On characters of $L_{\frak{sl}_n}(-\Lambda_0)$-modules

TL;DR

This paper analyzes the characters of modules for the vertex operator algebra associated with rak{sl}_n at level -mbda_0, establishing their asymptotic behavior, quantum dimensions, and modular properties.

Contribution

It provides the first detailed asymptotic analysis and modular decomposition of characters for these modules, confirming predictions about quantum dimensions.

Findings

Quantum dimensions are equal to one.

Asymptotic expansions of characters are established.

Decomposition into theta and false theta functions is achieved.

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.