# A remark on matrix product operator algebras, anyons and subfactors

## TL;DR

This paper establishes a deep connection between matrix product operator algebras, subfactor theory, and anyon models, showing their equivalence under certain conditions and extending the framework to non-flat symmetric bi-unitary connections.

## Contribution

It demonstrates that the structures used in recent tensor network and anyon models are mathematically equivalent to those in subfactor theory, broadening the understanding of their interrelations.

## Key findings

- Flat symmetric bi-unitary connections correspond to subfactor tube algebras.
- Matrix product operator algebras from non-flat connections are isomorphic to those from flat connections.
- The algebraic structures in tensor networks and subfactors are fundamentally the same.

## Abstract

We show that the mathematical structures in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, a system of flat symmetric bi-unitary connections arising from a subfactor with finite index and finite depth satisfies all their requirements for tensors and the tube algebra for such a subfactor and the anyon algebra for such tensors are isomorphic up to the normalization constants. Furthermore, the matrix product operator algebras arising from tensors corresponding to possibly non-flat symmetric bi-unitary connections are isomorphic to those arising from flat symmetric bi-unitary connections for subfactors.