiyah-Singer Index Theorem If we define N ( λ ) := (cid:8) λ n ∈ spec( D ) (cid:12)(cid:12) | λ n | ≤ λ (cid:9) , λ ≥ , then for some constant C > N ( λ ) ∼ Cλ m/d , as λ → ∞ . (1.12)(v) If D has a positive definite leading symbol, then the spectrum spec( D ) is bounded frombelow. The Heat Kernel. We now specialize to a second order operator, H ∈ P s,e ( M, E ), and assumethat H has positive definite leading symbol. Clearly, the model we are having in mind is ageneralized Laplacian, i.e., an operator H ∈ P s,e ( M, E ) such that its principal symbol σ ( H )satisfies σ ( H )( x, ξ ) = | ξ | g id E x , ξ ∈ T ∗ x M. Recall that the heat equation with initial condition ϕ ∈ L ( M, E ) in terms of H is the partialdifferential equation (cid:0) ddt + H (cid:1) ϕ ( t ) = 0 , t ≥ , ϕ (0) = ϕ. (1.13)Formally, if { λ n } n ≥− n denotes the set of eigenvalues of H with eigenvectors ϕ n , the solution to(1.13) is ϕ ( t ) = e − tH ϕ = X n ≥− n e − tλ n ϕ n (cid:0) ϕ n , ϕ (cid:1) L = Z M X n ≥− n e − tλ n ϕ n ( x ) h ϕ n ( y ) , ϕ ( y ) i vol M ( y ) . Thus, e − tH is an integral operator with kernel k t ( x, y ) = e − tH ( x, y ) = X n ≥− n e − tλ n ϕ n ( x ) ⊗ ϕ n ( y ) ∗ ∈ C ∞ (cid:0) M × M, E ⊠ E ∗ (cid:1) . (1.14)Here, for vector bundles E → M and F → N , we employ the standard notation E ⊠ F := π ∗ M E ⊗ π ∗ N F → M × N, (1.15)where π M and π N are the natural projections. The formal expression (1.14) can be made preciseusing the following basic estimate. Lemma 1.2.3. Let < λ ≤ λ ≤ . . . denote the positive eigenvalues of H , and let λ − ≤ . . . ≤ λ − n denote the finite number of eigenvalues of H which are less or equal than 0. Let { ϕ n } n ≥− n be a basis of smooth eigenvectors and for N ∈ N consider k Nt ( x, y ) := N X n = − n e − tλ n ϕ n ( x ) ⊗ ϕ ∗ n ( y ) ∈ C ∞ (cid:0) M × M, E ⊠ E ∗ (cid:1) . Then for every k ∈ N and t > there exists a constant C such that for every N ∈ N and t ≥ t we can estimate (cid:13)(cid:13)(cid:13) k Nt ( x, y ) − X n< e − tλ n ϕ n ( x ) ⊗ ϕ ∗ n ( y ) (cid:13)(cid:13)(cid:13) C k ≤ Ce − tλ / . 1. The Signature Operator and the Rho Invariant Proof. Sobolev embedding [49, Lem. 1.3.5] and the elliptic estimate (1.11) imply that for l >k + m/ C , C such that for all n ≥ k ϕ n k C k ≤ C k ϕ n k L l ≤ C (cid:0) k ϕ n k L + k H l/ ϕ n k L (cid:1) = C (cid:0) λ l/ n (cid:1) . Thus, for some other constants C and C , (cid:13)(cid:13) e − tλ n ϕ n ( x ) ⊗ ϕ ∗ n ( y ) (cid:13)(cid:13) C k ≤ C e − tλ n (cid:0) λ ln (cid:1) ≤ C e − tλ n / (1 + t − l ) , where we have used that for x = λ n t > x l ≤ C l e x/ and thus, x l e − x ≤ C l e − x/ . Now for each n ≥ λ n ≥ λ and thus for t ≥ t e − tλ n / (1 + t − l ) ≤ e − t λ n / (1 + t − l ) e − ( t − t ) λ / = Ce − t λ n / e − tλ / , where the constant C depends only on t and l . Putting the pieces together we find that thereexists a constant C such that for every N ≥ t ≥ t (cid:13)(cid:13)(cid:13) N X n =0 e − tλ n ϕ n ( x ) ⊗ ϕ ∗ n ( y ) (cid:13)(cid:13)(cid:13) C k ≤ Ce − tλ / N X n =0 e − t λ n / Now the eigenvalue asymptotics (1.12) shows that P Nn =0 e − t λ n / is absolutely convergent for N → ∞ . Absorbing this to the constant, we get the desired result.The above result shows that the kernels k Nt ( x, y ) converge for N → ∞ to the expression(1.14), uniformly with respect to all C k . Hence, we can define e − tH for t > e − tH ϕ )( x ) = Z M k t ( x, y ) ϕ ( y ) vol M ( y ) , ϕ ∈ L ( M, E ) . In Appendix D.1 we give an expression for e − tH using the spectral theorem. It is then easyto check that the collection e − tH forms a strongly continuous semi-group in each Sobolev space L s ( M, E ), i.e., e − ( s + t ) H = e − sH e − tH , and lim t → k e − tH ϕ − ϕ k L s = 0 for each ϕ ∈ L s ( M, E ) , see also Proposition D.1.2. Moreover, e − tH is smooth in t and does indeed solve the heat equation(1.13). In addition, each e − tH is trace class, andTr e − tH = Z M tr E (cid:2) k t ( x, x ) (cid:3) vol M ( x ) , where tr E : C ∞ (cid:0) M, End( E ) (cid:1) → C ∞ ( M ) denotes the fiberwise trace.More generally, given an auxiliary differential operator D : C ∞ ( M, E ) → C ∞ ( M, E ) of order d ≥ 0, the uniform bound of Lemma 1.2.3 ensures that we can apply D under the integral to get De − tH ϕ ( x ) = D (cid:16) Z M k t ( x, y ) ϕ ( y ) vol M ( y ) (cid:17) = Z M D x k t ( x, y ) ϕ ( y ) vol M ( y ) , .2. Dirac Operators and the Atiyah-Singer Index Theorem D x means applying D with respect to the x variable. Thus, the operator De − tH has asmooth kernel D x k t ( x, y ) ∈ C ∞ (cid:0) M × M, E ⊠ E ∗ (cid:1) , so that De − tH is trace class. The estimate in Lemma 1.2.3 implies the following basic estimateon Tr( De − tH ). Proposition 1.2.4. For every t > there exists a constant C such that for all t ≥ t (cid:12)(cid:12) Tr (cid:0) De − tH P (0 , ∞ ) (cid:1)(cid:12)(cid:12) ≤ Ce − tλ / , where λ is the smallest positive eigenvalue of H and P (0 , ∞ ) is the spectral projection of H associated to the interval (0 , ∞ ) .Proof. The kernel of e − tH P (0 , ∞ ) is given by˜ k t ( x, y ) := k t ( x, y ) − X n< e − tλ n ϕ n ( x ) ⊗ ϕ ∗ n ( y ) . Then we find that for t ≥ t (cid:12)(cid:12) Tr (cid:0) De − tH P (0 , ∞ ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z M tr E (cid:2) D x ˜ k t ( x, y ) (cid:3) y = x vol M ( x ) (cid:12)(cid:12)(cid:12) ≤ rk( E ) vol( M ) (cid:13)(cid:13) D x ˜ k t ( x, y ) (cid:13)(cid:13) C ≤ C (cid:13)(cid:13) ˜ k t ( x, y ) (cid:13)(cid:13) C d ≤ C e − tλ / , where in the last line we have used that D is a differential operator of order d and then Lemma1.2.3. The McKean-Singer Formula. We now turn our attention to first order differential operators.To be able to restrict to the formally self-adjoint case, we use the following construction: Let D + : C ∞ ( M, E + ) → C ∞ ( M, E − ) be an elliptic differential operator of first order, acting betweenHermitian vector bundles E + and E − . We define E := E + ⊕ E − , and consider D := (cid:18) D − D + (cid:19) : C ∞ ( M, E ) → C ∞ ( M, E ) , where D − := ( D + ) t . (1.16)Certainly, an operator of this form is formally self-adjoint and elliptic. Definition 1.2.5. Let E → M be a Hermitian vector bundle endowed with a splitting E = E + ⊕ E − .(i) Let σ : E → E be the involution on E given by σ | E ± = ± id. Then σ is called the gradingoperator of E .(ii) An operator D ∈ P s,e ( M, E ) is called Z -graded if { D, σ } = Dσ + σD = 0 . Note that—unless stated otherwise—we are using commutators and anti-commutators inan ungraded sense. Clearly, D is Z -graded if and only if it is of the form (1.16).2 1. The Signature Operator and the Rho Invariant (iii) For T ∈ C ∞ (cid:0) M, End( E ) (cid:1) , we define fiberwise supertrace of T asstr E ( T ) := tr E ( σT ) ∈ C ∞ ( M ) . (iv) If D ∈ P e,s ( M, E ) is Z -graded, and M is closed, then the heat supertrace associated to D is defined as Str( e − tD ) := Tr (cid:0) σe − tD (cid:1) . Note that in (iv) the operator H = D is positive and splits as H = H + ⊕ H − , where H ± areformally self-adjoint, positive operator as well. Thus, we are in the situation considered beforeso that the respective heat traces exist, andStr( e − tD ) = Tr( e − tH + ) − Tr( e − tH − ) = Tr( e − t ( D + ) t D + ) − Tr( e − tD + ( D + ) t ) . Now, as an elliptic operator on a closed manifold D + has a well-defined Fredholm indexind( D + ) = dim (cid:0) ker D + (cid:1) − dim (cid:0) ker( D + ) t (cid:1) = Str( P ) , where P : L ( M, E ) → ker( D ) is the orthogonal projection on the kernel of D . The famousMcKean-Singer formula [69] relates this index and the heat supertrace. Theorem 1.2.6 (McKean-Singer) . Let M be a closed manifold, E → M a Hermitian vectorbundle, and let D ∈ P e,s ( M, E ) be Z -graded. Then for all t > D + ) = Str( e − tD ) . Proof. Since D has no negative eigenvalues, we have P + P (0 , ∞ ) = Id. Then the estimate inProposition 1.2.4 implies that there exist constants c and C such that for large t (cid:12)(cid:12) Str( e − tD ) − Str( P ) (cid:12)(cid:12) = (cid:12)(cid:12) Str( e − tD P (0 , ∞ ) ) (cid:12)(cid:12) ≤ Ce − ct . Thus, lim t →∞ Str( e − tD ) = Str( P ) = ind( D + ) . It remains to check that Str( e − tD ) is independent of t > 0. For this, we note that the heatequation yields ddt e − tD = − D e − tD . The basic trace estimate then implies that Str( e − tD ) is differentiable in t with ddt Str( e − tD ) = − Str( D e − tD ) . However, since D anti-commutes with σ , we infer from the trace property thatStr( D e − tD ) = Tr( σD e − tD ) = − Tr( DσDe − tD ) = − Tr( σe − tD D ) = − Str( D e − tD ) , so that indeed Str( D e − tD ) = 0. .2. Dirac Operators and the Atiyah-Singer Index Theorem Heat Kernel Asymptotics. So far, the treatment of the heat kernel has been of a functionalanalytic nature. The only input are the basic properties of elliptic differential operators on closedmanifolds as in Theorem 1.2.1. However, one important missing piece is the analysis of the heattrace as t → 0, which requires further work. We summarize the following from [49, Sec.’s 1.8 &1.9]. Theorem 1.2.7. Let M be a closed manifold, and let H ∈ P s,e ( M, E ) have positive definiteleading symbol. Let D : C ∞ ( M, E ) → C ∞ ( M, E ) be an auxiliary differential operator of order d ≥ , and let k t ( x, y ) denote the kernel of De − tH . (i) There exists an asymptotic expansion k t ( x, x ) ∼ ∞ X n =0 t n − m − d e n ( x ) , as t → , with e n ∈ C ∞ (cid:0) M, End( E ) (cid:1) such that e n ( x ) is locally computable from the total symbols of H and D near x . If n + d is odd, then e n = 0 . (ii) The trace of De − tH admits an asymptotic expansion Tr( De − tH ) = Z M tr E (cid:2) k t ( x, x ) (cid:3) vol M ( x ) ∼ ∞ X n =0 t n − m − d a n ( D, H ) , as t → . The asymptotic expansion can be differentiated in t , and the a n are given by a n ( D, H ) = Z M tr E (cid:2) e n (cid:3) vol M . The Index Density. As a consequence of the McKean-Singer formula and the asymptoticexpansion of the heat trace, we get the following result of [69] and [5]. Theorem 1.2.8. Let e n ( x ) be the coefficient appearing in the asymptotic expansion of e − tD ( x, x ) as in Theorem 1.2.7. Then Z M str E [ e n ] vol M = ( ind( D + ) , if n = dim M, , if n < dim M. Remark. (i) For aparent reasons, the differential form str E [ e m ] vol M with m = dim M is called the indexdensity of D + .(ii) As mentioned in Theorem 1.2.7, the coefficients e n vanish if n is odd. This implies that theindex of D + vanishes if M is odd dimensional.We also want to note an important consequence of the local nature of the e n . Assume thattwo operators D ∈ P e,s ( M, E ) and D ′ ∈ P e,s ( M, E ′ ) are locally equivalent . This means that forevery x ∈ M , there exists a neighbourhood U of x and a local isometry Φ : E | U → E ′ | U suchthat Φ ◦ D ◦ Φ − = D ′ over U . (1.17)Since the sections e n and e ′ n in the respective asymptotic expansions of the heat kernels arecomputable from the total symbols over U , relation (1.17) implies that str E [ e n ] | U = str E ′ [ e ′ n ] | U .This has the following consequence.4 1. The Signature Operator and the Rho Invariant Corollary 1.2.9. Let M be a closed manifold, E → M a Hermitian vector bundle, and let D ∈ P e,s ( M, E ) be a Z -graded operator. (i) If π : c M → M is a k -fold regular cover, and b D : C ∞ ( c M , π ∗ E ) → C ∞ ( c M , π ∗ E ) is the natural lift of D to c M , then ind( b D + ) = k · ind( D + ) . (ii) If A is a flat connection on a Hermitian vector bundle F of rank k over M , and if D A isthe operator D twisted by F and A , then ind (cid:0) D + A (cid:1) = k · ind( D + ) . Proof. For (i) one notes that the b e n are the lifts to c M of the e n . Thus, Z c M str π ∗ E [ b e n ] vol c M = Z c M π ∗ (cid:0) str E [ e n ] vol M (cid:1) = k · Z M str E [ e n ] vol M , where we have used that vol( c M ) = k · vol( M ). For (ii) note that choosing local trivializationsfor F which are parallel with respect to A , one finds that the flat bundle F is locally isomorphicto M × C k endowed with the trivial connection. This yields that D A is locally equivalent to D ⊕ k .As we have seen in Proposition 1.1.8, the signature of a closed manifold equals the index ofan elliptic differential operator of first order. Hence, Corollary 1.2.9 proves our earlier assertionthat the twisted signatures do not contain new information other than the rank of the twistingbundle. Corollary 1.2.10. Let M be a closed, even dimensional manifold. (i) If π : c M → M is a k -fold regular cover, then Sign( c M ) = k · Sign( M )(ii) If A is a flat connection on a Hermitian vector bundle E of rank k over M , then Sign A ( M ) = k · Sign( M ) . Theorem 1.2.8 is the starting point for local index theory. Since the coefficients e n are—inprincipal—locally computable, a strategy to prove the Atiyah-Singer Index Theorem is toidentify the index density str E [ e m ] vol M with a Chern-Weil representative of an appropriatecharacteristic class. Note, however, that Chern-Weil classes are expressions in the curvature,whereas the e n a priori contain higher order derivatives of the connection. Nevertheless, this .2. Dirac Operators and the Atiyah-Singer Index Theorem Geometric Dirac Operators. Let ( M, g ) be an oriented, Riemannian manifold, and let E → M be a Hermitian vector bundle. E is called a Clifford module if it is endowed with a bundle map c : T ∗ M → End( E ) such that for every ξ ∈ T ∗ Mc ( ξ ) = −| ξ | g id E and c ( ξ ) ∗ = − c ( ξ ) . (1.18)A Clifford module is called Z -graded if E is Z -graded and if c ( ξ ) is an odd element of End( E )for all ξ ∈ T ∗ M . Provided that E is endowed with suitable connection, we can construct anatural first order differential operator: Definition 1.2.11. Let E be a Clifford module over a Riemannian manifold ( M, g ).(i) A connection ∇ E on E which is compatible with the metric is called a Clifford connection if for all e ∈ C ∞ ( M, E ) and ξ ∈ Ω ( M ) (cid:2) ∇ E , c ( ξ ) (cid:3) e = ∇ E (cid:0) c ( ξ ) e (cid:1) − c ( ξ ) ∇ E e = c ( ∇ g ξ ) e, where ∇ g is the Levi-Civita connection acting on forms.(ii) If ∇ E is a Clifford connection, we define the associated geometric Dirac operator by D := c ◦ ∇ E : C ∞ ( M, E ) → C ∞ ( M, E ) . Here, we are viewing the Clifford structure as a bundle map c : T ∗ M ⊗ E → E .(iii) A geometric Dirac operator is called Z -graded if E is a Z -graded Clifford module, and theClifford connection respects the splitting E = E + ⊕ E − . Remark 1.2.12. One often defines a Dirac operator to be a formally self-adjoint elliptic operatorwhose square is a generalized Laplacian. It is straightforward to check that geometric Diracoperators have this property. However, not every Dirac operator is a geometric one. In Section3.2 we sketch how to associate Dirac operators to generalized connections, in particular Clifford superconnections , thereby obtaining a larger class of Dirac operators, see also [13, Prop. 3.42]. Canonical Structures on Clifford Modules. As pointed out, the index density is a purelylocal object. The reason for studying geometric Dirac operators is that Clifford modules have acanonical local structure which we now describe briefly. For proofs we refer to [13, Sec.’s 3.2 &3.3].We assume from now on that m = dim M is even. Some aspects of the odd dimensionalcase are contained in Section 1.2.3 and Section 1.5.2 in a special case. Let us further assumefor the moment that M is spin . Without going into the details of the definition and the topo-logical restrictions that this imposes on M , we note that it implies that there exists a uniqueirreducible Clifford module S of rank 2 m/ over M , called the spinor module . It follows from therepresentation theory of Clifford algebras thatEnd( S ) = Cl( T ∗ M ) . (1.19)The Clifford module S is naturally Z -graded, and (1.19) is an isomorphism of Z -graded algebras.Here, the grading on Cl( T ∗ M ) is the one induced via the symbol map from the even/odd grading6 1. The Signature Operator and the Rho Invariant on differential forms. Now, every Z -graded Clifford module E can be decomposed as E = S ⊗ W ,where W carries a trivial Clifford structure. Moreover,End( W ) = End Cl ( E ) := n T ∈ End( E ) (cid:12)(cid:12)(cid:12) [ T, c ( α )] s = 0 for all α ∈ T ∗ M o , (1.20)and End( E ) = Cl( T ∗ M ) ⊗ s End Cl ( E ) . (1.21)Here, [ ., . ] s and ⊗ s are the commutator and the tensor product in the Z -graded sense. If σ denotes the grading operator of E , there exists a unique decomposition σ = τ ⊗ σ W ∈ Cl( T ∗ M ) ⊗ s End Cl ( E ) , (1.22)where τ := i m/ c (vol M ) is the chirality operator, and σ W is a grading operator on W . Now, if T ∈ End( W ), then one verifies thatstr W ( T ) = tr W [ σ W T ] = 1rk( S ) tr E (cid:2) ( τ ⊗ σ W ) T (cid:3) = 12 m/ str E [ τ T ] . (1.23)Moreover, the spinor module S comes equipped with a canonical Clifford connection ∇ S whichis induced by the Levi-Civita connection ∇ g via (1.19). From this one gets a 1-1 correspondencebetween Clifford connections ∇ E on E and connections of the form ∇ S ⊗ ⊗ ∇ W , where ∇ W is a Hermitian connection on W . The curvature F ∇ W satisfies F ∇ W = F ∇ E − R S ∈ Ω (cid:0) M, End Cl ( E ) (cid:1) , (1.24)where for any orthonormal frame { e i } of T MR S := g (cid:0) R g ( e i , e j ) e k , e l (cid:1) e i ∧ e j ⊗ c ( e k ) c ( e l ) . (1.25)Here, R g ∈ Ω (cid:0) M, End( T M ) (cid:1) is the curvature tensor of the Levi-Civita connection.We now note that the right hand sides of (1.20), (1.23) and (1.24) can be defined globally on M without referring to the spinor module S . Thus, we can introduce corresponding objects alsoin the case that M is not spin. In particular, the definition of End Cl ( E ) in (1.20) is meaningfulfor every Clifford module E . Definition 1.2.13. Let M be an m -dimensional manifold, where m is even, and let E be a Z -graded Clifford module over M , endowed with a Clifford connection ∇ E .(i) Let R S be defined as in (1.25). Then we can decompose F ∇ E = R S + F E/S , where F E/S ∈ Ω (cid:0) M, End Cl ( E ) (cid:1) . The term F E/S is called the twisting curvature of E .(ii) If T ∈ C ∞ (cid:0) M, End Cl ( E ) (cid:1) , then its relative supertrace is defined asstr E/S ( T ) := 12 m/ str E ( τ T ) . .2. Dirac Operators and the Atiyah-Singer Index Theorem relative Chern character form of E is given bych E/S ( E, ∇ E ) := str E/S (cid:2) exp (cid:0) i π F E/S (cid:1)(cid:3) ∈ Ω ev ( M ) . (1.26) Remark. It follows from (1.23) and (1.24) that if M is spin so that E = S ⊗ W , thench E/S ( E, ∇ E ) = str W (cid:2) exp (cid:0) i π F W (cid:1)(cid:3) . In particular, if W is ungraded this coincides with the Chern character form of W as in DefinitionA.1.3. In general, if W = W + ⊕ W − , the relative Chern character is the difference of the Cherncharacters of W + and W − , see also (3.16). The Local Index Theorem. We can now state a version of the local index theorem as in [13,Thm. 4.2]. Theorem 1.2.14 (Patodi, Gilkey) . Let M be a closed, oriented Riemannian manifold of evendimension m , and let E → M be a Z -graded Clifford module with Clifford connection ∇ E andDirac operator D . Let e n ( x ) be the coefficient appearing in the asymptotic expansion of e − tD ( x, x ) as in Theorem 1.2.7. Then str E [ e n ] vol M ( x ) = ((cid:0) b A ( T M, ∇ g ) ∧ ch E/S ( E, ∇ E ) (cid:1) [ m ] , if n = dim M, , if n < dim M, (1.27) where the Hirzebruch b A -form is as in Definition A.1.4, and ( . . . ) [ m ] means taking the m -formpart of a differential form. Remark 1.2.15. There is a stronger version of the local index theorem due to E. Getzler, see[44] and [13, Thm. 4.1], which we also want to recall. Let σ : Cl( T ∗ M ) → Λ • T ∗ M be the symbolmap (1.4), and use this to endow Cl( T ∗ M ) with a Z -grading. With respect to this let Cl n ( T ∗ M )be the subbundle of Cl( T ∗ M ) of elements of degree ≤ n . Then it can be shown that e n ∈ C ∞ (cid:0) M, Cl n ( T ∗ M ) ⊗ s End Cl ( E ) (cid:1) . The stronger version of the local index theorem is the formula(4 π ) m/ X n ≤ m σ ( e n ) = det / (cid:18) R g / R g / (cid:19) ∧ exp( − F E/S ) , where R g is the Riemann curvature tensor. For the definition of the right hand side, see AppendixA, in particular (A.1) and Definition A.1.4. Now the supertrace of elements in Cl( T ∗ M ) vanishesaway from degree m . Hence, Theorem 1.2.14 follows by computing the supertrace of τ and takinginto account the powers of i π appearing in our definition of b A and ch E/S . In Section 1.5.2, wewill sketch a proof of a variation formula for the Eta invariant based on this more general localindex theorem.A direct consequence of the local index theorem 1.2.14 and Theorem 1.2.8 is the famousAtiyah-Singer Index Theorem for geometric Dirac operators in its cohomological version.8 1. The Signature Operator and the Rho Invariant Theorem 1.2.16 (Atiyah-Singer) . Let M be a closed, oriented Riemannian manifold of evendimension m , and let E → M be a Z -graded Clifford module with Clifford connection ∇ E andDirac operator D . Then ind( D + ) = Z M b A ( T M, ∇ g ) ∧ ch E/S ( E, ∇ E ) . In terms of characteristic classes, ind( D + ) = (cid:10) b A ( T M ) ∪ ch E/S ( E ) , [ M ] (cid:11) . A special case of the Atiyah-Singer Index Theorem is one of its predecessors, the HirzebruchSignature Theorem. It arises if the geometric Dirac operator in question is a twisted signatureoperator. Therefore, we now collect some details about the structure of the exterior algebra as aClifford module. Clifford Structures on the Exterior Algebra. Let M be an m -dimensional closed, orientedRiemannian manifold. For the moment we do not assume that m is even. We consider theClifford structure c : T ∗ M → End (cid:0) Λ • T ∗ M (cid:1) , c ( ξ ) = e( ξ ) − i( ξ ) , see (1.3). The Levi-Civita connection ∇ g acting on forms is a Clifford connection, and we get ageometric Dirac operator d + d t = c ◦ ∇ g : Ω • ( M ) → Ω • ( M ) . There are two natural gradings on Λ • T ∗ M , one given by the even/odd grading and one given bythe chirality operator τ . We know from Lemma 1.1.6 that if m is even, Clifford multiplication isodd with respect to both gradings. However, if m is odd, Clifford multiplication commutes with τ so that in this case we do not get a Z -graded Clifford module. Moreover, if M is spin andeven dimensional, then Λ ∗ T ∗ M ∼ = Cl( T ∗ M ) = End( S ) = S ⊗ S ∗ , which means that the twisting bundle is isomorphic to the dual bundle S ∗ of S . This motivatesthe following Definition 1.2.17. Let M be a manifold of dimension m , not necessarily even. We define a transposed Clifford multiplication b c : T ∗ M → End (cid:0) Λ • T ∗ M (cid:1) , b c ( ξ ) := e( ξ ) + i( ξ ) . The transposed Clifford multiplication has the following properties. Lemma 1.2.18. Let { e i } be a local orthonormal frame for T M . (i) With the obvious abbreviations, we have b c i b c j + b c j b c i = 2 δ ij , and c i b c j + b c j c i = 0 . .2. Dirac Operators and the Atiyah-Singer Index Theorem If we define b τ := i [ m +12 ] b c . . . b c m , then b τ = ( − m , and b τ = τ ◦ ( − ν , where ν : Λ • T ∗ M → N is the number operator given by ν ( ω ) = k if ω ∈ Λ k T ∗ M . (iii) Let b Cl( T ∗ M ) denote the subbundle of End (cid:0) Λ • T ∗ M (cid:1) generated by transposed Clifford mul-tiplication. Then b Cl( T ∗ M ) = End Cl (cid:0) Λ • T ∗ M (cid:1) , where End Cl is defined as in (1.20) with respect to the even/odd grading. We shall not include the proof which is straightforward but a bit tedious. Part (i) and (iii)of Lemma 1.2.18 can be found in [13, p. 144]. Part (ii) can be easily proved by induction on m .However, we want to point out that part (iii) implies that we have an isomorphism of Z -gradedalgebras, End (cid:0) Λ • T ∗ M (cid:1) = Cl( T ∗ M ) ⊗ s b Cl( T ∗ M ) , (1.28)which is (1.21) translated to the case at hand. Moreover, part (ii) of Lemma 1.2.18 shows thatthe decomposition (1.22) in the case at hand is( − ν = τ ⊗ b τ . Remark. In the case that m is even, one might expect that the decomposition (1.21) forEnd (cid:0) Λ • T ∗ M (cid:1) with respect to the τ -grading is given by (1.28) together with the grading op-erator τ ⊗ 1. However, with respect to this, the endomorphism b c ( α ) for α ∈ T ∗ M is even, andthis is incompatible with part (i) of Lemma 1.2.18. To stay in the Z -graded formalism, onewould have to consider yet another kind of Clifford multiplication, namely e c := b c ◦ ( − ν : T ∗ M → End (cid:0) Λ • T ∗ M (cid:1) . This generates a subalgebra e Cl( T ∗ M ) of End (cid:0) Λ • T ∗ M (cid:1) of purely even degree with respect to τ so that End (cid:0) Λ • T ∗ M (cid:1) = Cl( T ∗ M ) ⊗ s e Cl( T ∗ M ) . Fortunately, in the discussion to follow, we are interested only in elements of e Cl( T ∗ M ), respec-tively b Cl( T ∗ M ), which are of even with respect to the even/odd grading. For elements of thisform, e c and b c coincide up to sign. More precisely, if { e i } is a local frame for T M , then for all k ≤ m/ e c i . . . e c i k = ( − k b c i . . . b c i k . Hence, even if it is incorrect from a formal point of view, we use the transposed Clifford multi-plication b c also in the case that Λ • T ∗ M is graded by τ . Traces of the Exterior Algebra. Let π : Cl( T ∗ M ) → C be the projection onto the subalgebra C ⊂ Cl( T ∗ M ). One easily verifies that (cid:2) Cl( T ∗ M ) , Cl( T ∗ M ) (cid:3) ∩ C = { } , so that we can define atrace on Cl( T ∗ M ) by tr Cl := 2 m/ π : Cl( T ∗ M ) → C , see [44, Thm. 1.8]. In the same way, we get a trace d tr Cl on b Cl( T ∗ M ). Note that in the case that m is even, the natural supertrace of [13, Prop. 3.21] is given bystr Cl := tr Cl ◦ τ : Cl( T ∗ M ) → C . 1. The Signature Operator and the Rho Invariant Proposition 1.2.19. With respect to the decomposition (1.28) we have tr Λ • = tr Cl ⊗ d tr Cl , where tr Λ • is the natural trace on End(Λ • T ∗ M ) . Remark. We include a proof, since the treatment in [13] considers only the case that m iseven. There are some non-trivial sign difficulties involved, since (1.28) involves the graded tensorproduct, whereas tr Cl and d tr Cl are traces rather than supertraces. Yet, for elements of puredegree, tr Cl ⊗ d tr Cl (cid:0) ( a ⊗ ˆ a )( b ⊗ ˆ b ) (cid:1) = ( − | ˆ a || b | tr Cl ( ab ) d tr Cl (ˆ a ˆ b )= ( − | ˆ a || b | tr Cl ( ba ) d tr Cl (ˆ b ˆ a )= ( − | ˆ a || b | + | ˆ b || a | tr Cl ⊗ d tr Cl (cid:0) ( b ⊗ ˆ b )( a ⊗ ˆ a ) (cid:1) . (1.29)Moreover, for tr Cl ( ab ) and d tr Cl (ˆ a ˆ b ) to be non-zero it is necessary that | a | = | b | and | ˆ a | = | ˆ b | . Inthis case the sign in (1.29) is always +1. Hence, tr Cl ⊗ d tr Cl is indeed a trace. Proof of Proposition 1.2.19. Since the assertion is local, it suffices to consider an m -dimensionalEuclidean vector space V . Let us first consider the case V = R , and let e be a unit vector. ThenΛ • R = C ⊕ C e ∼ = C . Consider the following elements of End( C ) c := (cid:18) − 11 0 (cid:19) , b c := (cid:18) (cid:19) , and n := (cid:18) − (cid:19) . Then Cl( R ) ⊂ End( C ) is the algebra generated by Id and c , and b Cl( R ) is generated by Id and b c . Moreover, c b c = − b cc = − n , which implies that all monomials in c and b c have vanishing traceexcept c b c = Id. In this case,tr( c b c ) = tr(Id) = 2 = tr Cl ( c ) d tr Cl ( b c ) , which yields the claimed formula in the case that V = R .We now assume that the claim holds for and m -dimensional vector space V , and want toprove it for V ⊕ R . Let { e i } be an orthonormal basis for V , and let e be a unit vector in R . Foran ordered multi-index A = ( i < . . . < i k ) we consider the following sets of generators e A := e i ∧ . . . ∧ e i k ∈ Λ • V, c A := c ( e i ) . . . c ( e i k ) ∈ Cl( V ) , and for α ∈ { , } , e A,α := e A ∧ e α ∈ Λ • ( V ⊕ R ) , c A,α := c A c α ∈ Cl( V ⊕ R ) , where c := c ( e ). In the same way we define b c A ∈ b Cl( V ) and b c A,α ∈ b Cl( V ⊕ R ). Then a shortcomputation shows thattr Λ • ( V ⊕ R ) ( c A,α b c B,β ) = X | C |≤ m X γ ∈{ , } (cid:10) c A,α b c B,β e C,γ , e C,γ (cid:11) = X | C |≤ m X γ ∈{ , } ( − | B || α | (cid:10) ( c A b c B )( c α b c β ) e C,γ , e C,γ (cid:11) = X | C |≤ m ( − | B || α | + | C | ( | α | + | β | ) (cid:10) ( c A b c B ) e C , e C (cid:11) tr Λ • R ( c α b c β ) . .2. Dirac Operators and the Atiyah-Singer Index Theorem m = 1 we know that tr Λ • R ( c α b c β ) = 0 if ( α, β ) = (0 , Λ • ( V ⊕ R ) ( c A,α b c B,β ) = tr Λ • V ( c A b c B ) tr Λ • R ( c α b c β ) . (1.30)Now by induction we havetr Λ • V = tr Cl( V ) ⊗ \ tr Cl( V ) , tr Λ • R = tr Cl( R ) ⊗ \ tr Cl( R ) . (1.31)Moreover, as in (1.29) one checks that with respect toCl( V ⊕ R ) ⊗ s b Cl( V ⊕ R ) ∼ = (cid:0) Cl( V ) ⊗ s Cl( V ) (cid:1) ⊗ s (cid:0) b Cl( R ) ⊗ s b Cl( R ) (cid:1) one has tr Cl( V ⊕ R ) ⊗ \ tr Cl( V ⊕ R ) = (cid:0) tr Cl( V ) ⊗ \ tr Cl( V ) (cid:1) ⊗ (cid:0) tr Cl( R ) ⊗ \ tr Cl( R ) (cid:1) , which together with (1.30) and (1.31) proves the assertion for V ⊕ R . Local Index Density and the Signature Theorem. As in the above proof let V := T ∗ x M for some x ∈ M , and let R be an element in the Lie algebra so ( V ) ⊂ End( V ). Let V C be thecomplexification of V . Then iR ∈ End( V C ) is a self-adjoint endomorphism, and we can definecosh( iR ) ∈ End( V C ) via the spectral theorem. Since the eigenvalues of iR are real, and cosh is apositive function on R , we can definedet / (cid:0) cosh( iR ) (cid:1) := q det (cid:0) cosh( iR ) (cid:1) . It follows from the spectral theorem thatdet / (cid:0) cosh( iR ) (cid:1) = exp (cid:0) tr (cid:2) log cosh( iR ) (cid:3)(cid:1) , which agrees with the definition in (A.1). Note, however, that the context here is slightly differentsince we are considering elements in so ( V ) whereas in (A.1) we are considering elements of thealgebra (Λ ev C m ) ⊗ End( V C ).We then have the following version of [13, Lem. 4.5] Lemma 1.2.20. Let V be an m -dimensional oriented Euclidean vector space, and let R ∈ so ( V ) .Define b R S := − h Re i , e j i b c i b c j ∈ b Cl( V ∗ ) , where { e i } is any orthonormal basis for V , and b c i = b c ( e i ) . Then d tr Cl (cid:2) exp( i b R S ) (cid:3) = 2 m/ det / (cid:0) cosh( iR/ (cid:1) . Proof. Let k ∈ N be such that m = 2 k or m = 2 k + 1. Since R ∈ so ( V ) we can find anorthonormal basis { e i } such that R ( e j − ) = θ j e j , R ( e j ) = − θ j e j − , j = 1 , . . . , k, and R ( e k +1 ) = 0 , (1.32)where the last condition has to be considered as empty if m is even. Then b R S = − X j ≤ k θ j b c j − b c j . 1. The Signature Operator and the Rho Invariant Since c i − b c i and c j − b c j commute for i = j , one findsexp( i b R S ) = Y j ≤ k exp (cid:0) ( − iθ j / b c j − b c j (cid:1) = Y j ≤ k (cid:16) cosh( θ j / − i sinh( θ j / b c j − b c j (cid:17) , where the last equality follows from the relation ( b c j − b c j ) = − 1. By definition of the trace on b Cl( V ∗ ), we find d tr Cl (cid:2) exp( i b R S ) (cid:3) = 2 m/ Y j ≤ k cosh( θ j / . On the other hand, for all z ∈ C cosh (cid:20)(cid:18) − zz (cid:19)(cid:21) = cos( z ) (cid:18) (cid:19) , cosh(0) = 1 , from which it follows that q det (cid:0) cosh( iR/ (cid:1) = Y j ≤ k q cosh( θ j / = Y j ≤ k cosh( θ j / . The next result can be found in [13, p. 145]. Although the treatment there is only for m even, one verifies without effort that it holds for odd m as well. Lemma 1.2.21. Let ( M, g ) be an oriented Riemannian manifold with Riemann curvature tensor R g ∈ Ω (cid:0) M, End( T M ) (cid:1) . In a local orthonormal frame { e i } for T M write R g = R lkij e i ∧ e j ⊗ ( e l ⊗ e k ) , and R lkij = g ln R nkij = g (cid:0) R ( e i , e j ) e k , e l (cid:1) . Define R S ∈ Ω (cid:0) M, Cl( T ∗ M ) (cid:1) as in (1.25) , and b R S := − R lkij e i ∧ e j ⊗ b c k b c l ∈ Ω (cid:0) M, b Cl( T ∗ M ) (cid:1) . Then, the curvature R Λ • T ∗ M of the induced connection on Λ • T ∗ M decomposes as R Λ • T ∗ M = R S + b R S . Lemma 1.2.20 extends to the case R ∈ Λ V ⊗ so ( V ), where (A.1) is used to define the righthand side, see [13, pp. 144–146]. Then Lemma 1.2.20 and Lemma 1.2.21 imply the following Proposition 1.2.22. Let ( M, g ) be an oriented Riemannian manifold of dimension m , and let R g be the Riemann curvature tensor. Then b A ( T M, ∇ g ) ∧ d tr Cl (cid:2) exp( i π b R S ) (cid:3) = 2 m/ b L ( T M, ∇ g ) . with the Hirzebruch b L -form, see Definition A.1.4. From Proposition 1.2.22 and Theorem 1.2.16, we obtain the index theorem for twisted signa-ture operators. Theorem 1.2.23 (Atiah-Singer, Hirzebruch) . Let M be a closed, oriented Riemannian manifoldof even dimension m . Let E → M be a Hermitian vector bundle with connection A . .3. Manifolds with Boundary and the Eta Invariant The index of the twisted signature operator D + A is given by ind( D + A ) = 2 m/ Z M b L ( T M, ∇ g ) ∧ ch( E, A ) . (ii) If A is a flat connection, and E has rank k , then Sign A ( M ) = k Z M L ( T M, ∇ g ) = k · (cid:10) L ( T M ) , [ M ] (cid:11) , (1.33) where L ( T M, ∇ g ) is the Hirzebruch L -form as in (A.4) . Let ( M, g ) be a closed, oriented Riemannian manifold of dimension m . Let E → M be a Hermitianvector bundle, and let D : C ∞ ( M, E ) → C ∞ ( M, E )be a formally self-adjoint elliptic differential operator of first order, i.e., D ∈ P s,e ( M, E ). As M is closed, the growth of the eigenvalues of D are controlled by (1.12). This allows us to make thefollowing definition. Definition 1.3.1. The Eta function of D is defined for s ∈ C with Re( s ) > m as η ( D, s ) := X = λ ∈ spec( D ) sgn( λ ) | λ | s . Via a Mellin transform, the Eta function is related to the heat operator e − tD in the followingway η ( D, s ) = 1Γ (cid:0) s +12 (cid:1) Z ∞ Tr (cid:0) De − tD (cid:1) t s − dt, Re( s ) > m, (1.34)where Γ( s ) is the Gamma function ,Γ( s ) := Z ∞ e − t t s − dt, Re( s ) > . Note that for (1.34) to exist we are using that De − tD is trace class, and that there exist constants c and C such that for large t | Tr( De − tD ) | ≤ Ce − ct . This follows from Proposition 1.2.4, because using the notation introduced there, we haveTr( De − tD ) = Tr (cid:0) De − tD P (0 , ∞ ) (cid:1) . Thus, (1.34) yields a holomorphic function in the half planeRe( s ) > m . See Appendix C.1 for some facts about the Gamma function which we will use freely. 1. The Signature Operator and the Rho Invariant As we have noted in Theorem 1.2.7, there is an asymptotic expansionTr (cid:0) De − tD (cid:1) ∼ ∞ X n =0 t n − m − a n ( D ) , as t → , (1.35)where a n ( D ) is an integral over a quantity locally computable from the total symbol of D .Substituting the asymptotic expansion into (1.34) and dividing the integration into R + R ∞ oneeasily verifies that for each N ≥ η ( D, s ) = 1Γ (cid:0) s +12 (cid:1) (cid:16) N X n =0 a n ( D ) n − m + s + h N ( s ) (cid:17) , (1.36)where h N ( s ) is holomorphic in the half plane Re( s ) > m − ( N + 1). Since Γ (cid:0) s +12 (cid:1) − is an entirefunction, one can use this to deduce Proposition 1.3.2. The Eta function η ( D, s ) extends uniquely to a meromorphic function onthe whole plane with possible simple poles for s ∈ { m − n | n ∈ N } . Regularity at s = 0. We note that Γ (cid:0) s +12 (cid:1) − has no zeros to cancel the possible poles. Thisis an important difference between the Eta function and the Zeta function of, say, a generalizedLaplacian, see e.g. [13, Prop. 9.35]. Therefore, the following result is very remarkable. Forreferences we refer to Remark 1.3.5 below. Theorem 1.3.3 (Atiyah-Patodi-Singer, Gilkey) . Let D be a formally self-adjoint elliptic dif-ferential operator of first order on a closed, Riemannian manifold M . Then the Eta function η ( D, s ) has no pole at s = 0 . If D is a geometric Dirac operator, η ( D, s ) is holomorphic for Re( s ) > − / . Definition 1.3.4. Using the result of Theorem 1.3.3, we can define the Eta invariant of D as η ( D ) := η ( D, . Moreover, we define the ξ -invariant and the reduced ξ -invariant by ξ ( D ) := η ( D ) + dim(ker D )2 , and [ ξ ( D )] := ξ ( D ) mod Z . Remark 1.3.5. To further stress the non-triviality of Theorem 1.3.3, we want to give somehistorical remarks.(i) Atiyah, Patodi and Singer first deduced the regularity of the Eta function at 0 from theirproof of the index theorem for elliptic differential operators of first order on manifolds withboundary, see [7, Thm. 3.10]. The improved regularity for geometric Dirac operators is [7,Thm. 4.2].(ii) Later, the same authors generalized the result to pseudo-differential operators of arbitraryorder on closed, odd dimensional manifolds using K -theoretic arguments and the regularityresults for geometric Dirac operators, see [9, Thm. 4.5]. .3. Manifolds with Boundary and the Eta Invariant local Eta function of an ellipticoperator D , η ( D, s, x ) := 1Γ (cid:0) s +12 (cid:1) Z ∞ tr (cid:0) k t ( x, x ) (cid:1) t s − dt, (1.37)where k t ( x, y ) is the kernel of De − tD . Studying various examples, Gilkey found that η ( D, s, x ) is in general not regular at s = 0.(v) Later Bismut and Freed were able to refine the result of Atiyah-Patodi-Singer for geometricDirac operators. They showed using local index theory techniques, that the local Etafunction η ( D, s, x ) of a geometric Dirac operator is holomorphic for Re( s ) > − 2, see [18,Thm. 2.6]. Their result implies that for a geometric Dirac operator one can define the Etainvariant directly by η ( D ) = 1 √ π Z ∞ t − / Tr (cid:0) De − tD (cid:1) dt. (1.38) Lemma 1.3.6. Let M and N be closed, oriented Riemannian manifolds, with Hermitian vectorbundles E over M and F over N . Let D ∈ P s,e ( M, E ) and B ∈ P s,e ( N, F ) . (i) Assume that there exists and isometry ϕ : M → N , and a unitary bundle map Φ : E → F covering ϕ such that Φ ◦ D = B ◦ Φ . Then η ( D ) = η ( B ) . In particular, if M = N , E = F and { D, Φ } = 0 , then η ( D ) = 0 . (ii) If M = N , then η ( D ⊕ B ) = η ( D ) + η ( B ) . (iii) Assume that D is Z -graded with grading operator σ on E . Consider the operator D ⊗ σ ⊗ B on C ∞ (cid:0) M × N, E ⊠ F (cid:1) , with the fiber product E ⊠ F as in (1.15) . Then η (cid:0) D ⊗ σ ⊗ B (cid:1) = ind( D + ) · η ( B ) . Sketch of proof. Part (i) and (ii) of the above result are immediate for η ( D, s ) for Re( s ) large,since the whole spectrum has the respective properties. By meromorphic continuation, theycontinue to hold for s = 0. We sketch a proof of (iii).First note that D ⊗ σ ⊗ B anti-commute as operators on C ∞ (cid:0) M × N, E ⊠ F (cid:1) . Therefore, (cid:0) D ⊗ σ ⊗ B (cid:1) e − t ( D ⊗ σ ⊗ B ) = ( D ⊗ e − t ( D ⊗ ⊗ B ) + ( σ ⊗ B ) e − t ( D ⊗ ⊗ B ) . 1. The Signature Operator and the Rho Invariant To compute traces one may choose an orthonormal basis of L (cid:0) M × N, E ⊠ F (cid:1) of the form { ϕ i ⊗ ψ j } with ϕ i ∈ C ∞ ( M, E ) and ψ j ∈ C ∞ ( N, F ). Then one easily finds thatTr (cid:16) ( D ⊗ e − t ( D ⊗ ⊗ B ) (cid:17) + Tr (cid:16) ( σ ⊗ B ) e − t ( D ⊗ ⊗ B ) (cid:17) = Tr L ( M,E ) (cid:0) De − tD (cid:1) Tr L ( N,F ) (cid:0) e − tB (cid:1) + Str L ( M,E ) (cid:0) e − tD ) Tr L ( N,F ) (cid:0) Be − tB (cid:1) . Since D is Z -graded, σDe − tD = − De − tD σ and thus,Tr (cid:0) De − tD (cid:1) = Tr (cid:0) σ De − tD (cid:1) = − Tr (cid:0) σDe − tD σ (cid:1) = − Tr (cid:0) De − tD (cid:1) , where we use the trace property in the last equality. Therefore, Tr (cid:0) De − tD (cid:1) = 0. Moreover,Theorem 1.2.6 asserts that for all t > (cid:0) e − tD ) = ind( D + ) . We conclude that for Re( s ) large, η (cid:0) D ⊗ σ ⊗ B, s (cid:1) = 1Γ (cid:0) s +12 (cid:1) Z ∞ Str (cid:0) e − tD ) Tr (cid:0) Be − tB (cid:1) t s − dt = ind( D + ) · η ( B, s ) . By meromorphic continuation, part (iii) follows. The Rho Function. We now want to use the local nature of the coefficients a n ( D ) appearingin (1.36) to introduce the Rho function—respectively, the Rho invariant. Definition 1.3.7. Let D ∈ P s,e ( M, E ), and let A be a flat connection on a Hermitian vectorbundle of rank k . Denote by D A the operator D twisted by A , and use the notation D ⊕ k for theoperator D twisted by the trivial flat bundle C k . We then define the Rho function of D A as ρ ( D A , s ) = η ( D A , s ) − η ( D ⊕ k , s ) , s ∈ C . Moreover, we define the Rho invariant of D A as ρ ( D A ) := ρ ( D A , . From Theorem 1.3.3 we know that the meromorphic functions η ( D A , s ) and η ( D ⊕ k , s ) have nopole in 0. Thus, the Rho invariant is well-defined. However, unlike in the case for the individualEta invariants, this already follows from the local nature of heat trace asymptotics. Proposition 1.3.8. Let D A and D ⊕ k , where A is a flat U( k ) -connection. Then the Rho functionis holomorphic on the whole plane, and for all s ∈ C , we have ρ ( D A , s ) = 1Γ (cid:0) s +12 (cid:1) Z ∞ (cid:2) Tr (cid:0) D A e − tD A (cid:1) − Tr (cid:0) D ⊕ k e − t ( D ⊕ k ) (cid:1)(cid:3) t s − dt, (1.39) in particular, ρ ( D A ) = 1 √ π Z ∞ t − / (cid:2) Tr (cid:0) D A e − tD A (cid:1) − Tr (cid:0) D ⊕ k e − t ( D ⊕ k ) (cid:1)(cid:3) dt. .3. Manifolds with Boundary and the Eta Invariant Proof. For Re( s ) > m , we already know from (1.34) that ρ ( D A , s ) is holomorphic and that (1.39)is the correct formula. Moreover, as for the Eta function, we can split up the integral and usethat R ∞ extends to a holomorphic function on C . Concerning R , we now make the followingobservation: As we have already seen in the proof of Corollary 1.2.9, the operators D A and D ⊕ k are locally equivalent. Then Theorem 1.2.7 implies that all coefficients of the asymptoticexpansions of Tr (cid:0) D A e − tD A (cid:1) and Tr (cid:0) D ⊕ k e − t ( D ⊕ k ) (cid:1) as t → N there exists a constant C such that as t → (cid:12)(cid:12) Tr (cid:0) D A e − tD A (cid:1) − Tr (cid:0) D ⊕ k e − t ( D ⊕ k ) (cid:1)(cid:12)(cid:12) ≤ Ct N . This shows that the integral Z (cid:2) Tr (cid:0) D A e − tD A (cid:1) − Tr (cid:0) D ⊕ k e − t ( D ⊕ k ) (cid:1)(cid:3) t s − dt exists and defines a holomorphic function for all s ∈ C with Re( s ) > − ( N + 1). Continuing inthis way, one finds that ρ ( D A , s ) is holomorphic for s ∈ C . While the Eta invariant is a spectral invariant and thus encodes global information about themanifold and the operator, its deformation theory turns out to be expressible in terms of heattrace asymptotics, thus being a local quantity. We have included further details in Appendix D,and summarize only briefly what we need here. Families of Operators. Let M be a closed manifold of dimension m , and let E → M be aHermitian vector bundle. Let U ∈ R p be open, and let ( D u ) u ∈ U be a p -parameter family ofdifferential operators on C ∞ ( M, E ) of order d . Choosing local frames for E we can write locally D u = X | α |≤ d a α ( x, u ) ∂ αx , (1.40)where α = ( α , . . . , α m ) is a multi-index, x = ( x , . . . , x m ) is a local coordinate chart on M , andthe a α ( x, u ) are matrix valued functions. Definition 1.3.9. The p -parameter family ( D u ) u ∈ U is called smooth if for every local frame andcoordinate chart, the functions a α ( x, u ) as in (1.40) are smooth, jointly in x and u .We also need a functional analytic consequence of the geometric notion of smoothness we havejust defined. Let s, s ′ ∈ R and denote by B ( L s , L s ′ ) the space of bounded linear operators L s → L s ′ endowed with the operator norm k . k s,s ′ . The proof of the following result is straightforward. Lemma 1.3.10. Let ( D u ) u ∈ U be a p -parameter family of differential operator of order d ≥ which is smooth in the sense of Definition 1.3.9. Then for all s ∈ R , we have a smooth map U → B ( L s + d , L s ) , u D u . For more details on the following results we refer to Appendix D, in particular PropositionD.2.4 and Proposition D.2.5.8 1. The Signature Operator and the Rho Invariant Proposition 1.3.11. Let ( D u ) u ∈ R be a smooth one-parameter family of formally self-adjointelliptic operators of first order on C ∞ ( M, E ) , and let a m ( dD u du , D u ) denote the constant term inthe asymptotic expansion of √ t Tr (cid:0) dD u du e − tD u (cid:1) , as t → . (1.41) Then the following holds. (i) Assume that dim(ker D u ) is constant. Then the meromorphic extension of η ( D u , s ) is con-tinuously differentiable in u , and ddu η ( D u ) = − √ π a m ( dD u du , D u ) . (1.42)(ii) Without the assumption on ker( D u ) , the reduced ξ -invariant [ ξ ( D u )] ∈ R / Z is continuouslydifferentiable in u , and ddu [ ξ ( D u )] = − √ π a m ( dD u du , D u ) . (1.43)An immediate consequence of the local nature of a m ( dD u du , D u ) is the following result, see [8,Thm. 3.3]. Corollary 1.3.12. Let ( D u ) u ∈ R be a smooth one-parameter family of operators in P s,e ( M, E ) ,and let A be a flat connection on a Hermitian vector bundle of rank k . Denote by ( D A,u ) u ∈ R theone-parameter family of operators obtained by twisting with A . (i) If the kernels of D u and D A,u are of constant dimensions, then the Rho invariant ρ ( D A,u ) is independent of u . (ii) In the general case, only the reduced Rho invariant, (cid:2) ρ ( D A,u ) (cid:3) := (cid:2) ξ ( D A,u ) (cid:3) − k · (cid:2) ξ ( D u ) (cid:3) , is independent of u .Proof. As in the proof of Proposition 1.3.8, the operators D A,u and D ⊕ ku are locally equivalentsmooth families in the sense of (1.17). Since the twisting connection is independent of u , thefamilies (cid:0) dD ⊕ ku du (cid:1) u ∈ R and (cid:0) dD A,u du (cid:1) u ∈ R are locally equivalent as well. Now, the a n in the asymptoticexpansion of (1.41) can be computed locally from the total symbols of the involved operators,and so a m (cid:0) dD ⊕ ku du , ( D ⊕ ku ) (cid:1) = a m (cid:0) dD A,u du , ( D A,u ) (cid:1) . Since η ( D ⊕ ku ) = k · η ( D u ), the result follows from (1.42) respectively (1.43). Remark. One can also use Proposition 1.3.8 and proceed as in Appendix D to differentiateunder the integral to proof part (i) in Corollary 1.3.12. This has the advantage that it is a bitless involved than the proof of the variation formula of the individual Eta invariant, since onedoes not have to deal with meromorphic continuations. However, the main steps remain thesame. .3. Manifolds with Boundary and the Eta Invariant Spectral Flow. The smoothness of [ ξ ( D u )] shows that the discontinuities of ξ ( D u ) as u variesare only integer jumps. Heuristically, this is due to the fact that the Eta invariant is a regularizedsignature so that whenever an eigenvalue of D u crosses 0, it changes by an integer multiple of 2.This can be made more precise using the notion of spectral flow , which we now introduce briefly.For a smooth one-parameter family of formally self-adjoint elliptic operators ( D u ) u ∈ [ a,b ] , itcan be shown that the associated family of compact resolvents varies smoothly with u in theoperator norm on L ( M, E ), see the proof of Theorem D.1.7 for some related ideas. This impliesthat the eigenvalues of D u can be arranged in such a way that they vary continuously with u . Inparticular, we can find a partition a = u < u < . . . < u n = b such that for each i ∈ { , . . . , n } there is c i > c i / ∈ [ u ∈ [ u i − ,u i ] spec( D u ) . (1.44)For u ∈ [ u i − , u i ] denote by P [0 ,c i ] ( u ) the finite-rank spectral projection associated to eigenvaluesin the interval [0 , c i ]. We then define the spectral flow in the spirit of [82]. Definition 1.3.13. Let ( D u ) u ∈ [ a,b ] be a smooth one-parameter family in P ds,e ( M, E ), and let a = u < u < . . . < u n = b be a partition such that there exist c i as in (1.44). Then the spectralflow between D a and D b is defined asSF( D u ) u ∈ [ a,b ] := n X i =1 rk (cid:0) P [0 ,c i ] ( u i ) (cid:1) − rk (cid:0) P [0 ,c i ] ( u i − ) (cid:1) . Figure 1.1: Spectral FlowWithout going into further details, we note that the spectral flow is well-defined and inde-pendent of the choices made. Moreover, as indicated in Figure 1.1, there is a built-in conventionof how to count zero eigenvalues at the end points. For more details and generalizations werefer to [20, 65, 82]. In the situation at hand, there is also the approach as in [89] using Kato’sselection theorem. The Variation Formula. With the notion of spectral flow at hand, we can now state a resulton the variation of the Eta invariant, due to Atiyah, Patodi and Singer in [9]. We sketch a proofin Corollary D.2.6, see also [61, Lem. 3.4].0 1. The Signature Operator and the Rho Invariant Proposition 1.3.14. Let ( D u ) u ∈ [ a,b ] be a smooth one-parameter family of operators in P e,s ( M, E ) , where M is closed. Then ξ ( D b ) − ξ ( D a ) = SF( D u ) u ∈ [ a,b ] + Z ba ddu [ ξ ( D u )] du. To relate the Eta invariant to the index theorem, we have to leave the realm of closed manifoldsbriefly. Product Structures at the Boundary. Let N be compact manifold with boundary ∂N = M .We equip N with a metric g N of product type near the boundary, i.e., we assume that a collarof the boundary is isometric to ( − , × M endowed with the metric g = dt + g M , where g M isa metric on M . If N is oriented, we get an induced orientation on ( − , × M which we use toorient M . This is the outward normal first convention, see Figure 1.2. Atiyah, Patodi and Singerin [7] use a different convention, which explains some sign differences.Figure 1.2: Collar and orientation convention Definition 1.3.15. A Z -graded, formally self-adjoint elliptic differential operator of first order D : C ∞ ( N, E ) → C ∞ ( N, E ) is called in product form if the following holds.(i) On a collar of M E + | ( − , × M = π ∗ E M , where π : ( − , × M → M is the projection, and E M is a Hermitian vector bundle over M .(ii) If γ is the bundle isomorphism E + | M → E − | M , given by applying the symbol of D + to theoutward normal unit vector, then D + = γ (cid:0) ddt − D M (cid:1) , over ( − , × M, where D M is a formally self-adjoint elliptic differential operator of first order on E M , calledthe tangential operator . .3. Manifolds with Boundary and the Eta Invariant closeddouble X = N ∪ M − N. Here, − N denotes N with the reversed orientation. One uses γ to glue E | ∂N to E | ∂ ( − N ) to get abundle E X → X , and the operator D extends naturally over X . For more details and a detailedproof of the following we refer to [21, Thm. 9.1]. Proposition 1.3.16. If D is in product form, then the natural extension D X : C ∞ ( X, E X ) → C ∞ ( X, E X ) is invertible and extends D in the sense that D X | C ∞ ( N,E ) = D . Boundary Conditions and the Index Theorem. To formulate an index theorem for D oneneeds to introduce suitable boundary conditions in order to render D Fredholm. As observedby Atiyah and Bott [4], most geometric Dirac operators do not admit local boundary conditionsthat set up a well-posed boundary value problem. Atiyah, Patodi and Singer [7] solved this byintroducing the following global boundary projection. Definition 1.3.17. Let D ∈ P s,e ( N, E ) be Z -graded and in product form, with tangentialoperator D M . The Atiyah-Patodi-Singer projection P ≥ ( D M ) : C ∞ ( M, E M ) → C ∞ ( M, E M )is defined as the spectral projection onto the subspace spanned by the eigenvectors of D M corre-sponding to eigenvalues ≥ Theorem 1.3.18 (Atiyah-Patodi-Singer) . Let N be compact, oriented Riemannian manifold ofdimension n with boundary ∂N = M , and let D : C ∞ ( N, E ) → C ∞ ( N, E ) be a Z -graded, formally self-adjoint elliptic differential operator of first order. Assume that themetric and D are in product form in a collar of M . Let C ∞ (cid:0) N, E + ; P ≥ (cid:1) := (cid:8) ϕ ∈ C ∞ ( N, E + ) (cid:12)(cid:12) P ≥ ( D M )( ϕ | M ) = 0 (cid:9) , where P ≥ ( D M ) is the projection of Definition 1.3.17. Then D + : C ∞ (cid:0) N, E + ; P ≥ (cid:1) → C ∞ (cid:0) N, E − (cid:1) has a natural Fredholm extension with ind (cid:0) D + ; P ≥ (cid:1) = Z N str E [ e e n ] vol N − ξ ( D M ) . (1.45) Here, ξ ( D M ) is the ξ -invariant as in Definition 1.3.4, and e e n ( x ) is the coefficient of the constantterm in the asymptotic expansion as t → of the heat kernel e − tD X ( x, x ) associated to theextension D X to the closed double X . 1. The Signature Operator and the Rho Invariant Remark. We are following [7] and use the operator D X on the closed double to define the indexdensity. Clearly, this is an ad hoc method which allows to use the asymptotic expansion of theheat trace in its version for closed manifolds as in Theorem 1.14. However, the trace expansioncan be formulated in a much more abstract functional analytic setting to incorporate the case ofmanifolds with boundary, see [24] for an expository account.If dim N is even and D is a Z -graded geometric Dirac operator, the index density can bemade explicit using the local index theorem for closed manifolds. One obtains the index theoremfor geometric Dirac operators on manifolds with boundary, see [7, Thm. 4.2]. Theorem 1.3.19 (Atiyah-Patodi-Singer) . Let N be compact, oriented Riemannian manifold ofeven dimension n with boundary ∂N = M , and let D : C ∞ ( N, E ) → C ∞ ( N, E ) be a Z -graded,geometric Dirac operator which is in product form near M . Then the index of D + : C ∞ (cid:0) N, E + ; P ≥ (cid:1) → C ∞ (cid:0) N, E − (cid:1) is given by ind (cid:0) D + ; P ≥ (cid:1) = Z N b A ( T N, ∇ g ) ∧ ch E/S ( E, ∇ E ) − ξ ( D M ) , (1.46) where b A ( T N, ∇ g ) is the b A -form of Definition A.1.4 with respect to the Levi-Civita connection ∇ g and ch E/S is the realtive Chern character (1.26) . In Proposition 1.1.8 we have seen that the signature of closed manifolds equals the index ofthe signature operator. One motivation that led Atiyah, Patodi and Singer to the discovery ofTheorem 1.3.18 was the search for a generalization of this to the case for manifolds with boundary.We thus briefly recall the definition of the signature of a manifold with boundary. Let N be a compact, oriented and connected manifold of dimension n with boundary ∂N . Then ∂N is closed and naturally oriented, but we allow that it consists of several connected components.Let α : π ( N ) → U( k ) be a representation of the fundamental group. Via the map induced bythe inclusion ∂N ֒ → N , the representation α restricts to π ( ∂N ). There is a relative version H • ( N, ∂N, E α ) of cohomology with local coefficients, whose construction we will not describein detail. We only note that the machinery of algebraic topology—like cup and cap products,Poincar´e duality, and exact sequence of pairs—extends to this context . In particular, there is arelative intersection pairing H p ( N, E α ) × H n − p ( N, ∂N, E α ) → C , ( a, b ) (cid:10) a ∪ b , [ N, ∂N ] (cid:11) , where the fundamental class [ N, ∂N ] is the generator of H n ( N, ∂N ) determined by the orientation.Via the de Rham isomorphism, the relative cohomology groups are isomorphic to de Rhamcohomology with compact support in the interior of N , H p ( N, ∂N, E α ) ∼ = H pc ( N, E α ) . However, one has to be careful with the Mayer-Vietoris sequence and excision, where one has to use vanKampen’s Theorem to relate the representations of the involved fundamental groups. .4. The Atiyah-Patodi-Singer Rho Invariant E α to denote the flat vector bundle determined by α . With respect to thisidentification, the intersection pairing is induced byΩ p ( N, E α ) × Ω n − pc ( N, E α ) C , ( ω, η ) Z N h ω ∧ η i . Now assume that n is even. There is a natural map H n/ ( N, ∂N, E α ) → H n/ ( N, E α ) which wecan combine with the intersection pairing to define a twisted intersection form Q α : H n/ ( N, ∂N, E α ) × H n/ ( N, ∂N, E α ) → C . As for closed manifolds, the intersection form is skew for ( n ≡ n ≡ Definition 1.4.1. Let N be a compact, connected manifold with boundary of even dimension n , and let α : π ( N ) → U( k ) be a representation of the fundamental group . Then the twistedsignature of M is defined as Sign α ( N ) := Sign( Q α ) , where we use the earlier convention regarding the signature of a skew endomorphism. Remark. To turn Q α into a non-degenerate form, one needs to restrict to b H n/ ( N, E α ) := im (cid:0) H n/ ( N, ∂N, E α ) → H n/ ( N, E α ) (cid:1) . Then Poincar´e duality ensures that we get a non-degenerate form b Q α : b H n/ ( N, E α ) × b H n/ ( N, E α ) → C . Since this process just eliminates the radical of Q α , it is immediate thatSign Q α = Sign b Q α . To state the relation of the signature on a manifold with boundary to the index of the signatureoperator, we first need to understand the structure of the signature operator near the boundary.Therefore, we now consider the model case of a cylinder, and derive the formula for the oddsignature operator.Let M be a closed, oriented manifold of odd dimension m and consider the cylinder N := R × M . We use the natural splitting T ∗ N = R ⊕ T ∗ M to orient N and make the identificationΦ : C ∞ (cid:0) R , Ω • ( M ) (cid:1) ⊕ C ∞ (cid:0) R , Ω • ( M ) (cid:1) ∼ = −→ Ω • ( N ) , Φ( ω , ω ) := dt ∧ ω + ω , where t denotes the R coordinate. Clearly, d N Φ (cid:0) ω , ω ) = dt ∧ (cid:0) ∂ t ω − d M ω (cid:1) + d M ω = Φ (cid:0) ∂ t ω − d M ω , d M ω (cid:1) . (1.47)We now endow N with a metric of product form g = dt + g M , and denote by τ N and τ M thechirality operators on Ω • ( N ) and Ω • ( M ), respectively. Then one checks that τ N Φ (cid:0) ω , ω ) = Φ (cid:0) τ M ω , τ M ω (cid:1) . 1. The Signature Operator and the Rho Invariant From this we obtain isomorphismsΦ ± : C ∞ (cid:0) R , Ω • ( M ) (cid:1) ∼ = −→ Ω ± ( N ) , ω Φ( ω, ± τ M ω ) . Let D + N : Ω + ( N ) → Ω − ( N ) be the signature operator. Then a short computation using (1.47)and the formula (1.7) for the adjoint differential yieldsΦ − − ◦ D + N ◦ Φ + = τ M (cid:0) ∂ t − τ M d M − d M τ M (cid:1) . The same continues to hold if E → M is a Hermitian vector bundle, and when we twist with aunitary connection A on π ∗ E in temporal gauge , i.e., a connection of the form A := π ∗ a . Here, π : N → M is the natural projection and a is a unitary connection on E . We summarize whatwe have observed so far. Proposition 1.4.2. Let D + A be the signature operator on the cylinder N = R × M twisted by aunitary connection A = π ∗ a in temporal gauge. Then D + A is isometric to τ M (cid:0) ∂ t − B a (cid:1) : C ∞ (cid:0) R , Ω • ( M, E ) (cid:1) → C ∞ (cid:0) R , Ω • ( M, E ) (cid:1) , where B a := τ M (cid:0) d a + d ta ) = τ M d a + d a τ M . (1.48) Definition 1.4.3. Let a be a unitary connection over an oriented Riemannian manifold M ofodd dimension. The operator B ev a := B a | Ω ev ( M,E ) : Ω ev ( M, E ) → Ω ev ( M, E )is called the odd signature operator on M twisted by a . Remark 1.4.4. (i) Note that the operator B a does indeed preserve the even/odd grading, B a = B ev a ⊕ B odd a : Ω ev ( M, E ) ⊕ Ω odd ( M, E ) → Ω ev ( M, E ) ⊕ Ω odd ( M, E ) . Moreover, B ev a and B odd a are conjugate via τ M . The kernels of B a and B ev a are of a topo-logical nature, sinceker( B a ) = ker( d a + d ta ) = H • ( M, E a ) , ker( B ev a ) = H ev ( M, E a ) . (ii) It is straightforward to check that the odd signature operator is a geometric Dirac operatorin the sense of Definition 1.2.11. Here, one defines the Clifford structure by c ev : T ∗ M ⊗ Λ ev T ∗ M → Λ ev T ∗ M, c ev ( ξ ) ω := τ M (cid:0) ξ ∧ ω − i( ξ ) ω (cid:1) . (1.49)Paying close attention to the various identifications made, one also verifies that the map τ M : Λ • T ∗ M → Λ • T ∗ M corresponds to Clifford multiplication c N ( dt ) : Λ + T ∗ N | M → Λ − T ∗ N | M . Hence, Proposition 1.4.2 shows that D + A is of product form in the sense of Definition 1.3.15. .4. The Atiyah-Patodi-Singer Rho Invariant m = 2 k − 1. Then one can check, using the formula (1.6)for τ M , that for all ω ∈ Ω p B a ω = − i k + p ( p +1) (cid:0) ( − p ∗ d a − d a ∗ (cid:1) ω. In particular for p = 2 q , B ev a ω = i k ( − q +1 ( ∗ d a − d a ∗ ) ω. Having identified the tangential operator, the APS projection sets up a well-defined index problemfor the signature operator on manifolds with boundary. Moreover, the Atiyah-Patodi-Singer IndexTheorem for geometric Dirac operators 1.3.19 and the index theorem for the twisted signatureoperator in Theorem 1.2.23 calculates its index. If the twisting bundle is flat, this index isrelated to the signature for manifolds with boundary as introduced in Section 1.4.1. However,this relation is more difficult to establish than in Proposition 1.1.8 for the case that the manifoldis closed. We only state the result and refer to [7, Sec. 4] for the proof. A concise discussion alsobe found in [21, Sec. 23]. Theorem 1.4.5 (Atiyah-Patodi-Singer) . Let N be a compact, oriented Riemannian manifold ofeven dimension n with boundary ∂N = M , and let E → N be a Hermitian vector bundle of rank k with a unitary connection A . Assume that the metric is in product form, and that A is intemporal gauge A = π ∗ a on a collar of M . Let D + A be the twisted signature operator on N , andlet P ≥ ( a ) : Ω • ( M, E | M ) → Ω • ( M, E | M ) be the APS projection in Definition 1.3.17 of the tangential operator B a . Then ind (cid:0) D + A ; P ≥ ( a ) (cid:1) = 2 n/ Z N b L ( T N, ∇ g ) ∧ ch( E, A ) − ξ ( B a ) . Moreover, if A is flat, then ind (cid:0) D + A ; P ≥ ( a ) (cid:1) = Sign A ( N ) − dim(ker B a ) , and therefore, Sign A ( N ) = k · Z N L ( T N, ∇ g ) − η ( B ev a ) . Remark. In the last equation, the occurrence of B ev a stems from the relation ξ ( B a ) = η ( B ev a ) + dim(ker B a ) , which follows from Remark 1.4.4 (i). Rho Invariants. Motivated by Theorem 1.4.5, Atiyah, Patodi and Singer [8] introduced theRho invariant, which we have already briefly considered in Proposition 1.3.8 and Corollary 1.3.12.We now treat the Rho invariant associated to the odd signature operator in more detail.6 1. The Signature Operator and the Rho Invariant Definition 1.4.6. Let M be a closed, oriented Riemannian manifold of odd dimension m , andlet A be a flat unitary connection on a Hermitian bundle E of rank k . Then the Rho invariantof A is defined as ρ A ( M ) := ρ ( B ev A ) = η ( B ev A ) − k · η ( B ev ) . We have the following immediate consequences of what we have discussed so far. Proposition 1.4.7. (i) If A ′ is a flat connection, unitarily equivalent to A , then ρ A ′ ( M ) = ρ A ( M ) . In particular,the Rho invariant depends only on the holonomy representation hol A : π ( M ) → U( k ) . For this reason, we also use the notation ρ α ( M ) if the focus is on representations of thefundamental group. (ii) The Rho invariant is independent of the metric used to define η ( B ev A ) and η ( B ev ) . Therefore,it is a smooth invariant of M and A . (iii) If N is a compact, oriented manifold with boundary ∂N = M , and the representation α : π ( M ) → U( k ) extends to a unitary representation β : π ( N ) → U( k ) , then ρ α ( M ) = Sign β ( N ) − k · Sign( N ) . Proof. Part (i) follows from the fact that the Eta invariant does not change, if we transformwith a unitary bundle isomorphism. Part (ii) is a consequence of Corollary 1.3.12 (i), sincethe dimensions of the kernels of B ev α and B ev are independent of the metric. Part (iii) followsimmediately from the signature formula of Theorem 1.4.5. Remark 1.4.8. (i) Part (i) of Proposition 1.4.7 shows that the Rho invariant can be interpreted as a map,defined on the moduli space of flat connections ρ : M (cid:0) M, U( k ) (cid:1) → R , [ A ] ρ A ( M ) . In the explicit examples in Section 2.3 and Chapter 4 we will use this and consider partic-ularly well-suited representatives for gauge equivalence classes of flat connections.(ii) Proposition 1.4.7 (iii) gives a negative answer to the question of whether Corollary 1.2.10continues to hold for the signature of manifolds with boundary. As mentioned in theintroduction, an explanation of this signature defect was one of the motivations leading tothe discovery of Theorem 1.4.5, and the Rho invariant is indeed an intrinsic characterizationof this.(iii) Rho invariants do in general give non-trivial invariants. As an easy example we consider M = S . We view S as a subset of C , endowed with the metric of length 2 π . A flatU(1)-connection on the trivial line bundle is determined by its holonomy e πia ∈ U(1) with a ∈ R . The corresponding odd signature operator is easily seen to be B ev a = − i ( L e − ia ) : C ∞ ( S ) → C ∞ ( S ) , .5. Rho Invariants and Local Index Theory ϕ ∈ C ∞ ( S ) and z ∈ S , L e ϕ ( z ) = ddt (cid:12)(cid:12) t =0 ϕ ( ze it ) . Therefore, B ev a ϕ = λϕ if and only if λ + a ∈ Z and ϕ ( z ) = z λ + a ϕ (1) . This implies η ( B ev a , s ) = X n ∈ Z n = a sgn( n − a ) | n − a | s , Re( s ) > . In Proposition C.1.2 (i), we have included a computation of the value of the meromorphiccontinuation of this expression at 0. The result is, see also Definition C.1.1, η ( B ev a ) = 2 P ( a ) = ( , for a ≡ Z , a − , for a ∈ (0 , 1) and a ≡ a mod Z .(iv) We want to point out that the Rho invariant is a true extension of the signature defect.More explicitly, there are Rho invariants which cannot be calculated using the formula ofProposition 1.4.7 (iii). As an example, consider a compact oriented surface Σ with oneboundary component ∂ Σ = S . Then the fundamental class of S is a commutator in π (Σ), see for example the discussion in Section 2.3.1. This implies that a non-trivialU(1)-representations of π ( S ) cannot extend to a representation of π (Σ). As seen in Proposition 1.4.7, the Rho invariant can be computed in a purely topological wayif the representation α extends over a bounding manifold. However, we have already pointedout that this situation is often too restrictive. Therefore, intrinsic methods to compute Rhoinvariants are of great interest. One observation for applying topological tools is the relation ofRho invariants to Chern-Simons invariants. We refer to Appendix A for definitions and basicproperties of Chern-Simons invariants, and make the relation more precise now. The followingresult goes back to [8, Sec. 4], see also [60, Sec. 7]. Proposition 1.5.1. Assume that M is a closed, oriented Riemannian manifold of odd dimension m . Let A t be a smooth path of connections on a fixed Hermitian vector bundle E → M . Thenthe reduced ξ -invariant satisfies Z ddt [ ξ ( B A t )] dt = 2 m +12 Z M b L ( T M, ∇ g ) ∧ cs( A , A ) , where cs( A , A ) is the transgression form of the Chern character, see Definition A.1.6. We postpone the proof and mention some consequences.8 1. The Signature Operator and the Rho Invariant Corollary 1.5.2. Let A t be a smooth path of connections on E . (i) The following variation formula holds ξ ( B A ) − ξ ( B A ) = SF( B A t ) t ∈ [0 , + 2 m +12 Z M b L ( T M, ∇ g ) ∧ cs( A , A ) . (ii) If A and A are flat, and M is 3-dimensional, then ρ A ( M ) = ρ A ( M ) + 4 CS( A , A ) mod Z . Here, CS( A , A ) is the Chern-Simons invariant associated to the Chern character, seeDefinition A.2.3 and (A.7) . (iii) Assume that A t is a path of flat connections and that either A and A reduce to SU( k ) -connections or (dim M ≡ . Then η ( B ev A ) − η ( B ev A ) = 2 SF( B ev A t ) t ∈ [0 , − dim ker B ev A + dim ker B ev A . In particular, ρ A ( M ) = ρ A ( M ) mod Z . Remark 1.5.3. (i) Corollary 1.5.2 (ii) shows that on a 3-manifold the reduction mod Z of the Rho invari-ant is up to a constant the Chern-Simons invariant of the corresponding flat connection.Therefore, the reduced Rho invariant is the integral over local invariants of the connections.Now, the unreduced version is essentially this “local” contribution plus a spectral flow termwhich encodes “global” topological information.(ii) Under the hypothesis of part (iii) the reduction mod Z of ρ A t ( M ) is constant. In otherwords, the Chern-Simons invariant is constant on connected components of the modulispace of flat connections. Therefore, unreduced Rho invariants associated to one-parameterfamilies of flat connections have only integer jumps, which occur precisely at the pointswhere the rank of the twisted cohomology groups changes. Independently, Farber-Levine[39] and Kirk-Klassen [58, 59] have developed powerful methods to compute this spectralflow term in purely cohomological terms.We will now give a proof of Proposition 1.5.1 based on the signature theorem for manifoldswith boundary in Theorem 1.4.5. We shall also sketch a different proof using Getzler’s approachto local index theory in Section 1.5.2 below. Proof of Proposition 1.5.1. Consider the cylinder N := [0 , × M , endowed with the productmetric g N = du + π ∗ g . Here, we are using u to denote the coordinate on [0 , π : N → M is the natural projection. We endow π ∗ E with the connection e A t := du ∧ ddu + π ∗ A tϕ ( u ) , where ϕ : [0 , → [0 , 1] is a smooth function such that for some ε > ϕ ( u ) = ( , if u < ε, , if u > − ε, (1.50) .5. Rho Invariants and Local Index Theory N and the cutoff function ϕ see Figure 1.3. For fixed t , the connection e A t is in temporal gauge on a collar of ∂N . Therefore,we can apply Theorem 1.4.5 for each t to the signature operator D + e A t and conclude that ξ ( B A t ) − ξ ( B A ) = 2 m +12 Z N b L ( T N, ∇ g N ) ∧ ch( π ∗ E, e A t ) mod Z . Since g N is the product metric on N it is straightforward to check that b L ( T N, ∇ g N ) = π ∗ b L ( T M, ∇ g ) . Then, if R N/M denotes integration along the fiber, see Proposition 2.1.12 below, we observe that Z N b L ( T N, ∇ g N ) ∧ ch( π ∗ E, e A t ) = Z M b L ( T M, ∇ g ) ∧ Z N/M ch( π ∗ E, e A t )= Z M b L ( T M, ∇ g ) ∧ cs( A tϕ ( u ) ) . Here, we have used Lemma A.2.1 in the second line to replace R N/M ch( π ∗ E, e A t ) with the trans-gression form of the Chern character computed with respect to the path u A tϕ ( u ) . If we choosea different path, the result will differ by an exact form on M , see Proposition A.2.2 for a proof.This allows us to remove the function ϕ and use the path u A tu . Thencs( A tϕ ( u ) ) = cs( A , A t ) mod d Ω ev ( M ) . Since M is closed, this shows that ξ ( B A t ) − ξ ( B A ) = 2 m +12 Z M b L ( T M, ∇ g ) ∧ cs( A , A t ) mod Z , which implies Proposition 1.5.1.0 1. The Signature Operator and the Rho Invariant Proof of Corollary 1.5.2. Part (i) is an immediate consequence of Proposition 1.5.1 and Propo-sition 1.3.14. Next, we note that ξ ( B A ) − ξ ( B A ) = η ( B ev A ) − η ( B ev A ) + dim ker( B ev A ) − dim ker( B ev A ) . (1.51)Using part (i) and reducing mod Z one finds that η ( B ev A ) − η ( B ev A ) = 2 m +12 Z M b L ( T M, ∇ g ) ∧ cs( A , A ) mod Z . Now, if we assume that m = 3, the b L -form equals 1, and we obtain part (ii). Concerning part(iii), we now consider the connection e A := dt ∧ ddt + A t on the cylinder [0 , × M . Since we areassuming that A t is a path of flat connections, we have F e A = dt ∧ ddt A t , and exp (cid:0) dt ∧ ddt A t (cid:1) = 1 + dt ∧ ddt A t . This implies Z M b L ( T M, ∇ g ) ∧ cs( A , A ) = Z [0 , × M π ∗ b L ( T M, ∇ g ) ∧ tr E (cid:2) exp( i π F e A ) (cid:3) = Z M b L ( T M, ∇ g ) ∧ i π tr E [ A − A ] . If A and A are SU( k )-connections, then tr E [ A − A ] = 0, so that the integrand vanishes.On the other hand, if ( m ≡ m part so that the integralvanishes again. In both cases, part (iii) follows from part (i) and (1.51). Variation of the Metric. As already pointed out in Remark 1.5.3, an important tool forstudying the Rho invariant is studying its variation under deformations of the flat connection.However, the moduli space of flat connections is often discrete or at least disconnected. Therefore,it is not always possible to find a path of flat connections which joins given endpoints. However,in many cases one can deform the geometry of the underlying manifold in such a way that twistedEta invariants become computable. The main concern of this thesis are Rho invariants of fiberbundles, and as already explained in the introduction there are powerful methods to deform thegeometry to a much simpler situation. One underlying result is the following, which is an analogof Proposition 1.5.1, see [8, Sec. 2]. Proposition 1.5.4. Assume that g t is a smooth path of Riemannian metrics on a closed, odddimensional manifold M , and denote by ∇ g t the associated one-parameter family of Levi-Civitaconnections. Moreover, let A be a flat connection on a Hermitian vector bundle E → M of rank k . Then η ( B ev A , g ) − η ( B ev A , g ) = k · Z M T L ( ∇ g , ∇ g ) , where T L ( ∇ g , ∇ g ) is the transgression form of the L -class of T M , see Remark A.1.7.Proof. As before, let N := [0 , × M be the cylinder, endowed with the bundle π ∗ E and theconnection e A := dt ∧ ddt + π ∗ A . In contrast to the proof of Proposition 1.5.1, the connection e A isflat, since A is independent of t . Let α : π ( N ) → U( k ) denote the holonomy representation of e A . Recall that Sign α ( N ) is defined using the homomorphism H • ( N, M, E α ) → H • ( N, E α ) . .5. Rho Invariants and Local Index Theory M is a deformation retract of N , and α iscompatible with the natural retraction. Therefore, Sign α ( N ) = 0.Now let ϕ be a cutoff function as in (1.50), and endow N with the metric g N := dt + π ∗ g ϕ ( t ) . Let ∇ g N be the Levi-Civita connection associated to g N . Since g N is in product form near theboundary and Sign A ( N ) = 0, Theorem 1.4.5 yields η ( B ev A , g ) − η ( B ev A , g ) = k · Z N L ( T N, ∇ g N ) . Moreover, ∇ N is in temporal gauge on a collar of the boundary, so that we can deduce fromProposition A.2.4 and Remark A.2.5 that Z N L ( T N, ∇ g N ) = Z M T L ( ∇ g , ∇ g ) . Remark 1.5.5. In the next chapter we will also encounter a one-parameter family of connections( ∇ t ) t ∈ [0 , on T M which is not associated to a family of Riemannian metrics. Nevertheless, wecan study the associated family of generalized odd signature operators D t := c ev ◦ ∇ t : Ω ev ( M ) → Ω ev ( M ) , where c ev is Clifford multiplication as defined in (1.49). Certainly, the operators D t will in generalnot be formally self-adjoint, even if all ∇ t are compatible with the metric. Without going intodetail, we note that a certain restriction on the torsion tensor of ∇ t guarantees that we getformally self-adjoint operators. Then we can use the variation formula of Proposition 1.3.14, ξ ( D ) − ξ ( D ) = SF( D t ) t ∈ [0 , + Z ddt (cid:2) ξ ( D t (cid:3) dt. However, since ( D t ) t ∈ [0 , is not a family of geometric Dirac operators, the local variation cannot be identified using Theorem 1.4.5. Yet, if we are interested in Rho invariants, it followsfrom Corollary 1.3.12 that for any flat U( k )-connection A , the local variations of ( D ⊕ kt ) t ∈ [0 , and( D A,t ) t ∈ [0 , agree. In particular, ρ ( D A, ) = ρ ( D A, ) + SF( D A,t ) t ∈ [0 , − k · SF( D t ) t ∈ [0 , . (1.52) In the remainder of this chapter we sketch a proof of Proposition 1.5.1 based on local indextheory techniques. Mainly, this is because the underlying ideas will be helpful in the discussionof Rho invariants of fiber bundles in Chapter 3. Aside from that, a proof of the variation formulain Proposition 1.5.1, which does not rely on the index theorem for manifolds with boundary,underlines the intrinsic nature of Rho invariants. The setup. Assume that M is a closed, oriented Riemannian manifold of odd dimension m , andlet A u be a smooth path of connections on a fixed Hermitian vector bundle E → M . We wantto have an explicit formula for the variation ddu [ ξ ( B A u )] of the reduced ξ -invariant, where B A u = τ (cid:0) d A u + d tA u (cid:1) : Ω • ( M, E ) → Ω • ( M, E ) . 1. The Signature Operator and the Rho Invariant Proposition D.2.5 shows that ddu [ ξ ( B A u )] = − √ π a m ( B A u ) , where a m ( B A u ) is the constant term in the asymptotic expansion of √ t Tr (cid:0) dB Au du e − tB Au (cid:1) , as t → . For brevity we will also use the common notation a m ( B A u ) = LIM t → √ t Tr (cid:0) dB Au du e − tB Au (cid:1) . (1.53)In the case at hand, it is immediate that dB Au du = τ c (cid:0) ddu A u (cid:1) . Now, to get a formula for a m ( B A u ), we can fix u . To keep the notation short, we thus extractthe following setup with which we will work Definition 1.5.6. Let A be a connection on E , and let ˙ A ∈ Ω (cid:0) M, End( E ) (cid:1) . Define D A := d A + d tA : Ω • ( M, E ) → Ω • ( M, E ) , and denote by k t ( x, x ) ∈ C ∞ (cid:0) M, End(Λ • T ∗ M ⊗ E ) (cid:1) the restriction to the diagonal of the kernel k t ( x, y ) := (cid:16) √ tc ( ˙ A ) e − tD A (cid:17) ( x, y ) . With this notation our goal is now to computeLIM t → tr Λ • T ∗ M ⊗ E (cid:2) τ k t ( x, x ) (cid:3) . According to (1.28) we can decomposeEnd(Λ • T ∗ M ⊗ E ) = Cl( T ∗ M ) ⊗ s End Cl (cid:0) Λ • T ∗ M ⊗ E (cid:1) , where End Cl (cid:0) Λ • T ∗ M ⊗ E (cid:1) = b Cl( T ∗ M ) ⊗ End( E ) . The local index theory proof of Proposition 1.5.1 will follow from the following odd dimensionalversion of [13, Thm. 4.1], see also Remark 1.2.15. Theorem 1.5.7. Let M be a Riemannian manifold of odd dimension m , and let k t ( x, x ) be asin Definition 1.5.6. There is an asymptotic expansion k t ( x, x ) ∼ (4 πt ) − m ∞ X n =0 t n +12 k n ( x ) , as t → , such that k n ( x ) ∈ C ∞ (cid:16) M, Cl n +1 ( T ∗ M ) ⊗ End Cl (cid:0) Λ • T ∗ M ⊗ E (cid:1)(cid:17) . .5. Rho Invariants and Local Index Theory With respect to the symbol map σ : Cl( T ∗ M ) → Λ • T ∗ M , one has m − X n =0 σ ( k n ) = det / (cid:18) R g / R g / (cid:19) ∧ (cid:0) ˙ A ∧ exp( − F ) (cid:1) , where R g is the Riemann curvature tensor, and F is the twisting curvature, defined in a localorthonormal frame { e i } for T M by F := F ∇ A,g − g (cid:0) R g ( e i , e j ) e k , e l (cid:1) e i ∧ e j ⊗ c k c l ∈ Ω (cid:0) M, End Cl (cid:0) Λ • T ∗ M ⊗ E (cid:1)(cid:1) . Here, ∇ A,g is the connection on Λ • T ∗ M ⊗ E induced by the Levi-Civita connection and theconnection A on E . Before we sketch how Theorem 1.5.7 can be proved along parallel lines as in [13, Ch. 4], wededuce the following consequence which gives an alternative proof of Proposition 1.5.1. Proposition 1.5.8. Assume that M is a closed, oriented Riemannian manifold of odd dimension m . Let A u be a smooth path of connections on a fixed Hermitian vector bundle E → M . Thenthe reduced ξ -invariant associated to the family B A u = τ (cid:0) d A u + d tA u (cid:1) satisfies ddu [ ξ ( B A u )] = 2 m +12 Z M b L ( T M, ∇ g ) ∧ i π tr E (cid:2) ddu A u ∧ exp( i π F A u ) (cid:3) , where F A u is the curvature or A u .Proof. For fixed u we use the notation of Definition 1.5.6, and let A := A u and ˙ A := ddu A u . ThenProposition D.2.5 shows that with the kernel k t ( x, x ) ddu [ ξ ( B A u )] = − √ π LIM t → Z M tr Λ • T ∗ M ⊗ E (cid:2) τ k t ( x, x ) (cid:3) vol M . (1.54)It follows from Theorem 1.5.7 thatLIM t → Z M tr Λ • T ∗ M ⊗ E (cid:2) τ k t ( x, x ) (cid:3) vol M = (4 π ) − m/ Z M tr Λ • T ∗ M ⊗ E (cid:2) τ k m − (cid:3) vol M . As in Proposition 1.2.19 we can decomposetr Λ • T ∗ M ⊗ E = tr Cl ⊗ d tr Cl ⊗ tr E . Now, the definition of tr Cl is in such a way that for a ∈ Cl( T ∗ M )tr Cl ( τ a ) = ( , if a ∈ Cl m − ( T ∗ M ) , m/ , if a = τ .This implies that for all κ ∈ C ∞ (cid:0) M, End( T ∗ M ⊗ E ) (cid:1)Z M tr Λ • T ∗ M ⊗ E (cid:2) τ κ (cid:3) vol M = 2 m/ Z M i − m +12 ( d tr Cl ⊗ tr E ) (cid:2) σ ( κ ) (cid:3) , 1. The Signature Operator and the Rho Invariant where the factor i − m +12 arises from the fact that σ ( τ ) = i m +12 vol M . Then, we can use Theorem1.5.7 again to infer that(4 π ) − m/ Z M tr Λ • T ∗ M ⊗ E (cid:2) τ k m − (cid:3) vol M = √ π (2 πi ) − m +12 Z M det / (cid:18) R g / R g / (cid:19) ∧ ( d tr Cl ⊗ tr E ) (cid:2) ˙ A ∧ exp( − F ) (cid:3) . Now, we decompose F = b R S + F A , where b R S is the twisting curvature of Λ • T ∗ M as in Lemma1.2.21, and F A ∈ Ω (cid:0) M, End( E ) (cid:1) is the curvature of A . Then one computes(2 πi ) − m +12 Z M det / (cid:18) R g / R g / (cid:19) ∧ ( d tr Cl ⊗ tr E ) (cid:2) ˙ A ∧ exp( − F ) (cid:3) = − Z M b A ( T M, ∇ g ) ∧ ( d tr Cl ⊗ tr E ) (cid:2) i π ˙ A ∧ exp( i π F ) (cid:3) = − Z M b A ( T M, ∇ g ) ∧ d tr Cl (cid:2) exp (cid:0) i π b R S (cid:1)(cid:3) ∧ tr E (cid:2) i π ˙ A ∧ exp (cid:0) i π F A (cid:1)(cid:3) = − m/ Z M b L ( T M, ∇ g ) ∧ tr E (cid:2) i π ˙ A ∧ exp (cid:0) i π F A (cid:1)(cid:3) , where we have used Proposition 1.2.22 in the last line. Using (1.54), the proof of Proposition1.5.8 is finished. Getzler’s Rescaling. We now want to motivate why Theorem 1.5.7 can be proved in the sameway as the local index theorem [13, Thm. 4.1]. Since it is also basic for the considerations in thenext chapters, we first want to extract one of the main ideas of Getzler’s approach in [45]. Thisis to consider an appropriate rescaling of the Riemannian metric.Let M be a closed manifold, and let g be a Riemannian metric. For t > g t := t − g . We define a rescaled Clifford multiplication by c t : (cid:0) T ∗ M, g t (cid:1) → (cid:0) Λ • T ∗ M, g (cid:1) , c t ( ξ ) := √ t (cid:0) e( ξ ) − i( ξ ) (cid:1) , (1.55)where i( ξ ) denotes inner multiplication by ξ with respect to the fixed metric g on Λ • T ∗ M . Oneeasily checks that c t ( ξ ) = − t · | ξ | g = −| ξ | g t , and c t ( ξ ) ∗ = − c t ( ξ ) w.r.t. g. (1.56)This means that c t defines a Clifford structure for ( T ∗ M, g t ) on the bundle (cid:0) Λ • T ∗ M, g (cid:1) . Sincethe Levi-Civita connection ∇ g is invariant under rescaling with a constant parameter, it definesa Clifford connection for every t , (cid:2) ∇ g , c t ( ξ ) (cid:3) = c t (cid:0) ∇ g ξ (cid:1) , ξ ∈ Ω ( M ) . We thus get a family of de Rham operators D t , defined in a local orthonormal frame { e j } for( T M, g ) by D t := c t ( e j ) ∇ ge j = √ t ( d + d t ) : Ω • ( M ) → Ω • ( M ) . (1.57)Here, d t is the adjoint differential with respect to the fixed metric g . .5. Rho Invariants and Local Index Theory Remark. It might be confusing that the de Rham differential d is also rescaled, although it isdefined without using a metric. This is due to the fact that we are fixing g as a reference metricon Λ • T ∗ M while varying the metric on T ∗ M .We wish to make this more precise, and take g t as a metric on Λ • T ∗ M rather than the fixedmetric g . Consider e c t : (cid:0) T ∗ M, g t (cid:1) → (cid:0) Λ • T ∗ M, g t (cid:1) , e c t ( ξ ) := e( ξ ) − i t ( ξ ) , (1.58)where now, i t ( ξ ) is inner multiplication with respect to g t . Note that i t ( ξ ) is related to innermultiplication with respect to g via i t ( ξ ) = t · i( ξ ). From this, one deduces that e c t ( ξ ) = − t · | ξ | g = −| ξ | g t , and c t ( ξ ) ∗ = − c t ( ξ ) w.r.t. g t . (1.59)Hence, e c t defines a Clifford structure for ( T ∗ M, g t ) on the bundle (cid:0) Λ • T ∗ M, g t (cid:1) . The relationbetween c t and e c t is as follows. Lemma 1.5.9. There is an isometry of vector bundles given by δ t : (cid:0) Λ • T ∗ M, g t (cid:1) → (cid:0) Λ • T ∗ M, g (cid:1) , δ t ( α ) := ( √ t ) | α | α, where α is homogenous of degree | α | . Moreover, the Clifford structures c t and e c t are related by e c t = δ − t ◦ c t ◦ δ t . The proof is straightforward and is left to the reader. We also note that if d tg t denotes theadjoint differential with respect to g t , then √ t ( d + d t ) = δ t ◦ (cid:0) d + d tg t (cid:1) ◦ δ − t . This explains in more detail why the de Rham differential in (1.57) is rescaled.Now, one of the ideas underlying Getzler’s approach in [45] can be expressed in the simpleidentity e − tD = e − sD t | s =1 , (1.60)where D is the de Rham operator with respect to g , and D t is the rescaled de Rham operator(1.57). The deep insight behind (1.60) is that the asymptotic expansion as t → e − tD can be related to the Euclidean heat kernel since—very roughly—the metric g t convergeslocally to a Euclidean metric as t → Remarks on the Proof of Theorem 1.5.7. We shall make the above idea only a bit moreprecise, and refer to [13, Sec. 4.3] for more details. Let x ∈ M , and let V := T x M . We choose aball U ⊂ V of radius less than the injectivity radius of M , so that exp x : U → M parametrizesa normal neighbourhood of x in M . Using parallel transport along geodesic rays with respect tothe flat connection A , we can identify the bundle E | U with the trivial bundle U × E x . Hence, for y ∈ U we can consider h ( t, y ) := e − tD A (cid:0) exp x ( y ) , x (cid:1) ∈ End(Λ • V ∗ ⊗ E x ) , (1.61)and k ( t, y ) := (cid:16) √ tc ( ˙ A ) e − tD A (cid:17)(cid:0) exp x ( y ) , x (cid:1) ∈ End(Λ • V ∗ ⊗ E x ) . (1.62)6 1. The Signature Operator and the Rho Invariant Using the symbol map σ : Cl( V ∗ ) → Λ • V ∗ , we get sections σ ( h ) , σ ( k ) ∈ C ∞ (cid:0) R + × U, Λ • V ∗ ⊗ End Cl (Λ • V ∗ ⊗ E x ) (cid:1) . Now Getzler’s rescaling method can be described as follows: We fix U as the coordinate space, butreplace the metric on M by g t := t − g . This implies that the new system of normal coordinatesis given by exp x ◦√ t : U → M, y exp( √ ty ) . On End Cl (Λ • V ∗ ⊗ E x ), we fix the reference metric given by g and the metric on E , but we usethe rescaled metric on Cl( V ∗ ). According to Lemma 1.5.9 this means that the symbol map hasto be replaced by σ t := δ − t ◦ σ : Cl( V ∗ ) → Λ • V ∗ . One checks that if ξ ∈ V ∗ ⊂ Cl( V ∗ ) and a ∈ Cl( V ∗ ), then σ t ( ξ · a ) = √ t ξ ∧ σ t ( a ) − √ t i( ξ ) σ t ( a ) , (1.63)where i( ξ ) is interior multiplication with respect to the fixed metric g . Definition 1.5.10. Let s ∈ R + be an auxiliary parameter as in (1.60). Then the rescaled heatkernel is defined as h ( s, t, y ) := t m/ σ t (cid:0) h ( st, √ ty ) (cid:1) ∈ Λ • V ∗ ⊗ End Cl (Λ • V ∗ ⊗ E x ) . As remarked in [13, p. 155] the extra factor t m/ enters because h ( t, y ) is a density in the y variable. Now Getzler’s local index theorem can be reformulated as follows, see [13, Thm. 4.21]. Theorem 1.5.11. The limit lim t → h ( s, t, y ) exists. For y = 0 and s = 1 , lim t → h (1 , t, 0) = (4 π ) − m/ det / (cid:18) R gx / R gx / (cid:19) ∧ exp( − F x ) , where R gx is the Riemann curvature tensor at x and F is the twisting curvature, defined inTheorem 1.5.7. Remark. In [13], Theorem 1.5.11 is proved under the assumption that m is even. However,a close examination of all intermediate steps shows that this restriction is not necessary. Inparticular, [13, Prop. 4.12] continues to hold for odd m as well. The argument is similar to (1.32)in the proof of Lemma 1.2.20.Using (1.62) we now define as in Definition 1.5.10 k ( s, t, y ) := t m/ σ t (cid:0) k ( st, √ ty ) (cid:1) ∈ Λ • V ∗ ⊗ End Cl (Λ • V ∗ ⊗ E x ) . For y ∈ U let ˙ A y ∈ Λ V ∗ ⊗ End( E x ) denote the pullback of ˙ A to U at the point y . Then (1.63)shows that k ( s, t, y ) = ˙ A √ ty ∧ h ( s, t, y ) − t i (cid:0) ˙ A √ ty (cid:1) h ( s, t, y ) , and Theorem 1.5.11 implies .5. Rho Invariants and Local Index Theory Corollary 1.5.12. The limit lim t → k ( s, t, y ) exists. For y = 0 and s = 1 , lim t → k (1 , t, 0) = (4 π ) − m/ det / (cid:18) R gx / R gx / (cid:19) ∧ (cid:0) ˙ A x ∧ exp( − F x ) (cid:1) . On the other hand—as in the even dimensional case—the rescaled kernel satisfieslim t → k (1 , t, 0) = (4 π ) − m m − X n =0 σ (cid:0) k n ( x ) (cid:1) , where the k n are the coefficients appearing in the asymptotic expansion in Theorem 1.5.7. Thistogether with Corollary 1.5.12 finishes the outline of the proof of Theorem 1.5.7.8 1. The Signature Operator and the Rho Invariant hapter 2 Rho Invariants of Fiber Bundles,Basic Considerations In this chapter we start to work on the main topic of this thesis. Our concern is to investigatehow the structure of an oriented fiber bundle of closed manifolds can be used to analyze Rhoinvariants of its total space.For this reason, we will first give a detailed summary of some geometric preliminaries as theyappear in the theory of Riemannian submersions and in Bismut’s local index theory for families.Since we are dealing with the odd signature operator, our emphasis is to understand the structureof the space of differential forms. In particular, we obtain descriptions of the exterior differential,the adjoint differential and the Levi-Civita connection, which account for the special situationarising in the context of the total space of a fiber bundle.After having established how the odd signature operator can be expressed in terms of asubmersion metric, we describe the idea of an adiabatic metric on a fiber bundle. This is themain tool on which our discussion of Rho invariants of fiber bundles relies. Very roughly, theidea is to rescale the metric on the base manifold in order to deform the geometry of the fiberbundle to an “almost product” situation. Using the variation formula, we shall see that the Etainvariant of the odd signature operator has a well-defined limit under this process, which is calledthe adiabatic limit . As the Rho invariant is independent of the underlying metric, we can thenreplace the Eta invariants in its definition by adiabatic limits.With this idea in mind, we will analyze the first class of examples, namely principal circlebundles over closed surfaces. Thanks to the low dimensions of fiber and base as well as theenhanced symmetry provided by the principal bundle structure, one can compute Rho invariantswithout having to use the more advanced theory of Chapter 3. Nevertheless, adiabatic metricswill already play a prominent role in the discussion.490 2. Rho Invariants of Fiber Bundles, Basic Considerations Let F ֒ → M π −→ B be an oriented fiber bundle, where all manifolds are assumed to be closed,connected and oriented. Let T v M := ker π ∗ be the vertical subbundle of T M . Then T v M is involutive , i.e., if [ ., . ] denotes the Lie bracket on C ∞ ( M, T M ), we have[ U, V ] ∈ T v M, U, V ∈ C ∞ ( M, T v M ) . We now assume that M is endowed with a connection , i.e., a vertical projection P v : T M → T v M. This induces a splitting T M = T h M ⊕ T v M, T h M := ker P v , and π ∗ : T h M → π ∗ T B is an isomorphism. In the following we usually identify T h M and π ∗ T B via this isomorphism. Given a vector field X ∈ C ∞ ( B, T B ) we can use the connection to lift X horizontally to a vector field X h ∈ C ∞ ( M, T h M ). We will frequently use the following easyresult, see [13, Lem. 10.7]. Lemma 2.1.1. Let V ∈ C ∞ ( M, T v M ) be a vertical vector field on M , and let X ∈ C ∞ ( B, T B ) be a vector field on B . Then [ X h , V ] ∈ C ∞ ( M, T v M ) . The horizontal distribution T h M is in general not involutive. The following quantity measuresthe failure of being so. Definition 2.1.2. The curvature form of the connection P v is given byΩ ∈ C ∞ ( M, Λ T h M ∗ ⊗ T v M ) , Ω( X, Y ) := − P v (cid:0) [ X, Y ] (cid:1) , where X, Y ∈ C ∞ ( M, T h M ). Riemannian Connections on Fiber Bundles. We now assume that the fiber bundle isequipped with a submersion metric . This means that with respect to the splitting T M = π ∗ T B ⊕ T v M induced by the connection, we take a metric of the form g = π ∗ g B ⊕ g v , (2.1)where g v is a Riemannian metric on T v M , and g B is a metric on B . We will frequently write g = g B ⊕ g v , the pullback and the identification T M = π ∗ T B ⊕ T v M being understood. Remark. Note that our point of view is somewhat reversed to the situation encountered indifferential geometry. Recall, see e.g. [81, pp. 212–214], that a Riemannian submersion is definedas a submersion π : ( M, g ) → ( B, g B ) such that the push-forward π ∗ : (ker π ∗ ) ⊥ → T B is anisometry. Then one deduces that M is a fiber bundle over B , endowed with a natural connectiongiven by the orthogonal projection onto ker π ∗ . Moreover, via the isomorphism (ker π ∗ ) ⊥ ∼ = π ∗ T B the metric g is of the form g B ⊕ g v . In contrast to this point of view, we start with a fiber bundle,choose a connection and then endow the vertical and the horizontal bundles with metrics. Thisis because we often want to fix only a vertical metric. .1. Fibered Calculus g v , the following result provides a natural connection ∇ v on T v M ,see [13, Prop. 10.2]. Proposition 2.1.3. Let g v be a metric on T v M , and let P v be a vertical projection. Then thereis a natural connection ∇ v on T v M defined by ∇ v := P v ◦ ∇ g ◦ P v , where ∇ g is the Levi-Civita connection associated to a metric of the form (2.1) . The connection ∇ v is independent of the choice of the metric g B on B . It is compatible with g v and torsion freewhen restricted to T v M , i.e., ∇ vU V − ∇ vV U = [ U, V ] , U, V ∈ C ∞ ( M, T v M ) . Remark 2.1.4. The connection ∇ v can be thought of as a family of Levi-Civita connectionsparametrized by B , when we consider every fiber F ⊂ M endowed with the metric induced by g v | F . This might give a good intuition why ∇ v is canonically associated to g v and independentof g B . Definition 2.1.5. Let g v be a vertical Riemannian metric. Then we define the Weingarten map W ∈ C ∞ (cid:0) M, T h M ∗ ⊗ End( T v M ) (cid:1) , W X ( V ) := ∇ vX V − P v (cid:0) [ X, V ] (cid:1) , where V ∈ C ∞ ( M, T v M ) and X ∈ C ∞ ( M, T h M ). The mean curvature of ∇ v with respect tothe vertical projection P v is defined as k v = k v ( P v , g v ) ∈ C ∞ ( M, T h M ∗ ) , k v ( X ) := tr v ( W X ) , where X ∈ C ∞ ( M, T h M ) and tr v : End( T v M ) → C is the fiberwise vertical trace. Remark. (i) If { e i } is a local orthonormal frame for T v M , we havetr v ( W X ) = X i g v (cid:0) ∇ vX e i − [ X, e i ] , e i (cid:1) = − X i g v (cid:0) [ X, e i ] , e i (cid:1) . (2.2)This is because ∇ v is a metric connection so that g v ( ∇ vX e i , e i ) = Xg v ( e i , e i ) − g v ( e i , ∇ vX e i ) = − g v ( ∇ vX e i , e i ) . (ii) In the literature, one finds different conventions of how to define the Weingarten map.First of all in Riemannian geometry, one usually defines it as the negative of what we havedefined. Moreover, one often normalizes the mean curvature by a factor of (dim F ) − . Wechose to follow the conventions of [13, Sec. 10.1].Associated to the metric g B on B , there is the Levi-Civita connection ∇ B . Apart from theLevi-Civita connection ∇ g on T M we can thus form the direct sum connection ∇ ⊕ := π ∗ ∇ B ⊕ ∇ v (2.3)with respect to the splitting T M = π ∗ T B ⊕ T v M . Note that if X h is a horizontal lift and if V is vertical, then ∇ ⊕ V X h = 0. Clearly, the connection ∇ ⊕ preserves the metric g = g B ⊕ g v .However, it is not necessarily torsion free. Hence, it does usually not coincide with the Levi-Civita connection of M . Following [14, Sec. I] we introduce the following natural tensors whichmeasure the difference of ∇ ⊕ and ∇ g .2 2. Rho Invariants of Fiber Bundles, Basic Considerations Definition 2.1.6. Let ∇ ⊕ be defined as in (2.3) and let ∇ g be the Levi-Civita connectionassociated to the metric (2.1). Define S := ∇ g − ∇ ⊕ ∈ C ∞ (cid:0) M, End( T M ) (cid:1) , and let θ be its metric contraction θ ( X )( Y, Z ) := g (cid:0) S ( X ) Y, Z (cid:1) , X, Y, Z ∈ C ∞ ( M, T M ) . Since both connections ∇ ⊕ and ∇ g preserve the metric, the tensor θ is antisymmetric in Y and Z . Therefore, it is a section of T ∗ M ⊗ Λ T ∗ M . Also note that the above tensors are relatedto the O’Neill tensors of the Riemannian submersion, see [81, pp. 212–214]. The following resultdescribes all non-trivial components of θ , see [16, Sec. 4 (a)]. Proposition 2.1.7. The tensor θ is independent of the chosen metric g B on B . Moreover, if X , Y are horizontal vector fields, and U, V are vertical vector fields, then θ ( X )( V, Y ) = θ ( V )( X, Y ) = g (cid:0) Ω( X, Y ) , V (cid:1) , θ ( U )( V, X ) = g (cid:0) ∇ gU V, X (cid:1) , where Ω is the curvature of the fiber bundle, see Definition 2.1.2. We also note the following formulæ for θ and the mean curvature k v . Lemma 2.1.8. Let g v be a vertical Riemannian metric. Then for U , V vertical and X horizontal θ ( U )( V, X ) = − L X ( g v )( U, V ) , where L X denotes the Lie derivative in the X direction. If k v ∈ C ∞ ( M, T h M ∗ ) is the meancurvature of g v , then k v ( X ) = tr v (cid:2) L X ( g v ) (cid:3) = − X i θ ( e i )( e i , X ) , where { e i } is an arbitrary local orthonormal frame for T v M .Proof. Let U, V be vertical and X horizontal. Then by definition of the Lie derivative L X ( g v )( U, V ) = Xg v ( U, V ) − g v (cid:0) [ X, U ] , V (cid:1) − g v (cid:0) U, [ X, V ] (cid:1) = g v (cid:0) ∇ vX U − [ X, U ] , V (cid:1) + g v (cid:0) U, ∇ vX V − [ X, V ] (cid:1) , (2.4)where we have used that ∇ v is a metric connection. This shows thattr v (cid:2) L X ( g v ) (cid:3) = 2 X i g v (cid:0) ∇ vX e i − [ X, e i ] , e i (cid:1) = 2 tr v ( W X ) = 2 k v ( X ) . Now choose a metric g B on B and endow M with the submersion metric g = g B ⊕ g v . Bydefinition of ∇ v , we can replace g v and ∇ v in (2.4) with g and ∇ g , respectively. Since ∇ g istorsion free, we find L X ( g v )( U, V ) = g (cid:0) ∇ gX U − [ X, U ] , V (cid:1) + g (cid:0) U, ∇ gX V − [ X, V ] (cid:1) = g (cid:0) ∇ gU X, V (cid:1) + g (cid:0) U, ∇ gV X (cid:1) = g (cid:0) X, −∇ gU V − ∇ gV U (cid:1) = − g (cid:0) X, ∇ gU V (cid:1) + g (cid:0) X, [ U, V ] (cid:1) . Now, [ U, V ] is vertical, so that we find using Proposition 2.1.7 that L X ( g v )( U, V ) = − g (cid:0) X, ∇ gU V (cid:1) = − θ ( U )( V, X ) . .1. Fibered Calculus Convention. At this point it is convenient to introduce a convention regarding local compu-tations. We will always denote by { e i } a local, oriented frame for T v M and by { f a } a local,oriented frame for T B . The horizontal lifts to a frame for T h M will be denoted with the sameletters. Upper indices denote the dual frames and the summation convention will be understood.Indices a, b, c, . . . always refer to horizontal directions and i, j, k, . . . to vertical ones. If we havechosen metrics g v and g B , we will always choose local orthonormal frames. Whenever we do notwant to distinguish horizontal and vertical directions, we use the notation { E I } for the frame { f , f , . . . , e , e , . . . } with uppercase indices I, J, K, . . . . Moreover, if ∇ is a connection on avector bundle over M , we will use the abbreviations ∇ a , ∇ i , ∇ I for ∇ f a , ∇ e i , ∇ E I .As an example regarding this convention, we write the tensor θ of Definition 2.1.6 as θ = θ IJK E I ⊗ E J ∧ E K . Then Proposition 2.1.7 shows that if we distinguish vertical and horizontal direction, the functions θ IJK satisfy the relations θ aib = − θ abi = θ iab = g (cid:0) Ω ab , e i (cid:1) ,θ ija = − θ iaj = θ jia = g (cid:0) ∇ gi e j , f a (cid:1) ,θ ajk = θ ijk = θ abc = 0 . (2.5) Another Natural Connection. The connection ∇ v associated to a vertical Riemannian met-ric is not the only natural connection on T v M . For X ∈ C ∞ ( M, T h M ) consider the verticalprojection of the Lie derivative L X , i.e., L vX : T v M → T v M, L vX ( V ) := P v [ X, V ] . Note that Lemma 2.1.1 yields that if X is a horizontal lift, then [ X, V ] is automatically vertical.For general X ∈ C ∞ ( M, T h M ), V ∈ C ∞ ( M, T v M ) and ϕ ∈ C ∞ ( M ) we have L vX ( ϕV ) = P v (cid:0) ϕ [ X, V ] + ( Xϕ ) V (cid:1) = ϕ L vX ( V ) + ( Xϕ ) V and, since X is horizontal, L vϕX ( V ) = P v (cid:0) ϕ [ X, V ] − ( V ϕ ) X (cid:1) = ϕ L vX ( V ) . Therefore, we can define a connection on T v M as follows: Definition 2.1.9. Let ∇ v be the natural connection associated to a vertical metric g v . We definethe connection e ∇ v on T v M by e ∇ vU := ∇ vU , U ∈ C ∞ ( M, T v M ) , and e ∇ vX := L vX , X ∈ C ∞ ( M, T h M ) . If g B is a metric on B with Levi-Civita connection ∇ B , we also define a connection on T M = π ∗ T B ⊕ T M via e ∇ ⊕ := π ∗ ∇ B ⊕ e ∇ v . Clearly, the definition of e ∇ vX for horizontal X is independent of any choice of metric. However,for vertical U we cannot use the Lie derivative to define e ∇ vU which is why the connection ∇ vU enters the definition.4 2. Rho Invariants of Fiber Bundles, Basic Considerations Lemma 2.1.10. (i) For X ∈ C ∞ ( M, T h M ) e ∇ vX ( g v ) = L X ( g v ) , i.e., e ∇ vX does in general not preserve the metric g v . Similarly e ∇ ⊕ , does in general notpreserve the metric g = g B ⊕ g v . (ii) The torsion of e ∇ ⊕ coincides with the curvature of the fiber bundle, T ( e ∇ ⊕ ) = Ω ∈ C ∞ (cid:0) M, Λ T h M ∗ ⊗ T v M (cid:1) . Proof. Part (i) is clear by definition. For part (ii) we use local orthonormal frames { f a } and { e i } for T B and T v M according to the convention on p. 53. Then one computes e ∇ ⊕ i e j − e ∇ ⊕ j e i = ∇ vi e j − ∇ vj e i = P v (cid:0) ∇ gi e j − ∇ gj e i (cid:1) = P v [ e i , e j ] = [ e i , e j ] , e ∇ ⊕ a e i − e ∇ ⊕ i f a = e ∇ va e i = P v [ f a , e i ] = [ f a , e i ] , and e ∇ ⊕ a f b − e ∇ ⊕ b f a = π ∗ (cid:0) ∇ Ba f b − ∇ Bb f a (cid:1) = π ∗ [ f a , f b ] = [ f a , f b ] h = [ f a , f b ] + Ω( f a , f b ) . Note that in the first and third row we have used that ∇ g respectively ∇ B are torsion free, andin the second row that [ f a , e i ] is vertical as f a is a horizontal lift, see Lemma 2.1.1. Let F ֒ → M π −→ B be an oriented fiber bundle as above. The natural exact sequence T v M ֒ → T M → π ∗ T B translates to exterior bundles as π ∗ Λ p T ∗ B ֒ → Λ p T ∗ M → Λ p ( T v M ) ∗ . In terms of differential forms, the pullback of forms gives a natural inclusion π ∗ : Ω p ( B ) → Ω p ( M ) . Definition 2.1.11. Let ω ∈ Ω • ( M ), and let i( . ) denote inner multiplication with a vector field.Then ω is called(i) horizontal if i( V ) ω = 0 for all V ∈ C ∞ ( M, T v M ),(ii) basic if ω = π ∗ α for some α ∈ Ω • ( B ), and(iii) vertical if i( X ) ω = 0 for all X ∈ C ∞ ( M, T h M ).We denote by Ω • v ( M ) the algebra of vertical differential forms, and by Ω • h ( M ) the algebra ofhorizontal differential forms. .1. Fibered Calculus Remark. (i) Note that the definition of vertical forms requires the choice of a vertical projection P v : T M → T v M , whereas horizontal and basic forms are defined independently of such achoice. Moreover, P v gives an identificationΦ : C ∞ (cid:0) M, Λ q ( T v M ) ∗ (cid:1) → Ω qv ( M ) , Φ( ω )( X , . . . , X q ) := ω ( P v X , . . . , P v X q ) . We will usually suppress this isomorphism from the notation and identify a section of( T v M ) ∗ with a vertical differential form.(ii) A horizontal form ω can always be written (non-uniquely) as a sum ω = X i ϕ i ( π ∗ α i ) , ϕ i ∈ C ∞ ( M ) , α i ∈ Ω • ( B ) . Thus, on a somewhat formal level,Ω • h ( M ) ∼ = π ∗ Ω • ( B ) ⊗ C ∞ ( M ) , where the tensor product is over π ∗ C ∞ ( B ).(iii) More generally, we can decompose every differential form ω on M as ω = X i α i ∧ β i , α i ∈ Ω • h ( M ) , β i ∈ Ω • v ( M ) , and so Ω k ( M ) ∼ = M p + q = k π ∗ Ω p ( B ) ⊗ s Ω qv ( M ) =: M p + q = k Ω p,q ( M ) . (2.6)Here, ⊗ s is a graded tensor product over π ∗ C ∞ ( B ). We will often write ⊗ instead of ⊗ s ,keeping in mind that there is a grading involved.(iv) So far we have only implicitly remarked on our orientation convention. We use the basisfirst orientation which can be described as follows: In (2.6) we consider the case k = dim M , p = dim B and q = dim F . Let vol B ( g B ) ∈ Ω p ( B ) and vol F ( g v ) ∈ Ω qv ( M ) be oriented volumeforms associated to metrics g B and g v . Then we orient M according to the prescriptionvol M ( g ) = π ∗ (cid:0) vol B ( g B ) (cid:1) ∧ vol F ( g v ) . (2.7) Integration Along the Fiber. If the fiber bundle is endowed with a connection and a verticalmetric, there is a natural right-inverse for the map π ∗ : Ω p ( B ) → Ω p ( M ). We recall the followingwell-known facts, see e.g. [22, Sec. 1.6]. Proposition 2.1.12. There is a natural homomorphism of C ∞ ( B ) modules, Z M/B : Ω • , dim F ( M ) → Ω • ( B ) , 2. Rho Invariants of Fiber Bundles, Basic Considerations called “integration along the fiber” , which is uniquely defined by the property that Z B α ∧ (cid:16) Z M/B ω (cid:17) = Z M π ∗ α ∧ ω, α ∈ Ω dim B − k ( B ) , ω ∈ Ω k, dim F ( M ) . Moreover, for α and ω as above, d B Z M/B ω = Z M/B d M ω and α ∧ Z M/B ω = Z M/B π ∗ α ∧ ω. Definition 2.1.13. Let vol F ( g v ) ∈ Ω dim Fv ( M ) be the vertical volume form associated to a verticalmetric, and let v F ( g v ) := Z M/B vol F ( g v ) ∈ C ∞ ( B )be the function which associates to a point y ∈ B the volume of the fiber over y . Then we definethe basic projection on horizontal forms asΠ B : Ω • h ( M ) → Ω • ( B ) , Π B ( ω ) := 1 v F ( g v ) Z M/B ω ∧ vol F ( g v ) . Here, the normalization factor enters since we want Π B ( π ∗ α ) = α for every α ∈ Ω • ( B ). Wealso want to point out that if we allow conformal changes of the vertical metric, we can easilyachieve that v F ( g v ) = 1. This is the content of the following simple result. Lemma 2.1.14. Let n := dim F , and let g v be a vertical metric. Define u := n log (cid:0) π ∗ v F ( g v ) (cid:1) ∈ C ∞ ( M ) . Then the metric e g v := e − u g v has unit volume along the fibers, i.e., v F ( e g v ) = 1 .Proof. Let { e i } be a local, oriented orthonormal frame for ( T v M, g v ). Let u be defined as above,and let e e i := exp( u ) e i . Then { e e i } is a local, oriented orthonormal frame with respect to themetric e g v , andvol F ( e g v ) = e e ∧ . . . ∧ e e n = ( e − nu ) e ∧ . . . ∧ e n = (cid:0) π ∗ v F ( g v ) (cid:1) − vol F ( g v ) . This yields that v F ( e g v ) = 1.One might expect, that there is a canonical description of the kernel of Π B . This is indeedtrue. However, the corresponding result is not completely straightforward. We give a proof inChapter 3, see Proposition 3.1.8. For the time being we need a better understanding of thecalculus for differential forms on fiber bundles. Let F ֒ → M π −→ B be an oriented fiber bundle of closed manifolds as before. Let E → M be aHermitian vector bundle which admits a flat connection A . We denote by Ω qv ( M, E ) the space ofvertical E -valued q -forms, i.e., the space of sections of Λ q T v M ∗ ⊗ E . The canonical connection ∇ v in Proposition 2.1.3 together with A induces a natural connection ∇ A,v : Ω • v ( M, E ) → Ω ( M ) ⊗ Ω • v ( M, E ) . .1. Fibered Calculus d A,v : Ω qv ( M, E ) → Ω q +1 v ( M, E ) , d A,v = e i ∧ ∇ A,vi , where { e i } is any local orthonormal frame for T v M . As in (2.6), we can split the space of E -valued k -forms on M as Ω k ( M, E ) = M p + q = k Ω p,q ( M, E ) . The vertical differential then extends to Ω • ( M, E ) by requiring that d A,v ( α ⊗ ω ) = ( − p α ⊗ d A,v ω, α ⊗ ω ∈ Ω p,q ( M, E ) . On the other hand, the connection A defines a total exterior differential d A on Ω • ( M, E ). Itinherits a bigrading d A = X i + j =1 d ij , where d ij : Ω p,q ( M, E ) → Ω p + i,q + j ( M, E ) for all p, q. We now want to describe this in terms of the data introduced in Section 2.1.1. Let e ∇ v be theconnection on T v M as in Definition 2.1.9. It induces a connection on vertical differential forms,which we denote by the same letter. Similarly, we obtain a connection e ∇ ⊕ on Λ • T ∗ M , and using A , we define e ∇ A, ⊕ on Λ • T ∗ M ⊗ E . Remark. There is a subtlety concerning the action on vertical differential forms of the connection e ∇ vX h , if X h is the horizontal lift of a vector field X on B . Since on T v M , the action of e ∇ vX h is given by the Lie derivative, one might expect that the same is true for its action on forms.However, it is in general not true that L X h ω is automatically vertical for a vertical differentialform ω , compare with Lemma 2.1.1. For example, if ω ∈ Ω v ( M ), then the Cartan formula yields L X h ( ω )( Y h ) = (i( X h ) ◦ dω )( Y h ) = − ω (cid:0) [ X h , Y h ] (cid:1) = ω (cid:0) Ω( X, Y ) (cid:1) . This is in general non-zero, so that L X h ω is in general not a vertical form. Thus, e ∇ vX h agrees ingeneral only with the vertical projection L vX h of the Lie derivative.As always let { f a } be a local frame for T B , and write Ω = f a ∧ f b ⊗ Ω ab for the curvatureof the fiber bundle. Proposition 2.1.15. The total exterior differential d A on Ω • ( M, E ) splits as d A = d A,v + d A,h + i(Ω) , where for ω ∈ Ω • ( M, E ) d A,h ω = f a ∧ e ∇ A, ⊕ a ω, and i(Ω) ω = f a ∧ f b ∧ i(Ω ab ) ω. Here, i(Ω ab ) denotes interior multiplication with Ω ab ∈ C ∞ ( M, T v M ) . For convenience we sketch a proof, although the result is well known, see [13, Prop. 10.1] or[19, Prop. 3.4]. Before we do so, let us point out that there is no d − , contribution to d A , whichis due to the fact that the vertical distribution is integrable, see [74, p. 58].8 2. Rho Invariants of Fiber Bundles, Basic Considerations Proof of Proposition 2.1.15. We assume for simplicity that A is the trivial connection on thetrivial line bundle. Recall, e.g. from [13, Prop. 1.22], that if ∇ is a torsion free connection on T M , then d can be expressed in terms of ∇ as the compositionΩ • ( M ) ∇ −→ Ω ( M ) ⊗ Ω • ( M ) e ◦ −→ Ω • +1 ( M ) . (2.8)Here, the second arrow means contraction with exterior multiplication. Let g be a metric of theform g B ⊕ g v . It follows from Lemma 2.1.10, that we can define a torsion free connection ∇ on T M by ∇ X := e ∇ ⊕ X − Ω( X, . ) , X ∈ C ∞ ( M, T M ) . Let { f a } and { e i } be local orthonormal frames for T B and T v M respectively. Since d satisfiesthe Leibniz rule it suffices to compute de i and df a . From (2.8) and the definition of e ∇ ⊕ we get de i = e j ∧ ∇ j e i + f a ∧ ∇ a e i = e j ∧ ∇ vj e i + f a ∧ (cid:0) − e i ( ∇ a e k ) e k − e i ( ∇ a f b ) f b (cid:1) = d v e i + f a ∧ (cid:0) − e i ([ f a , e k ]) e k − e i ( − Ω ab ) f b (cid:1) = d v e i + f a ∧ e ∇ ⊕ a e i + e i (Ω ab ) f a ∧ f b . This is the required formula for de i . On the other hand, since ∇ j acts trivially on basic forms,and Ω( f b , f c ) has no horizontal component, one easily finds that df a = f b ∧ ∇ b f a = f b ∧ e ∇ ⊕ b f a . Since A is flat we have d A = 0. This implies the following. Corollary 2.1.16. Let { ., . } denote the anti-commutator of two operators in the ungraded sense.Then d A,v = i(Ω) = 0 , d A,h + (cid:8) d A,v , i(Ω) (cid:9) = 0 , (cid:8) d A,v , d A,h (cid:9) = (cid:8) d A,h , i(Ω) (cid:9) = 0 . More on the Mean Curvature. From Proposition 2.1.15, we can also deduce the followingformula which relates the mean curvature with the differential of the vertical volume form, see[13, Lem. 10.4]. In the theory of foliations, this is known as Rummler’s formula , see e.g. [95, p.38]. Proposition 2.1.17. Let vol F ( g v ) be the volume form associated to a vertical metric. Let k v ∈ Ω , ( M ) be the mean curvature form. Then d M vol F ( g v ) = k v ∧ vol F ( g v ) + i(Ω) vol F ( g v ) . Proof. Since vol F ( g v ) has maximal vertical degree, Proposition 2.1.15 implies that d M vol F ( g v ) = d h vol F ( g v ) + i(Ω) vol F ( g v ) . Clearly, we could also use the Levi-Civita connection associated to the metric g . However, its explicit formulais more complicated (see also Remark 2.1.20 below). .1. Fibered Calculus { f a } and { e j } are local frames for T B and T v M , we compute d h vol F ( g v ) = f a ∧ e ∇ va (cid:0) vol F ( g v ) (cid:1) = f a ∧ e ∇ va ( e j ) ∧ i( e j ) (cid:0) vol F ( g v ) (cid:1) = − f a ∧ e j (cid:0) [ f a , e k ] (cid:1) e k ∧ i( e j ) (cid:0) vol F ( g v ) (cid:1) = − f a ∧ e j (cid:0) [ f a , e k ] (cid:1) δ kj ∧ vol F ( g v )= − X j g v (cid:0) [ f a , e j ] , e j (cid:1) f a ∧ vol F ( g v ) . Now, (2.2) identifies the last line with k v ∧ vol F ( g v ). Corollary 2.1.18. Let vol F ( g v ) be the volume form associated to a vertical metric g v , and let v F be the volume of the fiber as in Definition 2.1.13. Then the basic projection of the mean curvatureform is given by Π B ( k v ) = d B log( v F ) ∈ Ω ( B ) . Proof. We differentiate v F and use Proposition 2.1.17 to find that d B v F = d B Z M/B vol F ( g v ) = Z M/B d M vol F ( g v ) = Z M/B k v ∧ vol F ( g v ) = v F Π B ( k v ) . Corollary 2.1.18 shows that the basic projection of the mean curvature form gives a trivialelement in the cohomology of the base. Moreover, it vanishes if the metric g v has constantvolume along the fiber. As we have seen in Lemma 2.1.14, this can be achieved by a conformaldeformation of the vertical metric. In Section 3.1 we will transfer a result of [36] from the theoryof foliations to the situation at hand and show that one can always deform the vertical projectionand the vertical metric of the fiber bundle in such a way that not only the basic projection butthe mean curvature form itself vanishes. To study the de Rham operator on a fiber bundle F ֒ → M π −→ B , we also need to understand theadjoint differential d tA , where A is a flat connection on a Hermitian vector bundle E → M . Forthis we will use the local formula d tA = − i( E I ) ∇ A,gI , where { E I } is a local orthonormal frame for T M , and ∇ A,g is the Levi-Civita connection on formstwisted by A . We want to use this formula to split d tA in terms of its bidegrees with respect to thedecomposition (2.6). For this we need to relate the Levi-Civita connection ∇ A,g on forms withthe connection ∇ A, ⊕ . Recall that in Definition 1.2.17 we have introduced a transposed Cliffordas b c : T ∗ M → End(Λ • T ∗ M ) , b c ( ξ ) = e( ξ ) + i( ξ ) . Lemma 2.1.19. Let E → M be a Hermitian bundle which admits a flat connection A . Then,on Λ • T ∗ M ⊗ E , the difference of ∇ A,g and ∇ A, ⊕ is given by ∇ A,g = ∇ A, ⊕ + (cid:0) c ( θ ) − b c ( θ ) (cid:1) , where θ is the tensor defined in Definition 2.1.6. 2. Rho Invariants of Fiber Bundles, Basic Considerations Proof. Let { E I } be a local orthonormal frame of T M with dual coframe { E I } . Then by definitionof S and θ ,( ∇ A,gI − ∇ A, ⊕ I ) = ( ∇ A,gI − ∇ A, ⊕ I )( E J ) ∧ i( E J ) = − E J (cid:0) S ( E I ) E K (cid:1) e( E K ) i( E J )= − θ IKJ e( E K ) i( E J ) . On the other hand, (cid:0) c ( θ I ) − b c ( θ I ) (cid:1) = θ IJK (cid:0) c ( E J ) c ( E K ) − b c ( E J ) b c ( E K ) (cid:1) = − θ IJK (cid:0) e( E J ) i( E K ) − i( E J ) e( E K ) (cid:1) = − θ IKJ e( E K ) i( E J ) + θ IJK e( E K ) i( E J ) = − θ IKJ e( E K ) i( E J ) , where in the last line we have first renamed J and K in the first summand and then usedantisymmetry of θ IJK in J and K for the second summand, see (2.5). Remark 2.1.20. One could use the above result to give a different proof of Proposition 2.1.15by writing out locally d A = E I ∧ ∇ A,gI = E I ∧ ∇ A, ⊕ I − θ IKJ E I ∧ E K ∧ i( E J )Splitting this into horizontal and vertical contributions, and using (2.5), one verifies that d A = d A,v + f a ∧ (cid:0) ∇ A, ⊕ a − θ kja e k ∧ i( e j ) (cid:1) + i(Ω) . Comparing this with Proposition 2.1.15 we see that in particular, e ∇ va = ∇ va − θ kja e k ∧ i( e j ) . (2.9)A more invariant description of the term occurring here is as follows: Define a tensor field B ∈ C ∞ (cid:0) M, T h M ∗ ⊗ End(Λ • T v M ∗ ) (cid:1) by requiring that g v (cid:0) α, B ( X ) β (cid:1) = L vX ( g v )( α, β ) , X ∈ C ∞ ( M, T h M ) , α, β ∈ Ω • v ( M ) . Then Lemma 2.1.8 (or a direct computation) easily implies that B ( X ) = X a θ kja e k ∧ i( e j ) . (2.10)We now have the following analog of Proposition 2.1.15 for the adjoint differential. Proposition 2.1.21. Let E → M be a Hermitian vector bundle which admits a flat connection A . Then the twisted adjoint differential splits as d tA = d tA,v + d tA,h + i(Ω) t , where the terms are given in local orthonormal frames { e i } and { f a } by d tA,v = − i( e i ) ◦ ∇ A, ⊕ i : Ω p,q → Ω p,q − ,d tA,h = − i( f a ) ◦ (cid:0) ∇ A, ⊕ a + B ( f a ) + k v ( f a ) (cid:1) : Ω p,q → Ω p − ,q , i(Ω) t = − i( f a ) i( f b ) e(Ω ab ) : Ω p,q → Ω p − ,q +1 . Here, B ( f a ) is defined as in (2.10) , k v is the mean curvature form, and e(Ω ab ) denotes exteriormultiplication with the dual of Ω ab ∈ C ∞ ( M, T v M ) . .1. Fibered Calculus Remark 2.1.22. The definition of the connection e ∇ v and the relation (2.9) between ∇ v and e ∇ v shows that we can write alternatively d tA,v = − i( e i ) ◦ e ∇ A, ⊕ i : Ω p,q → Ω p,q − ,d tA,h = − i( f a ) ◦ (cid:0) e ∇ A, ⊕ a + 2 B ( f a ) + k v ( f a ) (cid:1) : Ω p,q → Ω p − ,q . Proof of Proposition 2.1.21. For convenience we drop again the reference to the flat connection.According to Lemma 2.1.19 and the local formula for d t we have d t = − i( E I ) ∇ ⊕ I + i( E I ) θ IJK e( E J ) i( E K ) . Splitting this into horizontal and vertical parts and checking bidegrees one finds d tv = − i( e j ) ∇ ⊕ j + i( e j ) θ jab e( f a ) i( f b ) + i( f a ) θ abj e( f b ) i( e j ) = − i( e j ) ∇ ⊕ j . Here, we have used the relation θ abj = − θ jab , see (2.5). Similarly, d th = − i( f a ) ∇ ⊕ a + i( e j ) θ jka e( e k ) i( f a ) = − i( f a ) (cid:0) ∇ ⊕ a − θ jka i( e j ) e( e k ) (cid:1) = − i( f a ) (cid:16) ∇ ⊕ a + θ kja (cid:0) e( e k ) i( e j ) − δ jk (cid:1)(cid:17) = − i( f a ) (cid:0) ∇ A, ⊕ a + B ( f a ) + k v ( f a ) (cid:1) , where we have used symmetry of θ jka in j and k , the definition of B ( f a ), and Lemma 2.1.8.Lastly, one has i(Ω) t = i( f a ) θ ajb e( e j ) i( f b ) = − i( f a ) i( f b ) e(Ω ab ) . Partial de Rham Operators. Having established the description of the adjoint differential inanalogy to Proposition 2.1.15, we can now split the twisted de Rham operator on M . Definition 2.1.23. We use the abbreviations D A,v := d A,v + d tA,v , D A,h := d A,h + d tA,h , and T := i(Ω) + i(Ω) t . The operators D A,v and D A,h are called the vertical respectively horizontal twisted de Rhamoperator on M . Remark 2.1.24. (i) Tautologically, the de Rham operator on M splits as D A = D A,v + D A,h + T : Ω • ( M, E ) → Ω • ( M, E ) . (2.11)(ii) The vertical de Rham operator is a first order differential operator acting fiberwise , i.e.,[ D A,v , π ∗ ϕ ] = 0 for all ϕ ∈ C ∞ ( B ). This means roughly, that it can be thought of as asmooth family of first order elliptic differential operators on Λ • T ∗ F ⊗ E | F parametrized by B , see Definition 1.3.9. We refer to Section 3.1 for some more details. Clearly, an analogousstatement cannot be formulated for the horizontal de Rham operator D A,h , unless thehorizontal distribution is integrable.2 2. Rho Invariants of Fiber Bundles, Basic Considerations (iii) We want to add some remarks about the effect the splitting (2.11) has on the spectrum of D A . Let us assume that A is the trivial connection. The appearance of the mean curvatureform in the formula for d th shows that in general, D h will not restrict to an operator onbasic forms, see Definition 2.1.11. Instead, if D B denotes the de Rham operator on B , D h ( π ∗ α ) = π ∗ ( D B α ) − i( k v ) π ∗ α, α ∈ Ω • ( B ) , (2.12)where i( k v ) denotes interior multiplication with the mean curvature form. However—asalready pointed out—we will see in Section 3.1 that upon changing the vertical metric andthe horizontal distribution, we can achieve that k v vanishes. In this case (2.12) shows thateigenforms of D B lift to eigenforms of D h . Since D v vanishes on basic forms, this produceseigenforms of D v + D h . In particular, spec( D B ) ⊂ spec( D v + D h ). However, the full deRham operator on M is given by (2.11), and T will in general not act trivially on π ∗ Ω • ( B ).This should give a hint at why—even in the case that k v vanishes—the relation betweenthe spectrum of D M and the spectra of D h and D v is non-trivial. We refer to [50, Ch.’s3&4] for a detailed study of related questions. Some Commutator Relations. The explicit description of the adjoint differential has thefollowing consequence, see also [1, Prop. 3.1] for a generalization. Proposition 2.1.25. Let { f a } be local orthonormal frame for T B , and define a bundle endo-morphism K := − i( f a ) ◦ (cid:0) B ( f a ) + k v ( f a ) (cid:1) : Λ • T ∗ M → Λ • T ∗ M. (2.13) Then d A,v d tA,h + d tA,h d A,v = d A,v K + Kd A,v . Proof. If { e i } is a local orthonormal frame for T v M , one checks that e ∇ A, ⊕ a ◦ d A,v = d A,v ◦ e ∇ A, ⊕ a . Since e ∇ A, ⊕ i f a = 0, this implies d A,v ◦ (cid:0) i( f a ) ◦ e ∇ A, ⊕ a (cid:1) + (cid:0) i( f a ) ◦ e ∇ A, ⊕ a (cid:1) ◦ d A,v = 0 . Now Remark 2.1.22 yields that d tA,h = − i( f a ) ◦ (cid:0) e ∇ A, ⊕ a + 2 B ( f a ) + k v ( f a ) (cid:1) . Then, with K as in (2.13), one easily verifies that indeed d A,v d tA,h + d tA,h d A,v = d A,v K + Kd A,v . Corollary 2.1.26. The anti-commutator { D A,v , D A,h } is a first order differential operator actingfiberwise.Proof. According to Corollary 2.1.16 and the corresponding statement for the formal adjoints,we have { d A,h , d A,v } = 0 and { d tA,h , d tA,v } = 0 . This implies { D A,v , D A,h } = { d A,v , d tA,h } + { d tA,v , d A,h } = { d A,v , K } + { K t , d tA,v } , which is C ∞ ( B ) linear and thus a first order differential operator acting fiberwise. .2. Rho Invariants and Adiabatic Metrics As before let F ֒ → M π −→ B be an oriented fiber bundle of closed manifolds, and let E → M bea Hermitian vector bundle endowed with a unitary flat connection A . We endow T v M with ametric g v and B with a Riemannian metric g B , and consider the associated submersion metric g := g B ⊕ g v .If dim M is odd the odd signature operator on M twisted by A is given in terms of the partialde Rham operators introduced in Definition 2.1.23 as B ev A = τ M D A,v + τ M D A,h + τ M T : Ω ev ( M, E ) → Ω ev ( M, E ) , (2.14)where τ M is the chirality operator on the total space M of the fiber bundle. It is useful to identifyΩ ev ( M, E ) in terms of the splitting T M = π ∗ T B ⊕ T v M . From (2.6) we see thatΩ ev ( M, E ) = X p + q ≡ π ∗ Ω p ( B ) ⊗ Ω qv ( M, E ) . Using this identification, we defineΦ : Ω ev ( M, E ) → π ∗ Ω • ( B ) ⊗ Ω • v ( M, E ) , Φ( α ⊗ ω ) = α e ⊗ ω e + τ M ( α o ⊗ ω o ) , where α e/o and ω e/o refer to the even/odd degree parts. Since it is straightforward, we skip theproof of the following result. Lemma 2.2.1. Assume that M is odd dimensional. (i) If the fiber F is even dimensional, then Φ gives rise to an isometry Φ : Ω ev ( M, E ) ∼ = −→ π ∗ Ω ev ( B ) ⊗ Ω • v ( M, E ) , and the odd signature operator is equivalent to Φ ◦ B ev A ◦ Φ − = D A,v + τ M D A,h + T (ii) If F is odd dimensional, then Φ : Ω ev ( M, E ) ∼ = −→ π ∗ Ω • ( B ) ⊗ Ω ev v ( M, E ) , and Φ ◦ B ev A ◦ Φ − = τ M D A,v + D A,h + τ M T. The Vertical Chirality Operator. For a more explicit formula for the odd signature oneneeds to understand how the chirality operator splits with respect to (2.6). In the general settingat hand we will not give a detailed account but add some remarks which will be used in theexamples below.4 2. Rho Invariants of Fiber Bundles, Basic Considerations Definition 2.2.2. Let M be endowed with a vertical metric g v , and let n := dim F . Thenthe vertical chirality operator τ v : Ω qv ( M ) → Ω n − qv ( M ) is defined with respect to an oriented,orthonormal frame { e i } for T v M by τ v = i [ n +12 ] c v ( e ) · . . . · c v ( e n ) . Here, c v : T v M ∗ → End (cid:0) Λ • T v M ∗ (cid:1) is the vertical Clifford multiplication, c v ( ξ ) = e( ξ ) − i( ξ ) , ξ ∈ Ω v ( M ) . The vertical Clifford multiplication extends naturally to vertical differential forms, and up tothe normalization factor, τ v is Clifford multiplication with the vertical volume form. In particular,it is independent of the chosen frame. We also recall the convention (2.7) that if g = g B ⊕ g v , weorient M using vol M ( g ) = π ∗ (cid:0) vol B ( g B ) (cid:1) ∧ vol F ( g v ) . Lemma 2.2.3. Let τ B be the chirality operator on Ω • ( B ) , and let ( π ∗ α ) ∧ ω ∈ Ω p,q ( M ) . (i) Assume that F is even dimensional. Then τ M ( π ∗ α ∧ ω ) = π ∗ ( τ B α ) ∧ τ v ω. (ii) If F is odd dimensional, then τ M ( π ∗ α ∧ ω ) = ( − p · ( π ∗ ( τ B α ) ∧ τ v ω, if B is even dimensional , − i · π ∗ ( τ B α ) ∧ τ v ω, if B is odd dimensional . The proof is a bit tedious but straightforward and shall be skipped. Remark. In Proposition 2.1.21 we have described the adjoint differential d tA using the localformula d tA = − i( E I ) ◦ ∇ A,gI . However, as in (1.7), we also have the description d tA = ( − m +1 τ M ◦ d A ◦ τ M , where m = dim M . Using this together with Lemma 2.2.3 and Proposition 2.1.15, one could givea different proof of Proposition 2.1.21. Clearly, the main point is then to compute τ M d A,h τ M ,which amounts to proof that τ v (cid:2) e ∇ vX , τ v (cid:3) = 2 B ( X ) + k v ( X ) , X ∈ C ∞ ( M, T h M ) . (2.15)Conversely, (2.15) can be verified using Proposition 2.1.21 and (2.9). In a similar way as in Section 1.5.2, we now want to rescale the metric on the fiber bundle F ֒ → M π −→ B . Yet, the important difference is that we only rescale the metric on the basemanifold. In order avoid square roots of ε , we are using ε rather than ε to rescale the metric. .2. Rho Invariants and Adiabatic Metrics Definition 2.2.4. Let g B be a metric on B and g v be a vertical metric. For ε > adiabatic metric g ε := ε g B ⊕ g v . (2.16)Associated to each g ε , we have a Levi-Civita connection ∇ g ε . Note that unlike in the caseof a single manifold, the family ∇ g ε is not independent of ε since we only scale the base metric.However, the direct sum connection ∇ ⊕ is independent of ε since both, ∇ B and ∇ v are, seeProposition 2.1.3. Similarly, the tensor θ as in Definition 2.1.6 is independent of ε . Adiabatic Families of Odd Signature Operators. Now let E → M be a flat unitary bundlewith connection A , and let ∇ A,g ε and ∇ A, ⊕ be the induced connections on Λ • T ∗ M ⊗ E . We canuse Lemma 2.1.19 to write ∇ A,g ε = ∇ A, ⊕ + (cid:0) c ε ( θ ) − b c ε ( θ ) (cid:1) . (2.17)Here, Clifford multiplication is defined with respect to the fixed metric g = g B ⊕ g v on Λ • T ∗ M ,i.e., c ε ( f a ) = εc ( f a ) , c ε ( e i ) = c ( e i ) , b c ε ( f a ) = ε b c ( f a ) , b c ε ( e i ) = b c ( e i ) , compare with (1.55). Lemma 2.2.5. For each ε > , the connection ∇ A,g ε on Λ • T ∗ M ⊗ E is compatible with the fixedmetric g = g B ⊕ g v . Moreover, it is a Clifford connection with respect to c ε , i.e., (cid:2) ∇ A,g ε , c ε ( ξ ) (cid:3) = c ε (cid:0) ∇ g ε ξ (cid:1) , ξ ∈ Ω ( M ) . Sketch of proof. Lemma 2.2.5 is basically [13, Prop. 10.10]. The main observation there is that ∇ A, ⊕ is compatible with g and satisfies (cid:2) ∇ A, ⊕ , c ε ( ξ ) (cid:3) = c ε (cid:0) ∇ ⊕ ξ (cid:1) , ξ ∈ Ω ( M ) . On the other hand, according to (2.17), we need to consider c ε (cid:0) θ ( X ) (cid:1) − b c ε (cid:0) θ ( X ) (cid:1) ∈ C ∞ (cid:0) M, End(Λ • T ∗ M ⊗ E ) (cid:1) , X ∈ C ∞ ( M, T M ) . Since c ε and b c ε are defined with respect to the fixed metric g , and since θ ( X ) is a 2-form, onefinds that c ε (cid:0) θ ( X ) (cid:1) and b c ε (cid:0) θ ( X ) (cid:1) are self-adjoint with respect to g . This implies that ∇ A,g ε iscompatible with the metric. Lemma 1.2.18 (i) shows that for ξ ∈ Ω v ( M ) (cid:2) c ε ( θ ) − b c ε ( θ ) , c ε ( ξ ) (cid:3) = (cid:2) c ε ( θ ) , c ε ( ξ ) (cid:3) As in [13, Prop. 10.10] one then finds that (cid:2) ∇ A, ⊕ + c ε ( θ ) , c ε ( ξ ) (cid:3) = c ε (cid:0) ∇ g ε ξ (cid:1) , which proves that ∇ A,g ε is indeed a Clifford connection.6 2. Rho Invariants of Fiber Bundles, Basic Considerations Remark 2.2.6. (i) Again, it might be confusing that all connections ∇ A,g ε are compatible with the fixed metric g on Λ • T ∗ M . This is due to the fact that we have defined ∇ A,g ε in such a way, that italready incorporates the isometry of Lemma 1.5.9, which in the case at hand takes the form δ ε : (cid:0) Ω • ( M, E ) , g ε (cid:1) → (cid:0) Ω • ( M, E ) , g (cid:1) , δ ε ( π ∗ α ∧ ω ) := ε | α | π ∗ α ∧ ω. (ii) We also want to point out that the chirality operator τ M on (cid:0) Ω • ( M, E ) , g (cid:1) does not changewith ε . Indeed, it is immediate that vol M ( g ε ) = ε − dim B vol M ( g ) from which it follows that c ε (cid:0) vol M ( g ε ) (cid:1) = c (cid:0) vol M ( g ) (cid:1) Definition 2.2.7. Let g ε be an adiabatic metric on M , and assume that m = dim M is odd. Wedefine the adiabatic family of odd signature operators as B ev A,ε := τ M D A,v + ε · τ M D A,h + ε · τ M T : Ω ev ( M, E ) → Ω ev ( M, E ) . (2.18)The definition is in such a way that B ev A,ε is given by Clifford contraction of ∇ A,g ε withrespect to the Clifford multiplication τ M ◦ c ε . Thus, all B ev A,ε are geometric Dirac operatorson Ω ev ( M, E ) which are formally self-adjoint with respect to the L -structure induced by thefixed reference metric g . The ε factors occur since each horizontal Clifford variable is scaled with ε . Adiabatic Limit of the Eta Invariant. The family of operators B ev A,ε converges pointwise to τ M D A,v , which is not an elliptic operator. Therefore, the following result is remarkable, see also[16, Prop. 4.3]. Proposition 2.2.8. Let g ε be an adiabatic metric on the total space of a fiber bundle F ֒ → M π −→ B , and assume that m = dim M is odd. Let A be a flat U( k ) -connection, and let η ( B ev A,ε ) be thefamily of Eta invariants associated to the adiabatic family of odd signature operators B ev A,ε . Thenthe “adiabatic limit of the Eta invariant” exists in R . More precisely, lim ε → η ( B ev A,ε ) = η ( B ev A ) + k · Z M T L ( ∇ g , ∇ ⊕ ) , where T L ( ∇ g , ∇ ⊕ ) is the transgression form of the L -class with respect to the connection ∇ g and ∇ ⊕ on T M .Proof. Fix ε ∈ (0 , η ( B ev A,ε ) = η ( B ev A ) + k · Z M T L ( ∇ g , ∇ g ε ) . (2.19)Moreover, we deduce from Proposition A.2.4 that Z M T L ( ∇ g , ∇ g ε ) = Z M T L ( ∇ g , ∇ ⊕ ) + Z M T L ( ∇ ⊕ , ∇ g ε ) . As in Definition 2.1.6 consider S = ∇ g − ∇ ⊕ and S ε = ∇ g ε − ∇ ⊕ . .3. The U (1) -Rho Invariant for S -Bundles over Surfaces P v S ε = P v S and P h S ε = ε P h S, where P v/h : T M → T v/h M is the vertical, respectively horizontal, projection of the fiber bundle.Hence, lim ε → ∇ g ε = ∇ ⊕ + P v S, and the limit is uniform in ε . Therefore,lim ε → Z M T L ( ∇ ⊕ , ∇ g ε ) = Z M T L ( ∇ ⊕ , ∇ ⊕ + P v S ) . Now, a consequence of Proposition 2.1.7 is that for fixed X ∈ C ∞ ( M, T M ), the only non-trivialcomponent of P v S ( X ) is P v S ( X ) : T h M → T v M. In particular, P v S ( X ) and all its powers are trace-free which implies that T L ( ∇ ⊕ , ∇ ⊕ + P v S ) = 0 . Hence, we can take the limit in (2.19) and getlim ε → η ( B ev A,ε ) = η ( B ev A ) + k · Z M T L ( ∇ g , ∇ ⊕ ) . Remark. So far Proposition 2.2.8 is not of particular value for explicit computations of η ( B ev A ).First of all, we do not yet know anything about the adiabatic limit lim ε → η ( B ev A,ε ). However, inChapter 3 we will describe how powerful methods of local families index theory give an alternativeexpression of the adiabatic limit in more topological terms. Another aspect worth mentioning isthat the Chern-Simons term R M T L ( ∇ g , ∇ ⊕ ) can be very difficult to compute, see e.g. [79] forvery explicit computations in the case of circle bundles over surfaces.Concerning Rho invariants we already know at this point that the transgression term doesnot play a role. This is because the transgression term is the same for B ev A and the untwistedodd signature operator B ev , since A is flat. Moreover, according to Proposition 1.4.7 the Rhoinvariant is independent of the metric, so that we have the freedom of choosing particular well-suited vertical and horizontal metrics. Summarizing these observations, we obtain the followingresult, which is the underlying idea for our discussion of Rho invariants of fiber bundles. Corollary 2.2.9. With respect to all adiabatic metrics g ε on M we have ρ A ( M ) = lim ε → η ( B ev A,ε ) − k · lim ε → η ( B ev ε ) S -Bundles over Sur-faces In this section we will see how the idea of Corollary 2.2.9 is already helpful for explicit compu-tations, even without employing more abstract theory we will encounter in Chapter 3. We give8 2. Rho Invariants of Fiber Bundles, Basic Considerations an elementary computation of Rho invariants for the simple but already non-trivial example ofa principal S -bundle over a closed surface. We content ourselves with the U(1)-Rho invariantsince all phenomena related to adiabatic limits appear. Some parts of our discussion are borrowedfrom [79]. The setup there is the Spin c Dirac operator which is, however, closely related to atwisted odd signature operator.Before we can start with the discussion of the odd signature operator on a principal S -bundleover a Riemannian surface, we need an explicit description of flat U(1)-connections. Let Σ be a closed, oriented surface of genus g , and let S ֒ → M π −→ Σ be an oriented principalcircle bundle. Since H (Σ , Z ) = Z , such a bundle is classified up to isomorphism by its degree l ∈ Z . Given that, there is a very explicit construction, which we describe now. Topological Description. Let D ⊂ Σ be an embedded disc, and let Σ := Σ \ D . Clearly, H ( D , Z ) = { } , and the long exact cohomology sequence of the pair (Σ , ∂ Σ ) implies that H (Σ , Z ) = { } as well. Since principal S -bundles are classified by their first Chern class, thisshows that the restriction of S ֒ → M π −→ Σ to D and Σ is trivializable. Fixing an identification ∂ D = − ∂ Σ = S as oriented manifolds, we conclude that—up to isomorphism—the bundle π : M → Σ is given by a glueing function of the form ϕ : ∂ ( D × S ) → ∂ (Σ × S ) , ϕ ( z, λ ) = ( z, z − l λ ) , (2.20)where z ∈ S = ∂ D = − ∂ Σ . We want to use this description to determine the fundamental groupof M . For elements a, b ∈ π (Σ) we write [ a, b ] = b − a − ba , which according to our conventionmeans to first follow the path a , then b and then the same again with the orientations reversed. Lemma 2.3.1. Let S ֒ → M π −→ Σ be an oriented principal circle bundle of degree l ∈ Z . Thenthe fundamental group of M has the presentation π ( M ) = D a , b , . . . , a g , b g , γ (cid:12)(cid:12)(cid:12) g Y i =1 [ a i , b i ] = γ l , γ central E , where a , b , . . . , a g , b g are lifts to M of the standard generators of π Σ and γ is the homotopyclass of the S -fiber.Proof. Let c be the homotopy class of ∂ Σ . Then the canonical generators of π (Σ ) are the onesindicated in Figure 2.1. It is well known, see e.g. [40, Sec. III.3.5], that π (Σ ) = D a , b , . . . , a g , b g , c (cid:12)(cid:12)(cid:12) g Y i =1 [ a i , b i ] = c − E . Write π (cid:0) ∂ ( D × S ) (cid:1) = (cid:10) ˜ c, ˜ γ (cid:12)(cid:12) [˜ γ, ˜ c ] = 1 (cid:11) , π (cid:0) ∂ (Σ × S ) (cid:1) = (cid:10) c, γ (cid:12)(cid:12) [ γ, c ] = 1 (cid:11) . Note that ˜ c is annihilated under the inclusion ∂ D ֒ → D . Moreover, the map (2.20) induces a mapon fundamental groups ϕ ∗ : π (cid:0) ∂ ( D × S ) (cid:1) → π (cid:0) ∂ (Σ × S ) (cid:1) , ϕ ∗ (˜ γ ) = γ, ϕ ∗ (˜ c ) = γ − l c − . .3. The U (1) -Rho Invariant for S -Bundles over Surfaces π (Σ )Van Kampen’s Theorem now shows that π ( M ) = D a , b , . . . , a g , b g , c, γ (cid:12)(cid:12)(cid:12) g Y i =1 [ a i , b i ] = c − , ∀ i [ a i , γ ] = [ b i , γ ] = 1 , c − = γ l E which by cancelling c coincides with the claimed presentation.Since H = π / [ π , π ], it follows immediately from the above Lemma that H ( M, Z ) = H (Σ , Z ) ⊕ Z l , (2.21)where we set Z l = Z if l = 0. As the first homology group H (Σ , Z ) is equal to Z g , we deducethat Hom (cid:0) H (Σ , Z ) , U(1) (cid:1) = U(1) g . The long exact coefficient sequence shows that this 2 g -dimensional torus can be identifiedwith H (Σ , R ) /H (Σ , Z ). From (2.21) one can now determine the moduli space of U(1)-representations, which is the topological version of moduli space of flat Hermitian line bundles,see Proposition B.1.8. Lemma 2.3.2. Let M → Σ be an oriented principal circle bundle of degree l . Then the modulispace of flat Hermitian line bundles on M is given by M (cid:0) M, U(1) (cid:1) ∼ = ( U(1) g × Z l , if l = 0 , U(1) g +1 , if l = 0 . Remark 2.3.3. Note that in the case l = 0 it follows from Poincar´e duality H ( M, Z ) ∼ = H ( M, Z ) and (2.21) that Tor H ( M, Z ) = Z l . Hence there are flat line bundles which are topo-logically non-trivial. This corresponds to the above decomposition of M (cid:0) M, U(1) (cid:1) into l differentcomponents. We have included some more details on flat line bundles which are topologicallynon-trivial in Appendix B.1, see in particular Lemma B.1.10.0 2. Rho Invariants of Fiber Bundles, Basic Considerations Flat Line Bundles over M . We now need a description of M (cid:0) M, U(1) (cid:1) in terms of flat linebundles. We will see that every flat line bundle over the total space M arises as the pullback ofa line bundle on the base Σ. Since M → Σ is a principal S -bundle, there exists an associatedHermitian line bundle L → Σ which clearly has to play a particular role. Much of the discussionto follow is inspired by [78, Sec. 3.3], although we include some more details and put moreemphasis on the explicit description of the U(1)-moduli space.We will work with respect to a fixed connection on the principal S -bundle. For this we firstendow Σ with a Riemannian metric g Σ of unit volume. As noted in Appendix B.3, this amountsto fixing a complex structure on Σ. We identify the Lie algebra of S with i R . Let e be thevector field on M associated to the S -action, e | p = ddt (cid:12)(cid:12) t =0 p · e it , p ∈ M. A connection on the principal bundle π : M → Σ is a 1-form iω ∈ Ω ( M, i R ) such that ω ( e ) = 1 and R ∗ e it ω = ω, where R e it denotes right-multiplication, compare with (B.1). Let F ω ∈ Ω (Σ , i R ) be the curvatureof iω . Since the cohomology class of i π F ω represents the rational first Chern class of the bundle π : M → Σ we can choose ω in such a way that − π dω = i π π ∗ F ω = l · π ∗ vol Σ . (2.22)Let L → Σ be the line bundle associated to the principal bundle structure. The connection ω induces a natural connection A ω on L . We write L ω for the line bundle L endowed with thisparticular connection A ω . As explained in Appendix B.3, this is the same as fixing a holomorphicstructure on L . The following simple observation relies only on the principal bundle structureand not on the particular structure group U(1) or the dimension of the base. Lemma 2.3.4. The pullback π ∗ L ω → M is canonically trivial and the pullback connection π ∗ A ω satisfies π ∗ A ω = d M + iω. Proof. Recall that the associated bundle L ω → Σ is defined by the pullback diagram M × C −−−−→ M y π y M × C / ∼ −−−−→ Σwhere ( p, v ) ∼ ( pz, z − v ) for all z ∈ S . Since π ∗ L ω is given by the same pullback diagram, wetautologically get M × C = π ∗ L ω . Under this identification, π ∗ C ∞ (Σ , L ω ) = (cid:8) ϕ : M → C (cid:12)(cid:12) ϕ ( pz ) = z − ϕ ( p ) (cid:9) . The pullback connection π ∗ A ω acts on equivariant functions ϕ as( π ∗ A ω ) X ϕ = X h ϕ = d M ϕ ( X ) − X v ϕ, .3. The U (1) -Rho Invariant for S -Bundles over Surfaces X h/v denotes the horizontal/vertical projection of X with respect to the connection iω .The latter is explicitly given by X vp = ddt (cid:12)(cid:12) p · exp (cid:0) tiω ( X p ) (cid:1) = ω ( X ) e | p . If ϕ is an equivariant function, then eϕ = − iϕ . Therefore, X vp ϕ = − iω ( X p ) ϕ ( p ) . Extending by the Leibniz rule to all functions on M , we get π ∗ A ω = d M + iω. Now let L A → Σ be an arbitrary Hermitian line bundle of degree k with a holomorphicstructure given by a unitary connection A , see Appendix B.3. It follows from Proposition B.3.5that upon transforming A with a complex gauge transformation f ∈ G c we may—and will—assume in the following that i π F A = k vol Σ . (2.23)In general, to achieve this, we really need to transform with a complexified gauge transformationand not just a unitary one, see Proposition B.3.5. Lemma 2.3.5. Assume that l = 0 , and let q := k/l . Then the connection A q := π ∗ A − q iω on π ∗ L A is flat. Moreover, the holonomy of A q along the S -fiber γ is given by hol A q ( γ ) = exp(2 πiq ) . Proof. By functoriality, we have F π ∗ A = π ∗ F A . Thus, it follows from assumptions (2.22) and(2.23) that the curvature of A q satisfies F A q = π ∗ F A − q idω = − πi (cid:0) k − ql (cid:1) π ∗ vol Σ = 0 . To compute the holonomy, let p ∈ M be arbitrary and let γ ( t ) = p · exp( it ) with t ∈ [0 , π ]parametrize the fiber containing p . Clearly, hol π ∗ A ( γ ) = 0 and ω γ ( t ) ( . γ ( t )) = 1. Therefore,hol A q ( γ ) = exp (cid:16) − Z γ − q iω (cid:17) = exp (cid:16) q i Z π ω γ ( t ) ( . γ ( t )) dt (cid:17) = exp(2 πiq ) . The Moduli Space of Flat Line Bundles. After this preparation, we can now give the geo-metric description of M (cid:0) M, U(1) (cid:1) . Recall that Pic(Σ) denotes the Picard group of holomorphicline bundles over Σ, see Definition B.3.4. Proposition 2.3.6. Let l = 0 and let M (cid:0) M, U(1) (cid:1) be the moduli space of flat line bundles over M . Then ω induces a natural surjection π ∗ : Pic(Σ) → M (cid:0) M, U(1) (cid:1) , [ L A ] (cid:2) π ∗ L A , A q (cid:3) , where A q is defined as in Lemma 2.3.5. There is a natural Z -action on Pic(Σ) given by ( L A , k ) L A ⊗ L ⊗ kω , and with respect to this, Pic(Σ) / Z ∼ = M (cid:0) M, U(1) (cid:1) . 2. Rho Invariants of Fiber Bundles, Basic Considerations Proof. Let L A → Σ be a holomorphic line bundle. Assume that B is another unitary connectionon L , satisfying condition (2.23) and inducing an equivalent holomorphic structure, i.e., B = A + u − du, for some u ∈ G c .Condition (2.23) means in particular that F A = F B , which implies that in fact u ∈ G . From thiswe obtain that B q = A q + π ∗ ( u − du ) , for some u ∈ G .i.e., the flat connections B q and A q on π ∗ L A are equivalent. This shows that the map in Propo-sition 2.3.6 is well-defined.To verify that it is surjective, let L → M be a flat Hermitian line bundle with connection˜ A . Let γ denote the generator of the S -fiber in π ( M ). Then Lemma 2.3.1 shows that γ l is acommutator. It follows that for some k ∈ Z ,hol ˜ A ( γ ) = exp(2 πik/l ) . (2.24)Now let L A → Σ be an arbitrary holomorphic line bundle of degree k . We infer from (2.24) andLemma 2.3.5 that π ∗ L A ⊗ L − , endowed with the connection A q ⊗ − ⊗ ˜ A , is a flat line bundleon M with trivial holonomy along the fiber γ . This easily implies that it is equivalent to thepullback π ∗ C B of the trivial line bundle over Σ endowed with a flat connection B . Thus, as linebundles with connection, L = π ∗ ( L A ⊗ C B ) , which proves surjectivity of the map in Proposition 2.3.6.As we have seen in Lemma 2.3.4, the pullback π ∗ L ω with connection π ∗ A ω − iω is the trivialflat line bundle. Using this one observes that the map π ∗ is invariant under the natural Z -actionon Pic(Σ). Assume now that π ∗ L A = π ∗ L B for two holomorphic line bundles over Σ of degree k and m respectively. Since their holonomies along γ agree, it follows that m − k = nl for some n ∈ Z . Thus, π ∗ ( L B ⊗ L − A ) = π ∗ L ⊗ nω and π ∗ B = π ∗ A + n · iω. We deduce that L B = L A ⊗ L ⊗ nω as holomorphic line bundles, which is what we needed toprove. Remark 2.3.7. Recall that the S -bundle π : M → Σ gives rise to the Gysin sequence ... → H (Σ) ∪ c −→ H (Σ) π ∗ −→ H ( M ) π ∗ −→ H (Σ) → , see [22, Prop. 14.33]. Here, c = c ( M ) ∈ H (Σ) is the first Chern class (or Euler class) ofthe oriented S -bundle. If we are assuming that l = 0, the map H (Σ) ∪ c −→ H (Σ) gives anisomorphism in de Rham cohomology. This implies that for cohomology with integer coefficients,the map π ∗ : H (Σ , Z ) → H ( M, Z ) appearing in the Gysin sequence surjects onto the torsionsubgroup of H ( M, Z ). It is related to the map π ∗ of Proposition 2.3.6 by the following diagramPic(Σ) −−−−→ c H ( M, Z ) π ∗ y π ∗ y M (cid:0) M, U(1) (cid:1) −−−−→ c Tor (cid:0) H ( M, Z ) (cid:1) . .3. The U (1) -Rho Invariant for S -Bundles over Surfaces c is equivariant with respect to the Z -action on Pic(Σ), c ( L A ⊗ L ⊗ kω ) = c ( L A ) + k · c ( M ) . Using Proposition B.3.5 one can now interpret the above diagram as the geometric version ofLemma 2.3.2 in the case l = 0.The structure result Proposition 2.3.6 excludes the case that the circle bundle is of degree l = 0, i.e., isomorphic to Σ × S . However, a geometric description in this case is easy to finddirectly. As in Remark 1.4.8 (iii), a flat line bundle L q over S is the trivial line bundle endowedwith the connection d − qz − dz for some q ∈ R . Here, we view S as a subset of C , and z − dz expresses the Maurer-Cartan form of S . Clearly, L q and L q ′ are unitarily equivalent if and onlyif q − q ′ ∈ πi Z . Without effort one verifies the following result. Lemma 2.3.8. If M = Σ × S is the trivial circle bundle over Σ , then M (cid:0) Σ , U(1) (cid:1) × M (cid:0) S , U(1) (cid:1) ∼ = M (cid:0) M, U(1) (cid:1) . Here, the isomorphism is given by (cid:0) [ L A ] , [ L q ] (cid:1) (cid:2) L A ⊠ L q (cid:3) , where L A ⊠ L q is the fiber product defined in (1.15) , endowed with its natural connection. We now want to identify the odd signature operator on the total space of a principal circle bundleover a closed, oriented surface. Certainly, the underlying principal bundle structure will playan important role, and many features generalize to arbitrary principal bundles with compactstructure group. However, we will not give many comments about these generalizations. Fibered Calculus on M . To start, we need to identify some of the quantities defined in Section2.1 in the case at hand. Let iω be a connection on the principal S -bundle π : M → Σ. Sincethe vector field e associated to the S -action gives a trivialization of the vertical tangent bundle,we get a vertical projection P v : T M → T v M, X ω ( X ) e. With respect to this, the curvature Ω in the sense of Definition 2.1.2 is related to the curvatureof iω by Ω( X h , Y h ) = − ω (cid:0) [ X h , Y h ] (cid:1) e = dω ( X h , Y h ) e = − iF ω ( X, Y ) e, where X h and Y h are horizontal lifts of vector fields X, Y on Σ. In particular, when we fix ametric g Σ of unit volume and require that ω satisfies (2.22), we haveΩ( X h , Y h ) = − πl vol Σ ( X, Y ) e. (2.25)We now endow T v M with the vertical metric g v := ω ⊗ ω , and consider the submersion metric g = g Σ ⊕ g v .4 2. Rho Invariants of Fiber Bundles, Basic Considerations Lemma 2.3.9. (i) With respect to the trivialization given by e , the canonical connection ∇ v on T v M is thetrivial connection, i.e., ∇ vX e = 0 , X ∈ C ∞ ( M, T M ) . (ii) If X ∈ C ∞ (Σ , T Σ) , we have L X h ( e ) = [ X h , e ] = 0 , and L vX h ( g v ) = 0 . In particular, the connection e ∇ v from Definition 2.1.9 agrees with ∇ v , and the mean cur-vature k v as well as the tensor B in (2.10) vanish.Proof. The connection ∇ v is compatible with g v . Hence,0 = g v ( ∇ vX e, e ) + g v ( e, ∇ vX e ) , which yields ∇ vX e = 0. This proves (i). Since iω is a connection, we have R ∗ e it ω = ω . This impliesthat the metric g v = ω ⊗ ω is invariant under the flow associated to the vector field e . Since π ∗ g Σ is constant along the fiber, we find that for all vector fields X on Σ0 = L e ( g )( X h , e ) = g (cid:0) [ e, X h ] , e (cid:1) + g (cid:0) X h , [ e, e ] (cid:1) . As [ e, e ] = 0 we conclude that g (cid:0) [ e, X h ] , e (cid:1) = 0. This implies that [ e, X h ] = 0, because Lemma2.1.1 ensures that the vector field [ e, X h ] is vertical. In particular, since L vX h g v ( e, e ) = X h (cid:0) g v ( e, e ) (cid:1) − g v (cid:0) [ X h , e ] , e (cid:1) , we deduce that the vertical Lie derivative of g v vanishes. Using its very definition, we see thatthe tensor B is indeed trivial. Moreover, we know from Lemma 2.1.8 that the mean curvature isgiven by the trace of L vX h ( g v ). Thus, it is also is zero. Moreover, using part (i) we find that thederivations L X h and ∇ vX h agree on e . Since both satisfy the Leibniz rule they are necessarilyequal. Hence, by definition, the connection e ∇ v agrees with ∇ v . Rho Invariants for Trivial Circle Bundles. Before we continue with the general discussion,we assume that l = 0 so that M = Σ × S . We endow M with the natural connection iω = z − dz given by the Maurer-Cartan form on S . Choose a flat line bundle L → M , i.e., L = L A ⊠ L q → Σ × S , where L A and L q are flat line bundles over Σ respectively S , see Lemma 2.3.8. We identifyΩ ev (Σ × S , L ) = Ω • (Σ , L A ) ⊗ C ∞ ( S ) . Using Lemma 2.2.1 and Lemma 2.2.3, we can write the odd signature operator as B A,q := B ev A,q = τ Σ ⊗ B q + D A ⊗ , where D A is the twisted de Rham operator on Σ and B q is the odd signature operator on S , B q = − i ( L e − iq ) : C ∞ ( S ) → C ∞ ( S ) . .3. The U (1) -Rho Invariant for S -Bundles over Surfaces B A,q is of the form considered in Lemma 1.3.6 (iii). According to the Hirzebruch SignatureTheorem, the index of D + A vanishes for all flat connections A on Σ and so η ( B A,q ) = ind( D + A ) · η ( B q ) = 0 . (2.26)Therefore, all Rho invariants for the trivial circle bundle Σ × S vanish. The Structure of B A,q in the General Case. We now assume that l = 0. Let L A → Σ be aline bundle of degree k endowed with a Hermitian connection A which satisfies the condition of(2.23), i π F A = k · vol Σ . We endow the pullback L := π ∗ L A → M with the flat connection of Lemma 2.3.5, i.e., A q = π ∗ A − iq ω, q := k/l. Since L is the pullback of L A , we alter the identification (2.6) slightly toΩ • ( M, L ) = π ∗ Ω • (Σ , L A ) ⊗ Ω • v ( M ) . As in Lemma 2.1.15 we write the twisted de Rham operator as d A q = d q,v + d A,h + i(Ω) , where d q,v = d v − iq e( ω ) , d A,h = e( f a ) e ∇ A q , ⊕ a = ( π ∗ d A ) ⊗ f a ) ⊗ e ∇ va . (2.27)As always e( . ) denotes exterior multiplication and { f , f } is a local orthonormal frame for T Σ.To describe the odd signature operator, we split the space of L -valued differential forms ofeven degree as in Lemma 2.2.1,Ω ev ( M, L ) = π ∗ Ω • (Σ , L A ) ⊗ C ∞ ( M ) . Proposition 2.3.10. With respect to the above identification, the odd signature operator is givenby B A,q = τ Σ ⊗ B q,v + D A,h + τ M T, where the individual terms are B q,v = − i ( L e − iq ) , D A,h = D A ⊗ c ( f a ) ⊗ L vf a , and τ M T = ( , ⊕ Ω , , − πl on Ω , . Moreover, we have the (anti-)commutator relations (cid:2) ⊗ B q,v , D A,h (cid:3) = 0 and (cid:8) τ Σ ⊗ B q,v , D A,h (cid:9) = 0 . (2.28)6 2. Rho Invariants of Fiber Bundles, Basic Considerations Proof. Let α ∈ π ∗ Ω p (Σ , L A ) and ϕ ∈ C ∞ ( M ). Then,( τ M D q,v )( α ∧ ϕ ) = ( − p τ M (cid:0) α ∧ ( D q,v ϕ ) (cid:1) = ( τ Σ α ) ∧ ( τ v D q,v ϕ ) , where we have used Lemma 2.2.3 in the last equality. Now, checking the factors of i in Definition2.2.2 one finds that τ v ( ω ) = − i . Thus,( τ v D q,v ) ϕ = ( τ v d q,v ) ϕ = τ v (cid:0) d v − iqω (cid:1) ϕ = τ v ( ω ) (cid:0) L e − iq (cid:1) ϕ = − i (cid:0) L e − iq (cid:1) ϕ. According to Lemma 2.2.1 the horizontal part of B A,q coincides with the horizontal de Rhamoperator D A,h = d A,h + d tA,h = e( f a ) e ∇ A, ⊕ a − i( f a ) (cid:0) e ∇ A, ⊕ a + 2 B ( f a ) + k v ( f a ) (cid:1) . Here, we have used Proposition 2.1.21 and (2.9). Hence, we deduce from Lemma 2.3.9 that D A,h = c ( f a ) e ∇ A, ⊕ a = D A ⊗ c ( f a ) ⊗ e ∇ va . For the last term appearing in the formula for B A,q note that T ( α ⊗ ϕ ) = e( f ) e( f ) i(Ω ) − i( f ) i( f ) e(Ω ) , where (2.25) shows that Ω = − πle . Therefore, T is non-zero only on Ω , , and( τ M T )(vol Σ ∧ ϕ ) = τ M (cid:0) − πl i( f ) i( f ) vol Σ ∧ ϕω (cid:1) = − πl (cid:0) τ Σ (1) ∧ ϕτ v ( ω ) (cid:1) = − πl vol Σ ∧ ϕ. In this computation we have used Lemma 2.2.3 to express τ M in terms of τ Σ and τ v . Also notethat τ Σ (1) = i vol Σ and τ v ( ω ) = − i .Lemma 2.3.9 (ii) implies that the bundle endomorphism K as defined in (2.13) vanishes.Thus, we deduce from Proposition 2.1.25 that d q,v d tA,h + d tA,h d q,v = 0 , and D q,v D A,h + D A,h D q,v = 0 . (2.29)Also D A and c ( f a ) anti-commute with τ Σ , since Σ is even dimensional. This yields the relationsin (2.28). The Spectrum of B q,v . The vertical odd signature operator B q,v is not elliptic, since itsprincipal symbol vanishes in all directions orthogonal to the fiber. Thus, we do not know muchabout the spectrum of B q,v by employing the general theory. However, due to the S -symmetry,we can determine its eigenvalues by hand. Remark. Before we state the next result, recall that L ω → Σ denotes the line bundle associatedto M endowed with the connection A ω induced by ω . As we have seen in Lemma 2.3.4, thepullback π ∗ L ω → M is isomorphic to the trivial line bundle. Under this identification, a function ϕ ∈ C ∞ ( M ) is the pullback of a section s ϕ ∈ C ∞ (Σ , L ω ) if and only if ϕ ( p · z ) = z − ϕ ( p ) , p ∈ M, z ∈ S . (2.30)Moreover, the Lie derivative is related to the connection A ω via s ( X h ϕ ) = A ω ( X ) s ϕ . (2.31)We refer to the proof of Lemma 2.3.4 for more details. .3. The U (1) -Rho Invariant for S -Bundles over Surfaces Lemma 2.3.11. Assume that l = 0 , and let L A be a holomorphic line bundle over Σ of degree k . Moreover, let q := k/l , and let L = π ∗ L A be the associated flat line bundle over M . Then ker (cid:0) B q,v − λ (cid:1) = { } if and only if λ + q ∈ Z . Moreover, if λ + q ∈ Z , then ker (cid:0) B q,v − λ (cid:1) ∼ = π ∗ C ∞ (Σ , L B λ ) , where L B λ := L A ⊗ L − ( λ + q ) ω . The operator D A,h restricted to Ω • (Σ) ⊗ ker (cid:0) B q,v − λ (cid:1) corresponds under the above isomorphismto D B λ : Ω • (Σ , L B λ ) → Ω • (Σ , L B λ ) , where B λ = A ⊗ ⊗ − ( λ + q ) A ω is the natural connection on L B λ .Proof. Let ϕ ∈ ker( B q,v − λ ). According to Proposition 2.3.10 this means that L e ϕ = i (cid:0) q + λ (cid:1) ϕ. For t ∈ R and p ∈ M let b ϕ t ( p ) := ϕ ( p · e it ). Then, since e is the vector field generated by the S -action, ddt b ϕ t = L e ( b ϕ t ) = i ( q + λ ) · b ϕ t , i.e., b ϕ t = e i ( q + λ ) t ϕ. This implies that q + λ ∈ Z or ϕ = 0. Rewriting the result in terms of z = e it we see that ϕ ( p · z ) = z q + λ · ϕ ( p ) , z ∈ S . As in (2.30) this means that we can identify ϕ with a section ϕ ∈ π ∗ C ∞ (cid:0) Σ , L A ⊗ L − q − λω (cid:1) = π ∗ C ∞ (Σ , L B λ ) . Tracing the proof backwards shows that conversely every such element gives an eigenvector of B q,v . The assertion about D A,h easily follows from (2.31) and Proposition 2.3.10. Remark 2.3.12. Without going into details, we want to mention that we have actually deter-mined the full spectrum of B q,v . We note without proof that B q,v is essentially self-adjoint in π ∗ Ω • (Σ , L A ) ⊗ C ∞ ( M ) and that (2.28) implies that it commutes with the formally self-adjointelliptic operator D h + B q,v . This suffices to guarantee that spec( B q,v ) consists only of eigenvalues—though, with infinite multiplicities. Then Lemma 2.3.11 implies thatspec( B q,v ) = { λ | λ + q ∈ Z } . Moreover, as in the case of eigenvalues with finite multiplicity, we can decompose π ∗ Ω • (Σ , L A ) ⊗ C ∞ ( M ) = M λ ∈ spec( B q,v ) Ω • (Σ) ⊗ ker (cid:0) B q,v − λ (cid:1) ∼ = M λ ∈ spec( B q,v ) Ω • (Σ , L B λ ) . We also want to note that this decomposition is essentially the decomposition of the infinite di-mensional S -module π ∗ Ω • (Σ , L A ) ⊗ C ∞ ( M ) into its irreducible components. A similar situationshould occur for general Lie groups.8 2. Rho Invariants of Fiber Bundles, Basic Considerations The fact that the commutators in (2.28) in Proposition 2.3.10 are zero allows us to give anelementary computation of Eta invariants, see [79, App. C] for a related treatment. However,the Eta invariant of the full signature operator is not directly tractable. Therefore, we introducethe following. Definition 2.3.13. Let L A → Σ be a line bundle of degree k , and let L := π ∗ L A be thecorresponding flat line bundle over M . We call the operator B ⊕ A,q := τ Σ ⊗ B q,v + D A,h on π ∗ Ω • (Σ , L A ) ⊗ C ∞ ( M )the truncated odd signature operator twisted by L . Remark 2.3.14. The connection ∇ ⊕ from (2.3) together with A q induces a connection ∇ A q , ⊕ on Λ • T ∗ M ⊗ L . Then the truncated odd signature operator is given by Clifford contraction of ∇ A q , ⊕ . Therefore, it is almost as good as a geometric Dirac operator. However, ∇ A q , ⊕ is not aClifford connection since it is compatible with ∇ ⊕ and not with the Levi-Civita connection ∇ g .As remarked earlier an operator of this type is in general not formally self-adjoint. However, inthe situation at hand, B ⊕ A,q is clearly formally self-adjoint, since B q,v and D A,h are.Since B ⊕ A,q is an formally self-adjoint elliptic differential operator on a closed manifold, itsEta function is well-defined and for Re( s ) large, η ( B ⊕ A,q , s ) = 1Γ (cid:0) s +12 (cid:1) Z ∞ Tr (cid:2) B ⊕ A,q exp (cid:0) − t ( B ⊕ A,q ) (cid:1)(cid:3) t s − dt. Moreover, Theorem 1.3.3 implies that the meromorphic extension of η ( B ⊕ A,q , s ) has no pole in0. Our strategy is now to compute the Eta invariant of the truncated odd signature operatorexplicitly and determine its kernel, see Proposition 2.3.15 and Proposition 2.3.16. Then in Section2.3.4 we will use these results to determine the Rho invariant of the full odd signature operator B A,q . For this we want to use the variation formula of Proposition 1.3.14 for Eta invariants sothat we will need to understand the spectral flow between B ⊕ A,q and B A,q . However, the difference B A,q − B ⊕ A,q might be “too large” compared to B ⊕ A,q to get a good control of the spectrum nearzero. As we will see the solution to this problem is to study an adiabatic metric. After this shortdigression on our strategy let us now investigate the truncated odd signature operator. Proposition 2.3.15. Let S ֒ → M π −→ Σ be a principal circle bundle of degree l = 0 . Let L A → Σ be a line bundle over Σ of degree k , and let L := π ∗ L A be the corresponding flat line bundle over M . Then, with q = k/l , η ( B ⊕ A,q ) = 2 lP ( q ) , where P is the second periodic Bernoulli function, i.e., if q − [ q ] = q with q ∈ [0 , and [ q ] ∈ Z ,then P ( q ) = q − q + , see Definition C.1.1. In particular, η ( B ⊕ A,q ) is independent of the metric g Σ , and the connection A involved in its definition. .3. The U (1) -Rho Invariant for S -Bundles over Surfaces Proof. Formula (2.28) in Proposition 2.3.10 shows that τ Σ ⊗ B q,v anti-commutes with D A,h .Hence, we can split B ⊕ A,q e − t ( B ⊕ A,q ) = D A,h e − tD A,h − t ( τ Σ ⊗ B q,v ) + (cid:0) τ Σ ⊗ B q,v (cid:1) e − tD A,h − t ( τ Σ ⊗ B q,v ) . Since D A,h anti-commutes with τ Σ one finds as in the proof of Lemma 1.3.6 thatTr (cid:2) D A,h e − tD A,h − t ( τ Σ ⊗ B q,v ) (cid:3) = 0 . Now let λ ∈ spec( B q,v ). Then according to Lemma 2.3.11, the operator (cid:0) τ Σ ⊗ B q,v (cid:1) e − tD A,h − t ( τ Σ ⊗ B q,v ) on Ω • (Σ) ⊗ ker (cid:0) B q,v − λ (cid:1) is unitarily equivalent to (cid:0) τ Σ e − tD Bλ (cid:1) λe − tλ : Ω • (Σ , L B λ ) → Ω • (Σ , L B λ ) , where B λ = A − ( λ + q ) A ω . It follows from the McKean-Singer formula and the index theoremfor the signature operator thatTr (cid:2) τ Σ e − t ( D Bλ ) (cid:3) = ind (cid:0) D + B λ (cid:1) = 2 (cid:0) k − l ( λ + q ) (cid:1) = − lλ. Hence, using the decomposition from Remark 2.3.12 we findTr (cid:2) B ⊕ A,q exp (cid:0) − t ( B ⊕ A,q ) (cid:1)(cid:3) = X λ ∈ spec( B q,v ) Tr (cid:2) τ Σ e − t ( D Bλ ) (cid:3) λe − tλ = − l X λ ∈ spec( B q,v ) λ e − tλ , Hence, we find that for Re( s ) large, η ( B ⊕ A,q , s ) = 1Γ (cid:0) s +12 (cid:1) Z ∞ Tr (cid:2) B ⊕ A,q exp (cid:0) − t ( B ⊕ A,q ) (cid:1)(cid:3) t s − dt = 1Γ (cid:0) s +12 (cid:1) Z ∞ − l X λ ∈ spec( B q,v ) λ e − tλ t s − dt = − l X λ ∈ spec( B q,v ) | λ | − s (cid:0) s +12 (cid:1) Z ∞ e − x x s − dx = − l X λ ∈ spec( B q,v ) | λ | − s , where we have substituted x = tλ . Also note that interchanging summation and integration canbe justified by the large and small time estimates on P λ ∈ spec( B q,v ) λ e − tλ , see Proposition 1.2.4and Theorem 1.2.7. Now, sincespec( B q,v ) = (cid:8) λ ∈ R (cid:12)(cid:12) λ + q ∈ Z (cid:9) , 2. Rho Invariants of Fiber Bundles, Basic Considerations we find that η ( B ⊕ A,q , s ) = − l X n ∈ Z n = q | n − q | − s , Re( s ) > . We have included a computation of the value at s = 0 in Proposition C.1.2 (ii). The result is theclaimed formula η ( B ⊕ A,q ) = 2 lP ( q ) . Proposition 2.3.16. The kernel of the truncated odd signature operator is given by ker( B ⊕ A,q ) = ker(1 ⊗ B q,v ) ∩ ker( D A,h ) ∼ = ( { } , if q / ∈ Z ,H • (Σ , L B ) , if q ∈ Z . Here, L B is the trivial line bundle endowed with the flat connection B = A − qA ω . Moreover, if q ∈ Z , and if g denotes the genus of Σ , then H • (Σ , L B ) ∼ = ( C ⊕ C g ⊕ C , if B is the trivial connection , { } ⊕ C g − ⊕ { } , otherwise.Proof. Since B ⊕ A,q is formally self adjoint, we haveker( B ⊕ A,q ) = ker( B ⊕ A,q ) = ker (cid:0) ⊗ B q,v + D A,h (cid:1) , where we have used that τ Σ ⊗ B q,v anti-commutes with D A,h . Since both, 1 ⊗ B q,v and D A,h , areformally self-adjoint we getker (cid:0) ⊗ B q,v + D A,h (cid:1) = ker(1 ⊗ B q,v ) ∩ ker( D A,h ) . Now Lemma 2.3.11 shows thatker(1 ⊗ B q,v ) ∼ = ( { } , if q / ∈ Z , Ω • (Σ , L B ) , if q ∈ Z , (2.32)where L B = L A ⊗ L − qω , endowed with the connection B = A ⊗ ⊗ qA ω . If q / ∈ Z the proofis finished, so that we assume from now on that q ∈ Z . Since L A is of degree k and L ω of degree l , we find that L A ∼ = L k/lω = L qω . Thus, L B is isomorphic to the trivial line bundle and B is flat.Moreover, Lemma 2.3.11 identifies the restriction of the operator D A,h to ker(1 ⊗ B q,v ) with thede Rham operator D B on Ω • (Σ , L B ). Using this and (2.32) we deduce from the Hodge-de-Rhamisomorphism that ker(1 ⊗ B q,v ) ∩ ker( D A,h ) ∼ = ker( D B ) ∼ = H • (Σ , L B ) . Now if B is isomorphic to the trivial connection, we have the well-known cohomology groups ofa surface H • (Σ) ∼ = C ⊕ C g ⊕ C , where g is the genus of Σ. In the case that B is non-trivial, the index theorem for the twisted deRham operator shows that ind( D B ) is independent of B as long as B is flat. Hence, X p ( − p dim H p (Σ , L B ) = ind( D B ) = ind( D ) = χ (Σ) = 2 − g. .3. The U (1) -Rho Invariant for S -Bundles over Surfaces H (Σ , L B ) = dim H (Σ , L B ) . However, if B is a non-trivial flat connection, then H (Σ , L B ) = { } . One way to see this is as follows : Let β : π (Σ) → U(1) be the holonomy representation of B .Then according to Proposition 1.1.5 and [33, Prop. 5.14], H (Σ , L B ) = (cid:8) z ∈ C (cid:12)(cid:12) β ( c ) z = z for all c ∈ π (Σ) (cid:9) . Therefore, for H (Σ , L B ) is trivial unless β ≡ 1. Putting these observations together, we findthat for non-trivial B we have indeed H • (Σ , L B ) ∼ = { } ⊕ C g − ⊕ { } . After having calculated the Eta invariant for the truncated signature operator B ⊕ A,q , we turn ourattention to the Rho invariant of the odd signature operator B A,q . As before, let S ֒ → M π −→ Σbe a circle bundle of degree l = 0 over a closed surface Σ. We let g Σ be a metric on Σ of unitvolume, and g v be the metric on T v M such that the vector field e has length 1. For ε > g ε := ε g Σ ⊕ g v , and let ∇ g ε be the Levi-Civita connection associated to g ε . For each t ∈ [0 , 1] and ε > T M by ∇ ε,t := (1 − t ) ∇ ⊕ + t ∇ g ε , where ∇ ⊕ is the direct sum connection of (2.3), which is independent of the scaling parameter ε .Now let L A → Σ be a holomorphic line bundle of degree k , and let L := π ∗ L A be thecorresponding flat line bundle over M . Contracting the connection ∇ ε,t ⊗ ⊗ A q on π ∗ Λ • ( T ∗ Σ) ⊗ π ∗ L A with the natural Clifford multiplication, we obtain a 2-parameter family of formally self-adjointelliptic operators B ε,tA,q := τ Σ ⊗ B q,v + εD A,h + tε τ M T on π ∗ Ω • (Σ , L A ) ⊗ C ∞ ( M ) , which connects the truncated odd signature operator with the full odd signature operator asso-ciated to g ε . Proposition 2.3.17. There exists ε such that for all ε < ε the following holds. See also Corollary 4.3.6 below for a different proof of this fact. 2. Rho Invariants of Fiber Bundles, Basic Considerations (i) If q / ∈ Z , then for all t ∈ [0 , (cid:0) B ε,tA,q ) = { } , and SF (cid:0) B ε,tA,q (cid:1) t ∈ [0 , = 0 . (ii) If q ∈ Z and A q is non-trivial, then for all t ∈ [0 , (cid:0) B ε,tA,q ) ∼ = C g − , and SF (cid:0) B ε,tA,q (cid:1) t ∈ [0 , = 0 . (iii) For the trivial connection we have ker (cid:0) B ε,t (cid:1) ∼ = ( C g +1 , if t = 0 , C g +2 , if t = 0 . and SF (cid:0) B ε,t (cid:1) t ∈ [0 , = ( , if l < , − , if l > Proof. To keep the notation simple we abbreviate B ε,tA,q = τ Σ ⊗ B q,v + εD A,h + tε τ M T =: B + εD + tε S. According to (2.28) we have { B, D } = 0, and so (cid:0) B ε,tA,q (cid:1) = B + ε D + ε t S + ε t { B, S } + ε t { D, S } . (2.33)By definition the operators B , D and S are formally self-adjoint. Thus,( ε / B + ε / tS ) ≥ ε / D + ε / tS ) ≥ . This implies ε t { B, S } ≥ − εB − ε t S and ε t { D, S } ≥ − ε D − ε t S . Using this in (2.33) we can estimate that for ε < / (cid:0) B ε,tA,q (cid:1) ≥ ε (cid:0) B + D − εt S (cid:1) . Now B + D is an elliptic operator, and since B and D are both formally self-adjoint, it hasnon-negative spectrum. Hence, its non-zero eigenvalues are bounded from below by some λ > 0. Moreover, S is an operator of order 0 so that S is a bounded operator. Letting ε < min { , λ || S || } we find that for all ε < ε and t = 0ker (cid:0) B + D − εt S (cid:1) = ker B ∩ ker D ∩ ker S, where we have used that ker( B + D ) = ker B ∩ ker D .We now switch back to the usual notation. Using Proposition 2.3.16 we can reformulate whatwe have observed so far:(i) If q / ∈ Z and ε < ε , then ker( B ε,tA,q ) = { } . (ii) If q ∈ Z and ε < ε , then for all t = 0ker( B ε,tA,q ) = ker( τ M T ) ∩ H • (Σ , L B ) , L B = L A ⊗ L − qω . .3. The U (1) -Rho Invariant for S -Bundles over Surfaces q / ∈ Z the proof is finished and we assume henceforth that q ∈ Z . FromProposition 2.3.10 we know that τ M T = ( , ⊕ Ω , , − πl on Ω , . On the other hand, we have seen in Proposition 2.3.16 that H • (Σ , L B ) ∼ = ( C ⊕ C g ⊕ C , if B is the trivial connection , { } ⊕ C g − ⊕ { } , otherwise.Therefore, since l = 0,ker( τ M T ) ∩ H • (Σ , L B ) ∼ = ( C ⊕ C g ⊕ { } , if B is the trivial connection , { } ⊕ C g − ⊕ { } , otherwise.This implies part (ii) and the first assertion of (iii). To finish the proof we still have to computethe spectral flow in the case that B is the trivial connection. Since the kernels of B ε,t are ofconstant dimension for t = 0 the only possible spectral flow contribution is at t = 0. As we haveseen ker( B ε, ) contains H (Σ) as a summand, whereas ker( B ε,t ) for t = 0 does not. Now H (Σ)is spanned by the cohomology class of vol Σ and we can explicitly compute that B ε,t (vol Σ ) = (cid:0) τ Σ B v + εD h + tε τ M T (cid:1) (vol Σ ) = − πlε t vol Σ . (2.34)According to the convention in Definition 1.3.13 of how to count eigenvalues at the endpoints wefind that SF (cid:0) B ε,t (cid:1) t ∈ [0 , = ( , if l < , − , if l > Remark. Note that (2.34) also shows that the odd signature operator associated to the metric g ε has a non-trivial eigenvalue − πlε which is of order ε . Eigenvalues of this type play a specialrole in the general adiabatic limit formula of [30]. We will discuss this in more detail in Sections3.3.2 and 3.3.3, see in particular Proposition 3.4.3.We now have collected all ingredients to compute the U(1)-Rho invariant for circle bundlesover surfaces. Theorem 2.3.18. Let S ֒ → M π −→ Σ be a principal circle bundle of degree l = 0 . Let L A → Σ bea line bundle over Σ of degree k , and let L := π ∗ L A be the corresponding flat line bundle over M .Write q := k/l and assume that the flat connection A q = π ∗ A − iqω is not the trivial connection.Then ρ A q ( M ) = 2 l (cid:0) P ( q ) − (cid:1) + sgn( l ) . If M = Σ × S is the trivial circle bundle, then all Rho invariants vanish.Proof. The Rho invariant associated to the odd signature operator is independent of the metric.In particular, ρ A q ( M ) = η ( B ε, A,q ) − η ( B ε, ) , ε > . 2. Rho Invariants of Fiber Bundles, Basic Considerations Hence, the variation formula (1.52) implies that for all ε > ρ A q ( M ) = η ( B ε, A,q ) − η ( B ε, ) + 2 SF (cid:0) B ε,tA,q (cid:1) t ∈ [0 , − (cid:0) B ε,t (cid:1) t ∈ [0 , − dim(ker B ε, A,q ) + dim(ker B ε, ) + dim(ker B ε, A,q ) − dim(ker B ε, ) . As we have seen in Proposition 2.3.15, the Eta invariant associated to the truncated odd signatureoperator does not change if the metric on the base is rescaled. Thus, η ( B ε, A,q ) − η ( B ε, ) = 2 l (cid:0) P ( q ) − (cid:1) . From Propositions 2.3.16 and 2.3.17 we see that if A q is non-trivialdim(ker B ε, A,q ) = dim(ker B ε, A,q ) . On the other hand, in the untwisted casedim(ker B ε, ) − dim(ker B ε, ) = (2 g + 1) − (2 g + 2) = − . Lastly, we have seen in Proposition 2.3.17 that for ε small enoughSF (cid:0) B ε,tA,q (cid:1) t ∈ [0 , = 0 and SF (cid:0) B ε,t (cid:1) t ∈ [0 , = ( , if l < , − , if l > ρ A q ( M ) = 2 l (cid:0) P ( q ) − (cid:1) + sgn( l ) . The triviality of Rho invariants for Σ × S follows from (2.26). hapter 3 Rho Invariants of Fiber Bundles,Abstract Theory This chapter forms the main theoretical part of the thesis. After having introduced the idea ofadiabatic metrics on fiber bundles and seen their effect in the computation of the Rho invariantfor principal circle bundles, we now want to describe how powerful tools of local index theory leadto a general formula for the adiabatic limit of Eta invariants. Since there exists a wide range ofliterature on this subject, the ideas presented here are not new. Nevertheless, we give a detailedaccount, including some proofs if feasible.The treatment starts with the bundle of vertical cohomology groups over the base of the fiberbundle. To relate it to the kernel of the vertical de Rham operator, we discuss a fibered versionof the Hodge decomposition theorem. As a byproduct of this we can prove a result about howto achieve that the mean curvature of a fiber bundle vanishes. Continuing with the main line ofargument, we give a detailed discussion of the natural flat connection that exists on the bundleof vertical cohomology groups. It is precisely this topological nature of the kernel of the verticalde Rham operator which will make the adiabatic limit formula accessible for computationalpurposes. In particular, we will need to discuss a version of the odd signature operator on thebase twisted by the bundle of vertical cohomology groups.As the formulation of the general adiabatic limit formula relies on Bismut’s local index theoryfor families, we continue with a brief survey of the main constructions and necessary results. Inparticular, we include a short discussion of superconnections and associated Dirac operators.Returning to the context of fiber bundles, we introduce the Bismut superconnection and recallhow it appears in the local index theorem for families.With these notions at hand, we will give a heuristic derivation of the adiabatic limit formulafor families of odd signature operators. Then, referring to the literature for rigorous proofs, wefinally state the general adiabatic limit formula for the Eta invariant due to Dai. One of the termsappearing there has a topological interpretation in terms of the Leray-Serre spectral sequence,and we discuss this briefly.We finish this chapter using the adiabatic limit formula to derive again the formula for theRho invariant of a principal S -bundle over a closed surface. Although we have obtained theformula already in the last chapter, it is illuminating to observe how the abstract theory leadsto a shorter a more conceptual proof. 856 3. Rho Invariants of Fiber Bundles, Abstract Theory In Remark 2.1.24 we have pointed out without further comments that there is a relationshipbetween differential operators acting fiberwise and families of differential operators in the senseof Definition 1.3.9. We want to make this a bit more precise now. Let F ֒ → M π −→ B be anoriented fiber bundle, where as before all manifolds are assumed to be closed, connected andoriented, and let E → M be a Hermitian over M . Definition 3.1.1. Let D : Ω • ( M, E ) → Ω • ( M, E ) be a differential operator. Then we call D a fiberwise differential operator, if[ D, π ∗ ϕ ] = 0 , for all ϕ ∈ C ∞ ( B ) . We call D fiberwise elliptic if in addition its principal symbol σ ( D )( x, ξ ) : E x → E x , x ∈ M, is invertible for every non-vanishing ξ ∈ T vx M ∗ .Certainly, if T v M is endowed with a metric, and A is a flat connection on E , the vertical deRham operator D A,v as in Definition 2.1.23 is a fiberwise elliptic operator in the sense of thisdefinition. Local Trivializations and Families. To relate fiberwise differential operators with familiesof differential operators as in Definition 1.3.9, we describe a particular way to construct localtrivializations of the fiber bundle, see [50, Lem. 1.3.3]. Lemma 3.1.2. Let g = g B ⊕ g v be a submersion metric as in Section 2.1. Let y ∈ B and F := π − ( y ) . Then for every sufficiently small geodesic neighbourhood U around y there existsan isomorphism of fiber bundles Φ : U × F → π − ( U ) , such that for all ( u, x ) ∈ U × F and every vector v ∈ T y U ⊂ T ( y,x ) U × F , Φ( y, x ) = x, π ◦ Φ( u, x ) = u, and Φ ∗ v = v h , where v h refers to the horizontal lift of v .Proof. Let b = dim B , and let U ⊂ B be a geodesic ball centered in y . We identify U with anopen ball in R b in such a way that y = 0. We can use the horizontal projection P h : T M → T h M to lift the coordinate vector fields ∂ a to horizontal vector fields ∂ ha on π − ( U ). Identifying a point u ∈ U with the vector field u a ∂ a we get a vector field u h = u a ∂ ha on π − ( U ), and hence a flowΦ t ( u, . ) : M → M, u ∈ U. It follows from the construction that for small t , the flow Φ t ( u, . ) maps the fiber F diffeomorphi-cally onto the fiber π − ( tu ). Moreover, for all x ∈ F we have Φ st ( u, x ) = Φ t ( su, x ), so that wecan choose U small enough to defineΦ : U × F → π − ( U ) , ( u, x ) Φ ( u, x ) . The claimed properties all follow immediately from this definition. .1. The Bundle of Vertical Cohomology Groups U × F → π − ( U ) of the form just described, we can transferall geometric structures on π − ( U ) to U × F : First of all, it is straightforward to check thatΦ ∗ T v M = U × T F. Therefore, the pullback Φ ∗ ( g v ) of the vertical metric is the same as a family of Riemannian metrics g F,u on F . In the same way the restriction of ∇ v to T v M induces a family ∇ F,u of covariantderivatives on T F , and Proposition 2.1.3 shows that each ∇ F,u is the Levi-Civita connection on F with respect to the metric g F,u , compare with Remark 2.1.4. Similarly, we pull the horizontaldistribution T h M back to U × F and use this to identifyΩ • v (cid:0) π − ( U ) (cid:1) ∼ = C ∞ (cid:0) U, Ω • ( F ) (cid:1) , and Ω p,q (cid:0) π − ( U ) (cid:1) ∼ = Ω p (cid:0) U, Ω q ( F ) (cid:1) . (3.1)Note, however, that Φ ∗ T h M will in general not coincide with T U × F , unless the curvature Ω ofthe fiber bundle is trivial. Remark. We want to give a note about the definition of Ω p (cid:0) U, Ω q ( F ) (cid:1) . A naive way—which issufficient for our purposes—is to define elements of C ∞ (cid:0) U, Ω q ( F ) (cid:1) to be locally of the form X | I | = q f I ( y, x ) dx I , y ∈ U, with local coordinates x i for F , multi indices I , and smooth functions f I ( y, x ) satisfying theappropriate transformation laws with respect to changes of the coordinate chart. Similarly onetreats elements of Ω p (cid:0) U, Ω q ( F ) (cid:1) . From a more invariant perspective, one could endow Ω q ( F )with its natural Fr´ech´et topology and consider smooth maps with respect to this.Under the identification of (3.1), a vertical differential operator D on Ω • ( M ) can be writtenover U × F as D = X j K j ( u ) ⊗ D j ( u ) , where each K j ( u ) is a bundle endomorphism of Λ • T ∗ U and each D j ( u ) is a smooth b -parameterfamily of differential operators on Ω • ( F ) in the sense of Definition 1.3.9. Vertical de Rham Operators. We use the above digression to give a description of the verticalde Rham operator. For this we first need to incorporate a bundle E over M , endowed with a flatconnection A . Let π F : U × F → F be the projection onto the second factor, and denote by E | F be the restriction of E to the fiber F . Lemma 3.1.3. There exists a natural lift of the bundle isomorphism Φ : U × F → π − ( U ) to an isomorphism of flat Hermitian vector bundles Φ E : π ∗ F ( E | F ) → E | π − ( U ) . 3. Rho Invariants of Fiber Bundles, Abstract Theory Proof. Let u ∈ U , and let F u be the fiber over u . Then Φ( u, . ) maps F diffeomorphically to F u .Using parallel transport with respect to A along the flow lines of Φ t ( u, . ) we can lift this to anisomorphism Φ E ( u, . ) : E | F → E | F u . Now, since A is a flat Hermitian connection, the bundles E | F and E | F u are naturally endowed withflat Hermitian connections induced by A . Since we are using parallel transport with respect to A , a locally constant, unitary frame for E | F will be mapped by Φ E to a locally constant, unitaryframe for E | F u . This implies that Φ E is, in fact, an isomorphism of flat Hermitian bundles.In a similar way as in (3.1), we can use Lemma 3.1.3 to identifyΩ p,q (cid:0) π − ( U ) , E | π − ( U ) (cid:1) ∼ = Ω p (cid:0) U, Ω q ( F, E | F ) (cid:1) , (3.2)where E | F is endowed with a fixed flat Hermitian connection A F . We then let D A F ,u be thefamily of de Rham operators on Ω • ( F, E | F ) associated to the metric g F,u and the flat connection A F . Then under the identification (3.2) we can write D A,v = ( − p ⊗ D A F ,u on Ω p (cid:0) U, Ω • ( F, E | F ) (cid:1) . (3.3) Proposition 3.1.4. The C ∞ ( B ) -module ker( D A,v ) ∩ Ω • v ( M, E ) is isomorphic to the space ofsmooth sections of a vector bundle, which we denote by H • A,v ( M ) → B . Moreover, ker( D A,v ) ∼ = Ω • (cid:0) B, H • A,v ( M ) (cid:1) . Sketch of proof. It suffices to show that the assertion is true locally, i.e., that for sufficiently smallopen subsets U ⊂ M ker( D A,v ) ∩ Ω • (cid:0) π − ( U ) , E (cid:1) is isomorphic to the space of differential forms over U with values in a vector bundle. For thislet U ⊂ B be as in Lemma 3.1.2 such that π − ( U ) ∼ = U × F , and write D A,v as in (3.3). Since D A,v acts as ± Id on Ω • ( U ), it suffices to consider D A F ,u acting onΩ • v (cid:0) π − ( U ) , E (cid:1) ∼ = C ∞ (cid:0) U, Ω • ( F, E | F ) (cid:1) . For fixed u , the Hodge-de-Rham theorem for D A F ,u implies that ker( D A F ,u ) is isomorphic to H • ( F, E A | F ), where E A | F is short for E | F endowed with the flat connection A F . Since we knowfrom Lemma 3.1.3 that A F does not vary with u , we infer that dim ker( D A F ,u ) is constant for u ∈ U . Hence, we are precisely in the situation of Proposition D.1.8—respectively Remark D.1.9.Therefore, the family of projections P u : Ω • ( F, E | F ) → ker (cid:0) D A F ,u (cid:1) , u ∈ U, is a smooth family of finite rank smoothing operators. Using this, it is straightforward to checkthat the collection H • A,v ( U ) := [ u ∈ U ker (cid:0) D A F ,u (cid:1) → U forms a smooth vector bundle over U , see [13, Lem. 9.9] for a detailed proof. Then the assertionof Proposition 3.1.4 easily follows. .1. The Bundle of Vertical Cohomology Groups We now want to use Proposition 3.1.4 to prove the following fibered version of the de Hodgedecomposition theorem. Theorem 3.1.5. Let E → M be a Hermitian vector bundle over the total space of an orientedfiber bundle of closed manifolds F ֒ → M π −→ B . Assume that E admits a flat connection A . Withrespect to every submersion metric, there is an L -orthogonal splitting of smooth forms Ω • ( M, E ) = ker( D A,v ) ⊕ im( D A,v )= (cid:0) ker d A,v ∩ ker d tA,v (cid:1) ⊕ im( d A,v ) ⊕ im( d tA,v ) . Moreover, the splitting is independent of the chosen metric g B on B .Proof. We start with a local consideration. With the same notation as in the proof of Proposition3.1.4 we consider the family of de Rham operators D A F ,u on Ω • ( F, E | F ). We know from the proofof Proposition 3.1.4 that dim ker( D A F ,u ) is constant for u ∈ U , so that the family of projections P u onto the kernels depends smoothly on u . According to Proposition D.1.8 (ii) the same is truefor the family of Green’s operators, G u : Ω • ( F, E | F ) → Ω • ( F, E | F ) . Let ω u ∈ C ∞ (cid:0) U, Ω • ( F, E | F ) (cid:1) . For fixed u we can decompose ω u = P u ω u + D u ◦ G u ◦ (Id − P u ) ω u , and both summands depend smoothly on u as P u and G u do so. Writing D A,v | π − ( U ) for therestriction of D A,v to Ω • (cid:0) π − ( U ) , E | π − ( U ) (cid:1) , one readily concludes thatΩ • (cid:0) π − ( U ) , E | π − ( U ) (cid:1) = ker (cid:0) D A,v | π − ( U ) (cid:1) ⊕ im (cid:0) D A,v | π − ( U ) (cid:1) . (3.4)Now let { ϕ i } be a partition of unity on B , subordinate to a finite covering B = S i U i ,such that (3.4) holds for every Ω • (cid:0) π − ( U i ) , E | π − ( U i ) (cid:1) . For ω ∈ Ω • ( M, E ) and every i we candecompose( π ∗ ϕ i ) ω = α i + D A,v β i , α i ∈ ker (cid:0) D A,v | π − ( U i ) (cid:1) , β i ∈ Ω • (cid:0) π − ( U i ) , E | π − ( U i ) (cid:1) . Then, since D A,v is C ∞ ( B ) linear, ω = X i ( π ∗ ϕ i ) α i + X i ( π ∗ ϕ i ) D A,v β i = X i ( π ∗ ϕ i ) α i + D A,v (cid:16) X i ( π ∗ ϕ i ) β i (cid:17) , which is a decomposition of ω in terms of ker( D A,v ) ⊕ im( D A,v ). Clearly, the decomposition is L -orthogonal, since D A,v is formally self-adjoint. The equalitiesker( D A,v ) = (cid:0) ker d A,v ∩ ker d tA,v (cid:1) and im( D A,v ) = im( d A,v ) ⊕ im( d tA,v )follow as in the unparametrized case. Finally, the assertion that the vertical Hodge decompositionis independent of the metric g B on B is immediate from the fact that D A,v is independent of g B .0 3. Rho Invariants of Fiber Bundles, Abstract Theory Before we continue the discussion of the bundle H • A,v ( M ) → B , we want to give an interestingapplication of the fibered Hodge decomposition theorem. The corresponding result for foliationsis [36, Thm. 4.18]. However, in the case of fiber bundles, the proof can be simplified considerably,and the author of this thesis is not aware of a corresponding treatment in the literature. Thissubsection is not essential for the line of thoughts in the later sections. Yet, it might be helpfulin more complicated examples. Theorem 3.1.6. Let g v be a metric on T v M of unit volume. Then there exists a verticalprojection P v : T M → T v M such that the associated mean curvature form k v ( g v , P v ) vanishes. Since we have seen in Lemma 2.1.14 that we can deform a vertical metric conformally to ametric of unit volume, Theorem 3.1.6 implies Corollary 3.1.7. Every oriented fiber bundle of closed manifolds admits a connection and avertical metric such that the mean curvature form vanishes. Before we give the proof of Theorem 3.1.6, we extract the part where we will use the verticalHodge decomposition of Theorem 3.1.5. Recall that we have introduced the basic projection Π B in Definition 2.1.13. Proposition 3.1.8. There is an L -orthogonal splitting of smooth horizontal forms, Ω • h ( M ) = π ∗ Ω • ( B ) ⊕ d tv (cid:0) Ω • , ( M ) (cid:1) . Moreover, the kernel of the basic projection is given by ker Π B = d tv (cid:0) Ω • , ( M ) (cid:1) . Proof. First of all, as the 0th cohomology group of the fiber consists only of constant functions,one deduces from Proposition 3.1.4 thatker D v ∩ Ω • h ( M ) = π ∗ Ω • ( B ) . Since Ω • h ( M ) ⊥ im( d v ), the vertical Hodge decomposition in Theorem 3.1.5 yieldsΩ • h ( M ) = π ∗ Ω • ( B ) ⊕ d tv (cid:0) Ω • , ( M ) (cid:1) . For the second assertion we note that for all α ∈ Ω • ( B ) and ω ∈ Ω • h ( M ) (cid:10) α , Π B ( ω ) (cid:11) = v − F Z M/B h π ∗ α, ω i vol F , where v F is the function which associates to a point y ∈ B the volume of the fiber over y ,see Definition 2.1.13. This implies that ker Π B ⊥ π ∗ Ω • ( B ). On the other hand, as in theunparametrized case, one finds that for ω ∈ Ω • , ( M ),Π B ( d tv ω ) = v − F Z M/B d tv ( ω ) ∧ vol F = 0 . Therefore, d tv (cid:0) Ω • , ( M ) (cid:1) ⊂ ker Π B , which finishes the proof. .1. The Bundle of Vertical Cohomology Groups Proof of Theorem 3.1.6. Let g v be a vertical metric such that v F ( g v ) = 1. We choose an arbitraryvertical projection P v : T M → T v M , and let g be a submersion metric on M satisfying T v M ⊥ = ker P v and g (cid:12)(cid:12) T v M × T v M = g v . According to Corollary 2.1.18, the assumption that v F ( g v ) = 1 implies that the basic projection ofthe mean curvature k v vanishes. Form Proposition 3.1.8 we deduce that there exists η ∈ Ω , ( M )such that d tv η = k v ∈ Ω , ( M ) . Define h ∈ C ∞ ( M, T ∗ M ⊗ T ∗ M ) by h ( X, Y ) := η ( P h X, P v Y ) + η ( P h Y, P v X ) , X, Y ∈ C ∞ ( M, T M ) . Here, P h = Id − P v is the horizontal projection. Then h is a symmetric 2-tensor, and h ( X, X ) = 0for all X ∈ C ∞ ( M, T M ). Thus, we can define a new metric on T M by letting e g := g + h. Note that the restriction of e g to T v M still coincides with g v . Let e P v/h denote the verticalrespectively horizontal projection associated to e g . Then, if { e i } is any local orthonormal framefor T v M with respect to g v , we have e P v ( X ) = P v ( X ) + X i η ( X, e i ) e i , e P h ( X ) = P h ( X ) − X i η ( X, e i ) e i . Let e k v denote the mean curvature form associated to g v and e P v , and let { f a } be a local orthonor-mal frame for T h M with respect to the original metric g . Then, according to formula (2.2) forthe mean curvature, e k v ( f a ) = − X i g v (cid:0) [ e P h f a , e i ] , e i (cid:1) = k v ( f a ) + X ij g v (cid:0) [ η ( f a , e j ) e j , e i ] , e i (cid:1) . Now, using standard arguments involving the Lie bracket and the fact that ∇ v is metric andtorsion-free as a connection on T v M , one gets g v (cid:0) [ η ( f a , e j ) e j , e i ] , e i (cid:1) = − e i (cid:2) η ( f a , e j ) (cid:3) g v ( e j , e i ) + η ( f a , e j ) g v (cid:0) [ e j , e i ] , e i (cid:1) = − e i (cid:2) η ( f a , e i ) (cid:3) + η ( f a , e j ) g v ( e j , ∇ ve i e i ) . On the other hand, according to Proposition 2.1.21, d tv η ( f a ) = X i ( ∇ ⊕ e i η )( f a , e i ) = X i (cid:16) e i (cid:2) η ( f a , e i ) (cid:3) − η ( ∇ ⊕ e i f a , e i ) − η ( f a , ∇ ⊕ e i e i ) (cid:17) . Since ∇ ⊕ e i f a = 0 and ∇ ⊕ e i e i = ∇ ve i e i we conclude that X ij g v (cid:0) [ η ( f a , e j ) e j , e i ] , e i (cid:1) = − d tv η ( f a ) . Employing the definition of η we have thus achieved that e k v = k v − d tv η = 0 . This shows that the mean curvature associated to e P v and g v vanishes.2 3. Rho Invariants of Fiber Bundles, Abstract Theory Remark. The statement of Theorem 3.1.6 for foliations is not true without changes. The un-derlying reason is that Corollary 2.1.18 does not generalize, i.e., the basic projection of the meancurvature does not necessarily give a trivial cohomology class. Note that the definition of coho-mology requires extra work for the possibly singular leaf space of a foliation. But even if oneuses the basic cohomology as the correct substitute, Corollary 2.1.18 does not carry over, and onefinds topological obstructions to the vanishing of the mean curvature form. In the language offoliation theory, Corollary 3.1.7 asserts that the vertical distribution of a fiber bundles is a taut foliation. For a detailed discussion of the aspects mentioned here, in particular the differencebetween tense and taut foliations, we refer to [36] and references given therein. Let E → M be a Hermitian vector bundle over the total space of the fiber bundle F ֒ → M π −→ B ,and assume that E admits a flat connection A . As we have seen in Corollary 2.1.16 there is avertical differential d A,v : Ω • v ( M, E ) → Ω • +1 v ( M, E ) , d A,v = 0 . Hence, we can form the quotient ker d A,v / im d A,v . If M is endowed with a vertical metric, it isan immediate consequence of the fibered Hodge decomposition theorem thatker d A,v / im d A,v ∼ = ker( D A,v ) ∩ Ω • v ( M, E ) , (3.5)which is the space of sections of the bundle H • A,v ( M ) → B of Proposition 3.1.4. This is notsurprising, since if F ⊂ M is a fiber of π : M → B we can restrict d A,v as in Section 3.1.1 to anoperator d A,v | F : Ω • ( F, E | F ) → Ω • +1 ( F, E | F ) . Then ker( d A,v | F ) / im( d A,v | F ) is just the de Rham cohomology of F with values in the flat bundle E | F , so that ker d A,v / im d A,v is roughly the union of the cohomology groups of all fibers. Theseobservations and Proposition 3.1.4 imply Proposition 3.1.9. The space ker d A,v / im d A,v is isomorphic to the space of sections of a finiterank vector bundle H • A,v ( M ) → B , which we call the “bundle of vertical cohomology groups” . Itsfiber over a point y ∈ B is isomorphic to the de Rham cohomology of π − ( y ) with values in theflat bundle E | π − ( y ) . Remark. (i) Although the bundles H • A,v ( M ) and H • A,v ( M ) are isomorphic, we will usually not identifythem, since the latter is only defined when we have chosen a vertical metric. This is whywe have introduced the term “bundle of vertical cohomology groups” only here rather thanalready in Proposition 3.1.4.(ii) As for the bundle H • A,v ( M ) we consider also differential forms on B with values in thebundle of vertical cohomology groups. Then it is immediate thatΩ p (cid:0) B, H qA,v ( M ) (cid:1) ∼ = ker (cid:0) d A,v : Ω p,q ( M, E ) → Ω p,q +1 ( M, E ) (cid:1) im (cid:0) d A,v : Ω p,q − ( M, E ) → Ω p,q ( M, E ) (cid:1) . (3.6) .1. The Bundle of Vertical Cohomology Groups The Natural Flat Connection. Recall from Corollary 2.1.16, that on Ω • ( M, E ), d A,h + (cid:8) d A,v , i(Ω) (cid:9) = 0 , and (cid:8) d A,v , d A,h (cid:9) = (cid:8) d A,h , i(Ω) (cid:9) = 0 , where Ω is the curvature of the fiber bundle. Now the fact that d A,h anti-commutes with d A,v implies that d A,h descends to a well-defined map¯ d A,h : Ω p (cid:0) B, H • A,v ( M ) (cid:1) → Ω p +1 (cid:0) B, H • A,v ( M ) (cid:1) . Moreover, if ω ∈ Ω p,q ( M, E ) satisfies d A,v ω = 0, then it follows from the relation d A,h ω = − d A,v ◦ i(Ω) ω that d A,h ω is a d A,v -exact element of ker (cid:0) d A,v : Ω p +2 ,q ( M, E ) → Ω p +2 ,q +1 ( M, E ) (cid:1) . This impliesthat ( ¯ d A,h ) = 0. In other words, we have found a natural flat connection on the bundle ofvertical cohomology groups which is induced by d A,h . Definition 3.1.10. We denote by ∇ H A,v : C ∞ (cid:0) B, H • A,v ( M ) (cid:1) → Ω (cid:0) B, H • A,v ( M ) (cid:1) the flat connection defined by ¯ d A,h . More precisely, for all X ∈ C ∞ ( B, T B ) and ω ∈ Ω • v ( M, E )with d A,v ω = 0 we define ∇ H A,v X [ ω ] := (cid:2) i( X h ) d A,h ω (cid:3) ∈ C ∞ (cid:0) B, H • A,v ( M ) (cid:1) . Relation to the Leray-Serre Spectral Sequence. We want to point out that we have justconstructed the term ( E • , • , d ) of the spectral sequence associated to the complex (cid:0) Ω • ( M, E ) , d A (cid:1) .To explain this—and also for later use—we make a short digression on the Leray-Serre spectralsequence.Recall, e.g. from [67, Sec 2.2], that a complex endowed with a decreasing filtration gives riseto a spectral sequence. The appropriate filtration in the case at hand is the Serre filtration givenby F k Ω • := X p ≥ k Ω p, • ( M, E ) . (3.7)Then, if b = dim B and d = d A , we have { } = F b +1 Ω • ⊂ F b Ω • ⊂ . . . ⊂ F Ω • = Ω • , d (cid:0) F k Ω • (cid:1) ⊂ F k Ω • . Note that the latter follows from Proposition 2.1.15, since each of the terms appearing in d A , d A = d A,v + d A,h + i(Ω) , preserves F k Ω • . In the same way one verifies that the de Rham cohomology H • ( M, E A ) inheritsa filtration defined by F k H n := im (cid:0) H n ( F k Ω • , d ) → H n ( M, E A ) (cid:1) . (3.8)4 3. Rho Invariants of Fiber Bundles, Abstract Theory One now constructs a spectral sequence as follows, see the proof of [67, Thm. 2.6]. For r ∈ N define Z p,qr := F p Ω p + q ∩ d − (cid:0) F p + r Ω p + q +1 (cid:1) ,B p,qr := F p Ω p + q ∩ d (cid:0) F p − r Ω p + q − (cid:1) ,E p,qr := Z p,qr Z p +1 ,q − r − + B p,qr − , E p,q := F p Ω p + q F p +1 Ω p + q ∼ = Ω p,q ( M, E ) . (3.9)Then the differential d naturally defines on each bigraded module E • , • r a differential d r of bidegree( r, − r ) in such a way that E p,qr +1 ∼ = ker (cid:0) d r : E p,qr → E p + r,q +1 − rr (cid:1) im (cid:0) d r : E p − r,q + r − r → E p + r,q +1 − rr (cid:1) . The general theory of spectral sequences now implies the following, see again [67, Thm. 2.6]. Theorem 3.1.11. The spectral sequence (cid:0) E • , • r , d r (cid:1) collapses for r = b + 1 and converges to H • ( M, E A ) . More precisely, for all r ≥ b + 1 E p,qr ∼ = F p H p + q F p +1 H p + q , where F p H • is the filtration of H • ( M, E ) given by (3.8) . Now we can interpret the bundle of vertical cohomology groups in terms of the Leray-Serrespectral sequence. Certailnly, the term E • , • in (3.9) coincides with Ω • , • ( M, E ). Moreover, oneeasily verifies that the natural construction of the differential in the proof of [67, Thm. 2.6]coincides with d A,v . Thus, (cid:0) E • , • , d (cid:1) = (cid:0) Ω • , • ( M, E ) , d A,v (cid:1) . In the discussion following (3.6) we have constructed a natural differential ¯ d A,h on the cohomologyof (cid:0) E • , • , d (cid:1) , and again, one can easily check that it coincides with the differential on E • , • abstractly constructed from (3.9). Without giving more details we summarize that Lemma 3.1.12. The Leray-Serre spectral sequence satisfies E p,q ∼ = (cid:0) Ω p (cid:0) B, H qA,v ( M ) (cid:1) , ¯ d A,h (cid:1) and E p,q ∼ = H p (cid:0) Ω • (cid:0) B, H qA,v ( M ) (cid:1) . We can also use the vertical Hodge decomposition of Theorem 3.1.5 to give a Hodge theoreticdescription of the flat connection ∇ H A,v . We fix a vertical metric g v and use this to identify H • A,v ( M ) with H • A,v ( M ) using the vertical Hodge-de-Rham isomorphism in (3.5). From Section2.1.3 we know that Ω • v ( M, E ) is endowed with the natural connection e ∇ A,v , induced by thevertical Lie derivative and the connection A . Then we have the following, see also [19, Prop3.14]. Proposition 3.1.13. Under the vertical Hodge-de-Rham isomorphism the flat connection ∇ H A,v coincides with the connection defined by ∇ H A,v X := P ker( D A,v ) ◦ e ∇ A,vX h , X ∈ C ∞ ( B, T B ) . .1. The Bundle of Vertical Cohomology Groups Proof. For convenience we drop the reference to the flat connection A . Denote byΨ : ker d v ∩ Ω • v ( M ) → C ∞ (cid:0) B, H • v ( M ) (cid:1) the quotient map. Then, according to Definition 3.1.10, ∇ H v X (cid:0) Ψ( ω ) (cid:1) = Ψ ◦ i( X h ) ◦ d h ( ω ) , ω ∈ ker d v ∩ Ω • v ( M ) , X ∈ C ∞ ( B, T B ) . Using Proposition 3.1.9 and Theorem 3.1.5, we can explicitly describe the isomorphism H • v ( M ) ∼ = H • v ( M ) in terms of sections by the composition C ∞ (cid:0) B, H • v ( M ) (cid:1) = ker D v ∩ Ω • v ( M ) ֒ → ker d v ∩ Ω • v ( M ) Ψ −→ C ∞ (cid:0) B, H • v ( M ) (cid:1) . This implies that ∇ H v X Ψ( ω ) ∈ C ∞ (cid:0) B, H • v ( M ) (cid:1) corresponds to P ker D v ◦ i( X h ) ◦ d h (cid:0) P ker D v ω (cid:1) ∈ C ∞ (cid:0) B, H • v ( M ) (cid:1) . Finally, Proposition 2.1.15 shows that on Ω • v ( M )i( X h ) ◦ d h = e ∇ vX h , from which we obtain the claimed formula. Metrics on the Bundle of Vertical Cohomology Groups. The C ∞ ( B )-module Ω • v ( M, E )is endowed with the pairing( ω, η ) M/B := Z M/B h ω, η i vol F ( g v ) ∈ C ∞ ( B ) , ω, η ∈ Ω • v ( M, E ) , (3.10)where the scalar product in the integrand is induced by g v together with the Hermitian metricon E . Definition 3.1.14. Let g v be a vertical metric. We define h ω, η i H A,v := ( ω, η ) M/B , ω, η ∈ C ∞ (cid:0) B, H • A,v ( M ) (cid:1) , and, if τ v is the vertical chirality operator, Q A,v ( ω, η ) := h ω, τ v η i H A,v . We also use h ., . i H A,v and Q A,v to the corresponding objects induced by the vertical Hodge-de-Rham isomorphism on H • A,v ( M ).Clearly, h ., . i H A,v is a Hermitian metric on the vector bundle H • A,v ( M ) → B , which throughthe vertical Hodge-de-Rham isomorphism depends on the vertical metric g v . In contrast, Q A,v isindependent of g v since it is related to the vertical intersection form via Q A,v (cid:0) [ α ] , [ β ] (cid:1) = i k Z M/B h α ∧ β i , [ α ] , [ β ] ∈ H • A,v ( M ) , (3.11)where k depends only on the degrees of α and β . Furthermore, one easily checks that Q A,v is anindefinite Hermitian form with signatureSign( Q A,v ) = rk (cid:0) H + A,v ( M ) (cid:1) − rk (cid:0) H − A,v ( M ) (cid:1) . Here, H ± A,v ( M ) denotes the ± τ v . This implies that Q A,v has signature 0 unlessthe dimension of the fiber is divisible by 4 in which case Sign( Q A,v ) = Sign( F ).6 3. Rho Invariants of Fiber Bundles, Abstract Theory Proposition 3.1.15. The flat connection ∇ H A,v is compatible with the indefinite Hermitianmetric Q A,v . It is compatible with the Hermitian metric h ., . i H A,v if and only if for all X ∈ C ∞ ( M, T h M ) 2 P ker D v ◦ B ( X ) ◦ P ker D v + k v ( X ) = 0 , (3.12) where B ( X ) is the tensor as in (2.10) , and k v is the mean curvature form.Proof. For the first part we use the description (3.11) for Q A,v . Let α, β ∈ Ω • v ( M, E ) be chosenin such a way that h α ∧ β i is of maximal vertical degree. Then d B Z M/B h α ∧ β i = Z M/B d h h α ∧ β i = Z M/B h d A,h α ∧ β i + ( − | α | h α ∧ d A,h β i . Hence, if d A,v α = d A,v β = 0 we have d B Q A,v (cid:0) [ α ] , [ β ] (cid:1) = Q A,v (cid:0) [ d A,h α ] , [ β ] (cid:1) + ( − | α | Q A,v (cid:0) [ α ] , [ d A,h β ] (cid:1) , so that, according to Definition 3.1.10, XQ A,v (cid:0) [ α ] , [ β ] (cid:1) = Q A,v (cid:0) ∇ H A,v X [ α ] , [ β ] (cid:1) + Q A,v (cid:0) [ α ] , ∇ H A,v X [ β ] (cid:1) , X ∈ C ∞ ( B, T B ) . This shows that ∇ H A,v is indeed compatible with Q A,v . Now let g v be a vertical metric, and let ω, η ∈ C ∞ (cid:0) B, H • A,v ( M ) (cid:1) . Then d B Z M/B h ω, η i vol F ( g v ) = Z M/B d h (cid:0) h ω, η i (cid:1) ∧ vol F ( g v ) + Z M/B h ω, η i d h vol F ( g v ) . It follows from Proposition 2.1.17 that d h vol F ( g v ) = k v ∧ vol F ( g v ) . Since the connection ∇ A,v is compatible with the metric on Ω • v ( M, E ) we know that for all X ∈ C ∞ ( B, T B ) X h h ω, η i = (cid:10) ∇ A,vX h ω , η (cid:11) + (cid:10) ω , ∇ A,vX h η (cid:11) . From (2.9) and (2.10) we then deduce X h h ω, η i = (cid:10) e ∇ A,vX h ω , η (cid:11) + (cid:10) ω , e ∇ A,vX h η (cid:11) + (cid:10) B ( X h ) ω , η (cid:11) + (cid:10) ω , B ( X h ) η (cid:11) . Now, B ( X h ) is easily seen to be self-adjoint with respect to the metric h ., . i on Ω • v ( M, E ). Puttingall pieces together, we find that for the connection ∇ H A,v X of Proposition 3.1.13 X h ω, η i H A,v = (cid:10) ∇ H A,v X ω , η (cid:11) H A,v + (cid:10) ω , ∇ H A,v X η (cid:11) H A,v + (cid:10) ω , B ( X h ) η (cid:11) H A,v + (cid:10) ω , k v ( X h ) η (cid:11) H A,v , which proves that ∇ H A,v is compatible with h ., . i H A,v if and only if (3.12) holds. Since the metricsas well as the connections ∇ H A,v and ∇ H A,v on H • A,v ( M ) and H • A,v ( M ) coincide under the verticalHodge-de-Rham isomorphism, the proof of Proposition 3.1.15 is finished. .1. The Bundle of Vertical Cohomology Groups Remark. If we denote by p and q the maximal ranks of subbundles of H • A,v ( M ) on which Q A,v is positive respectively negative definite, we can rephrase the first part of Proposition 3.1.15by saying that ∇ H A,v is a flat U( p, q )-connection. The choice of a vertical metric reduces thestructure group of H • A,v ( M ) to the subgroup U( p ) × U( q ). However, the connection does notnecessarily reduce to a flat U( p ) × U( q )-connection, the geometric obstruction being (3.12). Aswe have seen in Theorem 3.1.6 we can always arrange that the mean curvature form vanishes.For arbitrary fiber bundles, the tensor B ( X ) is, however, a non-trivial obstruction. It would beinteresting to find a topological condition which guarantees that there exists a vertical metricsuch that (3.12) holds. Definition 3.1.16. Let D A,v and D A,h be the vertical and horizontal de Rham operators as inDefinition 2.1.23. If dim M is odd, we define the odd signature operator on B with values in thebundle of vertical cohomology groups , D B ⊗ ∇ H A,v : Ω • (cid:0) B, H • A,v ( M ) (cid:1) → Ω • (cid:0) B, H • A,v ( M ) (cid:1) , by D B ⊗ ∇ H A,v := P ker D A,v ◦ τ M D A,h ◦ P ker D A,v . Here, τ M is the chirality operator associated to a fixed submersion metric on M . Remark 3.1.17. (i) Certainly, D B ⊗ ∇ H A,v is a formally self-adjoint elliptic differential operator and thus has awell-defined Eta invariant. This will play an important role in Dai’s general adiabatic limitformula for the Eta in Section 3.3.(ii) We note that if dim B is odd, and (3.12) is satisfied, then D B ⊗ ∇ H A,v is actually isometricto two copies of the odd signature operator on B twisted by ∇ H A,v . This is because wehave not restricted to forms on the base of even degree, compare with Remark 1.4.4 (i). p, q )-Connections Before we continue with the general discussion, we briefly want to digress on the Eta invariant ofthe operator D B ⊗ ∇ H A,v introduced above. Without any effort, we can treat the more generalcase that E → B is a complex vector bundle, endowed with an indefinite Hermitian metric Q anda connection ∇ , not necessarily flat, but compatible with Q . We choose a splitting E = E + ⊕ E − into subbundles where Q is positive respectively negative definite, and define τ E to be ± id E on E ± . Then we can define a Hermitian metric on E via h ( e, f ) := Q ( e, τ f ) , e, f ∈ E, compare with Definition 3.1.14. Note that the splitting E = E + ⊕ E − is orthogonal with respectto h . We now define an End( E )-valued 1-form on Bω ∇ ,τ E ( X ) := τ E (cid:2) ∇ X , τ E (cid:3) = τ E ◦ ∇ X ◦ τ E − ∇ X , X ∈ C ∞ ( B, T B ) . Recall that U( p, q ) denotes the isometry group of the quadratic from p X j =1 | z j | − p + q X j = p +1 | z j | . 3. Rho Invariants of Fiber Bundles, Abstract Theory Then we have the following simple result. Lemma 3.1.18. For all X ∈ C ∞ ( B, T B ) , the endomorphism ω ∇ ,τ E ( X ) is self-adjoint withrespect to h . It interchanges the subbundles E + and E − . Moreover, the connection ∇ u := ∇ + ω ∇ ,τ E , is unitary with respect to h .Proof. Let e, f ∈ C ∞ ( B, E ). Since ∇ X is compatible with Q , one verifies—using in particularthat τ E = id E and that Q ◦ τ E = Q , h (cid:0) ω ∇ ,τ E ( X ) e, f (cid:1) = Q (cid:0) τ E (cid:2) ∇ X , τ E (cid:3) e, τ E f (cid:1) = Q (cid:0) ∇ X ( τ E e ) , f (cid:1) − Q (cid:0) ∇ X e, τ E f (cid:1) = − Q (cid:0) τ E e, ∇ X f (cid:1) + Q (cid:0) e, ∇ X ( τ E f ) (cid:1) = Q (cid:0) e, (cid:2) ∇ X , τ E (cid:3) f (cid:1) = h (cid:0) e, ω ∇ ,τ E ( X ) f (cid:1) . Hence, ω ∇ ,τ E ( X ) is self-adjoint with respect to h . Now, let P E ± := (id E ± τ E ) denote theprojection onto E ± . Then one easily obtains that P E + ◦ ∇ X ◦ P E − + P E − ◦ ∇ X ◦ P E + = − ω ∇ ,τ E ( X ) . On the one hand, this implies that ω ∇ ,τ E ( X ) interchanges the subbundles E + and E − . Bydefinition of ∇ u , we can deduce on the other hand, that ∇ u preserves E + and E − from which iteasily follows that ∇ u is unitary with respect to h . Remark. In the case that E = H • A,v ( M ) is the bundle of vertical cohomology groups, Q = Q v is the vertical intersection form and ∇ = ∇ H A,v is the natural flat connection, the 1-form ω ∇ ,τ E is precisely the 1-form appearing in (3.12), compare also with (2.15). This gives a more abstractexplanation of Proposition 3.1.15. The Odd Signature Operator with values in E . To define the analog of D B ⊗ ∇ H A,v inthe case at hand, we choose a metric g B on B , and let τ B the associated chirality operator onΩ • ( B, E ). Let b := dim B , and extend τ E to Ω • ( B, E ) by requiring that τ E ( α ⊗ e ) = ( − p ( b +1) α ⊗ τ E e, α ∈ Ω p ( B ) , e ∈ C ∞ ( B, E ) . (3.13)Checking signs one finds that τ B τ E = τ E τ B . We then define τ := τ B τ E : Ω • ( B, E ) → Ω b −• ( B, E ) , which takes the place of the total chirality operator τ M , compare with Lemma 2.2.3. Note thatmore explicitly, if α ⊗ e ∈ Ω p ( B, E ), then τ ( α ⊗ e ) = τ B (cid:0) ( − p ( b +1) α ⊗ τ E e (cid:1) = ( − p ( b +1) ( τ B α ) ⊗ τ E e. Then the analog of D B ⊗ ∇ H A,v is given by D B ⊗ ∇ := τ d ∇ + d ∇ τ : Ω • ( B, E ) → Ω • ( B, E ) , (3.14)where d ∇ is the exterior differential on B twisted by the connection ∇ on E . We also define D B ⊗ ∇ u := τ d ∇ u + d ∇ u τ, and denote by ∇ u, ± the restriction of ∇ u to E ± . .1. The Bundle of Vertical Cohomology Groups Lemma 3.1.19. With respect to the splitting E = E + ⊕ E − , the operator D B ⊗ ∇ u is of theform D B ⊗ ∇ u = D ∇ u, + B − D ∇ u, − B ! , where D ∇ u, ± B := τ B ( − b +1 d ∇ u, ± + d ∇ u, ± τ B , b := dim B. Moreover, if we define V := D B ⊗ ∇ − D B ⊗ ∇ u , then V is a self-adjoint operator on Ω • ( B, E ) of order 0 which interchanges Ω • ( B, E + ) and Ω • ( B, E − ) . In particular, D B ⊗ ∇ and D B ⊗ ∇ u are formally self-adjoint.Proof. It follows from Lemma 3.1.18 that τ E commutes with ∇ u . Using the sign convention in(3.13) it is immediate that τ E d ∇ u = ( − b +1 d ∇ u τ E . Hence, τ d ∇ u + d ∇ u τ = (cid:0) ( − b +1 τ B d ∇ u + d ∇ u τ B (cid:1) τ E , which proves the first assertion. The other assertions are a simple consequence of the correspond-ing properties of ω ∇ ,τ E in Lemma 3.1.18, since by definition d ∇ u = d ∇ + e (cid:0) ω ∇ ,τ E (cid:1) , where e( . ) is exterior multiplication. Difference of Eta Invariants. Roughly, Lemma 3.1.19 asserts that D ⊗ ∇ is the direct sumof two geometric Dirac operator plus a lower order perturbation which interchanges the twistingbundles. This leads to a simple relation between the Eta invariants of D ⊗ ∇ and D ⊗ ∇ u . Thefollowing result is a reformulation of [16, Thm.’s 2.7 & 2.35]. We formulate it in terms of the ξ -invariant, see Definition 1.3.4. Theorem 3.1.20. As before, let V = D B ⊗ ∇ − D B ⊗ ∇ u . Then ξ ( D B ⊗ ∇ ) − ξ ( D B ⊗ ∇ u ) = SF (cid:0) D B ⊗ ∇ u + xV (cid:1) x ∈ [0 , . Remark. By comparison with the variation formula of Corollary D.2.6, we see that Theorem3.1.20 asserts that the contribution coming from the variation of the reduced ξ -invariant vanishes,i.e., Z ddx [ ξ ( D x )] dx = 0 . In fact, this is precisely what Bismut and Cheeger prove, see [16, Lem. 2.11]. Recall fromProposition D.2.5 that ddt [ ξ ( D x )] dx = − √ π a n ( V, D x ) , where a n ( V, D x ) is the constant term in the asymptotic expansion of √ t Tr (cid:0) V e − tD x (cid:1) , as t → . 3. Rho Invariants of Fiber Bundles, Abstract Theory In the case that dim B is even, Theorem 1.2.7 shows that there are no half integer powers of t in the asymptotic expansion of Tr (cid:0) V e − tD x (cid:1) , so that Theorem 3.1.20 is a consequence of thegeneral theory for elliptic operators. However, the odd dimensional case requires considerablymore work. In [16, Lem. 2.11], Bismut and Cheeger prove the corresponding result for operatorsof the form we are considering here. Their proof uses Getzler’s local index theory techniques fortwisted Dirac operators, adapted to odd dimensional base spaces, in a similar way as we havedescribed in Section 1.5.2. To discuss Dai’s adiabatic limit formula, we need to recall some aspects of Bismut’s local indextheory for families. We will be rather sketchy and refer to the original article [14] and thetreatment in [13, Ch.’s 9 & 10] for more details. The survey article [15] is also recommended.For convenience and since we will not need a greater generality, we restrict to the case of thesignature operator. The predecessor of local index theorem for families is the K -theoretic version by Atiyah andSinger [10], which we briefly recall. As announced we consider only the case of the signatureoperator. Let F ֒ → M π −→ B be an oriented fiber bundle of closed manifolds, where F is assumedto be even dimensional. We choose a vertical projection and a vertical metric g v . Let ∇ v bethe associated connection on T v M , and let D + v be the vertical signature operator defined by thevertical chirality operator. As in Proposition 3.1.4 we can view ker D + v and coker D + v as (spacesof sections of) finite dimensional vector bundles over B . Definition 3.2.1. The index bundle associated to D + v is defined byInd D + v := [ker D + v ] − [coker D + v ] ∈ K ( B ) . Note that as we are considering only families of signature operators we do not need thebeautiful construction for the case of varying dimensions as in [10]. The Chern character definesa map in K -theory, ch : K ( B ) ⊗ C → H ev ( B ) . Then cohomological version of the families index theorem as in [10, Thm. 5.1] is Theorem 3.2.2 (Atiyah-Singer) . The Chern character of the index bundle associated to thesignature operator is given by ch(Ind D + v ) = h Z M/B L ( T v M, ∇ v ) i ∈ H ev ( B ) , where L ( T v M, ∇ v ) is the Hirzebruch L -form of T v M defined via Chern-Weil theory as in (A.4) in terms of ∇ v . .2. Elements of Bismut’s Local Index Theory for Families Quillen [84] introduced superconnections to study Chern-Weil theory for the Chern character ofa difference bundle. We briefly recall the basic definitions, and refer to [13, Sec. 1.4] and [84] fordetails. Let B be a closed, oriented manifold, and let E → B be a complex vector bundle. Definition 3.2.3. A differential operator A on Ω • ( B, E ) is called a generalized connection on E if it satisfies the Leibniz rule A ( α ∧ β ) = dα ∧ β + ( − | α | α ∧ A β, where α ∈ Ω • ( B ) and β ∈ Ω • ( B, E ). The curvature of A is defined as A ∈ Ω • (cid:0) B, End( E ) (cid:1) . If E is Z -graded and A is of odd parity with respect to the total grading on Ω • ( B, E ), then A is called a superconnection . Remark. (i) If E = E + ⊕ E − is Z -graded, the total grading of Ω • ( B, E ) referred to above is defined byΩ( B, E ) ± := Ω ev ( B, E ± ) + Ω odd ( B, E ∓ ) . This should not be confused with the grading Ω ± ( B, E ), induced by the chirality operator τ B .(ii) The fact that the curvature A is indeed given by the action of an element in Ω • (cid:0) B, End( E ) (cid:1) works as in the case of a usual connection, see [13, Prop. 1.38].(iii) A generalized connection A is determined by its homogeneous components A = A [0] + A [1] + A [2] + . . . , where A [ p ] ∈ Ω p (cid:0) B, End( E ) (cid:1) for p = 1 and A [1] is a connection on E . In the case that A isa superconnection, the connection part A [1] preserves the splitting E = E + ⊕ E − . Definition 3.2.4. Let E be Z -graded, and let A be a superconnection on E . Then we definethe Chern character form of A asch s ( E, A ) := str E (cid:0) γ exp( − A ) (cid:1) . Here, γ : Ω • ( B ) → Ω • ( B ) is a normalization function, defined for forms of homogeneous degreeby γ ( α ) := (cid:0) √ πi (cid:1) | α | α, where √ i = e iπ . (3.15)The discussion in Appendix A generalizes to the case of superconnections. In particular,ch s ( E, A ) is a closed differential from on B whose cohomology class is independent of the super-connection A . Since the supertrace vanishes on endomorphisms of odd parity, one can also checkthat ch s ( E, A ) ∈ Ω ev ( B, E ). We also note that if A = ∇ is a connection in the usual sense, whichdecomposes with respect to the splitting E = E + ⊕ E − into ∇ = ∇ + ⊕ ∇ − , thench s ( E, ∇ ) = ch( E + , ∇ + ) − ch( E − , ∇ − ) , (3.16)where the right hand side is as in Definition A.1.3. Then the main idea of [84] can be summarizedas02 3. Rho Invariants of Fiber Bundles, Abstract Theory Theorem 3.2.5 (Quillen) . Let E → B be a Z -graded Hermitian bundle over B . Let [ E + ] − [ E − ] be the induced element in K ( B ) , and let A be a superconnection on E . Then ch (cid:0) [ E + ] − [ E − ] (cid:1) = (cid:2) ch s ( E, A ) (cid:3) ∈ H ev ( B ) . Generalized Clifford Connections and Dirac Operators. Whenever E is endowed withthe structure of a Clifford module, one can associate a Dirac operator to a generalized connection A . Let σ − : Λ • T ∗ B → Cl( T ∗ B )be the quantization map. If c : Cl( T ∗ B ) → End( E ) denotes Clifford multiplication, we get anatural Clifford contractionΩ • ( B, E ) σ − −−→ C ∞ (cid:0) B, Cl( T ∗ B ) ⊗ E (cid:1) c −→ C ∞ ( B, E ) . Since A maps C ∞ ( B, E ) to Ω • ( B, E ), we can define D A := c ◦ σ − ◦ A : C ∞ ( B, E ) → C ∞ ( B, E ) . (3.17)Clearly, this defines an elliptic operator of first order. In order for D A to be formally self-adjoint,we certainly have to require that the connection part A [1] of A is a Clifford connection in senseof Definition 1.2.11. Furthermore, some condition has to be imposed on the other homogeneouscomponents of A , which we derive now, see also [13, p. 117]. For p = 1 and a local orthonormalframe { f a } for T B we can write locally A [ p ] = p ! f a ∧ . . . ∧ f a p ∧ T a ...a p , with T a ...a p ∈ C ∞ (cid:0) B, End( E ) (cid:1) . The contribution to D A is then given by c ◦ σ − ◦ A [ p ] = p ! c a . . . c a p T a ...a p , where c a j is short for Clifford multiplication with f a j . For e, e e ∈ C ∞ ( B, E ) one computes that (cid:10) c a . . . c a p T a ...a p e , e e (cid:11) = ( − p (cid:10) e , T ∗ a ...a p c a p . . . c a e e (cid:11) = ( − p ( p +1)2 (cid:10) e , T ∗ a ...a p c a . . . c a p e e (cid:11) . This motivates the following Definition 3.2.6. Let E be a Hermitian vector bundle. A generalized connection A is called unitary if its connection part is a unitary connection and if for p = 1 A ∗ [ p ] = ( − p ( p +1)2 A [ p ] . Here, taking the adjoint is meant with respect to the endomorphism part only. A is is called a generalized Clifford connection , if in addition, its connection part A [1] is a Clifford connection,and if for p = 1 and ξ ∈ Ω ( B ) A [ p ] c ( ξ ) = ( − p c ( ξ ) A [ p ] , where again the product is to be understood in the endomorphism part. .2. Elements of Bismut’s Local Index Theory for Families Proposition 3.2.7. Let E → B be a Hermitian vector bundle endowed with a Clifford structure,and let A be a generalized Clifford connection. Then D A is formally self-adjoint and D A is ageneralized Laplacian. The symbol of A is given by Clifford multiplication c : T ∗ B ⊗ E → E . Ifin addition E is Z -graded and A is a superconnection, then D A is Z -graded. Remark. As we have pointed out in Remark 1.2.12, not every Dirac operator arises as a geometricDirac operator associated to a Clifford connection. However, it is shown in [13, Prop. 3.42] thatthere is a 1-1 correspondence between Clifford superconnections and Z -graded Dirac operatorswith symbol being the given Clifford structure. Going through the proof of loc.cit. one sees thatthe same statement is true for ungraded Dirac operators and generalized Clifford connections asdefined above. Note, however, that in contrast to [13], we require Dirac operators to be formallyself-adjoint. Generalizing Quillen’s construction to infinite dimensional bundles, Bismut [14] found a heatequation formula for the Chern character form of the index bundle. We describe the setupbriefly, again restricting to the case of the untwisted signature operator. Bismut’s Superconnection. Let F ֒ → M π −→ B be an oriented fiber bundle of closed manifolds.We choose a vertical projection, and let Ω • v ( M ) be the C ∞ ( B )-module of vertical differentialforms. We formally interpret this as the space of sections of an infinite dimensional bundle E over B , where the fiber E y over y ∈ B is given by the space of differential forms over π − ( y ).Since this picture has only motivational character, we do not give any details of how this bundleof Fr´ech´et spaces is defined rigorously. We can then view the space of all differential forms Ω • ( M )as Ω • ( M ) ∼ = Ω • ( B, E ) , compare with (2.6). Proposition 2.1.15 shows that the total exterior differential d M on Ω • ( M )splits as d M = d v + d h + i(Ω) , (3.18)where with respect to any choice of metric g B in a local orthonormal frame { f a } for T Bd h = f a ∧ e ∇ ⊕ a , i(Ω) = f a ∧ f b ∧ i(Ω ab ) . (3.19)Recall that when restricted to Ω • v ( M ), the connection e ∇ ⊕ a is defined as e ∇ va , see Definition 2.1.9.We view the latter as a natural connection ∇ E on the infinite dimensional bundle E . Then(3.18) and (3.19) express d M as a generalized connection on E with connection part ∇ E . It isa superconnection with respect to the even/odd grading on Ω • v ( M ). In this interpretation, theproperty d M = 0 states that d M is a flat superconnection on the bundle of vertical differentialforms, an observation which is due to [19, Sec. III (b)]. For this reason, d M together with itsinterpretation as a superconnection is sometimes called the Bismut-Lott superconnection .04 3. Rho Invariants of Fiber Bundles, Abstract Theory Definition 3.2.8. Assume that F ֒ → M π −→ B is endowed with a vertical metric g v and a verticalprojection. Let ∇ v be the associated canonical connection, and let D v be the vertical de Rhamoperator. With respect to any choice of g B and a local orthonormal frame { f a } on B define ∇ E ,u := f a ∧ (cid:0) ∇ va + k v ( f a ) (cid:1) : Ω • v ( M ) → Ω , • ( M ) , (3.20)where k v is the mean curvature form. Then, the Bismut superconnection is defined as B := D v + ∇ E ,u − c v (Ω) : Ω • v ( M ) → Ω • ( M ) , where c v denotes the vertical Clifford multiplication on Ω • ( M ), and locally c v (Ω) = f a ∧ f b ∧ c v (Ω ab ) . The Bismut superconnection naturally extends to an operator Ω • ( M ) → Ω • ( M ), if we replace ∇ v with ∇ ⊕ in (3.20). For this note that for α ∈ Ω p ( B ) and ω ∈ Ω • v ( M ), f a ∧ ∇ ⊕ a (cid:0) ( π ∗ α ) ∧ ω ) (cid:1) = π ∗ ( d B α ) ∧ ω + ( − p π ∗ α ∧ f a ∧ ∇ va ω. This relation also shows that replacing ∇ v with ∇ ⊕ is the same as extending B from Ω • v ( M ) = C ∞ ( B, E ) to Ω • ( M ) = Ω • ( B, E ) by requiring the Leibniz rule. Moreover, we find that the term f a ∧ ∇ ⊕ a is independent of the chosen metric g B , since this is true for the connection ∇ v , seeProposition 2.1.3. Remark 3.2.9. We want to point out that the definition of B can be motivated by an infinitedimensional version of Lemma 3.1.18. If we choose a vertical metric g v on T v M , we can view thepairing ( ., . ) M/B in (3.10) as a metric on the bundle E . As in Proposition 3.1.15, the connectionpart ∇ E = f a ∧ e ∇ va of d M is compatible with the vertical intersection pairing, but not necessarilywith ( ., . ) M/B . If we proceed as in Lemma 3.1.18—using in particular (2.9) and (2.15)—we seethat the unitary connection associated to ∇ E is given by f a ∧ e ∇ va + f a ∧ τ v (cid:2) e ∇ va , τ v (cid:3) = f a ∧ (cid:0) e ∇ va + B ( f a ) + k v ( f a ) (cid:1) = f a ∧ (cid:0) ∇ va + k v ( f a ) (cid:1) . This is precisely the connection part ∇ E ,u of the Bismut superconnection. In order to get theunitary superconnection associated to d M we proceed as in Definition 3.2.6 and replace the otherhomogeneous components d v and i(Ω) of d M with ( d v + d tv ) = D v and (i(Ω) − i(Ω) t ) = − c v (Ω) , which explain the remaining terms in the definition of B .Since Remark 3.2.9 is the underlying motivation for large parts of the treatment in thissection, we extract the following result, adding some observations which are immediate. Proposition 3.2.10. The Bismut superconnection is the generalized unitary connection associ-ated to the flat superconnection d M . It is a superconnection with respect to the even/odd gradingon Ω • v ( M ) . If the fiber is even dimensional, it is also a superconnection with respect to the gradinginduced by the vertical chirality operator τ v . .2. Elements of Bismut’s Local Index Theory for Families Remark 3.2.11. In the context of the signature operator we are interested in the grading givenby the vertical chirality operator. However, we get a superconnection with respect to this gradingonly if the fiber is even dimensional. In the case that the fiber is odd dimensional, one can turn B into a superconnection by adding an auxiliary Grassmann variable, see [18, Sec.’s II (b) & (f)].This is based on Quillen’s ideas in the finite dimensional case as in [84, Sec. 5]. The Chern Character of the Bismut Superconnection. As in Definition 3.2.3, the curva-ture of the Bismut superconnection is defined as the differential operator B : Ω • ( M ) → Ω • ( M ) . In analogy with the finite dimensional situation in Section 3.1.6, we cannot expect that theBismut superconnection is flat. Proposition 3.2.12. The curvature of B is a fiberwise elliptic operator. It is of second orderwith leading term given by the vertical Laplacian D v : Ω • ( M, E ) → Ω • ( M, E ) . Proof. According to Definition 3.1.1, we have to check first that B is C ∞ ( B ) linear. For all ϕ ∈ C ∞ ( B ), [ B , π ∗ ϕ ] = e( π ∗ d B ϕ ) , and thus, (cid:2) B , π ∗ ϕ (cid:3) = B ◦ e( π ∗ d B ϕ ) + e( π ∗ d B ϕ ) ◦ B . Now, the operators D v , c v (Ω) and f a ∧ k v ( f a ) all anti-commute with exterior multiplication witha horizontal 1-form. Hence, (cid:2) B , π ∗ ϕ (cid:3) = f a ∧ ∇ ⊕ a ( π ∗ d B ϕ ) = π ∗ d B ϕ = 0 , so that B is indeed a fiberwise differential operator. Now, B contains D v as term of second orderbut a priori there might be other contributions coming from ( ∇ E ,u ) and the anti-commutator of D v and ∇ E ,u . To see that this is not the case we note that ∇ E ,u agrees with d h up to a term oforder 0. Moreover, we know from Corollary 2.1.16 that d h = − (cid:8) d v , i(Ω) (cid:9) and (cid:8) D v , d h (cid:9) = (cid:8) d tv , d h (cid:9) , where both terms are of order ≤ 1, see Proposition 2.1.25 for the second term. This implies that( ∇ E ,u ) and { D v , ∇ E ,u } are also of order ≤ Theorem 3.2.13. The operator e − B : Ω • ( M ) → Ω • ( M ) is a well-defined, fiberwise smoothingoperators with coefficients in Ω • ( B ) . Using the fiberwise supertrace, this result allows us to study the Chern character of thesuperconnection B . To keep the motivational part of this section reasonably short we skip thediscussion of the odd dimensional case and assume henceforth that the fiber is even dimensional.06 3. Rho Invariants of Fiber Bundles, Abstract Theory Definition 3.2.14. Let F ֒ → M π −→ B have even dimensional fiber F , and let g v be a verticalmetric. We define the Chern character of the Bismut superconnection asch s ( E , B ) := Str v (cid:0) γe − B (cid:1) ∈ Ω • ( B ) , where γ is the normalization function as in (3.15). Transgression and the Rescaled Superconnection. One of Bismut’s main observations in[14] is that ch s ( E , B ) is the right candidate for the Chern character form for the index bundle asin Definition 3.2.1. This becomes apparent when rescaling the vertical metric, see [14, Sec. III(c)].For t ∈ (0 , ∞ ) we rescale the vertical metric with a factor of t − . As in (1.55) this means thatClifford multiplication with a vertical 1-form ξ has to be replaced with c v,t ( ξ ) = √ t (cid:0) e( ξ ) − i( ξ ) (cid:1) , where inner multiplication is with respect to the fixed metric g v . Also, if { e i } is a local orthonor-mal frame for T v M with respect to g v , then {√ te i } is orthonormal with respect to the rescaledmetric. Since ∇ v and k v are independent of t , this motivates the following Definition 3.2.15. For t ∈ (0 , ∞ ) define the rescaled Bismut superconnection by B t := √ t D v + ∇ E ,u − √ t c v (Ω) . Now, the infinite dimensional version of the transgression formula for the Chern character isthe following, see [13, Thm. 9.17]. Theorem 3.2.16. For all t ∈ (0 , ∞ ) , the differential form ch s ( E , B t ) is closed and satisfies thetransgression formula ddt ch s ( E , B t ) = − γ d B Str v (cid:16) d B t dt e − B t (cid:17) . Since this transgression form will be of importance in the next section, we make the followingabbreviation. Definition 3.2.17. The transgression form associated to the rescaled Bismut superconnectionis given by α ( B t ) := γ √ πi Str v (cid:16) d B t dt e − B t (cid:17) ∈ Ω odd ( B ) . Bismut’s Local Index Theorem for Families. With the above ingredients, we can nowsummarize Bismut’s main results [14, Thm.’s 3.4, 4.12 & 4.16] in the case of the signatureoperator, see also [13, Thm.’s 10.21, 10.23 & 10.32]. Theorem 3.2.18 (Bismut) . Let F ֒ → M π −→ B be endowed with a vertical metric g v and avertical projection, and assume that F is even dimensional. Then the rescaled superconnection B t satisfies (cid:2) ch s ( E , B t ) (cid:3) = ch(Ind D + v ) ∈ H ev ( B ) . (3.21) Moreover, with respect to the C ∞ -topology on Ω • ( B ) and for t → t → ch s ( E , B t ) = Z M/B L ( T v M, ∇ v ) and α ( B t ) = O (1) , (3.22) .2. Elements of Bismut’s Local Index Theory for Families where L ( T v M, ∇ v ) is the Hirzebruch L -form of T v M with respect to the connection ∇ v . Inparticular, ch s ( E , B t ) = Z M/B L ( T v M, ∇ v ) − d Z t α ( B s ) ds. (3.23) Remark. (i) An immediate consequence of Bismut’s local index theorem for families is Atiyah andSinger’s earlier result as stated in Theorem 3.2.2. This is because (3.21) and (3.23) im-ply that ch(Ind D + v ) = h Z M/B L ( T v M, ∇ v ) i ∈ H ev ( B ) . (ii) We also want to remark that (3.21) is a generalization of the McKean-Singer formula inTheorem 1.2.6. In the case that B is a point, the term ch(Ind D + v ) coincides with thenumerical index. On the other hand, e − B t = e − tD v , since the higher degree terms in theBismut superconnection vanish. Hence, if B is a point, the equation in (3.21) is equivalentto the usual McKean-Singer formula.To understand why (3.21) is related to the limit t → ∞ also in higher dimensions, we sum-marize some results from [13, Sec. 9.3]. Let P ker( D v ) be the projection E → ker D v , and define ∇ H v ,u := P ker( D v ) ◦ ∇ E ,u ◦ P ker( D v ) . (3.24)According to Lemma 3.1.18 and Proposition 3.2.10, the connection ∇ H v ,u is a unitary connectionon the bundle H • v ( M ) → B , compatible with the grading given by τ v . We can thus define theassociated Chern character form ch s (cid:0) H • v ( M ) , ∇ H v ,u (cid:1) as in (3.16). Then there is the followingresult, see [13, Thm.’s 9.19 & 9.23]. Theorem 3.2.19 (Berline-Getzler-Vergne) . With respect to the C ∞ -topology on Ω • ( B ) and for t → ∞ lim t →∞ ch s ( E , B t ) = ch s (cid:0) H • v ( M ) , ∇ H v ,u (cid:1) , and α ( B t ) = O ( t − / ) . In particular, ch s (cid:0) H • v ( M ) , ∇ H v ,u (cid:1) = ch s ( E , B t ) − d Z ∞ t α ( B s ) ds. (3.25)It is immediate from Definition 3.2.1, the definition of ∇ H v ,u and (3.16) that (cid:2) ch s (cid:0) H • v ( M ) , ∇ H v ,u (cid:1)(cid:3) = ch(Ind D + ) . so that (3.25) is an extension to differential forms of (3.21), and explains why the latter is relatedto taking the limit t → ∞ . Remark. We want to point out that Theorem 3.2.19 holds for more general superconnectionsbut relies on the fact that the dimensions of the kernels of its homogeneous component of degree0 do not jump. In contrast, Theorem 3.2.18 continues to hold without any assumption on thekernels but only for the Bismut superconnection associated to a family of Dirac operators.08 3. Rho Invariants of Fiber Bundles, Abstract Theory We now want to relate the discussion in the last section to adiabatic limits of Eta invariants andgive a motivation for Dai’s general adiabatic limit formula. A large part of this subsection willbe heuristic without rigorous arguments. We hope that nevertheless, our discussion helps to givesome intuition underlying the sophisticated theory. Statement of the Reduced Adiabatic Limit Formula. Let F ֒ → M π −→ B be an orientedfiber bundle of closed manifolds, where M is odd dimensional. Since this section has only mo-tivational character, we assume for simplicity that F is even dimensional and that B is odddimensional. We endow the fiber bundle with a vertical projection and a submersion metric g = g B ⊕ g v . Let α ( B t ) be the transgression form associated to the rescaled Bismut supercon-nection, see Definition 3.2.17. Theorem 3.2.18 and Theorem 3.2.19 show that α ( B t ) = O (1) as t → α ( B t ) = O ( t − / ) as t → ∞ . This implies that we can make the following definition,which goes back to [16]. Definition 3.3.1. If F is even dimensional and B is odd dimensional, we define the Bismut-Cheeger Eta form b η := Z ∞ α ( B t ) dt. Now let g ε be the adiabatic metric (2.16) on M , and consider the associated adiabatic familyof de Rham operators, D M,ε = D v + ε · D h + ε · T : Ω • ( M ) → Ω • ( M ) . Moreover, recall that in Definition 3.1.16 we have introduced the odd signature operator on B with values in the bundle of vertical cohomology groups, D B ⊗ ∇ H v : Ω • (cid:0) B, H • v ( M ) (cid:1) → Ω • (cid:0) B, H • v ( M ) (cid:1) . Then we have the following version of [30, Thm.’s 0.1 & 4.4]. Theorem 3.3.2 (Dai) . Assume that b := dim B is odd and that dim F is even. Then the adiabaticlimit of the reduced ξ -invariant is given by lim ε → ξ (cid:0) τ M D M,ε (cid:1) = ξ (cid:0) D B ⊗ ∇ H v (cid:1) + 2 b +12 Z B b L ( T B, ∇ B ) ∧ b η mod Z . This formula has a long history which started with Witten’s famous holonomy formula in[97]. The first mathematically rigorous treatments were given by Bismut and Freed in [18] aswell as Cheeger in [26], both for the case that B = S . Bismut and Freed emphasize the relationto Bismut’s local index theory, whereas Cheeger gives an independent proof based on Duhamel’sformula and finite propagation speed methods. Later in [16], Bismut and Cheeger generalizedthe adiabatic limit formula to higher dimensional base spaces under the assumption that thevertical Dirac operator is invertible. Note that in this case the twisted Eta invariant on the basedoes not appear. Dai’s remarkable generalization in [30] allows the vertical Dirac operators tohave non-trivial kernels—which, however, need to form a vector bundle over B . This is why his .3. A General Formula for Rho Invariants The Total Dirac Operator. For the rest of this subsection we want to give a heuristic derivationof Theorem 3.3.2. Recall that we have seen in Theorem 3.2.16 that the form α ( B t ) plays the roleof the transgression form associated to the Chern character of the superconnection B t . The roughidea is now that the adiabatic limit formula of Theorem 3.3.2 is an infinite dimensional analog ofthe variation formula for the Eta invariant as in Proposition 1.5.1, respectively Corollary 1.5.2.This idea does not apply directly to the adiabatic family of odd signature operators, and asa tool we have to introduce another natural Dirac operator acting on Ω • ( M ). Let c : T ∗ M → End (cid:0) Λ • T ∗ M (cid:1) denote the natural Clifford structure, and define a connection on Λ • T ∗ M via ∇ S := ∇ ⊕ + c ( θ ) , where θ is the tensor as in Definition 2.1.6. One easily checks that ∇ S is a Clifford connectionas in Definition 1.2.11, see e.g. [13, Prop. 10.10]. Definition 3.3.3. The total Dirac operator D S on M is the geometric Dirac operator associatedto ∇ S , D S = c ◦ ∇ S : Ω • ( M ) → Ω • ( M ) . Remark. We recall from Lemma 2.1.19 that on Λ • T ∗ M , the difference of the Levi-Civita con-nection ∇ g and the connection ∇ ⊕ is given by ∇ g = ∇ ⊕ + (cid:0) c ( θ ) − b c ( θ ) (cid:1) , where b c is the transposed Clifford structure. Therefore, the total Dirac operator D S does ingeneral not coincide with the de Rham operator D M . Relation to the Bismut Superconnection. As in Section 3.2.3, we interpret Ω • ( M ) as thespace of differential forms on B with values in the infinite dimensional vector bundle E → B ofvertical differential forms. Then the total Dirac operator is a differential operator D S : Ω • ( B, E ) → Ω • ( B, E ) . Using the local description (2.5) for the tensor θ , one then checks that with respect to a localorthonormal frame { f a } for T B , D S = D v + c a ∇ E ,ua − c a c b c v (Ω ab ) , (3.26)where ∇ E ,u is defined as in (3.20). The fact that the terms appearing in this formula are remi-niscent of the terms appearing in the definition of the Bismut superconnection seems to be oneof the main ideas which lead Bismut to the definition of B , compare with [15, Thm. 2.24] and[14, Sec. V]. We want to make this relation more precise now.In the interpretation of Proposition 3.2.10, the Bismut superconnection is a superconnectionon Λ • T ∗ B ⊗ E . Comparing with Definition 3.2.6, one finds that it is a Clifford connection withrespect to the Clifford structure induced by c : T ∗ B → Λ • T ∗ B . Now formally using (3.17), wecan associate a Dirac operator to this, which is locally given by D v + c a ∇ E ,ua − c a c b c v (Ω ab ) . 3. Rho Invariants of Fiber Bundles, Abstract Theory Up to factors of , this is coincides with (3.26). Hence, the total Dirac operator is essentiallythe Dirac operator on Ω • ( B, E ) associated to the Bismut superconnection. To state the relationmore precisely, consider a metric of the form g B ⊕ t g v , where t ∈ (0 , ∞ ), and let D S,t be theassociated total Dirac operator. As in the discussion preceding Definition 3.2.15 one infers that D S,t = √ tD v + c a ∇ E ,ua − √ t c a c b c v (Ω ab ) . Then we have as in [13, Thm. 10.19] Proposition 3.3.4. Consider the Dirac operator on Ω • ( B, E ) associated to the rescaled Bismutsuperconnection B t on Λ • T ∗ B ⊗ E , i.e., D B t := c ◦ σ − ◦ B t : Ω • ( B, E ) → Ω • ( B, E ) Then, under the identification Ω • ( M ) = Ω • ( B, E ) , we have D B t = D S,t . Remark. (i) The factor 4 occurs because we have defined the Bismut superconnection in a slightlydifferent way compared to [14]. This is because we wanted Proposition 3.2.10 to holdwithout any constants appearing in B , see also [19, Rem. 3.10].(ii) We also want to stress that Proposition 3.3.4—in the same way as Proposition 3.2.10earlier—is more a formal interpretation than a mathematical statement. Nevertheless, thishelps in understanding some of the ideas underlying the technicalities of local families indextheory. Variation of the Eta Invariant of the Total Dirac Operator. After having introduced thetotal Dirac operator and its interpretation in term of the rescaled Bismut superconnection ouraim is now to give a heuristic explanation of Theorem 3.3.2, in the case that the adiabatic familyof de Rham operators is replaced with a corresponding family of total Dirac operators.We first note that since D S,t is a path of formally self-adjoint elliptic operators of first orderon Ω • ( M ), the general variation formula for the ξ -invariant in Proposition D.2.5 shows that ξ (cid:0) τ M D S,t (cid:1) − ξ (cid:0) τ M D S,t (cid:1) = Z t t √ π LIM u → √ u Tr h τ M dD S,t dt e − uD S,t i dt mod Z , (3.27)where LIM u → means taking the constant term in the asymptotic expansion as u → 0. Giventhe interpretation of Proposition 3.3.4, we now formally apply Proposition 1.5.8 to the case athand and get 1 √ π LIM u → √ u Tr h τ M dD S,t dt e − uD S,t i = − b +12 Z B b L ( T B, ∇ B ) ∧ Tr v h τ v γ √ πi (cid:0) d B t dt e − B t (cid:1)i = 2 b +12 Z B b L ( T B, ∇ B ) ∧ α ( B t ) . (3.28) .3. A General Formula for Rho Invariants Remark. Note that in contrast to Proposition 1.5.8 we are not only in the situation that thetwisting bundle E is infinite dimensional, but are also dealing with a path of superconnectionsrather than usual connections. This requires special considerations already in the case of a finitedimensional twisting bundle, see [46, Sec. 2].Despite this apparent lack of mathematical rigor, we assume for the rest of this motivationalpart that the formula in (3.28) is valid. The Limit as t → ∞ . Note that for the total Dirac operator, the component c a ∇ E ,ua is theanalog of the horizontal de Rham operator D h in Definition 2.1.23. In analogy to Definition3.1.16 we thus define an operator D B ⊗ ∇ H v ,u : Ω • (cid:0) B, H • v ( M ) (cid:1) → Ω • (cid:0) B, H • v ( M ) (cid:1) ,D B ⊗ ∇ H v ,u := P ker D v ◦ τ M (cid:0) c a ∇ E ,ua (cid:1) ◦ P ker D v , (3.29)compare with (3.24). Then the remarkable result [30, Thm. 1.6] guarantees the following Theorem 3.3.5 (Dai) . Abbreviate D S, ∞ := D B ⊗ ∇ H v ,u . Then, for N large enough, there existpositive constants C and C such for all s > as t → ∞ (cid:12)(cid:12)(cid:12) Tr ′ (cid:0) τ M D S,t e − sD S,t (cid:1) − Tr (cid:0) D S, ∞ e − sD S, ∞ (cid:1)(cid:12)(cid:12)(cid:12) ≤ C √ ts N e − C s . Here, Tr ′ indicates taking the trace over those eigenvalues of D S,t which are bounded away from0 as t → ∞ . It is shown in [30, Thm. 1.5] that there are only finitely many eigenvalues of D S,t whichconverge to zero as t → ∞ . We also note that the formulation in [30] is slightly different fromwhat is stated here. This is due to the fact that we are using a different scaling and a differentparameter, see Remark 3.3.8 (iii) below. For the time being, we only use that—very roughly—Theorem 3.3.5 means that lim t →∞ ξ (cid:0) τ M D S,t (cid:1) = ξ (cid:0) D B ⊗ ∇ H v ,u (cid:1) mod Z . Together with (3.27) and (3.28) we arrive at the following heuristic formula ξ (cid:0) τ M D S (cid:1) = ξ (cid:0) D B ⊗ ∇ H v ,u (cid:1) + 2 b +12 Z B b L ( T B, ∇ B ) ∧ Z ∞ α ( B t ) dt mod Z . (3.30) The Adiabatic Limit Formula for D S . To understand how an adiabatic limit comes intoplay, we consider a metric of the form g ε,t = ε g B ⊕ t g v . Let D S,ε,t be the associated total Dirac operator. Since multiplying g ε,t by ε − does not changethe ξ -invariant, we have ξ (cid:0) D S,ε,t (cid:1) = ξ (cid:0) √ εD S,ε,t (cid:1) . We note that explicitly, √ εD S,ε,t = √ εtD v + √ ε c a ∇ E ,ua − ε √ εt c a c b c v (Ω ab ) . 3. Rho Invariants of Fiber Bundles, Abstract Theory As we have seen in Section 1.5, the proof of Proposition 1.5.8 uses Getzler’s rescaling by √ u .Since (3.28) is a formal transition to the case at hand, it is reasonable to expect that the samerescaling is involved there. Without going into detail, we note that using the rescaling √ uε instead, one formally obtains1 √ π LIM u → √ u Tr h τ M dD S,ε,t dt e − uD S,ε,t i = 2 b +12 Z B b L ( T B, ∇ B ) ∧ εα ( B εt ) . Inserting this in (3.30) and substituting s = 4 εt , we arrive at ξ (cid:0) τ M D S,ε (cid:1) = ξ (cid:0) D B ⊗ ∇ H v ,u (cid:1) + 2 b +12 Z B b L ( T B, ∇ B ) ∧ Z ∞ ε α ( B s ) ds mod Z , (3.31)where we have used that the term ξ (cid:0) D B ⊗∇ H v ,u (cid:1) is independent of ε . This is because D B ⊗∇ H v ,u is an operator over B and rescaling the full metric does not change the ξ -invariant. Now letting ε → D S,ε . Remark. We want to point out again, that our considerations leading to (3.31) are heuristicand have no rigorous mathematical foundation. In fact, we do not even expect (3.27), (3.30) and(3.31) to be correct without any changes. However, it would be interesting to find estimates onthe correction term in (3.31) along the line of thoughts we have presented to give an alternativeproof of Theorem 3.3.2 for the total Dirac operator. Relation between D S,ε and D M,ε . What is still missing in our heuristic explanation ofTheorem 3.3.2 is why we can replace D S,ε with D M,ε and D B ⊗ ∇ H v ,u with D B ⊗ ∇ H v . First ofall, the relation between the latter two operators fits exactly into the framework of Section 3.1.6.In particular, Theorem 3.1.20 yields that ξ (cid:0) D B ⊗ ∇ H v ,u (cid:1) = ξ (cid:0) D B ⊗ ∇ H v (cid:1) mod Z . Formally, the relation between D S,ε and D M,ε is an infinite dimensional analog of the samesituation, where the odd signature operator τ M D M,ε plays the role of D B ⊗ ∇ , whereas τ M D S,ε is an analog of the operator D B ⊗ ∇ u , compare with Lemma 3.1.19 and Proposition 3.2.10. Thenthe heuristic analog of Theorem 3.1.20 is that we can make the required substitution in Theorem3.3.2. For a technically precise explanation, see [30, p. 304] and [17, p. 374]. After this heuristic digression, we now want to state Dai’s general adiabatic limit formula forthe Eta invariant of the odd signature operator in its full generality. From now on we will alsoinclude the case that dim B is even and dim F is odd. Hence, we first need the analog of theBismut-Cheeger Eta form in Definition 3.3.1 for the case of odd dimensional fibers, see [16, Def.4.93 & Rem. 4.100]. Definition 3.3.6. Assume that dim F is odd and dim B is even, and let B t be the rescaledBismut superconnection as in Definition 3.2.15. Then we define b η := 1 √ π Z ∞ γ Tr ev v (cid:2) τ v d B t dt e − B t (cid:3) dt ∈ Ω ev ( B ) , where γ is the normalization function as in (3.15), and Tr ev v indicates taking the even form partof Tr v . .3. A General Formula for Rho Invariants Remark. (i) The vertical chirality operator enters in Definition 3.3.6, since in contrast to [16] we aredealing with the operator τ M D M rather than the spin Dirac operator. Nevertheless, Tr v ◦ τ v should not be viewed as a supertrace, since in the case that dim F is odd, τ v commuteswith vertical Clifford multiplication.(ii) The fact that the integral defining b η is indeed convergent relies on the odd dimensionalanalogs of the small and large time behaviour of α ( B t ) in Theorem 3.2.18 and Theorem3.2.19, i.e.,Tr ev v (cid:2) τ v d B t dt e − B t (cid:3) = O (1) as t → , Tr ev v (cid:2) τ v d B t dt e − B t (cid:3) = O ( t − / ) as t → ∞ . It is difficult to find an explicit proof in the literature, since the treatment is usually inthe superconnection formalism which does not apply to the operator in question. However,as pointed out in Remark 3.2.11 one can overcome this difficulty by introducing an extraGrassmann variable to turn B t into a superconnection of the required form, see again [18,Sec.’s II (b) & (f)], and also [25, Sec. 5.2.2] for additional remarks and references. Thereader who feels uncomfortable with this can equally well consider b η ( s ) := 1 √ π Z ∞ γ Tr ev v (cid:2) τ v d B t dt e − B t (cid:3) t s dt, Re( s ) large , and define the Eta form as the constant term in the Laurent series around s = 0 of themeromorphic continuation. Then the whole discussion to follow goes through with onlyminor changes—the sole problem being a more awkward notation.(iii) In contrast to the case that F is even dimensional, the Bismut-Cheeger Eta form is now adifferential form on B of even degree. One easily checks that the degree 0 term is given by b η [0] = 12 √ π Z ∞ u − / Tr v (cid:2) τ v D v e − uD v (cid:3) ds, where we have substituted 4 u = t . Hence, if B ev v = τ v D v | Ω ev v ( M ) is the family of verticalodd signature operators, we see that b η [0] = η ( B ev v ) ∈ C ∞ ( B ) , (3.32)which is the function that associates to each point y ∈ B the Eta invariant of the fiber π − ( y ). Note that we have used (1.38) which is possible since B ev v is a family of geo-metric Dirac operators so that the Eta function can be defined without making use of ameromorphic extension. Preliminary Adiabatic Limit Formula. The starting point for the rigorous treatment of theadiabatic limit formula is [30, Prop. 1.4], which in the case of the odd signature operator reads Proposition 3.3.7 (Dai) . Let D ε be the family of de Rham operator associated to the adiabaticmetric (2.16) . Then there exists a small positive number α such that lim ε → η ( τ M D ε ) = 2 [ b +12 ]+1 Z B b L ( T B, ∇ B ) ∧ b η + lim ε → √ π Z ∞ ε − α u − / Tr (cid:2) τ M D ε e − uD ε (cid:3) du, provided either one of the limits exists. 3. Rho Invariants of Fiber Bundles, Abstract Theory Remark 3.3.8. (i) Dai deduces Proposition 3.3.7 from the main result in [16] which—translated to the situationat hand—gives an explicit formula forlim ε → Tr (cid:2) τ M D ε e − uD ε (cid:3) , together with remainder estimates, which are uniform in ε for compact u -intervals, see [16,(4.40)]. We want to point out, that Bismut’s and Cheeger’s proof is rather involved, onedifficulty being the presence of the term Tr (cid:2) τ M εD h e − uD ε (cid:3) , which behaves “singular” withrespect to Getzler’s rescaling, see [16, Rem. 3.4]. This part should simplify if one couldfind a proof along the lines of the heuristic discussion in Section 3.3.1.(ii) Note that for the odd signature operator, we already know from Proposition 2.2.8 that thelimit on the left hand side in Proposition 3.3.7 exists. By comparison with Theorem 3.3.2we see that the second term on the right hand side will produce the twisted Eta invariantof the base as well as the integer contribution which we omitted so far.(iii) To relate this term to the discussion in Section 3.3.1, we substitute s = ε u , which corre-sponds to rescaling the metric g ε to ε g ε . Then Z ∞ ε − α u − / Tr (cid:2) τ M D ε e − uD ε (cid:3) du = Z ∞ ε − α s − / Tr (cid:2) τ M ( ε D ε ) e − s ( ε D ε ) (cid:3) ds. We now note that if we rename √ t = ε − , then ε D ε becomes √ tD ( √ t ) − = √ tD v + D h + √ t T. where the individual terms are defined as in Definition 2.1.23. This means that the operator ε D ε plays essentially the role of the operator D S,t in the discussion of Section 3.3.1, andthat taking the limit ε → t → ∞ in our heuristic explanation of Theorem 3.3.2. We also note without givingthe details that this also explains why Theorem 3.3.5 as we have stated it is indeed areformulation of [30, Thm. 1.6]. Behaviour of Small Eigenvalues. The limit on the right hand side of the formula in Propo-sition 3.3.7 is closely related to eigenvalues of τ M D ε which approach zero as ε → 0. This isroughly because of their presence, there is no uniform bound of the form Ce − cu for the termTr (cid:2) τ M D ε e − uD ε (cid:3) as ε → u , compare with Lemma D.2.1. The following result providesthe essential analysis of the spectrum of D ε as ε → 0, see [30, Thm. 1.5]. Theorem 3.3.9 (Dai) . For ε > the eigenvalues of τ M D ε depend analytically on ε . There existanalytic functions { λ iε | i ∈ Z } such that spec( τ M D ε ) = S i ∈ Z λ iε for all ε > . Moreover, thefunctions λ iε have the following properties. (i) There exists a positive constant λ such that either for some ε | λ iε | ≥ λ > , for ε ≤ ε , .3. A General Formula for Rho Invariants or λ iε has a complete asymptotic expansion λ iε ∼ X k ≥ µ ik ε k as ε → , where µ i ∈ spec (cid:0) D B ⊗ ∇ H v (cid:1) , see Definition 3.1.16. The latter gives a bijective correspon-dence n λ ε ∈ spec( τ M D ε ) (cid:12)(cid:12)(cid:12) λ ε = O ( ε ) as ε → o ←→ spec (cid:0) D B ⊗ ∇ H v (cid:1) . (ii) Assume that λ iε = O ( ε ) as ε → , and that µ i = 0 . Then there is a uniform remainderestimate of the form λ iε = µ i ε + ε C ( ε )( µ i ) , | C ( ε ) | ≤ const . (iii) For every K > , n i ∈ Z (cid:12)(cid:12)(cid:12) λ iε = O ( ε ) as ε → , and | µ i | ≤ K o < ∞ . Remark. As one might expect, the proof of Theorem 3.3.9 in [30] relies on standard perturbationtheory as in [56], and this part is in fact not very difficult. However, to prove that the eigenvalueshave a complete asymptotic expansion, Dai makes use of Melrose’s theory of degenerate ellipticproblems as used in [66]. We refer to [30, Sec. 2] for more details and references. The General Adiabatic Limit Formula. For r ∈ N defineΛ( ε r ) := (cid:8) λ ε ∈ spec( τ M D ε ) (cid:12)(cid:12) λ ε = O ( ε r ) , as ε → (cid:9) . (3.33)Note that part (iii) of Theorem 3.3.9 shows that ε r ) < ∞ for all r ≥ 2. Then Dai’s mainresults can be stated in the following way, see [30, Cor. 1.6 & Prop. 1.8]. The formal butilluminating outline of the proof in [30, p. 275] is also recommended. Theorem 3.3.10 (Dai) . With the notation of Proposition 3.3.7, lim ε → √ π Z ∞ ε − α u − / Tr (cid:2) τ M D ε e − uD ε (cid:3) du = η (cid:0) D B ⊗ ∇ H v (cid:1) + lim ε → X λ ε ∈ Λ( ε ) sgn( λ ε ) . In particular, lim ε → η ( τ M D ε ) = 2 [ b +12 ]+1 Z B b L ( T B, ∇ B ) ∧ b η + η (cid:0) D B ⊗ ∇ H v (cid:1) + lim ε → X λ ε ∈ Λ( ε ) sgn( λ ε ) . This separates the computation of the adiabatic limit of the Eta invariant on the total spaceof the fiber bundle into three terms, which are all of a very different nature. Intuitively, the firstterm contains global information about the fiber, but is local on B . The second term is globalon the base and contains cohomological information about the fiber. The third term is global onboth, the base and the fiber. It fits into the heuristic discussion of Section 3.3.1 as an analog ofthe spectral flow term in Corollary 1.5.2. Following again [30], we will see in the next subsectionthat for the odd signature operator, this term is expressible in completely topological terms.16 3. Rho Invariants of Fiber Bundles, Abstract Theory Let ∆ ε be the Laplace operator associated to an adiabatic metric g ε on an oriented fiber bundle F ֒ → M π −→ B of closed manifolds. In [66], Mazzeo and Melrose analyze the space of harmonicforms H • ε ( M ) = ker(∆ ε ) as ε → 0, and show that it has a basis which extends smoothly to ε = 0. Using a Taylor series analysis to determine which forms lie in this limiting space, theyfind a Hodge theoretic version of the Leray-Serre spectral sequence. We review the result briefly,and refer to [66] as well as [41] and [30, Sec. 4.2] for details. As in the latter reference, we re-strict to the odd dimensional case and use the formulation in terms of the odd signature operator. The Hodge Theoretic Spectral Sequence. Assume that dim M is odd. For Λ( ε r ) as in(3.33) we define G Λ( ε r ) := X λ ε ∈ Λ( ε r ) ker (cid:0) τ M D ε − λ ε (cid:1) . We view this as a family of subspaces of Ω • ( M ), parametrized by ε ∈ (0 , ∞ ). Note that Theorem3.3.9 implies that for r ≥ G Λ( ε r ) is finite dimensional. Now the analysis of [66] adapted tothe case at hand yields the following, see [30, Thm. 0.2 & Prop. 4.2]. Theorem 3.3.11 (Mazzeo-Melrose, Dai) . For r ≥ the family G Λ( ε r ) depends smoothly on ε down to ε = 0 , i.e., there exists a smooth family of orthonormal bases for G Λ( ε r ) , parametrizedby ε ∈ (0 , ∞ ) , that extends smoothly to ε = 0 . One can then define G r := lim ε → G Λ( ε r ) , d r := lim ε → ε − r d ε : G r → G r , (3.34) where d ε = d v + εd h + ε i(Ω) . This defines a spectral sequence that is isomorphic to the Leray-Serrespectral sequence, i.e., ( G r , d r ) ∼ = ( E • , • r , d r ) , r ≥ . Remark. (i) Note that the spectral sequence ( G r , d r ) is not defined as a spectral sequence associated toa filtered complex as in (3.9). Nevertheless, as subspaces of Ω • ( M ) = L p,q Ω p,q ( M ), thespaces G r are naturally bigraded.(ii) Since we are working on Ω • ( M ) with the fixed reference metric g we are using the modifiedde Rham differential d ε , compare with Remark 2.2.6.(iii) It is immediate that each d ε maps G Λ( ε r ) to itself, since d ε commutes with τ M D ε . However,note that for an element ω ε of a basis of G Λ( ε r ) as in Theorem 3.3.11, one has( τ M D ε ) ω ε = λ ε ω ε = O ( ε r ) , as ε → . Since τ M D ε = τ M d ε + d ε τ M , this implies that d ε ω ε = O ( ε r ) as well. This should serve as amotivation for the factor ε − r in (3.34). The case r = 1. Because of the extra difficulty which arises from the fact that G Λ( ε ) is notfinite dimensional the case r = 1 is excluded from Theorem 3.3.11. Nevertheless, for motivationalpurposes, we now want to make a formal digression on this case, since it might give an idea of .3. A General Formula for Rho Invariants λ ε be a family of eigenvalues of τ M D ε , which is of order ε as ε → 0. Since for ε ∈ (0 , ∞ ), the family τ M D ε depends analytically on ε , standard perturbation theory ensures that λ ε depends analytically on ε and that there exists an analytic family of eigenforms ω ε ∈ Ω • ( M )with eigenvalue λ ε . We now assume without justification that λ ε and ω ε extend analytically to[0 , ∞ ) so that we can write λ ε = X k ≥ λ k ε k , and ω ε = X k ≥ ω k ε k , as ε → . From Definition 2.2.7 we know that τ M D ε = τ M D v + ετ M D h + ε τ M T : Ω • ( M ) → Ω • ( M ) . Then comparing ε -powers in the identity ( τ M D ε ) ω ε = λ ε ω ε one finds that D v ω = 0 , τ M (cid:0) D v ω + D h ω (cid:1) = λ ω . In particular, ω ∈ ker( D v ) = Ω • (cid:0) B, H • v ( M ) (cid:1) and thus also d v ω = 0. Then the second equationyields P ker( D v ) ⊥ (cid:0) τ M (cid:0) D v ω + D h ω (cid:1)(cid:1) = 0 . (3.35)On the other hand, when we compare ε -powers in the identity (cid:0) τ M d ε ω ε , d ε τ M ω ε (cid:1) L = 0, we candeduce that (cid:0) τ M d v ω + τ M d h ω , d v τ M ω + d h τ M ω (cid:1) L = 0 . Using this and (3.35) one infers that P ker( D v ) ⊥ (cid:0) d v ω + d h ω (cid:1) = 0 . Hence, lim ε → ε − d ε ω ε = lim ε → ε − (cid:0) d v ω + ε ( d v ω + d h ω ) + ε ( . . . ) (cid:1) = d v ω + d h ω = P ker( D v ) ( d h ω ) . This shows at least formally that the construction of Theorem 3.3.11 extends to the case r = 1and gives G = Ω • (cid:0) B, H • v ( M ) (cid:1) , d = P ker( D v ) ◦ d h . Now Proposition 3.1.13 shows that via the Hodge-de-Rham isomorphism,( G , d ) ∼ = (cid:0) Ω p (cid:0) B, H • v ( M ) (cid:1) , ¯ d h (cid:1) , with the differential ¯ d h on Ω p (cid:0) B, H • v ( M ) (cid:1) associated to the flat connection ∇ H v , see Definition3.1.10. According to Lemma 3.1.12 this means that ( G , d ) is isomorphic to the E -term of theLeray-Serre spectral sequence of the fiber bundle. Remark. In principle, one could now continue and give a formal derivation of Theorem 3.3.11along the lines just presented. However, already in the case r = 2, the Hodge theoretical expres-sion of the differential becomes very unpleasant. Therefore, we end the digression and refer to[66, 30, 41] for more details.18 3. Rho Invariants of Fiber Bundles, Abstract Theory We also want to point out that Theorem 3.3.11 will not be explicitly used later on. Yet, wehave included the above discussion to motivate why it is reasonable to expect that the termlim ε → X λ ε ∈ Λ( ε ) sgn( λ ε )appearing in Dai’s general adiabatic limit formula has a topological interpretation in terms ofthe Leray-Serre spectral sequence. We shall make this more precise now. Multiplicative Structure on the Leray-Serre Spectral Sequence. From now on we alsoincorporate a flat twisting bundle E over M with connection A . The entire treatment of Section3.3 carries over verbatim; we have omitted it so far only for notational convenience. However,in the discussion to follow, there are some small distinctions to be made. We use the notation( E • , • A,r , d A,r ) for the spectral sequence associated to the flat bundle, and reserve the notation( E • , • r , d r ) from (3.9) for the spectral sequence associated to the trivial connection.Recall that there is a multiplicative structure of the form · : E p,qA,r × E s,tA,r → E p + s,q + tr , which is canonically induced by the wedge-product and the metric h : E ⊗ E → C on the bundle E , see [22, pp. 174–177], [67, Thm. 5.2] or [33, Thm. 9.24]. For brevity, we introduce thenotation E kA,r := M p + q = k E p,qA,r , · : E kA,r × E lA,r → E k + lr . Then the differential and the multiplicative structure satisfy the relation d r ( ω · η ) = d A,r ( ω ) · η + ( − k ω · d A,r ( η ) , ω ∈ E kA,r , η ∈ E lA,r . (3.36)Denote m = dim M and b = dim B . Since we are assuming that the fiber bundle is oriented,the bundle H m − bv ( M ) → B is trivializable, where each vertical volume form gives a canonicaltrivialization. We then have E m = E b,m − b = H b (cid:0) B, H m − bv ( M ) (cid:1) ∼ = C , , where a natural basis ξ ∈ E m is induced by any volume form on M . Since M is closed H m ( M ) ∼ = C , which implies that E mr for all r ≥ 2. Moreover, as the isomporphism H m ( M ) ∼ = C is alsocanonically induced by any volume form on M , we obtain natural bases ξ r for all E mr with r ≥ E • , • A,r , we can thus define a natural pairing Q A,r : E kA,r × E m − kA,r → C , (3.37)by requiring that Q A,r ( ω, η ) ξ r = ω · η, for all ( ω, η ) ∈ E kA,r × E m − kA,r . Remark. Due to the presence of the pairing E × E → C in its definition, the pairing (3.37) iscomplex anti-linear in the first variable. For even dimensional manifolds, the above pairing hasbeen analyzed already by [27] in the context of the signature of fiber bundles. .3. A General Formula for Rho Invariants P A,r : E kA,r × E m − k − A,r → C , P A,r ( ω, η ) := Q A,r (cid:0) ω, d A,r η (cid:1) . (3.38) Lemma 3.3.12. Let ω ∈ E kA,r and η ∈ E m − k − A,r . Then P A,r ( ω, η ) = ( − ( k +1)( m − k ) P A,r ( η, ω ) . In particular, if k = m − , then P A,r is Hermitian if m = 4 n − and skew Hermitian if m = 4 n − .Proof. Since E mr ∼ = C for each r ≥ d r | E m − r ≡ 0. Then (3.36) implies that Q A,r (cid:0) ω, d A,r η (cid:1) ξ r = ω · d A,r η = ( − k +1 ( d A,r ω ) · η = ( − ( k +1)( m − k ) η · d A,r ω = ( − ( k +1)( m − k ) Q A,r (cid:0) η, d A,r ω (cid:1) ξ r . This implies the first assertion. Setting k = m − , the exponent becomes ( m +12 ) , which yieldsthe second assertion. Dai’s Correction Term. Using the pairing P A,r introduced in (3.38), we now come to thetopological interpretation of the third term in the general adiabatic limit formula of Theorem3.3.10. Definition 3.3.13. For each r ≥ σ A,r := Sign (cid:0) P A,r : E kA,r × E kA,r → C (cid:1) , with k := m − , where as before the signature of a skew form is defined as the number of positive imaginaryeigenvalues minus the number of negative imaginary ones. Moreover, we write σ A := X r ≥ σ A,r . Note that the sum defining σ A is finite since the spectral sequence collapses after finitelymany steps so that for large r the number σ A,r is always 0. This is due to the presence of thedifferential d A,r in the definition of Q A,r . Now we can state [30, Thm. 4.4]. Theorem 3.3.14 (Dai) . Let E be a flat Hermitian bundle with connection A over the odddimensional total space of an oriented fiber bundle F ֒ → M π −→ B of closed manifolds. For anyadiabatic metric on M , let Λ A ( ε ) be defined in analogy to (3.33) with respect to the operator τ M D A,ε . Then lim ε → X λ ε ∈ Λ A ( ε ) sgn( λ ε ) = 2 σ A . Consequently, the following adiabatic limit formula holds lim ε → η ( B ev A,ε ) = 2 [ b +12 ] Z B b L ( T B, ∇ B ) ∧ b η A + η (cid:0) D B ⊗ ∇ H A,v (cid:1) + σ A . Here, b η A ∈ Ω • ( B ) is defined in analogy to the untwisted case in Definition 3.3.1, respectivelyDefinition 3.3.6. 3. Rho Invariants of Fiber Bundles, Abstract Theory Note that we have divided the adiabatic limit formula in Theorem 3.3.10 by a factor of 2 toget the corresponding formula for the odd signature operator, see Remark 1.4.4. An Adiabatic Limit Formula for the Rho Invariant. An immediate consequence of Theo-rem 3.3.14 is the result we were aiming at in this section. In analogy to Definition 1.4.6 we firstmake the following Definition 3.3.15. Let E be a flat bundle of rank k with connection A over the odd dimensionaltotal space of an oriented fiber bundle of closed manifolds. For every submersion metric, we definethe Bismut-Cheeger Rho form as b ρ A := b η A − k · b η ∈ Ω • ( B ) , and the Rho invariant of the bundle of vertical cohomology groups as ρ H A,v ( B ) := η (cid:0) D B ⊗ ∇ H A,v (cid:1) − k · η (cid:0) D B ⊗ ∇ H v (cid:1) ∈ R . We include the factor in the definition of ρ H A,v ( B ) since the operator D B ⊗ ∇ H A,v is essen-tially two copies of a usual odd signature operator, see Remark 3.1.17 (ii). Now an immediate—yetinteresting—consequence of Theorem 3.3.14 and Corollary 2.2.9 is Theorem 3.3.16. Let E be a flat Hermitian bundle with connection A over the odd dimensionaltotal space of an oriented fiber bundle F ֒ → M π −→ B of closed manifolds. Then with respect toevery submersion metric ρ A ( M ) = 2 [ b +12 ] Z B b L ( T B, ∇ B ) ∧ b ρ A + ρ H A,v ( B ) + σ A − k · σ, where σ A and σ refer to Dai’s correction term as in Definition 3.3.13. We now want to use Theorem 3.3.16, to compute the U(1)-Rho invariant for principal circlebundles over Riemann surfaces thus giving a different proof of Theorem 2.3.18. Related resultsare due to Zhang [98] and Dai-Zhang [32]. In the first reference, the case of the untwisted spinDirac operator for circle bundles over general even dimensional spin manifolds is studied, seealso [51, Sec. 3 a)]. Dai and Zhang use the main result of [98] and a computation of the Etainvariant for the untwisted odd signature operator to determine the Kreck-Stolz invariant forcircle bundles. We want to point out that the strategy in [32] for the untwisted odd signatureoperator is to apply the signature theorem for manifolds with boundary to the disk bundleassociated to the given circle bundle. Related discussions can be found in [11, 63]. However,this approach does not carry over to non-trivial flat twisting bundles, since one would need anextension to a flat bundle over the disk bundle, see also Remark 1.4.8 (iv). In this respect ourstrategy is of a purely intrinsic nature. The Bismut-Cheeger Rho Form. As in Section 2.3 let Σ be a closed, oriented Riemannsurface of unit volume, and let S ֒ → M π −→ Σ a principal S -bundle over Σ of degree 0 = l ∈ Z .Choose a connection iω ∈ Ω ( M, i R ) on M , and use this and the metric on Σ to equip M with .4. Circle Bundles Revisited L A → Σ be a holomorphic line bundle of degree k , the holomorphicstructure being induced by a unitary connection A . As before, we assume that i π F ω = l · vol Σ , and i π F A = k vol Σ . We endow L := π ∗ L A → M with the flat connection of Lemma 2.3.5, i.e., A q = π ∗ A − iq ω, q := k/l. Proposition 3.4.1. The Bismut-Cheeger Rho form associated of the flat line bundle L = π ∗ L A is given by b ρ A q = 2 P ( q ) + l (cid:0) P ( q ) − (cid:1) vol Σ ∈ Ω ev (Σ) . Here, P and P are the first and second periodic Bernoulli functions of Definition C.1.1.Proof. First, we infer from (3.32) that (cid:0)b ρ A q (cid:1) [0] = η ( B q,v ) − η ( B ,v ) , where B q,v is the vertical odd signature operator B q,v = − i ( L e − iq ) : C ∞ ( M, L ) → C ∞ ( M, L ) , see Proposition 2.3.10. Hence, we can use Remark 1.4.8 (iii) to deduce that (cid:0)b ρ A q (cid:1) [0] = 2 P ( q ) . To identify the 2-form part of the Rho form, we recall from Definition 3.2.15 that the rescaledBismut superconnection is given by B t = √ t D q,v + ∇ E ,u − √ t c v (Ω) . First, we need to identify the terms appearing here. In the same way as in Proposition 2.3.10one finds that the vertical de Rham operator associated to the connection A q is given by D q,v = − iτ v ( L e − iq ) : Ω • v ( M, L ) → Ω • v ( M, L ) . Moreover, Lemma 2.3.9 shows that the mean curvature of the fiber bundle vanishes, and thatthe connections ∇ v and e ∇ v coincide. This implies that the connection part of B coincides withthe horizontal part of the exterior differential, ∇ E ,u = d A q ,h : Ω • v ( M, L ) → Ω , • ( M, L ) . Lastly, one easily checks—again as in Proposition 2.3.10—that c v (Ω) = 2 πil vol Σ ∧ τ v : Ω • v ( M, L ) → Ω , • ( M, L ) . (3.39)We now claim that in the case at hand B t = tD q,v + c v (Ω) D q,v . (3.40)22 3. Rho Invariants of Fiber Bundles, Abstract Theory Proof of (3.40) . From (2.29) and Corollary 2.1.16 we have the anti-commutator relations (cid:8) ∇ E ,u , D q,v (cid:9) = (cid:8) ∇ E ,u , c v (Ω) (cid:9) = 0 . Moreover, the explicit formulæ above easily yield c v (Ω) D q,v = D q,v c v (Ω) = 2 πl vol Σ ∧ ( L e − iq ) . Since ∇ E ,u agrees with d A q ,h we infer from Corollary 2.1.16 that( ∇ E ,u ) = − (cid:8) d q,v , i(Ω) (cid:9) , and one verifies that in the case at hand, (cid:8) d q,v , i(Ω) (cid:9) = − πl vol Σ ∧ ( L e − iq ) = − c v (Ω) D q,v . Lastly as Σ is 2-dimensional, we have c v (Ω) = 0. Putting all pieces together, we obtain B t = tD q,v − c v (Ω) D q,v − (cid:8) d q,v , i(Ω) (cid:9) = tD q,v + c v (Ω) D q,v . Having established the formula in (3.40), we continue with the proof of Proposition 3.4.1.Since D q,v and c v (Ω) D q,v commute, one can use (3.40) and Duhamel’s formula to show thatTr v (cid:2) τ v d B t dt e − B t (cid:3) = Tr v h √ t τ v (cid:0) D q,v + c v (Ω) t (cid:1) e − tD q,v (cid:0) − c v (Ω) D q,v (cid:1)i . According to Definition 3.3.6, one thus finds that the 2-form part of b η A q is given by( b η A q ) [2] = 12 πi √ π Z ∞ Tr v h √ t τ v c v (Ω) (cid:0) − D q,v + t (cid:1) e − tD q,v i dt = l vol Σ √ π Z ∞ (cid:16) − u − / Tr v (cid:2) D q,v e − uD q,v (cid:3) + u − / Tr v (cid:2) e − uD q,v (cid:3)(cid:17) du. (3.41)Note that in the second equality we have used (3.39) to replace τ v c v (Ω) with 2 πil vol Σ , and thenmade the substitution t = 4 u . We now introduce a complex parameter s with Re( s ) ≥ b η A q ) [2] ( s ) = l vol Σ (cid:0) s +12 (cid:1) Z ∞ (cid:16) − u s − Tr v (cid:2) D q,v e − uD q,v (cid:3) + u s − Tr v (cid:2) e − uD q,v (cid:3)(cid:17) du, see Remark 3.4.2 (i) below. Now for Re( s ) large enough, we can split up the integral and computeusing the Mellin transform that1Γ (cid:0) s +12 (cid:1) Z ∞ u s − Tr v (cid:2) D q,v e − uD q,v (cid:3) du = X λ ∈ spec( D q,v ) λ − s − , and 1Γ (cid:0) s +12 (cid:1) Z ∞ u s − Tr v (cid:2) e − uD q,v (cid:3) du = X λ ∈ spec( D q,v ) (cid:0) s +12 (cid:1) Z ∞ u s − e − uλ du = X λ ∈ spec( D q,v ) Γ (cid:0) s − (cid:1) Γ (cid:0) s +12 (cid:1) λ − s − = X λ ∈ spec( D q,v ) 2 s − λ − s − . .4. Circle Bundles Revisited τ v D q,v which corresponds to two copies ofthe operator considered there. We find, again for Re( s ) large enough, that X λ ∈ spec( D q,v ) λ − s − = X λ ∈ spec( τ v D q,v ) | λ | − s = 2 X n ∈ Z n = q | n − q | − s = 2 e ζ q ( s − , where e ζ q is the Zeta function in Lemma C.1.2. Hence,( b η A q ) [2] ( s ) = l vol Σ − ss − e ζ q ( s − . (3.42)We know the value of the meromorphic continuation e ζ q ( s ) to s = − b η A q ) [2] = lP ( q ) vol Σ , so that indeed ( b ρ A q ) [2] = ( b η A q ) [2] − b η [2] = l (cid:0) P ( q ) − (cid:1) vol Σ . Remark 3.4.2. (i) We want to point out that introducing a complex parameter to split up the sum in (3.41)is necessary. For this note that the individual terms do not give functions which areholomorphic for Re( s ) ≥ 0. However—at least in the case that q / ∈ Z —their sum isholomorphic for Re( s ) ≥ 0, since Lemma C.1.2 implies that the poles and zeros of − ss − and e ζ q ( s − 1) in (3.42) cancel each other out.(ii) Even though the above computations are very similar to the ones in the proof of Proposition2.3.15, we want to point out that there is a conceptual difference. Before, we had toincorporate the operator D A q ,h and work on Ω • ( M, L ), whereas now, we work only onΩ • v ( M, L ).Using Proposition 3.4.1 we can identify the first term in the general formula of Theorem3.3.16. Since Σ is 2-dimensional, we have b L ( T Σ , ∇ Σ ) = 1 so that2 Z Σ b L ( T Σ , ∇ Σ ) ∧ b ρ A q = 2 l (cid:0) P ( q ) − (cid:1) . (3.43) Dai’s Correction Term. We now want to understand the remaining terms appearing in The-orem 3.3.16 for the example at hand. Using (3.43) it is then immediate that our second proof ofTheorem 2.3.18 will follow from the following result. Proposition 3.4.3. Let l = 0 be the degree of the principal circle bundle, and let A q be the flatconnection on L = π ∗ L A as before. Then ρ H Aq,v (Σ) = 0 , and σ A q = ( − sgn( l ) , if A q is the trivial connection , , if A q is non-trivial. 3. Rho Invariants of Fiber Bundles, Abstract Theory Proof. Recall that L ω → Σ denotes the line bundle associated to the principal bundle π : M → Σ,endowed with the connection A ω naturally induced by ω . As in (2.32) we can identifyker( D q,v ) ∼ = ( { } , if q / ∈ Z , Ω • (Σ , L B ) ⊕ (cid:0) Ω • (Σ , L B ) ⊗ C [ ω ] (cid:1) , if q ∈ Z . (3.44)Here, L B = L A ⊗ L − qω , which is endowed with the connection B = A ⊗ ⊗ qA ω . Note that inaddition to (2.32), the term Ω • (Σ , L B ) ⊗ C [ ω ] appears since we do not restrict D q,v to Ω • , ( M, L ).If q / ∈ Z , we deduce from (3.44) that the bundle of vertical cohomology groups vanishes, whichcertainly implies that η (cid:0) D Σ ⊗ ∇ H A,v (cid:1) and σ A q are both zero. If q ∈ Z , we conclude as in the proofof Proposition 2.3.16 that L B is isomorphic to the trivial line bundle and that the connection B is flat. This implies that the operator D Σ ⊗ ∇ H A,v is unitarily equivalent to several copies of atwisted odd signature operator on Σ, see Remark 3.1.17 (ii) and Lemma 3.1.19. Since Σ is evendimensional, we thus infer that η (cid:0) D Σ ⊗ ∇ H A,v (cid:1) = 0. To finish the proof, it remains to compute σ A q for q ∈ Z .First of all, we explicitly describe the E -term of the Leray-Serre spectral sequence. UsingLemma 3.1.12 we proceed as in Proposition 2.3.16 to conclude from (3.44) that for q ∈ Z , E • , A q , ∼ = H • (Σ , L B ) , E • , A q , ∼ = H • (Σ , L B ) ⊗ C [ ω ] . (3.45)Now, according to Definition 3.3.13, we have to compute the signature of the pairing P A q , = Q A, (cid:0) . , d A q , ( . ) (cid:1) : E , A q , × E , A q , → C , (3.46)see (3.38). Here, we have used that concerning the signature of P A q , we can neglect the spaces E , A q , , since the differential d A q , is of bidegree (2 , − 1) and thus zero on E , A q , , see Figure 3.1.Figure 3.1: The E -term of the spectral sequenceNow, if A q is a non-trivial connection, the line bundle L B = L A ⊗ L − qω with its naturalconnection is not isomorphic to the trivial flat line bundle. According to (3.45) this implies that E , A q , = { } , see also Proposition 2.3.16. Hence, in this case σ A q = 0. Now we assume that A q is the trivial connection, and drop the subscripts A q from the notation. Using (3.44), one checksdirectly that E , = C [ ω ] and E , = C [vol Σ ] . .4. Circle Bundles Revisited Q in (3.46) is induced by the wedge product,followed by evaluation on the fundamental class. Using this, one easily verifies that Q satisfies Q : E , × E , C , Q (cid:0) [ ω ] , [vol Σ ] (cid:1) = 1 . Moreover, since the differential d is naturally induced by exterior differentiation, one obtainsfrom Proposition 2.1.15 that d : E , → E , , d [ ω ] = [i(Ω) ω ] = − πl [vol Σ ] . According to (3.46), this means P (cid:0) [ ω ] , [ ω ] (cid:1) = Q (cid:0) [ ω ] , d [ ω ] (cid:1) = − πl. This yields that Sign( P ) = − sgn( l ). Since the spectral sequence collapses at the E -stage, thereare no higher signatures. This finishes the computation of σ A q in the case that A q is the trivialconnection.26 3. Rho Invariants of Fiber Bundles, Abstract Theory hapter 4 After having described the abstract theory leading to a general formula for Rho invariants of afiber bundle, we now want to use this to discuss a second class of examples in detail. In contrastto the previous class considered, we now reverse the role of fiber and base, and consider fiberbundles over S with fiber a closed, oriented surface.We start this chapter to describe a particular way of constructing submersion metrics. Here, itis convenient to use a formulation in terms of symplectic forms and almost complex structures. Wethen identify the geometric objects in this setting, most notably the bundle of vertical cohomologygroups and the transgression form of the Bismut superconnection.Under the assumption that the mapping torus is of finite order, we derive a formula whichexpresses the Rho invariant in terms of Hodge-de-Rham cohomology. Here, we can treat the caseof a higher dimensional gauge group and arbitrary genus of the surface fiber without effort.From then on we restrict to the case that the fiber is a 2-dimensional torus. We describe insome detail the geometric setup, including a discussion of the spectrum of the Laplace operatoron a torus twisted by a flat U(1)-connection. Using this, we obtain a formula for the Rho form.We then employ ideas related to the classical Kronecker limit formula, to cast this expressioninto a different form which is more accessible for direct computations.To obtain explicit formulæ for the Rho invariant, we now have to distinguish between the casesthat the monodromy of the mapping torus is elliptic, parabolic or hyperbolic. In the elliptic case,we can specialize the previous result about finite order mapping tori to obtain a simple formulafor the Rho invariant. The parabolic case turns out to be more involved. Most notably, the Etainvariant of the odd signature operator with values in the bundle of vertical cohomology groupshas to be analyzed carefully. Yet, we shall arrive again at a very explicit formula for U(1)-Rhoinvariants.The last part of our discussion will be concerned with the case of a hyperbolic mapping torus.Here, the difficulty lies in identifying the Rho form. Generalizing considerations by Atiyah [3],we relate this term to the logarithm of a generalized Dedekind Eta function. We will find thatthe transformation property of this logarithm under the action of the modular group determinesthe value of the Rho invariant of a hyperbolic mapping torus. Using a result due to Dieter [35],we can then express the Rho invariant as the difference of certain Dedekind sums. Simplifyingthis expression we finally arrive at a general formula for U(1)-Rho invariants in the hyperboliccase as well. 12728 4. 3-dimensional Mapping Tori Let Σ be a closed, oriented surface, and let f ∈ Diff + (Σ) be an orientation preserving diffeomor-phism of Σ. Consider the mapping torusΣ f := (cid:0) Σ × R (cid:1) / Z , (4.1)where Z acts on Σ × R via k · ( x, t ) = (cid:0) f − k ( x ) , t + k (cid:1) , ( x, t ) ∈ Σ × R , k ∈ Z . (4.2)Then Σ f is naturally the total space of a fiber bundle over S , see Appendix B.2. Moreover,according to Lemma B.2.1, the diffeomorphism type of the mapping torus depends only on theisotopy class of f , i.e., on the image of f in the mapping class groupDiff + (Σ) / Diff (Σ) . We now describe the geometric structure we need on a mapping torus in some detail. Relatedmaterial can be found in the context of Seiberg-Witten equations in [90, Sec. 8]. Moser’s Trick. The freedom of varying f in its isotopy class allows us to fix particularlyconvenient choices for the monodromy map. To exhibit such a choice we need the followingresult of [76] which is sometimes called Moser’s trick , see also [68, Sec. 3.2]. Proposition 4.1.1. Let ω ∈ Ω (Σ) be a symplectic form on the closed, oriented surface Σ , andlet f ∈ Diff + (Σ) . Then there exists e f ∈ Diff + (Σ) , isotopic to f , with e f ∗ ω = ω .Sketch of proof. We first note that since Σ is of dimension 2, a symplectic form ω on Σ is simplya 2-form which is non-degenerate in the sense that R Σ ω = 0. Moreover, as f is orientationpreserving, we have [ f ∗ ω ] = [ ω ] ∈ H (Σ , R ) . Thus, there exists α ∈ Ω (Σ) with f ∗ ω = ω + dα. We now define a time dependent vector field X : R → C ∞ (Σ , T Σ) by requiring thati( X t )( ω + tdα ) = − α, t ∈ R . This is well-defined because ω is non-degenerate. Let Φ : R → Diff + (Σ) be the flow uniquelydefined by the initial value problem ddt Φ t = X t ◦ Φ t , Φ = id Σ . Then one checks using Cartan’s formula that ddt Φ ∗ t ( ω + tdα ) = Φ ∗ t dα + Φ ∗ t (cid:0) L X t ( ω + tdα ) (cid:1) = Φ ∗ t dα + Φ ∗ t (cid:0) d ◦ i( X t )( ω + tdα ) (cid:1) = Φ ∗ t dα − Φ ∗ t dα = 0 , .1. Geometric Preliminaries X t in the last line. Hence, Φ ∗ t ( ω + tdα ) is independent of t and so ω = Φ ∗ ω = Φ ∗ ( ω + dα ) = Φ ∗ ( f ∗ ω ) . Now the claim follows with e f := f ◦ Φ . Metrics on Mapping Tori. We can use Proposition 4.1.1 to define particular Riemannianmetrics on a mapping torus Σ f . As remarked before, a symplectic form ω ∈ Ω (Σ) is the sameas a volume form on Σ. The space of metrics on Σ with ω as a volume form has the followingdescription, see e.g. [68, Sec. 4.1].Recall that an almost complex structure is an endomorphism J ∈ C ∞ (cid:0) Σ , End( T Σ) (cid:1) with J = − Id. It is called ω -compatible if for all v, w ∈ T Σ with v = 0, ω ( J v, J w ) = ω ( v, w ) , and ω ( v, J v ) > . Then we get a Riemannian metric with volume form ω by letting g J ( v, w ) := ω ( v, J w ) , v, w ∈ T Σ . (4.3)Moreover, the space of metrics with volume form ω is naturally isomorphic to J ω := (cid:8) J ∈ C ∞ (cid:0) Σ , End( T Σ) (cid:1) (cid:12)(cid:12) J is an ω -compatible almost complex structure (cid:9) . Given a mapping torus Σ f , we now fix a symplectic form ω ∈ Ω (Σ) of unit volume. InvokingProposition 4.1.1 and Lemma B.2.1, we may assume that f ∗ ω = ω . Note that this implies that J ω is invariant under conjugation with f ∗ . Since J ω is easily seen to be path connected—infact, even contractible—we can choose a path J t : R → J ω of ω -compatible almost complexstructures on Σ satisfying J t +1 = f − ∗ ◦ J t ◦ f ∗ . Let g t be the path of Riemannian metrics defined by J t and ω as in (4.3), and define a metric onΣ × R by g := dt ⊗ dt + g t . (4.4)It is immediate from the fact that f ∗ ω = ω and the convention (4.2) of how to define the map-ping torus as a quotient of Σ × R that g descends to a Riemannian metric on the mapping torus Σ f . Calculus on Mapping Tori. We now want to identify the various quantities introduced inSection 2.1. Since the base of the fiber bundle is 1-dimensional, the curvature form Ω vanishes.Clearly, each vertical vector field on Σ f is induced by a path V : R → C ∞ (Σ , T Σ) , V t +1 = f ∗ V t , (4.5)and each horizontal vector field X can be identified with X = ϕ t ∂ t , with ϕ : R → C ∞ (Σ) , ϕ t +1 = ϕ t ◦ f. Lemma 4.1.2. Let U and V be vertical vector fields on Σ . With respect to a metric g as in (4.4) , the Levi-Civita connection ∇ g on Σ f is given by ∇ g∂ t V (cid:12)(cid:12) t = ∂ t V t + ˙ J t J t V t , ∇ gU V (cid:12)(cid:12) t = ∇ g t U t V t − ω ( U t , ˙ J t V t ) ∂ t , ∇ g∂ t ∂ t = 0 , (4.6)30 4. 3-dimensional Mapping Tori where ∇ g t is the Levi-Civita connection on Σ with respect to the metric g t and ˙ J t = [ ∂ t , J t ] .Moreover, the natural vertical connection ∇ v is given by ∇ vU V (cid:12)(cid:12) t = ∇ g t U t V t , ∇ v∂ t V (cid:12)(cid:12) t = ∂ t V t + ˙ J t J t V t . (4.7) In particular, the difference tensor S of Definition (2.1.6) is given by S ( U, V ) (cid:12)(cid:12) t = − ω ( U t , ˙ J t V t ) ∂ t , and the mean curvature form k v vanishes.Sketch of proof. The description of the Levi-Civita connection easily follows from the explicitformula, see e.g. [13, Sec. 1.2],2 g (cid:0) ∇ gX Y, Z (cid:1) = g (cid:0) [ X, Y ] , Z (cid:1) − g (cid:0) [ Y, Z ] , X (cid:1) + g (cid:0) [ Z, X ] , Y (cid:1) + Xg ( Y, Z ) + Y g ( Z, X ) − Zg ( X, Y ) . For example, 2 g (cid:0) ∇ g∂ t V, U (cid:1) = g (cid:0) [ ∂ t , V ] , U (cid:1) + g (cid:0) [ U, ∂ t ] , V (cid:1) + ∂ t g ( V, U )= g (cid:0) ∂ t V, U (cid:1) − g (cid:0) V, ∂ t U (cid:1) + ∂ t ω ( V, J U )= 2 g (cid:0) ∂ t V, U (cid:1) + ω ( V, ˙ J U )= 2 g (cid:0) ∂ t V, U (cid:1) + g ( J V, ˙ J U ) = 2 g (cid:0) ∂ t V + ˙ J J V, U (cid:1) , where we have used that ˙ J t is self-adjoint with respect to g t for each t . The second equationin (4.6) is proven similarly, while the third is clear. Then (4.7) follows by taking the verticalprojection of ∇ g . Taking differences yields the formula for S . If { e , e } is a vertical localorthonormal frame, Lemma 2.1.8 implies that k v ( ∂ t ) = g (cid:0) ˙ J J e , e (cid:1) + g (cid:0) ˙ J J e , e (cid:1) , which is zero since ˙ J t J t is easily seen to be skew-adjoint with respect to g t . Flat Connections and the Bundle of Vertical Cohomology Groups. In Appendix B.2 wehave included a detailed description of the moduli space of flat U( k )-connections over mappingtori. According to Proposition B.2.12, a flat Hermitian vector bundle over Σ f is given—up toisomorphism—by a pair ( a, u ), where u ∈ C ∞ (cid:0) Σ , U( k ) (cid:1) is a gauge transformation, and a is a flatU( k )-connection over Σ satisfying a = u − ( f ∗ a ) u + u − du. (4.8)We briefly recall from Section B.2 how this data defines a flat vector bundle over Σ f . First, let b f u : Σ × C k → Σ × C k , b f u ( x, z ) = (cid:0) f ( x ) , u ( x ) z (cid:1) , be the automorphism of the trivial bundle over Σ defined by u , see Remark B.2.5. Then we definea vector bundle E u → Σ f as the mapping torus (cid:0) (Σ × C k ) × R (cid:1) / ∼ , (cid:0) ( x, z ) , t + 1 (cid:1) ∼ (cid:0) b f u ( x, z ) , t (cid:1) . Viewing a as a constant path of Lie algebra valued 1-form, a ∈ C ∞ (cid:0) R , Ω (Σ , u ( k ) (cid:1) , condition(4.8) ensures that we can define a connection A on E u . Since a is flat, the same is true for A , see(B.23) and (B.24). .1. Geometric Preliminaries Lemma 4.1.3. Let u ∈ C ∞ (cid:0) Σ , U( k ) (cid:1) be a gauge transformation defining a bundle E u → Σ f ,and let A be a flat U( k ) -connection over T M defined by a pair ( a, u ) satisfying (4.8) . (i) The bundle automorphism b f u : Σ × C k → Σ × C k induces an isomorphism b f ∗ u : H • (Σ , E a ) → H • (Σ , E a ) . (ii) Let ∇ H A,v be the natural flat connection on the bundle H • A,v (Σ f ) → S of vertical cohomol-ogy groups, see Definition 3.1.10. Then its holonomy representation is given by hol ∇ HA,v : π ( S ) → GL (cid:0) H • (Σ , E a ) (cid:1) , hol ∇ HA,v ( γ ) = b f ∗ u , where γ ∈ π ( S ) is the canonical generator.Proof. The map b f u acts on Ω • (Σ , C k ) via b f ∗ u α = u − f ∗ α, α ∈ Ω • (Σ , C k ) . Condition (4.8) is easily seen to be equivalent to b f ∗ u ◦ d a = d a ◦ b f ∗ u . This implies that b f ∗ u descendsto an isomorphism on cohomology, which proves part (i). As in (B.22), we have the followingidentification of the space of vertical, E u -valued differential formsΩ • v (Σ f , E u ) = (cid:8) α t : R → Ω • (Σ , C k ) (cid:12)(cid:12) α t +1 = b f ∗ u α t (cid:9) . (4.9)With respect to this identification, the vertical differential d A,v coincides with d a applied pointwisefor each t . Moreover, since the connection A has no dt component, the horizontal differential d A,h : Ω • v (Σ f , E u ) → Ω , • (Σ f , E u )is given with respect to (4.9) by d A,h α t = dt ∧ ∂ t α t . From these observations one finds that the space of sections of H • A,v (Σ f ) → S can be describedas C ∞ (cid:0) S , H A,v (Σ f ) (cid:1) = (cid:8) [ α ] t : R → H • (Σ , E a ) (cid:12)(cid:12) [ α ] t +1 = f ∗ u [ α ] t (cid:9) . Now it is immediate from the definition of the connection ∇ H A,v that the holonomy representationhas the claimed form. The Bismut Superconnection. To identify the terms appearing in Dai’s adiabatic limitformula, we first need to understand the connection ∇ A,v acting on forms, and then the Bismutsuperconnection in the setting at hand. Lemma 4.1.4. With respect to the identification (4.9) , the connection ∇ A,v acting on Ω • v (Σ f , E u ) is given by ∇ A,v∂ t = ∂ t − ˙ τ v τ v , ∇ A,vV (cid:12)(cid:12) t = ∇ a,g t V t , where V is a vertical vector field, and ∇ a,g t denotes the connection on Ω • (Σ , C k ) induced by g t and a , acting pointwise for each t . 4. 3-dimensional Mapping Tori Proof. The assertion about ∇ A,vV is true by definition of ∇ A,v and the fact that A is independentof t . Concerning ∇ A,v∂ t , we observe that d A,h = dt ∧ ∂ t , which implies that e ∇ A,v∂ t = ∂ t . (4.10)Since the mean curvature form vanishes, we can use (2.9) and (2.15) to deduce that ∇ A,v∂ t = ∂ t + τ v [ ∂ t , τ v ] = ∂ t − ˙ τ v τ v , where we have used that ˙ τ v = [ ∂ t , τ v ] and ˙ τ v τ v = − τ v ˙ τ v .Using Lemma 4.1.4 we can now find a more explicit expression for the Bismut superconnectionand its transgression form. Proposition 4.1.5. The rescaled Bismut superconnection associated to A is given for s ∈ (0 , ∞ ) by B s = √ s D A,v + dt ∧ (cid:0) ∂ t − ˙ τ v τ v (cid:1) : Ω • v (Σ f , E u ) → Ω • (Σ f , E u ) , where D A,v is the vertical de Rham operator. Moreover, α ( B s ) = i π dt ∧ Tr v (cid:16) ˙ τ v (cid:0) d tA,v d A,v − d A,v d tA,v (cid:1) e − s D A,v (cid:17) ∈ Ω ( S ) . Proof. The formula for B s follows immediately from Lemma 4.1.4 and the fact that k v and Ω arezero, see (3.20) and Definition 3.2.15. For the second assertion, we drop the connection A fromthe notation. Since the base is 1-dimensional, and dt anti-commutes with D v , one finds that B s (cid:12)(cid:12) s =4 r = rD v + √ rdt ∧ (cid:2) ∇ v∂ t , D v (cid:3) . As in [13, Lem. 9.42] and [18, Thm. 3.3] an application of Duhamel’s formula then yields that e − rD v −√ rdt ∧ [ ∇ v∂t ,D v ] = e − rD v − √ r dt ∧ Z e − r ′ rD v (cid:2) ∇ v∂ t , D v (cid:3) e − (1 − r ′ ) rD v dr ′ . where the higher correction terms vanish, again since the base is 1-dimensional. Therefore,Str v (cid:16) d B s ds e − B s (cid:17)(cid:12)(cid:12)(cid:12) s =4 r = √ r Str v (cid:16) D v e − rD v −√ rdt ∧ [ ∇ v∂t ,D v ] (cid:17) = √ r Str v (cid:0) D v e − rD v (cid:1) + dt ∧ Str v (cid:16) D v [ ∇ v∂ t , D v ] e − rD v (cid:17) . Note that a factor of − dt and D v . Since τ v anti-commutes with D v , the first term vanishes. To simplify the second term, wealso drop the sub- and subscripts v from the notation. Then one verifies using Lemma 4.1.4 andthe relations D = d − τ dτ , [ ∂ t , τ ] = ˙ τ as well as ˙ τ τ = − τ ˙ τ that τ [ ∇ ∂ t , D ] = − (cid:0) τ [ ˙ τ τ, d ] − [ ˙ τ τ, d ] τ (cid:1) = (cid:0) ˙ τ d + τ d ˙ τ τ + ˙ τ τ dτ − d ˙ τ (cid:1) Now, since τ D = − Dτ , we can use the trace property and the fact that e − rD is a semi-group ofsmoothing operators to find thatTr (cid:0) τ d ˙ τ τ De − rD (cid:1) = − Tr (cid:0) d ˙ τ De − rD (cid:1) , Tr (cid:0) ˙ τ τ dτ De − rD (cid:1) = Tr (cid:0) ˙ τ dDe − rD (cid:1) . .2. Finite Order Mapping Tori (cid:0) D [ ∇ ∂ t , D ] e − rD (cid:1) = Str (cid:0) [ ∇ ∂ t , D ] De − rD (cid:1) = − Tr (cid:0) ( d ˙ τ − ˙ τ d ) De − rD (cid:1) = − Tr (cid:0) ˙ τ ( d t d − dd t ) e − rD (cid:1) . Recalling the normalization factor in Definition 3.2.17, we arrive at the claimed formula for thetransgression form. Proposition 4.1.5 shows that if we can achieve that ˙ τ v ≡ 0, then the Bismut-Cheeger Eta formassociated to a flat connection over Σ f vanishes. From the discussion in Section 4.1 we knowthat ˙ τ v ≡ f -invariant metric g Σ on Σ. Clearly, such a metric willnot exist for arbitrary f ∈ Diff + (Σ). Lemma 4.2.1. If f ∈ Diff + (Σ) is of finite order n , there exists a metric g Σ of unit volume with f ∗ g Σ = g Σ .Proof. Choose an arbitrary metric g Σ on Σ of unit volume and define e g Σ := n n − X j =0 ( f j ) ∗ g Σ . Then e g Σ is again a metric of unit volume. Moreover, since f n = id Σ , one finds that indeed f ∗ e g Σ = e g Σ . Remark 4.2.2. (i) A metric g Σ defines an almost complex structure on Σ which is integrable, see SectionB.3. If g Σ is f -invariant for some f ∈ Diff + (Σ), then f is holomorphic with respect to thecomplex structure defined by g Σ .(ii) It can be shown that if there exists an f -invariant metric g Σ , then the mapping class[ f ] ∈ Diff + (Σ) / Diff (Σ) is necessarily of finite order. If the genus of Σ is 0 or 1, this can bechecked directly, see Proposition 4.4.3 below for Σ = T . For higher genera, one can use forexample the holomorphic description, and invoke the Riemann-Hurwitz formula, see [40,Ch. V]. Rho Invariants of a Finite Order Mapping Torus. Assume from now on that f ∈ Diff + (Σ)is of finite order, and that an f -invariant metric g Σ has been chosen. We also fix a flat connection A on a Hermitian vector bundle E u → Σ f defined by a pair ( a, u ) satisfying b f ∗ u a = a as in (4.8).In Lemma 4.1.3 we have given a description of the bundle of vertical cohomology groups in termsof de Rham cohomology. However, the Rho invariant of the bundle of vertical cohomology groupsin Definition 3.3.15—which appears in the formula for the Rho invariant in Theorem 3.3.16—isdefined using the Hodge theoretic description. In the case of a finite order mapping torus, wehave the following extension of Lemma 4.1.3. Lemma 4.2.3. 4. 3-dimensional Mapping Tori (i) With respect to the induced metric on H • (Σ , E a ) , the bundle map b f u defines an isometry b f ∗ u : H • (Σ , E a ) → H • (Σ , E a ) , H • (Σ , E a ) = ker( d a + d ta ) ⊂ Ω • (Σ , C k ) . The splitting into ± -eigenspaces of τ Σ , H • (Σ , E a ) = H + (Σ , E a ) ⊕ H − (Σ , E a ) , is invariant with respect to b f ∗ u . (ii) The flat connection ∇ H A,v on the bundle of vertical cohomology groups is compatible withthe metric h ., . i H A,v of Definition 3.1.14, and its holonomy representation is given by hol ∇ HA,v ( γ ) = b f ∗ u ∈ GL (cid:0) H • (Σ , E a ) (cid:1) , where γ ∈ π ( S ) is the canonical generator.Proof. Since f is an isometry with respect to the metric g Σ , the pullback f ∗ : Ω • (Σ) → Ω • (Σ)commutes with the chirality operator τ Σ . Moreover, as b f ∗ u = u − f ∗ , the same is true for b f ∗ u .This, and the fact that f ∗ u ◦ d a = d a ◦ b f ∗ u implies part (i). Since ˙ τ ≡ 0, we can deduce from (2.15)and Proposition 3.1.15 that the connection ∇ H A,v is indeed compatible with the metric h ., . i H A,v .Moreover, if Ψ : H • (Σ , E a ) → H • (Σ , E a )denotes the Hodge-de-Rham isomorphism, the diagram H • (Σ , E a ) b f ∗ u −−−−→ H • (Σ , E a ) Ψ y Ψ y H • (Σ , E a ) b f ∗ u −−−−→ H • (Σ , E a ) , is commutative. This is because it is naturally induced by b f ∗ u : Ω • (Σ , C k ) → Ω • (Σ , C k ). Hence,the description of the holonomy representation follows from Lemma 4.1.3. Theorem 4.2.4. Let f ∈ Diff + (Σ) be of finite order. Let A be a flat U( k ) -connection over Σ f ,defined by a pair ( a, u ) of flat connection and gauge transformation over Σ satisfying b f ∗ u a = a .Then with respect to every f -invariant metric g Σ on Σ ρ A (Σ f ) =2 tr log (cid:2) b f ∗ u | H + (Σ ,E a ) ∩ Ω (cid:3) − rk (cid:2) ( b f ∗ u − Id) | H + (Σ ,E a ) ∩ Ω (cid:3) − (cid:2) b f ∗ u | H − (Σ ,E a ) ∩ Ω (cid:3) + rk (cid:2) ( b f ∗ u − Id) | H − (Σ ,E a ) ∩ Ω (cid:3) − k tr log (cid:2) f ∗ | H + (Σ) ∩ Ω (cid:3) + 2 k rk (cid:2) ( f ∗ − Id) | H + (Σ) ∩ Ω (cid:3) . Here, “ tr log ” is defined for a unitary map T ∈ U( n ) as tr log T := n X j =1 θ j ∈ R , where e πiθ j are the eigenvalues of T , and where we require θ j ∈ [0 , . .2. Finite Order Mapping Tori Proof. As we have already noted before, Proposition 4.1.5 implies that the Bismut-Cheeger Rhoform vanishes, because g Σ is f -invariant. Moreover, the Leray-Serre spectral sequence asso-ciated to A —respectively the trivial connection—collapses at the E -stage, since the base is1-dimensional, see Theorem 3.1.11. This implies that Dai’s correction term in Definition 3.3.13vanishes. Hence, Theorem 3.3.16 yields ρ A (Σ f ) = ρ H A,v ( S ) = η (cid:0) D S ⊗ ∇ H A,v (cid:1) − k · η (cid:0) D S ⊗ ∇ H v (cid:1) , (4.11)where D S ⊗ ∇ H A,v and D S ⊗ ∇ H v are as in Definition 3.1.16. We have seen in Lemma 4.2.3that the flat connection ∇ H v is unitary with respect to the metric h ., . i H v . Hence, we deducefrom Lemma 3.1.19 that D S ⊗ ∇ H v = (cid:18) D S ⊗ ∇ H v , + − D S ⊗ ∇ H v , − (cid:19) , (4.12)where D S ⊗ ∇ H v , ± = τ S d ∇ H v, ± + d ∇ H v, ± τ S , (4.13)and ∇ H v , ± denotes the restriction of ∇ H v to H ± v (Σ f ). The same formula holds for H • v (Σ f )replaced with H • A,v (Σ f ). Let us abbreviate B ± := τ S d ∇ H v, ± : C ∞ (cid:0) S , H ± v (Σ f ) (cid:1) → C ∞ (cid:0) S , H ± v (Σ f ) (cid:1) , (4.14)and define B A, ± correspondingly. Then (4.11) and (4.12) show that ρ A (Σ f ) = η ( B A, + ) − η ( B A, − ) − k · η ( B + ) + k · η ( B − ) . (4.15)Note that the factors disappear as the operators in (4.13) are equivalent to two copies of theoperators in (4.14), see Remark 3.1.17 (ii). We deduce from Lemma 4.2.3 (i) that b f ∗ u restricts toa unitary map on H ± (Σ , E a ). Hence, we can find a basis of H ± (Σ , E a ) such that b f ∗ u | H ± (Σ ,E a ) = diag (cid:0) e πiθ ± , . . . , e πiθ ± n (cid:1) , where θ ± j ∈ [0 , n = dim H + (Σ , E a ) = dim H − (Σ , E a ). Note that the equality ofdimensions follows from the fact that Sign a (Σ) = 0, which in turn is a consequence of the signatureformula, see Theorem 1.2.23. Now Lemma 4.2.3 (ii) yields that the restriction b f ∗ u | H ± (Σ ,E a ) definesthe holonomy representation of the connection ∇ H A,v , ± . From this we deduce as in Remark 1.4.8(iii) that η ( B A, + ) − η ( B A, − ) = X θ + j =0 (2 θ + j − − X θ − j =0 (2 θ − j − (cid:2) b f ∗ u | H + (cid:3) − rk (cid:2) ( b f ∗ u − Id) | H + (cid:3) − (cid:2) b f ∗ u | H − (cid:3) + rk (cid:2) ( b f ∗ u − Id) | H − (cid:3) , where for convenience we have abbreviated H ± = H ± (Σ , E a ) in the last equality. Continuingwith obvious abbreviations, we now decompose H ± = H ± ∩ (Ω ⊕ Ω ) ⊕ H ± ∩ Ω . 4. 3-dimensional Mapping Tori Each element of H ± ∩ (cid:0) Ω ⊕ Ω (cid:1) is of the form ϕ ± τ Σ ϕ with ϕ ∈ H . Hence, we have a naturalisomorphism (compare also with the proof of Proposition 1.1.8), H + ∩ (cid:0) Ω ⊕ Ω (cid:1) ∼ = −→ H − ∩ (cid:0) Ω ⊕ Ω (cid:1) , ϕ + τ Σ ϕ ϕ − τ Σ ϕ. Since b f ∗ u commutes with τ Σ , we can conclude that2 tr log (cid:2) b f ∗ u | H + ∩ (Ω ⊕ Ω ) (cid:3) − rk (cid:2) ( b f ∗ u − Id) | H + ∩ (Ω ⊕ Ω ) (cid:3) = 2 tr log (cid:2) b f ∗ u | H − ∩ (Ω ⊕ Ω ) (cid:3) − rk (cid:2) ( b f ∗ u − Id) | H − ∩ (Ω ⊕ Ω ) (cid:3) . Therefore, η ( B A, + ) − η ( B A, − ) =2 tr log (cid:2) b f ∗ u | H + ∩ Ω (cid:3) − rk (cid:2) ( b f ∗ u − Id) | H + ∩ Ω (cid:3) − (cid:2) b f ∗ u | H − ∩ Ω (cid:3) + rk (cid:2) ( b f ∗ u − Id) | H − ∩ Ω (cid:3) . This identifies the twisted terms appearing in the formula of Theorem 4.2.4. In the case that a isthe trivial connection and u ≡ 1, we can simplify this further. Since we are considering complexvalued forms, we have a conjugation H (Σ) → H (Σ) , α ¯ α. The chirality operator τ Σ is readily seen to anti-commute with conjugation. This yields an anti-linear isomorphism H + (Σ) ∩ Ω (Σ) ∼ = −→ H − (Σ) ∩ Ω (Σ) . Since f ∗ is the complex linear extension of a real automorphism, it commutes with conjugation.From this one readily deduces that the eigenvalues of f ∗ | H − ∩ Ω are complex conjugate to theeigenvalues of f ∗ | H + ∩ Ω . By checking the definition of “tr log” carefully, one conlcudestr log (cid:2) f ∗ | H − ∩ Ω (cid:3) = rk (cid:2) ( f ∗ − Id) | H + ∩ Ω (cid:3) − tr log (cid:2) f ∗ | H + ∩ Ω (cid:3) . Now, as rk (cid:2) ( f ∗ − Id) | H − ∩ Ω (cid:3) = rk (cid:2) ( f ∗ − Id) | H + ∩ Ω (cid:3) , we finally get η ( B + ) − η ( B − ) = 4 tr log (cid:2) f ∗ | H + ∩ Ω (cid:3) − (cid:2) ( f ∗ − Id) | H + ∩ Ω (cid:3) , which is precisely the untwisted term in the claimed formula. Remark. (i) The last step of the above proof is essentially equivalent to observing that f ∗ acting on H (Σ) is the complexification of a symplectic map. This explains why the eigenvaluescome in conjugate pairs. Clearly, this is no longer true if we consider u − f ∗ with u ∈ U(1),which also defines a flat connection over Σ f , see also Theorem 4.4.4 below.(ii) In a similar direction, if the connection a is non-trivial, complex conjugation gives rise toan anti-linear isomorphism H + (Σ , E a ) ∼ = −→ H − (Σ , E ¯ a ) . From this, one can relate the eigenvalues of b f ∗ u acting on H + (Σ , E a ) with the eigenvaluesof b f ∗ ¯ u acting on H − (Σ , E ¯ a ). However, this only simplifies the formula of Theorem 4.2.4 inthe case that a and u are real, in the sense that they arise from an O( k )-structure. .3. Torus Bundles over S , General Setup f in terms of Hodge-de-Rhamcohomology of Σ, it is only an intermediate step to an expression in completely topologicalterms. The next step would be to use the ideas of the Atiyah-Bott fixed point formula—see[13, Sec. 6.2]—to relate the traces appearing in Theorem 4.2.4 to the fixed point data of f .This, in turn, can be expressed in terms of the Seifert invariants of the finite order mappingtorus. We refer to [2, Sec. 5] and [43, Sec. 2.2] for a discussion of these ideas in the contextof the determinant line bundle over the moduli space of flat connections associated to afinite order mapping torus.(iv) In [75] an interpretation of the untwisted Eta invariant for finite order mapping tori is givenin terms of Meyer’s cocycle for the mapping class group, see [72]. It follows from the proofof Theorem 4.2.4 that the adiabatic limit of the untwisted Eta invariant is given by4 tr log (cid:2) f ∗ | H + (Σ) ∩ Ω (Σ) (cid:3) − (cid:2) ( f ∗ − Id) | H + (Σ) ∩ Ω (Σ) (cid:3) . (4.16)It would be interesting to relate this to the main result of [75]. S , General Setup We now consider the case that Σ = T is the 2-dimensional torus. In the same setting, Atiyah[3] studies a rich interplay between the untwisted Eta invariant and other topological invariants,the Dedekind Eta function and also number theoretical L -series. As a tool Atiyah also makesintensive use of the idea of adiabatic limits, and much of our discussion is influenced by thetreatment in [3]. We shall restrict to the case of U(1)-connections, which already containsmany important ideas. However, in view of the computations of Chern-Simons invariants fortorus bundles in [54, 57] the generalization to higher gauge groups would be extremely interesting. S Complex Structures on T . We fix the standard torus T = R / Z , endowed with the volumeform induced by ω = dx ∧ dy . As in Section 4.1 we are interested in the space of all metrics whichhave ω as a volume form. Equivalently, we need to understand the space J ω of all ω -compatiblealmost complex structures. It is well known that J ω is the Teichm¨uller space of T , i.e., theupper half plane H := (cid:8) σ = σ + iσ ∈ C (cid:12)(cid:12) v > (cid:9) , see [55, Thm. 2.7.2]. For definiteness, we will use the following explicit isomorphism. Note thateach almost complex structure J ∈ J ω can be identified with a matrix in M ( R ) because thetangent space of T is canonically isomorphic to R . Lemma 4.3.1. The map Φ : H → J ω , Φ( σ ) = 1 σ (cid:18) − σ −| σ | σ (cid:19) , σ = σ + iσ , We are using the letter σ for elements in H rather than the more common letter τ to avoid confusion with thechirality operator. 4. 3-dimensional Mapping Tori is a bijection. The metric on T defined by Φ( σ ) as in (4.3) is given with respect to the standardcoordinate basis as g σ = 1 σ (cid:16) dx ⊗ dx + σ ( dx ⊗ dy + dy ⊗ dx ) + | σ | dy ⊗ dy (cid:17) . Proof. If we identify J ∈ J ω with an element in M ( R ), one easily checks that J = − Id isequivalent to det( J ) = 1 and tr( J ) = 0. Hence there exist r, s, t ∈ R such that J = (cid:18) − r ts r (cid:19) , r + st = − . (4.17)Let J be the almost complex structure which induces the standard scalar product on R , i.e., J := (cid:18) − 11 0 (cid:19) , g = ω ( ., J . ) = dx ⊗ dx + dy ⊗ dy. Then one verifies that J is ω -compatible if and only if − J J is positive definite and symmetric.Now − J J = (cid:18) s rr − t (cid:19) , (4.18)and so (4.17) implies that − J J is positive definite if and only if s > 0. Using this one finds thatΦ is well-defined with inverse given byΦ − ( J ) = 1 s ( r + i ) . Moreover, (4.18) relates Φ( σ ) and the associated metric g σ and easily yields the second claim. Remark 4.3.2. Viewed from a complex analytic perspective, the above identification mightseem a bit cumbersome. As in [55, Sec. 2.7], any σ ∈ H defines a latticeΛ( σ ) := (cid:8) m + nσ (cid:12)(cid:12) ( m, n ) ∈ Z (cid:9) ⊂ C , and the quotient torus C / Λ( σ ) is naturally a complex manifold, with complex structure inducedby the one of the complex plane, and metric induced by σ ( dx + dy ). Note that one has todivide by σ to get a metric of unit volume. It is easy to check that ψ σ : R → C , ( x, y ) x + σy, descends to an isometry ψ σ : ( T , g σ ) → C / Λ( σ ) , (4.19)where the metric g σ is defined as in Lemma 4.3.1. The complex analytic description is bettersuited for explicit computations if σ is fixed. However, if σ varies the underlying manifold variesas well. This is sometimes inconvenient in the study of families. In the following, we will useboth descriptions. To avoid confusion we will reserve T for the standard 2-torus and use thenotation C / Λ( σ ) whenever we prefer to think in complex analytic terms. .3. Torus Bundles over S , General Setup C / Λ( σ ) has the advantage that the metric is up to a constant factor induced bythe standard metric. In particular, we will consider the 1-forms dz = dx + idy and d ¯ z = dx − idy .Note that (cid:0) dz , d ¯ z (cid:1) L = 0 , k dz k L = k d ¯ z k L = √ σ , and, with the chirality operator τ , τ dz = dz, τ d ¯ z = − d ¯ z, and τ ( dz ∧ d ¯ z ) = − σ . Using the isometry ψ σ of (4.19), one translates this easily to ( T , g σ ). More precisely, we define ω σ := ψ ∗ σ ( dz ) = dx + σdy, and ω ¯ σ := ψ ∗ σ ( d ¯ z ) = dx + ¯ σdy. (4.20)Then we have a natural basis ( ω σ , ω ¯ σ ) for the C ∞ ( T )-module Ω ( T ) satisfying( ω σ , ω ¯ σ ) L = 0 , k ω σ k L = k ω ¯ σ k L = √ σ ,τ ω σ = ω σ , and τ ω ¯ σ = − ω ¯ σ . (4.21) Flat Connections and Dolbeault Operators. The moduli space of flat U(1)-connectionsover T has a simple structure. Since T is the quotient of R by the standard lattice Z , wehave π ( T ) ∼ = Z e ⊕ Z e ⊂ R , where ( e , e ) is the standard basis of R . Then, since U(1) is abelian, M (cid:0) T , U(1) (cid:1) ∼ = Hom (cid:0) π ( T ) , U(1) (cid:1) ∼ = U(1) × U(1) . As we are usually working with connections rather than representations of the fundamental group,we summarize how the above isomorphism works explicitly. Lemma 4.3.3. (i) Up to gauge equivalence, a flat U(1) -connection over T is induced by a Z -invariant 1-formwith constant coefficients, a ν = − πi ( ν dx + ν dy ) ∈ Ω ( R , i R ) , ν = ( ν , ν ) ∈ R . (4.22) Two connections a ν and a ν ′ are gauge equivalent if and only if ν − ν ′ ∈ Z . (ii) In terms of the generators e , e of π ( T ) , the holonomy representation of a ν is given by hol a ν : π ( T ) → U(1) , hol a ν ( e j ) = e πiν j . Proof. A flat connection a over T is a Z -invariant 1-form, satisfying da = 0. Therefore, it givesan element in de Rham cohomology. Since H ( T , R ) = R , we can find a Z -invariant function f : R → R and ν = ( ν , ν ) ∈ R such that a − idf = − πi ( ν dx + ν dy ) . 4. 3-dimensional Mapping Tori Defining u := exp( if ), we get a gauge transformation on T which brings a into the claimed form.If a ν and a ν ′ are gauge equivalent, there exists a Z -invariant function u : R → U(1) satisfying2 πi (cid:0) ( ν − ν ′ ) dx + ( ν − ν ′ ) dy ) (cid:1) = u − du. This easily implies that u is of the form u = C · exp (cid:0) πi (cid:10) ν − ν ′ , ( xy ) (cid:11)(cid:1) , C ∈ U(1) , and this is Z -invariant precisely if ν − ν ′ ∈ Z . This proves part (i). Concerning (ii), we computeusing Definition B.1.2 hol a ν ( e j ) = exp (cid:16) − Z a ν (cid:12)(cid:12) se i ( e i ) ds (cid:17) = exp (cid:0) πiν i (cid:1) . As pointed out in Remark 4.3.2 it is often convenient to work on C / Λ( σ ) rather than T . Wecollect the following formulæ, for definitions see Appendix B.3, in particular (B.28). Proposition 4.3.4. Let a ν be a flat U(1) -connection over T as in Lemma 4.3.3, and let σ = σ + iσ ∈ H . (i) The pullback of a ν to C / Λ( σ ) is given by a = ( ψ − σ ) ∗ a ν = − ¯ w ν dz + w ν d ¯ z, where w ν := πσ ( ν − σν ) . (ii) Let ∂ a and ¯ ∂ a be the Dolbeault operators associated to a . Then the twisted Laplace operatoron C ∞ (cid:0) C / Λ( σ ) (cid:1) is given by ∆ a = 2 ∂ ta ∂ a = 2 ¯ ∂ ta ¯ ∂ a = − σ (cid:16) ∂ ∂z∂ ¯ z − ¯ w ν ∂∂ ¯ z + w ν ∂∂z − | w ν | (cid:17) . (iii) If ϕ ∈ C ∞ ( C ) is Λ( σ ) -invariant, then ∂ a ∂ ta ( ϕdz ) = ¯ ∂ ta ¯ ∂ a ( ϕdz ) = ( ∆ a ϕ ) dz, ¯ ∂ a ¯ ∂ ta ( ϕd ¯ z ) = ∂ ta ∂ a ( ϕd ¯ z ) = ( ∆ a ϕ ) d ¯ z, and ¯ ∂ a ∂ ta ( ϕdz ) = − ∂ ta ¯ ∂ a ( ϕdz ) = − σ (cid:16) ∂ ϕ∂ ¯ z + 2 w ν ∂ϕ∂ ¯ z + w ν ϕ (cid:17) d ¯ z,∂ a ¯ ∂ ta ( ϕd ¯ z ) = − ¯ ∂ ta ∂ a ( ϕd ¯ z ) = − σ (cid:16) ∂ ϕ∂z − w ν ∂ϕ∂z + ¯ w ν ϕ (cid:17) dz. In particular, ∆ a ( ϕdz ) = (∆ a ϕ ) dz and ∆ a ( ϕd ¯ z ) = (∆ a ϕ ) d ¯ z. Sketch of proof. Although Proposition 4.3.4 is a standard exercise in complex analysis, we wantto give some remarks on the proof to clarify the sign conventions we are using. For part (i), weuse the 1-forms ω σ and ω ¯ σ of (4.20) to express dy = iσ ( ω σ − ω ¯ σ ) , dx = iσ ( σω ¯ σ − ¯ σω σ ) . .3. Torus Bundles over S , General Setup ω σ = ψ ∗ σ ( dz ) and ω ¯ σ = ψ ∗ σ ( d ¯ z ) we deduce that( ψ − σ ) ∗ a ν = − πσ (cid:0) ν ( σd ¯ z − ¯ σdz ) + ν ( dz − d ¯ z ) (cid:1) = − ¯ w ν dz + w ν d ¯ z, with w ν = πσ ( ν − σν ) as claimed. For part (ii) and (iii) we note that ∂ a = ∂ − e( ¯ w ν dz ) , ¯ ∂ a = ¯ ∂ + e( w ν d ¯ z ) , and, since we are using the complex linear chirality operator, ∂ ta = − τ ◦ ¯ ∂ a ◦ τ, ¯ ∂ ta = − τ ◦ ∂ a ◦ τ. Then one computes that for every ϕ ∈ C ∞ ( C ), ∂ ta ∂ a ϕ = − τ ¯ ∂ a τ (cid:16) ∂ϕ∂z − ¯ w ν ϕ (cid:17) dz = − τ (cid:16) ∂ ϕ∂z∂ ¯ z − ¯ w ν ∂ϕ∂ ¯ z + w ν ∂ϕ∂z − | w ν | ϕ (cid:17) d ¯ z ∧ dz, where we have used that τ dz = dz . Since τ ( d ¯ z ∧ dz ) = 2 σ , this yields the claimed formula for ∂ ta ∂ a ϕ . In a similar way one computes ¯ ∂ ta ¯ ∂ a ϕ . Then part (ii) follows since∆ a ϕ = ( ∂ a + ¯ ∂ a ) t ( ∂ a + ¯ ∂ a ) ϕ = ∂ ta ∂ a ϕ + ¯ ∂ ta ¯ ∂ a ϕ. Using the same ideas one easily verifies part (iii).Using Proposition 4.3.4 it is straightforward to determine the spectrum of ∆ a . Proposition 4.3.5. Let σ ∈ H . For n = ( n , n ) ∈ Z define ϕ n := e ¯ w n z − w n ¯ z : C → U(1) , w n = πσ ( n − σn ) . Then ϕ n is Λ( σ ) -invariant, and { ϕ n | n ∈ Z } is a orthonormal basis for L (cid:0) C / Λ( σ ) (cid:1) . If a = − ¯ w ν dz + w ν d ¯ z is a flat U(1) -connection as before, then ∆ a ϕ n = λ n,ν ϕ n , where λ n,ν = 4 σ | w n − ν | ϕ n . Moreover, n ϕ n dz √ σ , ϕ n d ¯ z √ σ (cid:12)(cid:12)(cid:12) n ∈ Z o gives an orthonormal basis for the space of 1-forms consisting of eigenforms for ∆ a with respectto the same eigenvalues λ n,ν . Since Proposition 4.3.5 is well known, we shall proceed without further comments on theproof. However, we want to note that we can use Proposition 4.3.5 and the Hodge-de-Rhamisomorphism to determine the twisted cohomology groups of T . Recall that in the proof ofProposition 2.3.16 we have already done this using topological methods. Corollary 4.3.6. Let a ν be a flat U(1) -connection over T , and let σ ∈ H determine the metric g σ . Then the cohomology of T with values in the line bundle L a ν is given in terms of harmonicforms by H • ( T , L a ν ) = ( C ⊕ ( C ω σ ⊕ C ω ¯ σ ) ⊕ C dx ∧ dy, if ν ∈ Z , { } , if ν / ∈ Z , where ω σ and ω ¯ σ are as in (4.20) . 4. 3-dimensional Mapping Tori Mapping Tori with Fiber T . It is well known that the mapping class group of T is iso-morphic SL ( Z ), see [53, Sec. 2.9]. Here, the action of an element M ∈ SL ( Z ) on T is the oneinduced by matrix multiplication R → R , (cid:0) xy (cid:1) M (cid:0) xy (cid:1) . As every M ∈ SL ( Z ) has determinant 1, it preserves the volume form dx ∧ dy and we do nothave to invoke Proposition 4.1.1. On the other hand, SL ( Z ) acts on the Riemann sphere b C byfractional linear transformations, and this restricts to an action on H , M : H → H , M σ := aσ + bcσ + d , M = (cid:18) a bc d (cid:19) ∈ SL ( Z ) , (4.23)see for example [93, p. 6]. Unfortunately, the isometry (4.19) does not behave equivariantlywith respect to these two SL ( Z )-actions. We can remedy this, using the following involution onSL ( Z ), SL ( Z ) → SL ( Z ) , M = (cid:18) a bc d (cid:19) M op := (cid:18) d bc a (cid:19) . Note that ( M M ) op = M op2 M op1 , so that we can use the involution to turn a left action of SL ( Z )into a right action. Lemma 4.3.7. Let Φ : H → J ω be the map of Lemma 4.3.1. Then for M ∈ SL ( Z )Φ( M σ ) = ( M op ) − Φ( σ ) M op . Sketch of proof. The group SL ( Z ) is generated by the elements S := (cid:18) − 11 0 (cid:19) , T := (cid:18) (cid:19) , S = ( ST ) , S = Id , see [93, pp. 16–17]. As fractional linear transformations they act as S ( σ ) = − σ , T ( σ ) = σ + 1 . One then computes that for σ = σ + iσ ∈ H ,Φ( Sσ ) = Φ (cid:16) − ¯ σ | σ | (cid:17) = 1 σ (cid:18) σ − | σ | − σ (cid:19) = 1 σ (cid:18) − (cid:19) (cid:18) − σ −| σ | σ (cid:19) (cid:18) − 11 0 (cid:19) = S − Φ( σ ) S, and Φ( T σ ) = Φ( σ + 1) = 1 σ (cid:18) − σ − −| σ + 1 | σ + 1 (cid:19) = . . . = T − Φ( σ ) T. Now one has to verify that the formula of Lemma 4.3.7 holds for all words in S and T . This isalmost tautologically true. For example, if we consider M = ST , thenΦ( M σ ) = S − Φ( T σ ) S = ( T S ) − Φ( σ ) T S = ( M op ) − Φ( σ ) M op . .3. Torus Bundles over S , General Setup M M op we should either redefine theaction of SL ( Z ) as the mapping class group or turn the natural left action (4.23) of SL ( Z ) intoa right action. We opt for the latter, although this leads to an unfortunate difference in notationcompared to the literature. However, redefining the action of SL ( Z ) as the mapping class groupseems more unnatural. Definition 4.3.8. Let M ∈ SL ( Z ), and let σ ( t ) : R → H be M -invariant in the sense that M op σ ( t ) = σ ( t + 1). Then we define ( T M , g σ ) to be the mapping torus (cid:0) T × R (cid:1) / ∼ , (cid:0) M (cid:0) xy (cid:1) , t (cid:1) ∼ (cid:0)(cid:0) xy (cid:1) , t + 1 (cid:1) , (cid:0)(cid:0) xy (cid:1) , t (cid:1) ∈ T × R , endowed with the metric induced by g σ := dt ⊗ dt + g σ ( t ) . Here, for each t ∈ R the metric g σ ( t ) on T is defined as in Lemma 4.3.1. Flat U(1)-connections over T M . We now give an explicit description of flat U(1)-connectionsover T M up to gauge equivalence. Proposition 4.3.9. Let M ∈ SL ( Z ) . Every flat U(1) -connection A over the mapping torus T M is equivalent to one induced by a flat connection a ν over T as in Lemma 4.3.3 and a gaugetransformation u ∈ C ∞ (cid:0) T , U(1) (cid:1) satisfying (cid:18) m m (cid:19) := (Id − M t ) (cid:18) ν ν (cid:19) ∈ Z , (4.24) and u = exp (cid:2) − πi (cid:0) m x + m y + λ (cid:1)(cid:3) for some λ ∈ [0 , .Proof. Let a ν = − πi ( ν dx + ν dy ) be a connection over T as in Lemma 4.3.3. Since M acts bymatrix multiplication on R , the pullback of a ν by M is given by M ∗ a ν = − πi ( µ dx + µ dy ) , (cid:18) µ µ (cid:19) = M t (cid:18) ν ν (cid:19) . Now the condition c M ∗ u a ν = a ν of (4.8) means that connection a ν is the restriction of a connectionover T M if and only if there exists a Z -invariant function u : R → U(1) such that (cid:18) m m (cid:19) = (cid:0) Id − M t (cid:1) (cid:18) ν ν (cid:19) = i π (cid:18) u − ∂ x uu − ∂ y u (cid:19) . (4.25)Clearly, a function u : R → U(1) satisfying (4.25) is necessarily of the form u = exp (cid:2) − πi (cid:0) m x + m y + λ (cid:1)(cid:3) , λ ∈ [0 , , and this is Z -invariant precisely if ( m , m ) ∈ Z . Remark 4.3.10. 4. 3-dimensional Mapping Tori (i) Note that the gauge transformation u is—up to the number λ ∈ [0 , λ = 0 and neglect the gaugetransformation from the notation.(ii) In Remark B.2.10 (ii) we have pointed out that the bundle L → T M on which the flatconnection A is defined is not necessarily trivializable. In the case at hand this topologicaldata is encoded in (4.24). One verifies—using for example Proposition B.2.9—that L istrivializable if and only if (cid:18) m m (cid:19) ∈ im (cid:0) Id − M t : Z → Z (cid:1) . For example, if M = ( ), then Id − M t = ( ) and so (cid:18) (cid:19) / ∈ im (cid:0) Id − M t : Z → Z (cid:1) . This implies that in this case, the gauge transformation u = exp( − πiy ) defines a flatbundle L → T M which is not trivializable. We now want to use Proposition 4.1.5 to express the Eta form in terms of the data introducedin the last paragraphs. First of all, we need to understand the vertical chirality operator and itsvariation.Let σ ( t ) = σ ( t ) + iσ ( t ) be an M -invariant path in H in the sense of Definition 4.3.8, andlet ω σ ( t ) and ω ¯ σ ( t ) be the associated path of 1-forms as defined in (4.20). Differentiating withrespect to t easily yields that˙ ω σ ( t ) = ˙ σ ( t )2 iσ ( t ) ( ω σ ( t ) − ω ¯ σ ( t ) ) and ˙ ω ¯ σ ( t ) = ˙¯ σ ( t )2 iσ ( t ) ( ω σ ( t ) − ω ¯ σ ( t ) ) . Let τ t be the associated path of chirality operators on Ω • ( T ). Since the volume form on T does not vary with t the action of τ t on Ω ( T ) and Ω ( T ) is independent of t . On Ω ( T ) itis determined by τ t ω σ ( t ) = ω σ ( t ) and τ t ω ¯ σ ( t ) = − ω ¯ σ ( t ) , see (4.21). This readily implies that thederivative of τ t with respect to t is given by˙ τ t | Ω ( T ) = 0 , ˙ τ t | Ω ( T ) = 0 , ˙ τ t ω σ ( t ) = i ˙ σ ( t ) σ ( t ) ω ¯ σ ( t ) , ˙ τ t ω ¯ σ ( t ) = i ˙¯ σ ( t ) σ ( t ) ω σ ( t ) . (4.26) Proposition 4.3.11. Let a ν be a flat connection over T as in Proposition 4.3.9. Denote by A be the associated flat connection on the line bundle L → T M , and let σ ( t ) be an M -invariant pathin H . Then the Bismut-Cheeger Eta form is given by b η A = 12 π Re (cid:16) ˙ σ ( t ) Z ∞ F ν (cid:0) σ ( t ) , u (cid:1) du (cid:17) dt, where for every σ = σ + iσ ∈ H and u ∈ R + F ν ( σ, u ) = X n ∈ Z π σ (cid:0) n − ν − ¯ σ ( n − ν ) (cid:1) e − u π σ | n − ν − σ ( n − ν ) | . (4.27) The sum in (4.27) converges absolutely and there are estimates, locally uniform in σ , | F ν ( σ, u ) | ≤ Ce − cu as u → ∞ , | F ν ( σ, u ) | ≤ Ce − cu as u → . .3. Torus Bundles over S , General Setup Proof. According to Proposition 4.1.5, the Eta form associated to the connection A and the path σ ( t ) is given by b η A = i π dt ∧ Z ∞ Tr v (cid:16) ˙ τ v (cid:0) d tA,v d A,v − d A,v d tA,v (cid:1) e − u D A,v (cid:17) du. It follows from Lemma 4.1.4 that under the identificationΩ • v ( T M , L ) = (cid:8) α t : R → Ω • ( T ) (cid:12)(cid:12) α t +1 = c M ∗ α t (cid:9) , the operator d A,v coincides with d a ν applied pointwise for each t . The same is true for d tA,v and d ta ν , where the transpose has to be taken pointwise for each t with respect to the metric inducedby σ ( t ). Now, the operators d a ν d a ν , d a ν d ta ν and e − u D aν all preserve the decompositionΩ • ( T ) = Ω ( T ) ⊕ Ω ( T ) ⊕ Ω ( T ) . Moreover, we know from (4.26) that the operator ˙ τ v acts trivially on Ω ⊕ Ω . Therefore,Tr v (cid:16) ˙ τ v (cid:0) d tA,v d A,v − d A,v d tA,v (cid:1) e − u D A,v (cid:17) = Tr t (cid:16) ˙ τ t (cid:0) d ta ν d a ν − d a d ta ν (cid:1) e − u D aν (cid:12)(cid:12) Ω ( T ) (cid:17) , where the subscripts t indicate that we consider the right hand side as a function of t . Instead ofworking over ( T , g σ ) we now switch to C / Λ( σ ) to be able to use Proposition 4.3.4 and Proposition4.3.5. Using the notation introduced used there, we write a = ( ψ − σ ) ∗ a ν , w n = πσ ( n − σn ) , ϕ n = e ¯ w n z − w n ¯ z : C → U(1) , where n = ( n , n ) ∈ Z . Note that for notational convenience, we have now dropped thereference to the t dependence. Then Proposition 4.3.4 yields that (cid:0) d ta d a − d a d ta (cid:1) ( ϕ n dz ) = 2 ¯ ∂ a ∂ ta ( ϕ n dz ) = − σ ( w n − w ν w n + w ν ) ϕ n d ¯ z = − σ ( w n − ν ) ϕ n d ¯ z. Under the isometry ψ σ : ( T , σ ) → C / Λ( σ ), the pair ( dz, d ¯ z ) pulls back to ( ω σ , ω ¯ σ ). Hence, wededuce from (4.26) that ˙ τ (cid:0) d ta d a − d a d ta (cid:1) ( ϕ n dz ) = − i ˙¯ σ ( w n − ν ) ϕ n dz. Similarly, one finds that ˙ τ (cid:0) d ta d a − d a d ta (cid:1) ( ϕ n d ¯ z ) = − i ˙ σ ( ¯ w n − ν ) ϕ n d ¯ z. The Laplace operators ∆ a and D a ν coincide via ψ σ : ( T , σ ) → C / Λ( σ ). Using Proposition 4.3.5and the fact that e − u ∆ a is an operator with smooth kernel, one then concludes that i π Tr (cid:16) ˙ τ (cid:0) d ta d a − d a d ta (cid:1) e − u ∆ a (cid:17) = 12 π X n ∈ Z Re (cid:0) ˙ σ ( ¯ w n − ν ) (cid:1) e − uσ | w n − ν | = 12 π Re (cid:0) ˙ σF ν ( σ, u ) (cid:1) , 4. 3-dimensional Mapping Tori where in the last step we have simply used the definition of F ν ( σ, u ). Concerning the absoluteconvergence of F ν ( σ, u ) and the estimate as u → ∞ we assume for simplicity that ν = 0. Thegeneral case requires only minor changes. Define r σ := min (cid:8) | x − σx | (cid:12)(cid:12) ( x , x ) ∈ R , | x | + | x | = 1 (cid:9) ,R σ := max (cid:8) | x − σx | (cid:12)(cid:12) ( x , x ) ∈ R , | x | + | x | = 1 (cid:9) . Clearly, r σ and R σ depend continuously on σ . For n ∈ Z and some constants c and C , notdepending on n and σ , we have | w n | ≥ cr σ | n | , | w n | ≤ CR σ | n | , so that (cid:12)(cid:12) ¯ w n e − uσ | w n | (cid:12)(cid:12) ≤ C | n | e − uc | n | , where the constants c and C now depend continuously on σ . This implies absolute convergenceof the series in (4.27). Concerning the estimate as u → ∞ one now proceeds exactly as in theproof of Lemma 1.2.3 and we will skip the details. However, the estimate for u → F ν ( σ, u ) is bounded as u → f on R d X n ∈ Z d f ( n ) = X n ∈ Z d b f ( n ) , b f ( n ) = Z R d f ( x ) e − πi h n,x i dx, (4.28)see e.g., [64, Sec. 20.1]. We now let ν ∈ R be arbitrary again and use the Poisson summationformula to bring F ν ( σ, u ) into a different form. Since we will need the formula only to obtain theestimate as u → 0, we give only some intermediate steps F ν ( σ, u ) = X n ∈ Z ( ¯ w n − ν ) e − uσ | w n − ν | = X n ∈ Z Z R ( ¯ w x − ν ) e − uσ | w x − ν | e − πi h x,n i dx = X n ∈ Z e − πi h ν,n i σ u Z R ( x + ix ) e − π | x | e − πi h x,ξ n i dx (cid:12)(cid:12)(cid:12) ξ n = √ πuσ ( σ n n + σ n ) , where the last line follows from a suitable substitution. For arbitrary ξ = ( ξ , ξ ) ∈ R onecomputes that Z R ( x + ix ) e − π | x | e − πi h x,ξ n i dx = (cid:0) − iξ + ξ (cid:1) e − π | ξ | . Moreover, with ξ = √ πuσ ( σ n n + σ n ) as above, − iξ + ξ = 1 √ πuσ ( n + ¯ σn ) , and thus, F ν ( σ, u ) = 1 πσ X n ∈ Z e − πi h ν,n i ( ¯ w ∗ n ) u − e − | w ∗ n | uσ , w ∗ n := n + σn . (4.29)Since ¯ w n = 0 for n = (0 , u → | F ν ( σ, u ) | ≤ Ce − cu as u → 0, with constants c and C depending continuously on σ . .3. Torus Bundles over S , General Setup b η A further and get an expression for the integralof the Eta form b η A over the base. We note that the function F ν in (4.27) depends on ν ∈ R onlymodulo Z , see also (4.29). Hence, we will often assume in the following that 0 ≤ ν < ν ∈ [0 , . Theorem 4.3.12. For ν = ( ν , ν ) ∈ R with ≤ ν < and σ ∈ H write z = ν σ − ν . (i) Employing the notation q σ = e πiσ , and q z = e πiz , we define E ν ( σ ) := X n > X n > n ( q z + q − z ) n q n n σ , if ν = 0 , and E ν ( σ ) := X n > n q n z + X n > X n > n ( q z + q − z ) n q n n σ , if ν = 0 . Then the sum defining E ν ( σ ) converges absolutely to a function which is holomorphic on H . (ii) Let F ν ( σ, u ) be as in (4.27) . Then π Z ∞ F (cid:0) σ, u (cid:1) du = 16 − πσ + iπ ∂∂σ E ( σ ) , and for ν / ∈ Z π Z ∞ F ν (cid:0) σ, u (cid:1) du = P ( ν ) + iπ ∂∂σ E ν ( σ ) , where P is the second periodic Bernoulli function, see Definition C.1.1. Remark 4.3.13. (i) The function E ν ( σ ) is related to the logarithm of a generalized Dedekind Eta function, see(4.61) and Lemma 4.4.18 below. As such it appears in the constant term of the Laurentseries at s = 1 of certain Eisenstein series. Without going into details about the exactrelation, we recall that the determination of this constant term is classically referred to asa Kronecker limit formula , see [64, Ch. 20] and [88, Sec. 4]. In the proof of Theorem 4.3.12below, we mimic a combined proof of the first and the second Kronecker limit formula asin [64].(ii) Since the sum defining E ν ( σ ) converges absolutely, we can interchange the summation over n and n . Since | q n σ | < n > X n > q n n σ = q n σ − q n σ = q n / σ q − n / σ − q n / σ = i (cid:0) cot( πn σ ) + i (cid:1) . Hence, for ν = 0 E ν ( σ ) = i X n > n cos(2 πzn ) (cid:0) cot( πn σ ) + i (cid:1) . In the case that ν = 0 one can combine the two sums over n . Then E ν ( σ ) = i X n ≥ n (cid:0) cos(2 πn z ) cot( πn σ ) + sin(2 πn z ) (cid:1) . 4. 3-dimensional Mapping Tori Proof of Theorem 4.3.12. We can assume for simplicity that 0 ≤ ν < ν only modulo Z . As an auxiliary tool we define G ν ( σ, s ) := 12 π Γ( s ) Z ∞ u s − F ν ( σ, u ) du. (4.30)The estimates in Proposition 4.3.11 ensure that G ν ( σ, s ) is a holomorphic function for all s ∈ C .Clearly, G ν ( σ, 1) = 12 π Z ∞ F ν ( σ, u ) du. Since the sum over n = ( n , n ) ∈ Z defining F ν ( σ, u ) converges absolutely, we can first extractpossible terms with n = ν = 0, and then sum up the remaining terms. More precisely, define F ν ( σ, u ) := π σ P n ∈ Z | n − ν | e − u π σ | n − ν | , if ν = 0 , , if ν = 0 , and F ν ( σ, u ) := π σ X n = ν X n ∈ Z (cid:0) n − ¯ σn + ¯ z (cid:1) e − u π σ | n − σn + z | , where z = ν σ − ν . Accordingly, we can split G ν ( σ, s ) for Re( s ) large enough as G ν ( σ, s ) = 12 π Γ( s ) Z ∞ u s − F ν ( σ, u ) du + 12 π Γ( s ) Z ∞ u s − F ν ( σ, u ) du =: G ν ( σ, s ) + G ν ( σ, s ) . Now, again for Re( s ) large enough, we can interchange summation and integration, so that for ν = 0 the substitution u π σ | n − ν | u yields G ν ( σ, s ) = π σ X n ∈ Z | n − ν | s ) Z ∞ u s − e − u π σ | n − ν | du = σ s − π s − X n ∈ Z | n − ν | − s = σ s − π s − e ζ ν ( s − , where e ζ ν ( s − 1) is the periodic Zeta function in Proposition C.1.2. This implies that G ν ( σ, s )admits a meromorphic continuation to the whole s -plane, and G ν ( σ, 1) = ( , if ν = 0 , − (2 πσ ) − , if ν = ν = 0 . (4.31)To identify G ν ( σ, s ) we assume again that Re( s ) is large enough, so that we can freely interchangesummation and integration. Then, with the substitution u π σ u , we get G ν ( σ, s ) = σ s − π s − X n = ν s ) Z ∞ u s − X n ∈ Z (cid:0) n − ¯ σn + ¯ z (cid:1) e − u | n − σn + z | du. .3. Torus Bundles over S , General Setup X n ∈ Z (cid:0) n − ¯ σn + ¯ z (cid:1) e − u | n − σn + z | = X n ∈ Z Z R (cid:0) x − ¯ σn + ¯ z (cid:1) e − u | x − σn + z | e − πixn dx = X n ∈ Z e − πi Re( σn − z ) n e − u (Im( σn − z )) Z R (cid:0) x + i Im( n σ − z ) (cid:1) e − ux e − πixn dx, where we have separated the real and imaginary parts and then made the substitution x x − Re (cid:0) σn − z (cid:1) . Clearly, the integral in the last expression decays exponentially as | n | → ∞ .Hence, the sum converges absolutely, and we can rearrange the order of summation again. Write G ν ( σ, s ) = G ν ( σ, s ) + G ν ( σ, s ) , (4.32)where G ν ( σ, s ) is the contribution coming from n = 0, i.e., G ν ( σ, s ) = σ s − π s − s ) X n = ν Z ∞ u s − e − u (Im( σn − z )) Z R (cid:0) x + i Im( n σ − z ) (cid:1) e − ux dx. Setting a := Im( n σ − z ) we have Z R ( x + ia ) e − ux dx = √ π (cid:0) u − / − a u − / (cid:1) . Therefore, standard manipulations involving the Gamma function yield Z ∞ u s − e − ua Z R ( x + ia ) e − ux dx = √ π | a | − s (cid:0) Γ( s − ) − Γ( s − ) (cid:1) = √ π | a | − s Γ( s − ) 4 − s s − . Recalling that a = Im( n σ − z ) we find that for Re( s ) large enough, G ν ( σ, s ) = √ π σ s − π s − Γ( s − )Γ( s ) 4 − s s − X n = ν (cid:12)(cid:12) Im( n σ − z ) (cid:12)(cid:12) − s = √ π σ − s π s − Γ( s − )Γ( s ) 4 − s s − e ζ ν (2 s − , where we have used that Im( n σ − z ) = σ ( n − ν ). Hence we have found an expression for G ν ( σ, s ) which can be extended to a meromorphic function on the whole s -plane. It follows fromProposition C.1.2 that s = 1 is not a pole, and that G ν ( σ, 1) = P ( ν ) = ν − ν + . (4.33)Now we have to consider the general term G ν ( σ, s ) in (4.32). Write a n = | Im( n σ − z ) | and b n = π | n | . Note that a n , b n > n = ν and n = 0. Then, for Re( s ) large, G ν ( σ, s ) = σ s − π s − s ) X n = ν X n =0 e − πi Re( σn − z ) n Z ∞ u s − e − ua n Z R (cid:16) x + ia n sgn( n − ν ) (cid:17) e − ux e − ix sgn( n ) b n dxdu. (4.34)50 4. 3-dimensional Mapping Tori To compute the integrals in the sum, we replace a n and b n by real parameters a, b > 0. Then Z R (cid:16) x + ia sgn( n − ν ) (cid:17) e − ux e − ix sgn( n ) b dx = − (cid:16) ∂ b sgn( n ) + a sgn( n − ν ) (cid:17) Z R e − ux e − ix sgn( n ) b dx = − (cid:16) ∂ b sgn( n ) + a sgn( n − ν ) (cid:17) r πu e − b /u . Therefore, the u -integral in (4.34) in terms of the parameters a and b is given by − √ π (cid:16) ∂ b sgn( n ) + a sgn( n − ν ) (cid:17) K s − ( a, b ) , (4.35)where K s ( a, b ) is the Bessel K-function [64, Sec. 20.3] K s ( a, b ) = Z ∞ u s − e − ( a u + b /u ) du. Moreover, for fixed s , one has ∂ b K s ( a, b ) = − bK s − ( a, b ) , so that (4.35) is actually a sum of Bessel K -functions for different s -parameters. We also collectfrom [64, Sec. 20.3] that K s ( a, b ) is holomorphic on the whole s -plane and satisfies estimates,locally uniform in s , of the form | K s ( a, b ) | ≤ C (cid:16) ba (cid:17) s e − ab , ab → ∞ . This implies that the summand in (4.34) decays exponentially as | ( n , n ) | → ∞ , locally uniformin s . From this one deduces that G ν ( σ, s ) can be extended holomorphically to the whole s -plane,and that we can simply put s = 1 to find the value we are interested in. Now, K ( a, b ) = √ πa e − ab ,see [64, p. 271]. Using this, one verifies without effort that the value of (4.35) at s = 1 is equalto − πa (cid:0) − sgn( n ( n − ν )) (cid:1) e − ab . Using this one finds that G ν ( σ, 1) = − X n = ν X n =0 | n − ν | (cid:0) − sgn( n ( n − ν )) (cid:1) e − π | n || Im( σn − z ) | e − πi Re( σn − z ) n . Since n = 0 and n = ν , the above sum converges absolutely. Moreover, all the terms withsign( n ) = sign( n − ν ) drop out. Using the notation q σ = e πiσ and q z = e πiz , one obtains G ν ( σ, 1) = − X n > ν q n z − X n > X n > h ( n + ν )( q z q n σ ) n + ( n − ν )( q − z q n σ ) n i . (4.36)Note that for ν = 0 the first term is equal to zero. Now, q z and q σ are holomorphic as functionsof σ , and ν q n z = 12 πin ∂∂σ q n z , ( n ± ν )( q z q n σ ) n = 12 πin ∂∂σ ( q ± z q n σ ) n , .3. Torus Bundles over S , General Setup ν = 0 we have z ∈ H . This yields that n q n z decays exponentially as n →∞ . For arbitrary ν and n > n ( q ± z q n σ ) n decays exponentially in both, n and n . Moreover, this decay is certainly locally uniform in σ . This implies that in (4.36) we caninterchange summation and differentiation to find that G ν ( σ, 1) = iπ ∂∂σ E ν ( σ ) , (4.37)where E ν ( σ ) is defined as in part (i) of the theorem. Since the sums converge absolutely andlocally uniform in σ , we conclude that E ν ( σ ) defines a holomorphic function on H , which provespart (i). Moreover, we have split the auxiliary function in (4.30) for Re( s ) large as G ν ( σ, s ) = G ν ( σ, s ) + G ν ( σ, s ) + G ν ( σ, s ) . As we have seen, the terms on the right hand side extend to meromorphic function on the s -plane,and thus, the above equality continues to hold for all s . Therefore, we can insert the values at s = 1, which we have computed in (4.31), (4.33) and (4.37), and deduce that G ν ( σ, 1) = P ( ν ) + iπ ∂∂σ E ν ( σ ) + ( , if ν = 0 , − (2 πσ ) − , if ν = ν = 0 , which proves part (ii) of Theorem 4.3.12.As a consequence of Theorem 4.3.12, we obtain the expression for the integral of Bismut-Cheeger Eta form we were aiming at. We know from Proposition 4.3.11 that b η A = 12 π Re (cid:16) ˙ σ ( t ) Z ∞ F ν (cid:0) σ ( t ) , u (cid:1) du (cid:17) dt Hence, integrating the formula in Theorem 4.3.12 (ii) with respect to t one easily arrives at Theorem 4.3.14. Let M ∈ SL ( Z ) , let σ ( t ) be an M -invariant path in H , and use σ ( t ) to endowthe mapping torus T M with a metric. (i) The untwisted Eta form b η satisfies Z b η = 1 π Re h πσP (0) + iE ν ( σ ) i σ (1) σ (0) − π Z ˙ σ ( t ) σ ( t ) dt, where we use the abbreviation [ f ( σ )] σ (1) σ (0) = f ( σ (1)) − f ( σ (0)) . (ii) Let ν ∈ R \ Z with ≤ ν < satisfy (Id − M t ) ν ∈ Z , and let A be the corresponding flat U(1) -connection over the mapping torus T M . Then Z b η A = 1 π Re h πσP ( ν ) + iE ν ( σ ) i σ (1) σ (0) . Here, we encounter a similar situation as in Remark 3.4.2 (i). The terms G ν ( σ, s ) and G ν ( σ, s ) are notnecessarily holomorphic, whereas G ν ( σ, s ) and G ν ( σ, s ) are. This implies that the poles of G ν ( σ, s ) and G ν ( σ, s )have to cancel each other out. With some effort one can check this directly. We will not go into further details,since the value we are interested in is s = 1, which is no pole for any of the summands. 4. 3-dimensional Mapping Tori In particular, the Rho form b ρ A satisfies Z b ρ A = 1 π Re h πσ (cid:0) P ( ν ) − (cid:1) + i (cid:0) E ν ( σ ) − E ( σ ) (cid:1)i σ (1) σ (0) + 12 π Z ˙ σ ( t ) σ ( t ) dt. Remark 4.3.15. The forms b η A and b ρ A depend only on ν modulo Z . Also, by its very definition, P ( ν ) depends on ν only modulo Z . Therefore, it is reasonable—and convenient—to extend thedefinition of E ν ( σ ) to arbitrary ( ν , ν ) ∈ R by letting E ( ν ,ν ) ( σ ) := E ( ν − [ ν ] ,ν ) ( σ ) , where [ ν ] is the largest integer less or equal than ν . Then Theorem 4.3.14 (ii) continues to holdwithout the assumption on ν . S , Explicit Computations In this section, we want to give a more explicit formula for the Rho invariants of a mappingtorus T M with M ∈ SL ( Z ). The result depends considerably on whether M is elliptic, parabolicor hyperbolic—see Definition 4.4.1 below—and we have to treat all three cases separately.Explicit formulæ for the untwisted Eta invariant have been obtained in [3] and [26, App. 3].Both references make use of adiabatic limits, and much of our treatment parallels their dis-cussion. In [26] the focus is on the hyperbolic case, and the Eta invariant is identified withthe value of certain number theoretical L -series, see also [6, 17, 77]. This has its origin inHirzebruch’s work [52], where a topological interpretation of the aforementioned L -series wasconjectured. A similar relation can also be found for twisted Eta invariants. However, our aimis to get a simple formula for the Rho invariant, and values of L -series are certainly not easyto compute. Fortunately, Atiyah [3] found a number of very different ways to express the un-twisted Eta invariant of T M , and we shall derive a formula for the Rho invariant along those lines. Rough Classification of Elements in SL ( Z ). For explicit computations we now have tofind M -invariant paths in H . For this we will use that elements in SL ( Z ) split into three naturalclasses. Definition 4.4.1. Let M ∈ SL ( Z ), and let ∆ := (tr M ) − 4. Then M is called(i) elliptic , if ∆ < parabolic , if ∆ = 0, and(iii) hyperbolic , if ∆ > Remark. Recall that according to Lemma B.2.1, the diffeomorphism type of T M depends onlyon the conjugacy class of M in SL ( Z ). Moreover, T M − and T M are related by an orientationreversing diffeomorphism. In addition, one verifies that for all M ∈ SL ( Z ) S − M t S = M − , S = (cid:18) − 11 0 (cid:19) . .4. Torus Bundles over S , Explicit Computations T M ∼ = T M t . Since Rho invariants depends only on the oriented diffeomorphism type of themapping torus T M , and the relation among Rho invariants for different orientations is determinedby Lemma 1.3.6 (ii), we are interested in elements of SL ( Z ) only up to conjugation, takinginverses and transposes. Note that ∆ in Definition 4.4.1 is invariant under these operations sothat M , M − and M t all belong to the same class.We first collect some well-known facts, see for example [93, Sec. 1.4]. Proposition 4.4.2. Let M ∈ SL ( Z ) . (i) M is parabolic if and only if M is conjugate in SL ( Z ) to ± (cid:0) l (cid:1) with l ∈ Z . (ii) M is elliptic if and only if M it is of finite order with M = ± Id . In this case, M is oforder 3,4 or 6, and conjugate in SL ( Z ) to an element of the form ± (cid:18) − 11 0 (cid:19) , ± (cid:18) − (cid:19) , ± (cid:18) − 11 0 (cid:19) . Sketch of proof. The eigenvalues of M are easily seen to be κ = (cid:0) tr M + √ ∆ (cid:1) , κ − = (cid:0) tr M − √ ∆ (cid:1) , (4.38)where we fix the complex square root with √− i . By definition, M is parabolic if and only if κ = κ − = ± 1, so that the “if” part of (i) is clear. To prove the “only if” part, write M = (cid:0) a bc d (cid:1) ,and assume that M is parabolic with M = ± Id. We can then assume—modulo conjugation by S —that c = 0. Replacing M with − M if necessary we can also achieve that tr M = a + d = 2.Define g := gcd (cid:0) a − d, c (cid:1) , p := a − dg , q := 2 cg . It follows from a + d = 2 and ad − bc = 1 that ap + bq = p, cp + dq = q. Moreover, gcd( p, q ) = 1 so that we can find r, s ∈ Z with pr − qs = 1. Then ( p sq r ) ∈ SL ( Z ), andone verifies that (cid:18) p sq r (cid:19) − (cid:18) a bc d (cid:19) (cid:18) p sq r (cid:19) = (cid:18) l (cid:19) , for some l ∈ Z . This proves part (i).Concerning part (ii), we first note that part (i) implies that the only parabolic elements offinite order are ± Id. Hence, we can assume that κ = κ − . Then M is conjugate in GL ( C ) to (cid:0) κ κ − (cid:1) . Hence, M is of finite order if and only if κ is a root of unity. Then κ − = ¯ κ , and M iselliptic because | tr M | = 2 | Re( κ ) | < , since κ = ¯ κ. For the reverse direction, we only note that if M is elliptic, then tr M ∈ {− , , } and one easilychecks by hand that κ is a root of unity—in fact, κ = e i π , e i π or e i π . From this it is not difficultto determine explicitly all conjugacy classes of elliptic elements. We refer to [93, pp. 14–15].54 4. 3-dimensional Mapping Tori We now analyze the action of SL ( Z ) on H in some more detail. Recall from Definition 4.3.8that we let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) act on H by the restriction of the fractional linear transformation M op : b C → b C , M op z = dz + bcz + a . The following results are well-known but to fix notation we sketch the proof. .4. Torus Bundles over S , Explicit Computations Proposition 4.4.3. Let M ∈ SL ( Z ) with M = ± Id . (i) If M is parabolic of the form M = ± (cid:0) l (cid:1) with l = 0 , then M op has no fixed points in C ,and horizontal lines { σ ∈ H | Im( σ ) = const } are invariant under the action of M op . (ii) If M is elliptic, the fractional linear transformation given by M op has exactly one fixedpoint in H . (iii) M is hyperbolic, if and only if the fractional transformation given by M op has two distinctfixed points α, β ∈ R ⊂ C , and the circle n σ ∈ H (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) σ − α + β (cid:12)(cid:12) = (cid:12)(cid:12) α − β (cid:12)(cid:12)o is invariant under the action of M op .Sketch of proof. Part (i) is immediate since for all z ∈ C we have M op z = z + l . If M = (cid:0) a bc d (cid:1) isnot parabolic, one easily verfies that the eigenvalues κ and κ − as in (4.38) cannot be integers.Hence, M is not in diagonal or triangular form, which implies that b, c = 0. Then the fixed pointsof M op acting on b C are easily seen to be α = κ − ac and β = κ − − ac . If M is elliptic, then Im( κ ) > α = β . Thus, the unique fixed point of M op as claimedin part (ii) is given by α ∈ H if c > 0, and by β ∈ H if c < 0. Let us now assume that M ishyperbolic. Then the eigenvalues are real and κ > κ − . If we also assume for simplicity that c > 0, we get β < α . Then one verifies using elementary linear algebra that for all σ ∈ H (cid:12)(cid:12)(cid:12) σ − α + β (cid:12)(cid:12)(cid:12) = α − β ⇐⇒ Re (cid:16) σ − ασ − β (cid:17) = 0 . (4.39)On the other hand, M op σ is uniquely defined by the normal form of the fractional linear trans-formation M op σ − αM op σ − β = κ − σ − ασ − β . (4.40)Since M is hyperbolic we have κ ∈ R . Hence, one finds from (4.39) and (4.40) that the circle n σ ∈ H (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) σ − α + β (cid:12)(cid:12) = α − β o is indeed invariant under the action of M op . This proves part (iii). Assume that M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) is elliptic. Then, according to Proposition 4.4.2, M is of finiteorder, so that we are in the situation considered in Section 4.2. As in Proposition 4.4.3 there areprecisely two fixed points for M op acting on C , one of which lies in H , explicitly given by σ = κ − ac , ¯ σ = ¯ κ − ac , where, κ = (cid:0) tr M + i p − (tr M ) (cid:1) = e πiθ , θ ∈ (0 , ) . (4.41)Actually, one easily checks that θ ∈ { , , } .56 4. 3-dimensional Mapping Tori Theorem 4.4.4. Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) be elliptic, and let A be a flat connection over T M defined by a pair ( a ν , u ) as in Proposition 4.3.9. (i) If ν / ∈ Z , and θ is as in (4.41) , then ρ A ( T M ) = (2 − θ ) sgn( c ) . (ii) If a ν = 0 is the trivial connection, so that u ≡ e − πiλ ∈ U(1) , then ρ A ( T M ) = , if Re( u ) < Re( κ ) , sgn( c ) , if Re( u ) = Re( κ ) , c ) , if Re( u ) > Re( κ ) . Proof. If ν / ∈ Z , it follows from Lemma 4.3.3 (iii) that H • ( T , L a ν ) = { } . Hence, Theorem 4.2.4yields that in this case the only contribution to the Rho invariant comes from the Eta invariantof the trivial connection. More precisely, ρ A ( T M ) = − (cid:2) M ∗ | H + ( T ) ∩ Ω (cid:3) + 2 rk (cid:2) ( M ∗ − Id) | H + ( T ) ∩ Ω (cid:3) . Let σ respectively ¯ σ be the fixed point of M op in H as above, depending on whether c > c < 0. Use this to define an M -invariant metric on T as in Lemma 4.3.1. With the notation of(4.20), it follows from Corollary 4.3.6 that if c > 0, then H + ( T ) ∩ Ω = C ω σ , H − ( T ) ∩ Ω = C ω ¯ σ , and, if c < 0, then H + ( T ) ∩ Ω = C ω ¯ σ , H − ( T ) ∩ Ω = C ω σ . Moreover, it is immediate that M ∗ ω σ = κ · ω σ , and M ∗ ω ¯ σ = ¯ κ · ω ¯ σ . This implies that M ∗ | H ± ( T ) ∩ Ω = κ, M ∗ | H ∓ ( T ) ∩ Ω = ¯ κ, if ± c > . (4.42)Hence, with θ as in (4.41) and the definition of “tr log” in Theorem 4.2.4, one finds that if ν / ∈ Z , ρ A ( T M ) = ( − θ + 2 , if c > , − − θ ) + 2 , if c < . This proves part (i) of Theorem 4.4.4. Now assume that a ν is the trivial connection. Lemma4.3.3 and Proposition 4.3.9 then imply that we can choose ν = 0 and u to be the constant gaugetransformation e − πiλ , with λ ∈ [0 , A is given by2 tr log (cid:2) u − M ∗ | H + ( T ) ∩ Ω (cid:3) − rk (cid:2) ( u − M ∗ − Id) | H + ( T ) ∩ Ω (cid:3) − (cid:2) u − M ∗ | H − ( T ) ∩ Ω (cid:3) + rk (cid:2) ( u − M ∗ − Id) | H − ( T ) ∩ Ω (cid:3) + (2 − θ ) sgn( c ) . .4. Torus Bundles over S , Explicit Computations M ∗ in (4.42) with u − M ∗ . We assume forsimplicity that c > 0; the other case works analogously. Then u − M ∗ | H + ( T ) ∩ Ω = u − κ, u − M ∗ | H − ( T ) ∩ Ω = u − ¯ κ. Now if Re( u ) < Re( κ ), then λ ∈ [0 , θ ) or λ ∈ (1 − θ, (cid:2) ( u − M ∗ − Id) | H + ( T ) ∩ Ω (cid:3) = rk (cid:2) ( u − M ∗ − Id) | H − ( T ) ∩ Ω (cid:3) , and 2 tr log (cid:2) u − M ∗ | H + ( T ) ∩ Ω (cid:3) − (cid:2) u − M ∗ | H − ( T ) ∩ Ω (cid:3) = 4 θ − . This implies that if Re( u ) < Re( κ ), then ρ A ( T M ) = 0. Similarly, if Re( u ) > Re( κ ), one computesthat rk (cid:2) ( u − M ∗ − Id) | H + ( T ) ∩ Ω (cid:3) = rk (cid:2) ( u − M ∗ − Id) | H − ( T ) ∩ Ω (cid:3) , and 2 tr log (cid:2) u − M ∗ | H + ( T ) ∩ Ω (cid:3) − (cid:2) u − M ∗ | H − ( T ) ∩ Ω (cid:3) = 4 θ, so that ρ A ( T M ) = 2. Lastly, if Re( u ) = Re( κ ), then either λ = θ or λ = 1 − θ . In the first case,rk (cid:2) ( u − M ∗ − Id) | H + ( T ) ∩ Ω (cid:3) = 1 , rk (cid:2) ( u − M ∗ − Id) | H − ( T ) ∩ Ω (cid:3) = 0 , and 2 tr log (cid:2) u − M ∗ | H + ( T ) ∩ Ω (cid:3) = 4 θ, (cid:2) u − M ∗ | H − ( T ) ∩ Ω (cid:3) = 0 . This yields ρ A ( T M ) = 1. In a similar way one deals with the case λ = 1 − θ . If c < 0, one has toreplace κ with ¯ κ in the above computations, and one easily verifies that the result is the negativeof what we computed in the case c > ε → η ( B ev ε ) = (4 θ − 2) sgn( c ) . Here, B ev ε is the adiabatic family of untwisted odd signature operators associated to the M -invariant metric on T induced by σ respectively ¯ σ . Now in the case at hand, the family η ( B ev ε )is independent of ε , see [3, p. 360]. Hence, we arrive at the following Corollary 4.4.5. Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) be elliptic, and endow T M with the metric inducedby an M -invariant metric on T . Then, with θ is as in (4.41) , η ( B ev ) = (4 θ − 2) sgn( c ) . To check consistency with previous results, we now use Corollary 4.4.5 to compute the Etainvariant for the examples considered in [3, p. 372], respectively [75, p. 48].(i) M = (cid:18) − 11 1 (cid:19) . Then κ = e πi , so that θ = . Hence, η ( B ev ) = − − .58 4. 3-dimensional Mapping Tori (ii) M = (cid:18) − − 11 0 (cid:19) . Then κ = e πi and η ( B ev ) = − − .(iii) M = (cid:18) − 11 0 (cid:19) . Then κ = e πi , so that η ( B ev ) = − − If M is parabolic, we know from Proposition 4.4.2 that M is conjugate to an element of the form ± (cid:0) l (cid:1) . Therefore, we can always choose σ ( t ) := tl + i as an M -invariant path in H . We firstcompute the integral of the Rho form using Theorem 4.3.14. Proposition 4.4.6. Let ν = ( ν , ν ) ∈ R with ν / ∈ Z satisfy ( M t − Id) ν ∈ Z . Let A be thecorresponding flat connection over the mapping torus T M . Then the integral of the Rho form withrespect to the metric induced by σ ( t ) := tl + i is given by Z b ρ A = l (cid:0) P ( ν ) − (cid:1) + l π , where P is the second periodic Bernoulli function.Proof. Since both sides of the equation in Proposition 4.4.6 depend on ν only modulo Z , we canassume that ν ∈ [0 , . Since σ (0) = i and σ (1) = l + i , we havecot( πσ (1) n ) = cot( πσ (0) n ) for all n ∈ N . This implies—using the notation of Theorem 4.3.12 and Remark 4.3.13 (ii)—that E (cid:0) σ (1) (cid:1) = E (cid:0) σ (0) (cid:1) . We now claim that also E ν (cid:0) σ (1) (cid:1) = E ν (cid:0) σ (0) (cid:1) . (4.43)Indeed, if M = (cid:0) l (cid:1) , the condition ( M t − Id) ν ∈ Z guarantees that lν ∈ Z , whereas ν isarbitrary. Thus, with z ( t ) = ν σ ( t ) − ν as in Theorem 4.3.12, we get z (1) = z (0) + lν ∈ z (0) + Z . This easily yields (4.43) in the case at hand. If M = − (cid:0) l (cid:1) , then ( M t − Id) ν ∈ Z means that2 ν ∈ Z , lν + 2 ν ∈ Z . Since we are assuming that ν ∈ [0 , , there are only a few possible values for ν . First of all, if ν = 0, then z (1) = z (0), so that (4.43) holds again. If ν = , then ν ∈ ( { , } , if l is even , { , } , if l is odd . .4. Torus Bundles over S , Explicit Computations z (1) = z (0) + l and socos (cid:0) πz (1) n (cid:1) = ( − nl cos (cid:0) πz (0) n (cid:1) , sin (cid:0) πz (1) n (cid:1) = ( − nl sin (cid:0) πz (0) n (cid:1) . (4.44)This implies that only summands such nl is odd contribute to E ν (cid:0) σ (1) (cid:1) − E ν (cid:0) σ (0) (cid:1) . Hence, if l is even we are done. Let us thus assume that l is odd and—for definiteness—that ν = . Thecase ν = is analogous. Then, for odd n ∈ N ,cos (cid:0) πz (0) n (cid:1) = cos (cid:0) πin − π n (cid:1) = i n − sin( πin ) , sin (cid:0) πz (0) n (cid:1) = − i n − cos( πin ) , and so cos (cid:0) πnz (0) (cid:1) cot( πni ) + sin (cid:0) πnz (0) (cid:1) = 0 . This, together with (4.44), implies (4.43) in this last case as well. We can now use (4.43) andTheorem 4.3.14 to deduce that in all cases Z b ρ A = l (cid:0) P ( ν ) − ) + 12 π Z ldt = l (cid:0) P ( ν ) − ) + l π . To conclude the computation of the Rho invariants of T M for parabolic M ∈ SL ( Z ), westill have to determine the Rho invariant of the bundle of vertical cohomology groups ρ H A,v ( S )appearing in Theorem 3.3.16. Recall that Dai’s correction term vanishes because the base is1-dimensional, see the proof of Theorem 4.2.4. Proposition 4.4.7. For ε = ± and l ∈ Z let T M be the mapping torus of M = ε (cid:0) l (cid:1) . Endow T M with the metric given by σ ( t ) = tl + i . Then, for all connections A as in Proposition 4.4.6, ρ H A,v ( S ) = , if l = 0, − lπ + sgn( l ) , if ε = 1, l = 0, − lπ , if ε = − T vanish except for the trivial connection.Hence, for A as in Proposition 4.4.6 we can argue as in the proof of part (i) of Theorem 4.4.4that ρ H A,v ( S ) = − η (cid:0) D S ⊗ ∇ H v (cid:1) , (4.45)where D S ⊗ ∇ H v is as in Definition 3.1.16. However, unlike in the case of elliptic elements, theconnection ∇ H v on the bundle of vertical cohomology groups is not unitary, so that it is difficultto compute the above Eta invariant directly. The idea of our proof is to study the differencebetween D S ⊗ ∇ H v and the odd signature operator associated to the unitary connection ∇ H v ,u ,see (3.24) and (3.29). More precisely, we will compute η (cid:0) D S ⊗ ∇ H v ,u (cid:1) and then use the variationformula of Proposition 1.3.14 to obtain η (cid:0) D S ⊗ ∇ H v (cid:1) . Here, the considerations of Section 3.1.6will play a role. Proof of Proposition 4.4.7. We split the bundle of vertical cohomology groups as H • v ( T M ) = H v ( T M ) ⊕ H v ( T M ) ⊕ H v ( T M ) . 4. 3-dimensional Mapping Tori It follows from Corollary 4.3.6, that H v ( T M ) and H v ( T M ) can be trivialized by the constantsections 1 respectively dx ∧ dy . With respect to this trivialization, the connection ∇ H v is thetrivial connection, see (4.10). According to Remark 1.4.8 (iii) the Eta invariant of the untwistedodd signature operator over S vanishes, so that we only have to compute the contribution to η (cid:0) D S ⊗ ∇ H v (cid:1) coming from H v ( T M ).Let ω σ ( t ) and ω ¯ σ ( t ) be as in (4.20) with respect to σ ( t ) = tl + i , and define α t := ω σ ( t ) + ω ¯ σ ( t ) , β t := ω σ ( t ) − ω ¯ σ ( t ) . It is immediate from Corollary 4.3.6 that for each t the pair ( α t , β t ) forms an orthogonal basis of H ( T M , g σ ( t ) ). However, it is not necessarily a trivialization of the bundle of vertical cohomologygroups. For this note that, with ε = ± M ∗ ω σ ( t ) = εω σ ( t +1) , M ∗ ω ¯ σ ( t ) = εω ¯ σ ( t +1) , so that also M ∗ α t = εα t +1 , M ∗ β t = εβ t +1 . Nevertheless, this means that we can write every section of H v ( T M ) → S as ϕ α ( t ) α t + ϕ β ( t ) β t , with functions ϕ α and ϕ β on R satisfying the condition ϕ α ( t + 1) = εϕ α ( t ) , ϕ β ( t + 1) = εϕ β ( t ) . (4.46)Now, note that ∂ t ω σ ( t ) = ∂ t ω ¯ σ ( t ) = ldy = l i (cid:0) ω σ ( t ) − ω ¯ σ ( t ) (cid:1) . Using this we see that the flat connection ∇ H v on H v ( T M ) is given by ∇ H v ∂ t α t = ∂ t α t = − ilβ t , ∇ H v ∂ t β t = ∂ t β t = 0 . (4.47)Moreover, one verifies using (4.21) and (4.26) that τ t α t = β t , τ t β t = α t , ˙ τ t α t = ilα t , ˙ τ t β t = − ilβ t . (4.48)According to Lemma 4.1.4, this means that the unitary connection ∇ H v ,u of (3.24) on H v ( T M )is given by ∇ H v ,u∂ t α t = − il β t , ∇ H v ,u∂ t β t = − il α t . (4.49)With the operators of Definition 3.1.16 and (3.29) we introduce the abbreviations D := D S ⊗ ∇ H v | C ∞ ( S , H v ( T M )) , D u := D S ⊗ ∇ H v ,u | C ∞ ( S , H v ( T M )) . Then, using the splitting of the total chirality operator as in Lemma 2.2.3 one verifies that D = − iτ t ∇ H v ∂ t , D u = − iτ t ∇ H v ,u∂ t . .4. Torus Bundles over S , Explicit Computations η ( D ) = η (cid:0) D S ⊗ ∇ H v (cid:1) , (4.50)where the factor enters for the same reason as in Remark 3.1.17 (ii) and Remark 1.4.4 (i).Similarly, η ( D u ) = η (cid:0) D S ⊗ ∇ H v ,u (cid:1) . Now let D s := D u + s ( D − D u ) , s ∈ [0 , . Then D s is a family of self-adjoint operators, which is precisely of the form considered in Theorem3.1.20. Hence, the “local variation” of the Eta invariant vanishes , so that the general variationformula of Proposition 1.3.14 reduces to η ( D ) = η ( D u ) + dim(ker D u ) − dim(ker D ) + 2 SF( D s ) s ∈ [0 , . (4.51)To determine the terms appearing in (4.51), we now explicitly compute spec( D s ). If ϕ α α t + ϕ β β t isa section of H v ( T M ), then (4.47), (4.48) and (4.49) imply that D s acts in terms of the coordinatefunctions ( ϕ α , ϕ β ) as D s (cid:18) ϕ α ϕ β (cid:19) = − i (cid:20)(cid:18) ∂ t ∂ t (cid:19) − il (cid:18) s 00 1 − s (cid:19)(cid:21) (cid:18) ϕ α ϕ β (cid:19) . Hence, if ϕ α α t + ϕ β β t is an eigenvector with eigenvalue λ ( s ) ∈ R , then ∂ t (cid:18) ϕ α ϕ β (cid:19) = i (cid:18) λ ( s ) + (1 − s ) l λ ( s ) + (1 + s ) l (cid:19) (cid:18) ϕ α ϕ β (cid:19) =: iT λ ( s ) (cid:18) ϕ α ϕ β (cid:19) , (4.52)which is an ordinary linear differential equation with constant coefficients. Let us assume fromnow on that l = 0. The case l = 0 will be dealt with separately at the end. The characteristicequation for the eigenvalues κ of T λ ( s ) is κ = (cid:0) λ ( s ) + l (cid:1) − s (cid:0) l (cid:1) . Therefore, unless (cid:0) λ ( s ) + l (cid:1) = s (cid:0) l (cid:1) , the matrix T λ ( s ) has two distinct eigenvalues κ and − κ ,and can be brought into diagonal form. Now a solution to (4.52) satisfies the condition (4.46) ifand only if e iκ = ε . Write ε = e iθ with θ ∈ { , π } . Then the condition e iκ = ε is equivalent to κ = 2 πn + θ with n ∈ N . Hence, λ ( s ) is an eigenvalue of D s if and only if λ ( s ) = λ ± n ( s ) := ± q κ n + s (cid:0) l (cid:1) − l , κ n = 2 πn + θ for some n ∈ N .Moreover, unless n = 0 and θ = 0, the standard procedure for solving (4.52) gives us two linearlyindependent solutions for both eigenvalues λ + n ( s ) and λ − n ( s ). In the special case n = 0 and θ = 0we have λ ± ( s ) = ( ± s | l | − l ) , and so T λ ± ( s ) = s (cid:18) ±| l | − l ±| l | + l (cid:19) . 4. 3-dimensional Mapping Tori If s = 0, this matrix vanishes so that the eigenvalues λ ± (0) have multiplicity 2. If s = 0, thismatrix is in triangular form. This implies that the eigenvalues λ ± ( s ) have multiplicity 1. To seethis, consider for example the case l > 0, and T = T λ +0 ( s ) . Then e itT = (cid:18) (cid:19) + it (cid:18) sl (cid:19) , so that the only solutions of (4.52) satisfying the condition (4.46) are constant multiples of( ϕ α , ϕ β ) = (0 , D u coincides with D we see in particular thatspec( D u ) = (cid:8) ± (2 πn + θ ) − l (cid:12)(cid:12) n ∈ N (cid:9) , where all eigenvalues have multiplicity 2. Hence, for Re( z ) > η ( D u , z ) = 2 X n ∈ Z sgn(2 πn + θ − l ) | πn + θ − l | z = 2(2 π ) z X n ∈ Z sgn( n − l − θ π ) | n − l − θ π | z , which up to a factor is the Eta function considered in Proposition C.1.2. Hence, the value at z = 0 of the meromorphic continuation of η ( D u , z ) is given as follows: Let m ∈ Z be such that l π − θ π − m ∈ (0 , . Then η ( D u ) = lπ − θπ − m − . (4.53)This identifies the first term in (4.51).To compute the spectral flow term, we first assume that l > 0. Then the zero eigenvalues of D s for s ∈ (0 , 1) are given by those λ + n ( s ) for which s and n are related by κ n = (1 − s )( l ) . The family λ + n ( s ) is strictly increasing with s and all eigenvalues have multiplicity 2. For thelatter note that since we are assuming that s = 1, the eigenvalues of multiplicity 1, which wehave found in the case θ = 0, are never zero. Therefore, each zero eigenvalue will contribute +2to SF( D s ) s ∈ [0 , . Since 1 − s maps (0 , 1) bijectively onto itself, we have to count the number of n ∈ N for which 0 < κ n < l , or—equivalently—for which − θ π < n < l π − θ π . Now, with m asin (4.53), it is immediate to check that (cid:8) n ∈ N (cid:12)(cid:12) − θ π < n < l π − θ π (cid:9) = ( m, if θ = 0, m + 1 , if θ = π .Note that since we are assuming that l > 0, we certainly have m ≥ θ = 0 and m ≥ − θ = π . Concerning the endpoints of the path, there are no zero eigenvalues for s = 0. For s = 1 we only have one if θ = 0, and this is the eigenvalue λ +0 (1) of multiplicity 1. Putting allinformation together, we find that for l > D u ) − dim(ker D ) + 2 SF( D s ) s ∈ [0 , = ( − m + 1) , if θ = 0,4( m + 1) , if θ = π . .4. Torus Bundles over S , Explicit Computations l > η ( D ) = ( lπ − , if θ = 0, lπ , if θ = π .Let us now assume that l < 0. Then the role of λ + n ( s ) in the preceding discussion is replacedby λ − n ( s ), which strictly decreases with s . Hence, the contribution to the spectral flow is − m as in (4.53) we now have m ≤ − θ (cid:8) n ∈ N (cid:12)(cid:12) − θ π < n < − l π − θ π (cid:9) = − m − . For l < θ = 0, the zero eigenvalue λ − (1) of multiplicity 1 does not contribute to the spectralflow. Therefore, we arrive atdim(ker D u ) − dim(ker D ) + 2 SF( D s ) s ∈ [0 , = ( − m + 1) , if θ = 0,4( m + 1) , if θ = π .Hence, we conclude that for l < η ( D ) = ( lπ + 1 , if θ = 0, lπ , if θ = π .Hence, using (4.45) and (4.50) we have proved Proposition 4.4.7 in the case that l = 0. If l = 0,one easily checks that spec( D ) = (cid:8) πn + θ (cid:12)(cid:12) n ∈ Z (cid:9) , where all eigenvalues have multiplicity 2. This implies that spec( D ) is symmetric, so that η ( D ) =0. We can now combine Propositions 4.4.6 and 4.4.7 to obtain the formula for U (1)-Rho invari-ants for mapping tori with parabolic monodromy. According to Theorem 3.3.16, we have ρ A ( T M ) = 2 Z b ρ A + ρ H A,v ( S ) . Therefore, we arrive at the following Theorem 4.4.8. Let ε = ± and l ∈ Z , and let T M be the mapping torus of the parabolicelement M = ε (cid:0) l (cid:1) . Let A be a flat connection over the mapping torus T M , defined by ν ∈ R with ν / ∈ Z , satisfying ( M t − Id) ν ∈ Z . If l = 0 , the Rho invariant ρ A ( T M ) vanishes. For l = 0 we have ρ A ( T M ) = 2 l (cid:0) P ( ν ) − (cid:1) + ( sgn( l ) , if ε = 10 , if ε = − 4. 3-dimensional Mapping Tori Remark 4.4.9. (i) We want to point out that the assumption that ν / ∈ Z excludes possibly non-trivial flatconnections on T M which restrict to the trivial connection over T . Note that for ellipticelements in Theorem 4.4.4 (ii) we included a discussion. However, in the case of parabolicelements—and also in the hyperbolic case below—the case ν / ∈ Z is much more interestingand a parallel treatment of the remaining case would lead to more notational inconvenienceand a tedious distinction between all cases. Since the insight gained seemed not worth theeffort, we opted to work under the assumption that ν / ∈ Z only.(ii) Note that if ε = 1 in Theorem 4.4.8, then ν l ∈ Z , so that ν = k/l for some k ∈ Z . Hence,the formula for the Rho invariant is the same as the formula for the Rho invariant for aprincipal circle bundle of degree l over T in Theorem 2.3.18. The underlying reason is thatfor ε = 1, the mapping torus T M is at the same time a principal S -bundle of degree l over T , see [91, p. 470]. Now we turn to the generic—and most interesting—case that M is hyperbolic. This section is lessself-contained than the previous sections, since we will deduce the main result from a well-knowntransformation formula for certain generalized Dedekind Eta functions. Since this would leadto far afield, we shall not attempt to give a detailed treatment but refer to the literature for proofs. M -invariant Paths in the upper half plane. Assume that M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) is hyperbolic.As in the proof of Proposition 4.4.3, we know that b, c = 0, and that the fixed points of M op acting on b C are given by α = κ − ac , β = κ − − ac , where κ = (cid:0) a + d + √ ∆ (cid:1) . (4.54)For t ∈ R define σ ( t ) := 1 | κ | t + | κ | − t (cid:0) α | κ | t + β | κ | − t + i | α − β | (cid:1) . (4.55) Lemma 4.4.10. The path σ ( t ) in H lies on the circle n σ ∈ H (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) σ − α + β (cid:12)(cid:12) = (cid:12)(cid:12) α − β (cid:12)(cid:12)o and satisfies M op σ ( t ) = σ ( t + 1) .Proof. We proof the second assertion first. By comparison with (4.40), we thus have to showthat σ ( t + 1) − ασ ( t + 1) − β = κ − σ ( t ) − ασ ( t ) − β (4.56)Now, σ ( t ) − α = 1 | κ | t + | κ | − t (cid:0) α | κ | t + β | κ | − t − α | κ | t − α | κ | − t + i | α − β | (cid:1) = 1 | κ | t + | κ | − t (cid:0) ( β − α ) | κ | − t + i | α − β | (cid:1) , .4. Torus Bundles over S , Explicit Computations σ ( t ) − β = 1 | κ | t + | κ | t (cid:0) ( α − β ) | κ | t + i | α − β | (cid:1) . Let t, t ′ ∈ R , and abbreviate ε := sgn( α − β ) = sgn( c ). Then (cid:0) σ ( t ′ ) − α (cid:1)(cid:0) σ ( t ) − β (cid:1)(cid:0) σ ( t ′ ) − β (cid:1)(cid:0) σ ( t ) − α (cid:1) = (cid:0) − | κ | − t ′ + iε (cid:1)(cid:0) | κ | t + iε (cid:1)(cid:0) | κ | t ′ + iε (cid:1)(cid:0) − | κ | − t + iε (cid:1) = −| κ | − t ′ − t ) − iε (cid:0) | κ | t − | κ | − t ′ (cid:1) −| κ | t ′ − t ) − iε (cid:0) | κ | t ′ − | κ | − t (cid:1) = | κ | − t ′ − t ) . In particular, for t ′ = t + 1 we obtain the formula in (4.56), which in turn shows that σ ( t ) is M -invariant. Moreover, σ ( t ) − ασ ( t ) − β = κ − t σ (0) − ασ (0) − β . Now, as σ (0) = α + β i | α − β | , this implies as in the proof of Proposition 4.4.3 that all points σ ( t ) lie on the circle n σ ∈ H (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) σ − α + β (cid:12)(cid:12) = (cid:12)(cid:12) α − β (cid:12)(cid:12)o . The Rho invariant of the bundle of vertical cohomology groups. Having found an M -invariant path in H we now need to compute the integral over the Rho form and the Rhoinvariant of the bundle of vertical cohomology groups. We start with the latter, which is morestraightforward than in the parabolic case. Proposition 4.4.11. Let T M be the mapping torus of a hyperbolic element M ∈ SL ( Z ) . Endow T M with the metric given by (4.55) , and let A be a flat connection determined by ν = ( ν , ν ) ∈ R with ν / ∈ Z and ( M t − Id) ν ∈ Z . Then ρ H A,v ( S ) = 0 . Proof. Again we know from Corollary 4.3.6 that the twisted cohomology groups of T vanishexcept for the case that the underlying connection is the trivial one. Thus, as in (4.45) ρ H A,v ( S ) = − η (cid:0) D S ⊗ ∇ H v (cid:1) . Moreover, as explained in the proof of Proposition 4.4.7, we only need to study the restriction of D S ⊗ ∇ H v to C ∞ (cid:0) S , H v ( T M ) (cid:1) .In view of the rather complicated formula for σ ( t ) it is inconvenient to work directly with thebasis ( ω σ ( t ) , ω ¯ σ ( t ) ) of H ( T M , g σ ( t ) ), given by Corollary 4.3.6. Instead we define ω α ( t ) := | κ | t − iε | κ | − t ω σ ( t ) + | κ | t + iε | κ | − t ω ¯ σ ( t ) , and ω β ( t ) := | κ | − t + iε | κ | t ω σ ( t ) + | κ | − t − iε | κ | t ω ¯ σ ( t ) , 4. 3-dimensional Mapping Tori where as before ε = sgn( c ). Then (cid:0) ω α ( t ) , ω β ( t ) (cid:1) is a clearly basis of H ( T M , g σ ( t ) ) for each t .Moreover, (4.21) implies that τ t ω α ( t ) = − iω β ( t ) , τ t ω β ( t ) = iω α ( t ) , (4.57)where τ t is the chirality operator defined by σ ( t ). Now, a straightforward calculation—which weskip—shows that ω α ( t ) = | κ | t ( dx + αdy ) , ω β ( t ) = | κ | − t ( dx + βdy ) . (4.58)Thanks to this identity, we obtain—without having to compute the derivatives of ω σ ( t ) and ω ¯ σ ( t ) explicitly—that ∂ t ω α ( t ) = log | κ | · ω α ( t ) , ∂ t ω β ( t ) = − log | κ | · ω β ( t ) . (4.59)From the definition of α and β in (4.54) we immediately see that cα + a = κ and cβ + a = κ − .Moreover, using that ad − bc = 1 and that κ + κ − = a + d , one computes dα + b = dκ − adc + b = κ d − κ − c = κα, dβ + b = . . . = κ − β. This means that (1 , α ) and (1 , β ) are eigenvectors of M t = ( a cb d ) with eigenvalues κ and κ − ,respectively. Therefore, it follows from (4.58) that M ∗ ω α ( t ) = | κ | t (cid:0) ( a + cα ) dx + ( b + dα ) dy (cid:1) = | κ | t (cid:0) κdx + ακdy (cid:1) = sgn( κ ) ω α ( t + 1) , and similarly, M ∗ ω β ( t ) = sgn( κ ) ω β ( t + 1) . Hence, any section of H v ( T M ) → S can be written as ϕ α ( t ) ω α ( t ) + ϕ β ( t ) ω β ( t ) , where ϕ α , ϕ β ∈ C ∞ ( R ) satisfy ϕ α ( t + 1) = sgn( κ ) ϕ α ( t ) , ϕ β ( t + 1) = sgn( κ ) ϕ β ( t )We deduce from (4.57) and (4.59) that the operator D := − iτ t ∂ t on C ∞ (cid:0) S , H v ( T M ) (cid:1) acts interms of the coordinate functions ( ϕ α , ϕ β ) as D (cid:18) ϕ α ϕ β (cid:19) = (cid:18) ∂ t − log | κ |− ∂ t − log | κ | (cid:19) (cid:18) ϕ α ϕ β (cid:19) Hence, it becomes clear that if ( ϕ α , ϕ β ) defines an eigenvector of D with eigenvalue λ , then( ϕ α , − ϕ β ) gives rise to an eigenvector with eigenvalue − λ . This means that spec( D ) is sym-metric, so that η ( D ) = 0. Since the operator D is precisely the restriction of D S ⊗ ∇ H v to C ∞ (cid:0) S , H v ( T M ) (cid:1) , we obtain the desired result. The Logarithm of the Dedekind Eta Function. The discussion of the Rho form is moretransparent, if we consider the twisted and the untwisted Eta forms separately. We start with theuntwisted case. This case has already received a far-reaching treatment in the beautiful article[3], from which we borrow the main ideas. .4. Torus Bundles over S , Explicit Computations Dedekind Eta function is defined as η ( σ ) := q σ ∞ Y n =1 (1 − q nσ ) , σ ∈ H , q σ := e πiσ , see [64, Sec. 18.5]. As in [3] we use the bold symbol η to avoid confusion with an Eta functionin the sense of Definition 1.3.1. Using the power series expansionlog(1 − z ) = − ∞ X m =1 z m m , | z | < , one can define a logarithm of η ( σ ) bylog η ( σ ) := πiσ − X n> X m> q mnσ n . (4.60)The sum in (4.60) is of the same form as the one in defining E ( σ ) in Theorem 4.3.12 (i). Hence,we can make Remark 4.3.13 (i) more precise and note thatlog η ( σ ) = πiσ − E ( σ )2 = 12 (cid:0) πiσP (0) − E ( σ ) (cid:1) , (4.61)where as always, P is the second periodic Bernoulli function. Therefore, we can reformulateTheorem 4.3.14 (i) as Z b η = 2 π Im (cid:2) log η (cid:0) M op σ (0) (cid:1) − log η (cid:0) σ (0) (cid:1)(cid:3) − π Z ˙ σ ( t ) σ ( t ) dt, (4.62)where σ ( t ) = σ ( t )+ iσ ( t ) is an M -invariant path in H . Hence, the Eta invariant of T M is relatedto the transformation property of log η under modular transformations.The study of this has a long history, starting with Dedekind’s work [34]. There are severaldifferent proofs of the following theorem, see for example [92] for references and a beautifullysimple proof. A short discussion of the Dedekind sums appearing below is included in AppendixC.2. Theorem 4.4.12 (Dedekind) . Let σ ∈ H , and let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) with c = 0 . Then log η ( M op σ ) − log η ( σ ) = 12 log (cid:16) cσ + a sgn( c ) i (cid:17) + πi (cid:16) a + d c − sgn( c ) s ( a, c ) (cid:17) , where the logarithm on the right hand side is the standard branch on C \ R − , and s ( a, c ) is theclassical Dedekind sum, see (C.21) , s ( a, c ) = | c |− X k =1 P (cid:0) akc (cid:1) P (cid:0) kc (cid:1) . Remark. Note that since we have defined the action of SL ( Z ) on H using the involution M M op as in Lemma 4.3.7, we have to interchange a and d in the classical formula. However, s ( a, c )is not affected by this, see (C.22).68 4. 3-dimensional Mapping Tori The Untwisted Eta Invariant. From Theorem 4.4.12 we can deduce the formula for the Etainvariant of T M for hyperbolic M . The formula we shall obtain appears as a signature cocycle forthe mapping class group the formula already in [72], and as a signature defect in [52]. However,its derivation using Theorem 4.4.12 and the adiabatic limit formula as well as an explanation ofthe relation among these different invariants are due to Atiyah [3]. Theorem 4.4.13 (Atiyah, Hirzebruch, Meyer) . Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) by hyperbolic. Let g be the metric on T M defined by by σ ( t ) as in (4.55) , and let B ev be the associated odd signatureoperator on T M . Then η ( B ev ) = a + d c − c ) s ( a, c ) − sgn (cid:0) c ( a + d ) (cid:1) . Proof. Let g ε be the adiabatic metric associated to g , and denote by B ev ε the correspondingadiabatic family of odd signature operators. It follows from Proposition 4.4.11 and its proof thatthe Eta invariant of the bundle of vertical cohomology groups vanishes. We thus deduce fromTheorem 3.3.14 thatlim ε → η ( B ev ε ) = 2 Z b η = 4 π Im (cid:2) log η (cid:0) M op σ (0) (cid:1) − log η (cid:0) σ (0) (cid:1)(cid:3) − π Z ˙ σ ( t ) σ ( t ) dt, where we have used (4.62) for the last equality. Hence, Theorem 4.4.12 implies thatlim ε → η ( B ev ε ) = a + d c − c ) s ( a, c ) + 1 π h (cid:16) cσ (0) + a sgn( c ) i (cid:17) − Z ˙ σ ( t ) σ ( t ) dt i . (4.63)We now note that it follows from Lemma 4.4.10 that σ ( t ) = α + β + | α − β | e iϕ ( t ) , ϕ ( t ) = arg (cid:0) σ ( t ) − α + β (cid:1) . Here, the argument function is such that for z ∈ C \ R − , one has arg( z ) ∈ ( − π, π ). We obtain Z ˙ σ ( t ) σ ( t ) dt = − arg (cid:0) σ (1) − α + β (cid:1) + arg (cid:0) σ (0) − α + β (cid:1) . Using the explicit formula (4.55) for σ ( t ), one finds that arg (cid:0) σ (0) − α + β (cid:1) = π , and σ (1) − α + β α − β )( κ − κ − ) κ + κ − + i | α − β | κ + κ − . Hence, using the abbreviations x := sgn( c )( κ + κ − ) , y := ( κ − κ − ) , (4.64)we find that arg (cid:0) σ (1) − α + β (cid:1) = arg (cid:0) xy + i (cid:1) . Note that y = x − 1, so that 2 xy + i = i ( x − iy ) . Moreover, y > y < | x | so thatarg( x + iy ) ∈ ( (cid:0) , π (cid:1) , if x > , (cid:0) π , π (cid:1) , if x < . .4. Torus Bundles over S , Explicit Computations Z ˙ σ ( t ) σ ( t ) dt = arg (cid:0) − i ( x + iy ) (cid:1) + π = 2 arg( x + iy ) − ( , if x > , π, if x < . (4.65)On the other hand, it follows from the definition of α and β that cσ (0) + a = c (cid:0) α + β + i | α − β | (cid:1) + a = (cid:0) κ + κ − + i sgn( c )( κ − κ − ) (cid:1) = sgn( c )( x + iy ) , with x and y as in (4.64). Since we are using the standard branch of the logarithm, we haveIm log (cid:16) cσ (0) + a sgn( c ) i (cid:17) = arg (cid:0) − i ( x + iy ) (cid:1) = arg( x + iy ) − π . Combining this with (4.65) we find that2 Im log (cid:16) cσ (0) + a sgn( c ) i (cid:17) − Z ˙ σ ( t ) σ ( t ) dt = − sgn( x ) π. As κ + κ − = a + d we have sgn( x ) = sgn (cid:0) c ( a + d ) (cid:1) so that using (4.63), we finally arrive atlim ε → η ( B ev ε ) = a + d c − c ) s ( a, c ) − sgn (cid:0) c ( a + d ) (cid:1) . Hence, it remains to argue that in the case at hand, η ( B ev ) = lim ε → η ( B ev ε ) , This is precisely [3, Lem. 5.56] and we will not repeat the argument here. Remark. The proof of Theorem 4.4.13 in [3] is along different lines than our discussion. In[3, Thm. 5.60], the Eta invariant of T M is seen to be equal to a large number of quantities,including a signature defect. Then in [3, Sec. 6], the formula for η ( B ev ) is obtained by explicitlyconstructing a bounding manifold and a computation of the signature defect. In particular, thetransformation formula of the Dedekind Eta function in Theorem 4.4.12 is not used. However,since our focus is the application of the adiabatic limit formula to compute Eta respectively Rhoinvariants, we have to use Theorem 4.4.12 in some form. The Generalized Dedekind Eta Function. To obtain the formula for U(1)-Rho invariantsof T M in the spirit of the discussion of the untwisted case, we now need a twisted version of η ( σ )and a transformation formula for its logarithm. Fortunately, a corresponding treatment can befound in [35]. As in [35, p. 38] we make the following Definition 4.4.14. For g, h ∈ R and σ ∈ H let z := gσ + h , q z = e πiz and q σ = e πiσ . Define η g,h ( σ ) := ξ ( g, h ) q P g )2 σ (1 − q z ) ∞ Y n =1 (1 − q z q mσ )(1 − q − z q mσ ) , where ξ ( g, h ) := ( e πi ( g − ) P ( h ) , if g ∈ Z e πi [ g ] P ( h ) , if g / ∈ Z . As always, P is the first periodic Bernoulli function, and [ g ] is the largest integer less or equalthan g .70 4. 3-dimensional Mapping Tori Remark 4.4.15. (i) Since σ ∈ H , the term q mσ decays exponentially with m . This implies that η g,h ( σ ) iswell-defined.(ii) Definition 4.4.14 might look slightly different than the formula in [35]. Yet, writing g and h as ˜ g/f and ˜ h/f with integers ˜ g, ˜ h and f , the function η g,h ( σ ) is easily seen to be equalto what is denoted η ˜ g, ˜ h ( σ ) in loc.cit.(iii) The reason for the factor ξ ( g, h ) is to achieve that η g,h ( σ ) depends on g and h only modulo Z , see [35, p. 39].(iv) The Dedekind Eta function η ( σ ) is not equal to η , ( σ ), since the latter obviously vanishes.Dropping the factor 1 − q z from the definition of η g,h ( σ ) one would get a direct generalizationof η ( σ ) . However, Definition 4.4.14 allows us to use the results of [35] without too manychanges.One defines log η g,h ( σ ) in analogy to (4.60), see [35, p. 40]. Definition 4.4.16. Let ( g, h ) ∈ Q \ Z . If 0 ≤ g < 1, we definelog η g,h ( σ ) := πi (cid:0) ϕ ( g, h ) + P ( g ) (cid:1) − X n> n q nz − X m> X n> n ( q z + q − z ) n q mnσ , where ϕ ( g, h ) = ( − P ( h ) , if g = 0 , , if g = 0.For general g we define log η g,h ( σ ) := log η g − [ g ] ,h ( σ ) . The transformation formula of log η g,h ( σ ) is then given by [35, Thm. 1], Theorem 4.4.17 (Dieter) . Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) with c = 0 , let ( g, h ) ∈ Q \ Z , and define (cid:18) g ′ h ′ (cid:19) := (cid:18) a − c − b d (cid:19) (cid:18) gh (cid:19) . Then for all σ ∈ H log η g ′ ,h ′ ( M op σ ) − log η g,h ( σ ) = πi (cid:16) ac P ( g ) + dc P ( g ′ ) − c ) s g ′ ,h ′ ( d, c ) (cid:17) , where s g ′ ,h ′ ( d, c ) is the generalized Dedekind sum, see Definition C.2.5, s g ′ ,h ′ ( d, c ) = | c |− X k =0 P (cid:0) d k + g ′ c + h ′ (cid:1) P (cid:0) k + g ′ c (cid:1) . Remark. (i) As for the transformation formula for log η ( σ ), we have formulated Theorem 4.4.17 in termsof M op acting on H , which means that a and d have been interchanged in comparison to[35, Thm. 1]. .4. Torus Bundles over S , Explicit Computations η ( σ ) carries over with minor changes inthe case that g ∈ Z . It would be interesting to know if there is a proof for the general caseof Theorem 4.4.17 along the lines of [92]. Application to the Rho Form. As in the untwisted case, the structure of the formula inDefinition 4.4.16 resembles what we have encountered in Theorem 4.3.14 (ii). In fact, Lemma 4.4.18. With the notation of Theorem 4.3.12 and Remark 4.3.15, we have for all ν ∈ Q \ Z . Im (cid:0) log η ν , − ν ( σ ) (cid:1) = Im (cid:0) πiσP ( ν ) − E ν ( σ ) (cid:1) . Proof. Both sides of the equation are defined in terms of ν − [ ν ] and are Z -periodic in ν Hence,we can assume that ν ∈ [0 , . Then, if ν = 0, the relation is immediate—and clearly holds forthe real parts as well. If ν = 0, one observes thatIm (cid:16) πiP ( ν ) − X n> n e − πiν (cid:17) = − Im (cid:16) X n> n cos(2 πν ) (cid:17) = 0 , (4.66)where the first equality follows from the Fourier series expansion P ( ν ) = ν − = − πi X n> n (cid:0) e πiν − e − πiν (cid:1) , ν / ∈ Z , whose proof is a standard exercise. Then (4.66) implies that the imaginary parts of the extraterms in Definition 4.4.16 cancel each other out so that the result is indeed the right hand sideof the formula in Lemma 4.4.18. Proposition 4.4.19. Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) be hyperbolic, and ν ∈ R \ Z with (Id − M t ) ν ∈ Z . Let A be the corresponding flat U(1) -connection over the mapping torus T M , and use (4.55) to define a metric. Then Z b η A = a + dc P ( ν ) − c ) s ν ,ν ( a, c ) . Proof. Certainly, we want to use Lemma 4.4.18 to apply Theorem 4.4.17 to part (ii) of Theorem4.3.14. We first note that (Id − M t ) ν ∈ Z implies that ν ∈ Q , so that we are precisely in thesituation of Lemma 4.4.18. Abbreviate ν ′ := M t ν , so that by assumption ν − ν ′ ∈ Z . Thus,according to Definition 4.4.16,log η ν ′ , − ν ′ ( σ ) = log η ν , − ν ( σ ) , for all σ ∈ H . Moreover, P ( ν ′ ) = P ( ν ), and (cid:18) ν ′ − ν ′ (cid:19) = (cid:18) a − c − b d (cid:19) (cid:18) ν − ν (cid:19) . 4. 3-dimensional Mapping Tori Hence, Lemma 4.4.18, Theorem 4.4.17 and Theorem 4.3.14 (ii) imply that Z b η A = a + dc P ( ν ) − c ) s ν ′ , − ν ′ ( d, c ) . Now, s ν ′ , − ν ′ ( d, c ) = | c |− X k =0 P (cid:0) d k + ν ′ c − ν ′ (cid:1) P (cid:0) k + ν ′ c (cid:1) = | c |− X k =0 P (cid:0) dk + dν ′ − cν ′ c (cid:1) P (cid:0) k + aν + cν c (cid:1) = | c |− X k =0 P (cid:0) k + ν c (cid:1) P (cid:0) a k + ν c + ν (cid:1) = s ν ,ν ( a, c ) , where we have rewritten ν ′ in terms of ν , and then used that { ak | k = 0 , . . . , | c | − } is arepresentation system of Z modulo c , see Appendix C.2 for more details. This implies the desiredresult. Rho Invariants of Hyperbolic Mapping Tori. After this preparation, we finally arrive atthe main result of this section. Theorem 4.4.20. Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) be hyperbolic, and let ν ∈ R \ Z satisfy (cid:18) m m (cid:19) = (Id − M t ) (cid:18) ν ν (cid:19) ∈ Z . Let A be the corresponding flat U(1) -connection over the mapping torus T M , and define r ∈{ , . . . | c | − } by requiring that m ≡ r ( c ) . Then ρ A ( T M ) = a + d ) − c (cid:0) P ( ν ) − (cid:1) − | c |− r X k =1 P (cid:0) dkc (cid:1) + sgn (cid:0) c ( a + d ) (cid:1) − sgn( c ) δ ( ν ) (cid:0) − δ ( m c ) (cid:1) − P (cid:0) dm c (cid:1) − δ ( ν ) (cid:16) P (cid:0) m c (cid:1) − P (cid:0) dm c (cid:1)(cid:17) , where δ is the characteristic function of R \ Z .Proof. According to Proposition 4.4.11, the Rho invariant of the bundle of vertical cohomologygroups vanishes. Also, since the base is 1-dimensional, Dai’s correction term is zero. Hence, wecan use the general formula for Rho invariants in Theorem 3.3.16, together with the formulæ ofTheorem 4.4.13 and Proposition 4.4.19, to deduce that ρ A ( T M ) = a + d ) c (cid:0) P ( ν ) − (cid:1) − c ) (cid:0) s ν ,ν ( a, c ) − s ( a, c ) (cid:1) + sgn (cid:0) c ( a + d ) (cid:1) . (4.67)A formula for the difference of s ν ,ν ( a, c ) and s ( a, c ) is given in Proposition C.2.7. With r ∈{ , . . . | c | − } such that m ≡ r ( c ), we have s ν ,ν ( a, c ) − s ( a, c ) = | c | (cid:0) P ( ν ) − (cid:1) + | c |− r X k =1 P (cid:0) dk | c | (cid:1) + P (cid:0) dm | c | (cid:1) + δ ( ν ) (cid:16) P (cid:0) m | c | (cid:1) − P (cid:0) dm | c | (cid:1)(cid:17) + δ ( ν ) (cid:0) − δ ( m c ) (cid:1) . We now insert this into (4.67). Since P is odd, the factor sgn( c ) in front of s ν ,ν ( a, c ) − s ( a, c )cancels the norms in the denominators. Then we arrive at the formula of Theorem 4.4.20. .4. Torus Bundles over S , Explicit Computations Immediate Applications. The main formula in Theorem 4.4.20 might look more complicatedthan the intermediate formula (4.67). Yet, it is more satisfactory from a computational point ofview, since the sum P | c |− rk =1 P (cid:0) dkc (cid:1) is much easier to compute than the individual Dedekind sums.For concreteness, let us use Theorem 4.4.20 for some explicit computations. Example. (i) Consider M = (cid:18) − − (cid:19) , so that Id − M t = (cid:18) − − (cid:19) . Since det(Id − M t ) = 5, a pair ν = ( ν , ν ) ∈ R with m = (Id − M t ) ν ∈ Z has to consistof rational numbers with denominator 5. Recall that we exclude the case ν ∈ Z and mayrestrict to ν ∈ [0 , . One then verifies that to obtain a full set of representatives for theflat connections on T M we are interested in, we need to consider pairs ν and m with ν = ( , ) ( , ) ( , ) ( , ) m = (0 , 1) (1 , 0) (1 , 1) (2 , . As c = 1 and ν = 0, the formula of Theorem 4.4.20 reduces to ρ A ( T M ) = 2 (cid:0) ( a + d ) − (cid:1) ( ν − ν ) − sgn (cid:0) c ( a + d ) (cid:1) − sgn( c ) = − ν − ν ) . Hence, one computes ν = 15 25 35 45 ,ρ A ( T M ) = 85 125 125 85 . (ii) As a further example, let us consider M = (cid:18) (cid:19) , so that Id − M t = (cid:18) − − − − (cid:19) . Now, one easily verifies that we can represent the conjugacy classes of flat connections ofinterest by ν = (0 , ) ( , 0) ( , ) ,m = ( − , − 1) ( − , − 1) ( − , − ,r = 2 3 1 . Then ρ A ( T M ) = 2( ν − ν ) − − r X k =1 P (cid:0) k (cid:1) + 1 − P (cid:0) r (cid:1) − δ ( ν ) (cid:16) P (cid:0) r (cid:1) − P (cid:0) r (cid:1)(cid:17) . For ν = (0 , ), we have r = 2, and so ρ A ( T M ) = − (cid:0) P ( ) + P ( ) (cid:1) + 1 − P ( ) = 0 , for ν = ( , 0) with r = 3, ρ A ( T M ) = 2 (cid:0) − (cid:1) − P ( ) + 1 − P ( ) = 0 , 4. 3-dimensional Mapping Tori and lastly, for ν = ( , ) with r = 1, ρ A ( T M ) = 2 (cid:0) − (cid:1) + 1 − P ( ) = 1 , where we have used that P k =1 P (cid:0) k (cid:1) = 0, see (C.18).Recall from Corollary 1.5.2 (ii) that the non-integer part of the Rho invariant on a 3-dimensional manifold is essentially the Chern-Simons invariant associated to the Chern character.More precisely, ρ A ( T M ) ≡ A ) mod Z . (4.68)In the case of torus bundles over surfaces, computations for Chern-Simons invariants are containedin [42, 54, 57]. For the case of U(1)-connections, see for example [42, Thm. 7.22]. Corollary 4.4.21. Under the assumptions of Theorem 4.4.20 we have ρ A ( T M ) ≡ (cid:0) ν m − ν m (cid:1) mod Z . Proof. First note that for all k ∈ Z P (cid:0) kc (cid:1) ≡ kc mod Z . In particular, 4 | c |− r X k =1 P (cid:0) dkc (cid:1) ≡ dc ( | c |− r )( | c |− r +1)2 ≡ (cid:0) dm c − dm c (cid:1) mod Z . Here, we have used that by definition r ≡ m ( c ). We also note that (Id − M t ) ν = m meansexplicitly that (cid:18) m m (cid:19) = (cid:18) − a − c − b − d (cid:19) (cid:18) ν ν (cid:19) , (cid:18) ν ν (cid:19) = a + d − (cid:18) d − − c − b a − (cid:19) (cid:18) m m (cid:19) (4.69)Let us assume now that ν ∈ Z . Then ρ A ( T M ) ≡ − (cid:0) dm c − dm c (cid:1) − dm c ≡ − dm c mod Z . It follows from (4.69) that m c ≡ − ν modulo Z , and dm ≡ m modulo Z . Therefore, ρ A ( T M ) ≡ ν m mod Z , which is the claim of Corollary 4.4.21 in the case that ν ∈ Z . If ν / ∈ Z , we have ρ A ( T M ) ≡ a + d ) − c ( ν − ν ) − (cid:0) dm c − dm c (cid:1) − m c mod Z . (4.70)From (4.69) we know that ( a + d ) − c ν = ( d − m c − m . Inserting this into (4.70), one finds that ρ A ( T M ) ≡ (cid:16)(cid:0) ( d − m c − m (cid:1) ( ν − 1) + ( d − m c − dm c (cid:17) mod Z ≡ (cid:0) − ν m + dm c ( ν − m ) − m c ν (cid:1) mod Z ≡ (cid:0) − ν m + dm ν + ( ad − m c ν (cid:1) mod Z , .4. Torus Bundles over S , Explicit Computations ν − m = aν + cν , see (4.69). Now using that ad − bc and observing that bν + dν ≡ ν modulo Z , we arrive at ρ A ( T M ) ≡ (cid:0) − ν m + m ν (cid:1) mod Z . Remark. The formula of Corollary 4.4.21 also holds in the parabolic case: Let ε (cid:0) l (cid:1) with l ∈ Z and ε = ± 1, and let ν = ( ν , ν ) ∈ R \ Z satisfy m = (Id − M t ) ν ∈ Z . Then, if ε = 1, − lν = m ∈ Z , m = 0 , so that 2 (cid:0) − ν m + m ν (cid:1) = 2 lν . According to Theorem 4.4.8, this is congruent to ρ A ( T M ) modulo Z . If ε = − 1, then2 ν = m ∈ Z , − lν + 2 ν = m ∈ Z , so that again 2 (cid:0) − ν m + m ν (cid:1) = 2 lν − ν ν + 2 ν ν = 2 lν . Jeffrey’s Conjecture. We end the main discussion of this thesis with a remark concerninga possible perspective for further research. According to (4.68), Corollary 4.4.21 identifies theChern-Simons invariant only modulo Z , which might seem a bit disappointing. Moreover, themethods of [42, 54, 57] to obtain the formula for the Chern-Simons invariant are much lessinvolved than what we have presented. However, this is precisely the real strength of Theorem4.4.20. It can be used to compute the difference ρ A ( T M ) − A ) ∈ Z .Recall from Corollary 1.5.2 (i) that this is essentially the spectral flow of the odd signatureoperator between the trivial connection and A . For this reason it is promising that a generalizationof Theorem 4.4.20 to higher gauge groups might be a way to prove Jeffrey’s conjecture about themod 4 reduction of this spectral flow term, see [54, Conj. 5.8].76 4. 3-dimensional Mapping Tori ppendix A Characteristic Classes andChern-Simons Forms Although we assume that the reader is familiar with the theory of characteristic classes, weinclude a short survey of Chern-Weil theory in the way we will use it. We closely follow [13,Sec. 1.5] and [99, Ch. 1] to which we also refer for more details. We place some emphasis ontransgression forms and formulate the results about Chern-Simons invariants, which we use inSection 1.5. A.1 Chern-Weil Theory A.1.1 Connections and Characteristic Forms We start with a short algebraic preliminary. Let V be a complex vector space. For m ∈ N consider (Λ ev C m ) ⊗ V as module over the commutative algebra Λ ev C m . Then any element T ∈ Λ ev C m ⊗ End( V ) may be viewed as a module endomorphism. Upon choosing a basis for V ,this is a matrix with entries in Λ ev C m . In this way we can define expressions like T n and det T .We extend the trace tr V : End( V ) → C on V in the natural way to a tracetr V : (Λ ev C m ) ⊗ End( V ) → Λ ev C m . Let f ( z ) = P n ≥ a n z n be a formal power series with coefficients a n in C , and assume that T ∈ (Λ • +2 C m ) ⊗ End( V ). An endomorphism T of this form is nilpotent, so that we can define f ( T ) = X n ≥ a n T n ∈ (Λ ev C m ) ⊗ End( V ) . The following algebraic result is the main tool we use for defining the characteristic forms weneed. We skip the easy proof. Lemma A.1.1. For every T ∈ (Λ • +2 C m ) ⊗ End( V ) , det (cid:0) T (cid:1) = exp (cid:0) tr V (cid:2) log(1 + T ) (cid:3)(cid:1) , Appendix A. Characteristic Classes and Chern-Simons Forms where the exp( . ) is taken in the algebra Λ ev C m and the logarithm is defined using the formalpower series log(1 + z ) = X n ≥ ( − n n + 1 z n +1 . It follows from Lemma A.1.1 that if f ( z ) = 1 + P n ≥ a n z n is a normalized formal powerseries, then for all T ∈ (Λ • +2 C m ) ⊗ End( V )det (cid:0) f ( T ) (cid:1) = exp (cid:0) tr V (cid:2) log f ( T ) (cid:3)(cid:1) . Motivated by this we also definedet / (cid:0) f ( T ) (cid:1) := exp (cid:0) tr V (cid:2) log f ( T ) (cid:3)(cid:1) . (A.1) Characteristic Forms of Complex Vector Bundles. Now let E → M be a complex vectorbundle over an m -dimensional manifold M . Let ∇ be a connection on E with curvature F ∇ ∈ Ω (cid:0) M, End( E ) (cid:1) . In this context Ω ev (cid:0) M, End( E ) (cid:1) plays the role of Λ ev C m ⊗ End( V ) in the aboveconsiderations.If T ∈ C ∞ (cid:0) M, End( E ) (cid:1) , then the commutator [ ∇ , T ] is an element of Ω (cid:0) M, End( E ) (cid:1) . Wecan extend this to a derivation[ ∇ , · ] : Ω • (cid:0) M, End( E ) (cid:1) → Ω • +1 (cid:0) M, End( E ) (cid:1) , by requiring that for α ∈ Ω • ( M ) of pure degree | α | , and T ∈ Ω • (cid:0) M, End( E ) (cid:1) ,[ ∇ , α ∧ T ] = ( dα ) ∧ T + ( − | α | α ∧ [ ∇ , T ] . Then it is easy to check that for every such T ,tr E [ ∇ , T ] = d (cid:0) tr E T (cid:1) . (A.2)Let f ( z ) = P n ≥ a n z n be a formal power series. We define f ( ∇ ) := X n ≥ a n (cid:0) i π F ∇ (cid:1) n ∈ Ω ev (cid:0) M, End( E ) (cid:1) . From (A.2) and the fact that [ ∇ , F ∇ ] = 0, we obtain d tr E (cid:0) f ( ∇ ) (cid:1) = tr E (cid:2) ∇ , f ( ∇ ) (cid:3) = 0 . (A.3) Definition A.1.2. Let E be a complex vector bundle over M with connection ∇ , and let f ( z ) = P n ≥ a n z n be a formal power series. Then we define the characteristic form of ∇ associated to f by tr E (cid:2) f ( ∇ ) (cid:3) = tr E (cid:2) X n ≥ a n (cid:0) i π F ∇ (cid:1) n (cid:3) ∈ Ω ev ( M ) . Definition A.1.3. Let E be a complex vector bundle over M with connection ∇ .(i) The characteristic form associated to exp( z ) is called the Chern character form ch( E, ∇ ) := tr E (cid:2) exp (cid:0) i π F ∇ (cid:1)(cid:3) ∈ Ω ev ( M ) . .1. Chern-Weil Theory c ( E, ∇ ) := det (cid:0) i π F ∇ (cid:1) = exp (cid:0) tr E (cid:2) log(1 + i π F ∇ ) (cid:3)(cid:1) ∈ Ω ev ( M )is called the total Chern form .(iii) The j -th Chern form c j ( E, ∇ ) ∈ Ω j ( M )is defined as the component of degree 2 j of the total Chern form, i.e., c ( E, ∇ ) = [ m/ X j =0 c j ( E, ∇ ) = 1 + c ( E, ∇ ) + c ( E, ∇ ) + . . . . Remark. (i) Note that it follows form Lemma A.1.1 that the total Chern form fits into the frameworkof Definition A.1.2 if we take f ( z ) = log(1 + z ) and exponentiate in Ω ev ( M ) after takingtr E (cid:2) f ( ∇ ) (cid:3) . Since d (cid:0) exp ◦ tr E (cid:2) f ( ∇ ) (cid:3)(cid:1) = d (cid:0) tr E (cid:2) f ( ∇ ) (cid:3)(cid:1) ∧ (cid:0) exp ◦ tr E (cid:2) f ( ∇ ) (cid:3)(cid:1) , it follows from (A.3) that this construction also gives closed forms.(ii) When we decompose the Chern character form into its homogeneous componentsch( E, ∇ ) = [ m/ X j ch j ( E, ∇ ) , then one easily finds relations between ch j and the Chern forms c j for small j . Here, weare dropping the reference to E and ∇ for the moment. For example,ch = rk E, ch = c , ch = c − c , ch = (3 c − c c + c ) , . . . . (iii) If E and E are two complex vector bundles over M endowed with connections ∇ and ∇ , the following relations are immediate.a) The Chern character form satisfiesch( E ⊕ E , ∇ ⊕ ∇ ) = ch( E , ∇ ) + ch( E , ∇ )and ch( E ⊗ E , ∇ ⊗ ⊗ ∇ ) = ch( E , ∇ ) ∧ ch( E , ∇ )b) The total Chern form satisfies c ( E ⊕ E , ∇ ⊕ ∇ ) = c ( E , ∇ ) ∧ c ( E , ∇ )80 Appendix A. Characteristic Classes and Chern-Simons Forms (iv) If E is equipped with a metric and ∇ is a compatible connection, then the associatedChern forms and the Chern character form are R valued forms. Moreover, assume that E is of rank k and admits an SU( k )-structure. The latter means that the determinant linedet( E ) = Λ k E is trivial. Then for every compatible connection ∇ ch j +1 ( E, ∇ ) = 0 , which is due to the fact that the trace of elements in the Lie algebra su ( k ) vanishes. Inparticular, the first Chern form c ( E, ∇ ) is trivial for SU( k )-bundles. Characteristic Forms of Real Vector Bundles. For our purposes it is enough to definethe characteristic forms which are obtained by complexifying the bundle and the connection. Inparticular, we need not restrict to orthogonal connections as one would need to define the Eulerclass. Definition A.1.4. Let M be an m -dimensional manifold, and let ∇ be a connection on a realvector bundle E → M . Let E C := E ⊗ C be endowed with the induced connection ∇ C .(i) We call the characteristic form p ( E, ∇ ) := det / (cid:0) i π F ∇ C ) (cid:1) = exp (cid:16) tr V h log (cid:0) i π F ∇ C ) (cid:1)i(cid:17) the total Pontrjagin form of ∇ .(ii) The j -th Pontrjagin form p j ( E, ∇ ) ∈ Ω j ( M )is defined as the component of degree 4 j of the total Pontrjagin form, i.e., p ( E, ∇ ) = [ m/ X j =0 p j ( E, ∇ ) = 1 + p ( E, ∇ ) + p ( E, ∇ ) + . . . . (iii) We define the Hirzebruch b L -form as b L ( E, ∇ ) := det / i π F ∇ C tanh (cid:0) i π F ∇ C (cid:1) ! ∈ Ω • ( M ) . (iv) Moreover, the b A -form is defined as b A ( E, ∇ ) := det / i π F ∇ C sinh (cid:0) i π F ∇ C (cid:1) ! ∈ Ω • ( M ) . Remark. (i) The definition of the characteristic forms above varies in the literature. First of all, someauthors, e.g. [13], drop the normalizing constants i π from the definition. We include themto get integer valued characteristic classes. Moreover, the b L -form is related to the classicalHirzebruch L -form via 2 n · b L ( E, ∇ ) [4 n ] = L ( E, ∇ ) [4 n ] , (A.4)where ( . . . ) [ n ] means taking the n -form component of a differential form. .1. Chern-Weil Theory b L -form and the b A -form are well-defined. We give somebrief remarks and refer to [73, App. B] for more details. Recall that the Bernoulli numbers B n can be defined by the following generating function: ze z − ∞ X n =0 B n z n n ! , | z | < π, (A.5)see [29, Sec. 9.1]. With respect to this sign convention, the first non-trivial B n are givenby B = 1 , B = − , B = , B = − , B = , . . . Using (A.5), one finds that for | z | < πz/ z/ 2) = 1 + X n ≥ n )! B n z n = 1 + z − z + . . . and z/ z/ 2) = 1 + X n ≥ n − − n − (2 n )! B n z n = 1 − z + z + . . . . This shows that both are normalized power series, which implies that the Hirzebruch b L -formand the b A -form are well-defined. Moreover, if dim M = 4, then b L ( E, ∇ ) = 1 + p ( E, ∇ ) , b A ( E, ∇ ) = 1 − p ( E, ∇ ) . (A.6)(iii) Note that if E is endowed with a bundle metric, and compatible connection ∇ , then ∇ C on E C satisfies F t ∇ C = − F ∇ C . Thus, (cid:0) i π F ∇ C (cid:1) t = 1 − i π F ∇ C . Hence also, det / (cid:0) i π F ∇ C (cid:1) = det / (cid:0) − i π F ∇ C (cid:1) , and so [ m/ X l =0 c l ( E C , ∇ C ) = det (cid:0) i π F ∇ C (cid:1) = det / (cid:0) i π F ∇ C (cid:1) det / (cid:0) i π F ∇ C (cid:1) = det / (cid:0) − ( i π F ∇ C ) (cid:1) = [ m/ X j =0 ( − j p j ( E, ∇ ) . From this we deduce that c j ( E C , ∇ C ) = ( − j p j ( E, ∇ ) , and c j − ( E C , ∇ C ) = 0 . (iv) If E can be written as E = E ⊕ E , and ∇ decomposes as ∇ = ∇ ⊕∇ , the total Pontrjaginform p and the forms introduced in Definition A.1.4 satisfy p ( E ) = p ( E ) ∧ p ( E ) , b L ( E ) = b L ( E ) ∧ b L ( E ) , b A ( E ) = b A ( E ) ∧ b A ( E ) , where we are dropping the references to the connections.82 Appendix A. Characteristic Classes and Chern-Simons Forms A.1.2 Transgression and Characteristic Classes As we have seen in (A.3), characteristic forms associated to a formal power series f as in DefinitionA.1.2 are closed. Therefore, they define de Rham cohomology classes. The famous Chern-Weiltheorem states that the difference tr E (cid:2) f ( ∇ ) (cid:3) − tr E (cid:2) f ( ∇ ) (cid:3) for two connections ∇ and ∇ on E is an exact form. Therefore, the cohomology class isindependent of the connection. We refer to [13, Prop. 1.41] for a proof of the following result. Theorem A.1.5. Let E → M be a complex vector bundle over a manifold M , and let f ( z ) = P n ≥ a n z n be a formal power series. If ∇ t is a smooth path of connections on E , then ddt tr E (cid:2) f ( ∇ t ) (cid:3) = d tr E (cid:2) i π ( ddt ∇ t ) ∧ f ′ ( ∇ t ) (cid:3) . In particular, if a := ∇ − ∇ ∈ Ω (cid:0) M, End( E ) (cid:1) is the difference of two connections, then tr E (cid:2) f ( ∇ ) (cid:3) − tr E (cid:2) f ( ∇ ) (cid:3) = d Z i π tr E (cid:2) a ∧ f ′ ( ∇ + ta ) (cid:3) dt Therefore, we have an equality of cohomology classes h tr E (cid:2) f ( ∇ ) (cid:3)i = h tr E (cid:2) f ( ∇ ) (cid:3)i ∈ H ev ( M ) . Definition A.1.6. Let E → M be a complex vector bundle over a manifold M , and let f ( z ) = P n ≥ a n z n be a formal power series.(i) Let ∇ be an arbitrary connection on E . Then the cohomology class c f ( E ) := h tr E (cid:2) f ( ∇ ) (cid:3)i ∈ H ev ( M )is called the f -class of E or the characteristic class of E associated to f .(ii) If M is closed and oriented, the number (cid:10) c f ( E ) , [ M ] (cid:11) = Z M c f ( E ) ∈ R is called the characteristic number of E associated to f . If all characteristic numbersassociated to f are integers, the f -class is called integer valued .(iii) If ∇ t is a path of connections, we call T c f ( ∇ t ) := Z i π tr E (cid:2) ( ddt ∇ t ) ∧ f ′ ( ∇ t ) (cid:3) dt ∈ Ω odd ( M )the transgression form of the f -class associated to ∇ t . If ∇ t = ∇ + ta we also use thenotation T c f ( ∇ , ∇ ) := Z i π tr E (cid:2) a ∧ f ′ ( ∇ + ta ) (cid:3) dt ∈ Ω odd ( M ) . .1. Chern-Weil Theory Chern-Simons form of ∇ withrespect to ∇ , cs( ∇ , ∇ ) = Z i π tr E (cid:2) a ∧ exp( ∇ + ta ) (cid:3) dt ∈ Ω odd ( M ) . Remark A.1.7. (i) Theorem A.1.5 also applies to characteristic forms of the form exp (cid:0) tr E (cid:2) f ( ∇ ) (cid:3)(cid:1) . For thisnote that ddt exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) = ddt (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) ∧ exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) = d (cid:0) i π tr E (cid:2) ( ddt ∇ t ) ∧ f ′ ( ∇ t ) (cid:3)(cid:1) ∧ exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) . This form is exact, since exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) is closed. Hence, the transgression form in thiscase is Z i π tr E (cid:2) ( ddt ∇ t ) ∧ f ′ ( ∇ t ) (cid:3) ∧ exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) dt (ii) When considering the cohomology class of one of the particular characteristic forms in-troduced in the last section, we will call them Chern character , Chern class , b L -class , etc.The distinction between forms and classes is done by incorporating the connection in thenotation. For example,ch( E, ∇ ) ∈ Ω ev ( M ) , but ch( E ) ∈ H ev ( M ) . (iii) The Chern and Pontrjagin classes are integer valued due to the normalization factor of i π ,see [73, App. C]. The other characteristic classes we have defined are in general only Q valued.(iv) Often the term Chern-Simons form is reserved for the degree 3 part of what we have calledthe Chern-Simons form. Due to its importance in 3-manifold topology, we want to derivean explicit formula for it. We abbreviate ∇ := ∇ and let F t denote the curvature of ∇ t := ∇ + ta . Then F t = F ∇ + t ( ∇ a ) + t a ∧ a. For the component of degree 4 of the Chern character form we have f ( z ) = z / 2, so that f ′ ( z ) = z . According to Definition A.1.6,cs( ∇ , ∇ ) [3] = − π Z tr E (cid:2) a ∧ (cid:0) F ∇ + t ∇ a + t a ∧ a (cid:1)(cid:3) dt. Integrating this expression we getcs( ∇ , ∇ ) [3] = − π tr E (cid:2) a ∧ F ∇ + a ∧ ∇ a + a ∧ a ∧ a (cid:3) . (A.7)In particular, if ∇ is a flat connection we get the well-known expressioncs( ∇ , ∇ ) [3] = − π tr E (cid:2) a ∧ ∇ a + a ∧ a ∧ a (cid:3) . Appendix A. Characteristic Classes and Chern-Simons Forms A.2 Chern-Simons Invariants There is also a different description of transgression forms, which we shall describe now. Let E → M be a complex vector bundle over a manifold M , endowed with a path ∇ t of connections.Over the cylinder N := [0 , × M , we consider the vector bundle π ∗ E → N , where π : N → M is the natural projection. The path ∇ t defines a connection on π ∗ E via e ∇ := dt ∧ ddt + π ∗ ∇ t . (A.8)Its curvature is easily seen to be given by F e ∇ = dt ∧ (cid:0) ddt π ∗ ∇ t (cid:1) + π ∗ F ∇ t . Since dt ∧ dt = 0, one deduces from the trace property that for all n ≥ π ∗ E (cid:2) F n e ∇ (cid:3) = dt ∧ π ∗ tr E (cid:2) n (cid:0) ddt ∇ t (cid:1) ∧ F n − ∇ t (cid:3) + π ∗ tr E (cid:2) F n ∇ t (cid:3) . This implies that for any formal power series f ( z ) = P n a n z n ,tr π ∗ E (cid:2) f ( e ∇ ) (cid:3) = i π dt ∧ π ∗ tr E (cid:2)(cid:0) ddt ∇ t (cid:1) ∧ f ′ ( ∇ t ) (cid:3) + π ∗ tr E (cid:2) f ( ∇ t ) (cid:3) . (A.9)Now, consider integration along the fiber as in Proposition 2.1.12, Z N/M : Ω • ( N ) → Ω •− ( M ) . Then, comparing (A.9) and the definition of the transgression form in Definition A.1.6, onereadily obtains Lemma A.2.1. If ∇ t is a path of connections over M , and e ∇ denotes the associated connection (A.8) over the cylinder N := [0 , × M , then T c f ( ∇ t ) = Z N/M tr π ∗ E (cid:2) f ( e ∇ ) (cid:3) ∈ Ω odd ( M ) . Using this result we can now derive the following important property of transgression forms,see [28, Sec. 3]. Proposition A.2.2. Let E → M be a complex vector bundle over a manifold M . If ∇ t is aclosed path of connections on E , then T c f ( ∇ t ) ∈ d Ω ev ( M ) . Proof. Since the space of connections on E is contractible, we can find a smooth two-parameterfamily ∇ s,t of connections which gives a homotopy relative endpoints from ∇ t to the constantpath. On the cylinder N we consider the one-parameter family e ∇ s := dt ∧ ddt + ∇ s,t , .2. Chern-Simons Invariants π from the notation. Using Theorem A.1.5 and (A.9)one finds that dds tr E (cid:2) f ( e ∇ s ) (cid:3) = − π d N tr π ∗ E h(cid:0) dds ∇ s,t (cid:1) ∧ dt ∧ (cid:0) ddt ∇ s,t (cid:1) ∧ f ′′ ( ∇ s,t ) i + d N tr E h i π (cid:0) dds ∇ s,t (cid:1) ∧ f ′ ( ∇ s,t ) i =: d N (cid:0) dt ∧ α ( s, t ) (cid:1) + d N β ( s, t ) , where α ( s, t ) and β ( s, t ) are two-parameter families of differential forms on M . Then dds Z N/M tr E (cid:2) f ( e ∇ s ) (cid:3) = Z N/M d N (cid:0) dt ∧ α ( s, t ) (cid:1) + Z N/M d N β ( s, t )= d M Z N/M dt ∧ α ( s, t ) + Z N/M dt ∧ (cid:0) ddt β ( s, t ) (cid:1) = d M Z N/M dt ∧ α ( s, t ) + β ( s, − β ( s, . By assumption, ∇ s, = ∇ s, is constant for all s . Checking the explicit formula for β ( s, t ) onefinds that β ( s, 1) = β ( s, dds T c f ( ∇ s,t ) = dds Z N/M tr E (cid:2) f ( e ∇ s ) (cid:3) ∈ d Ω ev ( M ) , from which the result follows. Chern-Simons Invariants. The last result shows that transgression forms can be used todefine numerical invariants associated to pairs of connections on odd dimensional manifolds. Inthis respect they are odd analogues of characteristic numbers. Definition A.2.3. Let M be a closed manifold, and let f ( z ) = P n ≥ a n z n be a formal powerseries. If ∇ and ∇ are two connections on a complex vector bundle E → M we define the Chern-Simons invariant of ∇ with respect to ∇ associated to f asCS f ( ∇ , ∇ ) := Z M T c f ( ∇ , ∇ ) . Proposition A.2.4. Let M be a closed manifold, and let f be a formal power series. Considertwo connections ∇ and ∇ on a complex vector bundle E → M . (i) If ∇ t is any path connecting ∇ and ∇ , then CS f ( ∇ , ∇ ) = Z M T c f ( ∇ t ) . (ii) Let ∇ be a third connection on E , then CS f ( ∇ , ∇ ) = CS f ( ∇ , ∇ ) + CS f ( ∇ , ∇ ) . Appendix A. Characteristic Classes and Chern-Simons Forms (iii) Assume that ∇ N is a connection over the cylinder N = [0 , × M such that on a collar ofthe boundary it is of the form (A.8) . Then CS f ( ∇ , ∇ ) = Z N tr E (cid:2) f ( ∇ N ) (cid:3) . (A.10)(iv) If W is a compact manifold with boundary M , and E , ∇ and ∇ extend to E W , ˆ ∇ and ˆ ∇ , then CS f ( ∇ , ∇ ) = Z W tr E (cid:2) f ( ˆ ∇ ) (cid:3) − Z W tr E (cid:2) f ( ˆ ∇ ) (cid:3) . (v) Assume that f gives an integer valued characteristic class, and let Φ : E → E be a bundleisomorphism. Then for every connection ∇ on E , CS f ( ∇ , Φ ∗ ∇ ) ∈ Z . Sketch of proof. Since M is assumed to be closed, part (i) follows from Proposition A.2.2. Part(ii) is an immediate consequence of (i). For (iii) let ∇ t be a path connecting ∇ and ∇ such thaton a collar of the boundary, ∇ N and e ∇ as in (A.8) agree. Theorem A.1.5 implies that ∇ N − e ∇ is the differential of a form on N with compact support away from the boundary. Then Stokes’Theorem readily yeilds (iii). Part (iv) also follows from Theorem A.1.5 and Stokes’ Theorem. For part (v) denote by ϕ the map covered by Φ. Then the mapping torus E Φ := (cid:0) [0 , × E (cid:1) / ∼ , (1 , x ) ∼ (0 , Φ( x ))is a Hermitian vector bundle over the mapping torus M ϕ . Endow E Φ with a connection ∇ Φ ,induced by connecting ∇ and Φ ∗ ∇ over M . Then one easily finds thatCS f ( ∇ , Φ ∗ ∇ ) = Z M ϕ tr E Φ (cid:2) f ( ∇ Φ ) (cid:3) . The right hand side is integer valued as M ϕ is closed. Remark A.2.5. For a characteristic class of the form exp (cid:0) tr E (cid:2) f ( ∇ ) (cid:3)(cid:1) we have seen in RemarkA.1.7 that the transgression form is given by Z i π tr E (cid:2) ( ddt ∇ t ) ∧ f ′ ( ∇ t ) (cid:3) ∧ exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) dt. Lemma A.2.1 extends to this context: Let e ∇ be the connection over N = [0 , × M , and writetr E (cid:2) f ( e ∇ ) (cid:3) = dt ∧ α ( t ) + β ( t ) , where α ( t ) and β ( t ) contain no dt -factor. Thenexp (cid:0) tr E (cid:2) f ( e ∇ ) (cid:3)(cid:1) = exp (cid:0) dt ∧ α ( t ) (cid:1) ∧ exp (cid:0) β ( t ) (cid:1) = (cid:0) dt ∧ α ( t ) (cid:1) ∧ exp (cid:0) β ( t ) (cid:1) . Alternatively, one could glue a cylinder to the boundary of W to interpolate between ˆ ∇ and ˆ ∇ , and thenuse Lemma A.2.1 as well as the additivity under cutting and pasting of characteristic numbers. .2. Chern-Simons Invariants Z N/M exp (cid:0) tr E (cid:2) f ( e ∇ ) (cid:3)(cid:1) = Z N/M dt ∧ α ( t ) ∧ exp (cid:0) β ( t ) (cid:1) = Z N/M tr E (cid:2) f ( e ∇ ) (cid:3) ∧ exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) = Z i π tr E (cid:2) ( ddt ∇ t ) ∧ f ′ ( ∇ t ) (cid:3) ∧ exp (cid:0) tr E (cid:2) f ( ∇ t ) (cid:3)(cid:1) dt where we have used (A.9) in the last line. Similarly, one checks that Proposition A.2.2 andProposition A.2.4 continue to hold in this context.88 Appendix A. Characteristic Classes and Chern-Simons Forms ppendix B Remarks on Moduli Spaces In this appendix we include some details concerning the moduli space of flat connections andthe moduli space of holomorphic line bundles over a Riemann surface. Since the Rho invariantdepends only on the gauge equivalence class of the underlying flat connection, understandingthe moduli space of flat connections is a prerequisite for the computation of Rho invariants.Moreover, the interplay between flat connections and representations of the fundamental groupis often used in the main body of this thesis. Therefore, we start with a detailed discussion ofthese topics, in particular including some remarks on the question of whether a given flat bundleis trivializable or not.We proceed with a discussion of the moduli space of flat connections associated to a mappingtorus. Here the objective is to prove the facts we have used in Chapter 4. After this, we addsome remarks about the moduli space of holomorphic line bundles over a Riemann surface andits relation to the moduli space of (flat) connections. This will establish some facts we have freelyused in Section 2.3. B.1 The Moduli Space of Flat Connections B.1.1 Flat Connections and Representations of the Fundamental Group Since many features become more transparent in a more general setup, we start working witha principal G -bundles, where G is an arbitrary connected matrix Lie group. Ultimately we areinterested in flat Hermitian vector bundles and restrict to G = U( k ). A general reference for thecontents of this section are [37, Sec. 2.1] and [62, Ch. II].Denote by g the Lie algebra of G . Since we are assuming that G is a matrix Lie group, g isa matrix Lie algebra. We use the notation “Ad” for the adjoint action of G on itself and “ad”to denote the adjoint action of G on g . Connections and Curvature. Let M be a connected manifold, and let P π −→ M be a principal G -bundle. Let R g denote the right-action of g ∈ G on P . Recall that a G -connection on P is aLie algebra valued 1-form A ∈ Ω ( P, g ) satisfying R ∗ g A = g − Ag, and A (cid:0) ddt (cid:12)(cid:12) t =0 p · exp( tX ) (cid:1) = X, p ∈ P, X ∈ g . (B.1)18990 Appendix B. Remarks on Moduli Spaces We denote the space of all G -connections on P by A ( P ). The curvature of A is defined as F A = dA + A ∧ A ∈ Ω ( P, g ) , where A ∧ A stands for taking the exterior product in the form part and matrix multiplicationin the Lie algebra part. The curvature is easily seen to be ad-equivariant and horizontal, i.e., R ∗ g F A = g − F A g, i (cid:0) ddt (cid:12)(cid:12) t =0 p · exp( tX ) (cid:1) F A = 0 . This implies that F A can also be viewed as a 2-form on M with values in the bundle ad( g ) = P × ad g . A connection A is called flat if F A = 0, and we denote by F ( P ) = { A ∈ A ( P ) | F A = 0 } the space of flat G -connections on P . Gauge Transformations and the Moduli Space. We also recall that a gauge transformation is a G -equivariant bundle isomorphism,Φ : P → P, Φ( p · g ) = Φ( p ) · g. If one defines u : P → G by requiring that Φ( p ) = p · u ( p ), then u is Ad-equivariant, u : P → G, u ( p · g ) = g − u ( p ) g. Conversely, it is easy to see that every gauge transformation arises this way. Hence, one of severalequivalent ways to define the group of gauge transformations is G ( P ) := C ∞ (cid:0) M, Ad( P ) (cid:1) , where Ad( P ) = P × Ad G. The pullback of a connection by a gauge transformation gives a natural action of G ( P ) on A ( P ).In terms of an Ad-equivariant map u : P → G this takes the form A · u = u − Au + u − du, A ∈ A ( P ) , u ∈ G ( P ) . We point out that u − du is the pullback of the Maurer-Cartan form on G via u . The curvaturebehaves equivariantly with respect to this action, F A · u = u − F A u, A ∈ A ( P ) , u ∈ G ( P ) . In particular, the action of G ( P ) on A ( P ) leaves the space F ( P ) of flat connections invariant.One can thus define the moduli space of flat connections on P as M ( P ) := F ( P ) / G ( P ) . Remark B.1.1. We will often encounter the situation that P is trivializable. If we fix a trivial-ization P ∼ = M × G , we can identify A ( M × G ) ∼ = Ω ( M, g ) , G ( M × G ) ∼ = C ∞ ( M, G ) . .1. The Moduli Space of Flat Connections The Holonomy Representation. Fix a base point p ∈ P , and let x := π ( p ). Consider aclosed loop based at x , i.e., c : I → M, I = [0 , , with c (0) = c (1) = x . Since I is contractible, the pullback c ∗ P → I is trivializable. As we are assuming that G isconnected, we can fix a lift b c : I → P of c such that b c (0) = b c (1) = p . Now let A be a G -connection—not necessarily flat for the moment—and let A t := ( b c ∗ A )( ∂ t ) ∈ C ∞ ( I, g ) . (B.2) Definition B.1.2. Let g t : I → G be the unique solution of the ordinary differential equation ∂ t g t = − A t g t , g = e, where e ∈ G is the identity element. Then the holonomy of A along c with respect to the basepoint p is defined by hol A ( c, p ) := g ∈ G. Note that the definition gives no reference to the lift b c we have fixed. The reason why we areallowed to do so is one of the contents of the following result. Lemma B.1.3. Let A be a connection on P , and let c : I → M be a closed loop, based at x . (i) If ϕ : I → I is an orientation preserving reparametrization, then hol A ( c ◦ ϕ, p ) = hol A ( c, p ) . (ii) For every gauge transformation u ∈ G ( P ) , hol A · u ( c, p ) = u ( p ) − hol A ( c, p ) u ( p ) . In particular, hol A ( c, p ) is independent of the lift b c chosen in its definition. (iii) Assume that e c is another loop, based at x , and denote by c ∗ e c the loop defined by firstrunning along c and then along e c . Then hol A ( c ∗ e c, p ) = hol A ( e c, p ) hol A ( c, p ) . (iv) Let p ∈ P be a different base point, and b c : I → P be a path connecting p with p . Thenthere exists g ∈ G such that hol A ( c − ∗ c ∗ c , p ) = g hol A ( c, p ) g − , where c := π ◦ b c .Proof. To prove part (i) let g t be as in Definition B.1.2. Then ∂ s g ϕ ( s ) = ϕ ′ ( s )( ∂ t g t ) | t = ϕ ( s ) = − ϕ ′ ( s ) A ϕ ( s ) g ϕ ( s ) = − ( b c ◦ ϕ ) ∗ A ( ∂ s ) g ϕ ( s ) . Although the assertions in Lemma B.1.3 is standard, we include a proof as the discussion to follow relies onsimilar arguments. Appendix B. Remarks on Moduli Spaces Moreover, we have g ϕ (0) = g = e and g ϕ (1) = g . Then part (i) is true by definition. To prove(ii) define u t := u ◦ b c ∈ C ∞ ( I, G ), and b c u := b c · u t . Then b c u is a lift of c with b c u (0) = b c u (1) = p · u ( p ) , and b c ∗ ( A · u )( ∂ t ) = b c ∗ u A ( ∂ t ) = A t · u t . If g t is as in Definition B.1.2, then ∂ t ( u − t g t u ) = ( ∂ t u − t ) g t u + u − ( ∂ t g t ) u = − u − t ( ∂ t u t )( u − t g t u ) − ( u − t A t u t )( u − t g t u )= − ( A t · u t )( u − t g t u ) . Since u − g u = e , this implies thathol A · u t ( c, p · g ) = u − g u = u − ( p ) hol A ( c, p ) u ( p ) . The second assertion of (ii) follows from the fact that every lift of c with base point p is of theform b c · u with u ( p ) = e . Concerning part (iii), we define e A t as in (B.2) with respect to a liftof e c and note without going into detail that hol A ( c ∗ e c, p ) is given by e g , where e g t is the uniquesolution to ∂ t e g t = − e A t e g t , e g = hol A ( c, p ) . This readily yields e g = hol A ( e c, p ) hol A ( c, p ). To prove (iv) we can solve the initial valueproblem ∂ t g t = − ( b c ∗ A )( ∂ t ) g t , g = e. Then one verifies without effort that the assertion holds with g := g .Let Ω( M, x ) be the based loop group of M . This is the set of all loops, based at x , moduloorientation preserving reparametrization. For reasons of functoriality we endow Ω( M, x ) withthe product c · e c := e c ∗ c , where e c ∗ c is as in Lemma B.1.3. Let p ∈ P with x = π ( p ). Usingthe above results, one obtains a well-defined homomorphismhol A : Ω( M, x ) → G, c hol A ( c, p ) (B.3) Definition B.1.4. Let A be a connection on P . Then the homomorphism (B.3) is called the holonomy representation of A with respect to the base point p . We also define the holonomygroup of A with respect to p as G A ( p ) := im (cid:0) hol A : Ω( M, x ) → G (cid:1) A connection A is called irreducible , if G A ( p ) = G . Otherwise, it is called reducible . Moreover,the isotropy group of A is defined as I ( A ) := (cid:8) u ∈ G ( P ) (cid:12)(cid:12) A · u = A (cid:9) . Lemma B.1.5. The conjugacy class of G A ( p ) is independent of p and the gauge equivalenceclass of A . Moreover, for fixed p ∈ P , the map I ( A ) → G, u u ( p ) , maps I ( A ) isomorphically to the centralizer of G A ( p ) in G . .1. The Moduli Space of Flat Connections Proof. The first assertion is immediate from Lemma B.1.3. Now fix p , and assume that u ∈ I ( A ).Lemma B.1.3 implies that for every loop c : I → M , based at x = π ( p ), we havehol A ( c, p ) = hol A · u ( c, p ) = u ( p ) − hol A ( c, p ) u ( p ) . Thus, u ( p ) lies in the centralizer of G A ( p ). Now let P be the set of all p ∈ P such that thereexists a horizontal path b c : I → P with b c (0) = p and b c (1) = p . Then P intersects every fiber of π : P → M . This is because if b c : I → P is an arbitrary path with b c (0) = p , we can solve ∂ t g t = − A t g t , g = e, with A t as in (B.2), to get a horizontal path b c · g t : I → P whose endpoint lies in the same fiberas the endpoint of b c .To prove injectivity of the map I ( A ) → G , u u ( p ), let u ∈ I ( A ) with u ( p ) = e . Since u is Ad-equivariant and P intersects every fiber of π : P → M , it suffices to show that u | P ≡ e .Let p ∈ P , and let b c a horizontal path connecting p with p . Then A · u = A implies that( u − ◦ b c ) ∂ t ( u ◦ b c )( t ) = u − du | b c ( t ) ( ddt b c ) = ( A − u − Au ) | b c ( t ) ( ddt b c ) = 0 , where we have used that ddt b c is horizontal. Hence u is constant along b c and thus, u ( p ) = e . Next,assume that g ∈ G lies in the centralizer of G A ( p ). We need to define u ∈ I ( A ) such that u ( p ) = g . We first define u | P to be the constant map g . To see that this defines a gaugetransformation, we need to check that u ( p · g ) = g − u ( p ) g , whenever p and p · g both lie in P .Let b c and b c g be horizontal paths connecting p with p , respectively with p · g . Then b c g ∗ ( b c − · g )is a horizontal path which connects p with p · g . This implies that g ∈ G A ( p ). Since we haveassumed that g lies in the centralizer of G A ( p ), we obtain u ( p · g ) = g − u ( p ) g = g − g g = g . Hence, u | P is Ad-equivariant and can be extended to a gauge transformation on P . To provethat u ∈ I ( A ) first note that the values of the 1-forms A · u and A on vertical vectors are bothprescribed by (B.1). To see that A · u and A also agree on horizontal vectors, it suffices to consider A | p and A · u | p for p ∈ P since both, A · u and A , are ad-equivariant. Now if v ∈ T p P is horizontal,there exists a horizontal path b c : ( − ε, ε ) → P with b c (0) = p and ddt b c (0) = v . By definition of P one easily checks that im( b c ) ⊂ P . Thus, u is constant along b c so that A · u | p ( v ) − A | p ( v ) = u − du | p ( v ) = u − ( p ) ddt (cid:12)(cid:12) t =0 u ◦ b c = 0 . Flat Connections and the Fundamental Group. After this technical preparation, we turnour attention to flat connections. The following result shows that they are of a topological nature. Proposition B.1.6. If A is flat, then the holonomy hol A ( c, p ) depends only on the homotopyclass [ c ] ∈ π ( M ) = π ( M, x ) . In particular, the holonomy representation defines a homomor-phism hol A ∈ Hom (cid:0) π ( M ) , G (cid:1) . Moreover, the assignment F ( P ) → Hom (cid:0) π ( M ) , G (cid:1) , A hol A gives well-defined map M ( P ) → Hom (cid:0) π ( M ) , G (cid:1) /G. Appendix B. Remarks on Moduli Spaces Proof. Consider a homotopy c : I × I → M, c ( s, 0) = c ( s, 1) = x . Since any fiber bundle has the homotopy lifting property, we can choose a lift b c : I × I → P, b c ( s, 0) = b c ( s, 1) = p Abusing notation we use the letter A also to denote the 1-form b c ∗ A ∈ Ω ( I × I, g ). The flatnesscondition dA + A ∧ A = 0 written out with respect to the coordinates ( s, t ) ∈ I × I is ∂ s A ( ∂ t ) − ∂ t A ( ∂ s ) + A ( ∂ s ) A ( ∂ t ) − A ( ∂ t ) A ( ∂ s ) = 0 . (B.4)For fixed s let g s = g s ( t ) : I → G denote the solution to ∂ t g s = − A ( ∂ t ) g s , g s (0) = e. (B.5)Since A depends smoothly on s and t , it follows from the standard theory of ordinary differentialequations that g s depends smoothly on s . We then compute ∂ t (cid:0) ∂ s g s + A ( ∂ s ) g s (cid:1) = ∂ s ( ∂ t g s ) + (cid:0) ∂ t A ( ∂ s ) (cid:1) g s + A ( ∂ s )( ∂ t g s )= − ∂ s (cid:0) A ( ∂ t ) g s (cid:1) + (cid:0) ∂ t A ( ∂ s ) (cid:1) g s − A ( ∂ s ) A ( ∂ t ) g s , where we have used (B.5). Then (B.4) implies that ∂ t (cid:0) ∂ s g s + A ( ∂ s ) g s (cid:1) = (cid:0) A ( ∂ s ) A ( ∂ t ) − A ( ∂ t ) A ( ∂ s ) (cid:1) g s − A ( ∂ t ) ∂ s g s − A ( ∂ s ) A ( ∂ t ) g s = − A ( ∂ t ) (cid:0) ∂ s g s + A ( ∂ s ) g s (cid:1) . The initial condition in (B.5) and the fact that b c ( s, 0) is constant for s ∈ I implies that ∂ s g s | t =0 =0 and A ( ∂ s ) | t =0 = 0. Hence, ∂ s g s = − A ( ∂ s ) g s , for all t ∈ I .Moreover, we have A ( ∂ s ) | t =1 = 0 so that g s (1) is independent of s . By definition of the holonomyand (B.5) this proves the first assertion of Proposition B.1.6. The other assertions are immediatefrom Lemma B.1.3. The Moduli Space of Representations. Let M ( M, G ) be the moduli space of flat principal G -bundles, i.e., the space of isomorphism classes of pairs ( P, A ) where P is a principal G -bundleand A is a flat connection on P . Our next goal is to show that the map in Proposition B.1.6induces an isomorphism M ( M, G ) ∼ = Hom (cid:0) π ( M ) , G (cid:1) /G. Remark B.1.7. Note that in general M ( M, G ) will be strictly larger than the moduli space M ( P ) for one fixed flat bundle P . This is because there might exist flat bundles such that theunderlying principal G -bundles are not isomorphic. In the case that G = U( k ) we will say moreabout this in Section B.1.2 below. .1. The Moduli Space of Flat Connections G -bundle to any representation α : π ( M ) → G , let f M be the universalcover of M . For definiteness we fix a base point x ∈ M and identify f M with the space ofhomotopy classes of paths in M starting at x . Then π ( M, x ) naturally acts on f M from theright. For α : π ( M, x ) → G we define the principal G -bundle P α := f M × α G = ( f M × G ) / ∼ where ( e x, g ) ∼ ( e x · c, α ( c ) − g ) , ( e x, g ) ∈ f M × G, c ∈ π ( M ) . (B.6)Pulling back the Maurer-Cartan form g − dg ∈ Ω ( G, g ) to f M × G defines a natural flat connectionon f M × G , which is invariant under the action (B.6) of π ( M ). In this way we get an inducedflat connection A α on P α . It is straightforward to check that with respect to the base point p := [ x , e ] ∈ P α , hol A α (cid:0) c, p (cid:1) = α ( c ) , c ∈ π ( M, x ) . More generally, we have Proposition B.1.8. Let P be a principal G -bundle with flat connection A , and let α : π ( M ) → G be a representation of the fundamental group. Then ( P, A ) is isomorphic to ( P α , A α ) if andonly if there exists g ∈ G with hol A = g − αg . In particular, we have a bijection M ( M, G ) ∼ = −→ Hom (cid:0) π ( M ) , G (cid:1) /G, [ P, A ] [hol A ] . Sketch of proof. The assertion that ( P, A ) ∼ = ( P α , A α ) implies hol A = g − αg is an immediategeneralization of Lemma B.1.3 (ii). For the reverse direction first consider a representation α and let e α := g − αg for some g ∈ G . Define f M × G → f M × G, ( e x, g ) ( e x, g g ) . This descend to a bundle map P α → P e α since (cid:0)e x · c, e α ( c ) − ( g g ) (cid:1) = (cid:0)e x · c, g ( α ( c ) − g ) (cid:1) . One verifies that this gives an isomorphism of flat bundles. Now assume that hol A = α , and fixa base point p ∈ P . For e x ∈ f M , let c e x : I → M be a path in M representing e x and starting at x = π ( p ). Let b c e x : I → P be the horizontal lift to P , starting at p . Using the same ideas as inProposition B.1.6 one finds that b c e x (1) depends only on the homotopy class of c e x . Hence, we geta well-defined map Φ : f M × G → P, ( e x, g ) b c e x (1) · g. The construction is in such a way that Φ is G -equivariant and surjective. Moreover, if c ∈ π ( M, x ), then a straightforward calculation shows that b c ( e x · c ) (1) = b c e x (1) · hol A ( c, p ) , and Φ − (cid:0)b c e x (1) (cid:1) = (cid:8)(cid:0)e x · c, hol A ( c, p ) − (cid:1) (cid:12)(cid:12) c ∈ π ( M, x ) (cid:9) . Since we are assuming that hol A ( c, p ) = α ( c ), this implies that Φ descends to a bundle iso-morphism P α → P . Concerning the relation between the flat connections A and A α we remarkwithout further comments that Φ is defined in such a way that it maps the horizontal distributionon the trivial bundle f M × G to the horizontal distribution on P given by A . This implies thatvia the isomorphism P α ∼ = P the connections A α and A agree.96 Appendix B. Remarks on Moduli Spaces B.1.2 Flatness and Triviality of U( k )-Bundles As mentioned in Remark B.1.7, a flat principal G -bundle is not necessarily trivializable. In thissection we give some details for the case G = U( k ). Via the standard representation of U( k ) on C k , a principal U( k )-bundle P defines a Hermitian vector bundle E → M and vice versa. Wewill freely switch between the two equivalent notions. Flat Line Bundles. We first recall that the space of all Hermitian line bundles over a givenmanifold can be described in terms of ˇCech cohomology, see [96, Sec. III.4] for details. Let M bea compact, connected manifold. For a Lie group G we denote by G the sheaf of locally smoothfunctions on M with values in G . Then H (cid:0) M, U(1) (cid:1) is isomorphic to the set of Hermitian linebundles over M up to isomorphism. Note that the group structure on the latter is given by thetensor products of line bundles. There is an exact sequence of sheaves.0 −→ Z −→ R e πix −−−→ U(1) −→ . (B.7)Since the sheaf R is fine (i.e., admits partitions of unity), the cohomology H • ( M, R ) vanishesaway from degree 0. The long exact sequence in cohomology then produces natural isomorphisms H p (cid:0) M, U(1) (cid:1) ∼ = H p +1 ( M, Z ) , p ≥ . For p = 1, this isomorphism coincides with the integral first Chern class c : H (cid:0) M, U(1) (cid:1) ∼ = −→ H ( M, Z ) . (B.8)On the other hand, we have seen in Proposition B.1.8 that flat Hermitian line bundles areclassified by representations of π ( M ) in U(1). Here, conjugation does not play a role here sinceU(1) is abelian. Therefore, the moduli space of flat Hermitian line bundles has the cohomologicaldescription Hom (cid:0) π ( M ) , U(1) (cid:1) = H (cid:0) M, U(1) (cid:1) . Note that in terms of ˇCech cohomology, H p (cid:0) M, U(1) (cid:1) refers to the sheaf of locally constant (ratherthan C ∞ ) functions with values in U(1). Similarly, we have to distinguish between H • ( M, R )and H • ( M, R ). We have a long exact coefficient sequence ... −→ H p ( M, Z ) −→ H p ( M, R ) −→ H p (cid:0) M, U(1) (cid:1) −→ H p +1 ( M, Z ) −→ ... Here, the integral first Chern class appears again as the map c : H (cid:0) M, U(1) (cid:1) −→ H ( M, Z ) . Moreover, the universal coefficient theorem shows thatker (cid:0) H ( M, Z ) → H ( M, R ) (cid:1) = Tor (cid:0) H ( M, Z ) (cid:1) . Together with (B.8) this easily yields Lemma B.1.9. A line bundle L → M admits a flat connection if and only if its integral firstChern class satisfies c ( L ) ∈ Tor (cid:0) H ( M, Z ) (cid:1) . .1. The Moduli Space of Flat Connections Remark. The above result also follows from Chern-Weil theory: The representative of the firstChern class in de Rham cohomology is given by (cid:2) tr( i π F A ) (cid:3) , where A is any U(1)-connection on L . Hence, L admits a flat connection if and only if c ( L ) vanishes in H ( M, R ). This is preciselythe case if the integral first Chern class is a torsion class.There is also a topological condition for triviality of a flat Hermitian line bundle. It isstraightforward to check that the natural map of sheaves U(1) → U(1) relates the two exactcoefficient sequences via0 −−−−→ H (cid:0) M, U (1) (cid:1) c −−−−→ H ( M, Z ) −−−−→ x x (cid:13)(cid:13)(cid:13) x H ( M, R ) −−−−→ H (cid:0) M, U (1) (cid:1) c −−−−→ H ( M, Z ) −−−−→ H ( M, R ) . Lemma B.1.10. Let L α → M be a flat Hermitian line bundle on M with holonomy α : H ( M, Z ) → U(1) . Then L α is trivializable if and only if the restriction of α to the torsionsubgroup Tor (cid:0) H ( M, Z ) (cid:1) is trivial.Proof. The line bundle L α is trivializable if and only if c ( L α ) = 0 in H ( M, Z ). The abovediagram shows that this is precisely if L α ∈ im (cid:2) H ( M, R ) → H (cid:0) M, U(1) (cid:1)(cid:3) . The latter means that α = exp(2 πi b α ) with b α : H ( M, Z ) → R . Since ( R , +) is a torsion freeabelian group, the restriction of b α to Tor (cid:0) H ( M, Z ) (cid:1) vanishes. Conversely, if α is trivial onTor (cid:0) H ( M, Z ) (cid:1) , it can be lifted to a homomorphism b α as above. Flat U( k )-undles. If we now consider U( k ) for k > 1, Lemma B.1.9 and Lemma B.1.10 do notgeneralize immediately, since we do not have a simple cohomological description of the space of(flat) U( k )-bundles. This is mainly due to the fact that H (cid:0) M, U( k ) (cid:1) is not a group since U( k ) isnon-abelian. However, in the case that dim M ≤ k ). The underlying reason is the following Proposition B.1.11. Let M be a manifold of dimension ≤ . Then every principal SU( k ) -bundle P → M is trivializable.Idea of proof. Let E SU( k ) → B SU( k ) be the universal SU( k )-bundle, see [33, Sec. 8.6]. It hasthe property that E SU( k ) is contractible, and the set of isomorphism classes of SU( k )-bundlesis isomorphic to the set of homotopy classes of maps M → B SU( k ). Since the total space of theuniversal SU( k )-bundle is contractible, the long exact homotopy sequence yields that π n (cid:0) B SU( k ) (cid:1) = π n − (cid:0) SU( k ) (cid:1) , n ≥ . (B.9)It is well known that SU( k ) is 2-connected, see [33, Sec. 6.14]. Hence, (B.9) implies that B SU( k )is 3-connected. Since we are assuming that dim M ≤ M → B SU( k )is homotopic to a constant map. Therefore, every SU( k )-bundle over M is trivializable.We then have the following generalization of Lemma B.1.9 and Lemma B.1.10.98 Appendix B. Remarks on Moduli Spaces Corollary B.1.12. Let M be a compact, connected manifold of dimension ≤ , and let E → M be a Hermitian vector bundle over M of rank k . (i) The bundle E admits a flat connection if and only if its integral first Chern class satisfies c ( E ) ∈ Tor (cid:0) H ( M, Z ) (cid:1) . (ii) Assume that E is flat with holonomy α : π ( M ) → U( k ) . Then E is trivializable if andonly if the restriction of det( α ) : H ( M, Z ) → U(1) to Tor (cid:0) H ( M, Z ) (cid:1) is trivial.Proof. Let det( E ) := Λ k E → M denote the determinant line bundle of E . One concludes fromthe exact sequence 0 −→ SU( k ) −→ U( k ) det −−→ U(1) −→ , that E ⊗ det( E ) − admits an SU( k )-structure. As we are assuming dim M ≤ 3, it followsfrom Proposition B.1.11 that E ⊗ det( E ) − is isomorphic to M × C k . This implies that E isflat/trivializable if and only if det( E ) is flat/trivializable. Since c ( E ) = c (cid:0) det( E ) (cid:1) ∈ H ( M, Z ) , part (i) readily follows from Lemma B.1.9, whereas (ii) is immediate from Lemma B.1.10. Remark B.1.13. If Σ is a closed, oriented surface, then H (Σ , Z ) ∼ = Z . Therefore, there areno torsion elements and every flat Hermitian vector bundle over Σ is isomorphic to the trivialbundle. For 3-manifolds there will in general exist non-trivial flat Hermitian bundles, see forexample Remark B.2.10 below or Section 2.3. B.2 Flat Connections over Mapping Tori We let M be a closed, oriented manifold, and let f ∈ Diff + ( M ) be an orientation preservingdiffeomorphism. Then we define the mapping torus M f of f asFigure B.1: The mapping torus of fM f := (cid:0) M × R (cid:1) / Z , .2. Flat Connections over Mapping Tori Z acts on M × R via k · ( x, t ) = (cid:0) f − k ( x ) , t + k (cid:1) , ( x, t ) ∈ M × R , k ∈ Z . (B.10)We use this definition rather than defining M f as a quotient of M × [0 , Z -equivariantly on M × R . Since we assume that f is orientation preserving, the productorientation of M × R defines an orientation on M f . The map M × R → S , ( x, t ) exp(2 πit )is invariant with respect to the action (B.10), and gives rise to a fiber bundle M ֒ → M f π −→ S . Let Diff ( M ) be the component of the identity in Diff + ( M ) with respect to the C ∞ -topology.We collect the following well-known material. Lemma B.2.1. The diffeomorphism class of M f depends only on the conjugacy class of f insidethe mapping class group Diff + ( M ) / Diff ( M ) . Moreover, there exists a diffeomorphism − M f ∼ = M f − , where − M f carries the reversed orienta-tion.Proof. To show that the diffeomorphism class of M f depends only on the isotopy class of f , let f t : [0 , → Diff + ( M ) be an isotopy. Possibly using a reparametrization of [0 , 1] we may assumethat f t is constant near 0 and 1. Define ϕ t := f − t ◦ f and extend ϕ t to a path ϕ t : R → Diff + ( M )by requiring that ϕ t +1 = f − ◦ ϕ t ◦ f . Then one easily checks thatΦ : M × R → M × R , Φ( x, t ) := (cid:0) ϕ t ( x ) , t (cid:1) , is Z -equivariant with respect to (B.10). Hence, it descends to a diffeomorphism Φ : M f → M f .Similarly, if g ∈ Diff + ( M ), then the mapΨ : M × R → M × R , Ψ( x, t ) := (cid:0) g ( x ) , t (cid:1) defines a diffeomorphism Ψ : M f → M gfg − . Thus, the diffeomorphism class of M f dependsonly on the conjugacy class of f in Diff + ( M ) / Diff ( M ). In a similar way one verifies that anorientation reversing diffeomorphism M f ∼ = M f − is induced by M × R → M × R , ( x, t ) ( x, − t ) . Remark B.2.2. We also want to point out that every oriented fiber bundle M ֒ → f M π −→ S arises in the way just described: Identify M with the fiber π − (1), and endow M with a verticalprojection P v : T M → T v M , see Section 2.1.1. For x ∈ M denote by c x : [0 , → M thehorizontal lift of the path [0 , → S , t e πit which starts at x . Then c x (1) ∈ M , so that we can define f : M → M, f ( x ) := c x (1) . Appendix B. Remarks on Moduli Spaces It follows from the standard theory of ordinary differential equations that f is a diffeomorphism.Moreover, since we have assumed that M ֒ → f M π −→ S is an oriented fiber bundle, it followsthat f is orientation preserving, i.e., f ∈ Diff + ( M ). One then verifies that the fiber bundle π : f M → S is indeed isomorphic to the fiber bundle given by the mapping torus M f . Thisidentification might seem to depend on the choice of P v and the identification of M as a fiber.However, since all vertical projections are homotopic, one can check that the conjugacy class of f in the mapping class group Diff + ( M ) / Diff ( M ) does not change, so that by Lemma B.2.1 theisomorphism class of the mapping torus is unambiguously defined. B.2.1 Algebraic Description of the Moduli Space Let G be a connected matrix Lie group. Ultimately G will be U( k ). We use Proposition B.1.8to identify M ( M f , G ) = Hom (cid:0) π ( M f ) , G (cid:1) /G, where G acts by conjugation. We fix a base point x ∈ M assume for simplicity that f ( x ) = x .This is possible as we can always find an element in the isotopy class of f ∈ Diff + ( M ) whichfixes x . Then the path γ : R → M × R , t ( x , t ) , descends to a closed path in M f , whose homotopy class we also denote by γ ∈ π ( M f ). Withoutincluding it in the notation, we are using [ x , 0] as a base point for M f . On the other hand, theinclusion M = M × { } ⊂ M × R induces a map i ∗ : π ( M ) → π ( M f ) . Then the following is easily verified. Lemma B.2.3. The fundamental group of M f with respect to the base point [ x , is given by π ( M f ) = (cid:10) π ( M ) , γ (cid:12)(cid:12) γ − cγ = f ∗ c, c ∈ π ( M ) (cid:11) , where f ∗ : π ( M ) → π ( M ) is the induced map on the fundamental group. The map i ∗ : π ( M ) → π ( M f ) gives rise to a natural homomorphism i ∗ : Hom (cid:0) π ( M f ) , G (cid:1) → Hom (cid:0) π ( M ) , G (cid:1) , α α ◦ i ∗ . This map is G -equivariant and we get an induced map[ i ∗ ] : M ( M f , G ) → M ( M, G ) , [ α ] [ α ◦ i ∗ ] . (B.11)Similarly, f ∗ : Hom (cid:0) π ( M ) , G (cid:1) → Hom (cid:0) π ( M ) , G (cid:1) is G -equivariant so that it descends to a map[ f ∗ ] : M ( M, G ) → M ( M, G ) , [ α ] [ α ◦ f ∗ ] . For the following result see also [2, Sec. 8]. Recall that for fundamental groups we are using the product [ c ] · [ e c ] := [ e c ∗ c ], where e c ∗ c means first e c andthen c . .2. Flat Connections over Mapping Tori Proposition B.2.4. The natural map (B.11) defines a surjection [ i ∗ ] : M ( M f , G ) → Fix[ f ∗ ] ⊂ M ( M, G ) . Moreover, if α ∈ Hom (cid:0) π ( M ) , G (cid:1) is such that [ α ] ∈ Fix[ f ∗ ] , then [ i ∗ ] − [ α ] ∼ = (cid:8) g ∈ G (cid:12)(cid:12) g − αg = f ∗ α (cid:9) /Z ( α ) (B.12) where Z ( α ) := { h ∈ G | h − αh = α } is the centralizer of α , and acts on G by conjugation.Proof. With γ as in Lemma B.2.3 defineΦ : Hom (cid:0) π ( M f ) , G (cid:1) → Hom (cid:0) π ( M ) , G (cid:1) × G, α (cid:0) α ◦ i ∗ , α ( γ ) (cid:1) . Then Lemma B.2.3 implies that Φ induces an isomorphismΦ : Hom (cid:0) π ( M f ) , G (cid:1) ∼ = −→ (cid:8) ( α, g ) ∈ Hom (cid:0) π ( M ) , G (cid:1) × G (cid:12)(cid:12) g − αg = f ∗ α (cid:9) . (B.13)The action of G by conjugation on Hom (cid:0) π ( M f ) , G (cid:1) translates to to the right hand side as G acting diagonally by conjugation, i.e., h ∈ G acts on an element ( α, g ) via( α, g ) · h := (cid:0) h − αh, h − gh (cid:1) . Then M ( M f , G ) ∼ = (cid:8) ( α, g ) ∈ Hom (cid:0) π ( M ) , G (cid:1) × G (cid:12)(cid:12) g − αg = f ∗ α (cid:9) /G. Under this identification, the map [ i ∗ ] in (B.11) is given by the projection onto the first factor.Now [ α ] = [ f ∗ α ] if and only if there exists g ∈ G with g − αg = f ∗ α , which guarantees that[ i ∗ ] : M ( M f , G ) → Fix[ f ∗ ] ⊂ M ( M, G ) , [ α, g ] [ α ] , is well-defined and surjective. The inverse image of [ α ] ∈ Fix[ f ∗ ] is given by[ i ∗ ] − [ α ] = (cid:8) ( e α, g ) ∈ Hom (cid:0) π ( M ) , G (cid:1) × G (cid:12)(cid:12) g − e αg = f ∗ e α, [ e α ] = [ α ] (cid:9) /G. Hence, if we fix a representative α , then[ i ∗ ] − [ α ] ∼ = (cid:8) g ∈ G (cid:12)(cid:12) g − αg = f ∗ α (cid:9) /Z ( α ) . Remark. (i) If we fix a different representative e α = h − αh in (B.12), then one easily checks that (cid:8)e g ∈ G (cid:12)(cid:12) e g − e α e g = f ∗ e α (cid:9) = (cid:8) h − gh ∈ G (cid:12)(cid:12) g − αg = f ∗ α (cid:9) . Moreover, Z ( e α ) = h − Z ( α ) h. This describes the relation among the isomorphisms in (B.12) for different choices of rep-resentatives for [ α ] ∈ Fix[ f ∗ ].(ii) We also want to point out that if we fix g with g − αg = f ∗ α , then (cid:8) g ∈ G (cid:12)(cid:12) g − αg = f ∗ α (cid:9) = Z ( α ) g. Furthermore, note that if α ∈ Hom (cid:0) π ( M ) , G (cid:1) is irreducible, then Z ( α ) coincides with thecenter of G , Z ( α ) = Z ( G ) = (cid:8) g ∈ G (cid:12)(cid:12) gh = hg for all h ∈ G (cid:9) . Appendix B. Remarks on Moduli Spaces B.2.2 Geometric Description of the Moduli Space We now turn to a more geometric description of M ( M f , G ). A related discussion, yet in adifferent context, can be found in [38, Sec. 5] and [90, Sec. 7]. Since we prefer to work withHermitian vector bundles over M f , we assume from now on that G = U( k ). Vector Bundles over Mapping Tori. Using the ideas of Remark B.2.2 one finds that everyHermitian vector bundle over M f is isomorphic to a mapping torus E b f := (cid:0) E × R (cid:1) / Z → (cid:0) M × R (cid:1) / Z = M f . (B.14)Here, b f is a bundle isomorphism covering f , and Z acts on E × R via k · ( e, t ) = (cid:0) b f − k ( e ) , t + k (cid:1) , ( e, t ) ∈ E × R , k ∈ Z . To keep the notation simple, we are identifying E × R with the pullback bundle π ∗ M E → M × R , where π M : M × R → M . For a fixed Hermitian vector bundle E → M , we denote by G ( E ) the group of gauge transformations and by G f ( E ) the set of isomorphism classes of bundleisomorphisms covering f . Note that there is a free and transitive action of G ( E ) on G f ( E ), givenby G f ( E ) × G ( E ) → G f ( E ) , ( b f , u ) b f ◦ u. Hence, upon fixing one particular b f ∈ G f ( E ), the space G f ( E ) is isomorphic to the group of gaugetransformations G ( E ). Remark B.2.5. If E = M × C k is the trivial bundle, the group G ( E ) coincides with C ∞ (cid:0) M, U( k ) (cid:1) , see Remark B.1.1. Then for u ∈ C ∞ (cid:0) M, U( k ) (cid:1) , we define b f u : M × C k → M × C k , b f u ( x, z ) := (cid:0) f ( x ) , u ( x ) z (cid:1) . This gives a canonical identification C ∞ (cid:0) M, U( k ) (cid:1) ∼ = −→ G f ( M × C k ) , u b f u . We then write E u for the bundle defined by b f u . Lemma B.2.6. If b f , b f : E → E are two bundle isomorphisms covering f , then E b f ∼ = E b f ifand only if there exists ϕ t ∈ C ∞ (cid:0) R , G ( E ) (cid:1) such that ϕ t +1 = b f − ◦ ϕ t ◦ b f . (B.15) Proof. If we define Φ : E × R → E × R , Φ( e, t ) := (cid:0) ϕ t ( e ) , t (cid:1) , then Φ (cid:0) b f − ( e ) , t + 1 (cid:1) = (cid:0) ϕ t +1 ◦ b f − ( e ) , t + 1 (cid:1) = (cid:0) b f − ◦ ϕ t ( e ) , t + 1 (cid:1) . This implies that Φ is a Z -equivariant bundle isomorphism and so E b f and E b f are isomorphic.Conversely, we can lift a bundle isomorphism E b f ∼ = E b f to a Z -equivariant map E × R → E × R and use this to define ϕ t with the required property. .2. Flat Connections over Mapping Tori Remark B.2.7. Assume that E = M × C k is a trivial bundle, and let u, v ∈ C ∞ (cid:0) M, U( k ) (cid:1) define vector bundles E u and E v as in Remark B.2.5. Then (B.15) takes the form ϕ t +1 = b f − v ◦ ϕ t ◦ b f u = v − ( ϕ t ◦ f ) u. Additional Remarks for Line Bundles. In the case that E = L is a line bundle, we canalso give a more topological interpretation of (B.15). Since U(1) is abelian, the group of gaugetransformations of L is given by G ( L ) = C ∞ (cid:0) M, U(1) (cid:1) , irrespectively of whether L is trivializableor not. From the long exact sequence associated to the coefficient sequence (B.7), we get an exactsequence 0 −→ Z −→ C ∞ ( M, R ) −→ C ∞ (cid:0) M, U(1) (cid:1) −→ H ( M, Z ) −→ . (B.16)Here, the map C ∞ (cid:0) M, U(1) (cid:1) → H ( M, Z ) is given by u u ∗ ∈ Hom (cid:0) π ( M ) , Z (cid:1) = H ( M, Z ) . Remark B.2.8. The de Rham theorem defines a mapΩ pcl ( M, R ) → H p ( M, R ) , where the left hand side denotes the space of closed p -forms. Since H ( M, Z ) has no torsion, wecan use this to identify H ( M, Z ) ∼ = Ω cl ( M, Z ) dC ∞ ( M, R ) . Here, Ω pcl ( M, Z ) denotes the space of closed p -forms with integral periods , i.e., the kernel of theprojection Ω pcl ( M, R ) → H p ( M, R ) /H p ( M, Z ) . Using this, the last map in (B.16) can be expressed as C ∞ (cid:0) M, U(1) (cid:1) → H ( M, Z ) , u h u − du πi i . (B.17)Now, u ∈ C ∞ (cid:0) M, U(1) (cid:1) is mapped to 0 if and only if u = exp(2 πig ) for some g ∈ C ∞ ( M, R ),because then u − du = 2 πidg. Also note that there exists g with u = exp(2 πig ) precisely if u is homotopic to a constant map.Then we have the following topological interpretation of (B.15). Proposition B.2.9. Let L → M be a Hermitian line bundle, and let b f , b f ∈ G f ( L ) . Define u ∈ G ( L ) by requiring that b f = b f ◦ u . Then the line bundles L b f and L b f are isomorphic if andonly if h u − du πi i ∈ im (cid:0) Id − f ∗ (cid:1) ⊂ H ( M, Z ) . (B.18) Proof. Assume first that L b f ∼ = L b f . Then there exists ϕ t ∈ C ∞ (cid:0) R , G ( L ) (cid:1) as in Lemma B.2.6such that ϕ t +1 = b f − ◦ ϕ t ◦ b f . Now, since U(1) is abelian, b f − ◦ ϕ t ◦ b f = ϕ t ◦ f . Hence, ϕ t +1 = u − ◦ b f − ◦ ϕ t ◦ b f = u − ( ϕ t ◦ f ) . Appendix B. Remarks on Moduli Spaces This yields ϕ − dϕ = f ∗ ( ϕ − dϕ ) − u − du (B.19)Moreover, since ϕ is homotopic to ϕ we can find g : M → R such that ϕ − dϕ = ϕ − dϕ + 2 πidg. From this and (B.19) we see that (B.18) is valid. Conversely, let us assume that (B.18) holds.This means that there exist ϕ ∈ C ∞ (cid:0) M, U(1) (cid:1) and g ∈ C ∞ ( M, R ) such that u − du = f ∗ ( ϕ − dϕ ) − ϕ − dϕ + 2 πidg. For t ∈ [0 , 1] define ϕ t := ϕ exp (cid:0) − πitg (cid:1) . Then, upon adding a constant to g , we conclude that u = ϕ − ( ϕ ◦ f ) . This allows us to extend ϕ t for all t ∈ R in such a way that ϕ t +1 = u − ( ϕ t ◦ f ) = b f − ◦ ϕ t ◦ b f , and we get the required isomorphism. Remark B.2.10. (i) As we have mentioned in Section B.1.2, line bundles over M f are classified by the group H ( M f , Z ). This group fits into an exact sequence associated to the fiber bundle M ֒ → M f → S . One can use for example the five-term exact sequence induced by the Leray-Serrespectral sequence to obtain an exact sequence . . . −→ H ( M ) Id − f ∗ −−−−→ H ( M ) −→ H ( M f ) i ∗ −→ H ( M ) Id − f ∗ −−−−→ H ( M ) −→ . . . (B.20)For higher dimensional spheres as the base this is usually referred as the Wang sequence ,see [33, p. 254].Using the discussion preceding Proposition B.2.9 we can give a geometric interpretationof the map H ( M ) → H ( M f ) in (B.20). First we use (B.17) to represent an elementof H ( M, Z ) by a gauge transformation u ∈ C ∞ (cid:0) M, U(1) (cid:1) , and let L u → M f be the linebundle defined by u as in Remark B.2.5. Since the isomorphism class of L u is independentof the homotopy class of u we get a well-defined map H ( M, Z ) → H ( M f , Z ) , h u − du πi i c ( L u ) . Now Proposition B.2.9 identifies the kernel of this map and gives a geometric explanationfor the exactness of the (B.20) at H ( M ). Also, from a geometric point of view, exactnessat H ( M f , Z ) is immediate; this simply means that a line bundle L → M f is of the form L = L u if and only if it restricts to the trivial line bundle over M . Concerning exactnessat H ( M ) we observe that the restriction L | M of a line bundle L → M f has to satisfy f ∗ c ( L | M ) = c ( L | M ). Conversely, this is precisely the condition which enables us to definea line bundle over M f . .2. Flat Connections over Mapping Tori M f whichare topologically non-trivial. A gauge transformation u ∈ C ∞ (cid:0) M, U(1) (cid:1) gives a flat bundle L u if and only if c ( L u ) is a torsion element in H ( M f , Z ). According to Proposition B.2.9and part (i) of this remark, this is precisely the case if there exists N ∈ N such that N · h u − du πi i = h u − N du N πi i ∈ im(Id − f ∗ ) ⊂ H ( M, Z ) . Already in the case that M is the 2-torus T one can easily find a diffeomorphism f : T → T such that coker (cid:0) Id − f ∗ : H ( M, Z ) → H ( M, Z ) (cid:1) contains torsion elements, see Remark 4.3.10 in Section 4.3. Flat Connections over Mapping Tori. For convenience we assume for the rest of this sectionthat a flat U( k )-bundle over M is necessarily trivializable. According to Remark B.1.13 this issatisfied for example if M = Σ is a closed surface, which is the case we are considering in Chapter4. Under this assumption a flat bundle E → M f restricts to the trivial bundle over the fiber, sothat it is of the form considered in Remark B.2.5, i.e., E = E u u ∈ C ∞ (cid:0) M, U( k ) (cid:1) , where E u is the mapping torus of the bundle isomorphism b f u : M × C k → M × C k . We can thenidentify the space of sections of E u as C ∞ ( M f , E u ) = (cid:8) ϕ t : R → C ∞ ( M, C k ) (cid:12)(cid:12) ϕ t +1 = u − ( ϕ t ◦ f ) (cid:9) . More generally, b f u induces a pullback b f ∗ u α = u − ( f ∗ α ) , α ∈ Ω • ( M, C k ) , (B.21)and Ω • ( M f , E u ) = (cid:8) α t ∈ Ω ( M × R , C k ) (cid:12)(cid:12) α t +1 = b f ∗ u α t (cid:9) . (B.22)In a similar way, a U( k )-connection A on E u can equivalently be described as a Lie algebra valued1-form A = a t + b t dt, a t ∈ C ∞ (cid:0) R , Ω ( M, u ( k ) (cid:1) , b t ∈ C ∞ (cid:0) R , C ∞ ( M, u ( k ) (cid:1) , which is Z -equivariant in the sense that a t +1 = b f ∗ u a t = u − ( f ∗ a t ) u + u − du, b t +1 = b f ∗ u b t = u − ( b t ◦ f ) u. (B.23)The curvature of A is given by F A = d M × R A + A ∧ A = d M a t + a t ∧ a t + (cid:0) d M b t − ∂ t a t + [ a t , b t ] (cid:1) ∧ dt. Hence, if F a t ∈ C ∞ (cid:0) R , Ω ( M, u ( k ) (cid:1) denotes the curvature associated to the path of connections a t over M , we have F A = F a t + (cid:0) d a t b t − ∂ t a t (cid:1) ∧ dt. Appendix B. Remarks on Moduli Spaces Therefore, A is flat if and only if F a t = 0 , and ∂ t a t = d a t b t . (B.24)Before we can describe the structure of the moduli space of flat U( k )-bundles on M f we needthe following technical result. Lemma B.2.11. Let A = a t + b t dt be a flat connection on E u . Then there exists a gaugetransformation v ∈ C ∞ (cid:0) M, U( k ) (cid:1) such that the constant path a defines a connection A on E v ,and there exists an isomorphism Φ : E v → E u , with Φ ∗ A = A . Proof. Let ϕ t : R → C ∞ (cid:0) M, U( k ) (cid:1) be the unique solution to ∂ t ϕ t = − b t ϕ t , ϕ ≡ e, where e ∈ U( k ) is the identity matrix. Define v := uϕ ∈ C ∞ (cid:0) M, U( k ) (cid:1) . We claim that ϕ t +1 = u − ( ϕ t ◦ f ) v and ∂ t ( a t · ϕ t ) = 0 . (B.25) Proof of (B.25) . For the first assertion we recall from (B.23) that b t satisfies b t +1 = u − ( b t ◦ f ) u. From this and the definition of ϕ t , one easily checks that both sides of the claimed equality aresolutions of the initial value problem ∂ t e ϕ t = − u − ( b t ◦ f ) u e ϕ t , e ϕ = u − v, and hence agree. Now, using the identity ∂ t ϕ − t = − ϕ − t ( ∂ t ϕ t ) ϕ − t , we compute that ∂ t ( a t · ϕ t ) = ∂ t ( ϕ − t a t ϕ t + ϕ − t d M ϕ t )= − ϕ − t ( ∂ t ϕ t ) ϕ − t a t ϕ t + ϕ − t ( ∂ t a t ) ϕ t + ϕ − t a t ( ∂ t ϕ t ) − ϕ − t ( ∂ t ϕ t ) ϕ − t d M ϕ t + ϕ − t d M ( ∂ t ϕ t )= ϕ − t ( ∂ t a t ) ϕ t + ϕ − t ( b t a t ) ϕ t − ϕ − t ( a t b t ) ϕ t + ϕ − t b t d M ϕ t − ϕ − t d M ( b t ϕ t )= ϕ − t (cid:0) ∂ t a t − d M b t − [ a t , b t ] (cid:1) ϕ t = ϕ − t (cid:0) ∂ t a t − d a t b t (cid:1) ϕ t . Since we are assuming that A is flat, condition (B.24) yields the second part of (B.25).Having established (B.25) we continue with the proof of Lemma B.2.11. First of all we canuse the first formula in (B.25) and Remark B.2.7, to deduce that the family ϕ t defines a bundleisomorphism Φ : E v ∼ = −→ E u . On the other hand, we can use (B.23) and the definition of ϕ to compute that f ∗ a = a · u − = ( a · ϕ ) · v − = a · v − , .2. Flat Connections over Mapping Tori a t · ϕ t is constant. Also note that a · ϕ = a · e = a . This implies that the constant path a defines a connection A on E v . Moreover, Φ ∗ A is given byΦ ∗ A = a t · ϕ t + (cid:0) ϕ − t b t ϕ t + ϕ − t ∂ t ϕ t (cid:1) dt, (B.26)and thus, Φ ∗ A = a t · ϕ t ≡ A . The Moduli Space of Flat Connections. We can now state geometric version of PropositionB.2.4. Recall that we are assuming that a flat bundle over M is necessarily trivial. We let F M denote the space of flat U( k )-connections over M , and define c M ( M f ) := (cid:8) ( a, v ) ∈ F M × C ∞ (cid:0) M, U( k ) (cid:1) (cid:12)(cid:12) b f ∗ v a = a (cid:9) . Here, b f ∗ v a is defined as in (B.21). Note the similarity of the definition of c M ( M f ) and (cid:8) ( α, g ) ∈ Hom (cid:0) π ( M ) , U( k ) (cid:1) × U( k ) (cid:12)(cid:12) g − αg = f ∗ α (cid:9) , which we considered in (B.13) in the context of the algebraic description of M (cid:0) M f , U( k ) (cid:1) . Thereis a natural action of C ∞ (cid:0) M, U( k ) (cid:1) on c M ( M f ), given by( a, v ) · ϕ := (cid:0) a · ϕ, ( ϕ − ◦ f ) vϕ (cid:1) , ( a, v ) ∈ c M ( M f ) , ϕ ∈ C ∞ (cid:0) M, U( k ) (cid:1) . This action is well-defined since for u := ( ϕ − ◦ f ) vϕ we have b f ∗ u ( a · ϕ ) = ( b f ∗ v a ) · v − u ( ϕ ◦ f ) = a · ϕ. For notational brevity we use the following abbreviations for the moduli spaces of flat bundles M ( M ) := M (cid:0) M, U( k ) (cid:1) , M ( M f ) := M (cid:0) M f , U( k ) (cid:1) . We then have the following analog of of Proposition B.2.4. Proposition B.2.12. With respect to the natural action of C ∞ (cid:0) M, U( k ) (cid:1) on c M ( M f ) we have c M ( M f ) (cid:14) C ∞ (cid:0) M, U( k ) (cid:1) ∼ = M ( M f ) . Moreover, the projection c M ( M f ) → F M onto the first factor induces a surjection [ i ∗ ] : M ( M f ) → Fix( f ∗ ) ⊂ M ( M ) . If we represent [ a ] ∈ Fix( f ∗ ) ⊂ M ( M ) by a ∈ F M , then [ i ∗ ] − [ a ] ∼ = (cid:8) v ∈ C ∞ (cid:0) M, U( k ) (cid:1) (cid:12)(cid:12) b f ∗ v a = a (cid:9)(cid:14) I ( a ) , where I ( a ) = { g ∈ U( k ) | g − ag = a } denotes the isotropy group of a in U( k ) . Appendix B. Remarks on Moduli Spaces Proof. Every element ( a, v ) ∈ M ( M f ) defines a flat U( k )-bundle ( E v , A ) over M f . Here, A is used as in Lemma B.2.11 to denote the flat connection on E v induced by the constant path a : R → Ω (cid:0) M, u ( k ) (cid:1) . We defineΨ : c M ( M f ) → M ( M f ) , ( a, v ) [ E v , A ] . If ( a, v ) ∈ c M ( M f ) and ϕ ∈ C ∞ (cid:0) M, U( k ) (cid:1) , then ϕ = v − ( ϕ ◦ f ) u, where u := ( ϕ − ◦ f ) vϕ. According to Lemma B.2.6 this implies that E u ∼ = E v . Moreover, the connection A on E v pullsback to the connection A · ϕ on E u , see (B.26). Thus,Ψ (cid:0) ( a, v ) · ϕ (cid:1) = Ψ( a, v ) , and we get a well-defined mapΨ : c M ( M f ) (cid:14) C ∞ (cid:0) M, U( k ) (cid:1) → M ( M f ) . To see that Ψ is surjective, let [ E u , A ] ∈ M ( M f ). Then Lemma B.2.11 implies that there exist a ∈ F M and v ∈ C ∞ (cid:0) M, U( k ) (cid:1) in such a way that E u is isomorphic to E v , and the connection A pulls back to the connection A defined by a . By definition, this gives an element ( a, v ) ∈ c M ( M f )such that Ψ( a, v ) = [ E u , A ]. To check injectivity, assume that Ψ( a, v ) = Ψ( e a, e v ). According toLemma B.2.6 and (B.26) this means that there exists ϕ t : R → C ∞ (cid:0) M, U( k ) (cid:1) such that ϕ t +1 = v − ( ϕ t ◦ f ) e v, a · ϕ t = e a, and ϕ − t ∂ t ϕ t = 0 . In particular, ϕ t ≡ ϕ is independent of t , and e a = a · ϕ, and e v = ( ϕ − ◦ f ) vϕ. Hence, ( e a, e v ) = ( a, v ) · ϕ , which establishes injectivity. The rest of the proof is formally the sameas the proof of Proposition B.2.4 and shall be omitted. B.3 Holomorphic Line Bundles over Riemann Surfaces. In this section we discuss some aspects of closed surfaces related to complex geometry. A generalreference is [96, Ch.’s I–III]. Moreover, a concise introduction can be found in [80, Sec. 1.4]. Complex Structures on Closed Surfaces. Let Σ be a closed, oriented surface. Endow Σwith a Riemannian metric g Σ , with volume form vol Σ of unit volume. The metric g Σ defines thestructure of a complex manifold on Σ in the following way: Let ∗ be the Hodge star operator onΣ. On Ω (Σ) it satisfies ∗ = − , = { α ∈ Ω | ∗ α = − iα } and Ω , = { α ∈ Ω | ∗ α = iα } . (B.27)Denote by P , and P , the associated projections and define the Dolbeault operators ∂ := P , ◦ d, ¯ ∂ := P , ◦ d. .3. Holomorphic Line Bundles over Riemann Surfaces. • → Ω • and satisfy ∂ = ¯ ∂ = 0 . This is because Ω , and Ω , are trivial on 2-dimensional almost complex manifolds. Therefore,the almost complex structure is integrable . This means that we can find local coordinates z = x + iy with holomorphic transition functions such that ∗ dx = dy . In these coordinates, the metric g Σ is conformal to the standard metric on C , i.e., g Σ = e u ( z, ¯ z ) dz ⊗ d ¯ z, u : U ⊂ Σ → R . For a proof see [94, Sec. 5.10] or [68, Thm 4.16]. The sheaf of holomorphic functions on Σ isgiven by O Σ ( U ) = ker ¯ ∂ | U ⊂ C ∞ ( U ) , U ⊂ Σ open . Holomorphic Line Bundles. Now let L → Σ be a Hermitian line bundle on Σ, and let A bea unitary connection on L with associated covariant derivative d A : Ω (Σ , L ) → Ω (Σ , L ) . Define the twisted Dolbeault operators by ∂ A := P , ◦ d A : Ω (Σ , L ) → Ω , (Σ , L ) , ¯ ∂ A := P , ◦ d A : Ω (Σ , L ) → Ω , (Σ , L ) . (B.28)As in the untwisted case the extension of ¯ ∂ A to Ω , • (Σ , L ) satisfies ¯ ∂ A = 0. We can then definea holomorphic structure on L by requiring that its sheaf of holomorphic sections is given by O ( U, L A ) := ker ¯ ∂ A | U . Since ¯ ∂ A ( f s ) = ( ¯ ∂f ) s + f ¯ ∂ A s , it is clear that O ( L A ) is a sheaf of O Σ -modules. Moreover, it followsfrom elliptic theory that the space of global sections O (Σ , L A ) := ker ¯ ∂ A is finite dimensional. Remark. According to the above definition, every unitary connection on a line bundle definesa holomorphic structure and one might ask whether this is a suitable definition. Indeed, onecan construct a complex structure on the total space of L such that the projection L → Σ isholomorphic. A section s ∈ C ∞ (Σ , L ) is then holomorphic as a map between Σ and L if and onlyif ¯ ∂ A s = 0, see [37, Thm. 2.1.53 & Sec. 2.2.2]. The Riemann-Roch Theorem. Although we will not need it in the main body of the thesis,we digress briefly on the famous Riemann-Roch Theorem in its version for line bundles, see [55,Thm 5.4.1] and [55, Sec. 5.6]. Let K := T Σ → Σ be the tangent bundle of Σ viewed as a complexline bundle. The metric g Σ and the Levi-Civita connection endow K with a Hermitian metric,respectively, a holomorphic structure. K together with this structure is called the canonical linebundle of Σ. Theorem B.3.1 (Riemann-Roch) . Let Σ be a closed Riemann surface of genus g . Let L A → Σ be a Hermitian line bundle endowed with the holomorphic structure given by a unitary connection A . Then dim O (Σ , L A ) − dim O (Σ , K ⊗ L − A ) = deg L A − g + 1 . Appendix B. Remarks on Moduli Spaces Remark. The left hand side of the Riemann-Roch Theorem is the index of the operator¯ ∂ A : Ω (Σ , L A ) → Ω , (Σ , L A ) . To see this, note first that by definition ker ¯ ∂ A = O (Σ , L A ). To identify the cokernel, we notethat ¯ ∂ tA = − ∗ ∂ A ∗ . Recall that we are using the complex linear ∗ operator. Therefore, the ∗ operator maps the kernel of ¯ ∂ tA toker (cid:0) ∂ A : Ω , (Σ , L A ) → Ω (Σ , L A ) (cid:1) . Observing that K − = ( T ∗ Σ) , , we can interpret ∂ A as an operator ∂ A : Ω (Σ , K − ⊗ L A ) → Ω , (Σ , K − ⊗ L A ) . Since anti-holomorphic sections of a line bundle are in 1-1 correspondence to holomorphic sectionsof the dual bundle we get that dim(ker ¯ ∂ tA ) = dim O (Σ , K ⊗ L − A )which identifies the left hand side of the Riemann-Roch Formula as an index. The right hand sideis then the integral over the corresponding index density as in the Atiyah-Singer Index Theorem1.2.16, see [13, Sec. 4.1]. Relation to the Signature Operator. Since we are usually dealing with the signature operatorrather than the Dolbeault operator we also want to mention how they can be related. It followsfrom the definition (B.27) of the almost complex structure on Σ thatΩ , = Ω + ∩ Ω and Ω , = Ω − ∩ Ω , We thus have isometries Φ + : Ω + → Ω ⊕ Ω , , α 7→ √ α [0] + α [1] , and Φ − : Ω − → Ω , ⊕ Ω , α α [1] + √ α [2] . Let ¯ ∂ A be the Dolbeault operator on Ω • (Σ , L A ). One checks thatΦ − ◦ ( d A + d tA ) ◦ Φ − = √ ∂ A : Ω ⊕ Ω , → Ω , ⊕ Ω and Φ + ◦ ( d + d t ) ◦ Φ − − = √ ∂ tA : Ω , ⊕ Ω → Ω ⊕ Ω , . This implies Lemma B.3.2. The de Rham operator d A + d tA on Σ with values in the line bundle L A is unitaryequivalent to √ 2( ¯ ∂ A + ¯ ∂ tA ) : Ω • (Σ , L A ) → Ω • (Σ , L A ) . Under this equivalence, the signature operator D + A corresponds to √ ∂ A : Ω (Σ , L A ) ⊕ Ω , (Σ , L A ) → Ω , (Σ , L A ) ⊕ Ω (Σ , L A ) .3. Holomorphic Line Bundles over Riemann Surfaces. Remark. With Lemma B.3.2 at hand one could derive the Riemann-Roch Theorem from theHirzebruch Signature Theorem and the Gauss-Bonnet Theorem, or vice versa. Certainly, therelation among these results becomes more complicated in higher dimensions. The Moduli Space of Holomorphic Line Bundles. We now want to give some remarks onthe notion of equivalence of holomorphic line bundles. First, recall from Section B.1.1 that thegroup of gauge transformations G = C ∞ (cid:0) Σ , U(1) (cid:1) acts on the space of Hermitian connections A ( L ) on L via A · u = A + u − du, u ∈ G , A ∈ A ( L )One easily checks that the associated twisted Dolbeault operators satisfy¯ ∂ A · u = ¯ ∂ A + u − ¯ ∂u = u − ¯ ∂ A u. Therefore, for every U ⊂ Σ we get an isomorphism of O Σ ( U )-modules O ( U, L A ) → O ( U, L A · u ) , s u − s. (B.29)Because of this, gauge equivalent connections on L give rise to equivalent holomorphic structures.The converse is certainly not true. For this note that we can use any f ∈ C ∞ (Σ , C ∗ ) to define aholomorphic structure on L via ( ¯ ∂ A ) f := f − ¯ ∂ A f = ¯ ∂ A + f − ¯ ∂f. (B.30)Via an isomorphism of the form (B.29), this holomorphic structure is equivalent to the oneinduced by A . However, the underlying connection A + f − df is in general not unitary.The relation between the moduli space of complex line bundles and the moduli space ofline bundles with connection can be made very explicit. Since we want to avoid dealing withisomorphic Hermitian line bundles which are not equal, we fix one Hermitian line bundle L → Σof degree 1 and let L k := L ⊗ k , where a negative exponent means taking the tensor product ofthe dual. We then let A k be the space of Hermitian connections on L k and A = S k A k .We now consider the complexified group G c := C ∞ (Σ , C ∗ ) of gauge transformations. It actson the set of holomorphic structures on L k via (B.30). We want to lift this to an action on A k .The underlying idea why this should be possible is a theorem of Chern that associates to everyholomorphic structure and Hermitian metric on a vector bundle a unique compatible Hermitianconnection, see [96, Thm. III.2.1]. In the simple case at hand we have the following: Lemma B.3.3. Let A ∈ A k with associated Dolbeault operator ¯ ∂ A , and let f ∈ G c . Define A · f := A + f − ¯ ∂f − ¯ f − ∂ ¯ f . The A · f is a Hermitian connection satisfying ¯ ∂ A f = f − ¯ ∂ A f. This defines an action of G c on A k which for G ⊂ G c coincides with the standard action.Proof. It is straightforward to check that f − ¯ ∂f − ¯ f − ∂ ¯ f is an imaginary valued 1-form on Σ.Thus, the connection A · f is Hermitian. Since P , (cid:0) ¯ f − ∂ ¯ f (cid:1) = 0, the second assertion also follows.12 Appendix B. Remarks on Moduli Spaces The Leibniz rule applied to ¯ ∂ and ∂ shows that A · ( f g ) = ( A · f ) · g so that we indeed get anaction of G c on A k . Moreover, a short computations shows that f − ¯ ∂f − ¯ f − ∂ ¯ f = f − df − ∂ log | f | (B.31)from which one readily finds that if f takes values in U(1), the action coincides with the usualone. Definition B.3.4. The moduli space of holomorphic structures on L k is defined as the quotient A k / G c . Moreover, we define the Picard group asPic(Σ) := [ k ∈ Z A k / G c = A / G c , where the group structure is induced by the tensor product of line bundles with connection. Remark. We note without further details that we have defined A k / G c and thus Pic(Σ) purelyin differential geometric language. The proof that the objects we obtain coincide with the onesdefined in holomorphic terms requires more work than sketched here. Relation to the Moduli Space of Unitary Connections. We now have the ingredients torelate the moduli space A / G of line bundles with connections and the moduli space Pic(Σ) ofholomorphic line bundles. We start by recalling a structure result for A / G , see [37, Sec. 2.2.1].Consider the map sending a connection to its Chern-Weil representative, CW : A → Ω (Σ , R ) , CW ( A ) := i π F A . Note that no trace is involved since we are dealing with line bundles. The image of CW is easilyseen to coincide with the space Ω cl (Σ , Z ) of closed 2-forms with integral periods, see RemarkB.2.8. Moreover, CW ( A + ia ) = CW ( A ) for every closed 1-form a . These considerations producea short exact sequence 0 → Ω cl (Σ , R ) → A CW −−→ Ω cl (Σ , Z ) → . We also note that the action of G on A changes a connection A by a closed 1-form with integralperiods. In particular, the map CW is invariant under the action of G . Moreover, in the case athand Ω cl (Σ , R ) / Ω cl (Σ , Z ) = H (Σ , R ) /H (Σ , Z ) = H (cid:0) Σ , U(1) (cid:1) . (B.32)Thus, taking quotients, we get the following exact sequence of groups0 → H (cid:0) Σ , U(1) (cid:1) → A / G CW −−→ Ω cl (Σ , Z ) → . Remark. This exact sequence generalizes to the case of an arbitrary closed manifold M . How-ever, the proof we sketched does not generalize immediately. For this note that in the case of asurface Σ, there are no line bundles which give torsion elements. Moreover, the first homologygroup H (Σ , Z ) is torsion-free, a fact we used in the second equality of (B.32). In the generalcase, the Chern-Weil map does not capture possible torsion. However, the universal coefficienttheorem shows that Tor (cid:0) H ( M, Z ) (cid:1) = Tor (cid:0) H ( M, Z ) (cid:1) . .3. Holomorphic Line Bundles over Riemann Surfaces. H (cid:0) M, U(1) (cid:1) = Hom (cid:0) H ( M, Z ) , U(1) (cid:1) which appears on the left hand side of the above sequence.Returning now to holomorphic line bundles over surfaces we state the following structureresult. Proposition B.3.5. The natural projections A / G → Pic(Σ) and Ω cl (Σ , Z ) → H (Σ , Z ) fit into the following commutative diagram with exact rows −−−−→ H (cid:0) Σ , U(1) (cid:1) −−−−→ A / G CW −−−−→ Ω cl (Σ , Z ) −−−−→ (cid:13)(cid:13)(cid:13) y y −−−−→ H (cid:0) Σ , U(1) (cid:1) −−−−→ Pic(Σ) c −−−−→ H (Σ , Z ) −−−−→ . Sketch of proof. We can write every element f ∈ G c uniquely as f = exp( ϕ ) u, u ∈ G , ϕ ∈ C ∞ (Σ , R ) . It follows from (B.31) that for every A ∈ A A · f = A + u − du + dϕ − ∂ϕ. Therefore, the moduli space Pic(Σ) = A / G c can alternatively be described as a quotient of A / G by the action [ A ] · ϕ := (cid:2) A + dϕ − ∂ϕ (cid:3) , [ A ] ∈ A / G , ϕ ∈ C ∞ (Σ , R ) . With respect to this, the map CW has the following equivariance property: CW (cid:0) [ A ] · ϕ (cid:1) = CW ([ A ]) + 1 πi ¯ ∂∂ϕ = CW ([ A ]) + 12 π (∆ ϕ ) vol Σ . The latter equality is a consequence of the K¨ahler identities but can also be checked directly inthis simple case. Therefore, we can equip the image of CW , i.e., the space Ω cl (Σ , Z ), with anatural action of C ∞ (Σ , R ) by defining ω · ϕ := ω + 12 π (∆ ϕ ) vol Σ , ϕ ∈ C ∞ (Σ , R ) , ω ∈ Ω cl (Σ , Z ) . It follows from the Hodge decomposition theorem that the quotient of this action coincides with H (Σ , Z ). The stabilizers consist of the constant functions and thus agree with the stabilizers ofthe action of C ∞ (Σ , R ) on A / G . Moreover, the action of C ∞ (Σ , R ) on the fiber H (cid:0) Σ , U(1) (cid:1) istrivial. Therefore, taking quotients in the equivariant exact sequence0 → H (cid:0) Σ , U(1) (cid:1) → A / G CW −−→ Ω cl (Σ , Z ) → → H (cid:0) Σ , U(1) (cid:1) → Pic(Σ) c −→ H (Σ , Z ) → Appendix B. Remarks on Moduli Spaces Remark. As in the case of Hermitian line bundles, there is a description of Pic(Σ) in terms ofˇCech cohomology. Let O ∗ Σ be the sheaf of nowhere vanishing holomorphic functions on Σ. ThenPic(Σ) = H (Σ , O ∗ Σ ) , see [96, Lem. III.4.4]. Moreover, the exponential sequence Z → O Σ → O ∗ Σ gives rise to a longexact sequence in cohomology ... → H p (Σ , Z ) → H p (Σ , O Σ ) → H p (Σ , O ∗ Σ ) → H p +1 (Σ , Z ) → ... The sheaf O Σ is not fine so that the above sequence contains much more information than itssmooth version discussed in Section B.1.2. In the case at hand, this sequence is essentiallythe second line of the diagram in Proposition B.3.5. To see this, note that as Σ is complex1-dimensional, we have H (Σ , O Σ ) = 0. Thus, the above sequence reads H (Σ , Z ) → H (Σ , O Σ ) → Pic(Σ) → H (Σ , Z ) → . Moreover, the space H (Σ , O Σ ) is isomorphic to the Dolbeault cohomology group H , (Σ). ViaHodge theory, the latter can be identified with H (Σ , R ). Using (B.32) we arrive at the sequenceof Proposition B.3.5. ppendix C Some Computations Here, we include a computational discussion which will be used in the main body of this thesis.We introduce basic Eta and Zeta functions, and derive some of their values. In the second partof this appendix, we briefly discuss the Dedekind sums and their generalization which we use inSection 4.4. In particular, we establish the relation among them which we need to prove Theorem4.4.20. C.1 Values of Zeta and Eta Functions C.1.1 The Gamma and the Hurwitz Zeta Function We will need some standard facts concerning the Gamma function and the generalization byHurwitz of the Riemann Zeta function, see for example [29, Ch. 9] as a general reference. Recallthat the integral representation of the Gamma function isΓ( s ) = Z ∞ e − t t s − dt, Re( s ) > . It satisfies the functional equation Γ( s + 1) = s Γ( s ) , (C.1)which can be used to extend Γ( s ) meromorphically to the whole plane. Then Γ( s ) has no zerosand only simple poles at s = 0 , − , − , . . . The residues are given byRes Γ( s ) (cid:12)(cid:12) s = − n = ( − n n ! , n ∈ N . (C.2)The Hurwitz Zeta function is defined for q ∈ R + as ζ q ( s ) := ∞ X n =0 n + q ) s , Re( s ) > . (C.3)In particular, ζ ( s ) is the Riemann Zeta function. Using the Mellin transform one can derivebasic properties of ζ q ( s ). First, we note that1( n + q ) s = 1Γ( s ) Z ∞ t s − e − t ( n + q ) dt. Appendix C. Some Computations The formula for the geometric series shows that for t > ∞ X n =0 e − t ( n + q ) = e − tq − e − t , (C.4)which clearly decays exponentially with t as t → ∞ . Bernoulli Polynomials. The behaviour as t → Bernoullipolynomials B n ( x ). Recall, e.g. from [29, Sec. 9.1], that they can be defined via the generatingfunction te xt e t − ∞ X n =0 B n ( x ) t n n ! , | t | < π, x ∈ R . (C.5)Comparing this with (A.5), we note that the Bernoulli numbers with respect to the normaliza-tion we are using are given by B n = B n (0). Moreover, expanding e xt in (C.5) and comparingcoefficients of t n yields that B n ( x ) = n X k =0 (cid:18) nk (cid:19) B k x n − k . In particular, B ( x ) = 1 , B ( x ) = x − , B ( x ) = x − x + . (C.6)From the definition (C.5), one easily finds that the Bernoulli polynomials have the symmetryproperty B n (1 − x ) = ( − n B n ( x ) . (C.7) Values of the Hurwitz Zeta Function. It now follows from (C.5) and (C.7) applied to (C.4)that for t ∈ (0 , π ) ∞ X n =0 e − t ( n + q ) = e (1 − q ) t e t − ∞ X n =0 ( − n B n ( q ) t n − n ! . In particular, P ∞ n =0 e − t ( n + q ) = O ( t − ) as t → 0. This implies that we can apply the Mellintransform to (C.3) and interchange summation and integration. Then splitting the integral into R + R ∞ one easily obtains that for Re( s ) > s ) ζ q ( s ) = h ( s ) + ∞ X n =0 ( − n B n ( q )( s + n − n ! , where h ( s ) can be extended to a holomorphic function of s ∈ C . Therefore, Γ( s ) ζ q ( s ) extends toa meromorphic function on the whole plane. It has simple poles at the points s = 1 , , − , − , . . . with residues Res (cid:0) Γ( s ) ζ q ( s ) (cid:1)(cid:12)(cid:12) s = − n +1 = ( − n B n ( q ) n ! , n ∈ N . (C.8)Now, since Γ( s ) has no zeros, we can deduce that ζ q ( s ) extends to a meromorphic function onthe whole plane which can have only simple poles. Using (C.2) and (C.8) one finds that ζ q ( s ) hasa simple pole at s = 1 with residue 1. The other poles are cancelled out by the zeroes of Γ( s ) − and ζ q ( − n ) = − B n +1 ( q ) n + 1 , n ∈ N . (C.9) .1. Values of Zeta and Eta Functions C.1.2 An Eta Function and a “Periodic” Zeta Function Definition C.1.1. For x ∈ R let [ x ] denote the largest integer less or equal than x . We definethe n -th periodic Bernoulli function as P n ( x ) := ( , if x ∈ Z ,B n (cid:0) x − [ x ] (cid:1) , if x / ∈ Z , for n odd,and P n ( x ) := B n (cid:0) x − [ x ] (cid:1) , for n even . Remark. The definition of P n ( x ) for odd n is a bit artificial. Note that (C.7) implies that B n (1) = ( − n B n (0), so that for n even, we have B n (1) = B n (0). We note without proof, thatfor odd n with n = 1, one has B n (1) = B n (0) = 0 so that a distinction is unnecessary. However,when working with P ( x ), the above convention is convenient. Using (C.6) we note that explicitly, P ( x ) = ( , if x ∈ Z ,x − [ x ] − , if x / ∈ Z , P ( x ) = (cid:0) x − [ x ] (cid:1) − (cid:0) x − [ x ] (cid:1) + . (C.10)Most of our computations of Eta invariants in the main body of this thesis will be based onthe following result. Proposition C.1.2. Let q ∈ R , and define for Re( s ) > η q ( s ) := X n ∈ Z n = q sgn( n − q ) | n − q | s , and e ζ q ( s ) := X n ∈ Z n = q | n − q | s . (i) The function η q ( s ) extends to a holomorphic function for all s ∈ C , and η q (0) = 2 P ( q ) . (ii) The function e ζ q ( s ) extends to a meromorphic function with only one simple pole at s = 1 .Moreover, e ζ q (0) = ( , if q / ∈ Z , − , if q ∈ Z , ζ q ( − 1) = − P ( q ) . Proof. Write q := q − [ q ] ∈ [0 , η q ( s ) and e ζ q ( s ) converge absolutelyfor Re( s ) > 1, we can change the order of summation. Then, if q ∈ Z so that q = 0, we find that η q ( s ) = ζ ( s ) − ζ ( s ) = 0 , and e ζ q ( s ) = 2 ζ ( s ) . If q / ∈ Z we get η q ( s ) = ∞ X n =1 (cid:16) n − q (cid:17) s − ∞ X n =0 (cid:16) n + q (cid:17) s = ζ − q ( s ) − ζ q ( s ) , and e ζ q ( s ) = ∞ X n =1 (cid:16) n − q (cid:17) s + ∞ X n =0 (cid:16) n + q (cid:17) s = ζ − q ( s ) + ζ q ( s ) . Appendix C. Some Computations Since ζ ( s ), ζ − q ( s ) and ζ q ( s ) extend to meromorphic functions on the whole plane, with onlyone simple pole at s = 1 of residue 1, we can extend η q ( s ) and e ζ q ( s ) meromorphically. One seesthat η q ( s ) has no pole, whereas e ζ q ( s ) has a simple pole at s = 1. Moreover, η q ( s ) vanishes if q ∈ Z . Otherwise, we deduce from (C.9), (C.6) and (C.7) that η q (0) = ζ − q ( s ) − ζ q ( s ) = − B (1 − q ) + B ( q ) = 2 B ( q ) . From the definition of P ( q ) and q , part (i) follows. Concerning part (ii), we first assume that q = 0. Then e ζ q (0) = − B (1) = − , and e ζ q ( − 1) = − B (1)2 = − B (0) . For q = 0, one finds that e ζ q (0) = − B (1 − q ) − B ( q ) = B ( q ) − B ( q ) = 0 , and e ζ q ( − 1) = − (cid:0) B (1 − q ) + B ( q ) (cid:1) = − B ( q ) . Remark. Clearly, one could go on without difficulty, and determine more values of η q ( s ) and e ζ q ( s ) in terms of the periodic Bernoulli functions. However, Proposition C.1.2 covers all the caseswe are interested in. C.2 Generalized Dedekind Sums When studying the Eta invariant for 2-dimensional torus bundles over the circle, one naturallyencounters versions of the Dedekind sums . In this section, we include some relevant definitionsand computations. We start to collect some facts about finite Fourier series, see [12, Ch. 7]. C.2.1 Some Finite Fourier Analysis Let c ∈ Z with c = 0. In this section we will always use the c -th root of unity ξ := exp (cid:0) πic (cid:1) . Definition C.2.1. Assume that f : Z → C is c -periodic , i.e., f ( k + c ) = f ( k ) for all k ∈ Z . The Fourier transform b f : Z → C is defined as b f ( p ) = | c |− X k =0 f ( k ) ξ − kp , p ∈ Z . Remark C.2.2. Since we allow c to be negative, one has to be a bit careful concerning signs.Let ε = sgn( c ) and denote by b f ε the Fourier transform of f with respect to the c -th root of unity ξ ε = exp (cid:0) πi | c | (cid:1) . Then for all p ∈ Z b f ε ( p ) = | c |− X k =0 f ( k )( ξ ε ) − kp = b f ( εp ) . .2. Generalized Dedekind Sums ξ − kp is c -periodic in p , the Fourier transform is again c -periodic. Moreover, we can shiftthe sum by any m ∈ Z , i.e., b f ( p ) = | c | + m − X k = m f ( k ) ξ − kp . This implies that if g ( k ) := f ( k + m ), then b g ( p ) = | c |− X k =0 f ( k + m ) ξ − kp = | c |− X k =0 f ( k ) ξ − ( k − m ) p = ξ mp b f ( p ) . (C.11)Furthermore, if a ∈ Z with gcd( a, c ) = 1, then { ak | k = 0 , . . . , | c | − } is a representation systemof Z modulo c , so that b f ( p ) = | c |− X k =0 f ( ak ) ξ − akp , gcd( a, c ) = 1 . Hence, if d ∈ Z is an inverse of a modulo c , i.e., ad ≡ c ), then g ( k ) := f ( ak ) satisfies b g ( p ) = | c |− X k =0 f ( ak ) ξ − kp = | c |− X k =0 f ( k ) ξ − kdp = b f ( dp ) . (C.12)The finite geometric series yields that | c |− X k =0 ξ kp = ( | c | , if p ≡ c ) , . From this one easily deduces the Fourier inversion formula f ( k ) = | c | | c |− X p =0 b f ( p ) ξ pk , (C.13)see [12, Thm. 7.2]. Moreover, if g is another c -periodic function, one has the convolution formulæ( f ∗ g )( k ) := | c |− X l =0 f ( l ) g ( k − l ) = | c | | c |− X p =0 b f ( p ) b g ( p ) ξ pk , (C.14)and ( b f ∗ b g )( p ) = | c | | c |− X k =0 f ( k ) g ( k ) ξ − pk , (C.15)see [12, Thm 7.10].The facts we have collected so far are sufficient for the application to generalized Dedekindsums in Section C.2.2 below. Yet, we need to compute the Fourier transform for one particularclass of functions, which form the building blocks of generalized Dedekind sums. First, we20 Appendix C. Some Computations introduce some notation. Note that fixing a pair ( a, c ) with gcd( a, c ) = 1, and an inverse d of a modulo c is the same as fixing a matrix M = (cid:18) a bc d (cid:19) ∈ SL ( Z ) , c = 0 . Here, b is uniquely determined by requiring that ad − bc = 1. Moreover, for x, y ∈ R we define (cid:18) x ′ y ′ (cid:19) := M t (cid:18) xy (cid:19) = (cid:18) ax + cybx + dy (cid:19) . (C.16) Proposition C.2.3. Let P be the first periodic Bernoulli function, see Definition C.1.1. Fix M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) with c = 0 , let x, y ∈ R and x ′ as in (C.16) . Define a c -periodic function by f x,y,a,c ( k ) := P (cid:18) a k + xc + y (cid:19) , k ∈ Z . Then, b f x,y,a,c ( p ) = sgn( c ) ( P ( x ′ ) , if p ≡ c ) , (cid:16) i cot (cid:0) πdpc (cid:1) − δ ( x ′ ) (cid:17) ξ d [ x ′ ] p , otherwise.Here, δ is the characteristic function of R \ Z , i.e., δ ( x ′ ) = 0 if x ′ ∈ Z and δ ( x ′ ) = 1 if x ′ / ∈ Z .Moreover, as in Definition C.1.1, the expression [ x ′ ] refers to the largest integer less or equalthan x ′ . Before we give the proof of Proposition C.2.3 let us collect some special cases. Corollary C.2.4. (i) If f ( k ) := f , , ,c ( k ) = P (cid:0) kc (cid:1) , then b f ( p ) = ( , if p ≡ c ) , i cot (cid:0) πp | c | (cid:1) , otherwise. (C.17) In particular, | c |− X k =1 P (cid:0) kc (cid:1) = 0 . (C.18)(ii) Let x ∈ R , and f x ( k ) := f x, , ,c ( k ) = P (cid:0) k + xc (cid:1) . Then b f ( p ) = sgn( c ) ( P ( x ) , if p ≡ c ) , (cid:16) i cot (cid:0) πpc (cid:1) − δ ( x ) (cid:17) ξ [ x ] p , otherwise. (C.19) Proof of Proposition C.2.3. The proof consists of proving the special cases (C.17) and (C.19)first. The general case then follows using (C.11) and (C.12). Formula (C.17) is standard, see [12,Lem 7.3]. Nevertheless, we sketch a proof, since most computations we encounter are deducedfrom this formula. We assume first that c > 0. For k ∈ { , . . . , c − } we have f ( k ) = P (cid:0) kc (cid:1) = ( , if k = 0, kc − , otherwise . .2. Generalized Dedekind Sums p ≡ c ), b f ( p ) = c − X k =1 (cid:0) kc − (cid:1) = c c ( c − − c − = 0 , which is also the claim in (C.18). Now, for p not divisible by c , we have b f ( p ) = c − X k =1 (cid:0) kc − (cid:1) ξ − kp = − c − X k =1 ξ − kp + ∂∂x (cid:12)(cid:12)(cid:12) x = − πip c − X k =1 exp (cid:0) kxc (cid:1) . For x / ∈ πic Z , the formula for the finite geometric series states that c − X k =1 exp (cid:0) kxc (cid:1) = 1 − e x − e x/c − . Using this one verifies that ∂∂x (cid:12)(cid:12)(cid:12) x = − πip c − X k =1 exp (cid:0) kxc (cid:1) = − − ξ − p = − ξ p/ ξ p/ − ξ − p/ = (cid:0) i cot (cid:0) πpc (cid:1) − (cid:1) . Moreover, for p not divisible by c , one has P c − k =1 ξ − kp = − 1. Therefore, in this case b f ( p ) = + (cid:0) i cot (cid:0) πpc (cid:1) − (cid:1) = i cot (cid:0) πpc (cid:1) , which proves (C.17) for c > 0. For general c = 0, let ε = sgn( c ). Then P (cid:0) kc (cid:1) = εP (cid:0) k | c | (cid:1) , so thatwe deduce from the case c > b f ( p ) = ( , if p ≡ c ), i ε cot (cid:0) πεp | c | (cid:1) , otherwise.Since the cotangent is an odd function, we obtain (C.17) for c < x ∈ [0 , c > 0. Then f x ( k ) = ( P (cid:0) xc (cid:1) , if k ≡ c ) ,P (cid:0) kc (cid:1) + xc , otherwise. (C.20)Therefore, for p ≡ c ), b f x ( p ) = P (cid:0) xc (cid:1) + c − X k =1 xc + c − X k =1 P (cid:0) kc (cid:1) = P (cid:0) xc (cid:1) + c − c x, where we have used (C.18). Now, one easily verifies that for x ∈ [0 , P (cid:0) xc (cid:1) + c − c x = P ( x ) , which implies (C.19) for the case p ≡ c ). If p is not divisible by c we use (C.20) and (C.17) todeduce that b f x ( p ) = P (cid:0) xc (cid:1) + xc c − X k =1 ξ − kp + c − X k =1 P (cid:0) kc (cid:1) ξ − kp = P (cid:0) xc (cid:1) − xc + i cot (cid:0) πpc (cid:1) . Appendix C. Some Computations Since P (cid:0) xc (cid:1) − xc = − δ ( x ), formula (C.19) follows for x ∈ [0 , x ∈ R we write x = [ x ] + x , so that f x ( k ) = f x (cid:0) k + [ x ] (cid:1) , x ∈ [0 , . Since we have already proved (C.19) for f x , we can use (C.11) to get the required formula for f x . For this note that in the case p ≡ c ) we have ξ [ x ] p = 1. The case c < f x is odd.Now for the general formula of Proposition C.2.3 observe that f x,y,a,c ( k ) = P (cid:0) a k + xc + y (cid:1) = P (cid:0) ak + ax + cyc (cid:1) = P (cid:0) ak + x ′ c (cid:1) = f x ′ ( ak ) , where x ′ = ax + cy . Then (C.12) implies that b f x,y,a,c ( p ) = b f x ′ ( dp ) , so that Proposition C.2.3 follows immediately from (C.19). C.2.2 Relation Among Some Dedekind Sums Let M = (cid:0) a bc d (cid:1) ∈ SL ( Z ) with c = 0. Recall that the classical Dedekind sums are defined by s ( a, c ) := | c |− X k =1 P (cid:0) akc (cid:1) P (cid:0) kc (cid:1) , (C.21)see e.g., [12, p. 128]. Since the first periodic Bernoulli function is odd, we can replace c with | c | in both denominators of (C.21). Moreover, using that { ap | p = 0 , . . . , | c | − } is a representationsystem for Z modulo c , and that ad ≡ c ), one can replace a with d , so that s ( a, c ) = s ( a, | c | ) = s ( d, c ) = s ( d, | c | ) . (C.22)There are many more relations among different Dedekind sums but their discussion would lead tofar afield. We only mention that (C.17) and (C.15) easily imply the classical cotangent formula s ( a, c ) = | c | | c |− X p =1 i cot (cid:0) dp | c | (cid:1) i cot (cid:0) − p | c | (cid:1) = | c | | c |− X p =1 cot (cid:0) dpc (cid:1) cot (cid:0) pc (cid:1) , The Dedekind sums s ( a, c ) were generalized in several ways. The generalization we are interestedin was considered in [35, 70, 85]. Definition C.2.5. For x, y ∈ R define s x,y ( a, c ) := | c |− X k =0 P (cid:0) a k + xc + y (cid:1) P (cid:0) k + xc (cid:1) . Again, there are several relations among generalized Dedekind sums for different values of( x, y ) and ( a, c ). For example, s x,y ( a, c ) depends on ( x, y ) only modulo Z . For y this is immediateand for x this is because for m ∈ Z , one has s x + m,y ( a, c ) = | c |− m − X k = − m P (cid:0) a k + xc + y (cid:1) P (cid:0) k + xc (cid:1) = | c |− X k =0 P (cid:0) a k + xc + y (cid:1) P (cid:0) k + xc (cid:1) . (C.23) .2. Generalized Dedekind Sums Lemma C.2.6. Let x ′ be as in (C.16) , and let δ be the characteristic function of R \ Z . Then s x,y ( a, c ) = | c | P ( x ) P ( x ′ ) − | c | | c |− X p =1 (cid:16) i cot (cid:0) πpc (cid:1) − δ ( x ) (cid:17) × (cid:16) i cot (cid:0) πdpc (cid:1) + δ ( x ′ ) (cid:17) ξ ([ x ] − d [ x ′ ]) p . We will now make a simplifying assumption, which renders the following formulæ a bit moretransparent. Assumption. From now on, ( x, y ) ∈ R will always be chosen in such a way that x ∈ [0 , 1) and (cid:18) mn (cid:19) := (Id − M t ) (cid:18) xy (cid:19) = (cid:18) x − x ′ y − y ′ (cid:19) ∈ Z . (C.24)Note that under this assumption we have x − x ′ = m ∈ Z and [ x ] = 0, which yields that[ x ′ ] = − m and x ′ − [ x ′ ] = x . Thus, the cotangent formula of Lemma C.2.6 simplifies to s x,y ( a, c ) = | c | P ( x ) − | c | | c |− X p =1 (cid:16) i cot (cid:0) πpc (cid:1) − δ ( x ) (cid:17)(cid:16) i cot (cid:0) πdpc (cid:1) + δ ( x ) (cid:17) ξ dmp . (C.25)We then have the following relation between the generalized Dedekind sums and the classicalDedekind sums, see also [70, Sec. 8] and [71, Sec. 4]. Proposition C.2.7. Under the above assumption, let r ∈ { , . . . | c | − } with m ≡ r ( c ) . Then s x,y ( a, c ) − s ( a, c ) = | c | (cid:0) P ( x ) − (cid:1) + | c |− r X k =1 P (cid:0) dk | c | (cid:1) + P (cid:0) dm | c | (cid:1) + δ ( x ) (cid:16) P (cid:0) m | c | (cid:1) − P (cid:0) dm | c | (cid:1)(cid:17) + δ ( x ) (cid:0) − δ ( mc ) (cid:1) . Proof. The cotangent formula (C.25) shows that s x,y ( a, c ) = | c | P ( x ) + | c | | c |− X p =1 cot (cid:0) πpc (cid:1) cot (cid:0) πdpc (cid:1) ξ dmp + | c | δ ( x ) | c |− X p =1 ξ dmp + | c | δ ( x ) | c |− X p =1 i cot (cid:0) πdpc (cid:1) ξ dmp − | c | δ ( x ) | c |− X p =1 i cot (cid:0) πpc (cid:1) ξ dmp . (C.26)Let ˜ x = 0 and ˜ y = − mc , so that in the notation of (C.16) we have ˜ x ′ = − m , and [˜ x ] − d [˜ x ′ ] = dm .Applying Lemma C.2.6 to s ˜ x, ˜ y ( a, c ) yields s ˜ x, ˜ y ( a, c ) = | c | | c |− X p =1 cot (cid:0) πpc (cid:1) cot (cid:0) πdpc (cid:1) ξ dmp , Appendix C. Some Computations which is exactly the first sum in (C.26). On the other hand, by definition, s ˜ x, ˜ y ( a, c ) = | c |− X k =1 P (cid:0) ak − mc (cid:1) P (cid:0) kc (cid:1) = | c |− X k =1 P (cid:0) ak + mc (cid:1) P (cid:0) kc (cid:1) = c − X k =1 P (cid:0) k + mc (cid:1) P (cid:0) dkc (cid:1) , where we have first used that P is odd and summed over − k , and then summed over ak insteadof k . Comparing the two expressions for s ˜ x, ˜ y ( a, c ) we find that | c | | c |− X p =1 cot (cid:0) πpc (cid:1) cot (cid:0) πdpc (cid:1) ξ dmp = | c |− X k =1 P (cid:0) k + mc (cid:1) P (cid:0) dkc (cid:1) (C.27)Using (C.17) and the Fourier inversion formula (C.13) we can immediately identify two otherterms in (C.26), namely | c | | c |− X p =1 i cot (cid:0) πdpc (cid:1) ξ dmp = P (cid:0) m | c | (cid:1) , | c | | c |− X p =1 i cot (cid:0) πpc (cid:1) ξ dmp = P (cid:0) dm | c | (cid:1) . (C.28)Moreover, | c |− X p =1 ξ dmp = | c | (cid:0) − δ ( dmc ) (cid:1) − | c | (cid:0) − δ ( mc ) (cid:1) − , where in the last equality we have used that gcd( d, c ) = 1. Employing this together with (C.27)and (C.28), we can rewrite (C.26) as s x,y ( a, c ) = | c | P ( x ) + c − X k =1 P (cid:0) k + mc (cid:1) P (cid:0) dkc (cid:1) + δ ( x ) (cid:16) − δ ( mc ) − | c | (cid:17) + δ ( x ) (cid:16) P (cid:0) m | c | (cid:1) − P (cid:0) dm | c | (cid:1)(cid:17) . (C.29)To find the claimed formula for s x,y ( a, c ) − s ( a, c ), let us first study the difference c − X k =1 P (cid:0) k + mc (cid:1) P (cid:0) dkc (cid:1) − s ( a, c ) = c − X k =1 P (cid:0) dk | c | (cid:1)(cid:16) P (cid:0) k + m | c | (cid:1) − P (cid:0) k | c | (cid:1)(cid:17) , (C.30)where we have used that we can replace c with | c | in the first term and that according to (C.22)we have s ( a, c ) = s ( d, | c | ). Now, with r ∈ { , . . . , | c | − } satisfying m ≡ r ( c ), we get P (cid:0) k + m | c | (cid:1) − P (cid:0) k | c | (cid:1) = r | c | , if k < | c | − r,P (cid:0) r | c | (cid:1) , if k = | c | − r, r | c | − , if k > | c | − r . .2. Generalized Dedekind Sums (cid:0) r | c | − (cid:1) | c |− X k =1 P (cid:0) dk | c | (cid:1) + | c |− r − X k =1 P (cid:0) dk | c | (cid:1) − (cid:0) r | c | − (cid:1) P (cid:0) d ( | c |− r ) | c | (cid:1) + P (cid:0) d ( | c |− r ) | c | (cid:1) P (cid:0) r | c | (cid:1) = | c |− r X k =1 P (cid:0) dk | c | (cid:1) + P (cid:0) d ( | c |− r ) | c | (cid:1)(cid:16) P (cid:0) r | c | (cid:1) − r | c | (cid:17) = | c |− r X k =1 P (cid:0) dk | c | (cid:1) + P (cid:0) dm | c | (cid:1) . Note that we have used (C.18) to see that P | c |− k =1 P (cid:0) dk | c | (cid:1) = 0. Combining this computation with(C.29), we arrive at s x,y ( a, c ) − s ( a, c ) = | c | P ( x ) − | c | δ ( x ) + | c |− r X k =1 P (cid:0) dk | c | (cid:1) + P (cid:0) dm | c | (cid:1) + δ ( x ) (cid:0) − δ ( mc ) (cid:1) + δ ( x ) (cid:16) P (cid:0) m | c | (cid:1) − P (cid:0) dm | c | (cid:1)(cid:17) , which is the claimed formula, since P ( x ) − δ ( x ) = x − x = P ( x ) − .26 Appendix C. Some Computations ppendix D Local Variation of the Eta Invariant We include some more analytical details concerning the heat operator. Specifically, we will givesome remarks concerning families of heat operators, and use this to give a proof of Proposition1.3.14. D.1 More Results on the Heat Operator In Chapter 1 we have defined the heat operator e − tH for a formally self-adjoint elliptic differentialoperator H of order 2 with positive definite leading symbol. We have done this by explicitlywriting it as the limit of integral operators with smooth kernels, which were defined in terms ofa spectral decomposition of H . In Lemma 1.2.3, we have derived a basic estimate for the heatkernel for large times. Concerning the small time behaviour of the heat operator, it is useful tohave a description of e − tH in terms of the resolvent ( H − z ) − for z / ∈ spec( H ). In particular,the proof of the asymptotic expansion in Theorem 1.2.7 as in [49] uses this description. D.1.1 Expression via the Resolvent Let E be a vector bundle over a closed manifold M of dimension m , and let H be a formally self-adjoint elliptic differential operator of order 2 with positive definite leading symbol. As before,for s, s ′ ∈ R , let B ( L s , L s ′ ) denote the space of bounded operators from L s to L s ′ endowed withthe operator norm k . k s,s ′ . Although this is not really necessary, we assume for simplicity that H is non-negative, i.e., spec( H ) ⊂ [0 , ∞ ), which is certainly true for an operator of the form H = D . Then the elliptic estimate (1.11) implies that for every s ≥ 0, there exists a constant C such that C − (cid:13)(cid:13) (Id + H ) s/ ϕ (cid:13)(cid:13) L ≤ k ϕ k L s ≤ C (cid:13)(cid:13) (Id + H ) s/ ϕ (cid:13)(cid:13) L , ϕ ∈ C ∞ ( M, E ) . (D.1)Consider the region Λ := (cid:8) z ∈ C (cid:12)(cid:12) Re( z ) + 1 ≤ | Im( z ) | (cid:9) . (D.2)Then there exists a constant C > z, spec( H )) ≥ C | z | for all z ∈ Λ . Appendix D. Local Variation of the Eta Invariant This, together with (D.1) and the spectral theorem, implies that for s ≥ z ∈ Λ, and ϕ ∈ C ∞ ( M, E ) k ( H − z ) − ϕ k L s ≤ C k (Id + H ) s/ ( H − z ) − ϕ k L ≤ C | z | − k (Id + H ) s/ ϕ k L ≤ C | z | − k ϕ k L s where we have used that (Id + H ) and ( H − z ) − commute. By duality, this also holds for s < s ∈ R , there exists a constant C such that k ( H − z ) − k s,s ≤ C | z | − , z ∈ Λ . (D.3)We now consider the contour Γ := ∂ Λ, oriented as the boundary, i.e., in such a way that [0 , ∞ )lies in the interior, see Figure D.1. Then the Cauchy formula implies that e − tH = 12 πi Z Γ e − tz ( H − z ) − dz. (D.4)Figure D.1: The contour ΓBecause of (D.3), this expression converges in the operator norm in L s ( M, E ) for every s ∈ R .To get norm estimates in B ( L s , L s + l ) for l > Proposition D.1.1. Let s ∈ R , k ∈ N , and ≤ l ≤ k . Then there exists a constant C suchthat for all z ∈ Λ k ( H − z ) − k k s,s + l ≤ C | z | l/ − k . (D.5) Proof. For z ∈ Λ, the spectrum of the operator ( H − z ) − is not contained in the negative realline, so that ( H − z ) − r can be defined for every r ≥ k ( H − z ) − r k s,s ≤ C | z | − r . .1. More Results on the Heat Operator k ( H − z ) − k k s,s + l ≤ C (cid:13)(cid:13) (Id + H ) l/ ( H − z ) − k (cid:13)(cid:13) s,s ≤ C (cid:13)(cid:13)(cid:0) (Id + H )( H − z ) − (cid:1) l/ (cid:13)(cid:13) s,s (cid:13)(cid:13) ( H − z ) l/ − k (cid:13)(cid:13) s,s ≤ C | z | l/ − k (cid:16) (cid:13)(cid:13) (Id + z )( H − z ) − (cid:13)(cid:13) ks,s (cid:17) ≤ C | z | l/ − k (cid:16) C | z +1 || z | (cid:17) k . The last factor can be bounded, independently of z ∈ Λ. This proves (D.5).An important consequence of (D.5) is that e − tH does indeed solve the heat equation. Wegive the following summary of well-known facts, see in particular [49, Lem. 1.7.5]. Proposition D.1.2. Let H be a non-negative operator in P s,e ( M, E ) and let s ∈ R . (i) The one-parameter family ( e − tH ) t ∈ (0 , ∞ ) is a smooth family of smoothing operators. If ϕ ∈ C ∞ ( M, E ) , then ( ddt + H ) e − tH ϕ = 0 , and lim t → k e − tH ϕ − ϕ k L s = 0 . (ii) The collection e − tH forms a semi-group, i.e., e − ( t + t ′ ) H = e − tH e − t ′ H , t, t ′ > . (iii) For l ≥ , there exists a constant C > such that for t ∈ (0 , k e − tH k s,s + l ≤ Ct − l/ . (D.6)(iv) Let c > be smaller than the smallest non-zero eigenvalue of H and let t > . Then thereexists a constant C > such that for all t ≥ t k e − tH − P k s,s ≤ Ce − ct , (D.7) where P is the projection onto ker H .Sketch of proof. The assertion (i) is [49, Lem. 1.7.5]. We skip the proof since it uses the sameideas as the proof of part (iii). Part (ii) is also standard: Let ε > 0, and consider the contourΓ ′ := Γ − ε . Then, using Cauchy’s formula and the resolvent equation, one finds that e − t ′ H e − tH = − π Z Γ ′ Z Γ e − t ′ z ′ e − tz ( H − z ) − ( H − z ′ ) − dzdz ′ = − π Z Γ ′ Z Γ e − tz − t ′ z ′ z − z ′ (cid:2) ( H − z ) − − ( H − z ′ ) − (cid:3) dzdz ′ . See Definition D.1.6 below. Appendix D. Local Variation of the Eta Invariant It now follows from standard “ C -valued” complex analysis, that first term is equal to e − ( t ′ + t ) H ,while the second term vanishes. To prove (D.6) choose k ∈ N with l ≤ k . We integrate (D.4)by parts to find that e − tH = 12 πi ( k − t k − Z Γ e − tz ( H − z ) − k dz. (D.8)Now let t ∈ (0 , 1) and substitute ζ = tz in (D.8). Then we can use Cauchy’s theorem to changeintegration over t Γ back to integration over Γ. This shows that for t ∈ (0 , e − tH = 12 πi ( k − t k Z Γ e − ζ ( H − ζ/t ) − k dζ. Now (D.5) applied to z = ζ/t ∈ Λ shows that k e − tH k s,s + l ≤ C t − k (cid:16) Z Γ | e − ζ | | ζ/t | l/ − k dζ (cid:17) ≤ C | t | − l/ . This proves (D.6). Concerning (iii) we only note that the large time estimate can be easily deducedfrom Lemma 1.2.3. Alternatively it can be proved using (D.8) by considering the contourΓ c := (cid:8) z ∈ C (cid:12)(cid:12) Re( z ) − c/ | Im( z ) | (cid:9) , and employing estimates corresponding to the ones in (D.5), see Figure D.2Figure D.2: The contour Γ c .1. More Results on the Heat Operator D.1.2 Perturbed Operators Let E be a vector bundle over a closed manifold M of dimension m , and let K be an integraloperator on L ( M, E ) with smooth kernel k ( x, y ) ∈ C ∞ ( M × M, E ⊠ E ∗ ). Then K is a smoothingoperator and thus, K ∈ B (cid:0) L s , L s ′ (cid:1) for all s, s ′ ∈ R . Moreover, the C k -norms of k ( x, y ) can be controlled by the operator norms of K . More precisely,for each k ∈ N , we can find a constant C as in [49, Lem. 1.2.7] such that k k ( x, y ) k C k ≤ C k K k − l,l , l > k + m/ . (D.9) Remark. Conversely, the Schwartz Kernel Theorem (see e.g. [94, Sec. 4.6]) ensures that anoperator K on L ( M, E ), which satisfies k K k − l,l < ∞ for l > k + m/ 2, has a kernel k ( x, y ) ofclass C k such that (D.9) holds.As before, assume that H ∈ P s,e ( M, E ) is non-negative. If K is a symmetric smoothingoperator on C ∞ ( M, E ), the operator H + K will in general not be an elliptic differential operator.Nevertheless, it follows from standard perturbation theory that H + K has all the properties, wehave obtained in Theorem 1.2.2 for the unperturbed case. In particular, we get a well-definedheat operator e − t ( H + K ) . We need to have a control on the difference e − t ( H + K ) − e − tH . Thefollowing result is basically [13, Prop. 9.46], only that we have changed it into a statement aboutoperators rather than kernels. Proposition D.1.3. Let K be a symmetric smoothing operator on C ∞ ( M, E ) , and let H ∈ P s,e ( M, E ) be non-negative. For k ≥ and t > define inductively K ( t ) := e − tH , K k ( t ) := Z t e − ( t − s ) H KK k − ( s ) ds. Then, if l ∈ N , there exists a constant C > such that for all k ≥ k K k ( t ) k − l,l ≤ C k t k k ! k K k k − l,l , Moreover, for each N ≥ there exists C > such that for all t ∈ (0 , (cid:13)(cid:13)(cid:13) e − t ( H + K ) − e − tH − N X k =1 ( − k K k ( t ) (cid:13)(cid:13)(cid:13) − l,l ≤ Ct N +1 . Remark. Before we sketch the proof we want to point out that for k ≥ K k ( t )can also be described as follows. Let∆ k := (cid:8) ( s , . . . , s k ) ∈ R k (cid:12)(cid:12) ≤ s ≤ . . . ≤ s k = 1 (cid:9) be the standard k -simplex. Then K k ( t ) = t k Z ∆ k e − t (1 − s k ) H Ke − t ( s k − s k − ) H . . . Ke − t ( s − s ) H Ke − ts H ds. (D.10)32 Appendix D. Local Variation of the Eta Invariant Sketch of proof. We use (D.10) to prove the estimate on K k ( t ). First it follows from (D.6) and(D.7) that k e − tH k s,s can be bounded, independently of t . Thus, k Ke − ts H k − l,l ≤ k K k − l,l k e − ts H k − l, − l ≤ C k K k − l,l , On the other hand, (cid:13)(cid:13) e − t (1 − s k ) H Ke − t ( s k − s k − ) . . . Ke − t ( s − s ) H (cid:13)(cid:13) l,l ≤ C k k K k k − l,l , Since k K k l,l ≤ C k K k − l,l we deduce, k K k ( t ) k − l,l ≤ t k vol(∆ k ) C k k K k k − l,l ≤ C k t k k ! k K k k − l,l , where we have used that vol(∆ k ) = k ! . Now the definition of K k ( t ) shows that ddt K k ( t ) = KK k − ( t ) − HK k ( t ) . Thus, for all N ≥ (cid:0) ddt + H + K (cid:1) N X k =0 ( − k K k ( t ) = ( − N KK N ( t ) . Using the estimate on k K k ( t ) k − l,l we see that (cid:13)(cid:13)(cid:13)(cid:0) ddt + H + K (cid:1) N X k =0 ( − k K k ( t ) (cid:13)(cid:13)(cid:13) − l,l ≤ C N t N Thus, P Nk =0 ( − k K k ( t ) is an approximate solution to the heat equation in terms of H + K inthe sense of [13, Sec. 2.4]. This implies that (cid:13)(cid:13)(cid:13) e − t ( H + K ) − e − tH − N X k =1 ( − k K k ( t ) (cid:13)(cid:13)(cid:13) − l,l ≤ Ct N +1 . We also need a result on the heat trace asymptotics when we perturb the operator H by asmoothing operator K . We first borrow the following from [13, Prop. 2.47]. Proposition D.1.4. Let K be a smoothing operator on L ( M, E ) . Then there exists an asymp-totic expansion of the form Tr( e − tH K ) ∼ Tr( K ) + ∞ X n =0 a n t n , as t → . Remark. The proof in [13] relies on the explicit description of the heat kernel by geometricallyconstructed approximations. This method to obtain the heat trace asymptotics is different fromthe one presented in [49], which is the one we are following in our presentation. However, a rough .1. More Results on the Heat Operator N ∈ N let K N ( t ) := N X n =0 ( − t ) n n ! H n K. Since K is a smoothing operator, K N ( t ) is a smoothing operator as well. Applying the heatequation to K N ( t ) yields (cid:0) ddt + H (cid:1) K N ( t ) = ( − t ) N N ! H N +1 K. Since we can bound k H N +1 K k − l,l for fixed N , this shows that K N ( t ) is an approximate solutionto the heat equation. For t → K which implies that (cid:13)(cid:13) e − tH K − K N ( t ) (cid:13)(cid:13) − l,l ≤ Ct N +1 . Using (D.9) and the expression of the trace in terms of kernels, we can then estimate (cid:12)(cid:12) Tr (cid:0) e − tH K − K N ( t ) (cid:1)(cid:12)(cid:12) ≤ Ct N +1 . From this one finds that the assertion holds with a n := ( − n n ! Tr( H n K ) . The following result is what we were aiming at in this section. We will give a version whichsuffices for our considerations, although a more general statement should be possible. Proposition D.1.5. Let H and K be as before, and assume in addition that K and H commute.Let D be an auxiliary formally self-adjoint differential operator of order d , and let K ′ be asymmetric smoothing operator on C ∞ ( M, E ) . Then there exists an asymptotic expansion Tr (cid:0) ( D + K ′ ) e − t ( H + K ) (cid:1) ∼ ∞ X n =0 t n − m − d a n , as t → . Moreover, if we denote by a n ( D, H ) the coefficients of the asymptotic expansion of Tr( De − tH ) ,we have a n = a n ( D, H ) , for n ≤ m. Sketch of proof. According to Theorem 1.2.7 it suffices to check thatTr (cid:0) ( D + K ′ ) e − t ( H + K ) (cid:1) − Tr (cid:0) De − tH (cid:1) has an asymptotic expansion as t → t . The assumption that K and H commute simplifies the situation considerably, since then e − t ( H + K ) = e − tH ∞ X n =0 ( − t ) n n ! K n , where the series converges in every L − l,l , because K is smoothing and thus bounded. This reducesthe claim to the study of terms of the formTr (cid:0) De − tH K n (cid:1) and Tr( K ′ e − tH K n ) , (D.11)where n ≥ 1. Now the trace property shows that both are of the form Tr (cid:0) e − tH ˜ K (cid:1) for somesmoothing operator ˜ K . We can thus apply Proposition D.1.4 to deduce that the terms in (D.11)are indeed asymptotic to power series in t .34 Appendix D. Local Variation of the Eta Invariant D.1.3 Variation of the Heat Operator Now let ( K u ) u ∈ U be a p -parameter family of smoothing operators. The constant C in the estimate(D.9) is independent of u . This implies that if K u depends smoothly on u with respect to k . k − l,l for each l ∈ N , then the family of kernels k u ( x, y ) will depend smoothly on u with respect to all C k -norms, where l > k + m/ 2. This motivates the following definition. Definition D.1.6. Let K u be a one-parameter family of operators with smooth kernels. Then K u is called a smooth family of smoothing operators , if for all l ∈ N , the assignment u K u issmooth in B (cid:0) L − l , L l (cid:1) .We can now formulate the following version of Duhamel’s formula, compare with [13, Thm.2.48]. Theorem D.1.7. Consider a one-parameter family H u in P s,e ( M, E ) of non-negative operators,and assume that H u is smooth in the sense of Definition 1.3.9. Then e − tH u is a smooth familyof smoothing operators, and ddu e − tH u = − Z t e − ( t − s ) H u ( ddu H u ) e − sH u ds. (D.12) Remark. Note that in contrast to the discussion in [13, Sec. 2.7] the smoothness in u doesnot follow from our description (1.14) of the heat kernel since in general the eigenvalues andeigenvectors of H u will not depend smoothly on u . However, we can use the description of e − tH u in terms of the resolvent to obtain the result. Proof. First note that the basic resolvent estimates (D.3) and (D.5) can be made uniform in u since the constants appearing there depend on u only through the elliptic estimate, which can bemade locally uniform in u .We now want to prove that ( H u − z ) − varies smoothly with u . Without loss of generalitywe consider an interval around 0. First of all, let Γ be the contour defined as the boundary of(D.2). Then for all z ∈ Γ, H u − z = ( H − z ) (cid:0) Id +( H − z ) − ( H u − H ) (cid:1) =: ( H − z )(Id + T u ) . (D.13)It follows from our smoothness assumption that T u is a bounded operator on each L l for everychoice of l ∈ Z , and that the assignment R → B ( L l , L l ) , u T u is smooth. Moreover, we can choose δ > u ∈ ( − δ, δ ) k T u k n,n ≤ for n ∈ {− l, . . . , l } . Using the Neumann series, this ensures that Id + T u is invertible in each B ( L n , L n ). Furthermore,the assignment u (Id + T u ) − is differentiable with ddu (Id + T u ) − = (Id + T u ) − (cid:0) ddu T u (cid:1) (Id + T u ) − . Inductively, one finds that (Id + T u ) − is smooth in u . Now (D.13) shows that for all z ∈ Γ( H u − z ) − = (Id + T u ) − ( H − z ) − . .1. More Results on the Heat Operator H u − z ) − depends smoothly on u as a map to B ( L n − , L n ), whereas before n ∈ {− l, . . . , l } . Hence, for k large enough, the assignment( − δ, δ ) → B ( L − l , L l ) , u ( H u − z ) − k is well-defined and smooth in u . Moreover, one easily finds that ddu ( H u − z ) − k = − k X n =1 ( H u − z ) − n (cid:0) ddu H u (cid:1) ( H u − z ) n − k − . (D.14)Since the basic resolvent estimate (D.5) can be made uniform in u , we deduce from (D.8) that e − tH u is differentiable and that we can differentiate under the integral using (D.14). Inductively,one finds that e − tH u is a smooth map to B ( L − l , L l ).Having established the smoothness of e − tH u in u , we can prove Duhamel’s formula (D.12)along the same lines as in [13, Sec. 2.7]: First, the heat equation implies that( ddt + H u ) ddu e − tH u = − ( ddu H u ) e − tH u . On the other hand, one finds that( ddt + H u ) Z t e − ( t − s ) H u ( ddu H u ) e − sH u ds = ( ddu H u ) e − tH u . Thus, ddu e − tH u + Z t e − ( t − s ) H u ( ddu H u ) e − sH u ds (D.15)solves the heat equation. We are thus left to show that (D.15) converges to 0 in L ( M, E ) as t → 0. Concerning the first term, we use ddu e − tH u = − πi Z Γ e − tz ( H u − z ) − (cid:0) ddu H u (cid:1) ( H u − z ) − dz and the resolvent estimate (D.3). Then, as in the proof of (D.6), one easily finds small timeestimates k ddu e − tH u ϕ k L ≤ Ct k ϕ k L , ϕ ∈ L ( M, E ) , which are uniform in u and t ∈ (0 , ddu e − tH u converges to 0 as t → B ( L , L ). For the second term in (D.15), we can argue as in the proof of Proposition D.1.3 toget uniform bounds on the integrand as t → 0. This implies our assertion.In a similar way one can show the following result. Proposition D.1.8. Let D u be a smooth family in P ds,e ( M, E ) and assume further that dim(ker D u ) is constant. (i) The projection P u onto ker( D u ) is a smooth family of smoothing operators. (ii) Denote by G u the family of Green’s operators of D u , defined by G u | ker( P u ) = (cid:0) D u | ker( P u ) (cid:1) − , G u | im( P u ) = 0 . Then, for every s ∈ R , the family G u is smooth in u as a map to B ( L s , L s + d ) . Appendix D. Local Variation of the Eta Invariant Sketch of proof. Since D u varies smoothly with u , one can verify—using the smooth dependenceof the resolvent on u —that for compact u -intervals, the non-zero eigenvalues are uniformly bondedaway from 0. Then, according to the spectral theorem, the projection P u onto ker D u is given by P u = 12 πi Z Γ ( D u − z ) − dz, (D.16)where Γ is the clockwise oriented boundary of a small disk B such that B ∩ spec( D u ) = { } , for all u .Integrating by parts, we get P u = 12 πi Z Γ ( − z ) k − ( D u − z ) − k dz. Resolvent estimates as in (D.3) and (D.5) and the formula (D.14) for the derivative of the resolventwith respect to u then yield that P u is a smooth family of smoothing operators, see [13, Prop.9.10] for a related discussion.Concerning the family of Green’s operators, we can take the same small disk B and orientthe boundary Γ counter-clockwise. Then G u = 12 πi Z Γ z − ( D u − z ) − dz. (D.17)Now the uniform bound k ( D u − z ) − k s,s + d ≤ C and the formula for the derivative of the resolventshow that G u is a smooth family of operators in B ( L s , L s + d ). Remark D.1.9. (i) Note that in (D.17) we cannot integrate by parts to increase the regularity of G u . This is,of course, already clear in the case of a single operator.(ii) If there exists a constant c such that c / ∈ spec( | D u | ) for all u in some interval [ − δ, δ ], wecan use (D.16)—with Γ being a circle of radius c around 0—to deduce that the spectralprojection onto all eigenvalues of norm less than c is smooth u , see also [13, Prop. 9.10].(iii) Theorem D.1.7 and Proposition D.1.8 carry over verbatim, if we consider a smooth p -parameter families of formally self-adjoint elliptic operators. This is important for thediscussion of fiber bundles.We also need the following consequence of Theorem D.1.7 Lemma D.1.10. Let H u be a smooth one-parameter family of non-negative operators in P s,e ( M, E ) , and let D u be an auxiliary smooth one-parameter family of formally self-adjointdifferential operators of order d . Then D u e − tH u is a differentiable family of trace-class operatorsand, if D u commutes with H u , ddu Tr (cid:0) D u e − tH u (cid:1) = Tr (cid:0) dD u du e − tH u (cid:1) − t Tr (cid:0) D u dH u du e − tH u (cid:1) . (D.18) .2. Parameter Dependent Eta Invariants Proof. Let K u := D u e − tH u / and L u := e − tH u / . Then Theorem D.1.7 implies that L u is asmooth family of smoothing operators, and the same is true for K u , since by Lemma 1.3.10 theoperators D u ∈ B ( L l + d , L l ) depend smoothly on u for every l ∈ Z . We now observe the following(i) K u and L u are smooth families of operators in B ( L , L ), and ddu ( K u L u ) = dD u du e − tH u + D u ddu e − tH u , where we are using the pairing B ( L d , L ) × B ( L , L d ) → B ( L , L ) , ( S, T ) ST, to differentiate D u e − tH u / .(ii) K u and L u are trace-class operators. Expressing the trace in terms of the kernels it followsfrom (D.9) that they depend continuously on u with respect to the trace norm.Using the H¨older inequality | Tr( ST ) | ≤ Tr | S | k T k , , these observations imply that Tr (cid:0) D u e − tH u (cid:1) is differentiable. If D u and H u commute, we can use (D.12) to compute that ddu Tr (cid:0) D u e − tH u (cid:1) = Tr (cid:0) dD u du e − tH u (cid:1) + Tr (cid:0) D u ddu e − tH u (cid:1) = Tr (cid:0) dD u du e − tH u (cid:1) − Z t Tr (cid:0) D u e − ( t − s ) H u ( ddu H u ) e − sH u (cid:1) ds = Tr (cid:0) dD u du e − tH u (cid:1) − Z t Tr (cid:0) D u ( ddu H u ) e − tH u (cid:1) ds. = Tr (cid:0) dD u du e − tH u (cid:1) − t Tr (cid:0) D u dH u du e − tH u (cid:1) . D.2 Parameter Dependent Eta Invariants As a consequence of the above discussion, we can now describe a proof of the variation formulafor the Eta invariant. D.2.1 Large Time Behaviour To study parameter dependent Eta invariants we let H u = D u , where D u is a smooth family in P s,e ( M, E ). We can then state the following parameter dependent version of Proposition 1.2.4,see also [31, Lem. A.14]. Lemma D.2.1. Under the assumptions of Theorem D.1.7, the one-parameter family D u e − tD u is a differentiable family of trace-class operators, and ddu Tr( D u e − tD u ) = (cid:0) t ddt (cid:1) Tr (cid:0) dD u du e − tD u (cid:1) . (D.19) Moreover, if dim(ker D u ) is constant, then for t > , there exist constants c and C such that for t ≥ t (cid:12)(cid:12) Tr( D u e − tD u ) (cid:12)(cid:12) ≤ Ce − ct , (D.20) locally uniform in u . Appendix D. Local Variation of the Eta Invariant Proof. By (D.18) we have ddu Tr( D u e − tD u ) = Tr (cid:0) dD u du e − tD u (cid:1) − t Tr (cid:0) D u dD u du e − tD u (cid:1) = Tr (cid:0) dD u du e − tD u (cid:1) − t Tr (cid:0) dD u du D u e − tD u (cid:1) = Tr (cid:0) dD u du e − tD u (cid:1) + 2 t ddt Tr (cid:0) dD u du e − tD u (cid:1) . Proposition 1.2.4 shows that for t > 0, we can find constants C ( u ) such that for t ≥ t (cid:12)(cid:12) Tr( D u e − tD u ) (cid:12)(cid:12) ≤ C ( u ) e − tλ ( u ) / , where λ ( u ) is the smallest non-zero eigenvalue of D u . The proof of Proposition 1.2.4 shows that C ( u ) depends continuously on u so that we can find C with C ( u ) ≤ C for compact u intervals.Moreover, the eigenvalues of D u vary continuously. Hence if dim(ker D u ) is constant, the non-zeroeigenvalues of D u have a uniform positive lower bound on compact u intervals. This proves thesecond assertion.The above lemma has the following consequence Corollary D.2.2. If dim(ker D u ) is constant, then ddu Z ∞ Tr (cid:0) D u e − tD u (cid:1) t s − dt = − (cid:0) dD u du e − D u (cid:1) − s Z ∞ Tr (cid:0) dD u du e − tD u (cid:1) t s − dt, and both sides are holomorphic for all s ∈ C .Proof. Let P u be the orthogonal projection onto ker D u . Since dim(ker D u ) is constant we knowfrom Proposition D.1.8 that P u depends smoothly on u . Moreover, ddu D u = ddu (cid:0) (Id − P u ) D u (Id − P u ) (cid:1) = − dP udu D u (Id − P u ) + (Id − P u ) dD u du (Id − P u ) − (Id − P u ) D u dP udu From this one deduces that Tr (cid:0) dD u du e − tD u (cid:1) satisfies an estimate of the form (D.20). Thus, forfixed T > ddu Z T t s − Tr (cid:0) D u e − tD u (cid:1) dt = Z T t s − (cid:0) t ddt (cid:1) Tr (cid:0) dD u du e − tD u (cid:1) dt = 2 h T s +12 Tr (cid:0) dD u du e − T D u (cid:1) − Tr (cid:0) dD u du e − D u (cid:1)i − s Z T Tr (cid:0) dD u du e − tD u (cid:1) t s − dt. The uniform estimates on Tr (cid:0) D u e − tD u (cid:1) and Tr (cid:0) dD u du e − tD u (cid:1) show that both sides are holomorphicfor s ∈ C and allow us to take T → ∞ . This proves the result. .2. Parameter Dependent Eta Invariants D.2.2 Small Times and Meromorphic Extension To extend Corollary D.2.2 to small times t ∈ [0 , Theorem D.2.3. Let A u be an auxiliary smooth family of formally self-adjoint operators of order a . There is an asymptotic expansion, locally uniform in u , such that Tr( A u e − tD u ) ∼ ∞ X n =0 t n − m − a a n ( A u , D u ) , as t → . (D.21) The a n ( A u , D u ) are integrals over quantities locally computable from the total symbols of A u and D u . Moreover, the functions a n ( A u , D u ) are smooth in u and (D.21) can be differentiated termby term, i.e., ddu Tr( A u e − tD u ) ∼ ∞ X n =0 t n − m − a ddu a n ( A u , D u ) , as t → . Proposition D.2.4. If dim(ker D u ) is constant, the meromorphic extension of η ( D u , s ) is con-tinuously differentiable in u , and ddu η ( D u ) = − √ π a m ( dD u du , D u ) , where a m ( dD u du , D u ) is the constant term in the asymptotic expansion of √ t Tr (cid:0) dD u du e − tD u (cid:1) , as t → . Proof. According to Theorem D.2.3 we have asymptotic expansions, which can be differentiatedin u , Tr (cid:0) D u e − tD u (cid:1) ∼ ∞ X n =0 t n − m − a n ( u ) , as t → , and Tr (cid:0) dD u du e − tD u (cid:1) ∼ ∞ X n =0 t n − m − a ′ n ( u ) , as t → . As remarked in Theorem 1.2.7 they can be differentiated in t as well so that (D.19) implies ddu a n ( u ) = ( n − m ) a ′ n ( u ) . (D.22)Let N ∈ N be fixed, and let r N ( t, u ) := Tr (cid:0) D u e − tD u (cid:1) − N X n =0 t n − m − a n ( u ) , and r ′ N ( t, u ) := Tr (cid:0) dD u du e − tD u (cid:1) − N X n =0 t n − m − a ′ n ( u ) . Appendix D. Local Variation of the Eta Invariant Then r N and r ′ N satisfy estimates, locally uniform in u , | r N ( t, u ) | ≤ Ct N , | r ′ N ( t, u ) | ≤ Ct N , as t → . (D.23)Moreover, (D.19) and (D.22) imply that ddu r N ( t, u ) = (1 + 2 t ddt ) r ′ N ( t, u ) . (D.24)Now for all s ∈ C with Re( s ) > m − ( N + 1) and s / ∈ { m − n | n ∈ N } Z t s − Tr (cid:0) D u e − tD u (cid:1) dt = N X n =0 a n ( u ) s + n − m + Z t s − r N ( t, u ) dt. Using (D.23) we can differentiate under the integral to deduce from (D.22) and (D.24) that ddu Z t s − Tr (cid:0) D u e − tD u (cid:1) dt = N X n =0 n − m ) a ′ n ( u ) s + n − m + Z t s − (1 + 2 t ddt ) r ′ N ( t, u ) dt = N X n =0 n − m ) a ′ n ( u ) s + n − m + 2 r ′ N (1 , u ) − s Z t s − r ′ N ( t, u ) dt = 2 Tr (cid:0) dD u du e − D u (cid:1) + N X n =0 − s a ′ n ( u ) s + n − m − s Z t s − r ′ N ( t, u ) dt, since 2( n − m ) a ′ n ( u ) s + n − m − a ′ n ( u ) = − s a ′ n ( u ) s + n − m . On the other hand, Z t s − Tr (cid:0) dD u du e − D u (cid:1) dt = N X n =0 a ′ n ( u ) s + n − m + Z t s − r ′ N ( t, u ) dt. This shows that the meromorphic extension to C of R t s − Tr (cid:0) D u e − tD u (cid:1) dt is continuously dif-ferentiable in u away from the poles, with derivative given by the meromorphic extension of2 Tr (cid:0) dD u du e − D u (cid:1) − s Z t s − Tr (cid:0) dD u du e − D u (cid:1) dt. Since dim(ker D u ) is constant we can apply Corollary D.2.2 to deduce that the meromorphicextension of the Eta function η ( D u , s ) is continuously differentiable in u . Moreover, for N > m and a suitable function h N ( u, s ), holomorphic for Re( s ) > m − ( N + 1), ddu η ( D u , s ) = − s Γ( s +12 ) (cid:16) N X n =0 a ′ n ( u ) s + n − m + h N ( u, s ) (cid:17) . In particular, ddu η ( D u ) = − ) − a ′ m ( u ) = − √ π a m ( dD u du , D u ) . .2. Parameter Dependent Eta Invariants D.2.3 The Case of Varying Kernel Dimension If we want to drop the assumption on dim(ker D u ), we have to study the reduced ξ -invariant[ ξ ( D u )] ∈ R / Z as in Definition 1.3.4. Proposition D.2.5. Let D u be a smooth family in P s,e ( M, E ) . Then the reduced ξ -function [ ξ ( D u )] ∈ R / Z is continuously differentiable in u , and ddu [ ξ ( D u )] = − √ π a m ( dD u du , D u ) , where a m ( dD u du , D u ) is the constant term in the asymptotic expansion of √ t Tr (cid:0) dD u du e − tD u (cid:1) , as t → . The result is the same as [31, Prop. A.17]. We include a proof for completeness and sketch afew more details. Proof. Choose δ small enough so that there exists c ∈ (0 , 1) with c / ∈ spec( | D u | ) for all u ∈ ( − δ, δ ).Denote by λ iu with i = 1 , . . . , i the finite number of eigenvalues of D u with | λ iu | < c , and let E u ( c ) := i M i =1 ker (cid:0) D u − λ iu (cid:1) . Let P u ( c ) be the projection onto E u ( c ). According to Remark D.1.9 P u ( c ) is a smooth familyof finite rank operators with smooth kernel. Thus D u ( c ) := D u (cid:0) Id − P u ( c ) (cid:1) + P u ( c )is a smooth perturbation of D u by finite rank operators with smooth kernel. Note that for fixed u we replace with 1 the finite number of eigenvalues of D u which are of norm smaller than c . Thus,the large eigenvalues of D u ( c ) are the same as those of D u . This implies that the Eta function iswell-defined for Re( s ) > m and satisfies η (cid:0) D u ( c ) , s (cid:1) = η ( D u , s ) + dim ker( D u ) + X λ iu =0 h − sgn( λ iu ) | λ iu | s i , Re( s ) > m. (D.25)Since the right hand side admits a meromorphic continuation to C , the same holds for η (cid:0) D u ( c ) , s (cid:1) .Moreover, s = 0 is no pole, and the reduced ξ -invariant satisfies (cid:2) ξ (cid:0) D u ( c ) (cid:1)(cid:3) = (cid:2) ξ ( D u ) (cid:3) . (D.26)We now need to understand the variation of ξ (cid:0) D u ( c ) (cid:1) . Since D u ( c ) is invertible we will study η (cid:0) D u ( c ) (cid:1) instead. Note that we cannot directly apply Proposition D.2.4 since D u ( c ) is in generalnot a family of differential operators. However, we have already done the major work and indicatethe changes to be made: Kato’s selection theorem ([56, Sec. II.6]), ensures that the eigenvalues λ iu can be ordered in such a way that theyare parametrized by continuously differentiable functions. Nevertheless, the total projection onto all eigenspacesspanned by the collection λ iu depends smoothly on u . Appendix D. Local Variation of the Eta Invariant (i) Since D u ( c ) is a smooth perturbation of D u by symmetric smoothing operators, the proofof Theorem D.1.7 goes through verbatim, showing that e − tD u ( c ) is a smooth family ofsmoothing operators which satisfies (D.12).(ii) Then, as in Lemma D.2.1, one finds that the one-parameter family D u ( c ) e − tD u ( c ) is adifferentiable family of trace-class operators satisfying (D.19). Clearly, the uniform large-time estimate (D.20) and Corollary D.2.2 also continue to hold.(iii) Since D u ( c ) is of the form D u + K u , where K u is a smooth family of smoothing operatorswhich commutes with D u , we can apply Proposition D.1.5 for fixed u to get asymptoticexpansions Tr (cid:0) D u ( c ) e − tD u ( c ) (cid:1) ∼ ∞ X n =0 t n − m − a n ( u ) , as t → , (D.27)and Tr (cid:16)(cid:0) ddu D u ( c ) (cid:1) e − tD u ( c ) (cid:17) ∼ ∞ X n =0 t n − m − a ′ n ( u ) , as t → , (D.28)where a ′ m ( u ) is equal to the constant term in the asymptotic expansion as t → √ t Tr (cid:0) dD u du e − tD u (cid:1) . Moreover, one deduces from Theorem D.2.3 and the proof of Propo-sition D.1.5 that (D.27) and (D.28) are locally uniform in u with coefficients dependingsmoothly on u .Now the proof of Proposition D.2.4 carries over to the situation at hand yielding that η (cid:0) D u ( c ) (cid:1) is continuously differentiable in u with ddu η (cid:0) D u ( c ) (cid:1) = − √ π a m ( dD u du , D u ) . Since (cid:2) ξ ( D u ) (cid:3) = η (cid:0) D u ( c ) (cid:1) , the proposition follows.From the proof of Proposition D.2.5 we can also easily deduce the variation formula for the ξ -invariant, as stated in Proposition 1.3.14. Corollary D.2.6. Let D u with u ∈ [ a, b ] be a smooth one-parameter family of operators in P s,e ( M, E ) . Then ξ ( D b ) − ξ ( D a ) = SF( D u ) u ∈ [ a,b ] + Z ba ddu [ ξ ( D u )] du. Proof. Without loss of generality we may assume that [ a, b ] = [ − δ, δ ] for δ as in the proof ofProposition D.2.5. Moreover, let D u ( c ) be the family of perturbed operators defined there. 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