$tt^*$ Geometry, Singularity Torsion and Anomaly Formulas

TL;DR

This paper explores the geometric and topological properties of Schr"odinger operators linked to polynomial deformations in complex space, revealing new anomaly formulas and torsion invariants related to singularity theory.

Contribution

It introduces a $tt^*$ geometry framework for the Hodge bundle associated with deformed Schr"odinger operators and derives new anomaly formulas for torsion invariants.

Findings

Established $tt^*$ geometry structure for the Hodge bundle.

Derived anomaly formulas for 2nd torsion type invariants.

Analyzed singularity torsion invariants in polynomial deformations.

Abstract

This paper is concerned with the Schr\"odinger operators $\Delta_{f_0}$ and $\Delta_f$ 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 $tt^*$ geometry structure on the Hodge bundle associated to $\Delta_f$, 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.