Recovering R-symbols from modular data

TL;DR

This paper demonstrates that the R-symbols of a premodular category can be reconstructed solely from its T-matrix, fusion rules, and Frobenius-Schur indicators, especially when the category is modular.

Contribution

It introduces a method to recover R-symbols from modular data, clarifying the relationship between these invariants in premodular and modular categories.

Findings

R-symbols can be reconstructed from T-matrix, fusion coefficients, and Frobenius-Schur indicators.

In modular categories, R-symbols are fully determined by modular data.

Provides a gauge-independent way to determine R-symbols in certain categories.

Abstract

Given a premodular category $\mathcal{C}$, we show that its $R$-symbol can be recovered from its $T$-matrice, fusion coefficients and some 2nd generalized Frobenius-Schur indicators. In particular, if $\mathcal{C}$ is modular, its $R$-symbols for a certain gauge choice are completely determined by its modular data.