Tinkertoys for the Twisted $E_6$ Theory

TL;DR

This work classifies twisted $E_6$ class-$\mathcal{S}$ theories using a detailed tinkertoy approach, enumerating twisted and untwisted punctures, fixtures, and gauging rules. It ties local Hitchin data and $k$-differentials to global symmetries and superconformal indices, uncovering that many Higgs branches are instanton moduli spaces and revealing new hyperKähler-quotient isomorphisms via S-duality. The paper further shows enhanced global symmetries controlled by Sommers-Achar groups, introduces two new realizations of $R_{2,5}$ in the twisted sector, and identifies numerous product SCFTs, supported by index factorization checks. Finally, it develops and tests conjectural Hilbert-series descriptions of instanton moduli spaces for exceptional groups, providing concrete isomorphisms and expanding the landscape of Higgs-branch dualities.

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 hyperKähler 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.