Ray-Singer Metrics Research Articles

Flat vector bundles and analytic torsion on orbifolds

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.

Morse-Smale flow, Milnor metric, and dynamical zeta function

We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.

Volterra calculus and the variation formulas for the equivariant Ray–Singer metric

In this paper, using the Greiner's approach to heat kernel asymptotics, we give new proofs of the equivariant Gauss–Bonnet–Chern formula and the variation formulas for the equivariant Ray–Singer metric, which are originally due to Bismut and Zhang.

Twisted analytic torsion and adiabatic limits

We study an analogue of the analytic torsion for elliptic complexes that are graded by $\mathbb{Z}_2$, orignally constructed by Mathai and Wu. Motivated by topological T-duality, Bouwknegt an Mathai study the complex of forms on an odd-dimensional manifold equipped with with the twisted differential $d_H = d+H$, where $H$ is a closed odd-dimensional form. We show that the Ray-Singer metric on this twisted determinant is equal to the untwisted Ray-Singer metric when the determinant lines are identified using a canonical isomorphism. We also study another analytical invariant of the twisted differential, the derived Euler characteristic $\chi'(d_H)$, as defined by Bismut and Zhang.

A Cheeger-Müller theorem for symmetric bilinear torsions on manifolds with boundary

In this paper, we extend Su-Zhang’s Cheeger-Muller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary. Our result also extends Bruning-Ma’s Cheeger-Muller type theorem for Ray-Singer metric on manifolds with boundary to symmetric bilinear torsions in product case. We also compare it with the Ray-Singer analytic torsion on manifolds with boundary.

Anomaly formulas for the complex-valued analytic torsion on compact bordisms

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 order to define the complex-valued analytic torsion in this situation, we study spectral properties of these generalized Laplacians. Then, as main results, we obtain so-called anomaly formulas for this 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 derived first by using the corresponding ones for the Ray–Singer metric, obtained by Brüning and Ma on manifolds with boundary, and then 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.

THE VARIATION FORMULAS FOR THE EQUIVARIANT RAY–SINGER METRIC

We give a new and detailed proof of the variation formulas for the equivariant Ray–Singer metric, which are originally due to J. M. Bismut and W. Zhang.

An anomaly formula for Ray–Singer metrics on manifolds with boundary

Using the heat kernel, we derive first a local Gauss–Bonnet–Chern theorem for manifolds with a non-product metric near the boundary. Then we establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary, not assuming that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary.

An anomaly formula for Ray–Singer metrics on manifolds with boundary

We establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary. We do not assume that the Hermitian metric on the flat vector bundle is flat, nor that the Riemannian metric has product structure near the boundary. To cite this article: J. Brüning, X. Ma, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 603–608.

Combinatorial invariants computing the Ray-Singer analytic torsion

It is shown that for any piecewise-linear closed orientable manifold K of odd dimension there exits an invariantly defined metric on the determinant line of cohomology det( H ∗(K; E) ), where E is an arbitrary flat bundle over K (here E is not required to be unimodular). The construction of this metric (called Poincaré-Reidemeister metric) is purely combinatorial; it combines the standard Reidemeister's construction with the Poincaré duality. The main properties of the Poincaré-Reidemeister metric consist in the following: (a) the Poincaré-Reidemeister metric can be computed starting from any polyhedral cell decomposition of the manifold in purely combinatorial terms; (b) the Poincaré-Reidemeister metric coincides with the Reidemeister metric when the latter is correctly defined (i.e., when the bundle E is unitary or unimodular). (c) The construction of Ray and Singer, which uses zeta-function regularized determinants of Laplacians, produces the metric on the determinant of cohomology, which coincides (via the De Rham isomorphisms) with the Poincaré-Reidemeister metric. This is the main result of the paper, showing that the Poincaré-Reidemeister metric computes combinatorially the Ray-Singer metric. (d) The Poincaré-Reidemeister metric behaves well with respect to natural correspondences between determinant lines which are discussed in the paper. It is shown also that the Ray-Singer metrics on some relative determinant lines can be computed combinatorially (including the even-dimensional case) in terms of the metrics determined by correspondences.

Milnor and ray-singer metrics on the equivariant determinant of a flat vector bundle

In this paper, we extend our previous results relating Milnor and Ray-Singer metrics on the determinant of the cohomology of a flat complex vector bundle to the equivariant case. Thus, we extend Lott and Rothenberg and Luck's theorems relating equivariant combinatorial and analytic torsions to flat vector bundles which are not necessarily unitarily flat.

Refined analytic torsion for twisted de Rham complexes

Let $E$ be a flat complex vector bundle over a closed oriented odd dimensional manifold $M$ endowed with a flat connection $\nabla$. The refined analytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler. Recently Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex with an odd degree closed differential form $H$, other than one form, as a flux and with coefficients in $E$. In this paper we generalize the construction of the refined analytic torsion to the twisted de Rham complex. We show that the refined analytic torsion of the twisted de Rham complex is independent of the choice of the Riemannian metric on $M$ and the Hermitian metric on $E$. We also show that the twisted refined analytic torsion is invariant (under a natural identification) if $H$ is deformed within its cohomology class. We prove a duality theorem, establishing a relationship between the twisted refined analytic torsion corresponding to a flat connection and its dual. We also define the twisted analogue of the Ray-Singer metric and calculate the twisted Ray-Singer metric of the twisted refined analytic torsion. In particular we show that in case that the Hermtitian connection is flat, the twisted refined analytic torsion is an element with the twisted Ray-Singer norm one.