Chiral algebras of class $\mathcal{S}$ and Moore-Tachikawa symplectic varieties

TL;DR

This paper constructs and classifies genus zero chiral algebras of class , showing they are simple, conformal, and linked to Moore-Tachikawa symplectic varieties via geometric methods.

Contribution

It provides a functorial construction of genus zero class  chiral algebras for any complex semisimple group and identifies their associated varieties with Moore-Tachikawa symplectic varieties.

Findings

Constructed unique family of simple, conformal vertex algebras for genus zero class .

Showed these vertex algebras' associated varieties are Moore-Tachikawa symplectic varieties.

Extended the construction to non-simply laced groups.

Abstract

We give a functorial construction of the genus zero chiral algebras of class $\mathcal{S}$, that is, the vertex algebras corresponding to the theory of class $\mathcal{S}$ associated with genus zero pointed Riemann surfaces via the 4d/2d duality discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees in physics. We show that there is a unique family of vertex algebras satisfying the required conditions and show that they are all simple and conformal. In fact, our construction works for any complex semisimple group G that is not necessarily simply laced. Furthermore, we show that the associated varieties of these vertex algebras are exactly the genus zero Moore-Tachikawa symplectic varieties that have been recently constructed by Braverman, Finkelberg and Nakajima using the geometry of the affine Grassmannian for the Langlands dual group.