Witten Deformation of the Analytic Torsion and the Spectral Sequence of a Filtration

Let F be a flat vector bundle over a compact Riemannian manifold M and let f : M → R be a Morse function. Let g be a smooth Euclidean metric on F , let g t = e g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g t . Assuming that ∇f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log ρ(t) for t → +∞ of the form a0 + a1t + b log ( t π ) + o(1), where the coefficient b is a halfinteger depending only on the Betti numbers of F . In the case where all the critical values of f are rational, we calculate the coefficients a0 and a1 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. 0. Introduction 0.1. The analytic torsion, ρ , introduced by Ray and Singer [RS], is a numerical invariant associated to a flat Euclidean vector bundle F over a compact Riemannian manifold M . It depends smoothly on the Riemannian metric g on TM and on the Euclidean metric g on F . Let f : M → R be a Morse function. By Witten deformation of the Euclidean bundle F we shall understand the family of metrics g t = e g , t > 0 (0.1) on F . Let ρ(t) be the Ray-Singer torsion of F associated to the metrics g and g t . Burghelea, Friedlander and Kappeler ([BFK3]) have shown (with some additional conditions on f , cf. Section 0.2) that the function log ρ(t) has an asymptotic expansion for t → +∞ of the form