Notes on two-dimensional pure supersymmetric gauge theories

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)/ℤk, SO(2k)/ℤ2, Sp(2k)/ℤ2, E6/ℤ3, and E7/ℤ2 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 description 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.

An introduction to decomposition

“…• Gauged linear sigma models (GLSMs): Decomposition has been checked in gauged linear sigma models via mirror symmetry and in quantum cohomology rings (the latter through Coulomb branch computations). For abelian GLSMs, this was described in [39,40] using Hori-Vafa mirrors [30]; for nonabelian GLSMs, this was checked in the papers describing nonabelian mirror constructions [10,18,20,21]. • In orbifolds, decomposition has been checked extensively [25,[42][43][44][45] in, for example, partition functions and massless spectra, as we will outline later in this article.…”

Higher Form Symmetries and M-theory

“…The last two terms correspond to (−1)-form and 4-form symmetries, which are somewhat more exotic from the field theory point of view, and we will ignore them in our analysis. (See [86][87][88] for recent work exploring such symmetries from the field theory point of view. )…”

Quantum symmetries in orbifolds and decomposition

“…One of the most important outcomes of that work was the discovery of decomposition, first described in [16], an equivalence between two-dimensional theories with one-form symmetries (and various generalizations) and disjoint unions of other two-dimensional theories. Decomposition has since been applied to a number of areas, including Gromov-Witten theory [17][18][19][20][21][22], gauged linear sigma models [23][24][25][26][27][28][29], mirror symmetry [16,[30][31][32][33] and heterotic string compactifications [34]. See e.g.…”