A proposal for nonabelian mirrors

Abstract

In this paper we propose a systematic construction of mirrors of nonabelian two dimensional (2,2) supersymmetric gauge theories. Specifically, we propose a construction of B-twisted Landau-Ginzburg orbifolds whose correlation functions match those of A-twisted supersymmetric gauge theories, and whose critical loci reproduce quantum cohomology and Coulomb branch relations in A-twisted gauge theories, generalizing the Hori-Vafa mirror construction. We check this proposal in a wide variety of examples. For instance, we construct mirrors corresponding to Grassmannians and two-step flag manifolds, as well as complete intersections therein, and explicitly check predictions for correlation functions and quantum cohomology rings, as well as other properties. We also consider mirrors to examples of gauge theories with U(k), U(k 1 ) × U(k 2 ), SU(k), SO(2k), SO(2k + 1), and Sp(2k) gauge groups and a variety of matter representations, and compare to results in the literature for the original two dimensional gauge theories. Finally, we perform consistency checks of conjectures of Aharony et al that a two dimensional (2,2) supersymmetric pure SU(k) gauge theory flows to a theory of k − 1 free twisted chiral multiplets, and also consider the analogous question in pure SO(3) theories. For one discrete theta angle, the SO(3) theory behaves the same as the SU(2) theory; for the other, supersymmetry is broken. We also perform consistency checks of analogous statements in pure supersymmetric SO and Sp gauge theories in two dimensions.

Topological operators, noninvertible symmetries and decomposition

“…(5.9) Furthermore, Υ acts as a dimension-zero field, with a ring relation Υ 2 = 1, the same structure we have seen previously in sigma models on disjoint unions and in other banded abelian gerbes. Similar structures arise in mirrors to other two-dimensional gauge theories with center-invariant matter, see for example [66][67][68][69][70]. As the stories are closely related, we will not describe them explicitly here.…”

“…Mirrors to such theories and their generalizations were discussed in [66][67][68][69][70]. For example, the mirror to the pure SU(2) theory is described by the superpotential [66, equ'n (12.3)]…”

Anomaly resolution via decomposition

“…[9,[13][14][15]), and has analogues in more general gauge theories with higher-form symmetries and various generalizations, as has been discussed in e.g. [16][17][18][19][20][21][22].…”

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.…”