Asymptotic expansion of the Witten deformation of the analytic torsion
Let \(M\) be a compact Riemannian manifold with a finite-dimensional representation \(V\) of the fundamental group. Let \(T(M;V)\) be the analytic torsion of \(M\) with coefficients in \(V\) as defined by Ray and Singer. Given a suitable Morse function \(h\), one can deform the de Rham complex by \(d^p(t) = e^{-th} d^pe^{th}\). Let \(\Delta_p(t)\) be the associated deformation of the Laplace operator and \(T(t)\) be the analytic Ray-Singer torsion obtained by a regularization of the zeta-function of \(\Delta_p(t)\). The asymptotic expansion of this function \(T(t)\) is investigated. For large \(t\) the de Rham complex splits into two parts, namely a part given by small eigenvalues and a part given by large eigenvalues of the Laplace operator because the spectrum of the Laplace operator splits into two pieces for large \(t\). The asymptotic expansion of the small part contains the combinatorial Reidemeister torsion, the one of the large part the analytic Ray-Singer torsion as constant coefficients. This allows to give new proofs of the results of Cheeger, Müller and Bismut-Zhang about the relation between the combinatorial and the analytic torsion.
scientific article; zbMATH DE number 880567