Witten deformation of the analytic torsion and the spectral sequence of a filtration

TL;DR

This paper studies the asymptotic behavior of the Ray-Singer analytic torsion for flat vector bundles over Riemannian manifolds, relating it to Morse functions and spectral sequences, with explicit calculations under rational critical values.

Contribution

It provides an asymptotic expansion of the analytic torsion and explicit formulas for coefficients using spectral sequences, extending previous results with new explicit computations.

Findings

Asymptotic expansion of log(ρ(t)) as t→+∞

Explicit calculation of coefficients when critical values are rational

Connection between analytic torsion and Morse spectral sequence

Abstract

Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^{-2tf}g and let \rho(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field grad(f) satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for \log(\rho(t)) for t\to +\infty of the form a_0+a_1t+b\log\left(\frac t\pi\right)+o(1), where the coefficient b is a half-integer depending only on the Betti numbers of F. In the case where all the critical values of f are rational, we calculate the coefficients a_0 and a_1 explicitly in terms of the spectral sequence of a filtration associated to the Morse function. These results are obtained as an applications of a theorem by Bismut and Zhang.