Technique in computing the characters of VOA modules using vector-valued functions for modular groups

For a rational vertex operator algebra (VOA) $V$ with a $\mathbb{Z}_{\ge 0}$ -graded simple $V$-module $M$ , there is a corresponding character \[\text{ch } M = \text{Tr}_Mq^{L^M_0 −c/24} = \sum_n \dim M_n q^{n−c/24} ,\] where $M_n$ is the subspace of $M$ on which $L^M_0$ acts by multiplication by $n$, $c$ is the central charge of $V$ and $q = e^{2\pi i \tau}.$

In this paper, we apply the notion of vector-valued functions for a modular group from the work of Peter Bantay and Terry Gannon to compute the $V$-module character $\text{ch } M.$ With the relation among simple $V$-modules $M$ and their corresponding simple objects of modular tensor category (MTC) $\mathcal{C},$ we can use the central charge and conformal weights of the MTC  $\mathcal{C}$ to compute the character $\text{ch } M.$ This technique can be used to compute $\text{ch } M$ up to central charge 24 by the restriction mentioned in P. Bantay paper.