N=1 conformal dualities

TL;DR

The paper presents a systematic algorithm to uncover conformal ${ m N}=1$ gauge-theory duals that live on the same conformal manifold ${ m M}_c$, by matching ${a}$, ${c}$, ’t Hooft anomalies, and protected indices between candidate pairs.Applying this method, the authors construct multiple explicit weakly coupled duals for simple gauge groups (notably ${ m G}_2$ and ${ m SU}(4)$) as well as for various class ${ m S}$ theories (including ${ m R}_{0,4}$, ${ m T}_4$, and ${ m R}_{2,5}$) and for ranks of ${ m E}_6$ MN theories, often in the form of quiver gauge theories.In each case, they verify duality by computing marginal deformation counts (dimension of ${ m M}_c$), matching conformal anomalies, and aligning supersymmetric indices, sometimes revealing dual frames with different breakings of global symmetry on generic points of ${ m M}_c$.They further extend the approach to four-dimensional Lagrangians for compactifications of the rank-one E-string theory on higher-genus Riemann surfaces, predicting emergent ${ m E}_8$ structure on subspaces and proposing geometric interpretations akin to crossing symmetry moves.Overall, the work argues that conformal dualities between ${ m N}=1$ gauge theories and certain strongly coupled SCFTs are plentiful, providing new, controllable Lagrangian descriptions for otherwise intricate fixed points and inviting deeper geometric and algebraic understanding.

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.

Figures (13)

• Figure 1: The dual of $G_2$ with $3\times {\bf 7}\oplus {\bf 27}$. In this and the following figures one should think of the models with all the possible gauge invariant cubic superpotentials turned on.
• Figure 2: The structure of the conformal manifold.
• Figure 3: The dual of $G_2$ with $12\times {\bf 7}$.
• Figure 4: The dual of $SU(4)$ with $4\times {\bf 6}\oplus8\times {\bf 4}\oplus 8\times {\bf \overline 4}$.
• Figure 5: The ${\cal N}=1$ conformal dual of $R_{0,4}$.