Elliptic genera of ALE and ALF manifolds from gauged linear sigma models

TL;DR

This paper computes the equivariant elliptic genera of ALE and ALF manifolds using gauged linear sigma models, revealing pole structures, wall crossing phenomena, and applications to BPS state counting in higher-dimensional theories.

Contribution

It introduces a novel localization method for calculating elliptic genera of ALE and ALF manifolds within gauged linear sigma models, connecting to wall crossing and BPS spectrum counting.

Findings

Elliptic genera exhibit interesting pole structures as functions of chemical potentials.

Decomposition into polar and universal terms reveals wall crossing phenomena.

Results apply to counting BPS states in 5d and 6d supersymmetric theories.

Abstract

We compute the equivariant elliptic genera of several classes of ALE and ALF manifolds using localization in gauged linear sigma models. In the sigma model computation the equivariant action corresponds to chemical potentials for U(1) currents and the elliptic genera exhibit interesting pole structure as a function of the chemical potentials. We use this to decompose the answers into polar terms that exhibit wall crossing and universal terms. We compare our results to previous results on the large radius limit of the Taub-NUT elliptic genus and also discuss applications of our results to counting of BPS world-sheet spectrum of monopole strings in the 5d N=2 super Yang-Mills theory and self-dual strings in the 6d N=(2,0) theories.