Asymptotic Expansion of the Witten deformation of the analytic torsion

Given a compact Riemannian manifold (Md, g), a finite dimensional representation �:�1(M) → GL(V) of the fundamental group �1(M) on a vector space V of dimension l and a Hermitian structure � on the flat vector bundle � → p M associated to �, Ray-Singer [RS] have introduced the analytic torsion T = T(M,�,g,�) > 0. Witten’s deformation dq(t) of the exterior derivative dq, dq(t) = e-htdqeht, 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 Σd+1j=0 ajtj + 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). © 1996 Academic Press, Inc. DOI: 10.1006/jfan.1996.0049 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-22539 Originally published at: Burghelea, D; Friedlander, L; Kappeler, T (1996). Asymptotic expansion of the Witten deformation of the analytic torsion. Journal of Functional Analysis, 137(2):320-363. DOI: 10.1006/jfan.1996.0049 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Asymptotic Expansion of the Witten deformation of the analytic torsion D. Burghelea L. Friedlander T. Kappler Vienna, Preprint ESI 44 (1993) August 12, 1993 Supported by Federal Ministry of Science and Research, Austria ASYMPTOTIC EXPANSION OF THE WITTEN DEFORMATION OF THE ANALYTIC TORSION D. Burghelea L. Friedlander T. Kappeler International Erwin Schrödinger Institute for Mathematical Physics