Classification of Module Categories for $SO(3)_{2m}$

TL;DR

This paper classifies $ ext{SO}(3)_{2m}$ module categories by analyzing nimrep graphs and cell systems, revealing the types of module categories and constructing related subfactors and Frobenius algebras.

Contribution

It provides a complete classification of $ ext{SO}(3)_{2m}$ module categories, including new constructions of subfactors and Frobenius algebras generalizing preprojective algebras.

Findings

Classified all $ ext{SO}(3)_{2m}$ module categories as $ ext{A}$, $ ext{E}$ types, no $ ext{D}$ type.

Constructed a subfactor with principal graph from $ ext{SO}(3)_{2m}$ fusion rules.

Introduced a Frobenius algebra generalizing higher preprojective algebras.

Abstract

The main goal of this paper is to classify $\ast$-module categories for the $SO(3)_{2m}$ modular tensor category. This is done by classifying $SO(3)_{2m}$ nimrep graphs and cell systems, and in the process we also classify the $SO(3)$ modular invariants. There are module categories of type $\mathcal{A}$, $\mathcal{E}$ and their conjugates, but there are no orbifold (or type $\mathcal{D}$) module categories. We present a construction of a subfactor with principal graph given by the fusion rules of the fundamental generator of the $SO(3)_{2m}$ modular category. We also introduce a Frobenius algebra $A$ which is an $SO(3)$ generalisation of (higher) preprojective algebras, and derive a finite resolution of $A$ as a left $A$-module along with its Hilbert series.