Inequivalent modular tensor categories with identical S, T and W
Abstract
It remains an open question whether the modular data (S,T) together with the Whitehead-link matrix W determine a modular tensor category up to ribbon equivalence. We answer negatively by constructing two braided-inequivalent Dijkgraaf--Witten modular categories for (ℤ/pℤ)3 for prime p≥5 with identical (S,T,W) and, more generally, identical Reshetikhin--Turaev invariants of all framed oriented links with at most two components under a single bijection of simple objects. The construction varies an alternating $3$-cocycle while fixing a quadratic cohomology class, using restriction to subgroups of rank at most two to match link invariants and the quadratic and alternating classes together to obstruct braided equivalence. A complete classification of modular tensor categories up to ribbon equivalence therefore requires invariants beyond those of colored framed links with at most two components. We also analytically evaluate an uncolored partition function on a closed oriented three-manifold that separates all (p−1)/2 equivalence classes in the family.
Related
BibTeX
@misc{luo2026inequivalent,
title = {{Inequivalent modular tensor categories with identical $S$, $T$ and $W$}},
author = {Ran Luo and Jiahua Tian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33231},
eprint = {2609.33231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33231},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Inequivalent modular tensor categories with identical S, T and W
AU - Ran Luo
AU - Jiahua Tian
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33231
UR - https://arxiv.org/abs/2609.33231
AB - It remains an open question whether the modular data (S,T) together with the Whitehead-link matrix W determine a modular tensor category up to ribbon equivalence. We answer negatively by constructing two braided-inequivalent Dijkgraaf--Witten modular categories for (ℤ/pℤ)^3 for prime p≥5 with identical (S,T,W) and, more generally, identical Reshetikhin--Turaev invariants of all framed oriented links with at most two components under a single bijection of simple objects. The construction varies an alternating $3$-cocycle while fixing a quadratic cohomology class, using restriction to subgroups of rank at most two to match link invariants and the quadratic and alternating classes together to obstruct braided equivalence. A complete classification of modular tensor categories up to ribbon equivalence therefore requires invariants beyond those of colored framed links with at most two components. We also analytically evaluate an uncolored partition function on a closed oriented three-manifold that separates all (p−1)/2 equivalence classes in the family.
ER -