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Asymptotic expansion of the Witten deformation of the analytic torsion


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.

Abstract

Given a compact Riemannian manifold (Md, 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 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.

Given a compact Riemannian manifold (Md, 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 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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1996
Deposited On:29 Nov 2010 16:28
Last Modified:05 Apr 2016 13:27
Publisher:Elsevier
ISSN:0022-1236
Publisher DOI:https://doi.org/10.1006/jfan.1996.0049
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1387514
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0858.57029
Permanent URL: https://doi.org/10.5167/uzh-22539

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