Chiral algebras of class $\mathcal{S}$ and Moore-Tachikawa symplectic varieties
Tomoyuki Arakawa
TL;DR
The paper constructs a unique family of genus-zero chiral algebras $\mathbf{V}^\mathcal{S}_{G,b}$ for any complex semisimple group $G$, providing a chiral quantization of the Moore– Tachikawa genus-zero symplectic varieties $W^b_G$ and proving their Higgs branches recover the associated Moore– Tachikawa varieties. It develops a comprehensive framework using BRST, Drinfeld–Sokolov reduction, the Feigin–Frenkel center, chiral differential operators, and equivariant $W$-algebras to realize these algebras as simple, conformal vertex algebras with explicit central charges and characters. The authors prove that $X_{\mathbf{V}^\mathcal{S}_{G,b}}\cong W^b_G$ and provide a robust glueing mechanism for higher-genus theories via the 2d TQFT paradigm, including explicit realizations for key low-rank examples such as $G=SL_2$ and $G=SL_3$. These results connect the 4d $\mathcal{N}=2$ class S/CFT data to precise vertex-algebraic objects, validating Higgs-branch correspondences and offering concrete tools for future investigations of quasi-lisse and other geometric properties.
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.