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arXiv · 2301.01990

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

Junrong Yan
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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.

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Junrong Yan. 2023-01-05. Witten deformation for non-Morse functions and gluing formula for analytic torsions. https://arxiv.org/abs/2301.01990

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