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.
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BibTeX
@misc{kawahigashi2026quantum,
title = {{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}},
author = {Yasuyuki Kawahigashi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02582},
eprint = {2611.02582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02582},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - 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
AU - Yasuyuki Kawahigashi
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02582
UR - https://arxiv.org/abs/2611.02582
AB - 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.
ER -