Torsion type invariants of singularities

Abstract

Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on ${\mathbb{C}}^n$ 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.

Cited by in corpus (3)

• Witten deformation for noncompact manifolds with bounded geometry
• $t{t}^{\ast }$ Geometry, Singularity Torsion and Anomaly Formulas
• Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem