Elliptic genera of pure gauge theories in two dimensions with semisimple non-simply-connected gauge groups

TL;DR

This work develops a systematic localization-based method to compute the elliptic genera of pure two-dimensional $(2,2)$ supersymmetric gauge theories with semisimple, non-simply-connected gauge groups $G/\Gamma$, including discrete theta angles. The authors compute explicit elliptic genera for low-rank examples (e.g., $SU(2)/\mathbb{Z}_2$, $SU(3)/\mathbb{Z}_3$, $SO(4)$, Spin$(4)/(\mathbb{Z}_2\times\mathbb{Z}_2)$, $SO(5)$, $Sp(6)/\mathbb{Z}_2$) by summing Jeffrey–Kirwan residues over the moduli spaces of flat connections in each sector labeled by the characteristic class $w$, and then weight these contributions with phases from discrete theta angles. They show consistency with decomposition due to finite one-form symmetries and with known SUSY-breaking patterns, and they extract general predictions for the elliptic genera of broader classes of groups by combining $w=0$ results, SUSY-breaking data, and decomposition. The results illuminate how global structure and discrete theta angles control IR dynamics and dualities in two dimensions and provide analytic benchmarks for future lattice studies and duality checks. The work also offers explicit formulas for wide families of groups, guiding higher-rank generalizations and deeper explorations of 2D gauge dynamics.

Abstract

In this paper we describe a systematic method to compute elliptic genera of (2,2) supersymmetric gauge theories in two dimensions with gauge group G/Gamma (for G semisimple and simply-connected, Gamma a subgroup of the center of G) with various discrete theta angles. We apply the technique to examples of pure gauge theories with low-rank gauge groups. Our results are consistent with expectations from decomposition of two-dimensional theories with finite global one-form symmetries and with computations of supersymmetry breaking for some discrete theta angles in pure gauge theories. Finally, we make predictions for the elliptic genera of all the other remaining pure gauge theories by applying decomposition and matching to known supersymmetry breaking patterns.