A characterization of a finite-dimensional commuting square producing a subfactor of finite depth

TL;DR

This paper characterizes finite-dimensional commuting squares of C*-algebras that generate hyperfinite type II_1 subfactors of finite depth, linking them to Morita equivalent fusion categories and tensor networks in topological order.

Contribution

It generalizes Sato's construction of commuting squares, providing a comprehensive characterization of those producing finite-depth subfactors and related 4-tensors.

Findings

Characterization of commuting squares producing finite-depth subfactors

Extension of Sato's construction to the general case

Connection to 4-tensors in topological order studies

Abstract

We give a characterization of a finite-dimensional commuting square of C*-algebras with a normalized trace that produces a hyperfinite type II_1 subfactor of finite index and finite depth in terms of Morita equivalent unitary fusion categories. This type of commuting squares were studied by N. Sato, and we show that a slight generalization of his construction covers the fully general case of such commuting squares. We also give a characterization of such a commuting square that produces a given hyperfinite type II_1 subfactor of finite index and finite depth. These results also give a characterization of certain 4-tensors that appear in recent studies of matrix product operators in 2-dimensional topological order.