Norm, trace, and formal codegrees of fusion categories

TL;DR

This paper investigates properties of fusion categories using Galois conjugates of formal codegrees, establishing finiteness results and rationality conditions, with implications for the structure of spherical braided fusion categories.

Contribution

It introduces new techniques involving norm and trace of formal codegrees to derive finiteness and rationality results in fusion categories.

Findings

Finitely many fusion categories have a fixed norm of global dimension.

Most formal codegrees in spherical fusion categories with square-free norm are rational integers.

Most spherical braided fusion categories with prime norm are pointed.

Abstract

We prove several results in the theory of fusion categories using the product (norm) and sum (trace) of Galois conjugates of formal codegrees. First, we prove that finitely-many fusion categories exist up to equivalence whose global dimension has a fixed norm. Furthermore, with two exceptions, all formal codegrees of spherical fusion categories with square-free norm are rational integers. This implies, with three exceptions, that every spherical braided fusion category whose global dimension has prime norm is pointed. The reason exceptions occur is related to the classical Schur-Siegel-Smyth problem of describing totally positive algebraic integers of small absolute trace.