Asymptotic expansion of the Witten deformation of the analytic torsion

Given a compact Riemannian manifold (M^d, g), a finite dimensional representation ρ:π₁(M) → GL(V) of the fundamental group π₁(M) on a vector space V of dimension l and a Hermitian structure μ on the flat vector bundle ℰ → ^p M associated to ρ, Ray-Singer [RS] have introduced the analytic torsion T = T(M,ρ,g,μ) > 0. Witten's deformation d_q(t) of the exterior derivative d_q, d_q(t) = e^-htd_qe^ht, with h: M → R a smooth Morse function, can be used to define a deformation T(h, t) > 0 of the analytic torsion T with T(h, 0) = T. The main results of this paper are to provide, assuming that grad _gh is Morse Smale, an asymptotic expansion for log T(h, t) for t → ∞ of the form Σ^d+1_j=0 a_jt^j + b log t + O(1/√t) and to present two different formulae for a₀. As an application we obtain a shorter derivation of results due to Ray-Singer [RS], Cheeger [Ch], Müller [Mu1, 2] which, in increasing generality, concern the equality for odd dimensional manifolds of the analytic torsion with the average of the Reidemeister torsion corresponding to the triangulation script capital T sign = (h, g) and the dual triangulation script capital T sign _{script D} = (d-h, g).