Modular categories are not determined by their modular data

TL;DR

This paper demonstrates that modular categories cannot be uniquely identified by their modular data, providing explicit examples of inequivalent categories sharing identical modular data, especially among integral modular categories.

Contribution

It constructs explicit examples of inequivalent modular categories with the same modular data, showing the limitations of modular data in classifying such categories.

Findings

Multiple inequivalent modular categories share identical modular data.

Examples are constructed over twisted Drinfeld doubles of finite groups.

The results apply to integral modular categories.

Abstract

Arbitrarily many pairwise inequivalent modular categories can share the same modular data. We exhibit a family of examples that are module categories over twisted Drinfeld doubles of finite groups, and thus in particular integral modular categories.