Torsion type invariants of singularities

Inspired by the LG/CY correspondence, we study the local index theory of the Schr\"odinger 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 Schr\"odinger 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.