Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua
Abstract
The modular data of the doubles of the Haagerup--Izumi categories is a grafting of two finite metric groups (conjectured by Evans--Gannon, proved for untwisted categories by Gannon--Schopieray--Yadav). Gang and Kim found non-unitary modular data of this shape in the topological twist of rank-$0$ S-fold theories, from the Bethe vacua of an abelianised gauge theory. The aim is to understand rigorously the significance of the handle-gluing and fibering operators, and to construct the Bethe vacua. For every hyperbolic once-punctured-torus monodromy φ, a word in L and R, the two groups are A∓=coker(φ∓1), with the torsion linking forms of the mapping tori of φ and −φ. Over all words of trace m2+2 these forms reproduce the Evans--Gannon table of Q-systems, one printed entry being corrected, and the distinct modular data in the list of Gannon--Schopieray--Yadav. The Bethe data are given in closed form by the word: the level matrix, the positive vacuum, ceff=#L, and the handle-gluing values 2#Lλ±1|A-||A+| at the identity classes, λ the stretch factor. Each kernel vector of φ∓1 modulo |A∓| at which an explicit trigonometric formula does not vanish gives a solution of the Bethe equations in closed form. Under the half-index dictionary, with the unit in the class of σV*, c≡#L−6#Rpmod24.
Related
- Haagerup conformal field theory at central charge eight
- Haagerup vertex operator algebras at central charge eight
- Haagerup symmetries of the E8 theory and conformal nets at central charge eight
- A vertex operator algebra for the S-fold realisation of a non-unitary Haagerup--Izumi modular fusion category
- Central Haagerup Tensor Products and Completely Bounded Maps under Strong Morita Equivalence
BibTeX
@misc{evans2026haagerupizumi,
title = {{Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua}},
author = {David E Evans},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18512},
eprint = {2610.18512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18512},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua
AU - David E Evans
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18512
UR - https://arxiv.org/abs/2610.18512
AB - The modular data of the doubles of the Haagerup--Izumi categories is a grafting of two finite metric groups (conjectured by Evans--Gannon, proved for untwisted categories by Gannon--Schopieray--Yadav). Gang and Kim found non-unitary modular data of this shape in the topological twist of rank-$0$ S-fold theories, from the Bethe vacua of an abelianised gauge theory. The aim is to understand rigorously the significance of the handle-gluing and fibering operators, and to construct the Bethe vacua. For every hyperbolic once-punctured-torus monodromy φ, a word in L and R, the two groups are A_∓=coker(φ∓1), with the torsion linking forms of the mapping tori of φ and −φ. Over all words of trace m^2+2 these forms reproduce the Evans--Gannon table of Q-systems, one printed entry being corrected, and the distinct modular data in the list of Gannon--Schopieray--Yadav. The Bethe data are given in closed form by the word: the level matrix, the positive vacuum, c_eff=#L, and the handle-gluing values 2^#Lλ^±1|A_-||A_+| at the identity classes, λ the stretch factor. Each kernel vector of φ∓1 modulo |A_∓| at which an explicit trigonometric formula does not vanish gives a solution of the Bethe equations in closed form. Under the half-index dictionary, with the unit in the class of σV^*, c≡#L−6#Rpmod24.
ER -