Witten deformation for non-Morse functions and gluing formula for analytic torsions

Abstract

This paper concentrates on analyzing Witten deformation for a family of non-Morse functions parameterized by $T\in \mathbb{R}_+$, resulting in a novel, purely analytic proof of the gluing formula for analytic torsions in complete generality due to Br\"unning-Ma. Intriguingly, the gluing formula in this article could be reformulated as the Bismut-Zhang theorem for non-Morse functions, and from the perspective of Vishik's theory of moving boundary problems, the deformation parameter $T$ parameterize a family of boundary conditions. Our proof also makes use of a connection between small eigenvalues of Witten Laplacians and Mayer-Vietoris sequences. Finally, these new techniques could be extended to analytic torsion forms and play key roles in the study of the higher Cheeger-M\"uller/Bismut-Zhang theorem for nontrivial flat bundles.