Tinkertoys for the Twisted $E_6$ Theory

TL;DR

This paper classifies and analyzes 4D N=2 superconformal theories derived from twisted compactifications of the 6D E6 (2,0) theory, exploring their properties, dualities, and Higgs branch structures.

Contribution

It provides a detailed classification of twisted E6 theories via punctured Riemann surfaces, including tables of twisted punctures and superconformal index calculations, revealing new dualities and Higgs branch phenomena.

Findings

Classification of twisted E6 theories using three-punctured spheres and cylinders.

Explicit superconformal index expressions for twisted fixtures.

Discovery of isomorphisms between Higgs branches and hyperKähler quotients.

Abstract

We study $4D$ $\mathcal{N}=2$ superconformal field theories that arise as the compactification of the six-dimensional $(2,0)$ theory of type $E_6$ on a punctured Riemann surface in the presence of $\mathbb{Z}_2$ outer-automorphism twists. We explicitly carry out the classification of these theories in terms of three-punctured spheres and cylinders, and provide tables of properties of the $\mathbb{Z}_2$-twisted punctures. An expression is given for the superconformal index of a fixture with twisted punctures of type $E_6$, which we use to check our identifications. Several of our fixtures have Higgs branches which are isomorphic to instanton moduli spaces, and we find that S-dualities involving these fixtures imply interesting isomorphisms between hyperK\"ahler quotients of these spaces. Additionally, we find families of fixtures for which the Sommers-Achar group, which was previously a Coulomb branch concept, acts non-trivially on the Higgs branch operators.