WITTEN DEFORMATION OF THE ANALYTIC TORSION AND THE SPECTRAL SEQUENCE OF A FILTRATION

Abstract. Let F be a flat vector bundle over a compact Rie-mannian manifold M and let f: M → R be a Morse function. Let gF be a smooth Euclidean metric on F, let gFt = e −2tfgF and let ρRS(t) be the Ray-Singer analytic torsion of F associated to the metric gFt. Assuming that ∇f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log ρRS(t) for t → + ∞ of the form a0 + a1t + b log t pi + 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 ra-tional, we calculate the coefficients a0 and a1 explicitly in terms of the spectral sequence of a filtration associated to the Morse func