The Costello–Yagi unification of integrability in supersymmetric gauge theories derives the R-matrices of the rational, trigonometric and elliptic lattice models from Wilson lines in four-dimensional Chern–Simons (4d CS) theory, and poses as an open problem the origin of the chiral Potts model — the canonical exactly solved model whose spectral parameter lives on an algebraic curve of genus greater than one. Moosavian–Yamazaki–Zhou (2025) substantially advanced this problem by engineering the generalized chiral Potts model within 4d CS: the higher-genus curve is realized as a branched cover of the Riemann sphere, the meromorphic one-form of the 4d CS action is essentially uniquely fixed by topological invariance, and the old Atiyah coincidence between the chiral Potts model and hyperbolic monopoles is traced to a ten-dimensional holomorphic Chern–Simons theory. However, the explicit computation of the chiral Potts R-matrix from the gauge theory remains open — the authors call it an outstanding challenge on multiple levels, since it requires a non-perturbative, root-of-unity formulation of 4d CS and line defects carrying minimal cyclic representations of quantum affine algebras. The problem posed here is precisely this surviving core: derive the (generalized) chiral Potts Boltzmann weights, and thereby the emergence of the higher-genus spectral curve, from first principles in 4d CS theory or its six- and ten-dimensional lifts.

Background

A two-dimensional lattice model is 'exactly solvable' when its Boltzmann weights satisfy the star-triangle relation (equivalently, the Yang–Baxter equation for the associated R-matrix), a local functional equation guaranteeing that transfer matrices built from the weights commute and are simultaneously diagonalizable. The weights depend on a spectral parameter attached to each rapidity line of the lattice. For the classical families this parameter lives on the Riemann sphere, the punctured complex plane, or an elliptic curve (a Riemann surface of genus one, i.e. with one handle), and the Belavin–Drinfeld classification shows these are the only possibilities for solutions with the rapidity-difference property, meaning the R-matrix depends on the rapidities only through their difference or ratio. The chiral Potts (chiral clock) model, discovered in the late 1980s by Au-Yang, McCoy, Perk, Tang and Yan and by Baxter, Perk and Au-Yang, evades this classification: its clock spins (ZN\mathbb{Z}_N-valued variables, n−1n-1 copies of them in the generalized model) interact through weights depending on rapidities living on an algebraic curve of genus greater than one (in the conventions of Moosavian–Yamazaki–Zhou, the curve of the generalized chiral Potts model of Bazhanov–Kashaev–Mangazeev–Stroganov has genus N2(n−1)(N(n−1)−n)+1N^{2(n-1)}(N(n-1)-n)+1, where NN is the number of clock states per spin variable and nn the label of the underlying quantum affine algebra sl^n\widehat{\mathfrak{sl}}_n (so that sln\mathfrak{sl}_n has rank n−1n-1, matching the n−1n-1 copies of the clock spin); the unique genus-one case, (n,N)=(2,2)(n,N)=(2,2), is the Ising model). The proof that its R-matrix satisfies the Yang–Baxter equation uses minimal cyclic representations — finite-dimensional representations on which the loop generators of the quantum group act cyclically — of the quantum affine algebra Uq(sl^n)U_q(\widehat{\mathfrak{sl}}_n) (a qq-deformation of the universal enveloping algebra of the affine Lie algebra sl^n\widehat{\mathfrak{sl}}_n, which extends sln\mathfrak{sl}_n by loop generators) at roots of unity qN=1q^N=1 (Date–Jimbo–Miki–Miwa; Kashaev–Mangazeev–Nakanishi); consequently the model has no rapidity-difference property and no classical R-matrix, placing it outside the standard quantization machinery. Four-dimensional Chern–Simons (4d CS) theory, introduced by Costello, is a holomorphic-topological gauge theory (invariant under holomorphic reparametrizations of one pair of directions and topological invariance in the other) on C×Σ\mathbb{C}\times\Sigma with action 12π∫ω∧CS(A)\frac{1}{2\pi}\int\omega\wedge\mathrm{CS}(A), where ω\omega is a meromorphic one-form on the Riemann surface Σ\Sigma. Wilson lines (gauge-invariant observables that transport a representation of the gauge group along a curve) stretched along the topological direction reproduce the R-matrices of essentially all lattice models whose spectral parameter lies on the sphere, the punctured plane or an elliptic curve. Costello and Yagi showed that this theory is a quasi-topological sector of six-dimensional maximally supersymmetric Yang–Mills theory, topologically twisted and Omega-deformed (an Omega-deformation is a superrotation of two coordinates coupled to a scalar, which effectively dimensionally reduces the theory), realized in string theory by D5-branes (extended membrane-like objects on which open strings end) in a background Ramond–Ramond flux (a flux of the higher-form gauge fields of string theory); applying dualities they unified the known appearances of the eight-vertex model and the XYZ spin chain in supersymmetric gauge theories. Surface operators (disorder operators supported on two-dimensional submanifolds, on which the gauge field has prescribed singular behavior) and monopoles (topological solitons carrying magnetic charge) play a central role: monopoles in the six-dimensional theory create surface operators, and crossings of surface operators produce a variant of the Bazhanov–Sergeev master solution of the star-triangle relation, which degenerates to the chiral Potts model when a parameter is specialized to a root of unity (Bazhanov–Sergeev; Kels–Yamazaki; Moosavian–Yamazaki–Zhou). Atiyah had already observed in 1990 a 'mysterious coincidence' between the chiral Potts model and magnetic monopoles, later connected to the identification between the chiral Potts spectral curve and spectral curves of SU(2) hyperbolic monopoles (monopoles on hyperbolic three-space). The question left open by Costello–Yagi is whether the chiral Potts model, its higher-genus curve and its R-matrix can be derived from this gauge-theoretic construction in the same way the elliptic models are.

Problem Statement

Derive the (generalized) chiral Potts model from supersymmetric gauge theory: starting from the four-dimensional Chern–Simons framework of Costello–Yagi, or its six-dimensional Omega-deformed / ten-dimensional holomorphic-Chern–Simons lifts, produce the model's Boltzmann weights and thereby demonstrate that its higher-genus spectral curve emerges as a geometric object of the underlying theory. The following deliverables all serve this single objective and are ordered by logical dependence rather than being independent questions. First, specify the complete 4d CS data: the surface Σ\Sigma realized, following Moosavian–Yamazaki–Zhou, as P1\mathbb{P}^1 with the meromorphic one-form ω\omega whose branch points encode the genus-≥2\ge2 spectral curve as a branched cover, the gauge algebra, and line defects carrying the minimal cyclic representations of Uq(sl^n)U_q(\widehat{\mathfrak{sl}}_n) at a root of unity qN=1q^N=1. Second, provide a controlled definition of the theory at the required root-of-unity coupling — non-perturbative, or via an explicit string/brane realization — precise enough to support computation, noting that the one-form has zeros on the curve so the standard expansion in the 4d CS coupling (equivalently, the classical/quantum loop expansion) is unavailable. Third, compute from this setup the R-matrix (equivalently the Boltzmann weights) of the (generalized) chiral Potts model and show that it satisfies the Yang–Baxter/star-triangle equation with spectral parameters on the genus-≥2\ge2 curve, coinciding with the known chiral Potts weights up to the conventional normalization and gauge freedom. A resolution would establish, in the sense Costello–Yagi asked for, that the higher-genus curve emerges in the low-energy physics in the way the Seiberg–Witten curve (the algebraic curve encoding the low-energy Coulomb-branch spectrum of a supersymmetric gauge theory) does when a D4–NS5 brane configuration is lifted to M-theory (the eleven-dimensional strong-coupling completion of string theory).

References

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