Notes on two-dimensional pure supersymmetric gauge theories

TL;DR

The paper investigates IR limits of pure two-dimensional (2,2) supersymmetric gauge theories with semisimple non-simply-connected groups and various discrete theta angles, analyzing both direct gauge-theory dynamics and nonabelian mirror descriptions. It shows that supersymmetric vacua exist for precisely one value of the discrete theta angle in each case, with the IR content containing as many twisted chiral multiplets as the rank, and that other theta angles lead to SUSY breaking. A central methodological contribution is a refined nonabelian-mirror framework that distinguishes mirrors of $G$ versus $G/K$ gauge theories and clarifies how discrete theta angles arise from weight- and root-lattice structures, via invariants computed from the lattice quotients. The work performs extensive consistency checks, including group-theoretic relations and decomposition, reinforcing the picture that pure $G$ or $G/K$ theories decompose into disjoint unions over theta angles with the IR endpoint shared among components. Overall, the results extend the landscape of known IR behaviors for 2d pure gauge theories and illuminate the deep connection between discrete theta angles and lattice-theoretic descriptions of centers.

Abstract

In this note we study IR limits of pure two-dimensional supersymmetric gauge theories with semisimple non-simply-connected gauge groups including SU(k)/Z_k, SO(2k)/Z_2, Sp(2k)/Z_2, E_6/Z_3, and E_7/Z_2 for various discrete theta angles, both directly in the gauge theory and also in nonabelian mirrors, extending a classification begun in previous work. We find in each case that there are supersymmetric vacua for precisely one value of the discrete theta angle, and no supersymmetric vacua for other values, hence supersymmetry is broken in the IR for most discrete theta angles. Furthermore, for the one distinguished value of the discrete theta angle for which supersymmetry is unbroken, the theory has as many twisted chiral multiplet degrees of freedom in the IR as the rank. We take this opportunity to further develop the technology of nonabelian mirrors to discuss how the mirror to a G gauge theory differs from the mirror to a G/K gauge theory for K a subgroup of the center of G. In particular, the discrete theta angles in these cases are considerably more intricate than those of the pure gauge theories studied in previous papers, so we discuss the realization of these more complex discrete theta angles in the mirror construction. We find that discrete theta angles, both in the original gauge theory and their mirrors, are intimately related to the descriptions of centers of universal covering groups as quotients of weight lattices by root sublattices. We perform numerous consistency checks, comparing results against basic group-theoretic relations as well as with decomposition, which describes how two-dimensional theories with one-form symmetries (such as pure gauge theories with nontrivial centers) decompose into disjoint unions, in this case of pure gauge theories with quotiented gauge groups and discrete theta angles.

Figures (1)

• Figure 1: Schematic illustration of weight and root lattice for $SU(2)$. Solid circles indicate points on both lattices; empty circles indicate points on the weight lattice that are not also on the root lattice.