On characters of $L_{{\mathfrak{sl}}_n}(-{\Lambda }_0)$-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 $L_{{\mathfrak{sl}}_{\ell }}(-{\Lambda }_0)$. We establish asymptotic behaviors of characters for the (ordinary) irreducible $L_{{\mathfrak{sl}}_{\ell }}(-{\Lambda }_0)$-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 ${\mathfrak{sl}}_3$. 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.

22 pages