Quantum 6j-symbols arising from a subfactor of finite index and finite depth and the zipper condition for 4-tensors in two-dimensional topological order

Abstract

We study 3-tensors satisfying the ``zipper condition,'' which play an important role in tensor network approaches to two-dimensional topological order. In our previous work, we identified the 4-tensors appearing in these approaches with bi-unitary connections in the Jones theory of subfactors in operator algebras. These bi-unitary connections give rise to subfactors of finite index. We prove that, when the associated subfactor has finite depth, the 3-tensors satisfying the zipper condition correspond precisely to the quantum $6j$-symbols, also known as $F$-symbols, of the unitary multi-fusion category arising from the subfactor. This realization of quantum $6j$-symbols differs from their realization through a flat bi-unitary connection arising from the higher relative commutants of a subfactor.