Slightly degenerate categories and $\mathbb{Z}$-modular data

TL;DR

This paper explores how slightly degenerate fusion categories naturally produce Z-modular data, extending the framework beyond spherical categories to pivotal categories, and interprets a categorification related to complex reflection groups.

Contribution

It introduces a method to derive Z-modular data from slightly degenerate fusion categories without assuming sphericality, and provides a new interpretation of a categorification linked to complex reflection groups.

Findings

Derived Z-modular data from slightly degenerate fusion categories.

Extended the framework to pivotal categories beyond spherical categories.

Connected the categorification to cyclic complex reflection groups.

Abstract

Given a slightly degenerate fusion category $\mathcal{C}$, we explain how it naturally gives rise to a Z-modular data. We do not restrict to spherical categories and work with pivotal categories instead. Finally, we give an interpretation in this framework of the Bonnaf\'e-Rouquier categorification of the $\mathbb{Z}$-modular datum associated to non trivial family of the cyclic complex reflection group.