Torsion type invariants of singularities

2016/03/21 by Huijun Fan, Hao Fang, Fan, Huijun +1 · 1 citation
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Spectral Theory in Mathematical Physics #math-ph #math.DG #math.MP

paper · pdf · doi:10.48550/arxiv.1603.06530

arxiv created 2016/03/21 · openalex publication_date 2016/03/21 · arxiv updated 2016/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on \mathbb Cⁿ by a quasi-homogeneous polynomial f. Under some mild assumption on f, we show that the small time heat kernel expansion of the corresponding Schrödinger operator exists and is a series of fractional powers of time t. Then we prove a local index formula which expresses the Milnor number of f by a Gaussian type integral. Furthermore, the heat kernel expansion provides spectral invariants of f. Especially, we define torsion type invariants associated to a singularity. These spectral invariants provide a new direction to study the singularity.

Citations

• The Witten equation and its virtual fundamental cycle

Cited by

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

Related

• tt^* Geometry, Singularity Torsion and Anomaly Formulas
• Nonlocal invariants in index theory
• A Local Index Theorem of Transversal Type on Manifolds with Locally Free \mathbbS¹-action
• Index of rigidity of differential equations and Euler characteristic of their spectral curves
• Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem