On Haagerup-Izumi fusion categories
Abstract
For every odd N, we construct a Haagerup-Izumi (HI) category for ℤ/N, using Barnes double-sine functions. Equivariantization gives near-group categories of type ((ℤ/N)2,N2) for every odd N. We classify connected separable algebras up to Morita equivalence in such HI categories and describe their dual categories. These duals are often HI for noncyclic groups with nontrivial pointed associators. We prove the Evans-Gannon conjecture that the modular data of the center is always a smashed sum involving a finite metric group of order N2+4 and recover this group and its quadratic form directly from the data of the HI category. We also give the HI equations and reconstruction in arbitrary characteristic and prove a Frobenius symmetry in characteristic two. Using this, we obtain the complete list of (untwisted, subfactor type) HI categories over ℂ for odd N≤ 43. Other applications include finding three pairwise non-Morita-equivalent ℤ/15 categories whose centers have the same modular data. We show that the center of a ℤ/5 HI category cannot be defined (i.e., admits no split ribbon form) over a cyclotomic field, disproving a conjecture of Davidovich-Hagge-Wang.
Related
- Cyclic Haagerup-Izumi fusion categories at every odd order
- A vertex operator algebra for the S-fold realisation of a non-unitary Haagerup--Izumi modular fusion category
- Vertex operator algebra realizations of untwisted Haagerup--Izumi centers
- Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua
- Pre-modular fusion categories of global dimension 2p
BibTeX
@misc{gannon2026haagerupizumi,
title = {{On Haagerup-Izumi fusion categories}},
author = {Terry Gannon and Andrew Schopieray and Harshit Yadav},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25185},
eprint = {2609.25185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25185},
}
RIS
TY - UNPB
TI - On Haagerup-Izumi fusion categories
AU - Terry Gannon
AU - Andrew Schopieray
AU - Harshit Yadav
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25185
UR - https://arxiv.org/abs/2609.25185
AB - For every odd N, we construct a Haagerup-Izumi (HI) category for ℤ/N, using Barnes double-sine functions. Equivariantization gives near-group categories of type ((ℤ/N)^2,N^2) for every odd N. We classify connected separable algebras up to Morita equivalence in such HI categories and describe their dual categories. These duals are often HI for noncyclic groups with nontrivial pointed associators. We prove the Evans-Gannon conjecture that the modular data of the center is always a smashed sum involving a finite metric group of order N^2+4 and recover this group and its quadratic form directly from the data of the HI category. We also give the HI equations and reconstruction in arbitrary characteristic and prove a Frobenius symmetry in characteristic two. Using this, we obtain the complete list of (untwisted, subfactor type) HI categories over ℂ for odd N≤ 43. Other applications include finding three pairwise non-Morita-equivalent ℤ/15 categories whose centers have the same modular data. We show that the center of a ℤ/5 HI category cannot be defined (i.e., admits no split ribbon form) over a cyclotomic field, disproving a conjecture of Davidovich-Hagge-Wang.
ER -