A Level-Depth Correspondence between Verlinde Rings and Subfactors

Abstract

We establish a correspondence between the levels of Verlinde rings and the depths of subfactors. Given the $l$-level Verlinde ring $R_l(G)$ of a simple compact Lie group $G$, the tensor products of fundamental representations give us the inclusion of a pair of $\text{II}_1$ factors $N\subset M$. For the depth $d$ of $N\subset M$, we first prove $d=l$ for type $A_n,C_n$ and $B_2$. More generally, the depth $d$ is shown to satisfy $\beta\cdot l\leq d\leq l$ with $\beta\in (0,1)$, where $\beta$ is uniquely determined by the simple type of $G$. We also show that the simple $N$-$N$-bimodules contained in $L^2(M)$ generate the Verlinde ring $R_l(G)$ as its fusion category.