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.
Highlights
Introduction and summaryOften different high energy theories are equivalent at low energies
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
An interesting possibility that can occur in two dimensions and not in higher dimensions is that a given high-energy supersymmetric theory can flow to more than one superconformal theory (SCFT) at low energies [3,4,5]
Summary
Often different high energy theories are equivalent at low energies. This universality has been observed to happen in many examples of supersymmetric gauge theories in various dimensions. To try to sieve through the large amount of cases to extract the essential properties, it is useful to understand what are the minimal sets of dualities from which all the rest can be derived. The goal is to understand the fate of the dualities in this reduction and try to derive known and new dualities in lower dimensions. It was possible to derive all the known (non-mirror) IR equivalences in three dimensions starting from four, and we found new dualities. We will discuss the step in the program, namely further reduction of three dimensional theories on a circle. In the rest of the introduction we will detail the new issues one encounters in two dimensions and in three dimensional theories on a circle, and briefly summarize our results
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