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

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Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let gₜ=e-2tfg and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric gₜ. Assuming that the vector field grad(f) satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for (ρ(t)) for t→ +∞ of the form a₀+a₁t+b(tπ)+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 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.