A proposal for nonabelian mirrors

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(k1) x U(k2), 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.