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

Abstract

In our article [arXiv:1511.05226], we studied the commutant ${\mathcal{C}}^{\prime }\subset \mathrm{Bim}(R)$ of a unitary fusion category $\mathcal{C}$, where $R$ is a hyperfinite factor of type $\mathrm{I}{\mathrm{I}}_1$, $\mathrm{I}{\mathrm{I}}_{\infty }$, or $\mathrm{II}{\mathrm{I}}_1$, 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 ${\mathcal{C}}^{\prime }$ are classified by connected finite depth unitary anchored planar algebras in $\mathcal{Z}(\mathcal{C})$. This extends the classification of finite depth objects of $\mathrm{Bim}(R)$ by connected finite depth unitary planar algebras.

20 pages, many figures