Curious Aspects of Three-Dimensional ${\cal N}=1$ SCFTs

TL;DR

This work analyzes 3d theories with ${\cal N}=1$ supersymmetry under time-reversal invariance, uncovering a web of dualities and IR phenomena. It introduces dual descriptions where time-reversal symmetry is emergent in the IR, and demonstrates symmetry enhancements such as $SU(3)$ in ${\cal N}=2$ QED, alongside ${\cal N}=2$ (and later ${\cal N}=1$) enhancements in Wess–Zumino models, all connected by flip operations and deformations. Core results include the duality $U(1)_0$ with a charge-2 chiral and its IR partner $U(1)_2\otimes[U(1)_{3/2}+\text{charge-1}]$, the emergence of TR at special fixed points, and the mapping between Abelian and non-Abelian TR-invariant theories via moduli and CS terms. These findings advance understanding of IR fixed points, moduli spaces, and symmetry enhancement in TR-invariant 3d SUSY systems, with potential implications for non-perturbative dualities and condensed-matter analogs.

Abstract

We study the dynamics of certain 3d ${\cal N}=1$ time reversal invariant theories. Such theories often have exact moduli spaces of supersymmetric vacua. We propose several dualities and we test these proposals by comparing the deformations and supersymmetric ground states. First, we consider a theory where time reversal symmetry is only emergent in the infrared and there exists (nonetheless) an exact moduli space of vacua. This theory has a dual description with manifest time reversal symmetry. Second, we consider some surprising facts about ${\cal N}=2$ $U(1)$ gauge theory coupled to two chiral superfields of charge 1. This theory is claimed to have emergent $SU(3)$ global symmetry in the infrared. We propose a dual Wess-Zumino description (i.e. a theory of scalars and fermions but no gauge fields) with manifest $SU(3)$ symmetry but only ${\cal N}=1$ supersymmetry. We argue that this Wess-Zumino model must have enhanced supersymmetry in the infrared. Finally, we make some brief comments about the dynamics of ${\cal N}=1$ $SU(N)$ gauge theory coupled to $N_f$ quarks in a time reversal invariant fashion. We argue that for $N_f<N$ there is a moduli space of vacua to all orders in perturbation theory but it is non-perturbatively lifted.

Figures (2)

• Figure 1: Phase diagram of the Wess-Zumino model and gauge theory. For the Wess-Zumino model, this is ($M_{11}$,$M_{22}$) plane and three half lines are given by $L_1: M_{11} -M_{22} = 0\ (M_{11}<0)$, $L_2: 2M_{11}+ M_{22}=0\ (M_{11}<0)$, and $L_3 : M_{11}+2M_{22} = 0 \ (M_{11}>0)$. For the gauge theory this is ($t$,$m_f$) plane and three lines are given by $L_1: m_f = 0\ (t>0)$, $L_2: t+m_f = 0\ (m_f>0)$, $L_3: t-m_f = 0\ (m_f<0)$. On the three solid half lines, we have an $SU(2)\times U(1)$ preserving deformation, and $\mathbb{C}\mathbb{P}^1$ worth of vacua due to spontaneous symmetry breaking. Otherwise, we have 2 isolated vacua due to time reversal symmetry breaking.
• Figure :