From 3d duality to 2d duality

TL;DR

This paper explores how 3d N=2 supersymmetric gauge theories reduce to 2d dualities upon circle compactification, revealing conditions under which dualities survive or fail, and reproducing known and new 2d dualities.

Contribution

It provides a detailed analysis of the reduction process from 3d to 2d dualities, including the role of non-compact branches and the conditions for duality survival.

Findings

Dualities with non-compact Higgs branches generally survive reduction.

Dualities with non-compact Coulomb branches typically fail upon reduction.

Many known 2d IR dualities are reproduced, and new dualities are identified.

Abstract

In this paper we discuss $3d$ ${\cal N}=2$ 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 $r\to 0$. The $2d$ limit depends on how the mass parameters are scaled as $r\to 0$, 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.