Witten deformation on non-compact manifolds: heat kernel expansion and local index theorem

Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as C n with (quasi-homogeneous) polynomials potentials, it’s already very complicated. Motivated by path integral formulation of the heat kernel, we...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 303; no. 1
Main Authors Dai, Xianzhe, Yan, Junrong
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2023
Springer Nature B.V
Subjects
Asymptotic series
Deformation
Kernels
Mathematics
Mathematics and Statistics
Operators (mathematics)
Parameters
Polynomials
Theorems
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Summary:Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as 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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-022-03150-0