Generalized $q$-MZVs and Characters of Vertex Algebras

Abstract

We analyze three related families of $q$-series arising from characters of vertex (super)algebras. More precisely, we consider: (i) graph series and characters of principal subspaces associated with arc algebras; (ii) characters, or indices, of class-$\mathcal{S}$ vertex operator algebras in the context of 4d $\mathcal{N}=2$ SCFT; and (iii) supercharacters of the $\mathcal{U}$ family of vertex superalgebras and their quasi-modular properties. Along the way, we introduce a family of multiple $q$-zeta values associated with simple Lie algebras and present their conjecturalproperties.