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

Technical Summary: Haagerup–Izumi modular data from hyperbolic words

Problem Statement
The paper addresses the rigorous construction and understanding of the modular data associated with the doubles of Haagerup–Izumi categories. While Evans and Gannon conjectured that these modular data arise from a "grafting" of two finite metric groups, and Gannon, Schopieray, and Yadav proved this for untwisted categories, the physical realization of these data via topological quantum field theories (TQFTs) remained to be fully understood. Specifically, Gang and Kim identified non-unitary modular data of this shape in the topological twist of rank-0 $S$-fold theories, derived from the Bethe vacua of an abelianized gauge theory. The central problem is to rigorously define the handle-gluing and fibering operators in this context, construct the Bethe vacua for general hyperbolic monodromies, and establish the precise correspondence between the algebraic data of the Haagerup–Izumi categories and the physical data of the $S$-fold TQFTs.

Methodology
The author develops a unified framework based on hyperbolic words $\phi$ in the generators $L$ and $R$ of $SL(2, \mathbb{Z})$, which correspond to the monodromy of once-punctured torus bundles. The methodology proceeds through three distinct "frames" that interpret the same underlying mathematical structure:

1. The Farey Frame: Utilizing the layered triangulation of the mapping torus $M_\phi$, the paper constructs Neumann–Zagier matrices $A$ and $B$. The level matrix $K$ is derived as the unique symmetric solution to the gluing equations $BK_{raw} = A$ satisfying a determinant law $\det K = 2^r$ (where $r$ is the word length). This frame defines the charge vectors and the Bethe equations.
2. The Nahm Frame: The Bethe equations are shown to be equivalent to Nahm's ladder system for a symmetric matrix $N = 2(I - K_{raw}^{-1})$. This equivalence allows for the explicit construction of solutions. The positive vacuum $V^*$ is identified as the solution propagated by the expanding eigenvector of $\phi$, expressed as ratios of heights along Farey slopes.
3. The Bethe Frame: This frame defines the physical observables. The handle-gluing operator $H_\alpha$ and fibering operator $F_\alpha$ are evaluated at the Bethe vacua. The paper employs a deformation of the Bethe equations (introducing a parameter $\epsilon$) to recover degenerate solutions that lie at infinity in the undeformed system.

Key mathematical tools include the theory of frieze rows (specifically $SL_2$ friezes) to constrain solutions within runs of the word, Hill's formula for periodic Jacobi matrices to compute the handle-gluing spectrum, and the use of torsion linking forms on the cokernels $A_\mp = \text{coker}(\phi \mp 1)$.

Key Contributions and Results

• Construction of Metric Groups: For every hyperbolic word $\phi$, the paper identifies two finite metric groups $A_\mp = \text{coker}(\phi \mp 1)$. It proves that $A_-$ carries the torsion linking form of the mapping torus $M_\phi$ and $A_+$ carries that of $M_{-\phi}$. For the diagonal family $L_m R_m$, these groups are shown to be $\mathbb{Z}_m \times \mathbb{Z}_m$ and $\mathbb{Z}_{m^2+4}$, respectively, matching the metric groups of the untwisted Haagerup–Izumi categories.
• Closed-Form Solutions: The paper provides closed-form expressions for the positive vacuum $V^*$ and its Galois conjugate $\sigma V^*$ for every hyperbolic word. These solutions are given as ratios of heights along the Farey slopes, propagated by the expanding eigenvector of the monodromy.
• Effective Central Charge: It is rigorously proved that the effective central charge is $c_{eff} = \#L$ (the number of $L$ letters in the word) for every hyperbolic word, derived via Rogers dilogarithm identities at the eigenvector solutions.
• Handle-Gluing Spectrum: The handle-gluing values $H_\alpha$ are computed for all non-degenerate classes.
  • For the two identity classes ($V^*$ and $\sigma V^*$), $H = 2^{\#L} \lambda^{\pm 1} |A_-| |A_+|$, where $\lambda$ is the stretch factor.
  • For non-identity non-degenerate classes associated with kernel vectors $v$ of $\phi \mp 1$, $H = 2^{\#L} |A_\mp|$.
  • The paper establishes that for words without singleton runs, these classes (plus the identity classes) constitute all non-degenerate classes.
• Modular Data and Grafting: Under specific physical inputs (the half-index dictionary and vacuum identification), the paper demonstrates that for words of odd trace, the resulting modular data $S$ and $T$ matrices correspond to the Evans–Gannon grafting. Specifically, for the diagonal family $L_m R_m$ where $m$ is prime and $m^2+4$ is prime, the data is a Galois conjugate of the unitary grafting, realizing the modular data of the double of the untwisted Haagerup–Izumi category.
• Spin and Linking Forms: The conformal spins of the vacua are shown to be quadratic refinements of the torsion linking forms on the groups $A_\mp$. The spin is given explicitly in terms of the kernel vectors and the linking form.

Significance and Scope
The paper claims to rigorously bridge the gap between the algebraic classification of Haagerup–Izumi categories and the physical construction of $S$-fold TQFTs. It provides a complete, closed-form description of the Bethe vacua and their associated invariants (handle-gluing, fibering, spin) for a broad class of words, confirming the Evans–Gannon grafting conjecture in the context of these TQFTs.

The work is modest regarding its scope:

• It proves results for all hyperbolic words without physical input regarding the structure of the level matrix and the existence of the positive vacuum.
• Results concerning the central charge $c$ and the identification of the unit object rely on the "half-index dictionary" and specific vacuum identification assumptions (Input (b)).
• The paper explicitly notes that it does not construct categories for the twisted modular data ($D^\omega H_{gm}$ with $\omega \neq 0$), nor does it fully resolve the degenerate classes for general words (though it handles the diagonal family $L_m R_m$ at prime $m$ via a separate analysis in Part IV).
• The sign of the $S_{00}$ entry and the completeness of the classification for words with singleton runs remain open or rely on numerical verification.

In summary, the paper establishes that the modular data of the doubles of Haagerup–Izumi categories can be systematically generated from the hyperbolic words of $SL(2, \mathbb{Z})$ via the Bethe vacua of associated abelianized gauge theories, with the handle-gluing and fibering operators playing the role of the key invariants linking the topological and algebraic structures.