Invariance of elliptic genus under wall-crossing

TL;DR

This paper establishes a topological criterion that determines when wall-crossing formulas for elliptic genus are trivial, with applications across GIT quotients, sheaf moduli, and gauge theories.

Contribution

It introduces a topological criterion that simplifies understanding of wall-crossing phenomena for elliptic genus across various geometric contexts.

Findings

Wall-crossing formulas can be trivial under certain topological conditions.

The criterion applies to GIT quotients, sheaf moduli, and gauge theories.

Provides a unified approach to wall-crossing in different geometric settings.

Abstract

Wall-crossing formulas for various flavors of elliptic genus can be obtained using master spaces. We give a topological criterion which implies that such wall-crossing formulas are trivial. Applications are given for: GIT quotients, following Thaddeus; moduli of sheaves, following Mochizuki; Donaldson-Thomas and Vafa-Witten theory, following Joyce and Tanaka-Thomas respectively.