Eigenvalues of rotations and braids in spherical fusion categories

TL;DR

This paper develops formulas to compute eigenvalues of rotation operators and braids in spherical fusion categories using Frobenius-Schur indicators, fusion rules, and modular data, advancing understanding of their algebraic and topological properties.

Contribution

It introduces explicit formulas linking eigenvalues of rotations and braids to fusion rules, Frobenius-Schur indicators, and modular data, providing new computational tools.

Findings

Eigenvalues of rotation operators can be computed from fusion rules and indicators.

Eigenvalues of braids in braided categories are determined by fusion data and modular matrices.

Rigidity of Frobenius-Schur indicators is established via planar algebra theory.

Abstract

We give formulae for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized Frobenius-Schur indicators in a semisimple spherical tensor category $\mathcal{C}$. In particular, this implies that the entire collection of rotation eigenvalues for a fusion category can be computed from the fusion rules and the traces of rotation at finitely many tensor powers. We also establish a rigidity property for FS indicators of fusion categories with a given fusion ring via Jones's theory of planar algebras. If $\mathcal{C}$ is also braided, these formulae yield the multiplicities of eigenvalues for a large class of braids in the associated braid group representations. When $\mathcal{C}$ is modular, this allows one to determine the eigenvalues and multiplicities of braids in terms of just the $S$ and $T$ matrices.