aa r X i v : . [ m a t h . DG ] M a r arXiv:0912.2184v2[math.DG] February, 2010 (revised)
TWISTED ANALYTIC TORSION
VARGHESE MATHAI AND SIYE WU
Abstract.
We review the Reidemeister torsion, Ray-Singer’s analytic torsion and the Cheeger-M¨uller theorem. We describe the analytic torsion of the de Rham complex twisted by a fluxform introduced by the current authors and recall its properties. We define a new twistedanalytic torsion for the complex of invariant differential forms on the total space of a principalcircle bundle twisted by an invariant flux form. We show that when the dimension is even,such a torsion is invariant under certain deformation of the metric and the flux form. Under T -duality which exchanges the topology of the bundle and the flux form and the radius of thecircular fiber with its inverse, the twisted torsions of invariant forms are inverse to each otherfor any dimension.Keywords: Analytic torsion, circle bundles, T -dualityMathematics Subject Classification (2000): Primary 58J52; Secondary 57Q10, 58J40, 81T30. Dedicated to Professor Yang Lo on the occasion of his 70th birthday
Introduction
Reidemeister torsion (or R -torsion) was introduced by Reidemeister [26] for 3-manifolds. Itwas generalized to higher odd dimensions by Franz [9] and de Rham [7]. As the first invariantthat could distinguish spaces which are homotopic but not homeomorphic, it can be used toclassify lens spaces [9, 8, 19]. Analytic torsion (or Ray-Singer torsion) is a smooth invariant ofcompact manifolds of odd dimensions defined by Ray and Singer [23, 24] as an analytic counter-part of the Reidemeister torsion. They conjectured that the two torsions are equal for compactmanifolds. This Ray-Singer conjecture was proved independently by Cheeger [6] and M¨uller[20]. Another proof of the Cheeger-M¨uller theorem with an extension was given by Bismutand Zhang [2] using Witten’s deformation. In [17], the current authors introduced an analytictorsion for the de Rham complex twisted by a flux form. It shares many properties with theRay-Singer torsion but has several novel features. In this paper, we review these developmentsand introduce a new twisted analytic torsion for the complex of invariant differential forms ona principal circle bundle.The paper is organized as follows. In Section 1, we review Reidemeister’s combinatorialtorsion, Ray and Singer’s analytic torsion and the Cheeger-M¨uller theorem. In Section 2, wedescribe the analytic torsion of the de Rham complex twisted by a flux form, its invarianceunder the deformation of the metric and flux form when the dimension is odd, the relationwith generalized geometry when the flux is a 3-form and the behavior under T -duality for 3-manifolds. In Section 3, we introduce a new twisted analytic torsion for the complex of invariantdifferential forms on the total space of a principal circle bundle twisted by an invariant fluxform. We show that when the dimension is even, such a torsion is invariant under certaindeformation of the metric and the flux form. Under T -duality, which exchanges the topology f the bundle with the flux form and the radius of circular fibers with its inverse, the twistedtorsions are inverse to each other for any dimensions.1. Reidemeister and Ray-Singer torsions
In algebraic topology, several groups can be assigned to a topological space such as itsfundamental group, homology and cohomology groups. For example, the dimension of p -thcohomology group H p ( X, R ) is, roughly speaking, the number of p -dimensional “holes” in thespace X . These groups are invariants in the sense that if two topological spaces are the same,or more precisely homeomorphic, then the corresponding groups are isomorphic. So if we findtwo spaces X and Y with H p ( X, R ) = H p ( Y, R ) for some integer p , then we can conclude that X is not homeomorphic to Y . In this way, we can distinguish different topological spaces.However, these groups are invariant not just under homeomorphisms, but under continuousdeformations, or homotopy. Two spaces can be homotopic but not homeomorphic to each other.As none of the above groups can distinguish two homotopic yet different topological spaces,we need additional invariants to achieve this. Reidemeister torsion is such an invariant that isinvariant under homeomorphisms but not under homotopy. It is a secondary invariant in thesense that it is an element in a certain space constructed from the above groups (which areregarded as the primary invariants).To recall its definition, we consider a topological space of dimension n with a finite triangu-lation K . The simplicial structure gives rise to a cochain complex ( C • ( K ) , δ ); we will take thereal coefficients unless otherwise indicated. Roughly, the Reidemeister torsion is the alternatingproduct of the determinants of the coboundary operators δ on various C p ( K ) ( p = 0 , , . . . , n ).More precisely, we take the adjoint δ † of δ : C p ( K ) → C p +1 ( K ) with respect to the inner productunder which the p -simplices form an orthonormal basis. We define the Laplacians ∆ p = δ † δ + δδ † on C p ( K ). Just like in Hodge theory, we have H p ( K ) ∼ = ker ∆ p . If H p ( K ) = 0 were true forany p , then all ∆ p would be invertible, and we could define the Reidemeister torsion as thenumber [26, 9, 7] τ ( K ) = n Y p =0 (det ∆ p ) ( − p +1 p/ . To take into account of the non-trivial cohomology groups, we choose a unit volume element η p of ker ∆ p ⊂ C p ( K ) for each p . Thus, we have η p ∈ det ker ∆ p ∼ = det H p ( K ). We define theReidemeister torsion τ ( K ) = n Y p =0 (det ′ ∆ p ) ( − p +1 p/ n O p =0 η ( − p p as an element of the line det H • ( K ) = N np =0 det H p ( K ) ( − p constructed from the cohomologygroups. Here, det ′ means taking the determinant of an operator in the subspace orthogonal toits kernel.We make two remarks here. First, in the above construction, the torsion is only defined upto a sign as the unit volume elements depend on the orientation on the cohomology groups. Amore intrinsic way is to define torsion as a norm on the determinant line bundle [2]. Second,the cochains on K can take values in a local system that comes from an orthogonal or unitaryrepresentation of the fundamental group of K or equivalently, a flat vector bundle over theunderlying topological space. The torsion defined would then depend on this data.It can be shown that τ ( K ) is invariant under the subdivision of K and is invariant underhomeomorphism. It also satisfy some functorial properties. Happily, it is not invariant underhomotopy. A celebrated example is the lens space, which is the quotient of the 3-sphere bya finite Abelian group. In fact, the lens spaces are classified by their cohomology groups and he Reidemeister torsion [9, 8, 19]. More recently, equivariant torsion has been used to classifyisometries and quotients of certain symmetric spaces up to diffeomorphisms [14].An alternative way to define the cohomology groups is by differential forms. If X is a smoothmanifold, let d : Ω p ( X ) → Ω p +1 ( X ) be the exterior differentiation of the space of p -forms, with d = 0. The p -th de Rham cohomology is H p dR ( X ) = ker( d : Ω p ( X ) → Ω p +1 ( X ))im( d : Ω p − ( X ) → Ω p ( X )) . The de Rham theorem states that there is a natural isomorphism H p dR ( X ) ∼ = H p ( K ), where K is a triangulation of X .A natural question is how to represent the Reidemeister torsion analytically. Suppose X iscompact, orientable and is equipped with a Riemannian metric. Then we define the Laplacians∆ p = d † d + dd † , where the adjoint is with respect to the inner product on Ω p ( X ) given by themetric. If X is compact, then H p dR ( X ) ∼ = ker ∆ p and is finite dimensional by Hodge theory.Taking the unit volume form η p of ker ∆ p ⊂ Ω p ( X ), the Ray-Singer analytic torsion is τ R S ( X ) = n Y p =0 (Det ′ ∆ p ) ( − p +1 p/ n O p =0 η ( − p p as an element, defined up to a sign, of the line det H • dR ( X ) [23]. The above constructiongeneralizes easily if the differential forms are valued in a flat Hermitian vector bundle. Byreplacing d in the Laplacian with the flat connection, we obtain Ray-Singer torsion that dependson a flat vector bundle.The determinants of the Laplacians, and of many other operators on infinite dimensionalspaces, are defined by regularization using zeta-functions. Let A be a self-adjoint semi-positiveoperator acting on a Hilbert space. Suppose the positive eigenvalues are listed as 0 < λ ≤ λ ≤ · · · , taking into account of multiplicities. Since the eigenvalues λ i typically go to infinityas i → ∞ , it does not make sense to consider their product in the usual sense. We define thezeta-function of A as ζ A ( s ) = ∞ X i =1 λ − si = tr ′ A − s , which is a function of one complex variable s . Here tr ′ means taking the trace in the subspaceorthogonal to ker A . If A is elliptic, then by the heat kernel method, ζ A ( s ) is analytic in s when ℜ s is sufficiently large and it can be extended meromorphically to the complex plane so that itis analytic at s = 0. We define Det ′ A = e − ζ ′ A (0) . When A acts on a finite dimensional space, the above definition reduces to the finite productof the positive eigenvalues of A .The Ray-Singer torsion is invariant under the deformation of the Riemannian metric on X when dim X is odd and therefore is a smooth invariant of odd-dimensional compact manifolds.Furthermore, it satisfies the same functorial properties as the Reidemeister torsion. Ray andSinger [23, 24] therefore conjectured that their torsion is equal to the Reidemeister torsionfor compact odd-dimensional manifolds. The case for lens spaces was verified in [22]. Theconjecture was proved independently by Cheeger [6] and M¨uller [20], when the local coefficientsare given by an orthogonal or unitary representation of the fundamental group. The result isalso true when the representation is unimodular [21]. In [2], Bismut and Zhang extended theCheeger-M¨uller theorem using the deformation of Witten [29]. They established the variationof the torsion under the deformations of the metric on the manifold of arbitrary dimensionsand the Hermitian form on the vector bundle with an arbitrary flat connection. . Analytic torsion of twisted complexes
Let X be a smooth manifold and H , a closed 3-form H called flux form, which has its originin supergravity and string theory [27, 1]. We have an operator d H = d + H ∧ · acting on Ω • ( X ),which squares to zero. So it can be used to define a twisted de Rham cohomology H • ( X, H ).The twisted de Rham complex is only Z -graded and so is the twisted de Rham cohomology. Let Ω ¯0 ( X ), Ω ¯1 ( X ) denote the space of differential forms on X of even, odd degrees, respectively.Then the operator d H acts between these two spaces.In fact, the above flux form can be a closed form of any odd degree with no 1-form component.In [17], we introduced the analytic torsion of the operator d H when X is compact, which we nowassume. The main difficulty was that the twisted de Rham complex does not have a Z -grading.Formally, given a Riemannian metric on X , the twisted analytic torsion is τ ( X, H ) = Det ′ Ω ¯0 ( X ) ( d † H d H ) / Det ′ Ω ¯1 ( X ) ( d † H d H ) − / η ¯0 ⊗ η − , where η ¯ k is a unit volume element of H ¯ k ( X, H ) ( k = 0 , H = d † H d H + d H d † H , the operator d † H d H on Ω ¯ k ( X ) is not elliptic and the usual heat kernelmethods seem inadequate. Instead, we use some properties of pseudo-differential operators. Let P ¯ k be the projection in Ω ¯ k ( X ) onto the image of d † H . Since P ¯ k is a pseudo-differential operator,we have [28] tr( P ¯ k ∆ − sH ) = c − s + c + o ( s ) , upon meromorphically continuing the left-hand side to s = 0. The coefficient c − is essentiallythe non-commutative residue [30, 12] of P ¯ k . It turns out that since P ¯ k is also a projection, c − = 0 [30]. Consequently, the zeta-function of the restriction of d † H d H to the image of d † H isregular at s = 0 and its determinant is defined. It is also possible to include a flat Hermitianvector bundle in the definition of twisted analytic torsion.Just as the usual de Rham cohomology groups, the twisted counterpart H • ( X, H ) is alsoinvariant under homotopies [17]. The twisted torsion τ ( X, H ) satisfies a similar set of functorialproperties as in [23]. Moreover, when X is compact and dim X is odd, τ ( X, H ) is invariantunder the deformation of the Riemannian metric on X , the inner product or Hermitian structureon the flat bundle and the flux form within its cohomology class [17]. Therefore the analytictorsion τ ( X, H ) is a secondary invariant in the same sense but in the twisted setting.When H is a 3-form on X , the deformation of the Riemannian metric g and that of the fluxform H within its cohomology class can be interpreted as a deformation of generalized metricson X [17]. Recall that a generalized metric on X is a splitting T X ⊕ T ∗ X = T + X ⊕ T − X suchthat the bilinear form h x + α, y + β i = ( α ( y ) + β ( x )) / , where x, y are vectors fields and α, β are 1-forms on X , is positive definite (negative definite,respectively) on T ± X and such that h T + X, T − X i = 0 [11]. The subspace T + X is the graph of g + B , where g is a usual Riemannian metric and B is a 2-form. Given a generalized metric,there is an inner product on Ω • ( X ) called the Born-Infeld metric [11]. It can be shown that theeffect of deformation H H − dB on torsion is equivalent to taking the adjoint of the operator d H with respect to the Born-Infeld metric [17]. This amounts to deforming a usual metric g toa generalized one. Thus analytic torsion should be defined for generalized metrics so that thedeformations of g and of H are unified.We consider a special case when H is a top form on an odd-dimensional orientable compactmanifold X . The cohomology class of H is [ H ] ∈ H top ( X, R ) ∼ = R . Then the twisted analytictorsion is [17] τ ( X, H ) = | [ H ] | τ RS ( X ) η ¯0 ⊗ η − , here τ RS ( X ) is the Ray-Singer torsion. This provides examples of twisted torsion for 3-manifolds, when H has to be a 3-form. The calculation of τ ( X, H ) is based on the work ofKontsevich and Vishik [15] on factorization of determinants in odd dimensions, although wehope that there are simpler methods which can be useful in more general cases as well. Thefactor | [ H ] | is also the torsion of the spectral sequence in [27]. It is not clear whether there is asimple relation between the twisted torsion and the classical Ray-Singer torsion in the generalsituation.Whereas Reidemeister’s combinatorial torsion precedes Ray-Singer’s analytic torsion, thesimplicial counterpart of the twisted analytic torsion is still missing in the general case. Thisis because the cup product in the simplicial cochain complex is associative but not gradedcommutative. Let h be a simplicial cocycle that represents the same cohomology class (underthe de Rham isomorphism) as H . Then ( δ + h ∪ · ) = ( h ∪ h ) ∪ · may not be zero. However,the situation simplifies if the degree of H or h is greater than dim X/
2, in which case h ∪ h = 0for dimension reason. In particular, this condition is satisfied if H is a top-degree form. In fact,a twisted version of the Cheeger-M¨uller theorem holds in this case [17].We consider the behavior of the twisted torsion under T -duality. Let T be the circle group and π : X → M , a principal T -bundle with Euler class e ( X ) ∈ H ( M, Z ). Suppose H is a closed3-form on X with integral periods. By the Gysin sequence, there is a dual circle fibrationˆ π : ˆ X → M , whose fiber is the dual circle group ˆ T , with a flux 3-form ˆ H on ˆ X such that [4] π ∗ [ H ] = e ( ˆ X ) , ˆ π ∗ [ ˆ H ] = e ( X )(modulo torsion elements). Thus T -duality for circle bundles is the exchange of background H -flux on the one side and the Chern class on the other. We have the following duality ontwisted cohomology groups [4]: H ¯0 ( ˆ X, ˆ H ) ∼ = H ¯1 ( X, H ) , H ¯1 ( ˆ X, ˆ H ) ∼ = H ¯0 ( X, H ) . Consequently, det H • ( ˆ X, ˆ H ) ∼ = (det H • ( X, H )) − . When dim X = 3, since H is a top-degreeform, we get [17] τ ( ˆ X, ˆ H ) = τ ( X, H ) − under the above identification. In general, it is not known whether there is such a conciserelation. Instead, we will explore an invariant version of the twisted torsion and its behaviorunder T -duality in the next section.Our method applies to other Z -graded complexes [18]. For example, suppose X is a complexmanifold and H ∈ Ω , ¯1 ( X ). Let ¯ ∂ H = ¯ ∂ + H ∧ · . If ¯ ∂H = 0, then ¯ ∂ H = 0 just as inthe de Rham case. Using the same argument, we can define an analytic torsion of the twistedDolbeault complex [18]. Alternatively, we can take H ∈ Ω , ( X ) with ¯ ∂H = 0. Let Ω − p,q ( X ) = Γ ( ∧ p T , ( X ) ⊗ ∧ q ( T , X ) ∗ ). Then H ∧ · : Ω − p,q ( X ) → Ω − ( p − ,q +2 ( X ) and again, ¯ ∂ H = 0.The cohomology of ¯ ∂ H contains information of the deformation of twisted generalized complexstructures [16]. A torsion as a secondary invariant in this case can also be defined. For example, affirmative answers to the following questions on functions of one complex variable wouldprovide a simpler calculation. Suppose a sequence of positive real numbers 0 < λ ≤ λ ≤ · · · goes to infinityfast enough so that the zeta-function ζ ( s ) = P ∞ i =1 λ − si is absolutely convergent and hence analytic when ℜ s issufficiently large. Suppose ζ ( s ) can be meromorphically continued so that it is regular at s = 0. We partitionthe set of positive integers to a disjoint union of finite sets I k ( k = 1 , , . . . ). Let Λ k = Q i ∈ I k λ i . Then Z ( s ) = P ∞ k =1 Λ − sk is absolutely convergent and hence analytic when ℜ s is sufficiently large. Can Z ( s ) bemeromorphically continued so that it is regular at s = 0? If so, is it true that Z ′ (0) = ζ ′ (0)? . T -duality for circle bundles and analytic torsion Analytic torsion for the complex of invariant forms.
Consider a principal T -bundle π : X → M . Suppose X is compact and H is a T -invariant closed 3-form on X . We consider the Z -graded complex ( Ω • ( X ) T , d H ) of T -invariant differential forms on X . At first sight, it seemsdifficult to define torsion, as the asymptotic expansion of the heat kernel with a group action [5],and hence the pole structure of the corresponding zeta-function, is rather complicated. However,given a connection on X , the space Ω • ( X ) T is isomorphic to Ω • ( M ) ⊕ Ω • ( M ). Through thisisomorphism, operators on Ω • ( X ) T acts on sections of bundles over M . The torsion is thendefined by the regularized determinants of elliptic operators on M .Suppose the metric g X on X is T -invariant and such that the length of every circular fiberis equal to some constant r >
0. Then g X = π ∗ g M + r A ⊙ A, where g M is a metric on M , A ∈ Ω ( X ) is a connection 1-form with the normalization R T A = 1and ⊙ stands for the symmetric tensor product. Since any ω ∈ Ω ¯ k ( X ) T can be written uniquelyas ω = r / π ∗ ω + r − / A ∧ π ∗ ω with ω ∈ Ω ¯ k ( M ) and ω ∈ Ω k − ( M ), there is an isomorphism φ : Ω ¯ k ( X ) T → Ω ¯ k ( M ) ⊕ Ω k − ( M )for k = 0 , φ ( ω ) = ( ω , ω ). Lemma 3.1.
The above isomorphism φ is an isometry under the inner products on Ω • ( X ) T and Ω • ( M ) defined by g X and g M , respectively.Proof. We have ∗ X ω = r − / A ∧ π ∗ ( ∗ M ω ) + r / π ∗ ( ∗ M ω ) and thus Z X ω ∧ ∗ X ω = Z M ω ∧ ∗ M ω + Z M ω ∧ ∗ M ω . The result follows. (cid:3)
Let d ¯ k be the restriction of the operator d H to Ω ¯ k ( X ) T and let ˜ d ¯ k = φd ¯ k φ − . If we write H = π ∗ H + A ∧ π ∗ H with H ∈ Ω ( M ) and H ∈ Ω ( M ) and denote by F ∈ Ω ( M ) thecurvature 2-form of A (that is, π ∗ F = dA ), then˜ d ¯ k = (cid:18) d H r − FrH − d H (cid:19) on Ω ¯ k ( M ) ⊕ Ω k − ( M ). Since φ is an isometry, we have ˜ d † ¯ k = φd † ¯ k φ − and ˜∆ ¯ k := ˜ d † ¯ k ˜ d ¯ k +˜ d k − ˜ d † k − = φ ∆ ¯ k φ − .Clearly, ˜∆ ¯ k is a second order elliptic operator on M whose leading symbol is the same as the(untwisted) Laplacian. The projection ˜ P ¯ k = φP ¯ k φ − onto the closure of im( ˜ d † ¯ k ) is a pseudo-differential operator on M of degree zero. By the same argument as in Theorem 2.1 of [17], thezeta-function ζ T ( s, d † ¯ k d ¯ k ) := Tr Ω ¯ k ( X ) T P ¯ k ∆ − s ¯ k = Tr ˜ P ¯ k ˜∆ − s ¯ k is holomorphic at s = 0 and hence we can define the determinantDet ′ T d † ¯ k d ¯ k := Det ′ ˜ d † ¯ k ˜ d ¯ k = e − ζ ′ T (0 ,d † ¯ k d ¯ k ) . The analytic torsion for the T -invariant part of the twisted de Rham complex is then defined,up to a sign, as τ T ( X, H, r ) := (Det ′ T d † ¯0 d ¯0 ) / (Det ′ T d † ¯1 d ¯1 ) − / η ¯0 ⊗ η − ∈ det H • ( X, H ) , here the unit volume elements η ¯ k ( k = 0 ,
1) are as before, since ker(∆ ¯ k ) ∼ = H ¯ k ( X, H ) areinvariant under T . In fact, if ˜ η ¯ k is the unit volume element of ker( ˜∆ ¯ k ), then since φ is anisometry, ˜ η ¯ k = (det φ )( η ¯ k ). Theorem 3.2. If X is compact and dim X is even, then τ T ( X, H, r ) is invariant under thedeformations of g X in the class of T -invariant metrics such that the length of every fiber is r and the deformations of H in the space of T -invariant -forms representing the same cohomologyclass.Proof. If the metric g M is deformed along a path parametrized by u ∈ R , then ∂ ˜ d ¯ k ∂u = 0 , ∂ ˜ d † ¯ k ∂u = − [ ˜ α, ˜ d † ¯ k ] , where ˜ α := ∂ ( ∗ M ) ∂u . If the 3-form H is deformed in its cohomology class with a parameter v ∈ R ,let B ∈ Ω ( X ) be given by ∂H∂v = − dB . Then ∂ ˜ d ¯ k ∂v = [ ˜ β, ˜ d ¯ k ] , ∂ ˜ d † ¯ k ∂v = − [ ˜ β † , ˜ d † ¯ k ] , where ˜ β = φβφ − and β = B ∧ · . Since dim M is odd, the method of § § ′ T ˜ d † ¯0 ˜ d ¯0 ) / (Det ′ T ˜ d † ¯1 ˜ d ¯1 ) − / ˜ η ¯0 ⊗ ˜ η − is invariant under these deformations. In both cases, the isomorphism φ remains fixed. So τ T ( X, H, r ) is also invariant.Finally, if the connection A is deformed with a parameter w ∈ R , let C = − ∂A∂w . Then ∂ ˜ d ¯ k ∂w = [˜ γ, ˜ d ¯ k ] , ∂ ˜ d † ¯ k ∂w = − [˜ γ † , ˜ d † ¯ k ] , where ˜ γ = (cid:0) r − C (cid:1) . Since dim M is odd, following the proof of Lemma 3.5 of [17], we get ∂∂w log[Det ′ ˜ d † ¯0 ˜ d ¯0 (Det ′ ˜ d † ¯1 ˜ d ¯1 ) − ] = 2 X k =0 , ( − k Tr(˜ γ ˜ Q ¯ k ) , where ˜ Q ¯ k = φQ ¯ k φ − is the projection onto ker( ˜∆ ¯ k ). On the other hand, the isomorphism φ varies as w . If φ (0) is the isomorphism at w = 0, then ∂∂w [ φ ◦ ( φ (0) ) − ] = ˜ γ . Since η ¯ k =(det φ ) − (˜ η ¯ k ), following the proof of Lemma 3.7 of [17], we get ∂∂w ( η ¯0 ⊗ η − ) = − X k =0 , ( − k Tr(˜ γ ˜ Q ¯ k ) η ¯0 ⊗ η − . Therefore τ T ( X, H, r ) is invariant under this deformation. (cid:3)
Behavior under T -duality. Recall that if π : X → M is a principal T -bundle and H isa T -invariant closed 3-form with integral periods, the T -dual bundle ˆ π : ˆ X → M is a principalbundle whose structure group is the dual circle group ˆ T . It is topologically determined by c ( ˆ X ) = π ∗ [ H ]. We now explain T -duality at the level of differential forms. We have the ommutative diagram X π (cid:31) (cid:31) ???????? X × M ˆ X ˆ p (cid:31) (cid:31) ??????? p (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) M ˆ X ˆ π (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) where X × M ˆ X denotes the correspondence space. Then p ∗ [ H ] = ˆ p ∗ [ ˆ H ] ∈ H ( X × M ˆ X, Z ).Choosing connection 1-forms A and ˆ A on the circle bundles X and ˆ X , respectively, the formula T ( ω ) = Z T e p ∗ A ∧ ˆ p ∗ ˆ A p ∗ ω, ω ∈ Ω • ( X )gives linear map T : Ω ¯ k ( X ) → Ω k +1 ( ˆ X ), k = 0 ,
1. Similarly, we define S : Ω ¯ k ( X ) → Ω k +1 ( ˆ X )by S (ˆ ω ) = Z ˆ T e − p ∗ A ∧ ˆ p ∗ ˆ A ˆ p ∗ ˆ ω, ˆ ω ∈ Ω • ( ˆ X ) . We next explain the construction of the T -dual flux form ˆ H on ˆ X . Let π ∗ F = dA andˆ π ∗ ˆ F = d ˆ A be the curvatures of the connections A and ˆ A , respectively. Since H − A ∧ ˆ F is abasic differential form on X , we have H = A ∧ π ∗ ˆ F − π ∗ Ω for some Ω ∈ Ω ( M ). Define the T -dual flux ˆ H byˆ H = ˆ π ∗ F ∧ ˆ A − ˆ π ∗ Ω . Then ˆ H is closed. Since d ( p ∗ A ∧ ˆ p ∗ ˆ A ) = − p ∗ H + ˆ p ∗ ˆ H, we have T ◦ d H = d ˆ H ◦ T, d H ◦ S = S ◦ d ˆ H . Therefore T -duality induces isomorphisms on twisted cohomology groups T ∗ : H ¯ k ( X, H ) → H k +1 ( ˆ X, ˆ H ) , k = 0 , S ∗ [4], and there is an isomorphismdet T ∗ : det H • ( X, H ) ∼ = (det H • ( b X, b H )) − . We will relate the twisted analytic torsions under this identification.Given the Riemannian metric g X on X or a triple ( g M , A, r ), we define the T -dual metric onˆ X as g ˆ X = ˆ π ∗ g M + r − ˆ A ⊙ ˆ A or given by the triple ( g M , ˆ A, r − ) so that g ˆ X is ˆ T -invariant and the length of every fiber is r − .We study the T -duality map on invariant differential forms. Lemma 3.3.
Under the above choices of Riemannian metrics, T : Ω ¯ k ( X ) T → Ω k +1 ( ˆ X ) ˆ T , S : Ω ¯ k ( ˆ X ) ˆ T → Ω k +1 ( X ) T are isometries for k = 0 , .Proof. For any ω = r / π ∗ ω + r − / A ∧ π ∗ ω ∈ Ω ¯ k ( X ) T , T ( ω ) = r − / ˆ π ∗ ω + r / ˆ A ∧ ˆ π ∗ ω .The result follows from applying formula in the proof of Lemma 3.1 to both ω and T ( ω ). S isthe inverse of T . (cid:3) heorem 3.4 ( T -duality and analytic torsion for circle bundles) . In the above notations, wehave, up to a sign, (det T ∗ )( τ T ( X, H, r )) = τ b T ( b X, b H ) − ∈ (det H • ( b X, b H , r − )) − . Proof.
We denote the restriction of d ˆ H to Ω ¯ k ( ˆ X ) ˆ T by ˆ d ¯ k . Since T is an isometry, we have T ◦ d † ¯ k = ˆ d † k +1 ◦ T and hence T ◦ ( d † ¯ k d ¯ k ) = ( ˆ d † k +1 ˆ d k +1 ) ◦ T . It follows that T isometrically mapsthe space of H -twisted even (odd) degree harmonic forms on X to the space of ˆ H -twisted odd(even) degree harmonic forms on ˆ X . So T maps the unit volume elements of H • ( X, H ) tothose of H • ( ˆ X, ˆ H ) up to a sign. T also maps isometrically on other eigenspaces, preserving the(positive) eigenvalues. We deduce that ζ T ( s, d † ¯ k d ¯ k ) = ζ ˆ T ( s, ˆ d † k +1 ˆ d k +1 )and the result follows. (cid:3) Acknowledgments.
We thank the referees for helpful comments.
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