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

TL;DR

The authors compute the equivariant elliptic genera of ALE and ALF spaces using localization in gauged linear sigma models, revealing meromorphic, pole-rich structures in the equivariant parameters and decomposing the results into polar (wall-crossing) and finite parts. They derive explicit expressions for Taub-NUT and multi-centered Taub-NUT, show how ALF reduces to ALE in the large-radius limit, and connect discrete BPS spectra to wall-crossing phenomena, including a 4d/6d interpretation via monopole strings and self-dual strings. The work also generalizes to ADE ALE spaces with two GLSM realizations yielding matching results and discusses non-holomorphic contributions from continuum states, along with applications to 1/4-BPS state counting in 5d ${ m N}=2$ SYM and $(0,4)$ elliptic genera. Overall, it provides a concrete, calculable framework for non-compact elliptic genera with implications for BPS spectra, black hole microstate counting, and potential links to mock modular forms and moonshine phenomena.

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.

Figures (3)

• Figure 1: The tension of a string that winds the Taub-NUT circle can be balanced by the angular momentum barrier, which leads to a "stable" BPS state contributing to the elliptic genus. However, as the size of the circle varies, this BPS state can become unstable.
• Figure 2: The IIA brane configuration of a $1/4$ BPS state carrying both instanton and electric charges in 5d ${\cal N}=2$ super Yang-Mills theory. These $1/4$ BPS states uplift to the BPS spectrum of M2 self-dual strings.
• Figure 3: $SU(3)$ self-dual string and a pair of distinct monopole strings.

Theorems & Definitions (1)

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