Six lectures on analytic torsion

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1 Six lectures on analytic torsion Ulrich Bunke May 26, 215 Abstract Contents 1 Analytic torsion - from algebra to analysis - the finite-dimensional case Torsion of chain complexes Torsion and Laplace operators Torsion and Whitehead torsion Zeta regularized determinants of operators - Ray-Singer torsion Motivation Spectral zeta functions Analytic torsion Ray-Singer torsion Torsion for flat bundles on the circle Morse theory, Witten deformation, the Müller-Cheeger theorem Morse theory Analytic torsion and torsion in Homology 22 5 L 2 -torsion 22 NWF I - Mathematik, Universität Regensburg, 934 Regensburg, GERMANY, ulrich.bunke@mathematik.uni-regensburg.de 1

2 1 Analytic torsion - from algebra to analysis - the finite-dimensional case 1.1 Torsion of chain complexes Let k be a field. If V is a finite-dimensional k-vector space and A : V! V an isomorphism, then we have the determinant det A 2 k. The torsion of a chain complex is a generalization of the determinant as we will explain next. If V is a finite-dimensional k-vector space of dimension n, thenwedefinethe determinant of V by det(v ):= n V. This is a one-dimensional k-vector space which functorially depends on V.By definition we have det{} := k. Note that det is a functor from the category of finite-dimensional vector spaces over k and isomorphisms to one-dimensional vector spaces over k and isomorphisms. If L is a one-dimensional vector space, then we have a canonical isomorphism Aut(L) = k. Under this identification the isomorphism det(a) :detv! detv and the functor det induced by an isomorphism A : V! V is exactly mapped to the element det(a) 2 k. We let L 1 := Hom k (L, k) denotethedualk-vector space. We now consider a finite chain complex over k, i.e.achaincomplex C :!C n 1! C n! C n+1!... of finite-dimensional k-vector spaces which is bounded from below and above. Definition 1.1. We define the determinant of the chain complex C to be the onedimensional k-vector space det C := O (det C n ) ( 1)n n The determinant det(c) only depends on the underlying Z-graded vector space of C and not on the di erential. The cohomology of the chain complex C can be considered as a chain complex H(C)with trivial di erentials. Hence the one-dimensional k-vector space det H(C) is well-defined. 2

3 Proposition 1.2. We have a canonical isomorphism C :detc =! det H(C). This isomorphism is called the torsion isomorphism. Proof. For two finite-dimensional k-vector spaces U, W we have a canonical isomorphism det(u W ) = det U det W. More generally, given a short exact sequence (with U in degree ) V :! U! V! W! W and therefore an isomor- we can choose a split. It induces a decomposition V = U phism det U det W = det V. The main observation is that this isomorphism does not depend on the choice of the split. The torsion of the short exact sequence V is the induced isomorphism V :detu (det V ) 1 det W! k. Finally, we can decompose a chain complex C into short exact sequences and construct the torsion inductively by the length of the chain complex. Assume that C starts at n 2 Z. We consider the two short exact sequences and the sequence A :! H n (C)! C n! B n+1!, B :! B n+1! C n+1! C n+1 /B n+1!, C :! C n+1 /B n+1! C n+2!!. Here B n+1 := d(c n ) C n+1 denotes the subspace of boundaries. Note that C starts in degree n +1. Byinduction,theisomorphism C is defined by 1O det(c) = (det C n ) ( 1)n (det C n+1 ) ( 1)n+1 (det C k ) ( 1)k k=n+2 A = (det H n (C)) ( 1)n (det B n+1 ) ( 1)n (det C n+1 ) ( 1)n+1 B = (det H n (C)) ( 1)n (det(c n+1 /B n+1 )) ( 1)n+1 = (det H n (C)) ( 1)n det C C = (det H n (C)) ( 1)n det H(C ) = det H(C). 1O k=n+2 1O k=n+2 (det C k ) ( 1)k (det C k ) ( 1)k 2 3

4 Example 1.3. If A : V! W is an isomorphism of finite-dimensional vector spaces, then we can form the acyclic complex A : V! W with W in degree. Its torsion is an isomorphism A :(detv ) 1 det W! k which corresponds to the generalization of the determinant of A as a morphism det(a) : det V! det W.OnlyforV = W we can interpret this as an element in k. Example 1.4. Let C be a finite chain complex and W be a finite-dimensional k-vector space. Then we form the chain complex W : W id! W W starting at and let n 2 Z. We say that the chain complex C := C from C by a simple expansion. W[n] is obtained If C is obtained from C by a simple expansion, then we have canonical isomorphisms det C = det C and H(C) = H(C ). Under these isomorphisms we have the equality of torsion isomorphisms C = C. 1.2 Torsion and Laplace operators We now assume that k = R or k = C. A metric h V on V is a (hermitean in the case k = C) scalarproductonv. It induces ametrich det V on det V. This metric is fixed by the following property. Let (v i ) i=1,...,n be an orthonormal basis of V with respect to h V,thenv 1 ^ ^v n is a normalized basis vector of det V with respect to h det V. Example 1.5. Let (V,h V )and(w, h W )befinite-dimensionalk-vector spaces with metrics and A : V! W be an isomorphism of k-vector spaces. Then we can choose an isometry U : W! V.Wehavethenumberdet(UA) 2 k which depends on the choice of U. We now observe that det A := det(ua) 2R + (1) does not depend on the choice of U. The analytic torsion generalizes this idea to chain complexes. Ametrich C on a chain complex is a collection of metrics (h Cn ) n.suchametricinduces ametricondetc and therefore, by push-forward, a metric C, h C on det H(C). 4

5 Definition 1.6. Let C be a finite chain complex and h C and h H(C) metrics on C and its cohomology H(C). Then the analytic torsion is defined by the relation T (C,h C,h H(C) ) 2 R + h det H(C) = T (C,h C,h H(C) ) C, h det C. Example 1.7. In general the analytic torsion depends non-trivially on the choice of metrics. For example, if t, s 2 R +,thenwehavetherelation T (C,sh C,th H(C) )=( t s ) (C), where (C) 2 Z denotes the Euler characteristic of C. But observe that if C is acyclic, then (C) =andt (C,h C,h H(C) )doesnotdependonthescaleofthemetrics. Example 1.8. In this example we discuss the dependence of the analytic torsion on the choice of h H(C). Assume that h H(C) i, i =, 1aretwochoices. Thenwedefinenumbers v k (h H(C),h H(C) 1 ) 2 R + uniquely such that v k (h H(C),h H(C) 1 ) h det Hk (C) 1 = h det Hk (C). We further set v(h H(C),h H(C) 1 ):= Y k v k (h H(C),h H(C) 1 ) ( 1)k. Then we have T (C,h C,h H(C) )=v(h H(C),h H(C) 1 ) T (C,h C,h H(C) 1 ). We consider the Z-graded vector space C := M n2z C n and the di erential d : C! C as a linear map of degree one. The metric h C induces a metric h C such that the graded components are orthogonal. Using h C we can define the adjoint d : C! C which has degree 1. We define the Laplace operator :=(d + d ) 2. Since d 2 =and(d ) 2 =wehave degree and therefore decomposes as =dd + d d. Hence the Laplace operator preserves = n2z n. 5

6 As in Hodge theory we have an orthogonal decomposition C = im(d) ker im(d ), ker(d) =im(d) ker. In particular, we get an isomorphism isomorphism of graded vector spaces H(C) = ker( ). This isomorphism induces the Hodge metric h H(C) Hodge on H(C). We let n be the restriction of n to the orthogonal complement of ker( n). Lemma 1.9. We have the equality T (C,h C,h H(C) Hodge )= s Y n2z det( n ) ( 1)n n. Proof. The di erential d induces an isomorphism of vector space d k : C k ker(d C k)? = im(d C k) C k+1. First show inductively that Y det d k ( 1)k+1 C :detc!det H(C) k is an isometry (see (1) for notation), hence T (C,h C,h H(C) Hodge )= " Y k det d k ( 1)k+1 #. (2) We repeat the construction of the torsion isomorphism. But in addition we introduce factors to turn each step into an isometry. Note that det(d k ) 1 det(d k ) : det(ker(d C k)? )! det(im(d C k)) is an isometry. This accounts for the first correction factor in the following chain of 6

7 isometries. det(c) = (det C n ) ( 1)n (det C n+1 ) ( 1)n+1 1 O k=n+2 (det C k ) ( 1)k det d n ( 1)n+1 A! (det H n (C)) ( 1)n (det B n+1 ) ( 1)n (det C n+1 ) ( 1)n+1 1O (det C k ) ( 1)k k=n+2 B = (det H n (C)) ( 1)n (det(c n+1 /B n+1 )) ( 1)n+1 = (det H n (C)) ( 1)n det(c ) Q 1 k=n+1 det d k ( 1)k+1 C = (det H n (C)) ( 1)n det H(C ) = We now observe that 1 Y k=n+1 det H(C). det d k =(detd kd k ) 1/2 =(detd k d k) 1/2. 1O k=n+2 (det C k ) ( 1)k Furthermore, we have This gives det k =det(d kd k )det(d k 1 d k 1). Y det( k2z k) ( 1)kk = Y k2z det(d kd k ) ( 1)k+1 = " # 2 Y det d k ( 1)k+1. (3) The Lemma now follows from (2). 2 k2z 1.3 Torsion and Whitehead torsion Let now G be a group and X be an acyclic based complex of free Z[G]-modules. Whitehead torsion is an element Its (X ) 2 Wh(G). Let : G! SL(N,C) be a finite-dimensional representation of G. It induces a ring homomorphism : Z[G]! End(C N ). Then we can form the complex C := X Z[G] C N. 7

8 Lemma 1.1. This complex is acyclic. Proof. The acyclic complex of free (or more generally, of projective) Z[G]-modules X admits a chain contraction. We get an induced chain contraction of C. 2 The basis of X together with the standard orthonormal basis of C N induces a basis of C which we declare to be orthonormal, thus defining a metric h C.SinceH(C) =wehavea canonical metric h H(C) on H(C) =C and the analytic torsion T (C,h C ):=T (C,h C,h H(C) ) is defined. The representation induces a homomorphism Under this homomorphism K 1 (Z[G])! K 1 (End(C N )) = K 1 (C) = C k.k! R +. K 1 (Z[G]) 3 [±g] 7! det(± (g)) =12 R +. Therefore, by passing through the quotient, we get a well-defined homomorphism : : Wh(G) =K 1 (Z[G])/(±[g])! R +. (4) Proposition The Whitehead torsion and the analytic torsion are related by T (C,h C )= ( (X )). Proof. We can define the Whitehead torsion of based complexes X over Z[G] again inductively by the length. We make the simplifying assumption that the complements of the images of the di erentials are free. Assume that the complex X starts with X n.we consider the short exact sequence of Z[G]-modules and set! X n! X n+1 p! Xn+1 /X n! X :! X n+1 /X n i! X n+2!.... Let c n be the chosen basis of X n.wechooseabasisc n+1 of X n+1 /X n.liftingitselements and combining it with the images of the elements of c n and get a basis b of X n+1. Note that X is again based (by c k := c k for k n +2andthebasisc n+1 chosen above) and starts at n + 1. Then by definition of the Whitehead torsion (X )=[c n+1 /b ] ( 1)n+1 (X ) 2 Wh(G). Note that where T (C,h C )=T (D,h D ) T (C,h C ), D :! X n Z[G] C N! X n+1 Z[G] C N p! X n+1 /X n Z[G] C N! 8

9 starting at n, andweusethemetricsinducedbyc n,c n+1 and c n+1. Here we use (2) and that d n+1 = i p and hence det d n+1 = det p det i. Thereforewemustcheckthat ([c n+1 /b ]) ( 1)n+1 = T (D,h D ). (5) The choice of the lift in the definition of b induces a split of this sequence D. On its middle vector space we have two metrics, one defined by the split, and the other defined by the basis c n+1. If we take the split metric, then its torsion is trivial. Hence T (D,h D ) is equal to the determinant of the base change from b to c n+1,i.e. (5)holdstrue,indeed.2 Example We consider the group Z/5Z and the complex X : Z[Z/5Z] 1 [1] [4]! Z[Z/5Z] starting at. This complex is acyclic since 1 [2] [3] is an inverse of the di erential. We consider the representation Z/5Z! U(1) which sends [1] to exp( 2 i ). Its Whitehead 5 torsion is represented by 1 [1] [4] 2 Z[Z/5Z].Then ([1 [1] [4]) = k1 exp( 2 i 5 ) exp(8 i 5 )k =2cos(2 5 ) 1 6= 1. 2 Zeta regularized determinants of operators - Ray- Singer torsion 2.1 Motivation Let (C,h C) be a finite chain complex over R or C with a metric. Then by Lemma 1.9 we have the following formula for its analytic torsion T (C,h C,h H(C) Hodge )= s Y k (det k ) ( 1)k k. Let now (M,g TM ) be a closed Riemannian manifold manifold. Then we can equip the de Rham complex (M) withaametrich (M) L given by 2 Z h (M) L (, )= ^ 2 g TM, where g TM is the Hodge- operator associated to the metric. More generally, let (V,r V,h V )beavectorbundlewithaflatconnectionandametric. Then we can form the twisted de Rham complex (M,V ). We consider the sheaf V of parallel sections of (V,r). The Rham isomorphism relates the sheaf cohomology of V with the cohomology of the twisted de Rham complex: M H(M,V) = H( (M,V )). 9

10 The metric h V together with the Riemannian metric g TM induce a metric h (M,V ) L on the 2 twisted de Rham complex. Note that to give (V,r) is,uptoisomorphism,equivalenttogivearepresentationofthe fundamental group 1 (M)! End(C dim(v ) ) (we assume M to be connected, for simplicity). Hence (M,V,r V ) is di erentialtopological data, whilethemetricsg TM and h V are additional geometric choices. In this section we discuss the definition of analytic torsion s Y T (M,r V,g TM,h V ):= (det k ) ( 1)k k which is essentially due to Ray-Singer [RS71]. To this end we must define the determinant of the Laplace operators properly. We will also discuss in detail, how the torsion depends on the metrics. 2.2 Spectral zeta functions We consider a finite-dimensional vector space with metric (V,h V )andalinear,invertible, selfadjoint and positive map : V! V. Then the endomorphism log( ) is defined by spectral theory and we have the relation e Tr log =det( ). k The spectral zeta function of is defined by (s) =Tr s, s 2 C. It is an entire function on C and satisfies () = Tr log( ). So we get the formula for the determinant of in terms of the spectral zeta function det = e (). is a di er- The idea is to use this formula to define the determinant in the case where ential operator. We now consider a closed Riemannian manifold (M,g TM )andavectorbundle(v,r V,h V ) with connection and metric. The metrics induce L 2 -scalar products on k (M,V )sothat we can form the adjoint r V, : 1 (M,V )! (M,V ) 1

11 of the connection r V.TheLaplace operator is the di erential operator :=r V, r V : (M,V )! (M,V ). It is symmetric with respect to the metric k.k L 2 := h (M,V ) L. 2 More generally, a second order di erential operator A on (M,V )iscalledof Laplace type of it is of the form A = +R, for definedforcertainchoicesofh TM, h V, r V such that R 2 (M,End(V )) is a selfadjoint bundle endomorphism with respect to the same metrics Assume that A is a Laplace type di erential operator on (M,V )andsymmetricwith respect to a metric k.k L 2.Weconsider(possiblydenselydefinedunbounded)operatorson the Hilbert space closure (M,V ) k.k L 2. The following assertions are standard facts from the analysis of elliptic operators on manifolds. 1. A is an elliptic di erential operator. Indeed, its principal symbol is that of the Laplace operator and given by A ( ) =k k 2 g TM. 2. A is essentially selfadjoint on the domain (M,V ). It is a general fact for a symmetric elliptic operator A on a closed manifold that its closure Ā coincides with the adjoint A. The proof uses elliptic regularity. 3. The spectrum of A is real, discrete of finite multiplicity and accumulates at +1. Since A is essentially selfadjoint the operator A + i is invertible on L 2 (M,V ). Using again elliptic regularity in the quantitative form k k H 2 apple C(kA k L 2 + k k L 2) we see that its inverse can be factored as a composition of a bounded operator and an inlcusion L 2 (M,V ) (A+i)! 1 H 2 (M,V ) incl! L 2 (M,V ). For a closed manifold the inclusion of the second Sobolev space into the L 2 -space is compact by Rellich s theorem. This shows that (A + i) 1 is compact as an operator on L 2. We conclude discreteness of the spectrum. Furthermore, using the positivity of the Laplace operator and the fact that R is bounded, we see that the spectrum accumulates at The number (with multiplicity) of eigenvalues of A less than 2 (, 1) grows as dim(m)/2. This is called Weyl s asymptotic. This follows from the heat asymptotics stated in Proposition A preserves an orthogonal decomposition of (M,V )=N P, such that dim(n) < 1, A N apple and A := A P >. This follows from 3. 11

12 6. A (s) :=Tr A, s is holomorphic for Re(s) > dim(m)/2. This is a consequence of Weyl s asymptotic stated in 4. In order to define A () we need an analytic continuation of the zeta function. Note that for >ands>wehavetheequality s = 1 (s) Z 1 e t t s 1 dt. (6) We define the Mellin transform of a measurable function of t 2 (, 1)! C for s 2 C by provided the integral converges. M( )(s) := Z 1 (t)t s 1 dt, Example 2.1. For 2 (, 1) wehavem(e t )(s) = (s) s. Lemma 2.2. Assume that is exponentially decreasing for t!1, and that it has an asymptotic expansion (t) t! X a n t n n2n for a monotoneously increasing, unbounded sequence ( n ) n2n in R. Then M( )(s) is defined for Re(s) > and has a meromorphic continuation to all of C with first order poles at the points s = n, n 2 N, such that res s= n M( )(s) =a n. Proof. This is an exercise. The idea is to split the integral in the Mellin transformation as R 1 + R 1. The second summand yields an entire function. In order to study the first 1 summand one decomposes (t) asasumofthefirstn terms of its expansion and a remainder. The integral of the asymptotic expansion term can be evaluated explicitly and contributes the singularities for Re(s) > n+1,andtheremaindergivesaholomorphic function on this domain. Since we can choose n arbitrary large we get the assertion. 2 In view of Weyl s asymptotics the heat trace of a Laplace-type operator A on a closed manifold is defined for t>as A (t) :=Tr e ta. Proposition 2.3. Assume that A is a Laplace-type operator on a closed manifold. 1. We have an asymptotic expansion A (t) t! X n a n (A)t n dim(m)/2. (7) 2. A (t) vanishes exponentially as t!1. 12

13 Proof. For a proof of 1. we refer e.g. to [?, Thm. 2.3]. Thesecondassertionisan exercise. 2 Note that the numbers a n (A) areintegralsoverm of local invariants of A. Furthernote that A N (t) issmoothatt =andtherefore A (t) = A (t) A N (t) also has an asymptotic expansion at t! whose singular part coincides with the singular part of (7). But also note that in the odd-dimensional case the positive part of the expansion for A (t) ingeneralhastermswitht m/2 for all m 2 N (not only for odd m). By (6) the spectral zeta function of A can be written in the form A (s) = 1 (s) M( A )(s). By Proposition 2.3 and Lemma 2.2 it has a meromorphic continuation. Since pole at s =wefurtherseethat A (s) isregularits =. (s) hasa Definition 2.4. We define the zeta-regularized determinant of a Laplace-type operator A on a closed manifold by det A := e A (). Remark 2.5. The value A () of the zeta function at zero can be calculated. It is given by the coe cient of the constant term of the asymptotic expansion of A (t). We get A () = a dim(m)/2 (A) dim(n). It is a combination of a locally computable quantity a dim(m)/2 and information about finitely many eigenvalues. Note that the determinant is a much more di cult quantity. Example 2.6. For R > letm be SR 1 A t 2. Lemma 2.7. We have det A = R 2. Proof. Then the eigenvalues of A are given by := R/RZ, i.e. thecircleofvolumer, and 4 2 R 2 n 2, n 2 Z. We can express the spectral zeta function in terms of the Riemann zeta function as A (s) =2 1 2s R 2s 2s (2s). We get A () = 4log(2 R 1 ) () + 4 (). 13

14 Using the formulas we get The final formula () = 1 2, () = log(2 ) 2 A () = 2 log(2 R 1 ) 2log(2 ) = 2log(R). det A = R 2 now follows. 2 Observe the dependence of the determinant on the geometry. 2.3 Analytic torsion We consider a closed Riemannian manifold (M,g TM )andavectorbundle(v,r V,h V ) with a flat connection and metric. The connection r V : (M,V )! 1 (M,V )extends uniquely to a derivation of (M)-modules d V : (M,V )! (M,V ) of degree one and square zero. Then the Laplace operator preserves degree and its components are of Laplace type. :=(d V, + d V ) 2 (8) k : k (M,V )! k (M,V ), k 2 N, (9) Definition 2.8. We define the analytic torsion of (M,r,h TM,h V ) by s Y T an (M,r V,h TM,h V ):= (det k ) ( 1)k k. It is the analog of T ( (M,V ),h (M,V ) L 2,h H(M,V ) Hodge ). As a consequence of Poincaré duality the analytic torsion for unitary flat bundles on even-dimensional manifolds is trivial. We say that a hermitean bundle with connection (V,r V,h V )isunitary, ifr V preserves h V. Unitary flat bundles correspond to unitary representations of the fundamental group. k2n Proposition 2.9. If M is even-dimensional and r V T an (M,r V,g TM,h V )=1. is unitary, then 14

15 Proof. We have k = d V, k dv k d V k 1d V, k 1 and therefore, with appropriate definitions and (3), s Y T an (M,r V,g TM,h V )= (det(d V, k dv k ) ) ( 1)k+1. (1) Using that r V is unitary we get the identity g TM(d V, k dv k ) 1 g TM = d V n k 1 dv, n k 1.Itimplies det(d V, k dv k ) =det(d V, n k 1 dv n k 1). If dim(m) iseven,thenweseethatthefactorsfork and dim(m) k 1in(1)cancel each other. 2 k 2.4 Ray-Singer torsion Let (M,g TM ) be a closed odd-dimensional Riemannian manifold, (V,r V,h V )beavector bundle on M with flat connection and metric, and h H(M,V) be a metric on the cohomology. Definition 2.1. We define the Ray-Singer torsion of (M,r V,h H(M,V) ) by T RS (M,r V,h H(M,V) ):=v(h H(M,V),h H(M,V ) Hodge ) T an (M,r V,g TM,h V ). (11) It is interesting because of the following theorem (which also justifies the omission of the metrics in the notation). Theorem The Ray-Singer torsion T RS (M,r V,h H(M,V) ) is independent of the choices of metrics g TM and h V. Proof. We give a sketch. We first assume that H(M,V) =. As a consequence all integrands below vanish exponentially at t! 1. Moreover, theray-singertorsion coincides with the analytic torsion. Let N denote the Z-grading operator on (M,V ). We define F (s) := 1 Z 1 Tr ( 1) N Ne t t s 1 dt. (s) This integral converges for Re(s) >> andbyproposition2.3andlemma2.2has,asa function of s, ameromorphiccontinuationtoc. Then log T an (M,r V,h TM,h V )= d ds s=f (s). Since any two metric data can be connected by a path, it su ces to discuss the variation formula. The derivative of F (s) withrespecttothemetricdataisgivenby F(s) = Z 1 Tr ( 1) N N (e t )t s 1 dt = 15 Z 1 Tr ( 1) N N ( )e t t s dt.

16 Here we use that ( ) commutes with N and the cyclicity of the trace. In order to calculate ( ) we encode the metric data into a duality map I : (M,V )! (M,V ), h,!i = I( )(!). We further define its logarithmic derivative L := I 1 (I) 2 (M,End( T M V )). We write d V, = I 1 d V, I, where the adjoint d V, of d V does not depend on the metrics. Consequently, (d V, )= [L, d V, ]. Inserting this into (8) we get ( ) = Ld V, d V + d V d V, L d V Ld V, + d V, Ld V. Using [,d V ]=,[,d V, ]=,[d V,N]= d V and [d V,,N]=d V, and the cyclicity of the trace we get Tr ( 1) N N ( )e t = Tr( 1) N L e t = d dt Tr ( 1)N Le t. Here are some more details of the calculation in which we move all di erential operators on the left of L to the right. In this process we must commute them with ( 1) N N,theheatoperator,andweusethecyclicityofthetrace. Tr ( 1) N N ( )e t = Tr ( 1) N N( Ld V, d V + d V d V, L d V Ld V, + d V, Ld V )e t = Tr ( 1) N N( Ld V, d V + Ld V d V, + Ld V, d V Ld V d V, )e t +Tr ( 1) N Ld V, d V e t + Tr ( 1) N Ld V d V, e t 2 We get by partial integration for Re(s) >> (inordertoavoidaboundarytermatt =) F(s) = = 1 (s) s (s) Z 1 Z 1 d Tr ( 1) N Le t t s dt dt Tr ( 1) N Le t t s 1 dt. We now use the asymptotic expansion (a generalization of Proposition 2.3, 1. to traces of the form Tr Le ta,wherel is some bundle endmorphism) Tr ( 1) N Le t t! X n2n b n t n dim(m)/2. (12) In particular it has no constant term. Therefore, by Lemma 2.2 we have F(s) = s (s) apple(s), where apple is meromorphic on C and regular at s =. derivative of the Ray-Singer torsion we must apply has a second order zero at s =weconcludethat Inordertogetthelogarithmic s to this function. Since d ds s= log T RS (M,r V,h H(M,V) )=. 16 (s)

17 In the presence of cohomology one uses a similar argument. In the definition of F (s) one replaces Tr by Tr(1 P ), where P is the projection onto ker( ). Then (12) has a constant term given by Tr ( 1) N PL. Using that res s= (s) =1weget log T an (M,r V,h TM,h V )= Tr( 1) N PL. This is exactly the negative of the logarithmic variation of the volume on the cohomology induced by the Hodge metric, i.e. log v(h H(M,V),h H(M,V ) Hodge )=Tr( 1) N PL. These two terms cancel in the product defining the Ray-Singer torsion. Remark The arguments (M,r V,h H(M,V) )oftheray-singertorsionaredi erentialtopological data. The right-hand side is of global analytic nature and apriori depends on additional geometric choices. The interesting fact is that its actually does not depend on these choices. This is a typical situation in which the natural question is now to provide an explicit description of this quantity in terms of di erential topology. 2.5 Torsion for flat bundles on the circle In this example we give an explicit calculation of the Ray-Singer torsion for M = S 1 and the flat line (V,r V )bundlewithholonomy16= 2 U(1). We have H(S 1, V) =sothat we can drop the metric on the cohomology from the notation. Proposition We have T RS (S 1, r V )= 1 2sin( q). Proof. We represent S 1 := R/Z in order to fix the geometry. In order to calculate the spectrum of the Laplace operator we work on the universal covering R and trivialize the bundle T R using the section dt. Wefurthertrivializethepull-backoftheflatlinebundle using parallel sections. Under these identifications 1 (S 1,L)={f 2 C 1 (R) (8t 2 R f(t +1)= f(t))}. The Laplace operator 1 acts t 2. We now calculate its spectrum. We choose q 2 (, 1) such that = e 2 iq.theeigenvectors of 1 are the functions t 7! e 2 i(q+n)t for n 2 Z, andthecorrespondingeigenvaluesare given by 4 2 (q + n) 2. The zeta function of the Laplace operator is now 1 =4 s 2s X n2z(q + n) 2s. 17

18 In order to calculate det function 1 we express this zeta function in terms of the Hurwitz zeta (s, q) := X n2n (q + n) s and then use known properties of the latter. We have 1 (s) =4 s 2s [ (2s, q)+ (2s, 1 q)]. We have the relation This q (s, q) = s (s q [ (s, q)+ (s, 1 q)] = [ (s+1, 1 q) (s+1,q)]+s[@ s (s+1, 1 s (s+1,q)]. We now use the expansion of the Hurwitz zeta function at s =1 with (s, q) =(s 1) 1 (q)+o(s 1) (q) := (q) (q). We see that the two di erences are regular at s =1. Theevaluationofthesecondterm at s =vanishesbecauseoftheprefactors. s q [ (s, q)+ (s, 1 q)] = (q) (1 q). Consequently, integrating from q =1/2 wegetusing that where (q) (1 q) = sin( s s= [ (s, q)+ (s, 1 q)] = +log( (q) (1 q)) = +log( sin q ), := 2@ s s= (s, 1/2) log( ). In order to determine this number we express the Hurwitz zeta function in terms of the Riemann zeta function (s, 1/2) = X (n +1/2) s =2 X s (2n +1) s n2n n2n = 2 X s (2n +1) s +2 X s (2n) s 2 X s (2n) s n2n n2n n2n + = (2 s 1) (s). 18

19 We get, using () = 1 s s= (, 1/2) = (log(2)2 s (s)+(2 s 1) (s)) s= = 1 2 log(2). Finally, = log(2) log( ). s s= [ (s, q)+ (s, 1 q)] = log(2) log(sin( q)). We have (,q)=1/2 q. This implies [ (2s, q)+ (2s, 1 q)] s= =. Wenowcalculate () 1 s s= 4 s 2s [ (2s, q)+ (2s, 1 q)] = 2log(2) 2log(sin( q)). We thus get We finally get det 1 =4sin 2 ( q). T RS (S 1, r V )= 1 2sin( q). 2 We now compare this result with an evaluation of the Reidemeister-Franz torsion (to be defined later) T RF (S 1, r V ). Using the standard cell decomposition S 1 = 1 /@ 1 the Reidemeister-Franz torsion is the analytic torsion T (C,h C )ofthechaincomplexc given by C : C 1! C starting at, where h C is the canonical metric. We have det 1 = (1 )(1 1 ) =2 1 =2 2cos(2 q). We calculate This gives We observe that 2 2cos(2 q) =2 2(1 2sin 2 ( q)) = 4 sin 2 ( q). T RF (S 1, r V )=T (C,h C )= 1 2sin( q). T RF (S 1, r V )=T RS (S 1, r V ). The equality T RF = T RS in general is the contents of the Cheeger-Müller theorem. 19