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

Abstract

LetF be a flat vector bundle over a compact Riemannian manifoldM and letf :M → ℝ be a Morse function. Letg F be a smooth Euclidean metric onF, letg =e −2tf g F , and letρ RS (t) be the Ray-Singer analytic torsion ofF associated with the metricg . Assuming that ∇f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for logρ RS (t) fort→+∞ of the forma 0+a 1 t+blog(t/π)+o(1), where the coefficientb is a half-integer depending only on the Betti numbers ofF. In the case where all the critical values off are rational, we calculate the coefficientsa 0 anda 1 explicitly in terms of the spectral sequence of a filtration associated with the Morse function. These results are obtained as applications of a theorem by Bismut and Zhang.