aa r X i v : . [ h e p - t h ] N ov Anomalies of E gauge theory on String manifolds Hisham Sati ∗ Department of MathematicsYale UniversityNew Haven, CT 06520USA
Abstract
In this note we revisit the subject of anomaly cancelation in string theory and M-theory on manifoldswith String structure and give three observations. First, that on String manifolds there is no E × E global anomaly in heterotic string theory. Second, that the description of the anomaly in the phase ofthe M-theory partition function of Diaconescu-Moore-Witten extends from the Spin case to the Stringcase. Third, that the cubic refinement law of Diaconescu-Freed-Moore for the phase of the M-theorypartition function extends to String manifolds. The analysis relies on extending from invariants whichdepend on the Spin structure to invariants which instead depend on the String structure. Along the way,the one-loop term is refined via the Witten genus. Contents D = 10, N = 1 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Refinement of the M-theory/IIA partition Function . . . . . . . . . . . . . . . . . . . . . . . . 133.3 The Cubic Refinement Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 ∗ Email: [email protected] . Current address: Department of Mathematics, University of Maryland, College Park, MD20742. Current Email: [email protected]
Introduction
The DMW anomaly [9] in the comparison of the partition functions of M-theory and type IIA string theory isgiven by the vanishing of the seventh integral Stiefel-Whitney class W ( X ) of spacetime X . The cancelationof this anomaly in [24] leads to the emergence of elliptic cohomology and other generalized cohomologytheories. The “String condition”, i.e. the vanishing of half the Pontrjagin class p ( X ) is stronger thanvanishing of W . There are also connections to generalized cohomology from the perspective of type IIB(target) string theory [25] [26]. It is natural then to ask how much of a role the String condition plays inglobal aspects of string theory. It is the purpose of this note to provide a few observations that provide onestep in shedding some light on this question.Manifolds satisfying the String condition are called “ String manifolds” and are characterized by havinga String structure, i.e. with a lifting of the structure group of the tangent bundle from Spin = O h i to its3-connected cover String = O h i . Generic examples of String manifolds occur when the manifold is highlyconnected. The simplest case is perhaps the spheres, which occur in the compactification of eleven- andten-dimensional supergravity (coupled to Yang-Mills) leading to gauged supergravity in lower dimensions(see [10]). While compact manifolds X with special holonomy G and Spin(7) require a non-vanishing p ( X ) = 0 ∈ H ( X ; Z ) [23], topological generalized such structures can be satisfied for manifolds withvanishing p [46]: S admits a topological generalized G structure and any manifold J of the form S × N ,where N is a Spin manifold, admits a topological generalized Spin(7) structure, since J (trivially) satisfies p ( J ) − p ( J ) = 0 and χ ( J ) = 0. Note that there is a difference between p being zero and p being zero,due to the possible existence of 2-torsion. Furthermore, there even exist flat manifolds of finite holonomygroups that have p = 0 ∈ H Z [29]. This is one justification for their use in models satisfying the equationsof motion.The goal of this note is to point out the following: (1) The global anomaly for heterotic string theory vanishes on String manifolds. (2)
The partition functions of M-theory and type IIA string theory match in the sense of [9] on Stringmanifolds. (3)
The description of the phase of the M-theory partition function as a cubic refinement [8] extends toString manifolds.One might wonder whether all what one needs to do to go from the Spin case to the string case is setthe first Pontrjagin class of the tangent bundle to zero, which would essentially make trivial the task we setout to achieve. This turns out to be naive because we are considering global questions. In particular, weare not guaranteed that the desired obstructions vanish. More precisely, we are trying to extend bundleson String manifolds rather that on Spin manifolds, and the corresponding cobordism groups may introducenew invariants and obstructions in going from the Spin to the String case. That we show that this is not thecase is not immediate and is in fact nontrivial. Mathematically, the results are possible due to the recentcalculations of M String n ( K ( Z , n ≤
14 in the companion paper [21]. That paper provides the maintechnical topological results and this note provides the analysis and geometry as well as physical motivationand the corresponding consequences.One might raise the following objection: If the anomaly vanishes for weaker condition then should it notvanish for the stronger one? The answer is not obvious as all without nontrivial calculations, and in fact thenotions of “weaker” and “stronger” might be misleading in this context. We know that Ω spin11 ( K ( Z , M String ( K ( Z , S [47].The objection would likewise be on both the cancelation of the DMW anomaly and the matching ofthe phases of the partition function. On the former there are indeed no subtleties, but the latter requiresthe analysis of subtle mod 2 invariants. In particular the question of extension of the Z invariant f ( a + b ) − f ( a ) − f ( b ) of [9] becomes a question of extension on String manifolds of M String ( Z ) rather than M Spin( Z ) for the Spin case, where Z is K ( Z , × K ( Z ,
4) or K ( Z , ∧ K ( Z , M String ( Z ) = M Spin ( Z ), so that there are no new invariants, hence obstructions, beyond the onecoming from the Spin case. Spin vs. String invariants.
The topological invariants in the heterotic or the M-theory partition func-tions depend explicitly on the choice of a Spin structure since Dirac operators are involved. That is whythe question of anomaly cancellation involves Spin bordism groups. However they do not depend by con-struction on the choice of a String structure. Therefore, for those original invariants, one simply does notneed to consider extension problems for manifolds with String structure. However, this would be needed fortopological invariants which depend explicitly on the choice of a String structure. To that end, we discuss acertain refinement of the M theory partition function relying on topological invariants which depend on thechoice of a String structure.We study the one-loop term in M-theory and provide a refinement to q -expansions via the Witten genus.The consequences for M-theory are as follows. The one-loop polynomial I appearing from the M5-braneworldvolume corresponds in the M-theory Lagrangian, via anomaly inflow, to the one-loop term I − loop = − I ∧ C [44] [12]. Thus, this topological term in M-theory will admit a similar elliptic refinement. As thetopological terms in M-theory, namely the Chern-Simons term I CS = C ∧ G ∧ G and the one-loop term I − loop , together with the Rarita-Schwinger field, make up the main contribution to the partition functionof the C-field, this leads to q -expansions in the M-theory action. We provide another interpretation of suchexpansions via replacing ordinary G bundles with corresponding loop bundles. The phase of the partitionfunction will also admit such a refinement. We study this via a refinement of the Atiyah-Patodi-Singer indextheorem [1] to the case of q -bundles in the sense of [6] [14]. The dimensional reduction, via the adiabaticlimit on the M-theory circle, of the elliptically refined eta invariant will lead to refined eta-forms in tendimensions, making connection to the discussion in [37]. In type IIA, the q -refinement of the mod 2 indexof the Dirac operator in relation to the partition function has already been discussed in [24] [38].Eleven-dimensional supersymmetry relates the one-loop term C ∧ I to a (curvature) -term [20] andto other (yet not fully determined) terms that enter in the effective action. While the effective action ofthe compactified theory depends nontrivially on moduli fields associated with the geometry of the com-pact manifold, this dependence is restricted by consistency with the duality symmetries in ten and lowerdimensions. For tori, this demonstrated in [15] [17]. The exact BPS (curvature) -couplings in M-theorycompactified on a torus are obtained in [34] [35] from the toroidal BPS membrane, using the point of viewthat the eleven-dimensional membrane gives the fundamental degrees of freedom of M-theory. This involvesmodularity coming from wrapped membranes.In section 2 we provide the invariants which depend on the String structure. Later in section 3 we givethe results listed above in the third paragraph of the introduction2 The Refined Invariants
In this section, we propose the invariants that would replace those invariants which depend on the choiceos Spin structure. The invariants we use will likewise depend (in general) on the choice of String structure.Explicit examples are given in [39], following the constructions of [36] [5] [6]. This can be seen most easilyfor the case of a three-dimensional sphere where the choice of framing is the same as the choice of Stringstructure, on which the invariants depend. In our current case we can, for instance, take our eleven-manifolds Y to be products of S with eight-dimensional factors. We start by providing motivation for refinement of the one-loop term in M-theory. I vs. Index and why String structure: from type IIA. Consider the topological part of the actionin type IIA string theory, which involves the B -field with its field strength H . Since I does not haveintegral periods then a coupling of the form exp[2 πi R X B ∧ I ] would not be well-defined in the presenceof topologically nontrivial B -fields. This led the authors of [3] to provide the following solution. The lastsentence in the previous paragraph means that the term exp[2 πi R X B ∧ b A ] is well-defined. However, inorder for the remaining term exp[2 πi R X B ∧ λ ] to be obtained from a gauge-invariant partition function,the condition [( λ + 2 ρ ) ∧ H ] dR = 0 is proposed in [3], where ρ is a closed 4-form with integral periods. Nowwe see that one simple way of satisfying the above condition is to require triviality of λ . This means thatwe have a String structure, that is our manifold X has a lifting of the structure group to the 3-connectedcover of the Spin(10). So what we have is an expression, which is essentially an index, valid when we imposethe String condition. This observation will be important for us. Anomaly cancellation via the elliptic genus: from heterotic string theory.
Consider heteroticstring theory on a Spin ten-manifold M . Here in addition to the tangent bundle there is also a gaugebundle with connection of curvature F . Because of modular invariance, the anomaly in this theory alwayshas a factorized form [42] I ( F, R ) = 12 π (Tr F − Tr R ) ∧ X ( F, R ) , (2.1)where X ( F, R ) is the Green-Schwarz anomaly polynomial. This anomaly can be cancelled by the followingterm in the effective action [18] [19] [30] S c = Z M B ∧ X ( F, R ) . (2.2)There is an interpretation in terms of the elliptic genus [30] [31]. The anomaly is given by the constant termof weight two of the 12-form in the elliptic genus I ( F, R ) = φ ell ( q, F, R ) | − form coeff . of q . (2.3)The terms which violate modular invariance can be factored out of the character-valued partition function φ ell ( q, F, R ) = exp (cid:18) − π E ( q )(Tr F − Tr R ) (cid:19) e φ ell ( q, F, R ) , (2.4)where e φ ell ( q, F, R ) is fully holomorphic and modular invariant of weight 4, that is it can be expressed in termsof the Eisenstein function E ( q ), and E ( q ) is the non-modular Eisenstein series of weight 2 (see expression32.9)). These can be written in terms of a parameter τ ∈ H in the complex upper half plane via q = e πiτ .Interestingly, the anomaly canceling terms are encoded in a τ -integral over the fundamental domain F ofthe weight zero terms of the modified elliptic genus [31] φ ell ( q, F, R ) = exp (cid:18) − π π Im τ (Tr F − Tr R ) (cid:19) e φ ell ( q, F, R ) , (2.5)namely − π Z F d τ (Im τ ) φ ell ( q, F, R ) | − forms = X ( F, R ) . (2.6)For comparison with our case, let us highlight some features of this [30] [31]:1. Essentially the same function that captures the anomaly also governs the cancellation of that anomaly.2. The integrands have a nontrivial dependence on the modular parameter τ , reflecting contributions notonly from physical massless states but also from an infinity of ‘unphysical’ modes. This is in contrastto what happens in field theory, signaling that string theory should not be thought of merely as asuperposition of an infinite number of point particle field theories.3. The interest in the structure of the anomaly canceling terms was not to prove that the anomaly cancels,but to understand how it cancels.Now in the case of M-theory, we can have essentially the same line of reasoning for all three points above,which refer to heterotic string theory. First, the anomaly involves I , which is the same term (up to a sign)which cancels that anomaly. Second, M-theory is more unlike field theory than is string theory, and hencewe also expect an important role for seemingly unphysical infinite modes. Third, indeed, understanding howthe anomaly cancels allows us to provide a generalization, namely the elliptic refinement. We will exploresimilar features for type IIA string theory. The case for similarity between heterotic string theory and typeIIA string theory is already made in [44], which we consider next to further motivate our proposal. The one-loop term in type IIA via modularity.
Duality, such as that between heterotic string theoryand type IIA string theory, tends to exchange worldsheet and spacetime effects (cf. [11]). Torsion incohomology shows up in the worldsheet global anomalies in the heterotic string, so that on the type IIA sidethis would appear in spacetime global anomalies. So we expect a similar situation as in [31] to take place onthe spacetime side in type IIA. Let us highlight an important aspect of this. In [44] it is shown that in ten-dimensional type IIA there is a one-loop contribution to the effective action of the form δS = R X B ∧ I .The computations that lead to this term are similar to those of [31] leading to the above anomaly cancellationin heterotic string theory, even though type IIA string theory in non-chiral. The one-loop computation on X = T × M involves integrating over the fundamental domain M corresponding to odd (even) Spinstructure for the left- (right-) moving fermions on the torus. The integral is δS = B · − i Z M d τ π Im τ · π Im τ φ ell ( q ) . (2.7)This is reduced to an integral over the boundary of the moduli space. Furthermore, the elliptic genus isreplaced with the massless contribution:1. In the (NS, R) sector, which is fermionic, the massless contribution of φ ell ( q ) computes the index n NS,R of the Dirac operator coupled to
T M given by n NS,R := R M I NS,R = R M I B + 3 I λ .4. In the (R, R) sector, which is bosonic, the massless contribution of φ ell ( q ) computes the index n R,R ofthe Dirac operator coupled to the Spin bundle SM given by n R,R := R M I R,R = R M − I B .Then the result in two dimensions is δS = iπ (2 n NS,R − n R,R ) B . Thus, the one-loop polynomial (and henceone loop term) in type IIA enters in the anomaly via the elliptic genus. Type IIB string theory.
This string theory is invariant under worldsheet parity. However, the term R B ∧ I is odd under B
7→ − B , so that such a term is absent in type IIB string theory. Indeed, this isdemonstrated explicitly in [7]. Topological String structures.
A topological String structure α top is by definition a “trivialization”of the Spin characteristic class p ( T M ) ∈ H ( M ; Z ) (see [5] [6]). A String manifold ( M, α top ) is a Spinmanifold M with a chosen topological String structure α top on the Spin bundle. The Thom spectrum MStringcan be realized as bordism classes of closed String manifolds [ M, α top ], i.e. manifolds with a String structureon their tangent bundle. The morphism i : MString ∗ → MSpin ∗ forgets the topological String structure, i.e.is given by i ([ M, α top ]) = [ M ]. Remark on connection to other proposals.
A proposal for refining field strength G of the C-field isgiven in [27], where the main idea is to replace G essentially with G E ( q ), where E ( q ) is the Eisenteinseries, a modular form of weight 4. This replacement is related to what we propose in this paper in thefollowing sense. Consider the one-loop term lifted to twelve dimensions, that is G ∧ I . This terms gets q -expansions via refining either G , or I (or both). What we do in this paper is the latter. Since thecorresponding refinement involves E ( q ), we see that both refinements are essentially the same, as far as theone-loop term is concerned. However, the method we provide is motivated by anomaly cancellations andhence seems more physically justified. The Witten genus is given by the series in the formal variables x and q (see [22]) φ W ( x, q ) := x sinh( x ) ∞ Y n =1 (1 − q n ) (1 − q n e x )(1 − q n e − x )= exp " ∞ X k =2 k )! E k ( q ) x k e E ( q ) x = θ W ( x, q ) e E ( q ) x ∈ Q [[ q ]][[ x ]] . (2.8)Here E k ( q ) := − B k k + P ∞ n =1 σ k − ( n ) q n ∈ Q [[ q ]] is the Eisenstein series, with the Bernoulli numbers B k and the sum σ k − ( n ) of the (2 k − n . For k ≥ E k ( q ) is the q -expansion of a modular form of weight 2 k . The constant term in E k ( q ) is rational while thehigher terms are integral. The Einsenstein function E = P m,n ∈ Z ( mτ + n ) − is not a modular function, buttransforms in an anomalous way E (cid:18) aτ + bcτ + d (cid:19) = ( cτ + d ) E ( τ ) − πic ( cτ + d ) . (2.9)A modular function ˆ E ( τ ) can be built out of E ( τ ) on the expense of losing holomorphicity in τ , ˆ E ( τ ) = E ( τ ) − π/ Im( τ ). Note that φ W ( x,
0) is the b A -genus, with the formal variable x giving the Pontrjagin classes5ia the generating formula P i ≥ p i = Q ∞ i =1 (1 + x i ). When x = 0, i.e. p = 0, the dependence of the Wittengenus on E is removed and hence modularity is restored [49]. Multiplicative series and q -expansions. For k ≥
1, consider the Newton polynomials N k ( p , · · · ) = P ∞ j =1 x kj ∈ Q [[ p , p , · · · ]], and consider the two series in Q [[ q ]][[ p , p , · · · ]] defined in [6] asΦ( p , · · · ) = exp " ∞ X k =2 k )! E k N k ( p , · · · ) e E p (2.10) e Φ( p , · · · ) = exp " ∞ X k =2 k )! E k N k ( p , · · · ) ∞ X j =1 E j p j − j ! = Θ e E p − p , (2.11)where Θ := Q ∞ i =1 θ W ( x i ) ∈ Q [[ q ]][[ p , p , · · · ]]. With N = P ∞ j =1 x j = p and N = P ∞ j =1 x j = p − p , wefind that the degree eight components of (2.10) and (2.11), respectively, areΦ = 112 E ( p − p ) + 12 E p , (2.12) e Φ = E (cid:20) E ( p − p ) + 13! E p (cid:21) . (2.13)Imposing the String condition, p = 0 (or, rationally, p = 0), givesΦ = − E p , e Φ = − E E p . (2.14)Using E = 1 + 240 P ∞ n =1 n q n − q n and E = − + q + · · · , the expressions (2.14) admit the following q -expansions Φ = − p + Ø( q ) , e Φ = 1144 p + Ø( q ) . (2.15)We therefore see that the q = 0 term of Φ is, up to a sign, the obstruction to having a Fivebrane structure[40] [41] and the q = 0 term of e Φ is three times the one-loop term I . Virtual bundles.
The above genera have another description, namely, one can view them as correspondingto ordinary Dirac operators coupled to infinite formal symmetric powers of an underlying finite-dimensionalbundle. Corresponding to a real l -dimensional bundle V , define the formal power series of virtual bundles R ( V ) := X n ≥ q n R n ( V ) = Y k ≥ (1 − q ) l O k ≥ Sym q k ( V ⊗ R C ) , (2.16)where Sym q ( V ) := L n ≥ q n Sym n ( V ) is the generating power series of the symmetric powers Sym n ( V ) ofthe bundle V . For a manifold M , the following topological index formula holds [6]Φ( p ( T M ) , · · · ) = b A ( T M ) ∪ ch( R ( T M )) . (2.17) Geometric String structures.
A geometric refinement α of the topological String structure α top trivial-izes this class at the level of differential forms, that is [36] [45] α gives rise to a form C α ∈ Ω ( Y ) such that dC α = p ( ∇ m ). This is essentially the C-field as explained in [39]. If ∇ V is a connection on a real vector6undle V → Y over some manifold Y , then the Chern-Weil representative of the i th Pontrjagin classwill be denoted p i ( ∇ V ) ∈ H i ( Y ; Z ). Let Y be a closed seven-manifold with a Spin structure SY anda topological String structure α top and Levi-Civita connection ∇ T Y . Let V → Y be a real vector bundlewith a metric h V and connection ∇ V . The twisted Dirac operator D Y ⊗ V acts on sections of the bundle SY ⊗ R V . Call a Riemannian Spin manifold a geometric manifold Y . In the presence of V , we also havea geometric bundle V := ( V, h V , ∇ V ), in the sense of [6]. The refined eta invariants require a taming [6].A taming of D Y ⊗ V is a self-adjoint operator Q acting on Γ( S ( Y ) ⊗ V ) and given by a smooth integralkernel such that D Y ⊗ V + Q is invertible. Such a taming always exists in eleven dimensions. In Physicalterms, we interpret a taming as the Pauli-Villars regularization imaginary mass term im . String structures and higher powers on Z vs. on Y . Consider the String manifold ( Z , β top )with topological String structure β top and boundary Y . The tangent bundle decomposes as T Z | Y ∼ = T Y ⊕ Ø R . The trivial bundle Ø R := Y × R has a canonical String structure. Then the decompositionimplies that Y has an induced String structure α top . Thus, starting from a String twelve-manifold we getan eleven-dimensional String boundary with an induced topological String structure. Let [ Z ] ∈ MSpin be a homotopy class represented by a closed 12-dimensional Spin manifold Z . The Witten genus in twelvedimensions is a morphism of ring spectra R : MSpin → KO [[ q ]] ∼ = Z [[ q ]], and is given by (see [22]) R ([ Z ]) = 12 Ind( D Z ⊗ R ( T Z )) = 12 h Φ , [ Z ] i . (2.18) The secondary invariants.
The secondary invariant s ( Y , α ) requires the existence of a Spin zero bor-dism Z of the String manifold ( Y , α ). This is always possible as the String cobordism group in elevendimensions is zero. Choose a connection ∇ Z extending the connection ∇ Y on the tangent bundle T Y of Y . Consider also the trivial bundle Ø R = Y × R with the trivial connection. resulting from the restriction T Z | Y ∼ = T Y ⊕ O R . Both connections induce a connection on the virtual bundles R n ( T Y ⊕ Ø R ) forall n ≥
0. At the level of differential forms, for a twelve-manifold Z , formula (2.17) givesΦ( p ( T Z ) , · · · ) = b A ( T Z ) ∪ ch( R ( T Z ) ∈ Ω ∗ ( Z )[[ q ]] . (2.19)Applying [6] to our case, the analytical and geometric secondary invariants are given by the formal powerseries s an ( Y , α, t ) := Z Y C α ∧ e Φ ( ∇ Y ) + 12 X n ≥ q n η ( R n ( T Y ⊕ Ø R )) ∈ R [[ q ]] , (2.20) s geom ( Y , α, ∇ Z ) := Z Y C α ∧ e Φ ( ∇ Y ) − Z Z Φ ( ∇ Z ) ∈ R [[ q ]] , (2.21)where the expression s an is a String cobordism class of the String manifold ( Y , α top ), and the cohomologyclasses match [ s an ] = [ s geom ]. Now, the integrands in (2.21) involves the expressions e Φ = 112 E (cid:2) − E p + (2 E + E ) p (cid:3) (2.22)Φ = 112 (cid:20) E E p ( p − p ) + 120 E ( p − p p + 3 p ) (cid:21) . (2.23)Applying the String condition, these give e Φ | λ =0 = − E E p = 12 · p + O ( q ) and Φ | λ =0 = 180 E p = 12 · p + O ( q ) , (2.24)7espectively. Note that in the zero mode case in twelve dimensions, the addition of the Chern-Simons term L CS = G ∪ G ∪ G to the one-loop term L − loop = − I ∧ G leads to a cancellation of the top Pontrjaginclass p . The same would happen here so that the two expressions coincide after taking cohomology classes, asin the case of the M5-brane. We see that the integrand in s an ( Y , α, t ) is of the form R Y C α ∧ I | λ =0 + O ( q ). We now consider Y to be a circle bundle over a ten-manifold X , over which type IIA string theory istaken. We consider the effects of the modes of this M-theory circle on analytical quantities which enter intothe definition of the phase of the partition function. Higher powers as higher modes of the 11th Rarita-Schwinger bundle.
Let S ( S ) be the Spinbundle associated to the vertical tangent bundle ( T S , g S ) on Y . The lift of the connection ∇| T S to T S is denoted by ∇ S . Let R be an integral power operation Z [Λ i , i = 0 , , , · · · ] or Z [Sym i , i = 0 , , , · · · ],where Λ and Sym denote antisymmetric and symmetric powers, respectively. The connection ∇ S liftsnaturally to R ( T Y ), which we denote ∇ R . For any fiber S , let D S , R be the Dirac operator acting onΓ (cid:0) S ( S ) ⊗ T S ⊗ R ( T Y | S ) (cid:1) defined as D S , R ( ψ z ⊗ r ) = c ( ξ ) ∇ ξ ψ z ⊗ r + c ( ξ ) ψ z ⊗ ∇ ξ r , (2.25)where ψ z ∈ Γ( S ( S )) and r ∈ Γ( R ( T Y )). Note that we are considering the derivative with respect tothe 11th direction, something that goes beyond previous treatments. Thus, in a sense, we are consideringthe vertical components of the Rarita-Schwinger operator, taking into account higher Fourier modes ’of thecircle in the eleventh direction.We now consider the zero modes of the operator D S , R . Applying [50] [51], the space ker( D S , R ) is ofconstant rank and forms a vector bundle over X withker( D S , R ) = R ( T X ⊕ O R ) . (2.26)Thus, the kernel of the vertical Dirac operator is given by the bundle of powers of the tangent bundle. Thiseffect, which improves what was considered in [33], shows that the Fourier modes for the M-theory circlecorrespond to powers of the split tangent bundle. Geometric representative of the Euler class of the circle bundle.
Let π S be the orthogonalprojection on T S with respect to g Y . In addition to the Levi-Civita connection ∇ LC , let ∇ be a connectionon T Y defined as (cf. [4]) ∇ z v z = π S ( ∇ LCz v z ) , ∇ µ v z = π S ( ∇ LCµ v z ) , ∇ z v µ = 0 , ∇ µ v ν = ∇ Xµ v ν , (2.27)for v a vector with z -component along the eleventh direction or µ -component along X . As indicated in [ ? ],this is more general than the usual Kaluza-Klein and Scherk-Schwarz settings for dimensional reduction inthe sense of allowing ∇ µ v z to be nonzero in general. Define a tensor S by S = ∇ LC − ∇ . Then, for ξ ∈ T S the unit vector field determined by g Y and the Spin structure on T S , the formula S ( ξ )( ξ ) = 0 holds [51]. For the Witten genus, we considered only symmetric powers in (2.16). T of ∇ defined by T µν = −S µν + S νµ . Let ξ ∗ ∈ T ∗ S be the dual of ξ . Then, since ∇ LC istorsion-free, we have the pairing hT ( U, V ) , ξ i = dξ ∗ ( U, V ) , U, V ∈ T X . (2.28)Therefore, T determines a 2-form in Λ ( T ∗ X ) such that π T represents the Euler class e of the line bundlecorresponding to the circle bundle and can be viewed as the Ramond-Ramond 2-form F [ ? ]. Adiabatic limit of eta invariant for the circle bundle.
Consider the metric g Y,t = g S ⊕ t π ∗ g X on Y with Levi-Civita connection ∇ L,t . The limit t → t → ∇ L,t = ∇ + π S S .Since ∇ preserves the splitting T Y = T S ⊕ T X , then π S S does not contribute to the characteristicforms of the power bundle R ( T Y ). Let S t Y be the Spin bundle of ( Y , g Y,t ) and S t X the Spin bundle of( X , t g X ). Then S t Y = S ( S ) ⊗ S t X . Consider spinors Ψ ∈ Γ( S t Y ) and polyvectors v ∈ Γ( R t ( T Y ))on Y . For any t >
0, the Dirac operator D tY, R is the differential operator acting on Γ( S t Y ⊗ R t ( T Y )), D tY, R (Ψ ⊗ v ) = c ( ξ ) ∇ L,tξ Ψ ⊗ v + c ( ξ )Ψ ⊗ ∇ L,tξ v + √ t X i =1 (cid:0) c ( e i ) ∇ L,te i Ψ ⊗ v + c ( e i )Ψ ⊗ ∇ L,te i v (cid:1) . (2.29)Then D tY, R is a self-adjoint elliptic first order differential operator on Γ( S t Y ⊗ R t ( T Y )). The limitlim t → η ( D tY, R ) exists in R / Z . The adiabatic limit of the eta invariant is interpreted in [33] [37] as the phaseof the partition function in the semiclassical limit of the dimensional reduction from M-theory to type IIAstring theory. Assuming the kernel is a vector bundle, as is the case above, then [50]lim t → η ( D tY, R ) ≡ η ( D X, R ( T X ⊕ R ) ) + 1(2 πi ) Z X b A ( i R X ) ∧ b η mod Z , (2.30)where b η are the eta-forms on X and b A ( i R X ) is the differential form representative of the b A -genus definedusing the curvature R X on X . Since X is even-dimensional, the eta invariant vanishes, and so we have η ( D X, R ( T X ⊕ R ) ) = dim ker( D X, R ( T X ⊕ R ) ). Then, from [50] [51], we getlim t → η ( D tY, R ) ≡
12 dim ker( D X, R ( T X ⊕ R ) ) + * b A ( T X )ch( R ( T X ⊕ R )) · tanh( e ) − e e tanh e , (cid:2) X (cid:3)+ mod Z . (2.31)This generalizes the discussion in [33] [37] to include higher modes of the M-theory circle as well as higherpowers of the bundles. Our discussion above on higher powers of bundles is related to our earlier discussion on q -expansions in thecontext of elliptic genera. We will make use of such a connection in this section, at least formally as ourmain interest is modularity. This complements the discussion in [37].The APS index theorem can be extended purely formally to graded bundles or q -bundles. Suppose that E r , r ≥
0, is a sequence of vector bundles and write E q = L r ≥ E r q r , where q is a formal variable. TheDirac operator can be twisted with the finite-dimensional E r to get D + E r = D + r . This can be extended to allof E q to get D + q = P r ≥ D + r q r : Γ( S + ⊗ E q ) → Γ( S − ⊗ E q ). The index of D + q is the formal power seriesind( D + q ) = X r ≥ ind( D + r ) q r = X r ≥ (dim ker( D + r ) − dim coker( D + r )) q r . (2.32)9he operators D + r and D + q restricted to the boundary Y of Z give the twisted Dirac operators D r,Y and D q,Y = P r D r,Y q r . The eta invariant of a twisted Dirac operator D q is the formal series η D q ( q ) = P r ≥ η D r q r . The q -analog of the number of zero modes is h ( q ) = dim ker( D q ) = X r ≥ dim ker( D r ) q r . (2.33)The formal Chern character can also be defined as ch( E q ) = P r ≥ ch( E r ) q r . Applying the APS indextheorem to each twisted Dirac operator and adding, keeping track of the grading, gives that the formaltwisted Dirac operator D + q with the APS boundary condition has index (see [14])ind( D + q ) = Z Z ch( E q ) b A ( Z ) − h D q,Y ( q ) + η D q,Y ( q )2 . (2.34)Of course the above works for any 4 k -dimensional manifold with boundary, with our main cases being k = 3for M-theory and k = 2 for the M5-brane.In [14] a new invariant η W ( q ) of framed manifolds Y k − is defined which, as a power series, is also amodular function (modulo the integers). For example, there are framings of the spheres S k − = ∂ D k forwhich η E ( q ) are expressed in terms of an Eisenstein series E k ( τ ). Eta invariants and the Witten genus.
Consider The Dirac operator twisted by S q T Z k . Its indexunder the global APS boundary condition is related to the Witten genus by [14] φ W ( ∂Z k ) η d ( τ ) − k = 12 ( h W + η W ) , (2.35)where η d ( τ ) = η d ( q ) := q / Q n ≥ (1 − q n ) is the Dedekind eta-function and h W and η W are the q -refinednumber of zero modes and the refined APS eta-invariant, respectively. Since the index and the kerneldimensions h are integers, then 12 η W ≡ φ W η d ( τ ) − k mod Z . (2.36) Example: Disks and spheres.
On ( D k , S k − ) the eta invariant is η W ( S k − ) ≡ E k ( τ ) η d ( τ ) − k mod Z . For k = 2 and k = 3, respectively, this gives12 η W ( S ) ≡ E ( τ ) η d ( τ ) − mod Z , (2.37)12 η W ( S ) ≡ E ( τ ) η d ( τ ) − mod Z . (2.38)We see that the Eisenstein series E and E appear in relation to the M5-brane and spacetime in M-theory,respectively. In addition, the former also appears when considering M-theory on decomposable manifolds,due to the composite nature of the topological parts of the action.We hope to consider in the future further relations between M-theory on one hand and modularity andthe Witten genus on the other. Having provided motivation and corresponding invariants to be used for the refinement of anomalies fromthe Spin to the String case, we now start with presenting the results listed in the introduction.10 .1 Global Anomalies in D = 10 , N = 1 Supergravity
We consider the following setting. An E × E bundle V ⊕ V on a ten-dimensional manifold M . We assume M to have a String structure, ie. that the spin bundle SM of M admits a lifting of the structure groupfrom Spin(10) to its 7-connected cover String(10). The condition for such a lift is given by λ ( T M ) = 0,where λ = p is half the Pontrjagin class.Due to the homotopy type of E , the E × E bundle on M is completely characterized by the degreefour class a ( V ⊕ V ) = a ( V ) + a ( V ) , (3.1)where a = p . The anomaly cancelation condition is given by λ ( T M ) − a ( V ⊕ V ) = 0 . (3.2)Assuming M to admit a String structure implies that a ( V ⊕ V ) = 0 . (3.3)Note that this does not necessarily imply that each factor is separately zero. We will thus take the conditionto be a ( V ) = − a ( V ) , (3.4)so that V has characteristic class a if and only if V has corresponding class − a . Note that every class in K ( Z ,
4) represents the characteristic class of an E bundle. We have the following fact from [48] Lemma 3.1.
The effective action of heterotic string theory is invariant under diffeomorphisms ϕ : M → M that admit a lift to the spin bundle of M and to V ⊕ V . Global anomalies are concerned with diffeomorphisms and/or gauge transformations that are not con-nected to the identity. The main question here is the string analog of the question raised in the spin case in[48]:
Is the effective action of N = 1 supergravity with group E × E invariant under such ϕ ? .The study of global anomalies requires considering the mapping cylinder X = (cid:0) M × S (cid:1) ϕ = (cid:0) M × I (cid:1) / ( x, ∼ ( ϕ ( x ) , , (3.5)and asking whether the bundles extend to X . Lemma 3.2. i. If M is a string manifold then so is X . ii. The bundle V ⊕ V can be extended to an E × E bundle over X . iii. X bounds a 12-dimensional string manifold B .Proof. The first part follows from the multiplicative behavior of the Spin characteristic classes under Whitneysum. For part (ii), note that the String analog of lemma (3.1) holds. Then the action of φ on V ⊕ V leads tothe identification of the fiber of V ⊕ V at ( φ ( x ) ,
1) with the fiber at ( x, M String (pt) = 0 and thus any 11-dimensional string manifold bounds a 12-dimensional stringmanifold.We now have Proposition 3.3.
The bundle V ⊕ V extends from X to an E × E bundle W ⊕ W over B . Fur-thermore, we can have a ( W ⊕ W ) = 0 . roof. V extends over B if and only if the cohomology class a ( V ) extends to a cohomology class in H ( B ; Z ). Considering the pair ( X , β ). This vanishes if X bounds a String manifold B over whichthe class β ∈ H ( X ; Z ) can be extended. This means that the pair is an element of the cobordism groupΩ ( K ( Z , M String ( K ( Z , B can always be chosen so that V extends to an E bundle W over B .Next we extend V . Corresponding to the inclusion ι : X ֒ → B , the tangent bundle of B decomposesas T B | X = T X ⊕ N , (3.6)where N is the normal bundle of X in B . Being a trivial real line bundle, N does not change the factthat the class λ of T X is zero, i.e. λ ( T X ) = λ ( T X ⊕ O ) = 0 . (3.7)extends to a trivial cohomology class in H ( B ; Z ). Since a ( V ) extends to a ( W ) ∈ H ( B ; Z ) then β = − a ( W ) is an element of H ( B ; Z ). Therefore, V and V both extend over B . Furthermore, theextension can be done in such a way that a ( W ) = − a ( W ). Theorem 3.4.
There are no global anomalies for E × E bundles V ⊕ V on string manifolds M .Proof. Given that ( M × S ) ϕ bounds a string manifold B over which V ⊕ V can be extended, the changein the effective action S under ϕ will be as in the spin case [47] [48]∆ S = 2 πi "Z B (cid:0) tr F + tr F − trR (cid:1) ∧ I − Z ( M × S ) ϕ H ∧ I , (3.8)where • F is the curvature of the E bundle so that tr i F is the Chern class of the bundle W i , i = 1 , • R is the curvature of the tangent bundle T B , so that trR is the Pontrjagin class, • I is the Green-Schwarz anomaly polynomial in characteristic classes of the gauge and tangent bundles,and hence satisfies dI = 0.Since V ⊕ V extends to W ⊕ W on B such that a ( W ) + a ( W ) = 0 = λ ( T B ) and hence (trivially)that a ( W ⊕ W ) + λ ( T B ) = 0, then the classes are trivial in cohomology and hence exact. Then H canbe defined to obey dH = tr F + tr F − trR (3.9)not just on the mapping cylinder ( M × S ) ϕ but also on the bounding manifold B . Inserting (3.9) in(3.8) we get ∆ S = 2 πi "Z B dH ∧ I − Z ( M × S ) ϕ H ∧ I . (3.10)Now using Stokes’ theorem for ( M × S ) ϕ = ∂B the invariance result ∆ S = 0 follows.12 .2 Refinement of the M-theory/IIA partition Function The setting for the comparison of M-theory and type IIA string theory is as follows. M-theory is ‘defined’on the eleven-dimensional total space Y of a circle bundle with base X , a ten-dimensional manifold onwhich type IIA string theory is defined. Both spaces X and Y are usually taken to be Spin manifoldsand they have additional structures on them. On Y there is an E bundle V . Due to the homotopy typeof E up to dimension 14, V over Y is completely characterized by a degree four class a as above. Variouskinds of spinors are defined on both Y and X . In particular, for our purposes, there are elements λ ofΓ( SY ⊗ V ), i.e. sections of the spin bundle SY coupled to the E vector bundle. On X , in additionto spinors, there are also the Ramond-Ramond (RR) fields of even degrees. In particular, there is a degreefour field F . Such fields are images in cohomology of K-theory elements x , as they obey the quantizationcondition [32] [13] F := P i =0 F i = q b A ( X ) ch( x ).The comparison of the E gauge theory description of M-theory to the K-theoretic description of typeIIA string theory was performed in [9] at the level of the partition functions and is shown to agree upondimensional reduction, i.e. integration over the S fiber. The comparison involves those cohomology classesthat lift to K-theory and the identification involve subtle torsion and mod 2 expressions. Definition 3.5.
The phase of the M-theory partition function on an eleven-dimensional spin manifold Y is [9] Φ a = ( − f ( a ), where f ( a ) is the mod 2 index of a Dirac operator coupled to V with characteristicclass a . Remarks1. f ( a ) = 0 for a = 0, in which case V = L O , i.e. 248 copies of a trivial bundle. The comparison to type IIA requires a corresponding mod 2 invariant in KO ( X ): For x ∈ K ( X ), j ( x ) is the mod 2 index with values in the KO class x ⊗ x . The refinement of the partition function to elliptic cohomology E in [24] introduces a mod 2 index withvalues in the EO ( X ) = Z [ q ] class x ⊗ x , where x is a class in E ( X ). f ( a ) cannot be a cubic function in a ∈ H Z since that would have dimension 12, which is greater thanthe dimensions of either X or Y .The comparison between M-theory on Y and type IIA string theory on X proceeds from the embed-ding SU (5) × SU (5) / Z ⊂ E (3.11)so that out of two SU (5) vector bundles E and E ′ of Chern classes c ( E ) = − a and c ( E ′ ) = − a ′ one buildsan E bundle whose characteristic class is a + a ′ . This requires a and a ′ to be elements of H ( X , Z ) thatlift to K-theory. The idea is then to compute f ( a + a ′ ) − f ( a ) − f ( a ′ ). Using the decomposition (3.11), thisis Z X c ( E ) c ( E ′ ) mod 2 = Z X c ( E ) c ( E ′ ) mod 2 . (3.12)Since c ( E ) = 0 then c ( E ) = Sq ( c ( E )) mod 2, and similarly for E ′ . Then the main result on f ( a ) is thatit is a quadratic refinement of a bilinear form, i.e. f satisfies [9] f ( a + a ′ ) = f ( a ) + f ( a ′ ) + Z X a ∪ Sq a ′ . (3.13)This has an interpretation in terms of cobordism as follows [9]. Remarks . A result of Atiyah and Singer [2] states that the mod 2 index of the Dirac operator coupled to a vectorbundle V on a Spin manifold X vanishes if X is the boundary of a Spin manifold over which V extends. f ( a )can be regarded as a Z -valued function which vanishes when a extends to a Spin manifold B bounding X [9]. The class a ∈ H ( X , Z ) can be extended to a bounding manifold B if and only if the map µ : X → K ( Z ,
4) can be extended to a map µ ′ : B → K ( Z , X , a ) is zero as an element ofthe cobordism group Ω spin10 ( K ( Z , f ( a ) can be viewed as an element of Hom (cid:16) Ω spin10 ( K ( Z , , Z (cid:17) .More precisely, since f ( a ) = 0 for a = 0 then [9] f ∈ Hom (cid:16)e Ω spin10 ( K ( Z , , Z (cid:17) . A result of Stong [43] states that e Ω spin10 ( K ( Z , Z × Z . Thus there are two invariants: one is linear[43] v ( a ) = Z X a ∪ w = Z X a ∪ Sq w , (3.14)with v ( a + a ′ ) = v ( a ) + v ( a ′ ), and another– the more relevant one– is nonlinear [28] Q ( a + a ′ ) = Z X a ∪ Sq a ′ = Z X ( Sq a ) ∪ a ′ . (3.15) Q ( a , a ) vanishes if both a and a can be extended to B or if either a or a is zero. This leads to [9]: Q ( a + a ) is a homomorphism from the bordism group Ω spin10 ( K ( Z , ∧ K ( Z , Z to Z . Thus, there isonly one cobordism invariant Q . The first observation is straightforward
Observation 3.6.
There is no DMW anomaly in the M-theory partition function for String manifolds.Proof.
The DMW anomaly for Spin manifolds is given by the vanishing of the seventh integral Stiefel-Whitney class W = 0 [9]. Since W = Sq λ then λ = 0 implies that W = 0 and hence no anomaly. Thisfact has also been observed in [24].The invariant in the case of String cobordism is still the Landweber-Stong invariant Q ( a , a ) [21]. Theanalysis follows that of [9]. The main result in this section is then Theorem 3.7.
The phases of M-theory and type IIA string theory coincide not just on Spin manifolds butalso on String manifolds, i.e. Φ M = Φ IIA . Consequently, the corresponding partition functions match.
In [8] the phase Φ of the M-theory partition function was interpreted as a cubic refinement of a trilinearform on the group ˇ H ( Y ) of degree four differential charactersΦ (123) Φ (1) Φ (2) Φ (3) Φ (12) Φ (13) Φ (23) Φ = (ˇ a ˇ a ˇ a )( Y ) ∈ U (1) , (3.16)where we denoted Φ ( ijk ) := Φ([ ˇ C ] + ˇ a i + ˇ a j + ˇ a k ) and so on. The validity of this formula requires that ˇ C and all ˇ a i , i = 1 , , ,
4, extend simultaneously on the same Spin twelve-dimensional manifold Z . In theSpin case, the obstruction to extending ( Y , a , · · · , a k ) is measured by Ω spin11 ( ∧ k K ( Z , k = 1 by Stong’s result. The result for k = 2 , , Y with the String condition, i.e. we will as-sume that p ( Y ) = 0. We know from [16] that Ω String11 = 0. Furthermore, we know from [21] that14
String11 ( ∧ k K ( Z , k = 1 ,
2. Extending the result to the case k = 3 , Proposition 3.8.
The cubic refinement law holds and is defined for String eleven-manifolds Y . Acknowledgements
The author thanks the American Institute of Mathematics for hospitality and the “Algebraic Topology andPhysics” SQuaRE program participants Matthew Ando, John Francis, Nora Ganter, Mike Hill, MikhailKapranov, Jack Morava, Nitu Kitchloo, and Corbett Redden for very useful discussions. Special thanks aredue to Mike Hill for performing the very nice computations in [21] and sharing them during the program. Theauthor would also like to thank the Hausdorff Institute for Mathematics (HIM) in Bonn and the organizersof the “Geometry and Physics” Trimester Program at HIM for hospitality. The first draft of this paper waswritten in 2008 and the author apologizes for taking so long to write the second version. The reason is thatthis version had to wait for some developments, both by the current author and other authors in order tocharacterize the elliptic invariants needed to replace the classical invariants.
References [1] M. F. Atiyah, V. K. Patodi and I. M. Singer,
Spectral asymmetry and Riemannian geometry , Bull.London Math. Soc. (1973), 229–234.[2] M. F. Atiyah and I. M. Singer, The index of elliptic operators: V , Ann. Math. (1971) 139.[3] D. Belov and G. W. Moore, Type II actions from 11-dimensional Chern-Simons theories ,[ arXiv:hep-th/0611020 ].[4] J.-M. Bismut, The index theorem for families of Dirac operators: two heat equation proofs , Invent. Math. (1986), 91–151.[5] U. Bunke, String structures and trivialisations of a Pfaffian line bundle , [ arXiv:0909.0846 ] [ math.KT ].[6] U. Bunke and N. Naumann,
Secondary Invariants for String Bordism and tmf , [ arXiv:0912.4875 ][ math.KT ].[7] K. Dasgupta and and S. Mukhi, A note on low-dimensional string compactifications , Phys. Lett.
B398 (1997), 285-290, [ arXiv:hep-th/9612188 ].[8] E. Diaconescu, D. S. Freed and G. Moore,
The M-theory 3-form and E gauge theory , Elliptic coho-mology, 44–88, London Math. Soc. Lecture Note Ser., 342, Cambridge Univ. Press, Cambridge, 2007,[ arXiv:hep-th/0312069 ].[9] D.-E. Diaconescu, G. Moore, and E. Witten, E gauge theory, and a derivation of K -theory fromM-theory , Adv. Theor. Math. Phys. (2002) 1031–1134, [ arXiv:hep-th/0005090 ].[10] M. J. Duff, B. E. W. Nilsson, and C. N. Pope, Kaluza-Klein supergravity , Phys. Rep. (1986), no.1-2, 1–142.[11] M. J. Duff and J. X. Lu,
Loop expansions and string/five-brane duality , Nucl. Phys.
B 357 (1991) 534.1512] M. J. Duff, J. T. Liu, and R. Minasian,
Eleven-dimensional origin of string-string duality: A one looptest , Nucl. Phys.
B452 (1995) 261-282, [ arXiv:hep-th/9506126 ].[13] D. S. Freed and M. J. Hopkins,
On Ramond-Ramond fields and K-theory , J. High Energy Phys. (2000) 044, [ arXiv:hep-th/0002027 ].[14] M. I. Galvez Carrillo,
Modular invariants for manifolds with boundary , PhD Thesis (2001), UniversitatAut`onoma de Barcelona.[15] M.B. Green, M. Gutperle, and P. Vanhove,
One loop in eleven dimensions , Phys. Lett.
B409 (1997)177–184, [ arXiv:hep-th/9706175 ].[16] V. Giambalvo, On h i -cobordism , Illinois J. Math. (1971) 533–541; erratum ibid (1972) 704.[17] M. B. Green, J. G. Russo, and P. Vanhove, Automorphic properties of low energy string amplitudes invarious dimensions , [ arXiv:1001.2535 ] [ hep-th ].[18] M.B. Green and J. Schwarz,
Anomaly cancellation in supersymmetric D = 10 gauge theory and super-string theory , Phys. Lett. (1984) 117-122.[19] M.B. Green, J. Schwarz and P.C. West, Anomaly free chiral theories in six-dimensions , Nucl. Phys.
B254 (1985) 327–348.[20] M.B. Green and P. Vanhove,
D-instantons, strings and M-theory , Phys. Lett.
B408 (1997) 122-134,[ hep-th/9704145 ].[21] M. Hill,
The String bordism of BE and BE × BE through dimension 14 , Illinois J. Math. (2009),183-196, [ arXiv:0807.2095 ] [ math.AT ].[22] Hirzebruch, T. Berger and R. Jung, Manifolds and Modular Forms, Aspects of Mathematics, Vieweg,Braunschweig/Wiesbaden, 1992.[23] D. D. Joyce, Compact manifolds with special holonomy, Oxford University Press, Oxford, 2000.[24] I. Kriz and H. Sati, M theory, type IIA superstrings, and elliptic cohomology , Adv. Theor. Math. Phys. (2004) 345-395, [ arXiv:hep-th/0404013 ].[25] I. Kriz and H. Sati, Type IIB string theory, S-duality and generalized cohomology , Nucl. Phys.
B715 (2005) 639, [ arXiv:hep-th/0410293 ].[26] I. Kriz and H. Sati,
Type II string theory and modularity , J. High Energy Phys. (2005) 038,[ arXiv:hep-th/0501060 ].[27] I. Kriz and H. Xing, On effective F-theory action in type IIA compactifications , Int. J. Mod. Phys.
A22 (2007),1279-1300, [ arXiv:hep-th/0511011 ].[28] P. S. Landweber and R. E. Stong,
A bilinear form for Spin manifolds , Trans. Amer. Math. Soc. (1987), no. 2, 625–640.[29] R. Lee and R. H. Szczarba,
On the integral Pontrjagin classes of a Riemannian flat manifold , GeometriaeDedicata (1974) 1–9. 1630] W. Lerche, B. E. W. Nilsson and A. N. Schellekens, Heterotic string-loop calculation of the anomalycancelling term , Nucl. Phys.
B289 (1987) 609 –627.[31] W. Lerche, B. E. W. Nilsson, A. N. Schellekens and N. P. Warner,
Anomaly cancelling terms from theelliptic genus , Nucl. Phys.
B 299 (1988) 91–116.[32] G. W. Moore and E. Witten,
Selfduality, Ramond-Ramond fields, and K theory , J. High Energy Phys. (2000) 032, [ arXiv:hep-th/9912279 ].[33] V. Mathai and H. Sati,
Some relations between twisted K-theory and E gauge theory , JHEP (2004) 016, [ arXiv:hep-th/0312033 ].[34] B. Pioline, H. Nicolai, J. Plefka, and A. Waldron, R couplings, the fundamental membrane and excep-tional theta correspondences , JHEP (2001) 036, [ arXiv:hep-th/0102123 ].[35] B. Pioline and A. Waldron, The automorphic membrane , JHEP (2004) 009,[ arXiv:hep-th/0404018 ].[36] C. Redden,
String structures and canonical 3-forms , [ arXiv:0912.2086 ] [ math.DG ].[37] H. Sati, E gauge theory and gerbes in string theory , Adv. Theor. Math. Phys. (2010), 1-39,[ hep-th/0608190 ].[38] H. Sati, O P bundles in M-theory , Commun. Number Theory Phys. (2009) 1-36, [ ] [ hep-th ].[39] H. Sati, Geometric and topological structures related to M-branes , Proc. Symp. Pure Math. (2010)181-236, [ arXiv:1001.5020 ] [ math.DG ].[40] H. Sati, U. Schreiber and J. Stasheff, Fivebrane structures , Rev. Math. Phys. (2009) 1–44,[ ] [ math.AT ].[41] H. Sati, U. Schreiber and J. Stasheff, Differential twisted String- and Fivebrane Structures , [ ][ math.AT ].[42] A.N. Schellekens and N.P. Warner, Anomalies, characters and strings
Nucl. Phys.
B287 (1987) 317–361.[43] R. Stong,
Calculation of Ω spin ( K ( Z, A one-loop test of string duality , Nucl. Phys.
B447 (1995), 261-270,[ arXiv:hep-th/9505053 ].[45] K. Waldorf,
String connections and Chern-Simons theory , [ arXiv:0906.0117 ] [ math.DG ].[46] F. Witt,
Special metric structures and closed forms , DPhil Thesis, University of Oxford, 2004,[ arXiv:math/0502443v2 ] [math.DG] .[47] E. Witten,
Global gravitational anomalies , Commun. Math. Phys. (1985) 197–229.[48] E. Witten,
Topological tools in ten-dimensional physics , in Unified String Theories, M. Green and D.Gross, eds., pp. 400-429, World Scientific, Singapore, 1986.[49] D. Zagier,
Note on the Landweber-Stong elliptic genus , in Elliptic Curves and Modular Forms in Alge-braic Topology, ed. P. Landweber, Springer 1326 (1988).1750] W. Zhang,
Eta invariants and Rokhlin congruences , C. R. Acad. Sci. Paris, Serie I (1992), 305-308.[51] W. Zhang,
Circle bundles, adiabatic limits of eta invariants and Rokhlin congruences , Ann. Inst. Fourier44