2023/01/05 by Junrong Yan, Yan, Junrong
Mathematics · #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2301.01990
openalex publication_date 2023/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper concentrates on analyzing Witten deformation for a family of non-Morse functions parameterized by T∈ ℝ+, resulting in a novel, purely analytic proof of the gluing formula for analytic torsions in complete generality due to Brünning-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-Müller/Bismut-Zhang theorem for nontrivial flat bundles.