Witten Index and Wall Crossing

TL;DR

This paper calculates the Witten index for certain supersymmetric quantum mechanics models, showing how it changes across phase boundaries and applying the results to BPS state counting in four-dimensional theories.

Contribution

It provides a residue integral formula for the Witten index and a wall crossing formula expressed as an integral at infinity, with applications to quiver quantum mechanics.

Findings

Derived a residue integral expression for the index.

Formulated a wall crossing formula as an integral at infinity.

Applied results to BPS state counting in 4D supersymmetric theories.

Abstract

We compute the Witten index of one-dimensional gauged linear sigma models with at least ${\mathcal N}=2$ supersymmetry. In the phase where the gauge group is broken to a finite group, the index is expressed as a certain residue integral. It is subject to a change as the Fayet-Iliopoulos parameter is varied through the phase boundaries. The wall crossing formula is expressed as an integral at infinity of the Coulomb branch. The result is applied to many examples, including quiver quantum mechanics that is relevant for BPS states in $d=4$ ${\mathcal N}=2$ theories.