A remark on boundary level admissible representations

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 $\vartheta_{11}(τ, z)$.

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