Asymptotic Expansion of the Witten Deformation of the Analytic Torsion

Given a compact Riemannian manifold (Md, g), a finite dimensional representationρ: π1(M)→GL(V) of the fundamental groupπ1(M) on a vector spaceVof dimensionland a Hermitian structureμon the flat vector bundle[formula]associated toρ, Ray–Singer [RS] have introduced the analytic torsionT=T(M, ρ, g, μ)>0. Wittenʹs deformationdq(t) of the exterior derivativedq,dq(t)=e−htdqeht, withh: M→Ra smooth Morse function, can be used to define a deformationT(h, t)>0 of the analytic torsionTwithT(h, 0)=T. The main results of this paper are to provide, assuming that grad ghis Morse Smale, an asymptotic expansion for log T(h, t) fort→∞ of the form[formula]and to present two different formulae fora0. 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 T=(h, g) and the dual triangulation TD=(d−h, g).