The zipper condition for $4$-tensors in two-dimensional topological order and the higher relative commutants of a subfactor arising from a commuting square

Abstract

Researchers in condensed matter physics recently study two-dimensional topological order in terms of tensor networks involving certain 3- and 4-tensors. Their 3-tensors satisfying the "zipper condition" play an important role there and such 3-tensors can be made into certain 2-tensors by combining two wires into one. We identify their 4-tensors with bi-unitary connections in Jones' subfactor theory in operator algebras with precise normalization constants. Then we prove that their 2-tensors satisfying the zipper condition are the same as flat fields of strings in subfactor theory which correspond to elements in the higher relative commutants of the subfactor arising from the bi-unitary connection. This is what we expect, since the zipper condition is a kind of pentagon relations, but we clarify what conditions are exactly needed for this -- we do not need the flatness or the finite depth condition for the bi-unitary connection. We actually generalize their 4-tensors so that the four index sets of the 4-tensors can be all different and work on a "half-version" of the zipper condition.