# N=1 conformal dualities

## TL;DR

This paper explores the existence of dual descriptions for certain 4D ${
m extbf{N}}=1$ superconformal theories, providing algorithms and concrete examples of weakly coupled dual gauge theories for strongly coupled models.

## Contribution

It introduces a simple algorithm to find conformal dualities and presents multiple explicit examples, including new duals for SQCD and class ${
m extbf{S}}$ theories.

## Key findings

- Identified conformal duals for ${
m extbf{N}}=1$ SQCD with $G_2$ and $SU(4)$ gauge groups.
- Found weakly coupled quiver duals for various class ${
m extbf{S}}$ theories.
- Derived conformal Lagrangians for theories from E-string compactifications.

## Abstract

We consider on one hand the possibility that a supersymmetric ${\cal N}=1$ conformal gauge theory has a strongly coupled locus on the conformal manifold at which a different, dual, conformal gauge theory becomes a good weakly coupled description. On the other hand we discuss the possibility that strongly coupled theories, e.g. SCFTs in class ${\cal S}$, having exactly marginal ${\cal N}=1$ deformations admit a weakly coupled gauge theory description on some locus of the conformal manifold. We present a simple algorithm to search for such dualities and discuss several concrete examples. In particular we find conformal duals for ${\cal N}=1$ SQCD models with $G_2$ gauge group and a model with $SU(4)$ gauge group in terms of simple quiver gauge theories. We also find conformal weakly coupled quiver theory duals for a variety of class ${\cal S}$ theories: $T_4$, $R_{0,4}$, $R_{2,5}$, and rank $2n$ Minahan-Nemeschansky $E_6$ theories. Finally we derive conformal Lagrangians for four dimensional theories obtained by compactifying the E-string on genus $g>1$ surface with zero flux. The pairs of dual Lagrangians at the weakly coupled loci have different symmetries which are broken on a general point of the conformal manifold. We match the dimensions of the conformal manifolds, symmetries on the generic locus of the conformal manifold, anomalies, and supersymmetric indices. The simplicity of the procedure suggests that such dualities are ubiquitous.