Classification of finite depth objects in bicommutant categories via anchored planar algebras

TL;DR

This paper links the classification of finite depth objects in bicommutant categories to anchored planar algebras, extending previous classifications in bimodule categories and providing a new framework for understanding these objects.

Contribution

It establishes a connection between finite depth objects in bicommutant categories and connected finite depth anchored planar algebras in the Drinfeld center.

Findings

Finite depth objects in $ ext{Bicom}(R)$ are classified by connected finite depth anchored planar algebras.

Extends classification results from bimodule categories to bicommutant categories.

Provides a new algebraic framework for understanding finite depth objects in these categories.

Abstract

In our article [arXiv:1511.05226], we studied the commutant $\mathcal{C}'\subset \operatorname{Bim}(R)$ of a unitary fusion category $\mathcal{C}$, where $R$ is a hyperfinite factor of type $\rm II_1$, $\rm II_\infty$, or $\rm III_1$, and showed that it is a bicommutant category. In other recent work [arXiv:1607.06041, arXiv:2301.11114] we introduced the notion of a (unitary) anchored planar algebra in a (unitary) braided pivotal category $\mathcal{D}$, and showed that they classify (unitary) module tensor categories for $\mathcal{D}$ equipped with a distinguished object. Here, we connect these two notions and show that finite depth objects of $\mathcal{C}'$ are classified by connected finite depth unitary anchored planar algebras in $\mathcal{Z}(\mathcal{C})$. This extends the classification of finite depth objects of $\operatorname{Bim}(R)$ by connected finite depth unitary planar algebras.