The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class $\mathcal{S}_k$

TL;DR

This paper explores the Coulomb and Higgs branches of a special class of four-dimensional $ calS_k$ $ calN=1$ SCFTs, defining and analyzing their BPS operators and providing a link to six-dimensional Little String theory.

Contribution

It introduces a method to precisely define Coulomb and Higgs branches in $ calS_k$ theories and connects these to six-dimensional Little String theory via deconstruction.

Findings

Defined Coulomb and Higgs branches for $ calS_k$ theories.

Analyzed BPS operators using superconformal index limits.

Demonstrated deconstruction of 6D Little String theory from $ calS_k$ quivers.

Abstract

Even though for generic $\mathcal{N}=1$ theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $\mathcal{N}=1$ SCFTs, these of Class $\mathcal{S}_k$ for which it is possible to define Coulomb and Higgs branches precisely as for the $\mathcal{N}=2$ theories of Class $\mathcal{S}$ from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional $(1,1)$ Little String theory can be deconstructed from a toroidal quiver in class $\mathcal{S}_k$