Projector matrix product operators, anyons and higher relative commutants of subfactors

Abstract

A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck–Mariën–Williamson–Şahinoğlu-Haegeman–Verstraete on two-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of length k is naturally identified with the kth higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between two-dimensional topological order and subfactor theory.