On invariants of Modular categories beyond modular data

TL;DR

This paper investigates new invariants of modular categories, specifically the $W$-matrix, which extend beyond traditional modular data, aiming to identify complete invariants for these categories.

Contribution

It introduces and analyzes the $W$-matrix as a novel invariant, demonstrating it is beyond the standard modular data $(S,T)$, and explores its potential as part of a complete invariant.

Findings

The $W$-matrix is strictly beyond the modular data $(S,T)$.

Punctured $S$-matrices also extend beyond the modular data.

The completeness of the triple $(S,T,W)$ as an invariant remains open.

Abstract

We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix--the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.