A Level-Depth Correspondence between Verlinde Rings and Subfactors

TL;DR

This paper establishes a mathematical correspondence between Verlinde rings associated with Lie groups and the depths of subfactors, revealing a precise relation for certain types and bounds for others, and linking bimodule categories to these structures.

Contribution

It introduces a novel connection between Verlinde rings and subfactor depths, providing bounds and explicit relations for simple Lie groups, and shows how bimodules generate the Verlinde ring.

Findings

Depth equals level for types A_n, C_n, B_2.

Depth bounds are proportional to the level with a specific constant.

Bimodules generate the Verlinde ring as a fusion category.

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.