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

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.