Non-Simply Laced Class-S Vertex Operator Algebras

TL;DR

This paper investigates non-simply laced class-S vertex operator algebras, analyzing their properties and reductions to determine if they correspond to 4d $ abla$2 SCFTs or are related to 3d gauge theories, providing insights into their physical relevance.

Contribution

It offers a detailed analysis of non-simply laced VOAs, their Drinfeld-Sokolov reductions, and proposes methods to distinguish VOAs originating from 4d theories from those that do not.

Findings

Many reduced VOAs correspond to 4d $ abla$2 SCFTs.

Identification of $F_4$ instanton VOAs related to 3d $ abla$4 quiver gauge theories.

Evidence that some non-simply laced VOAs do not originate from 4d theories.

Abstract

In arXiv:1811.01577 the VOAs associated to 4d $\mathcal{N}=2$ class-S theories were constructed in addition to a generalization for non-simply laced Lie algebras. However, 6d (2,0) theories have an ADE classification, and therefore class-S theories which are engineered by them come in ADE types. Thus, these non-simply laced VOAs are not thought to correspond to 4d physical theories. Regardless, we analyze these VOAs and their Drinfeld-Sokolov reductions in an effort to determine their properties. We find that many of these reduced VOAs appear to correspond to actual 4d $\mathcal{N}=2$ SCFTs. Additionally, we find what appear to be $F_4$ instanton VOAs, and hence from the proposal of arXiv:2201.13435 these correspond to 3d $\mathcal{N}=4$ quiver gauge theories whose Coulomb branches are $F_4$ instanton moduli spaces. Using a construction of twisted class-S VOAs from non-simply laced ones, we find additional evidence that the $F_4$ instanton VOAs do not correspond to four-dimensional field theories and outline an analogous argument for the $G_2$ instanton case. Our method appears to be quite general and may be a promising technique for ruling out 4d origins of VOAs that are similar to those of 4d $\mathcal{N}=2$ SCFTs, in addition to constructing VOAs of actual 4d theories.