Witten deformation of Ray-Singer analytic torsion

TL;DR

This paper studies the asymptotic behavior of Ray-Singer analytic torsion for flat vector bundles over compact Riemannian manifolds, using Witten deformation and Morse theory, providing explicit formulas for the expansion coefficients.

Contribution

It introduces explicit asymptotic formulas for the Ray-Singer torsion under Witten deformation, connecting Morse theory with analytic torsion in a novel way.

Findings

Asymptotic expansion of log( ho(t)) as t→∞ with explicit coefficients

Coefficient b in the expansion is a half-integer

Provides explicit formulas for coefficients a_0, a_1, and b

Abstract

Consider a flat vector bundle F over compact Riemannian manifold M and let f be a self-indexing Morse function on M. Let g be a smooth Euclidean metric on F. Set g_t=exp(-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(t)+o(1). We present explicit formulae for coefficients a_0,a_1 and b. In particular, we show that b is a half integer.