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 $\varphi$, a word in $L$ and $R$, the two groups are $A_\mp=\coker(\varphi\mp1)$, with the torsion linking forms of the mapping tori of $\varphi$ and $-\varphi$. Over all words of trace $m<sup>2+2$ 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 $2<sup>{#L}λ<sup>{\pm1}|A_-||A_+|$ at the identity classes, $λ$ the stretch factor. Each kernel vector of $\varphi\mp1$ modulo $|A_\mp|$ 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 $σV<sup>*$, $c\equiv#L-6#R\pmod{24}$.