Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua

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 $\phi$, a word in $L$ and $R$, the two groups are $A_{\mp }=\text{coker}(\phi \mp 1)$, with the torsion linking forms of the mapping tori of $\phi$ and $-\phi$. 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, $c_{\text{eff}}=\mathrm{\#}L$, and the handle-gluing values $2^{\mathrm{\#}L}{\lambda }^{\pm 1}\mathrm{\mid }{A}_{-}\mathrm{\mid }\mathrm{\mid }{A}_{+}\mathrm{\mid }$ at the identity classes, $\lambda$ the stretch factor. Each kernel vector of $\phi \mp 1$ modulo $\mathrm{\mid }{A}_{\mp }\mathrm{\mid }$ 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 $\sigma V^{\ast }$, $c\equiv \mathrm{\#}L-6\mathrm{\#}R(\mathrm{m}\mathrm{o}\mathrm{d}24)$.