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

Osmar Maldonado Molina

doi:10.1016/j.difgeo.2013.04.003

Abstract

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.

Highlights

In this paper, we denote by (M, ∂+M, ∂−M ) a compact Riemannian bordism
We study the complex-valued Ray–Singer torsion on (M, ∂+M, ∂−M )
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

Summary

Introduction

We denote by (M, ∂+M, ∂−M ) a compact Riemannian bordism. That is, M is a compact Riemannian manifold of dimension m, with Riemannian metric g, whose boundary ∂M is the disjoint union of two closed submanifolds ∂+M and ∂−M. 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. 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]. 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. 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

Heat trace asymptotic expansion and anomaly formulas

Complex-valued analytic torsion on compact Bordisms