2 4 Ja n 20 13 ANOMALY FORMULAS FOR THE COMPLEX-VALUED ANALYTIC TORSION ON COMPACT BORDISMS OSMAR

We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bilinear form are used to define non-selfadjoint Laplacians acting on vector-valued smooth forms under absolute and relative boundary conditions. In the process to define the complex-valued analytic torsion, we study spectral properties associated to these generalized Laplacians. As main results, we obtain anomaly formulas for the complex-valued analytic torsion. Our reasoning takes into account that the coefficients in the heat trace asymptotic expansion associated to the boundary value problem under consideration, are locally computable. The anomaly formulas for the complex-valued Ray--Singer torsion are obtained by using the corresponding ones for the Ray--Singer metric, obtained by Bruening and Ma on manifolds with boundary, and an argument of analytic continuation. In odd dimensions, our anomaly formulas are in accord with the corresponding results of Su, without requiring the variations of the Riemannian metric and bilinear structures to be supported in the interior of the manifold.


esponding one
for the Ray-Singer metric, obtained by Brüning and Ma on manifolds with boundary, and an argument of analytic continuation.In odd dimensions, our anomaly formulas are in accord with the corresponding results of Su, without requiring the variations of the Riemannian metric and bilinear structures to be supported in the interior of the manifold.

Introduction

In this paper, we denote by (M, ∂ + M, ∂ − M ) a compact Riemannian bordism.That is, M is a compact Riemannian manifold of dimensi n m, with Riemannian metric g, whose boundary ∂M is the disjoint union of two closed submanifolds ∂ + M and ∂ − M .For E a flat complex vector bundle over M , we consider generalized Laplacians acting on the space Ω(M ; E) of E-valued smooth differential forms on M satisfying absolute boundary conditions on ∂ + M and relative boundary conditions on ∂ − M .

We study the complex-valued Ray-Singer torsion on (M, ∂ + M, ∂ − M ).This torsion was introduced by Burghelea and Haller on closed manifolds, see [4] and [5], as a complex-valued version for the real-valued Ray-Singer torsion, originally studied by Ray and Singer in [21] for unitary flat vector b ndles on closed manifolds.Our main results are Theorem 2 and Theorem 3. In Theorem 3, we provide so-called anomaly formulas providing a logarithmic derivative for the complex-valued analytic torsion on compact Riemannian bordisms and its proof is based on the work by Brüning and Ma in [8] for the real-valued Ray-Singer torsion on manifolds with boundary.

The classical (real-valued) Ray-Singer analytic torsion, see [21], [17], [10], [19] and others, is defined in terms of a selfadjoint Laplacian ∆ E,g,h , constructed by using a Hermitian metric on the bundle, the Riemannian metric g and a flat connection ∇ E on E. In this paper ∆ E,g,h is referred as the Hermitian Laplacian.In [7], Bismut and Zhang interpreted the analytic torsion as a Hermitian metric in certain determinant line, and called it the Ray-Singer metric, see also [9].In this paper, we also adopt this approach.The Ray-Singer metri on manifolds with boundary has been intensively studied by several authors, among them [21], [10], [19], [20], [17], [11], [8] and [9].In particular, we are interested in the work of Brüning and Ma in [8], where the variation of the Ray-Singer metric, with respect to smooth variations on the underlying Riemannian and Hermitian metrics, was computed.

In order to define the complex-valued Ray-Singer torsion, we assume E admits a fiberwise nondegenerate symmetric bilinear form b and we proceed as in [4].The bilinear form b and the Riemannian metric g induce a nond generate symmetric bilinear form on Ω(M ; E) which is denoted by β g,b .With this data, one constructs generalized Laplacians ∆ E,g,b : Ω(M ; E) → Ω(M ; E), also referred as bilinear Laplacians.These generalized Laplacians are formally symmetric, with respect to β g,b on the space of smooth forms satisfying the boundary conditions specified above.

In Section 1, we use known theory on boundary value problems for differential operators to treat ellipticity, regularity and spectral properties for ∆ E,g,b .In particular, under the specified elliptic boundary conditions, ∆ E,g,b extends to a not necessarily selfadjoint closed unbounded operator in the L 2norm, it has compact resolvent and discrete spectrum, all its eigenvalues are of finite multiplicity, its (generalized) eigenspaces contain smooth differential forms only and the restriction of β g,b to each of these is also a nondegenerate bilinear form.Proposition 2 gives Hodge decomposition results in this setting, which are analog to the Hermitian situation, described for instance in [10], [19], [17] and more recently in [9].Section 1 ends with Proposition 3 stating that the 0-generalized eigenspace of ∆ E,g,b still computes relative cohomology H(M, ∂ − M ; E), without necessarily being isomorphic to it.

In Section 2, we recall generalities on the coefficients of the heat kernel asymptotic expansion for an elliptic boundary value problem.These coefficients are spectral invariants and locally computable as polynomial functions in the jets of the symbols of the operators under consideration, see [14], [22], [23] and [24].This fact provides the key ingredient in the proofs of Theorem 2, leading to Theorem 3. In [8], based on the computation of the coefficients of the constant terms in the heat trace asymptotic expansion for the Hermitian Laplacian under absolute boundary conditions, Brüning and Ma obtained anomaly formulas for the Ray-Singer metric.First, we use Poincaré duality in terms of Lemma 6, to infer from [8], the corresponding coefficients for the Hermitian Laplacian under relative boundary conditions and then we derive those corresponding to Hermitian Laplacian on the bordism (M, ∂ + M, ∂ − M ) under absolute and relative boundary conditions, see Proposition 5 and Theorem 1.We point out here that the anomaly formulas for the Ray-Singer metric in Theorem 1 were also obtained by Brüning and Ma in [9] continuing their work in [8].Next, in Lemma 10, we point out the holomorphic dependance of these coefficients on a complex parameter.Finally, an analytic continuation argument allows one to deduce the infinitesimal variation of these quantities for the bilinear Laplacian on the bordism (M, ∂ + M, ∂ − M ) from those corresponding to the Hermitian one, see Theorem 2.

In Section 3, we use the results from Section 1 and Section 2 to define the complex-valued analytic torsion on a com tain a nondegenerate bilinear form on the determinant line det(H(M, ∂ − M ; E)), denoted by τ E,g,b (0) and induced by the restriction of β g,b to the generalized 0-eigenspace of ∆ E,g,b .The (inverse square of) the complex-valued Ray-Singer torsion for manifolds with boundary is
τ RS E,g,b := τ E,g,b (0) • p det ′ (∆ E,g,b,p ) (−1) p p ,
where the product above is, in this situation, a non zero complex number with det ′ (∆ E,g,b,p ) being the ζ-regularized product of all non-zero eigenvalues of ∆ E,g,b,p .For closed manifolds, the variation of the complex analytic Ray-Singer torsion, with respect to smooth changes on the metric g and the bilinear form b, has been obtained in [4,Sections 7 and 8].Burghelea and Haller obtained in [4, Theorem 4.2] a geometric invariant by introducing appropriate correction terms.In [25], by using techniques from [26], [27], [10] and [19], Su generalized the complex-valued analytic Ray-Singer torsion to the situation in which
∂ + M = ∅ (or ∂ − M = ∅).
Also in [25], Su proved that in odd dimensions, the complex-valued analytic torsion does depend neither on smooth variations of the Riemannian metric nor on smooth variations of the bilinear form, as long as these are compactly supported in he interior of M .This section ends with Theorem 3, which gives formulas for the variation of the complex-valued analytic Ray-Singer torsion with respect to smooth variations of the metric and the bilinear form.In analogy with the re ults in [4], the anomaly formulas for the complex-valued Ray-Singer torsion are obtained by using the results for the coefficients of the constant term in the heat trace asymptotic expansion for the bilinear Laplacian obtained in Section 2.

In the Appendix, see Section 4, for the reader's convenience, we recall some formalism l ading to the characteristic forms appearing in the anomaly formulas stated in Proposition 4, Proposition 5, Theorem 1, Theorem 2 and Theorem 3.

The anomaly formulas given in Theorem 3 generalize the ones obtained by Burghe

a and Haller in the closed situation in [4], and also th
ones in [25] by Su in odd dimensions: they do not longer require g and b to be constant in a neighborhood of the boundary and both kind of boundary conditions are considered at the same time.

Ackowledgements.This paper has been written as part of a PhD thesis at the university of Vienna.I am deeply grateful to my supervisor Stefan Haller for useful discussions, his comments and important remarks on this work.


Bilinear Laplacians and Hodge decomposition on bordisms

1.1.Some background and notation.Let (M, ∂ + M, ∂ − M ) be a compact Riemannian bordism of dimension m.More precisely, M is a compact connected not necessarily orientable smooth manifold of dimension m with Riemannian metric g, whose boundary ∂M is the disjoint union of two closed submanifolds, ∂ + M and ∂ − M , and it inherits the Riemannian metric from M .We do not require the metric to satisfy any condition near the boundary.We denote by T M and T * M (resp.T ∂M and T * ∂M ) the tangent and cotangent bundle of M (resp.∂M ) respectively.We denote by ς in the geodesic unit inwards pointing normal vector field on the boundary.Let Θ M (resp.Θ ∂M ) be the orientation bundle of T M (resp.T ∂M ), considered as the flat real line bundle det(T * M ) → M (resp.det(T * ∂M ) → ∂M ) with transition functions {±1}, endowed with the unique flat connection specified by the de-Rham differential on (twisted) forms, see [3, page 88].For the canonical embedding i : ∂M ֒→ M , we write Θ M | ∂M := i * Θ M and, as real line bundles over ∂M , Θ M | ∂M and Θ ∂M are identified as follows: over the boundary, a section β of det(T * ∂ ) is identified with the section −ς in ∧ β of det(T * M )| ∂M , where ς in := g(•, ς in ) is the 1-form dual to ς in .For T M and T ∂M , the corresponding Levi-Cività connections are denoted by ∇ and by ∇ ∂ respectively.Recall the Hodge ⋆-operator ⋆ q := ⋆ g,q : Ω q (M ) → Ω m−q (M ; Θ M ), i.e., the linear isomorphism defined by α ∧ ⋆α ′ = α, α ′ g vol g (M ), for α, α ′ ∈ Ω q (M ) and 0 q m, where vol g (M ) ∈ Ω m (M ; Θ M ) is the volume form of M .

In this paper, we consider a flat complex vector bundle E over M , with a flat connection ∇ E , and denote by Ω(M ; E) be the space of E-valued smooth differential forms on M , endowed with the de-Rahm differential d E := d ∇ E .Moreover, assume E is endowed with a fiber-wise nondegenerate symmetric bilinear form b. We denote by E ′ the flat complex vector bundle dual to E with the induced flat connection ∇ E ′ and bilinear form b ′ dual to ∇ E and b respectively.Recal that one is always able to fix a (positive definite) Hermitian structure on E (in Section 2.3, we choose for instance a Hermitian structure compatible with the nondegenerate symmetric bilinear form).By choosing a Hermitian structure on E and using the Riemannian metric on M , consider the induced L 2 -norm on Ω(M ; E) and denote by L 2 (M ; E) its L 2 -completion.Recall that chosen Hermitian and Riemannian structures.

1.2.Generalized Laplacians on compact bordisms.As a first step to define the complex-valued anal eralized Laplacians which were introduced in [4] on closed manifolds.The nondegenerate symmetric bilinear form b on E and the Riemannian metric g on M permit to define a nondegenerate symmetric bilinear := ⋆ q ⊗ b : Ω q (M ; E) → Ω m−q (M ; E ′ ⊗ Θ M )
is defined by using the Hodge ⋆-operator ⋆ q and the isomorphism of vector bundles betwe ne defines d ♯ E,g,b,q : Ω q (M ; E)
→ Ω q−1 (M ; E) by (1) d ♯ E,g,b,q := (−1) q ⋆ b,q−1 −1 d E ′ ⊗Θ M ,m−q ⋆ b,q ,
where ⋆ b,q−1 −1 is the inverse of ⋆ b,q−1 and d E ′ ⊗Θ M is the de-Rham differential on Ω(M ; E ′ ⊗ Θ M ) induced by the dual connection on E ′ .It can easily be checked that d ♯ E,g,b is a cod principal symbol is a scalar positive real number, i.e, ∆ E,g,b is elliptic.For simplicity, the operator ∆ E,g,b in (2) will be called the bilinear Laplacian.A straightforward use of Stokes' Theorem leads to the Green's formulas:
β g,b (d E v,w)−β g,b (v,d ♯ E,g,b w) = ∂M i * (Tr(v∧⋆ b w)), β g,b (∆ E v,w)−β g,b )| B :=    w∈Ω(M ;E) i * + ⋆ b w=0, i * − w=0 i * + d ♯ E ′ ⊗Θ M ,g,b ⋆ b w=0, i * − d ♯ E,g,b w=0    .
For simpli on (M, ∂ + M, ∂ − M ).The integrants on the right of formulas in (3) vanish, on forms in Ω(M ; E)| B .The boundary conditions in (4) are an example of mixed boundary conditions, which provide elliptic boundary conditions for operators of Laplace type, see [13].Now we describe boundary operators implementing the boundary conditions in (4).Consider E ± := i * ± E and for 1 q m define ( 5)
B E,g,b : Ω q (M ; E) −→ Ω q−1 (∂ + M ; E + ) ⊕ Ω q (∂ + M ; E + ) ⊕ Ω q (∂ − M ; E − ) ⊕ Ω q−1 (∂ − M ; E − ) w → (B + w, B − w),
wh q−1 (∂ + M ; E + ) ⊕ := ⋆ ∂M b −1 i * + d ♯ E ′ ⊗Θ M ,g,b ′ v, d ♯ E,g,b w) = 0, (e) If v, w ∈ Ω(M ; E)| B 0 , then β g,b (d E v, w) = β g,b (v, d ♯ E,g,b w), (f ) If v, w ∈ Ω(M ; E)| B , then β g,b (∆ E,g,b v, w) = β g,b (v, ∆ E,g,b w).
Proof.The first assertion is obvious.The remaining assertions follow from (8), (4), the Green's formulas in (3) and straightforward manipulations coming from the definition of the operators and spaces above.

1.4.Boundary conditions and Poincaré duality.Consider the Riemannian bordism (M, ∂ + M, ∂ − M ).The boundary value problem specified by the operator ∆ E,g,b acting on the space Ω(M ; E)| B as defined by (4), will be denoted by (9) [∆, B] E,g,b
(M,∂ + M,∂ − M ) .
Let us denote by
(M, ∂ + M, ∂ − M ) ′ := (M, ∂ − M, ∂ + M ) the dual bordism to (M, ∂ + M, ∂ − M ). Then, we Θ M -valued forms (where the flat complex vector bundle E ′ is endowed with the dual connection ∇ E ′ and dual bilinear form b ′ ) under the boundary conditions specified by the vanishing of the boundary operator B ′ , i.e., the same operator from (5) but associated to (M, ∂ + M, ∂ − M ) ′ .The boundary value problem e by means of the Hodge ⋆-operator.Indeed, by the very definition of these operators, we have the equality
⋆ b d ♯ E,g,b d E = d E ′ ⊗Θ M d ♯ E ′ ⊗Θ M ,g,b ′ ⋆ b so that ⋆ b ∆ E,g,b = ∆ E ′ ⊗Θ M ,g,b ′ ⋆ b , and w ∈ Ω q (M ; E)| B ⇐⇒ ⋆ b w ∈ Ω m−q (M ; E ′ ⊗ Θ M )| B ′ .
That is, the Hodge-⋆ b -operator intertwines the roles of ∂ + M and ∂ − M in ( 9) and its du ∆, B] E,g,b (M,∅,∂M )
) is the boundary value problem where absolute (resp.relative) boundary conditions only are imposed on ∂M .1.5.Hermitian boundary value problems.We recall some facts for the Hermitian situation.By using a Hermitian structure h on E, instead of the bilinear form b, all over in the considerations above, one has ≪ v, w ≫ g,h := M Tr(v ∧ ⋆ h w) a Hermitian product on Ω(M ; E), where ⋆ h is in this ca elfadjoint with respect to ≪ v, w ≫ g,h , under abso- lute/relative boundary conditions on (M, ∂ + M, ∂ − M ). Let Ω(M ; E)| h
B be the space of E-valued smooth forms satisfying absolute/relative boundary co (4) but using instead the Hermitian form h. In order to distinguish this problem from the bilinear one, we refer to it as the Hermitian boundary value problem.

The Hermitian boundary value problem is an elliptic boundary value problem, see [12] and [13].This permits one to consider ∆ E,g,h , as an unbounded operator in the L 2 -norm and extend it to a selfadjoint operator with domain of definition being the H 2 -Sobolev closure of Ω(M ; E)| h B ; see [17], [10], [19], [12] and [13].In par ; E) is the space ker (∆ E,g,h ) ∩ Ω q (M ; E)| h
B of q-Harmonic forms satisfying boun ary conditions, then [17,Theorem 1.10] (see also [19, page 239
]) states that for each v ∈ Ω q (M ; E)| h B 0 , there exist unique v 0 ∈ H q ∆ B ( 2 ∈ d * E,g,h (Ω q+1 (M ; E)| h B 0 ) such that v = v 0 + v 1 + v 2 ,
where we have used the notation suggested in (8) associated to h.Moreover, the Hodge-De-Rham tells us that relative cohomology exactly coincides with the ∆ B (M ; E) ∼ = H q (M, ∂ − M ; E).
In the bil

and Ma), Proposition 5 and Lemma 7.More recently, B
, based on the methods developed in [8].

2.3.Involutions, bilinear and Hermitian forms.We fix a Hermitian structure compatible with the bilinear one as follows.Since E is endowed with a bilinear form b, there exists an anti-linear involution ν on E satisfying Remark that ∇ E ν = 0 is not required so that
h −1 ( ∇ E h) = ν −1 b −1 ( ∇ E b) ν + ν −1 ( ∇ E ν).
Therefore, this yields a Hermitian form on Ω(M ; E) compatible with β g,b in the sense that ≪ v, w ≫ g,h = β g,b (v, νw).for v, w ∈ Ω(M ; E).In [26] and [25], given a bilinear form b, this involution has been exploited to study the bilinear Laplacian in terms of the Hermitian one associated to the compatible Hermitian form in (36), in both cases with and without boundary.However, our approach is a little different since we do not use a Hermitian form globally compatible wit

β g,b on
(M ; E), but instead a local compatibility only, see section 2.4 below.We now study the situation where ν is parallel with respect to ∇ E .Lemma 8. Let us consider (M, ∂ + M, ∂ − M ) t e compact Riemannian bordism together with the complex flat vector bundle E as above.Suppose E admits a nondegenerate symmetric bilinear form.Moreover, suppose there exists a complex anti-linear nvolution ν on E, satisfying the conditions in (35) and ∇ E ν = 0. Let h be the (positive definite) Hermitian form on E compatible with b defined by (36).Then,
∆ E,g,b = ∆ E,g,h and B E,g,b = B E,g,h .
Proof.Consider ≪ •, • ≫ g,h the Hermitian product on Ω(M ; E), compatible with the bilinear form, and d * E,g,h , the formal adjoint to d E with respect to this product, which in terms of the Hodge ⋆-operator can be written up to a sign as
d * E,g,h = ±⋆ −1 h d E ⋆ h . Remark that ∇ E ν = 0 implies that d E ν = ν d E ; hence, with ⋆ h = ν • ⋆ b , we have (37) d * E,g,h = ± ⋆ −1 h d E ⋆ h = ± therefore the Hermitian and bilinear Laplacians coincide.We turn to the assertion for the corresponding boundary operators.On the one hand, the assertion is clear for B − E,g,b = B − E,g,h , because of (37) and (7).On the

her h
nd, for a form v ∈ Ω p (M ; E) and ι ς in , the interior product with respect to the dual form corresponding to ς in , the identity Proof.Since flat vector bundles are locally trivial, there exists a neighborhood V of x and a parallel complex anti-linear involution ν on E| V .Moreover, since b is nondegenerate and ν an involution, we can assume without loss of generality that ν can be chosen to be compatible with b at the fiber E x over x, such that b x (νe 1 , νe 2 ) = b x (e 1 , e 2 ) for all e i ∈ E (iii) For each s ∈ R with |s| 1, consider [∆, Ω B ] E,g,hs (M,∂ + M,∂ − M ) the corresponding Hermitian boundary value problem.Then, there exists a neighb rhood U of x such that
⋆ ∂M b i * ι ς in v = i * ⋆ M b v∆ E,g,b s−i | U = ∆ E,g,hs | U and B E,g,b s−i | U = B E,g,hs | U .
Proof.By Lemma 9.(i), for each x ∈ M , there exists a globally defined fiberwise symmetric bilinear form b on E such that the formula b z := b + z b in (38) defines a family of fiberwise nondegenerate symmetric bilinear forms on E, satisfying the required property in (i).In addition, we know that for each x ∈ M , there exist an open neighborhood V of x and a parallel complex anti-linear involution ν on E| V .By Lemma 9.(i)-(ii), we also know that we can find U ⊂ V a small enough open neighborhood of x, such that b s−i satisfies the conditions (i) and (ii) on E| U , for |s| 1.Hence, by using the formula in (36), we obtain a fiberwise positive definite Hermitian form compatible with b s−i on E| U given by h U s (e 1 , e 2 ) := b s−i (νe 1 , e 2 ).Now we extend h U s to a (positive definite) Hermitian form on E as follows.We take h ′ any arbitrary Hermitian form on E and consider the finite open covering
{U ′ 0 , U ′ 1 .