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

Junrong Yan

TL;DR

The work proves a purely analytic gluing formula for analytic torsions in full generality by employing a Witten deformation with non-Morse functions and novel coupling techniques that bridge large-time, small-time, and Mayer–Vietoris analyses. The approach reduces the problem to one-dimensional models and moving-boundary frameworks, and leverages spectral convergence of deformed Laplacians to derive the MV-torsion relation and the final gluing identity. Key innovations include the eigenvalue convergence results, a detailed small-time heat-trace analysis via one-dimensional reductions, and a precise accounting of universal constants through RS metrics on simple spaces. The methods promise extensions to analytic torsion forms and higher Cheeger–Müller/Bismut–Zhang-type theorems for nontrivial flat bundles, with potential applicability to other spectral invariants and moving-boundary problems.

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 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.

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

TL;DR

The work proves a purely analytic gluing formula for analytic torsions in full generality by employing a Witten deformation with non-Morse functions and novel coupling techniques that bridge large-time, small-time, and Mayer–Vietoris analyses. The approach reduces the problem to one-dimensional models and moving-boundary frameworks, and leverages spectral convergence of deformed Laplacians to derive the MV-torsion relation and the final gluing identity. Key innovations include the eigenvalue convergence results, a detailed small-time heat-trace analysis via one-dimensional reductions, and a precise accounting of universal constants through RS metrics on simple spaces. The methods promise extensions to analytic torsion forms and higher Cheeger–Müller/Bismut–Zhang-type theorems for nontrivial flat bundles, with potential applicability to other spectral invariants and moving-boundary problems.

Abstract

This paper concentrates on analyzing Witten deformation for a family of non-Morse functions parameterized by , 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 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.
Paper Structure (34 sections, 39 theorems, 248 equations, 3 figures)

This paper contains 34 sections, 39 theorems, 248 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. The gluing formulas
  4. Main ideas
  5. Coupling techniques
  6. Small eigenvalues and Mayer-Vietoris exact sequence
  7. Organization
  8. Preliminary
  9. Conventions
  10. A brief review on Hodge theory
  11. Analytic torsion for flat vector bundles
  12. On closed manifolds
  13. On manifolds with boundary
  14. Analytic torsion for Witten Laplacian and weighted Laplacian
  15. Witten Laplacian v.s. weighted Laplacian
  16. ...and 19 more sections

Key Result

Theorem 1.1

where $B\left(g^{TM}\right)$ is the secondary characteristic form introduced by Brüning-Ma in bruning2006anomaly, which is zero if $Y$ is totally geodesic in $\left(M, g^{T M}\right)$.

Figures (3)

  • Figure 1: Cutting $M$ along $Y$
  • Figure 2: Elongating
  • Figure 3: Graph of $f_T$

Theorems & Definitions (71)

  • Remark 1.1
  • Theorem 1.1: Brüning-Ma bruning2013gluing
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1: Hodge decomposition
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 61 more