A remark on boundary level admissible representations

TL;DR

This paper explains how characters of admissible representations of affine Kac-Moody and W-algebras decompose into products at boundary levels, linking recent advances in superconformal field theories and vertex algebras.

Contribution

It demonstrates that at boundary levels, characters of these algebras decompose into products, clarifying phenomena observed in recent studies.

Findings

Characters decompose into products at boundary levels

Decomposition involves Jacobi form ( au, z)

Connects superconformal theories with vertex algebra characters

Abstract

Recently a remarkable map between 4-dimensional superconformal field theories and vertex algebras has been constructed \cite{BLLPRV15}. This has lead to new insights in the theory of characters of vertex algebras. In particular it was observed that in some cases these characters decompose in nice products \cite{XYY16}, \cite{Y16}. The purpose of this note is to explain the latter phenomena. Namely, we point out that it is immediate by our character formula \cite{KW88}, \cite{KW89} that in the case of a \textit{boundary level} the characters of admissible representations of affine Kac-Moody algebras and the corresponding $W$-algebras decompose in products in terms of the Jacobi form $ \vartheta_{11}(\tau, z)$.