Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing

TL;DR

This paper extends the construction of vertex operator algebras associated with class S theories to include non-simply laced Lie algebras with outer automorphism twist lines, addressing previous limitations and proposing new definitions.

Contribution

It develops a framework for defining twisted chiral algebras of class S with outer automorphism twist lines, expanding the scope of Arakawa's uniform construction.

Findings

Proposed definitions for twisted chiral algebras with outer automorphism twists.

Performed consistency checks on the new definitions.

Identified open problems and future directions.

Abstract

The correspondence between four-dimensional $\mathcal{N}=2$ superconformal field theories and vertex operator algebras, when applied to theories of class $\mathcal{S}$, leads to a rich family of VOAs that have been given the monicker chiral algebras of class $\mathcal{S}$. A remarkably uniform construction of these vertex operator algebras has been put forward by Tomoyuki Arakawa in arXiv:1811.01577. The construction of arXiv:1811.01577 takes as input a choice of simple Lie algebra $\mathfrak{g}$, and applies equally well regardless of whether $\mathfrak{g}$ is simply laced or not. In the non-simply laced case, however, the resulting VOAs do not correspond in any clear way to known four-dimensional theories. On the other hand, the standard realisation of class $\mathcal S$ theories involving non-simply laced symmetry algebras requires the inclusion of outer automorphism twist lines, and this requires a further development of the approach of arXiv:1811.01577. In this paper, we give an account of those further developments and propose definitions of most chiral algebras of class $\mathcal S$ with outer automorphism twist lines. We show that our definition passes some consistency checks and point out some important open problems.