Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 087, 33 pages      arXiv:2203.15642      https://doi.org/10.3842/SIGMA.2026.087
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

Generalized $q$-MZVs and Characters of Vertex Algebras

Antun Milas
Department of Mathematics and Statistics, University at Albany, State University of New York, 1400 Washington Avenue, Albany NY 12222, USA

Received December 30, 2025, in final form August 20, 2026; Published online September 02, 2026

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.

Key words: $q$-zeta values; Lie algebras; vertex operator algebras.

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