D. Burghelea, L. Friedlander, T. Kappler
Asymptotic Expansion of the Witten deformation of the analytic torsion
The paper is published:
J. Funct. Anal. 137, 2 (1996) 320--363
- MSC:
- 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., See also {19B28}
- 58G26 Determinants and determinant bundles
Abstract: Given a compact Riemannian manifold $(M^d,g)$, a finite dimensional
representation $\rh:\pi_1(M)\to GL(V)$ of the fundamental group
$\pi_1(M)$ on a vectorspace $V$ of dimension $l$ and a Hermitian
structure $\mu$ on the flat vector bundle $\Cal E\overset p
\to\rightarrow M$ associated to $\rh$, Ray-Singer \cite {RS} have
introduced the analytic torsion $T=T(M,\rh,g,\mu)>0.$ Witten's
deformation $d_q(t)$ of the exterior derivative
$d_q,d_q(t)=e^{-ht}d_qe^{ht},$ with $h:M\to R$ 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
$\text{ grad }_gh$ is Morse Smale, an asymptotic expansion for
$\log T(h,t)$ for $t\to\infty$ of the form
$\sum_{j=0}^{d+1}a_jt^j+b\log t+O(\frac{1}{\sqrt t})$ and to
present two different formulae for $a_0.$ As an application we
obtain a shorter derivation of results due to Ray-Singer
\cite{RS}, Cheeger \cite{Ch}, M\"uller \cite{Mu1,2} and Bismut-Zhang
\cite{BZ} 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 $\Cal T=(h,g)$ and the dual
triangulation $\Cal T_{\Cal D}=(d-h,g).$
Keywords: combinatorial Reidemeister torsion; Witten's deformation; fundamental group; analytic torsion; Morse function; de Rham complex; Laplace operator; Ray-Singer torsion