A local families index formula for d-bar operators on punctured Riemann surfaces
aa r X i v : . [ m a t h . DG ] S e p A LOCAL FAMILIES INDEX FORMULA FOR ∂ -OPERATORS ON PUNCTURED RIEMANN SURFACES PIERRE ALBIN AND FR´ED´ERIC ROCHON
Abstract.
Using heat kernel methods developed by Vaillant, a localindex formula is obtained for families of ∂ -operators on the Teichm¨ulleruniversal curve of Riemann surfaces of genus g with n punctures. Theformula also holds on the moduli space M g,n in the sense of orbifoldswhere it can be written in terms of Mumford-Morita-Miller classes. Thedegree two part of the formula gives the curvature of the correspondingdeterminant line bundle equipped with the Quillen connection, a resultoriginally obtained by Takhtajan and Zograf. Contents
Introduction 11. Hyperbolic cusp operators 52. The boundary compactification of a Riemann surface withpuncture 93. The ∂ -operator as a Dirac-type hc-operator 114. The Teichm¨uller space and the Teichm¨uller universal curve 165. The canonical connection on the universal Teichm¨uller curve 176. A local formula for the family index 217. The spectral hc-zeta determinant 278. The curvature of the Quillen connection 39References 46 Introduction
Let X be a smooth even dimensional oriented compact manifold withboundary ∂X = ∅ . Assume that the boundary is the total space of a fibration(1) Z ∂X φ (cid:15) (cid:15) Y The first author was partially supported by a NSF postdoctoral fellowship. The secondauthor was supported by a postdoctoral fellowship of the Fonds qu´eb´ecois de la recherchesur la nature et les technologies. where Y and Z are compact oriented manifolds, Y being the base and Z being a typical fibre. Let x ∈ C ∞ ( X ) be a boundary defining function for X and(2) c : ∂X × [0 , ǫ ) x → N ⊂ X a corresponding collar neighborhood of ∂X in X . Let g hc be a metric on X \ ∂X which takes the form(3) c ∗ g hc = dxx + φ ∗ g Y + x g Z in the collar neighborhood (2), where g Z is a metric for the vertical tangentbundle of (1) and g Y is a metric on the base Y which is lifted to ∂X using achoice of connection for the fibration (1). Such a metric is called a productfibred hyperbolic cusp metric (product d -metric in the terminology of [31]).If the manifolds X , Y and Z are spin, one can construct a Dirac operatorassociated to the metric g hc . More generally, one can consider a Dirac typeoperator D constructed using the metric g hc and a Clifford module E → X with a choice of Clifford connection.In his thesis [31], Vaillant studied the index and the spectral theory ofsuch operators. To do so, he introduced the conformally related operator xD and defined the vertical family by D V := xD | ∂X , which is a family ofoperators on ∂X parametrized by the base Y and acting on each fibre of(1). Assuming that the rank of ker D V → Y is constant so that it is avector bundle over Y (constant rank assumption), Vaillant also introduceda horizontal operator(4) D H : C ∞ ( Y ; ker D V ) → C ∞ ( Y ; ker D V )which governs the continuous spectrum of D with bands of continuous spec-trum starting at the eigenvalues of D H and going out at infinity. In par-ticular, the operator D is Fredholm if and only if D H is invertible. In thatcase, Vaillant was able to obtain a formula for its index using heat kerneltechniques and Getzler’s rescaling along the lines of [22],(5) ind( D ) = Z X b A ( R hc ) Ch( F E/S hc ) − Z Y b A ( R g Y ) b η ( D V ) − η ( D H ) , the first term being the usual Atiyah-Singer integral, b η ( D V ) being the Bismut-Cheeger eta form of the vertical family and η ( D H ) being the eta invariantof D H .In [2], the authors, inspired by the work of Melrose and Piazza in [24]and [25], generalized the formula of Vaillant to families of Dirac type opera-tors. Via the use of Fredholm perturbations, a notion intimately related tospectral sections, it was also possible to study situations where the constantrank assumption is not satisfied, allowing among other things to generalizethe index theorem of Leichtnam, Mazzeo and Piazza [20].The present paper, which is a sequel to [2], intends to put into use theindex formula of [2] to study the following fundamental example arising in OCAL FAMILIES INDEX FOR ∂ -OPERATORS 3 Teichm¨uller theory. Assume that 2 g + n ≥ T g,n be the Teichm¨ullerspace of Riemann surfaces of genus g with n punctures. Let p : T g,n → T g,n be the Teichm¨uller universal curve whose fibre above [Σ] ∈ T g,n isthe corresponding Riemann surface Σ of genus g with n punctures. Let T i,jv T g,n → T g,n be the ( i, j ) vertical tangent bundle and let Λ i,jv be its dual.In particular, K v := Λ , v restricts on each fibre Σ to the correspondingcanonical line bundle K Σ := Λ , .For each ℓ ∈ Z , one can associate a family of ∂ -operators(6) ∂ ℓ : C ∞ ( T g,n ; K ℓv ) → C ∞ ( T g,n ; Λ , v ⊗ K ℓv )acting fibre by fibre on p : T g,n → T g,n and parametrized by the base T g,n . Bythe uniformization theorem for Riemann surfaces, each fibre Σ of p : T g,n → T g,n comes equipped with a hyperbolic metric g Σ . Compactifying each fibreby a compact Riemann surface with boundary, these metrics can be seen asproduct hyperbolic cusp metrics, the fibration structure on the boundarybeing the collapsing map onto a point. With these metrics, the family ∂ ℓ can be interpreted as a family of Dirac-type operators associated to a familyof product hyperbolic cusp metrics. Using the criterion of Vaillant [31], onecan check that each member of the family is Fredholm. The formula of [2]therefore applies.As described in [34], the fibration p : T g,n → T g,n is equipped with acanonical connection. This allows one to interpret the formula of [2] at thelevel of forms. In general, the eta forms involved in this formula are quitehard to compute. However, in this specific example, an explicit computationis possible using a result of Zhang [36], the vertical family being defined ona circle fibration. The main result of this paper , theorem 1, gives thefollowing local family index formula,(7) Ch(Ind( ∂ ℓ )) = Z T g,n /T g,n Ch( T − ℓv ( T g,n )) Td( T v T g,n ) + n − ℓ ) − n X i =1
12 tanh (cid:0) e i (cid:1) − e i ! − (cid:18) π √− (cid:19) N d Z ∞ Str ∂ A tD ℓ ∂t e − ( A tDℓ ) ! dt, where A tD ℓ is the Bismut superconnection and N is the number operatorin Λ T g,n . To define the form e i , let L i → T g,n be the complex line bundlewhich at [Σ] ∈ T g,n is given by the restriction of K v at the i th puncture(marked point) of Σ := p − ([Σ]). Then e i is the Chern form of L i as definedby Wolpert [35].Since the Teichm¨uller space T g,n is contractible, formula (7) only con-tains cohomological information in its degree zero part. However, sinceit is local and each of its terms is invariant under the action of the Te-ichm¨uller modular group Mod g,n , formula (7) also holds on the modulispace M g,n = T g,n / Mod g,n in the sense of orbifolds, where the fibration p : T g,n → T g,n is replaced by the forgetful map π n +1 : M g,n +1 → M g,n and where it acquires a topological meaning in higher degrees (see Corollary PIERRE ALBIN AND FR´ED´ERIC ROCHON M g,n , the Chern form e i representsthe Miller class ψ i = c ( L i ), while the first term on the left-hand side of (7)represents a linear combination of the Mumford-Morita classes(8) κ j = ( π n +1 ) ∗ ( c ( ψ j +1 n +1 )) , j ∈ N . This formula could be thought of as a local version of the Grothendieck-Riemann-Roch theorem applied to the forgetful map π n +1 : M g,n +1 → M g,n and a certain sheaf on M g,n +1 depending on ℓ (when ℓ = 0, it is the sheafof sections of the trivial line bundle). When ℓ = 0 or ℓ = 1, our formulaagrees modulo boundary terms with the one obtained by Bini [6] using theGrothendieck-Riemann-Roch theorem.Our results should be compared with the result of Takhtajan and Zograf[30] and Wolpert [35], who gave the two form part of (7) by interpreting it asthe first Chern form of the corresponding determinant line bundle equippedwith the Quillen connection. As in the compact case, the definition of theQuillen connection makes use of the determinant of the Laplacian. Howeverthe presence of cusps induces continuous spectrum for the Laplacian andthe usual definition of its determinant via zeta-regularization is necessarilymore delicate. Takhtajan and Zograf sidestepped this issue by defining thedeterminant in terms of the Selberg zeta function, in analogy with the com-pact case [13, 29]. The precise description of the heat kernel in [31] allowsus to proceed along the lines of [22, 28, 12] and extend the zeta functiondefinition to this context via renormalization. Unlike previous efforts (see,e.g., [14], [15] and [26]) this definition does not make use of the hyperbolicstructure of the underlying manifold and works more generally for the met-rics considered in [31, 2]. Furthermore we show that, for hyperbolic metricson surfaces with cusps, the resulting zeta-regularized determinant coincideswith that defined using the Selberg zeta function up to a universal constant(see theorem 2 and corollary 7.5)(9) det ′ (∆ ℓ ) = (cid:26) α ℓ,g,n Z Σ ( ℓ ) , ℓ ≥ α ℓ,g,n Z ′ Σ (1) , ℓ = 0 , ℓ ≥
0, where α ℓ,g,n is a constant only depending on ℓ , g and n . Withthis determinant and thanks to the fact ker ∂ ℓ is a holomorphic vector bundleon T g,n , the construction of the Quillen connection and the computation ofthe curvature are essentially as in [5], [8] with only minor changes. In thisway, we recover the index formula of [30] (see also [32] and [35]), √− π ( ∇ Q ℓ ) = 12 πi Z T g,n /T g,n Ch (cid:16) T − ℓ ( T g,n /T g,n ) (cid:17) · Td ( T ( T g,n /T g,n )) ! [2] − n X i =1 e i , see theorem 3 and corollary 8.5 below. OCAL FAMILIES INDEX FOR ∂ -OPERATORS 5 Our approach and the one of Takhtajan and Zograf [30] use substantiallythe fact that the dimension of the kernel of the family ∂ ℓ does not jump,so that these kernels fit together into a vector bundle on the Teichm¨ullerspace. More generally, one can ask if the work of Bismut, Gillet and Soul´e[9, 10, 11] for the determinant of ∂ -operators arising on (compact) K¨ahlerfibrations could be adapted to non-compact situations in order to deal withexamples where the rank of the kernel jumps.The paper is organized as follows. In §
1, we review the definition andmain properties of hyperbolic cusp operators. In §
2, we explain the pas-sage from a punctured Riemann surface to a compact Riemann surface withboundary. In §
3, we describe how the ∂ -operator on a punctured Riemannsurface can be seen as a Dirac type hyperbolic cusp operator. We also checkthat Vaillant’s formula (5) agrees with the Riemann-Roch theorem in thiscase. In § §
5, we make a quick review of Teichm¨uller theory fromour perspective. We then obtain our main result in § §
7, we studythe determinant of various Laplacians on Riemann surfaces of finite areaand relate them to Selberg’s zeta function following [12]. Finally, in §
8, weadapt the standard computation of the curvature of the Quillen connectionto our context and compare our result with those of Takhtajan-Zograf [30],Weng [32] and Wolpert [35].
Acknowledgement.
We would like to thank Leon Takhtajan and Peter Zo-graf for explaining to us their results. We are also grateful to Rafe Mazzeo,Richard Melrose, Gabriele Mondello and Sergiu Moroianu for helpful con-versations. Hyperbolic cusp operators
Let X be a smooth compact manifold with boundary ∂X = ∅ . Let x ∈C ∞ ( X ) be a boundary defining function, that is, x is a positive functionin the interior vanishing on the boundary such that its differential dx isnowhere zero on ∂X . For ǫ > ∂X in X ,(1.1) c : ∂X × [0 , ǫ ) x → N ǫ := { p ∈ X | x ( p ) < ǫ } ⊂ X. Consider a Riemannian metric g hc in the interior X \ ∂X taking the form(1.2) c ∗ g hc = dx x + x π ∗ L g ∂X in the collar neighborhood (1.1), where g ∂X is a Riemannian metric on ∂X and π L : ∂X × [0 , ǫ ) x → ∂X is the projection on the left factor. Such ametric is called a product hyperbolic cusp metric (or product d-metricin the terminology of Vaillant [31]). This is a complete metric on the interiorof X , hence the boundary ∂X is at infinity. Notice however that the volumeof X is finite with respect to the metric g hc . Following the philosophy of PIERRE ALBIN AND FR´ED´ERIC ROCHON
Melrose, one can get operators that are adapted to this geometry at infinityby considering the space of hyperbolic cusp vector fields V hc ( X ), thatis, the space of smooth vector fields on X with length uniformly boundedwith respect to the metric g hc ,(1.3) V hc ( X ) := { ξ ∈ C ∞ ( X ; T X ) | ∃ c > g hc ( ξ ( p ) , ξ ( p )) < c ∀ p ∈ X \ ∂X } . If z = ( z , . . . , z n − ) are local coordinates on ∂X , then in the collar neigh-borhood (1.1), a hyperbolic cusp vector field ξ takes the form(1.4) ξ = ax ∂∂x + n − X i =1 b i x ∂∂z i where a, b , . . . , b n − are smooth functions on X . It is possible to define avector bundle hc T X on X in such a way that its space of smooth sections iscanonically identified with hyperbolic cusp vector fields,(1.5) C ∞ ( X ; hc T X ) = V hc ( X ) . In the interior X \ ∂X , the vector bundle hc T X is isomorphic to the tangentbundle
T X . This identification does not extend to an isomorphism on theboundary of X . The metric g hc naturally induces a metric on hc T X whichis also well-defined on the boundary.A quick check indicates that V hc ( X ) is not closed under the Lie bracket.To define higher order hyperbolic cusp operators, it is convenient to considerthe conformally related metric(1.6) g cu := 1 x g hc . The metric g cu is called a product cusp metric . One can consider thecorresponding cusp vector fields (1.7) V cu ( X ) := x V hc ( X ) = { ξ ∈ C ∞ ( X ; T X ) | ∃ c > g cu ( ξ ( p ) , ξ ( p )) < c ∀ p ∈ X \ ∂X } . Alternatively, one can define cusp vector fields by(1.8) V cu ( X ) := { ξ ∈ C ∞ ( X ; T X ) | ξx ∈ x C ∞ ( X ) } , which makes it clear that the definition only depends on the choice of bound-ary defining function x and not on the choice of metric g cu . There is alsoan associated vector bundle cu T X over X whose space of smooth sections iscanonically identified with the space of cusp vector fields,(1.9) C ∞ ( X ; cu T X ) = V cu ( X ) . In the collar neighborhood (1.1), a cusp vector field ξ has to be of the form(1.10) ξ = ax ∂∂x + n − X i =1 b i ∂∂z i OCAL FAMILIES INDEX FOR ∂ -OPERATORS 7 with a, b , . . . , b n − ∈ C ∞ ( X ). As opposed to V hc ( X ), the space V cu ( X )is closed under the Lie bracket, so that it is naturally a Lie algebra. Itscorresponding universal enveloping algebra is the space Diff ∗ cu ( X ) of cuspdifferential operators . In the collar neighborhood (1.1), a cusp differentialoperator of order k , P ∈ Diff k cu ( X ), takes the form(1.11) P = X l + | α |≤ k p l,α (cid:18) x ∂∂x (cid:19) l (cid:18) ∂∂z (cid:19) α , p l,α ∈ C ∞ ( X ) . More generally, Mazzeo and Melrose in [21] defined the space of cusp pseu-dodifferential operators of order k , Ψ k cu ( X ). These operators are closedunder composition,(1.12) Ψ k cu ( X ) ◦ Ψ l cu ( X ) ⊂ Ψ k + l cu ( X ) . There is a corresponding cusp Sobolev space of order m ∈ N ,(1.13) H m cu ( X ) := { f ∈ L g cu ( X ) | P f ∈ L g cu ( X ) ∀ P ∈ Ψ m cu ( X ) } . One can also consider its weighted version x k H m cu ( X ) by some power x k of theboundary defining function. A cusp pseudodifferential operator P ∈ Ψ m cu ( X )then defines a bounded linear map(1.14) P : x k H l cu ( X ) → x k H l − m cu ( X ) . One interesting feature of the cusp operators is that if a cusp pseudodif-ferential P ∈ Ψ m ( X ) is invertible as a bounded linear map (1.14), then itsinverse is given by a cusp operator of order − m .Generalizing the relation V hc ( X ) = x V cu ( X ), one can define the space of hyperbolic cusp pseudodifferential operators of order m by(1.15) Ψ m hc ( X ) := x − m Ψ m cu ( X ) . A hyperbolic cusp operator P ∈ Ψ m hc ( X ) naturally induces a bounded linearmap(1.16) P : x k H l cu ( X ) → x k − m H l − m cu ( X ) . So far we have considered operators acting on functions on X , but if E → X and F → X are complex vector bundles on X , it is no more difficultto define the space of hyperbolic cusp operators Ψ ∗ hc ( X ; E, F ) acting fromsections of E to sections of F .In [21], Mazzeo and Melrose gave a very elegant criterion to determinewhen a cusp operator is Fredholm. They first introduced a notion of prin-cipal symbol adapted to the geometry at infinity, that is, involving the co-sphere bundle S ∗ ( cu T X ) of cu T X ,(1.17) σ k : Ψ k cu ( X ; E, F ) → C ∞ ( S ∗ ( cu T X ); hom( π ∗ E, π ∗ F ))where π : S ∗ cu T X → X is the bundle projection. A cusp operator A ∈ Ψ k cu ( X ; E, F ) is said to be elliptic if its principal symbol σ k ( A ) is invertible. PIERRE ALBIN AND FR´ED´ERIC ROCHON
In that case, by a standard construction, one can obtain a parametrix B ∈ Ψ k cu ( X ; F, E ) such that(1.18) BA − Id E ∈ Ψ −∞ cu ( X ; E ) , AB − Id F ∈ Ψ −∞ cu ( X ; F ) . However, since elements of Ψ −∞ cu ( X ; E ) are not compact in general, this doesnot insure that the operator A is Fredholm. One needs some extra decay atinfinity for the error term to be compact. Precisely, the subset of compactoperators in Ψ −∞ ( X ; E ) is given by x Ψ −∞ ( X ; E ). It is possible to insurethe error term is in that subset provided A is ‘invertible at infinity’. Thiscondition is determined by the normal operator map(1.19) N : Ψ k cu ( X ; E, F ) → Ψ k sus ( ∂X ; E, F )where Ψ k sus ( ∂X ; E, F ) is the space of suspended operators of order k in-troduced by Melrose in [23]. These are operators on ∂X × R which aretranslation invariant in the R direction. Essentially, the normal operator N ( A ) of A is its asymptotically translation invariant part at infinity. Thecriterion of Mazzeo and Melrose can now be stated as follows. Proposition 1.1 (Mazzeo-Melrose) . A cusp operator A ∈ Ψ k cu ( X ; E, F ) is Fredholm if and only if it is elliptic and its normal operator N ( A ) isinvertible. For hyperbolic cusp operators, the situation is much more delicate. Forsimplicity, let us restrict to a first order hyperbolic cusp differential oper-ator ð hc ∈ Ψ ( X ; E, F ). Then x ð hc ∈ Ψ ( X ; E, F ) is a cusp operator andwe can use proposition 1.1 to determine whether or not x ð hc is Fredholm.If it is Fredholm, then it is not hard to see that ð hc is Fredholm as well. Infact, in that case, the spectrum of ð hc is then necessarily discrete since itsparametrix in x Ψ − ( X ; F, E ) is a compact operator.However, even if x ð hc is not Fredholm, it is still possible for ð hc to beFredholm. Define the vertical family of ð hc to be(1.20) ð V hc := ( x ð hc ) | ∂X ∈ Ψ ( ∂X ; E, F ) . When ð hc is a self-adjoint Dirac type operator with E = F a Clifford bun-dle, the vertical family ð V hc is invertible if and only if the normal operator N ( x ð hc ) is invertible. In his thesis [31], Vaillant gave the following cri-terion to determine if a Dirac-type self-adjoint operator ð hc is Fredholm.The vertical family does not have to be invertible, but if it is not, Vaillantdefined another operator ð H hc acting on the finite dimensional vector space K := ker ð V hc and called the horizontal family . If Π denotes the projectionfrom L ( ∂X ; E ) onto K , then the horizontal family is defined by extendingan element ξ ∈ K into the interior to an element e ξ ∈ C ∞ ( X ; E ) and thenapplying ð hc and Π ,(1.21) ð H hc ξ := Π (cid:16) ð hc e ξ (cid:12)(cid:12)(cid:12) ∂X (cid:17) . In his thesis [31], Vaillant gave the following criterion.
OCAL FAMILIES INDEX FOR ∂ -OPERATORS 9 Proposition 1.2 (Vaillant [31], § . A Dirac type self-adjoint operator ð hc ∈ Ψ ( X ; E ) is Fredholm if and only if ð H hc is invertible. Moreover,the continuous spectrum of ð hc is governed by ð H hc with bands of continuousspectrum starting at the eigenvalues of ð H hc and going to infinity. The boundary compactification of a Riemann surface withpuncture
Let Σ be a Riemann surface of type ( g, n ), that is, Σ = Σ \ { x , . . . , x n } where Σ is a compact Riemann surface of genus g and x , . . . , x n are pairwisedistinct points on Σ. We will assume that 2 g + n ≥
3. The surface Σ isa compactification of Σ. An alternative way of compactifying the Riemannsurface Σ is to consider the radial blow up Σ b of Σ at the points { x , . . . , x n } with blow-down map(2.1) β : Σ b → Σ . This gives a compactification of Σ in which each puncture is replaced bya circular boundary. The Riemann surface with boundary Σ b also comesequipped with a natural choice of boundary defining function ρ ∈ C ∞ (Σ b )as we will see. This choice is dictated by the uniformization theorem forRiemann surfaces.Recall that, by the uniformization theorem, there is a canonical hyperbolicmetric g Σ on Σ obtained by taking the unique metric of constant scalar cur-vature equal to − H = { x + iy ∈ C | y > } equipped with the Poincar´e metric(2.3) g H := dx + dy y . Let Γ ∞ be the discrete Abelian group generated by the parabolic isometry z z + 1. The horn is the quotient(2.4) H := Γ ∞ \ H . Via the change of variable r = y , one sees that the horn is isometric to(0 , + ∞ ) r × R / Z equipped with the metric(2.5) drr + r dx . A cusp end is a subspace of H of the form (0 , a ] × R / Z . Near a puncture x i of Σ, the geometry of (Σ , g Σ ) is modeled on a cusp end. That is, aroundeach puncture x i , there exists a neighborhood N i ⊂ Σ and an isometry(2.6) ϕ i : N i → C i with a cusp end C i = (0 , y i ] × R / Z . Each cusp end has a natural compacti-fication(2.7) C i = (cid:20) , y i (cid:21) r i × R / Z where the coordinate r i can be seen as a boundary defining function forthe boundary { } × R / Z ⊂ C i . This boundary defining function can infact be defined intrinsically in terms of the hyperbolic metric (2.5). Indeed,we define a horocycle to be an embedded circle in a cusp end which isperpendicular to all geodesics emanating from the cusp. This definition isformulated purely in terms of the metric. On the other hand, as one cancheck, the horocycles are precisely given by the level sets of the function r i .Moreover, the value of the function r i on a horocycle γ = { u } × R / Z is alsodetermined by the hyperbolic metric. It is the area of the smaller cusp end(0 , u ) × R / Z , namely(2.8) r i ( u, v ) = area((0 , u ) × R / Z )) = Z u Z R / Z drdx = u. Thus, intuitively, the boundary defining function r i is the ‘area function’for the cusp end C i . The compactification C i induces a corresponding com-pactification N i via the isometry (2.6), and thus a compactification Σ hc ofΣ into a compact surface with boundary naturally diffeomorphic to Σ b . Toget a global boundary defining function, choose a smooth non-decreasingfunction χ ∈ C ∞ ([0 , + ∞ )) such that(2.9) χ ( x ) := (cid:26) x, if 0 ≤ x ≤ ;1 , if x ≥ , and consider χ ǫ ( x ) := ǫχ ( xǫ ) for 0 < ǫ < min { y , . . . , y n } . On each (com-pactified) cusp end C i , consider the function χ ǫ ( r i ). Then the function(2.10) ρ Σ ,ǫ ( σ ) := (cid:26) ϕ ∗ i ( χ ǫ ◦ r i )( σ ) , if σ ∈ N i , i ∈ { , . . . , n } ; ǫ, otherwise;is a boundary defining function for ∂ Σ hc in Σ hc . Since the choice of thenumber ǫ is not of primary importance, we will usually denote the func-tion ρ Σ ,ǫ simply by ρ Σ . With respect to this boundary defining function,the hyperbolic metric g Σ is a product hyperbolic metric. That is, in thecoordinates ( x, ρ Σ ) on N i , it is of the form(2.11) g Σ = dρ ρ + ρ dx near the boundary. OCAL FAMILIES INDEX FOR ∂ -OPERATORS 11 The ∂ -operator as a Dirac-type hc -operator Let K := Λ , denote the canonical line bundle on Σ. This line bundleand all of its tensor powers K ℓ have natural holomorphic structures. Inparticular, for each ℓ ∈ Z , there is a well-defined ∂ operator(3.1) ∂ ℓ : C ∞ (Σ; K ℓ ) → C ∞ (Σ; Λ , ⊗ K ℓ ) , where Λ , → Σ is the bundle of (0 , C i wherethe canonical line bundle is trivialized by the holomorphic section dz , ittakes the form(3.2) dz ∂∂z = ( dx − idy ) 12 (cid:18) ∂∂x + i ∂∂y (cid:19) = ( dx + i drr ) 12 (cid:18) ∂∂x − ir ∂∂r (cid:19) , r = 1 y , = 12 (cid:18) rdx + i drr (cid:19) (cid:18) r ∂∂x − ir ∂∂r (cid:19) . Thus, near the boundary ∂ Σ hc , the ∂ -operator is of the form(3.3) ∂ = 12 (cid:18) ρ Σ dx + i dρ Σ ρ Σ (cid:19) (cid:18) ρ Σ ∂∂x − iρ Σ ∂∂ρ Σ (cid:19) . Since ρ Σ ∂∂x − iρ Σ ∂∂ρ Σ is a hc-operator and ( ρ Σ dx + i dρ Σ ρ Σ ) is naturally asection of hc T ∗ Σ ⊗ R C , we see that the ∂ ℓ -operator naturally extends to givea hc-operator(3.4) ∂ ℓ : C ∞ (Σ hc ; hc K ℓ ) → ρ Σ C ∞ (Σ hc ; hc Λ , ⊗ hc K ℓ )where hc Λ , is the complex conjugate of hc K and hc K ⊂ hc T ∗ Σ ⊗ R C issuch that it is identified with K in the interior of Σ hc and it is trivializedby the section ρ Σ dz = ρ Σ dx − i dρ Σ ρ Σ near each connected component of theboundary. The metric g Σ induces a Hermitian metric on K ℓ and Λ , , aswell as on hc K ℓ and hc Λ , . We denote by H ℓ,i the Hilbert space of squareintegrable sections of hc K ℓ ⊗ ( hc Λ , ) i with respect to the natural scalarproduct(3.5) h f , f i H ℓ,i := Z Σ hc h f ( σ ) , f ( σ ) i g Σ dg Σ ( σ )where dg Σ is the natural extension of the volume form of g Σ on Σ hc .The operator ∂ ℓ is Fredholm. To see this, recall (see for instance propo-sition 3.67 in [5]) that(3.6) D ℓ := √ ∂ ℓ + ∂ ∗ ℓ )is a Dirac type operator induced by the Chern connection on K ℓ with Cliffordaction on ν ∈ C ∞ (Σ hc ; hc Λ Σ ) given by(3.7) c ( f ) ν = √ ε ( f , ) − ι ( f , )) ν, f ∈ C ∞ (Σ hc ; hc Λ Σ ) , where ε ( f , ) denotes exterior multiplication by f , . The operator D ℓ isformally self-adjoint. The vertical family D Vℓ of D ℓ is given by(3.8) c ( du ) ∂∂u acting on C ∞ ( R / Z ; hc K ℓ ⊕ hc Λ , ⊗ hc K ℓ ) on each circular boundary compo-nent of Σ hc , where u = − x is such that { ∂∂u (cid:12)(cid:12) σ } is an oriented orthonormalbasis of T σ ∂ Σ hc for each σ ∈ ∂ Σ hc . In particular, K = ker( D Vℓ ) is a complexvector space of dimension 2 n . By proposition 1.2, we need to show that thehorizontal family D Hℓ : K → K is invertible.
Proposition 3.1.
On each circular boundary component of Σ hc , the hori-zontal family is given by D Hℓ = (cid:18) ℓ − (cid:19) ic ( du ) . Proof.
The bundle on which D ℓ acts is (cid:0) C ⊕ Λ , Σ (cid:1) ⊗ K ℓ . Choose a spin structure on Σ and let S be the corresponding spinor bundle.It is well-known (see for instance [19]) that, seen as complex line bundle, S is a square root of the canonical line bundle K so that S ⊗ C S = K. Moreover, we have also that (cid:0) C ⊕ Λ , Σ (cid:1) ∼ = S ⊗ R S ∗ . Thus the operator D ℓ acts on S ⊗ R (cid:16) S ∗ ⊗ C K ℓ (cid:17) , which means that D ℓ is a Dirac operator twisted by the bundle S ∗ ⊗ C K ℓ .As a bundle with connection, the bundle S ∗ ⊗ K ℓ certainly does not have aproduct structure near the boundary since it has non-zero curvature. Thus,according to Proposition 3.15, p.44 in [31], the horizontal family D Hℓ at eachcusp is given by − iRc (cid:18) ∂∂u (cid:19) = − i (cid:18) − ℓ (cid:19) c (cid:18) ∂∂u (cid:19) where iRdg Σ is the curvature of the complex vector bundle S ∗ ⊗ C K ℓ (cf.(3.15)). Collecting the contributions at each cusp end, we get the desiredresult. (cid:3) This gives the following corollary.
OCAL FAMILIES INDEX FOR ∂ -OPERATORS 13 Corollary 3.2.
The operators D ℓ = √ (cid:18) ∂ ∗ ℓ ∂ ℓ (cid:19) , ∂ ℓ , and ∂ ∗ ℓ are Fredholm. Notice that proposition 3.1 is also consistent with the well-known factthat the band of continuous spectrum of the Hodge Laplacian D ℓ starts at (cid:0) − ℓ (cid:1) and goes to infinity.In his thesis [31], Vaillant obtained a general formula for the index of aDirac type operator on a fibred hyperbolic cusp operator. For the index ofthe operator ∂ ℓ , this formula is given by the usual Atiyah-Singer integrandtogether with two corrections coming from the boundary, namely the etainvariants associated to the vertical family of ∂ ℓ and the horizontal family D Hℓ ,(3.9) ind( ∂ ℓ ) = Z Σ hc Ch( hc K ℓ ) Td( hc K − ) − η ( D Vℓ ) − η ( D Hℓ ) . The eta invariant of the vertical family is easily seen to be zero. This isbecause modulo standard identifications, η ( D Vℓ ) corresponds to n times theeta invariant of the self-adjoint operator(3.10) 1 i ∂∂x = i ∂∂u : C ∞ ( R / Z ) → C ∞ ( R / Z ) . But the spectrum of i ∂∂x is 2 π Z and its eta functional(3.11) η ( 1 i ∂∂x , s ) = X k =0 πk | πk | − s , Re s >> s = 0 of the analytic continuation of η ( i ∂∂x , s ), is zero. Thecorresponding eta invariant η ( D Vℓ ) = nη ( i ∂∂x ) therefore vanishes. For thecomputation of the spectral asymmetry of D Hℓ , there is no regularizationinvolved since D Hℓ is just an endomorphism of a finite dimensional vectorspace. From proposition 3.1, we compute directly (see [2, (4.14)]) that(3.12) η ( D Hℓ ) = n sign (cid:18) ℓ − (cid:19) . The index is therefore given by(3.13) ind( ∂ ℓ ) = Z hc Σ Ch( hc K ℓ ) Td( hc K − ) + n (cid:18) − ℓ (cid:19) . The integral is also easy to compute. Let Θ Σ denote the curvature of hc T , Σ.Then the integrand is given by(3.14) Ch( hc K ℓ ) Td( hc T , Σ) = (cid:16) e − ℓi π Θ Σ (cid:17) i π Θ Σ − e − i π Θ Σ ! = 1 + (cid:18) − ℓ (cid:19) i π Θ Σ . By a standard computation (see for instance p.77 in [16]), we know that(3.15) i π Θ Σ = κ π dg Σ = − π dg Σ where κ = − g Σ . By the Gauss-Bonnet theo-rem applied to Σ, we get that(3.16) ind( ∂ ℓ ) = (cid:18) − ℓ (cid:19) Z Σ hc κ π dg Σ + n (cid:18) − ℓ (cid:19) = (cid:18) − ℓ (cid:19) χ (Σ) + n (cid:18) − ℓ (cid:19) = (cid:18) − ℓ (cid:19) (2 − g − n ) + n (cid:18) − ℓ (cid:19) . This gives the following formula.
Proposition 3.3.
The index of ∂ ℓ is given by ind( ∂ ℓ ) = (cid:26) (2 ℓ − g −
1) + ℓn, ℓ ≤ , (2 ℓ − g −
1) + ( ℓ − n, ℓ > . In fact, using the Riemann-Roch theorem on the compact Riemann sur-face Σ, it is also possible to compute explicitly the dimension of the kerneland the cokernel of ∂ ℓ (cf. p.404 in [30]). By definition, an element of f ∈ ker ∂ ℓ is a holomorphic section of K ℓ , so in each cusp end N j , it has aLaurent series expansion(3.17) f ( z ) = ∞ X k = −∞ a ( j ) k e πikz ( dz ) ℓ . When ℓ >
0, this expansion has to be of the form(3.18) f ( z ) = ∞ X k =1 a ( j ) k e πikz ( dz ) ℓ in order for f to be an element of H ℓ, . Such an f is said to be a cusp form of weight (2 ℓ, ℓ ≤
0, we can also have a constant coefficient in theseries,(3.19) f ( z ) = ∞ X k =0 a ( j ) k e πikz ( dz ) ℓ . OCAL FAMILIES INDEX FOR ∂ -OPERATORS 15 When ℓ = 0, using the coordinate ζ := e πiz near each puncture x j in Σ, wesee that such a f naturally extends to give a holomorphic function on Σ. Itis therefore constant, so that dim C ker ∂ = 1. When ℓ ≥
1, the section f takes the form(3.20) f ( z ) = ∞ X k =1 a ( j ) k ζ k (cid:18) dζ πiζ (cid:19) ℓ in the coordinate ζ near the puncture x j . Thus, it naturally extends to ameromorphic section of K ℓ → Σ with poles of order not exceeding ℓ − x , . . . , x n and holomorphic elsewhere. Conversely, sucha meromorphic section corresponds to an element of ker ∂ ℓ . We can thuscompute dim C ker ∂ ℓ by applying the Riemann-Roch theorem on Σ to theline bundle(3.21) L D ⊗ K ℓ where L D is the holomorphic line bundle associated to the divisor(3.22) D = n X i =1 ( ℓ − x i on Σ . This gives(3.23) dim C ker ∂ ℓ = h ( L D ⊗ K ℓ )= h ( K ⊗ ( L D ⊗ K ℓ ) − ) + deg( L D ⊗ K ℓ ) − g + 1= h ( K ⊗ ( L D ⊗ K ℓ ) − ) + n ( ℓ −
1) + (2 ℓ − g − , where h ( L ) denotes the dimension of the space of holomorphic sections ofthe holomorphic line bundle L . Now we compute that(3.24) deg( K ⊗ L − D ⊗ K − ℓ ) = − ( ℓ − g + n − . When ℓ = 1, K ⊗ ( L D ⊗ K ℓ ) − is the trivial line bundle, so h ( K ⊗ ( L D ⊗ K ℓ ) − ) = 1 in this case. When ℓ >
1, deg( K ⊗ L − D ⊗ K − ℓ ) < g + n ≥
3, and therefore h ( K ⊗ L − D ⊗ K − ℓ ) = 0. Finally,when ℓ <
0, elements of ker ∂ ℓ correspond to holomorphic sections of K ℓ Σ with zeros of degree at least − ℓ at each puncture. These in turn correspondto the holomorphic sections of a holomorphic line bundle of negative degree(since 2 g + n ≥ ∂ ℓ = 0 in that case. Hence, we see that thedimension of the kernel of ∂ ℓ is given by(3.25) dim ker ∂ ℓ = , ℓ < , , ℓ = 0 ,g, ℓ = 1(2 ℓ − g −
1) + n ( ℓ − , l ≥ . Comparing with the index (3.16), we also get that(3.26) dim ker ∂ ∗ ℓ = − (2 ℓ − g − − nℓ, ℓ < ,g, ℓ = 0 , , ℓ = 10 , l ≥ . These formulas are consistent with Kodaira-Serre duality, which asserts inthis case that ker ∂ ∗ ℓ ∼ = ker ∂ − ℓ .4. The Teichm¨uller space and the Teichm¨uller universal curve
So far we have assumed that the complex structure on Σ was fixed. Bychanging the complex structure, one can get instead a family of ∂ ℓ operators.The universal case is obtained by considering all at once the moduli spaceof all complex structures on a surface of type ( g, n ), two complex structuresbeing identified whenever there is a conformal transformation between themhomotopic to the identity. It is called the Teichm¨uller space of Riemannsurfaces of genus g with n punctures and is denoted T g,n . It is a complexmanifold of complex dimension 3 g − n which can be identified with an openset of C g − n . The Teichm¨uller space T g,n comes together with a universalbundle, the universal Teichm¨uller curve T g,n with bundle projection(4.1) p : T g,n → T g,n and fibre p − ([Σ]) the Riemann surface Σ of type ( g, n ) corresponding tothe point [Σ] ∈ T g,n . Denote by T i,jv T g,n → T g,n the vertical ( i, j ) tangentbundle of the fibration (4.1) for i, j ∈ { , } . On each fibre Σ := p − ([Σ]),the restriction of T i,jv T g,n is canonically identified with T i,j Σ. Denote byΛ i,jv → T g,n the dual of T i,jv . On each fibre we also have a ∂ -operator. Theseoperators fit together to give a family of operators(4.2) ∂ ℓ ∈ ρ − Ψ ( T g,n /T g,n ; (Λ , v ) ℓ , Λ , v ⊗ (Λ , v ) ℓ )where ρ is an appropriate boundary defining function (whose precise defi-nition we postpone to (5.16)). Each element of the family is a Fredholmoperator so that we have a family index in K ( T g,n ),(4.3) ind( ∂ ℓ ) ∈ K ( T g,n ) . Since the Teichm¨uller space is contractible, this families index really onlyencodes the numerical index of any member of the family under the identi-fication K ( T g,n ) ∼ = K (pt) ∼ = Z . Still, it is possible to exhibit an explicitrepresentative of the K -class ind( ∂ ℓ ) ∈ K ( T g,n ), providing in this way alocal description of the family index. This is because, according to (3.25)and (3.26), the dimensions of the kernel and the cokernel of elements of thefamily ∂ ℓ are always the same (they only depend on ℓ , g and n , not on thecomplex structure). This means that(4.4) ker ∂ ℓ → T g,n and ker ∂ ∗ ℓ → T g,n OCAL FAMILIES INDEX FOR ∂ -OPERATORS 17 form complex vector bundles on T g,n and the family index of ∂ ℓ can then beexpressed as the virtual difference of these two vector bundles,(4.5) ind ∂ ℓ = [ker ∂ ℓ ] − [ker ∂ ∗ ℓ ] ∈ K ( T g,n ) . In fact, as we will recall in a moment, these vector bundles both comeequipped with a natural connection. We can therefore express their respec-tive Chern characters at the level of forms. This provides a local descriptionof the Chern character of the family index(4.6) Ch(ind ∂ ℓ ) := Ch(ker ∂ ℓ ) − Ch(ker ∂ ∗ ℓ ) ∈ C ∞ ( T g,n , Λ ev ( T g,n )) . On the Teichm¨uller space itself, this local description of the index does notcontain more cohomological information than (3.16). However, the localdescriptions (4.5) and (4.6) are invariant under the action of the Teichm¨ullermodular group Mod g,n . This means that these local descriptions descendto the moduli space T g,n / Mod g,n (in the sense of orbifolds), which typicallyhas a non-trivial topology as well as singularities.5.
The canonical connection on the universal Teichm¨ullercurve
The fibration p : T g,n → T g,n comes together with a canonical connec-tion P . To describe this connection, one possible approach is to describeRiemann surfaces as certain quotients of the upper half-plane H . If Σ is aRiemann surface of genus g with n punctures, then it can be representedas a quotient Γ \ H of the upper half-plane by the action of a torsion-free finitely generated Fuchsian group Γ. The group Γ ⊂ PSL(2 , R ) is oftype ( g, n ), which is to say it is generated by 2 g hyperbolic transformations A , B , . . . , A g , B g and n parabolic transformations S , . . . , S n satisfying thesingle relation A B A − B − · · · A g B g A − g B − g S · · · S n = 1. Since H is sim-ply connected, in fact contractible, it is the universal cover of Σ under thequotient map H → Γ \ H . From this perspective, the canonical hyperbolicmetric g Σ associated to the (conformal structure of the) complex structureis precisely the metric on Γ \ H induced from the Poincar´e metric(5.1) g H := dx + dy y on H . The punctures of Σ then correspond to the image of the fixed points z , · · · , z n in R ∪ {∞} of the parabolic transformations S , . . . , S n under the quotientmap H → Γ \ H . Let Γ i be the cyclic subgroup of Γ generated by the para-bolic transformation S i for i = 1 , . . . , n . It can be identified with the cyclicgroup Γ ∞ by choosing σ i ∈ PSL(2 , R ) such that σ i ∞ = z i , so that(5.2) σ − i S i σ i = (cid:18) ±
10 1 (cid:19) , σ − i Γ i σ i = Γ ∞ . On Σ, sections of (Λ , ) ℓ ⊗ ((Λ Σ ) , ) m correspond to automorphic forms ofweight (2 ℓ, m ) with respect to the group Γ, that is, functions f : H → C such that(5.3) f ( γz ) γ ′ ( z ) ℓ γ ′ ( z ) m = f ( z ) ∀ z ∈ H , ∀ γ ∈ Γ . For instance, the natural K¨ahler metric associated to the hyperbolic metric g Σ , seen as a section of Λ , ⊗ Λ , , corresponds to the automorphic form ofweight (2 , y on H . In the correspondence between Riemann surfaces and quotients of H , achange of complex structure corresponds to a change of the Fuchsian groupΓ. This provides a canonical identification between the Teichm¨uller space T g,n of Riemann surfaces of type ( g, n ) and the Teichm¨uller space of Fuch-sian groups of type ( g, n ). Under this identification, the tangent space of T g,n at [Σ] can be identified with the subspace Ω − , (Σ) = ker ∂ ∗− ⊂ H − , of harmonic Beltrami differentials. Each element of µ ∈ Ω − , (Σ) has theform µ = y ϕ for a unique ϕ ∈ ker ∂ , so that dim C Ω − , (Σ) = 3 g − n .In particular, an element of Ω − , (Σ) decays exponentially fast as one ap-proaches a puncture (using the coordinates of (2.7)). The (holomorphic)cotangent space T ∗ [Σ] T g,n can be identified with ker ∂ on Σ, this space beingnaturally dual to Ω − , (Σ) via the pairing(5.5) ( µ, ϕ ) := Z Σ µϕ, µ ∈ Ω − , (Σ) , ϕ ∈ ker ∂ . To get complex coordinates on T g,n we can use the fact that to every µ ∈ Ω − , (Σ) satisfying(5.6) k µ k L ∞ = sup z ∈ Σ | µ ( z ) | < , one can associate a unique diffeomorphism f µ : H → H satisfying the Bel-trami equation(5.7) ∂f µ ∂z = µ ∂f µ ∂z and fixing the points 0 , , ∞ , where µ in (5.7) is seen as an automorphicform of weight ( − ,
2) on H . From this solution, one gets a new Fuchsiangroup by considering Γ µ := f µ Γ( f µ ) − , that is, a new complex structure byconsidering the Riemann surface Σ µ := Γ µ \ H . The diffeomorphism f µ alsonaturally descends to the quotient Γ \ H to give a diffeomorphism(5.8) f µ : Γ \ H → Γ µ \ H . Now, if one chooses a basis µ , . . . , µ g − n of Ω − , (Σ) and sets µ = ε µ + · · · + ε g − n µ g − n , then the correspondence ( ε , . . . , ε g − n ) [Σ µ ]defines complex coordinates in a neighborhood of [Σ] ∈ T g,n called Berscoordinates . In the overlapping of neighborhoods of two points [Σ] and[Σ µ ], the Bers coordinates transform complex analytically (see for instancep.409 in [30]), defining on T g,n a complex structure. The Bers coordinates OCAL FAMILIES INDEX FOR ∂ -OPERATORS 19 provide a local trivialization of the fibration p : T g,n → T g,n of the universalTeichm¨uller curve, in fact, of its universal cover, the Bers fibre space BF g,n (see p.138 in [34]). If U ⊂ T g,n is the open set where the Bers coordinates( ε , . . . , ε g − g + n ) associated to [Σ] are defined, then this trivialization isgiven by the commutative diagram(5.9) U × Σ ν / / pr % % JJJJJJJJJJJ p − ( U ) p (cid:15) (cid:15) U where pr is the projection on the first factor and ν is given by ν ( µ, σ ) = f µ ( σ ) ∈ p − ([Σ µ ]) where f µ denotes the map (5.8).This local trivialization also induces a lift of T [Σ] T g,n to T T g,n | p − ([Σ]) ,namely (see p.142 in [34]), a vector µ ∈ T [Σ] T g,n has a canonical lift pr ∗ µ ∈ T ( U × Σ) | { [Σ] }× Σ , and therefore a canonical lift ν ∗ (pr ∗ µ ) ∈ T T g,n | p − ([Σ]) .More generally, introducing Bers coordinates at each [Σ] ∈ T g,n , we canget in this way a canonical horizontal lift of T T g,n to T T g,n . In other words,associated to the fibration p : T g,n → T g,n , there is a canonical connection P , that is, P ⊂
T T g,n is a distribution of hyperplanes such that(5.10) p ∗ : P z → T p ( z ) T g,n is an isomorphism for every z ∈ T g,n . It is also possible to define a covariantderivative(5.11) ∇ P : C ∞ ( T g,n ; (Λ , v ) ℓ ⊗ (Λ , v ) m ) → C ∞ ( T g,n ; p ∗ ( T ∗ g,n ) ⊗ (Λ , v ) ℓ ⊗ (Λ , v ) m ) . This allows one to differentiate sections of (Λ , v ) ℓ ⊗ (Λ , v ) m with respect tovectors on the base T g,n . At [Σ] ∈ T g,n , the differentiation can be describedby using the Bers coordinates associated to T [Σ] T g,n ∼ = Ω − , (Σ) with thelocal trivialization (5.9) of p : T g,n → T g,n near [Σ]. In this trivialization,a section ω of (Λ , v ) ℓ ⊗ (Λ , v ) m corresponds to a section e ω of (pr ∗ Λ , ) ℓ ⊗ (pr ∗ Λ , ) m on U ×
Σ where pr : U × Σ → Σ is the projection on the secondfactor. Precisely, in terms of automorphic forms of weight (2 ℓ, m ), we havethat(5.12) e ω ( ε, σ ) = ω ◦ f µ (cid:18) ∂f µ ∂z (cid:19) ℓ (cid:18) ∂f µ ∂z (cid:19) m where µ = ε µ + · · · + ε g − n µ g − n . On Σ = p − ([Σ]) ⊂ T g,n , there isa canonical identification between (Λ , v ) ℓ ⊗ (Λ , v ) m and (Λ , ) ℓ ⊗ (Λ , ) m .Under this identification, the covariant derivative of ω takes the form (cf.p.409 in [30]),(5.13) ∇ P ∂∂εi ω (cid:12)(cid:12)(cid:12)(cid:12) p − ([Σ]) = ∂∂ε i e ω ( ε, σ ) (cid:12)(cid:12)(cid:12)(cid:12) ε =0 . An important example is given by the family of fibrewise hyperbolic areaforms dg Σ , which as was shown in [1] gives a parallel section of Λ , v withrespect to the connection P , ∇ P dg Σ = 0 . This corresponds to the fact that the automorphic form of weight (2 , y is parallel with respect to the connection P . However, notice that this doesnot imply the family of hyperbolic metrics g Σ , [Σ] ∈ T g,n is parallel withrespect to P as a section of T ∗ v T g,n ⊗ T ∗ v T g,n . In fact, they cannot be parallelwith respect to any connection, since otherwise this would mean that thesemetrics are all isometric, a contradiction since essentially by definition ofthe Teichm¨uller space, these metrics are not even conformal to one another.It is also possible to define the covariant derivative of families of operatorsusing the connection P . If A ε : H ℓ,m (Σ µ ) → H ℓ ′ ,m ′ (Σ µ ) is such family inthe trivialization (5.9) given by the Bers coordinates, then the covariantderivative of A ε at [Σ] is given by(5.14) ∇ P ∂∂εi A ε (cid:12)(cid:12)(cid:12)(cid:12) [Σ] = ∂∂ε i ( f µ ) ∗ A ε ( f µ ∗ ) − (cid:12)(cid:12)(cid:12)(cid:12) ε =0 , ∇ P ∂∂εi A ε (cid:12)(cid:12)(cid:12)(cid:12) [Σ] = ∂∂ε i ( f µ ) ∗ A ε ( f µ ∗ ) − (cid:12)(cid:12)(cid:12)(cid:12) ε =0 . For example, the covariant derivatives of ∂ ℓ and ∂ ∗ ℓ at [Σ] are given by (seeformula (2.6) in [30])(5.15) ∇ P µ ∂ ℓ = µ∂ ∗ ℓ +1 u, ∇ P µ ∂ ℓ = 0 , ∇ P µ ∂ ∗ ℓ = 0 , ∇ P µ ∂ ∗ ℓ = µ∂ ℓ − u − where u := y is seen as a section of Λ , ⊗ Λ , .As we have seen, each Riemann surface Σ of type ( g, n ) has a boundarycompactification Σ hc constructed using the metric g Σ . These compactifica-tions fit together to give a fibrewise boundary compactification hc T g,n of theuniversal Teichm¨uller curve. In terms of the local trivializations of (5.9),this is because the solution f µ to the Beltrami equation (5.7) is real ana-lytic (see for instance proposition 4.6.2 in [18]), it maps the fixed points ofΓ to the fixed points of Γ µ and, seen as a map f µ : Σ → Σ µ , it is asymptoti-cally holomorphic as one approaches any puncture of Σ. Since the canonicalconnection P is obtained by using Bers coordinates and infinitesimal defor-mations induced by the solutions of the Beltrami equation (5.7), we see thatit also naturally lifts to provide a canonical connection hc P to the fibration hc p : hc T g,n → T g,n . To get a natural boundary defining function for hc T g,n , we use the con-struction of (2.10) in each fibre. This definition depends on the choice of anumber ǫ > N i in a givensurface has area strictly greater than ǫ . To get a global definition hc T g,n , we OCAL FAMILIES INDEX FOR ∂ -OPERATORS 21 should replace the number ǫ by a smooth function a : T g,n → R + such thatin a given fibre Σ := p − ([Σ]), the area of each cusp end N i is strictly greaterthan a ([Σ]). We can then define our global defining function on hc T g,n to be(5.16) ρ ( σ ) = ρ Σ ,a ([Σ]) ( σ ) for σ ∈ Σ := p − ([Σ]) , [Σ] ∈ T g,n where ρ Σ ,ǫ : Σ hc → R is defined in (2.10) for the Riemann surface Σ and achoice of small ǫ > A local formula for the family index
The family of operators ∂ ℓ ∈ Ψ ( T g,n /T g,n ; hc K ℓv , hc Λ , v ⊗ hc K ℓv ) is a par-ticular example of the families of φ -hc operators considered in [2]. When weapply this local index theorem to our family ∂ ℓ with the canonical connec-tion hc P for the fibration hc p : hc T g,n → T g,n , we get the family version of(3.9),(6.1) Ch(Ind( ∂ ℓ )) = Z T g,n /T g,n Ch( T − ℓv ( T g,n )) Td( T v T g,n ) − b η ( D Vℓ ) − b η ( D Hℓ ) − (cid:18) π √− (cid:19) N d Z ∞ Str ∂ A tD ℓ ∂t e − ( A tDℓ ) ! dt, where the eta invariants of the vertical and horizontal families are replacedby the the corresponding eta forms of Bismut and Cheeger [7] (with non-standard Z grading for D H ). This is an equality at the level of forms.Notice that in [2] the first term is expressed in terms of the b A form. However,thanks to Theorem 5.5 in [34] and its reformulation in equation 5.3 of [34],the fibration p : T g,n → T g,n is K¨ahler fibration (see [10] for a defintion) sothat it is possible to rewrite the first term using the Todd form instead. Inthe last term, A tD ℓ is the rescaled Bismut superconnection while N is thenumber operator in Λ T g,n , that is, the action of N on forms of degree k on T g,n is multiplication by k . Remark 6.1.
In this paper, our convention for the Chern character differsfrom that of [5] . This is why we need to include these extra factors of πi in the last term. In principle, the eta forms would also require such factors,so really, by an eta form, we mean (2 πi ) − N times the eta form of Bismutand Cheeger (cf. equation 4.101 in [7] ). When we take the degree zero part of (6.1), we get back the numericalindex (3.9) by evaluating it at a given point [Σ] ∈ T g,n . In fact, as we haveseen, the degree zero part of b η ( D Vℓ ) is identically zero, while the degree zeropart of b η ( D Hℓ ) is n sign( ℓ − ). However, the higher degree components of b η ( D Hℓ ) vanish identically at the level of forms as we will see in a moment.Let ∂ T g,n := ρ − (0) be the union of the boundaries of the fibres of hc p : hc T g,n → T g,n . The map hc p induces a fibration structure(6.2) ∂p : ∂ T g,n → T g,n with typical fibre the disjoint union of n circles. In fact, the manifold ∂ T g,n has precisely n components,(6.3) ∂ T g,n = n [ i =1 ∂ i T g,n with ∂ i T g,n the component associated to the i th cusp. There is a correspond-ing fibration structure(6.4) ∂p i : ∂ i T g,n → T g,n . Recall that the vertical family D Vℓ decomposes as(6.5) D Vℓ = D V, − ℓ D V, + ℓ ! with respect to the Z grading of the Clifford bundle. Lemma 6.2.
The Chern form of ker D V, + ℓ → T g,n vanishes in positive de-grees, Ch(ker D V, + ℓ ) [2 k ] = 0 , k ∈ N . Proof.
Let D Vℓ,i be the vertical family of the i th component ∂ i T g,n of ∂ T g,n .Via the identification − c ( dρρ ) : hc Λ , v ⊗ hc K v −→ hc K v given by Clifford multiplication, the operator D V, + ℓ can be identified with(6.6) 1 i ∇ ∂∂x = i ∇ ∂∂u : C ∞ ( R / Z ; hc K ℓv ) → C ∞ ( R / Z ; hc K ℓv )where u = − x is such that ∂∂u is an oriented orthonormal basis of T σ ( ∂ i T g,n /T g,n )for each σ ∈ ∂ i T g,n . Thus, ker D V, + ℓ,i → T g,n defines a complex line bundleover the Teichm¨uller space T g,n and(6.7) ker D V, + ℓ = n M i =1 ker D V, + ℓ,i . For the corresponding Chern characters, this gives(6.8) Ch(ker D V, + ℓ ) = n X i =1 Ch(ker D V, + ℓ,i ) . To prove the lemma, it therefore suffices to show that Ch(ker D V, + ℓ,i ) [2] = 0for i ∈ { , . . . , n } . This will be true provided we can trivialize ker D V, + ℓ,i by OCAL FAMILIES INDEX FOR ∂ -OPERATORS 23 a parallel section. From the identification of D V, + ℓ,i with (6.6), a choice oftrivializing section is given by taking(6.9) s ℓ,i : [Σ] (cid:18) ρdx − i dρρ (cid:19) ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ∂ Σ hc ) i ∈ hc K ℓv (cid:12)(cid:12)(cid:12) ( ∂ Σ hc ) i where Σ = p − ([Σ]) and ρ is the boundary defining function of (5.16). No-tice that the section (6.9) is completely determined by the canonical familyof hyperbolic metrics g T g,n /T g,n . Conversely, for ℓ = 1, the section s ,i com-pletely determines the asymptotic behavior of g T g,n /T g,n as one approachesthe i th puncture. If the family of metrics g T g,n /T g,n were parallel with re-spect to the canonical connection P , we could conclude immediately thatthe section s ℓ,i is parallel. This is not the case, but at least the family ofmetrics g T g,n /T g,n is asymptotically parallel as one approaches a puncture.Indeed, from (5.15), we see that the parallel transport (along a path on T g,n ) defined by the canonical connection P is asymptotically holomorphicas one approaches a puncture. This is because the Beltrami differential µ in (5.7) vanishes exponentially fast as one approaches a puncture (using thecoordinates of (2.7)). Thus, parallel transport is asymptotically a conformaltransformation for the family of metrics g T g,n /T g,n . Since(6.10) ∇ P dg T g,n /T g,n = 0 , this means that the parallel transport defined by the connection P is asymp-totically an isometry as one approaches a puncture. That is, ∇ P g T g,n /T g,n isasymptotically zero as one approaches a puncture. In particular, this impliesthat for each i ∈ { , . . . , n } , the section s ℓ,i of (6.9) is parallel with respectto the connection P . (cid:3) Together with the boundary defining function ρ , the family of metric g T g,n /T g,n induces a natural family of metrics g i for each fibre of the fibration(6.4) in such a way that each fibre becomes isometric to the circle S := R / Z of length 1 (cf. [35]). With these identifications, we get a natural actionof S on each fibre, giving (6.4) the structure of a principal S -bundle. Byconstruction, the family of metrics g i is S -equivariant with respect to the S action. The canonical connection hc P naturally induces a connection P i on (6.4). Lemma 6.3.
The family of metrics g i is parallel with respect to the connec-tion P i , that is, the connection P i is unitary with respect to the metric g i .In particular, on the i th circular boundary component, the vector field ∂∂u isparallel with respect to the connection P i .Proof. By the proof of lemma 6.2, the family of metrics g T g,n /T g,n is asymp-totically parallel as one approaches a cusp, from which the result follows. (cid:3) We can now show that the eta form of D Hℓ vanishes in positive degrees. Lemma 6.4.
For each k ∈ N , the degree k part of the form b η ( D Hℓ ) vanishesidentically, b η ( D Hℓ ) [2 k ] = 0 , k > . Proof.
Since D Hℓ is just an endomorphism of ker D Vℓ , we see from proposi-tion 3.1, lemma 6.3 and the definition of the eta form that (see [2, (4.12)])(6.11) b η ( D Hℓ ) = 12 sign (cid:18) ℓ − (cid:19) Ch(ker D V, + ℓ ) . The result then follows from lemma 6.2. (cid:3)
On the other hand, the eta form of the vertical family gives a contributionin higher degrees. In fact, since the geometry of the boundary fibration isvery special, it is possible to compute the eta form explicitly. With respectto the decomposition (6.3), the vertical family D Vℓ admits a correspondingdecomposition(6.12) D Vℓ = n M i =1 D Vℓ,i where D Vℓ,i is a family of self-adjoint Dirac operators on the fibration (6.4).In terms of this decomposition, the eta form of D Vℓ can be expressed as(6.13) b η ( D Vℓ ) = n X i =1 b η ( D Vℓ,i ) . By (5.15) (see also the proof of lemma 8.1), the family of Dirac operators D ℓ is asymptotically parallel with respect to the canonical connection P asone approaches a cusp. This means that each of the vertical families D Vℓ,i isparallel with respect to the connection P i on (6.4). This fact, together withthe fact the family of metric g i is parallel with respect to the connection P i and is equivariant with respect to the circle action, means that we can applythe result of Zhang (Theorem 1.7 in [36]) to get an explicit formula for theeta form η ( D Vℓ,i ). Proposition 6.5 (Zhang, [36], Theorem 1.7) . The eta form of D Vℓ,i is givenby b η ( D Vℓ,i ) = 12 tanh (cid:0) e i (cid:1) − e i where e i := √− π Θ i is the curvature form of the circle bundle ∂p i : ∂ i T g,n → T g,n with connection P i and curvature Θ i , the Lie algebra of S being iden-tified with i R . Remark 6.6.
Notice in particular that this implies that the eta form is zeroin degree k for k = 0 modulo 2. Moreover, it is a closed form, an unusualfeature for a eta form. OCAL FAMILIES INDEX FOR ∂ -OPERATORS 25 Before stating our main theorem, let us give an alternate description ofthe Chern form e i . Namely, to the circle bundle (6.4) with connection P i and family of metrics (2 π ) g i , we can associate in a canonical way a complexline bundle L i → T g,n equipped with a Hermitian metric h i and a unitaryconnection ∇ L i in such a way that the curvature form of L i is precisely( − π √− e i . The line bundle L i is such that its unit circle bundle withinduced metric and connection is precisely the circle bundle (6.4) with familyof metrics 2 πg i and connection P i .Thinking of a fibre Σ := p − ([Σ]) as a punctured Riemann surface(6.14) Σ = Σ − { x , . . . , x n } , one can also define the line bundle L i by(6.15) L i, [Σ] := ( T , x i Σ) ∗ = K Σ (cid:12)(cid:12) x i , [Σ] ∈ T g,n . Moreover, from this perspective, the Hermitian metric h i and the unitaryconnection ∇ L i are easily seen to be the same as the one introduced byWolpert [35]. Thus, the form e i corresponds to the Chern form c ( k k can,i )of Corollary 7 in [35].Now, combining (6.1) with Lemma 6.4 and Proposition 6.5, we obtain thefollowing formula. Theorem 1.
The local family index of the family of operators D + ℓ := √ ∂ ℓ ∈ ρ − Ψ ( T g,n /T g,n ; hc K ℓv , hc Λ , v ⊗ hc K ℓv ) associated to the Teichm¨uller universal curve p : T g,n → T g,n and its canon-ical connection P is given by (6.16)Ch(ind ker D + ℓ ) = Z T g,n /T g,n Ch( T − ℓv ( T g,n )) Td( T v T g,n ) + n (cid:18) − ℓ (cid:19) − n X i =1
12 tanh (cid:0) e i (cid:1) − e i ! − (cid:18) π √− (cid:19) N d Z ∞ Str ∂ A tD ℓ ∂t e − ( A tDℓ ) ! dt, where A tD ℓ is the rescaled Bismut superconnection associated to the family D ℓ , e i is the canonical Chern form of the (holomorphic) cotangent bundlealong the i th cusp L i → T g,n and N is the number operator on Λ T g,n . As in [30], each of the terms in our formula is invariant under the actionof the Teichm¨uller modular group Mod g,n . Thus, formula (6.16) also holdson the moduli space M g,n := T g,n / Mod g,n in the sense of orbifolds with thefibration p : T g,n → T g,n replaced by the forgetful map π n +1 : M g,n +1 →M g,n . In fact, on the moduli space M g,n , the formula acquires a topologicalmeaning in higher degrees.To see this, define T g,n to be the space obtained from T g,n by fillingeach puncture of each fibre by a marked point. There is still a fibration p : T g,n → T g,n , but now with fibres being compact Riemann surfaces of genus g with n marked points. Let K v → T g,n denote the correspondingvertical canonical line bundle (the dual of the vertical (1 ,
0) tangent bundle).Let D i ⊂ T g,n be the divisor associated to the i th marked points and let L D be the line bundle associated to the divisor D := P ni =1 D i . Then, byanalogy with the discussion in §
3, we see that the family index of ∂ ℓ is thesame as the family index of the family of ∂ -operators(6.17) b ∂ ℓ : C ∞ ( T g,n ; K ℓv ⊗ L ℓ − D ) → C ∞ ( T g,n ; Λ , v ⊗ K ℓv ⊗ L ℓ − D )for ℓ > b ∂ ℓ : C ∞ ( T g,n ; K ℓv ⊗ L ℓD ) → C ∞ ( T g,n ; Λ , v ⊗ K ℓv ⊗ L ℓD )for ℓ ≤
0. On the fibration π n +1 : M g,n +1 → M g,n , this corresponds to thefollowing situation. Let ω π n +1 be the relative dualizing sheaf of this fibration,that is, the sheaf of sections of K v . Let ω π n +1 ( D ) be the logarithmic variantof ω π n +1 , which means that the local sections of ω π n +1 ( D ) are sections of ω π n +1 with possibly simple poles at the first n marked points. Then the linebundle K v ⊗ L D on T g,n corresponds to the sheaf ω π n +1 ( D ) on M g,n +1 .Going back to the formula of theorem 1, we see that the form e i thenrepresents the Miller class ψ i = c ( L i ). On the other hand, since the Millerclass ψ n +1 on M g,n +1 is given by ψ n +1 = c ( ω π n +1 ( D )) (see for instancep.254 in [33]), the first term in the right-hand side of (6.16) can be seen torepresent a linear combination of the Mumford-Morita classes(6.19) κ j := ( π n +1 ) ∗ ( ψ j +1 n +1 ) = h ( π n +1 ) ∗ ( e j +1 n +1 ) i , j ∈ N , where e n +1 is the Chern form of the vertical canonical line bundle K v ∼ = K v ⊗ L D . The precise formula involves the Bernouilli numbers B m and theBernouilli polynomials B m ( ℓ ), which are defined by the following identities,(6.20) xe x − X m ≥ B m x m m ! , e ℓx xe x − X m ≥ B m ( ℓ ) x m m ! . Thus, the first term in (6.16) is seen to represent the cohomology class(6.21) ( π n +1 ) ∗ (cid:18) e ℓψ n +1 ψ n +1 e ψ n +1 − (cid:19) = X m ≥ B m ( ℓ ) κ m − m ! . On the moduli space, theorem 1 therefore gives the following local formula(in the sense of orbifolds).
Corollary 6.7.
In the sense of orbifolds, the Chern character of the index ofthe family ∂ ℓ associated to the forgetful map π n +1 M g,n +1 → M g,n is given OCAL FAMILIES INDEX FOR ∂ -OPERATORS 27 at the form level by (6.22) Ch(ker D + ℓ ) − Ch(ker D − ℓ ) = X m ≥ B m ( ℓ ) m ! k m − + n (cid:18) − ℓ (cid:19) − n X i =1
12 tanh (cid:0) e i (cid:1) − e i ! − (cid:18) π √− (cid:19) N d Z ∞ Str ∂ A tD ℓ ∂t e − ( A tDℓ ) ! dt, where k m = ( π n +1 ) ∗ ( e m +1 n +1 ) and e i are canonical form representatives of theMorita-Mumford-Miller classes κ m and ψ i . If M g,n denote the Deligne-Mumford compactification of the moduli space M g,n , then theorem 1 can be intuitively interpreted as a local version ofthe Grothendieck-Riemann-Roch theorem applied to the morphism π n +1 : M g,n +1 → M g,n and the sheaf(6.23) e ω ℓ := (cid:26) ω π n +1 ( D ) ℓ − ⊗ ω π n +1 , ℓ > ,ω π n +1 ( D ) ℓ , ℓ ≤ . In this context, the Grothendieck-Riemann-Roch theorem was first studiedand used by Mumford [27] in the case n = 0 with formula given byCh(( π ) ∗ ω ℓπ ) = X m ≥ B m ( ℓ ) m ! κ m − + (terms coming from ∂ M g ) . When n >
0, a Grothendieck-Riemann-Roch formula was obtained for thesheaf ω ℓπ n +1 by Bini [6],(6.24)Ch(( π n +1 ) ∗ ω ℓπ n +1 ) = X m ≥ B m ( ℓ ) m ! e κ m − + (terms coming from ∂ M g,n ) . where e κ m := ( π n +1 ) ∗ (cid:0) c ( ω π n +1 ) m +1 (cid:1) . When ℓ = 0 and ℓ = 1, it makes senseto compare our formula with the one of Bini. In that case, using the relation κ m = e κ m + n X i =1 ψ mi proved by Arbarello and Cornalba [4] together with the identity x x = xe x − x , we can easily check that, as expected, our formula agrees with the interiorcontribution of (6.24).7. The spectral hc -zeta determinant On any geometrically finite hyperbolic surface Σ = Γ \ H , the Selberg’s zeta function is defined for Re ( s ) > Z Σ ( s ) = Y { γ } ∞ Y k =0 (cid:16) − e − ( s + k ) ℓ ( γ ) (cid:17) where the outer product goes over conjugacy classes of primitive hyperbolic elements of Γ and ℓ ( γ ) is the length of the corresponding closed geodesic.On closed hyperbolic surfaces, a well-known result of D’Hoker and Phong[13] says that the determinant of the Laplacian ∆ Σ ,ℓ acting on sections of K ℓ can be expressed in terms of special values of the Selberg’s Zeta function,(7.2) det(∆ Σ ,ℓ ) = Z Σ ( ℓ ) e − c ℓ − χ (Σ) , ℓ ≥ , det ′ (∆ Σ ,ℓ ) = Z ′ Σ (1) e − c χ (Σ) , ℓ = 0 , . where(7.3) c ℓ := X ≤ m<ℓ − (2 ℓ − m −
1) log(2 ℓ − m ) − (cid:18) ℓ + 12 (cid:19) + (cid:18) ℓ + 12 (cid:19) log 2 π + 2 ζ Riem ( − . Shortly after, it was shown by Sarnak [29] that for the geometric Laplacianwith non-negative spectrum ∆ Σ ,(7.4) det (∆ Σ + s ( s − Z Σ ( s ) = e E − s ( s − Γ ( s ) Γ ( s ) (2 π ) s ! − χ (Σ) where E = − − log 2 π + 2 ζ ′ Riem ( − is the Barnes double Gammafunction. As indicated in [29], the formula of D’Hoker and Phong can berecovered relatively easily from (7.4).On a Riemann surface with cusps, the Selberg Zeta function as definedabove still makes sense. However, since the Laplacian has a continuousspectrum, the definition of its determinant is more subtle. It was studiedby Efrat [14], [15] and by M¨uller [26] using scattering theory to under-stand the contribution from the continuous spectrum. In this paper, we userenormalized integrals to extend the usual definition of the determinant viazeta-regularization to these manifolds, with the advantage that this does notrequire the metric to have constant curvature. We then use the analysis of[12] to show that, on hyperbolic surfaces, our definition satisfies (7.4) withthe right-hand-side replaced by a meromorphic function depending only onthe genus and the number of punctures, an important feature for our pur-poses.7.1. The determinant of the Laplacian.
To relate the determinant with the Selberg Zeta function and get ananalog of formula (7.4), it is convenient to work first with the (positive)
OCAL FAMILIES INDEX FOR ∂ -OPERATORS 29 geometric Laplacian ∆ Σ instead of the ∂ -Laplacian. Recall that the two arethe same modulo a multiplicative constant,∆ ∂ = 12 ∆ Σ . Following [22, § § Σ usingthe renormalized trace (see e.g., [2]) ζ ∆ Σ ( z ) := 1Γ( z ) Z ∞ t z R Tr (cid:0) e − t ∆ Σ − P ker ∆ Σ (cid:1) dtt . Since ∆ Σ is Fredholm, zero is spectrally isolated and the integrand decaysexponentially for large times. Thus the integral defines a holomorphic func-tion for Re( z ) large enough. The small-times asymptotics of the integrand(whose existence follows from the construction of the heat kernel in [31]) al-low us to extend the function meromorphically to the whole complex plane.We denote the meromorphic extension by the same symbol and definelog det ∆ Σ := − ζ ′ ∆ Σ (0) . We can find a more explicit expression for the zeta function by subtractingthe first few terms in the expansion of the heat kernel at t = 0. The form ofthis expansion can be deduced for arbitrary φ -hc operators of Laplace-typefrom Vaillant’s construction (see the appendix of [3] for such an approach),but for the case at hand the expansion is well-known (see, e.g., (2.3) in [26])(7.5) R Tr( e − t ∆ Σ ) = a − t + e a − log t √ t + a − √ t + a + O ( √ t ) as t → + . Thus, writing f ( t ) = a − t − + e a − log t √ t + a − t − + a and choosing any C >
0, we have the expression ζ ∆ Σ ( z ) = z ) R C t z (cid:0) R Tr (cid:0) e − t ∆ Σ (cid:1) − f ( t ) (cid:1) dtt + z ) R ∞ C t z (cid:0) R Tr (cid:0) e − t ∆ Σ (cid:1) − dim ker − (∆ Σ ) (cid:1) dtt + C z Γ( z +1) ( a − dim ker − (∆ Σ )) + C z − a − ( z − ) Γ( z ) + e a − C z − Γ( z ) (cid:16) log Cz − − z − ) (cid:17) + C z − a − ( z − z ) . Differentiating and setting z = 0, we get ζ ′ ∆ Σ (0) = Z C (cid:0) R Tr (cid:0) e − t ∆ Σ (cid:1) − f ( t ) (cid:1) dtt + Z ∞ C (cid:0) R Tr (cid:0) e − t ∆ Σ (cid:1) − dim ker − (∆ Σ ) (cid:1) dtt + (log C + γ e ) ( a − dim ker − (∆ Σ )) − C − a − − C − a − + e a − C − ( − − C )(7.6) by using the fact that z ) ⇂ z =0 = 0, ∂ z ⇂ z =0 1Γ( z ) = 1 and ∂ z ⇂ z =0 1Γ( z +1) = γ e is Euler’s gamma constant.More generally, and to connect with (7.4), we can consider the determi-nant of ∆ Σ with its spectrum shifted by a complex number w , that is, thedeterminant of ∆ Σ + w . Just as before we have ζ ∆ Σ ( z ; w ) = 1Γ ( z ) Z ∞ t z (cid:0) R Tr (cid:0) e − t ∆ Σ (cid:1) − f ( t ) (cid:1) e − tw dtt + a w z + e a − w z − (cid:0) Γ log (cid:0) z − (cid:1) − log w Γ (cid:0) z − (cid:1)(cid:1) Γ( z )+ a − w z − Γ (cid:0) z − (cid:1) Γ ( z ) + a − w z − ( z − − (7.7)where the function Γ log ( z ) is defined to be(7.8) Γ log ( z ) := Z ∞ t z e − t log t dtt for Re z >
0. Since it satisfies the recurrence relationΓ log ( z + 1) = z Γ log ( z ) + Γ( z ) , it has a meromorphic continuation to the whole complex plane with polesat − N = 0 , − , − . . . . In particular, it has no pole at z = − . Taking thederivative of ζ ∆ Σ ( z ; w ) with respect to z and setting z = 0, we get ζ ′ ∆ Σ (0; w ) = − log det (∆ Σ + w )= Z ∞ (cid:0) R Tr (cid:0) e − t ∆ Σ (cid:1) − f ( t ) (cid:1) e − tw dtt − a log w − √ πa − √ w + a − w ( − w )+ e a − √ w (cid:18) Γ log ( −
12 ) − log w Γ( −
12 ) (cid:19) . (7.9)7.2. Relation with the Selberg Zeta function.
To relate the determinant with the Selberg Zeta function, we will follow [12]and use a description of the Selberg Zeta function in terms of the resolventof the Laplacian. Given a hyperbolic surface Σ of genus g with n cusps, wedenote by(7.10) R Σ ( s ) := (∆ Σ + s ( s − − the resolvent of the geometric Laplacian ∆ Σ with respect to the hyperbolicmetric. The Schwartz kernel of R Σ ( s ) is singular along the diagonal. Anatural way to remove this singular part is to subtract the resolvent of themodel hyperbolic space(7.11) R H ( s ) = (∆ H + s ( s − − . This resolvent has Schwartz kernel defined on H × H . Hence, thinking ofΣ as the quotient Γ \ H of the hyperbolic half-plane by some appropriate OCAL FAMILIES INDEX FOR ∂ -OPERATORS 31 discrete subgroup Γ ⊂ SL(2 , R ), there is a natural lift of the Schwartz kernel G Σ ( s ; z, w ) of R Σ ( s ) to H × H . Since locally R Σ ( s ) and R H ( s ) have the samefull symbol (being a parametrix for ∆ H + s ( s − ϕ Σ ( s ; z ) := (2 s −
1) [ G Σ ( s ; z, w ) − G H ( s ; z, w )] w = z will be smooth in z ∈ H and meromorphic in s . Because of the SL(2 , R )invariance of ∆ H , the Schwartz kernel G H will also be SL(2 , R ) invariantwhen restricted to the diagonal in H × H . This means in particular that thefunction ϕ Σ ( s ; z ) will be Γ-invariant and so will descend to give a functionon Σ = Γ \ H . Thus, we can define the function(7.13) φ Σ ( s ) := R Z Σ ϕ Σ ( s ; z ) dg Σ ( z )where dg Σ is the volume form associated to the hyperbolic metric, and theintegral is renormalized using ρ Σ , the boundary defining function of (2.10).This function has a meromorphic continuation to the complex plane withpossible poles at − N /
2. As a particular example, we can consider theHorn H := Γ ∞ \ H of (2.4). The end obtained as y → + ∞ is a cusp end, sowe can pick a boundary defining function as usual there, but the other endwhen y → + is not a cusp end, but a funnel, and for that end, one shouldtake a boundary defining function which is given by y near y = 0. With thischoice, we can make sense of the function φ H ( s ). The following propositionis due to Borthwick, Judge, and Perry [12]. Proposition 7.1 ([12], Proposition 4.3) . Let Σ be a Riemann surface ofgenus g and with n cusps. Then Z ′ Σ ( s ) Z Σ ( s ) = φ Σ ( s ) − nφ H ( s ) . The function φ H ( s ) can be computed explicitly [12, Proposition 2.4],(7.14) φ H ( s ) = − log 2 − Ψ (cid:18) s + 12 (cid:19) + 12 s − z ) is the digamma function Γ ′ ( z )Γ( z ) . Thus, to relate the logarithmicderivative of Z Σ ( s ) with the determinant we need to understand the function φ Σ ( s ) in terms of the heat kernel instead of the resolvent. Lemma 7.2.
In the sense of distributions, we have that (∆ Σ + s ) − = Z ∞ K Σ ( t ) e − st dt, for Re s > . Proof.
Let f ∈ C ∞ c (Σ) be a test function. Then by definition of the heatkernel, we have that ∂ t K Σ ( t ) f + ∆ Σ K Σ ( t ) f = 0 , K Σ (0) f = f. Using integration by part, this implies that(7.15) ∆ Σ (cid:18)Z ∞ K Σ ( t ) f e − st dt (cid:19) = Z −∞ − ∂ t K Σ ( t ) f e − st dt = − K Σ ( t ) f e − st (cid:12)(cid:12) ∞ − s Z ∞ K Σ ( t ) f e − st dt = f − s Z ∞ K Σ ( t ) f e − st dt, which shows that (∆ Σ + s ) (cid:18)Z ∞ K Σ ( t ) f e − st dt (cid:19) = f. This means that f Z ∞ K Σ ( t ) f e − st dt is a right inverse for (∆ Σ + s ). The same computation shows that it is a leftinverse since K Σ ( t )∆ Σ f = ∆ Σ K Σ ( t ) f by uniqueness of the solution for the heat equation. (cid:3) From this lemma we conclude that(7.16) ϕ Σ ( s ; z )2 s − Z ∞ ( K Σ ( t, z, z ) − K H ( t ; z, z )) e − s ( s − t dt. Since the left-hand side is smooth, the right-hand side is smooth as well,which means that K Σ ( t ; z, z ) and K H ( t ; z, z ) have the same term of order t − in their asymptotic expansions as t ց
0. Integrating (7.16) in z andtaking the finite part, we get(7.17) φ Σ ( s )2 s − R Z Σ Z ∞ ( K Σ ( t ; z, z ) − K H ( t ; z, z )) e − ts ( s − dtdg Σ ( z ) . The order of integration can be interchanged, since for Re( w ) ≫ Z Σ Z ∞ x w ( K Σ ( t ; z, z ) − K H ( t ; z, z )) e − ts ( s − dtdg Σ ( z ) = Z ∞ Z Σ x w ( K Σ ( t ; z, z ) − K H ( t ; z, z )) e − ts ( s − dtdg Σ ( z ) . Hence, we get that(7.19) φ Σ ( s )2 s − Z ∞ (cid:0) R Tr( K Σ ( t )) − R Tr Σ ( K H ( t )) (cid:1) e − ts ( s − dt OCAL FAMILIES INDEX FOR ∂ -OPERATORS 33 where R Tr Σ ( K H ( t )) := FP Z Σ K H ( t ; z, z ) dg Σ ( z ) . We recall that K H ( t ) itself does not descend to Σ × Σ, but its restriction tothe diagonal in H × H does descend to the diagonal in Σ × Σ. Indeed, it iswell-known that because the hyperbolic metric on H is SL(2 , R )-invariant, K H ( t ; z, z ) is constant in z , say equal to k H ( t ) dg H ( z ) for some function of t .Since the surfaces we are studying have finite area, we have R Tr( K H ( t )) = k H ( t ) area(Σ) . Corollary 7.3.
The function k H ( t ) has an asymptotic expansion given by k H ( t ) ∼ k − t − + o ( t − ) as t ց . Therefore, if Σ is a Riemann surface of genus g with n cusps, then theregularized trace of its heat kernel has the asymptotic expansion R Tr( K Σ ( t )) ∼ k − area(Σ) t − + O ( t − log t ) as t ց . Proof.
Consider the case where Σ has no cusp. Then it is well-known that R Tr( K Σ ( t )) = Tr( K Σ ( t )) has asymptotic expansion of the formTr( K Σ ( t )) ∼ αt − + β + O ( t ) as t ց , where α and β are some constants. By formula (7.19), R Tr( K H ( t )) has thesame term of order t − as t ց
0, hence we get that k − := α area(Σ) = 14 π is such that k H ( t ) ∼ k − t − + o ( t − ) as t ց . When we consider a surface with cusps, R Tr( K Σ ) will have the same termof order t − as R Tr Σ ( K H ( t )) as t ց
0, hence R Tr( K Σ ( t )) ∼ k − area(Σ) t − + O ( t − log t ) , as t ց . (cid:3) Consider the functional(7.20) D Σ ( s ) := det (∆ Σ + s ( s − (cid:0) − ζ ′ ∆ Σ (0; s ( s − (cid:1) . If we differentiate with respect s and use formula (7.9), we find that12 s − D ′ Σ ( s ) D Σ ( s ) = Z ∞ (cid:0) R Tr (cid:0) e − t ∆ Σ (cid:1) − a − t − (cid:1) e − ts ( s − dt − a − log ( s ( s − . (7.21) Combining formula (7.21) and (7.19) and using corollary 7.3, we get that(7.22) 12 s − (cid:18) D ′ Σ ( s ) D Σ ( s ) − Z ′ Σ ( s ) Z Σ ( s ) (cid:19) = Z ∞ (cid:0) R Tr( K Σ ( t )) − a − t − − R Tr( K Σ ( t )) + R Tr Σ ( K H ( t )) (cid:1) e − ts ( s − dt − k − area(Σ) log ( s ( s − n s − φ H ( s )= Z ∞ (cid:0) R Tr( K H ( t )) − k − area(Σ) t − (cid:1) e − ts ( s − dt − k − area(Σ) log ( s ( s − n s − φ H ( s ) , so that(7.23) 12 s − (cid:18) D ′ Σ ( s ) D Σ ( s ) − Z ′ Σ ( s ) Z Σ ( s ) (cid:19) = n s − φ H ( s )+ area(Σ) (cid:20)Z ∞ (cid:0) k H ( t ) − k − t − (cid:1) e − ts ( s − dt − k − log ( s ( s − (cid:21) . In particular, for fixed g and n , we see that the right hand side does notdepend on the Σ since the area is given by − πχ (Σ) by the Gauss-Bonnettheorem, a quantity that only depends on g and n .From (7.14), we see that(7.24) φ H ( s ) = dds (cid:18) − s log 2 − log Γ( s + 12 ) + 12 log(2 s − (cid:19) . Then, according to (7.23) and (7.4), there exists a constant C such that(7.25) D Σ ( s ) Z Σ ( s ) = C (cid:18) e E − s ( s −
1) Γ ( s ) Γ( s ) (2 π ) s (cid:19) − χ (Σ) √ s − s Γ( s + ) ! n where E := − − log 2 π + 2 ζ ′ ( − C can bedetermined by the asymptotic expansion of the logarithm of both sides of(7.25) as s approaches infinity. For the left side, it is clear from (7.1) thatlog Z Σ ( s ) has a trivial asymptotic expansion as s → + ∞ . Thus, from (7.9),we conclude that the asymptotic behavior of the logarithm of the left sideis given bylog (cid:18) D Σ ( s ) Z Σ ( s ) (cid:19) = a log s ( s −
1) + 2 √ πa − p s ( s − − e a − p s ( s − (cid:18) Γ log ( −
12 ) + 2 √ π log( s ( s − (cid:19) − a − s ( s −
1) ( − s ( s − o (1)(7.26) OCAL FAMILIES INDEX FOR ∂ -OPERATORS 35 as s → + ∞ . On the other hand, if we set(7.27) Z cu ( s ) = √ s − s Γ( s + ) = √ s q s − Γ( s − ) , we see using Stirling’s formula that its logarithm has the following asymp-totic behavior,(7.28)log ( Z cu ( s )) = −
12 log(2 π )+(1 − log 2) (cid:18) s − (cid:19) − (cid:18) s − (cid:19) log (cid:18) s − (cid:19) + o (1)as s → + ∞ . Since p s ( s −
1) = s −
12 + o (1) , (7.29) p s ( s −
1) log ( s ( s − (cid:18) s − (cid:19) log (cid:18) s − (cid:19) + o (1) , (7.30)as s → + ∞ , we can rewrite (7.26) aslog (cid:18) D Σ ( s ) Z Σ ( s ) (cid:19) = a log s ( s −
1) + (cid:18) √ πa − − e a − Γ log ( −
12 ) (cid:19) (cid:18) s − (cid:19) − √ π e a − (cid:18) s − (cid:19) log (cid:18) s − (cid:19) − a − s ( s −
1) ( − s ( s − o (1)(7.31)as s → + ∞ . Now, from [29] , we have also that(7.32) log (cid:18) e E − s ( s − Γ ( s ) Γ( s ) (2 π ) s (cid:19) = −
16 log s ( s −
1) + 12 s ( s − − s ( s − s ( s −
1) + o (1)as s → + ∞ . This asymptotic behavior only involves terms of the formlog ( s ( s − s ( s −
1) log ( s ( s − a − and e a − counterbalance the asymptotic behavior of (7.28) whilethe terms involving a − and a counterbalance the asymptotic behavior of(7.32). Comparing (7.31) with (7.32), we find(7.33) a − = g − − χ (Σ)2 , a = χ (Σ)6 . Comparing (7.31) with (7.28), we also get(7.34) e a − = n √ π , a − = n √ π − log 2 + Γ log ( − )4 √ π ! . In (2 .
19) of [29], the coefficient of log s ( s −
1) is , but it is supposed to be Now, recall that in [29], the constant E is chosen so that(7.35) log (cid:18) e E − s ( s − Γ ( s ) Γ( s ) (2 π ) s (cid:19) = − a − s ( s −
1) ( − s ( s − a log ( s ( s − . This means the constant C has to be chosen to compensate the constantterm of (7.28), that is,(7.36) log C = − n π = ⇒ C = (2 π ) − n . This gives the following result.
Theorem 2.
For a Riemann surface of genus g with n cusps satisfying g − n > and equipped with the hyperbolic metric, we have det (∆ Σ + s ( s − Z Σ ( s ) (cid:16) e E − s ( s −
1) Γ ( s ) Γ( s ) (2 π ) s (cid:17) − χ (Σ) (cid:16) s q π ( s − )Γ( s − ) (cid:17) n . As a consequence, we see that the ratiodet (∆ Σ + s ( s − Z Σ ( s )is a meromorphic function in s which only depends on the genus g andthe number of cusps n . This means that, up to a multiplicative constantdepending only on g and n , the determinant of ∆ Σ is given by Z ′ Σ (1). Theformula of theorem 2 can also be expressed in terms of the ∂ -Laplacian∆ ∂ = 12 ∆ Σ . For the heat kernel of the ∂ -Lapacian, we have the following short timeasymptotic expansion, R Tr (cid:0) e − t ∆ ∂ (cid:1) = R Tr (cid:16) e − t ∆ Σ (cid:17) = 2 a − t + √ e a − √ t log t + √ a − − √ e a − log 2 √ t + a + O ( √ t )(7.37)as t → + , where a − , e a − , a − , a are the coefficients in (7.5). From formula(7.7), we conclude that ζ ∆ ∂ (cid:18) z ; s ( s − (cid:19) = 2 z ζ ∆ Σ ( z ; s ( s − . Hence, ζ ′ ∆ ∂ (cid:18) s ( s − (cid:19) = (log 2) ζ ∆ Σ (0; s ( s − ζ ′ Σ (0; s ( s − , OCAL FAMILIES INDEX FOR ∂ -OPERATORS 37 which means thatdet (cid:18) ∆ ∂ + s ( s − (cid:19) = 2 − ζ ∆Σ (0; s ( s − det (∆ Σ + s ( s − . Now, ζ ∆ Σ (0; s ( s − ζ ∆ Σ (0; s ( s − a − a − s ( s − χ (Σ) (cid:18)
16 + s ( s − (cid:19) . (7.38)From theorem 2, we get the following formula. Corollary 7.4.
For a Riemann surface Σ of genus g with n cusps satisfying g − n > and equipped with the hyperbolic metric, we have det (cid:18) ∆ ∂ + s ( s − (cid:19) = Z Σ ( s ) (cid:16) + s ( s − e E − s ( s −
1) Γ ( s ) Γ( s ) (2 π ) s (cid:17) − χ (Σ) (cid:16) s q π ( s − )Γ( s − ) (cid:17) n . The determinant of ∆ ℓ for ℓ ≥ . As indicated in [29], it is possible to express the determinant of ∆ ℓ interms of Selberg Zeta function by using corollary 7.4. This is because thespectrum of ∆ ℓ is essentially given by a shifted version of the spectrum of∆ = ∂ ∗ ∂ .Recall first that these various Laplacians are related by the recurrencerelation (see for instance (1.3) in [30])(7.39) ∆ ℓ ∂ ∗ ℓ u = ∂ ∗ ℓ u (∆ ℓ − + ℓ − u := y is seen as a section of Λ , ⊗ Λ , on Σ. Taking the formaladjoint of (7.39), we get(7.40) u ∗ ∂ ℓ ∆ ℓ = (∆ ℓ − + ℓ − u ∗ ∂ ℓ where u ∗ is the conjugate u of u seen as a section of (Λ , ) − ⊗ (Λ , ) − .Since the operator ∂ ℓ is Fredholm, it has a well-defined parametrix ∂ − ℓ :(ker ∂ ∗ ℓ ) ⊥ → (ker ∂ ℓ ) ⊥ . Applying this parametrix to both sides of (7.40), weget(7.41) ∆ ℓ = (cid:0) u ∗ ∂ ℓ (cid:1) − (∆ ℓ − + ℓ − (cid:0) u ∗ ∂ ℓ (cid:1) . In the compact case, this directly implies that(7.42) det ′ (∆ ℓ ) = det(∆ ℓ − + ℓ − ℓ ≥ ∂ ℓ is surjective in that case. When ℓ = 1, the operator ∂ isnot surjective, but the cokernel of u ∗ ∂ ℓ ,ker ∂ ∗ u = ker ∂ = ker ∆ We have ℓ − ℓ − in [30] since we use the convention | dz | = 2. is precisely the kernel of ∆ , so that we have in that casedet ′ (∆ ) = det ′ (∆ ) . Using (7.41) once more and (3.25), we have on the other hand that for k > ℓ − + k ) = (cid:26) ( k ) g − det (∆ + k ) , ℓ = 2;( k ) (2 ℓ − g − det (∆ ℓ − + k + ℓ − , ℓ ≥ . Applying this recursively, we get(7.44) det ′ (∆ ℓ ) = det ′ (∆ ) , ℓ = 1;det (∆ + 1) , ℓ = 2; δ ℓ,g det (∆ + ℓ ( ℓ − , ℓ ≥ . where δ ℓ,g is a number depending only on ℓ and g . In the non-compact case,one has to be more careful since the regularized trace does not necessarilyvanish on a commutator. Taking this into account, the analog of (7.41) inthe non-compact case is(7.45) det ′ (∆ ℓ ) = D ℓ,n det (∆ ℓ − + ℓ − − log ( D ℓ,n ) = (cid:18) ddz z ) Z ∞ t zR Tr (cid:0) [( u ∗ ∂ ℓ ) − e − t ∆ ℓ − , u ∗ ∂ ℓ ] (cid:1) e − t ( ℓ − dtt (cid:19) z =0 regularizing as in (7.6). Although the term D ℓ,n might be hard to compute,what is clear is that it only depends on ℓ and the number n of cusps. Thisis because the regularized trace of a commutator [ A, B ] ‘localizes’ near theboundary in the sense that it only depends on the Taylor expansion of theintegral kernels at the boundary of the diagonal. Recall that to constructthe heat kernel (see [31, 2]) we start with a ‘parametrix’ for the heat equa-tion which solves a model equation at the cusp. The solution of this modelequation is then used iteratively to construct the Taylor expansion of theheat kernel as we approach the cusp, before finally solving away the remain-ing error in the interior. The upshot is that, since all cusps have isometricneighborhoods, the term D ℓ,n only depends on ℓ and n as required.Thus using recursively (7.45) and applying corollary 7.4, we get the fol-lowing. Corollary 7.5.
For a Riemann surface Σ of genus g ≥ with n cusp, wehave det ′ (∆ ℓ ) = (cid:26) α ℓ,g,n Z Σ ( ℓ ) , ℓ ≥ α ℓ,g,n Z ′ Σ (1) , ℓ = 0 , where each constant α ℓ,g,n > only depends on ℓ , g and n . A similarstatement holds of the determinant of D − ℓ D + ℓ = 2∆ ℓ . OCAL FAMILIES INDEX FOR ∂ -OPERATORS 39 The curvature of the Quillen connection
Recall that the determinant bundle of the family of operator ∂ ℓ is by defi-nition(8.1) λ ℓ := det ind ∂ ℓ = Λ max ker ∂ ℓ ⊗ (cid:0) Λ max coker ∂ ℓ (cid:1) − where ℓ ∈ Z and Λ max denotes the maximal exterior power of a vector space.The definition is particularly simple in this case because ker ∂ l is a vectorbundle over T g,n . The L -norm on ker ∂ ℓ defines a canonical metric on λ ℓ ,the L -metric, denoted k · k . An alternative metric which is more interestinggeometrically is the Quillen metric ,(8.2) k · k Q := (cid:0) det D − ℓ D + ℓ (cid:1) − k · k . Following the discussion of § k·k Q a compatibleconnection called the Quillen connection . In order to do that, considerover T g,n the Z -graded bundle(8.3) E ℓ = E + ℓ ⊕ E − ℓ , E + ℓ := Λ ℓ, ( T g,n /T g,n ) , E − ℓ := Λ ℓ, ( T g,n /T g,n ) . Let also π ∗ E ℓ → T g,n be the Fr´echet bundle whose fiber at [Σ] ∈ T g,n is(8.4) π ∗ E ℓ, [Σ] := ˙ C ∞ (cid:16) Σ; E l | Σ ⊗ | Λ Σ | (cid:17) , where | Λ Σ | is the density bundle on Σ and ˙ C ∞ (cid:16) Σ; E l | Σ ⊗ | Λ Σ | (cid:17) is the spaceof smooth sections of E l | Σ ⊗ | Λ Σ | with rapid decay at infinity. The familyof Dirac type operators(8.5) D ℓ := √ (cid:16) ∂ ℓ + ∂ ∗ ℓ (cid:17) , D + ℓ = √ ∂ ℓ , D − ℓ = √ ∂ ∗ ℓ , acts from π ∗ E ℓ to π ∗ E ℓ . One of the reasons that motivates the introductionof the fibre density bundle in the definition of π ∗ E is that in this way thecanonical connection on π : T g,n → T g,n induces a connection on π ∗ E ℓ ,denoted ∇ π ∗ E ℓ , which is automatically compatible with the metric of π ∗ E ℓ (cf. proposition 9.13 in [5]). Notice also that the density bundle | Λ Σ | iscanonically trivialized by the section | dg Σ | so that D ℓ acts on π ∗ E ℓ in anatural way. To the family of Dirac type operators D ℓ , we can associate asuperconnection(8.6) A ℓ := D ℓ + ∇ π ∗ E ℓ and its rescaled version(8.7) A sℓ := s D ℓ + ∇ π ∗ E ℓ . For s ∈ R + , we can define two differential forms α ± ℓ ∈ A ( T g,n )( s ), α ± ℓ ( s ) := R Tr π ∗ E ± ℓ (cid:18) ∂ A sℓ ∂s e − ( A sℓ ) (cid:19) = 12 s R Tr π ∗ E ± ℓ (cid:16) D ℓ e − ( A sℓ ) (cid:17) , (8.8)by taking the trace with respect to E + ℓ and E − ℓ respectively. The 0-formcomponent of ( A ℓ ) s is sD ℓ , while its 1-form component is s [ ∇ π ∗ E l , D ℓ ]. Onthe other hand, the 1-form component of e − ( A sℓ ) is given by (cid:16) e − ( A sℓ ) (cid:17) [1] = ( − s ) Z e − (1 − σ ) sD ℓ s − [ ∇ π ∗ E ℓ , D ℓ ] e − σsD ℓ dσ = − s Z e − (1 − σ ) sD ℓ [ ∇ π ∗ E ℓ , D ℓ ] e − σsD ℓ dσ. (8.9)The following observation will turn out to be very useful. Lemma 8.1.
The Schwartz kernel of [ ∇ π ∗ E ℓ , D ± ℓ ] vanishes to all order atthe front face. In particular, for P ∈ Ψ −∞ ( T g,n /T g,n ; E ℓ ) , R STr (cid:0) [[ ∇ π ∗ E ℓ , D ± l ] , P ] (cid:1) = 0 Proof.
Let [Σ] ∈ T g,n be given. If µ ∈ Ω − , (Σ) is a harmonic Beltramidifferential, let f µ : H → H be the unique diffeomorphism satisfying theBeltrami equation ∂f µ ∂z = µ ∂f µ ∂z and fixing the points 0 , , ∞ . In particular, since µ is a cusp form, it de-creases rapidly as z → ∞ . This means that f µ is asymptotically holo-morphic as z → ∞ . From the definition of the canonical connection on π : T g,n → T g,n , this means that the Schwartz kernel [ ∇ π ∗ E ℓ , D + ℓ ]( z, z ′ ) de-creases quickly as z and z ′ approaches a cusp in Σ. For D − ℓ = √ ∂ ∗ ℓ , the sameis true, but since ∂ ∗ ℓ = − u ℓ − ∂u − ℓ , we also need to use the fact that u = y is parallel with respect to the canonical connection on π : T g,n → T g,n . Now,we know that R STr (cid:0) [[ ∇ π ∗ E ℓ , D ± ℓ ] , P ] (cid:1) depends linearly on the asymptoticexpansions of [ ∇ π ∗ E ℓ , D ± ℓ ] and P at the corner of Σ × Σ. The asymptopticexpansion of [ ∇ π ∗ E ℓ , D ± ℓ ] being trivial, the result follows.Alternatively, the result also follows directly from the explicit formulas(5.15). (cid:3) With this lemma, the discussion of § Lemma 8.2.
The one form component of the differential forms α + ℓ ( s ) sat-isfies α + ℓ ( s ) [1] = α − ℓ ( s ) [1]OCAL FAMILIES INDEX FOR ∂ -OPERATORS 41 and has an asymptotic expansion of the form α + ℓ ( s ) [1] ∼ ∞ X − N s k ( a k + b k log s ) as s → + .Proof. The asymptotic expansion as s → + follows from the constructionof the heat kernel by Vaillant [31], its generalization in [2] and an applicationof the pushforward theorem for manifolds with corners. From (8.9), we havethat α + ℓ ( s ) [1] = − R Tr π ∗ E + ℓ (cid:18) D ℓ Z e − (1 − σ ) sD ℓ [ ∇ π ∗ E ℓ , D ℓ ] e − σsD ℓ dσ (cid:19) = − R STr π ∗ E ℓ (cid:18) D − ℓ Z e − (1 − σ ) sD ℓ [ ∇ π ∗ E l , D + ℓ ] e − σsD ℓ dσ (cid:19) . (8.10)Taking the complex conjugate and using the fact that ( D + ℓ ) ∗ = D − ℓ and that ∇ π ∗ E ℓ is a unitary connection, we have that(8.11) α + ℓ ( s ) [1] = − R Tr π ∗ E + ℓ (cid:18)Z e − σsD ℓ [ D − ℓ , ∇ π ∗ E ℓ ] e − (1 − σ ) sD ℓ D + ℓ dσ (cid:19) = 12 R STr π ∗ E ℓ (cid:18)Z e − (1 − σ ) sD ℓ [ D − ℓ , ∇ π ∗ E ℓ ] e − σsD ℓ D + ℓ dσ (cid:19) = 12 R STr π ∗ E ℓ (cid:18) D + ℓ Z e − (1 − σ ) sD ℓ [ D − ℓ , ∇ π ∗ E ℓ ] e − σsD ℓ dσ (cid:19) + 12 R STr π ∗ E ℓ (cid:18)(cid:20)Z e − (1 − σ ) sD ℓ [ D − ℓ , ∇ π ∗ E ℓ ] e − σsD ℓ dσ, D + ℓ (cid:21)(cid:19) = − R Tr π ∗ E − ℓ (cid:18) D + ℓ Z e − (1 − σ ) sD ℓ [ D − ℓ , ∇ π ∗ E ℓ ] e − σsD ℓ dσ (cid:19) + 0= α − ℓ ( s ) [1] , where Lemma 8.1 was used in the line before the last one. (cid:3) We would like to consider the one-forms(8.12) β ± ℓ ( z ) := 2 Z ∞ t z α ± ℓ ( t ) [1] dt. By Lemma 8.2, this integral is holomorphic for Re z >> β + ℓ ( z ) and β − ℓ ( z ) are well-defined meromorphic families of one-forms. Weare interested in their finite part at z = 0. More precisely, we would like toconsider the one-forms(8.13) β ± ℓ := ddz z ) β ± ℓ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z =02 PIERRE ALBIN AND FR´ED´ERIC ROCHON where the evaluation at zero means that we take the finite part at z = 0.More generally, we will use the notation Z ∞ γdt := (cid:18) ddz z ) Z ∞ t z γ ( t ) dt (cid:19) z =0 whenever the integral R ∞ t z − γ ( t ) dt varies meromorphically in z . Thus, inthis notation, β ± l = 2 Z ∞ α ± ℓ ( t ) [1] dt. Lemma 8.3.
Seen as a function on T g,n , the differential of ζ ′ (0; D − ℓ D + ℓ ) isgiven by dζ ′ (0; D − ℓ D + ℓ ) = − (cid:0) β + ℓ + β − ℓ (cid:1) .Proof. Using Duhamel’s formula, we have that dζ ′ (0; D − ℓ D + ℓ ) is given by(8.14) Z ∞ R Tr π ∗ E + ℓ (cid:18) − t Z t e − ( t − s ) D − ℓ D + ℓ [ ∇ π ∗ E ℓ , D − ℓ D + ℓ ] e − sD − ℓ D + ℓ ds (cid:19) dt = Z ∞ R Tr π ∗ E + ℓ (cid:18) − Z e − (1 − s ) tD − ℓ D + ℓ [ ∇ π ∗ E ℓ , D − ℓ D + ℓ ] e − stD − ℓ D + ℓ ds (cid:19) dt. On the other hand, we have β + ℓ = 2 Z ∞ α + ℓ ( t ) [1] dt = − Z ∞ R Tr π ∗ E + ℓ (cid:18) D − ℓ Z e − (1 − s ) tD ℓ [ ∇ π ∗ E ℓ , D + ℓ ] e − stD ℓ ds (cid:19) dt = − Z ∞ R Tr π ∗ E + ℓ (cid:18)Z e − (1 − s ) tD ℓ D − ℓ [ ∇ π ∗ E ℓ , D + ℓ ] e − stD ℓ ds (cid:19) dt, (8.15)while β − ℓ = − Z ∞ R Tr π ∗ E − ℓ (cid:18) D + ℓ Z e − (1 − s ) tD ℓ [ ∇ π ∗ E ℓ , D − ℓ ] e − stD ℓ ds (cid:19) dt = Z ∞ R STr π ∗ E ℓ (cid:18) D + ℓ Z e − (1 − s ) tD ℓ [ ∇ π ∗ E ℓ , D − ℓ ] e − stD ℓ ds (cid:19) dt = Z ∞ R STr π ∗ E ℓ (cid:18)Z e − (1 − s ) tD ℓ [ ∇ π ∗ E ℓ , D − ℓ ] D + ℓ e − stD ℓ ds (cid:19) dt + Z ∞ R STr π ∗ E ℓ (cid:18)(cid:20) D + ℓ , Z e − (1 − s ) tD ℓ [ ∇ π ∗ E ℓ , D − ℓ ] e − stD ℓ ds (cid:21)(cid:19) dt = Z ∞ R STr π ∗ E ℓ (cid:18)Z e − (1 − s ) tD ℓ [ ∇ π ∗ E ℓ , D − ℓ ] D + ℓ e − stD ℓ ds (cid:19) dt + 0 , (8.16)using Lemma 8.1 in the last step. The result then follows by combining(8.14), (8.15) and (8.16) and using the formula[ ∇ π ∗ E ℓ , D − ℓ D + ℓ ] = [ ∇ π ∗ E ℓ , D − ℓ ] D + ℓ − D − ℓ [ ∇ π ∗ E ℓ , D + ℓ ] . OCAL FAMILIES INDEX FOR ∂ -OPERATORS 43 (cid:3) If P : π ∗ E + ℓ → ker ∂ ℓ denotes the orthogonal projection onto the kernelof ∂ ℓ , then the connection(8.17) ∇ ker ∂ ℓ = P ∇ π ∗ E + ℓ P is compatible with the L -metric. It is holomorphic, so that ∇ ker ∂ ℓ is theChern connection of ker ∂ ℓ with respect to the L -metric k · k . It defines aconnection on det ∂ ℓ , ∇ det ∂ ℓ , which is the Chern connection of det ∂ ℓ withrespect to the L -metric. We define the Quillen connection on det ∂ ℓ tobe the connection given by(8.18) ∇ Q ℓ := ∇ det ∂ ℓ + β + ℓ . Proposition 8.4.
The Quillen connection is the Chern connection of det ∂ ℓ with respect to the Quillen metric k · k Q ℓ .Proof. We need to check that ∇ Q ℓ is holomorphic and is compatible withthe Quillen metric. To see that it is holomorphic, it suffices to check that β + ℓ is a (1 , D + ℓ = √ ∂ ℓ is a family of operators that variesholomorphically on T g,n , the form[ ∇ π ∗ E ℓ , D + ℓ ]has to be a (1 , β + ℓ ,we thus see it has to be a (1 , ∇ Q ℓ is compatible with the Quillen metric, notice that ingeneral, a connection which is compatible with the Quillen metric is of theform(8.19) ∇ det ∂ l − dζ ′ (cid:0) D − ℓ D + ℓ (cid:1) + ω where ω is any imaginary one-form. The result then follows by noticingthat, taking ω = β + ℓ − β + ℓ and using lemma 8.2 and lemma 8.3, we get theQuillen connection. (cid:3) We can now compute the curvature of the Quillen connection.
Theorem 3.
The curvature of the Quillen connection is given by √− π ( ∇ Q ℓ ) = Z T g,n /T g,n Ch (cid:16) T − ℓ ( T g,n /T g,n ) (cid:17) · Td ( T ( T g,n /T g,n )) ! [2] − n X i =1 e i Proof.
With respect to the connection ∇ det ∂ ℓ , we have(8.20) √− π (cid:16) ∇ det ∂ l (cid:17) = Ch (cid:16) ∇ ker ∂ l (cid:17) [2] . But by definition, since ∇ π ∗ E ℓ = A [1] for A the Bismut superconnection, (cf.Proposition 10.16 in [5]), we have by (8.17) that ∇ ker ∂ l is the connectionused in Theorem 1. Thus, Ch (cid:16) ∇ ker ∂ ℓ (cid:17) [2] is given by formula (6.16), so that √− π ( ∇ det ∂ ℓ ) = Z T g,n /T g,n Ch (cid:16) T − ℓ ( T g,n /T g,n ) (cid:17) · Td ( T ( T g,n /T g,n )) ! [2] − n X i =1 e i − π √− d Z ∞ R STr (cid:18) ∂ A t ∂t e − A t (cid:19) [1] dt. (8.21)On the other hand, from the definition of the Quillen connection, we have(8.22) (cid:0) ∇ Q ℓ (cid:1) = (cid:16) ∇ det ∂ l (cid:17) + dβ + ℓ . From lemma 8.3, d ( β + ℓ + β − ℓ ) = 0, hence(8.23) (cid:0) ∇ Q ℓ (cid:1) = (cid:16) ∇ det ∂ l (cid:17) + 12 d (cid:0) β + ℓ − β − ℓ (cid:1) . But using the fact the Bismut superconnection A is given by D ℓ + ∇ π ∗ E upto terms of degree 2, we have12 (cid:0) β + ℓ − β − ℓ (cid:1) = Z ∞ (cid:0) α + ℓ ( t ) [1] − α − ℓ ( t ) [1] (cid:1) dt = Z ∞ R STr (cid:18) ∂ A t ∂t e − A t (cid:19) [1] dt = Z ∞ R STr (cid:18) ∂ A t ∂t e − A t (cid:19) [1] dt. (8.24)In the last step, we have used the fact R STr (cid:16) ∂ A t ∂t e − A t (cid:17) is integrable in t ,so that there is no need to regularize. Combining (8.21), (8.23) and (8.24),the result follows. (cid:3) We should compare our result with the local index formula of Takhtajanand Zograf [30](8.25) ( ∇ Q ℓ ) = 6 ℓ − ℓ + 112 π ω W P − ω TZ , where ω W P is the Weil-Peterson K¨ahler form on T g,n and ω TZ is the K¨ahlerform on T g,n defined by Takhtajan and Zograf in terms of the cusp ends ofthe fibres of p : T g,n → T g,n . A well-known result of Wolpert [34] (see also OCAL FAMILIES INDEX FOR ∂ -OPERATORS 45 p. 424 in [30]) shows that(8.26) Z T g,n /T g,n Ch (cid:16) T − ℓ ( T g,n /T g,n ) (cid:17) · Td ( T ( T g,n /T g,n )) ! [2] =6 ℓ − ℓ + 112 π ω W P . Thus, comparing Theorem 3 with (8.25), we get the following relation.
Corollary 8.5 (Weng [32], Wolpert [35]) . For ℓ ≥ and n > , we have b η ( D Vℓ ) [2] = n X i =1 e i
12 = 19 ω TZ . The fact the Takhtajan-Zograf K¨ahler form is a rational multiple of thecurvature of a Hermitian line bundle was first obtained by Weng [32] usingArakelov theory. This was later improved and finalized by Wolpert [35], whoobtained more generally that e i = ω TZ ,i ( ω TZ ,i is defined in (8.31) below)via a natural intrinsic way to define metrics on the line bundles L i .For completeness, let us recall how the Takhtajan-Zograf K¨ahler form ω T Z is defined. Given a fibre Σ of p : T g,n → T g,n , identify it with a quotientof the upper half-plane, Σ ∼ = Γ \ H where Γ is the corresponding Fuchsiangroup of type ( g, n ). Let Γ , . . . , Γ n be the list of non-conjugate parabolicsubgroup of Γ as in (5.2) so that σ − i Γ i σ i = Γ ∞ for i ∈ { , . . . , n } . The Eisenstein-Mass series E i ( z, s ) associated to the i th cusp of the group Γ is defined for Re s > E i ( z, s ) := X γ ∈ Γ i \ Γ Im( σ − i γz ) s . The Eisenstein-Mass series naturally descends to define a function on thequotient Σ = Γ \ H . Recall that under the identification of T [Σ] T g,n with thespace of harmonic Beltrami differentials Ω − , (Σ), the Weil-Peterson K¨ahlermetric is defined by(8.28) h µ, ν i W P := Z Σ µνdg Σ = Z Σ h µ, ν i K − ⊗ Λ , dg Σ for µ, ν ∈ T [Σ] T g,n with corresponding K¨ahler form given by(8.29) ω W P ( µ, ν ) = √− h µ, ν i W P . To define their K¨ahler metric, Takhtajan and Zograf considered instead(8.30) h µ, ν i i = Z Σ µνE i ( · , dg Σ , i = 1 , . . . n. Each of these scalar products gives rise to a K¨ahler metric on T g,n withcorresponding K¨ahler form(8.31) ω TZ ,i ( µ, ν ) = √− h µ, ν i i , i = 1 , . . . , n. The sum of these metric is the Takhtajan-Zograf K¨ahler metric(8.32) h µ, ν i TZ := n X i =1 h µ, ν i i with corresponding K¨ahler form given by(8.33) ω TZ ( µ, ν ) = √− h µ, ν i TZ . We know from Corollary 8.5 that the eta form b η ( D Vℓ ) [2] is the K¨ahler formof a K¨ahler metric. This is consistent with Theorem 1 asserting that the etaform b η ( D Vℓ ) is closed. References [1] L. Ahlfors,
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