Near-group fusion categories and their doubles

TL;DR

This paper classifies near-group fusion categories, computes their doubles and modular data, and constructs over 40 new finite depth subfactors, linking these mathematical structures to conformal field theories.

Contribution

It provides a comprehensive classification of near-group fusion categories and explicitly constructs new subfactors with potential applications in conformal field theory.

Findings

Classified all near-group fusion categories.

Computed doubles and modular data for these categories.

Constructed over 40 new finite depth subfactors.

Abstract

A near-group fusion category is a fusion category C where all but 1 simple objects are invertible. Examples of these include the Tambara-Yamagami categories and the even sectors of the E6 and affine-D5 subfactors, though there are infinitely many others. We classify the near-group fusion categories, and compute their doubles and the modular data relevant to conformal field theory. Among other things, we explicitly construct over 40 new finite depth subfactors, with Jones index ranging from around 6.85 to around 14.93. We expect all of these doubles to be realised by rational conformal field theories.