On braided tensor categories of type BCD

TL;DR

This paper classifies all braided semisimple tensor categories related to orthogonal and symplectic groups, detailing their structure based on braiding eigenvalues and object dimensions, with implications for fusion categories.

Contribution

Provides a complete classification of braided semisimple tensor categories of types B, C, D, based on their Grothendieck semirings and braiding properties.

Findings

Classification of categories with non-symmetric braiding by eigenvalues.

Determination of categories with symmetric braiding by object dimensions.

Correction of a previous mistake without affecting main results.

Abstract

We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep(O(\infty) (formally), Rep(O(N), Rep(Sp(N) or of one of its associated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation. Note that the paper is followed by a brief erratum, which corrects a mistake, which does not affect the main results of the paper.