Reconstruction of modular data from $SL_2(\mathbb{Z})$ representations

Abstract

Modular data is the most significant invariant of a modular tensor category. We pursue an approach to the classification of modular data of modular tensor categories by building the modular $S$ and $T$ matrices directly from irreducible representations of $SL_2(\mathbb{Z}/n \mathbb{Z})$. We discover and collect many conditions on the $SL_2(\mathbb{Z}/n \mathbb{Z})$ representations to identify those that correspond to some modular data. To arrive at concrete matrices from representations, we also develop methods that allow us to select the proper basis of the $SL_2(\mathbb{Z}/n \mathbb{Z})$ representations so that they have the form of modular data. We apply this technique to the classification of rank-$6$ modular tensor categories, obtaining a classification up to modular data. Most of the calculations can be automated using a computer algebraic system, which can be employed to classify modular data of higher rank modular tensor categories.