Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem

TL;DR

This paper develops a new asymptotic expansion for heat kernels of Witten Laplacians on non-compact manifolds, leading to a local index theorem, advancing understanding of Landau-Ginzburg models in complex geometry.

Contribution

It introduces a parabolic distance for non-compact spaces, derives heat kernel asymptotics, and establishes a local index theorem for Witten deformation.

Findings

Derived pointwise heat kernel asymptotics with strong remainder estimates.

Established a local index theorem for Witten Laplacian on non-compact manifolds.

Connected heat kernel analysis to Landau-Ginzburg B-models.

Abstract

Asymptotic expansions of heat kernels and heat traces of Schr\"odinger operators on non-compact spaces are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li-Yau's famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time $t$, and obtain a local index theorem. This is the second of our papers in understanding Landau-Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray-Singer torsion for Witten deformation in the non-compact setting.