$t{t}^{\ast }$ Geometry, Singularity Torsion and Anomaly Formulas

This paper is concerned with the Schr"odinger operators ${\mathrm{\Delta }}_{f_0}$ and ${\mathrm{\Delta }}_f$ attached to a pair $({\mathbb{C}}^n,f_0)$ and its deformation $({\mathbb{C}}^n,f)$, where $f_0$ is a non-degenerate and quasi-homogeneous polynomial on ${\mathbb{C}}^n$ and $f$ is its relevant or marginal deformation. We give the $t{t}^{\ast }$ geometry structure on the Hodge bundle associated to ${\mathrm{\Delta }}_f$, which describes the genus 0 anomaly. Next we study the corresponding singularity torsion type invariants and give the anomaly formulas for the 2nd torsion type invariant.