Witten Deformation of Ray-singer Analytic Torsion

Let F be a flat vector bundle over a compact Riemannian manifold M and let f : M → R be a self-indexing 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 MorseSmale transversality conditions, we provide an asymptotic expansion for log ρ(t) for t → ∞ of the form a0 + a1t+ b log ( t π ) + o(1). We present explicit formulae for coefficients a0, a1 and b. In particular, we show that b is a half integer. 0. Introduction 0.1. The Ray-Singer analytic torsion. Let M be a compact manifold of dimension n and let F be a flat vector bundle on M . Let g and g be smooth metrics on F and TM respectively. In [RS] Ray and Singer introduced a numerical invariant of these data which is called the Ray-Singer analytic torsion of F and which we shall denote by ρ. 0.2. The Witten deformation. Suppose f : M → R is a Morse function. For t > 0, we denote by g t the smooth metric on F g t = e g . (0.1) Let ρ(t) be the Ray-Singer torsion on F associated to the metrics g t and g TM . Denote by ∇f the gradient vector field of f with respect to the metric g . Let B be the finite set of zeroes of ∇f . We shall assume that the following conditions are satisfied (cf. [BFK3, page 5]): (1) f : M → R is a self-indexing Morse function (i.e. f(x) = index(x) for any critical point x of f). 1991 Mathematics Subject Classification. Primary: 58G26.