From 3d duality to 2d duality

Abstract

In this paper we discuss 3d mathcal{N} = 2 supersymmetric gauge theories and their IR dualities when they are compactified on a circle of radius r, and when we take the 2d limit in which r → 0. The 2d limit depends on how the mass parameters are scaled as r → 0, and often vacua become infinitely distant in the 2d limit, leading to a direct sum of different 2d theories. For generic mass parameters, when we take the same limit on both sides of a duality, we obtain 2d dualities (between gauge theories and/or Landau-Ginzburg theories) that pass all the usual tests. However, when there are non-compact branches the discussion is subtle because the metric on the moduli space, which is not controlled by supersymmetry, plays an important role in the low-energy dynamics after compactification. Generally speaking, for IR dualities of gauge theories, we conjecture that dualities involving non-compact Higgs branches survive. On the other hand when there is a non-compact Coulomb branch on at least one side of the duality, the duality fails already when the 3d theories are compactified on a circle. Using the valid reductions we reproduce many known 2d IR dualities, giving further evidence for their validity, and we also find new 2d dualities.