Rigidity via Modular Properties of Theta Functions

TL;DR

This paper proves the rigidity of certain twisted Dirac operators on bundles by employing theta function properties, Lefschetz formulas, and localization techniques, revealing deep connections between geometry, topology, and modular forms.

Contribution

It introduces a novel approach using theta function modular properties to establish rigidity of Dirac operators on specific bundles, extending previous methods.

Findings

Lefschetz numbers are expressed via Jacobi theta functions.

Under certain conditions, the Lefschetz number remains constant, proving rigidity.

The approach links modular properties of theta functions to geometric operator invariance.

Abstract

In this paper we use methods of Liu to show that the twisted Dirac operators $D$ on certain bundles $\Phi$ considered by Guan and Wang are rigid. To do so, we use a Lefschetz formula and Atiyah-Bott localization to obtain formulas for the Lefschetz numbers $L$ of these operators $D$ in terms of Jacobi theta functions; then, using the translational and modular transformation properties of theta functions and the properties of their zeros, we prove $L$ is constant provided certain conditions on characteristic classes hold, thus showing the rigidity of $D$ on $\Phi$ under these conditions.