On characters of $L_{\frak{sl}_n}(-Λ_0)$-modules

Abstract: We use recent results of Rolen, Zwegers, and the first author to study characters of irreducible (highest weight) modules for the vertex operator algebra $L_{\frak{sl}\ell}(-\Lambda_0)$. We establish asymptotic behaviors of characters for the (ordinary) irreducible $L{\frak{sl}_\ell}(-\Lambda_0)$-modules. As a consequence we prove that their quantum dimensions are one, as predicted by representation theory. We also establish a full asymptotic expansion of irreducible characters for $\frak{sl}_3$. Finally, we determine a decomposition formula for the full characters in terms of unary theta and false theta functions which allows us to study their modular properties.