Ikhlef and Weston (2017) constructed quasi-local conserved currents, and from them parafermionic lattice operators of spin 1+iη/π1+i\eta/\pi, in the critical six-vertex model, and carried the construction to the trigonometric solid-on-solid (SOS) model through the vertex-face correspondence, identifying the operators with primary fields and screening operators of the associated c=1c=1 and c<1c<1 conformal field theories. These parafermions satisfy a discrete-integral (current-conservation) condition around lattice plaquettes - one discrete half of the Cauchy-Riemann equations - but not the second half, so they are not discretely holomorphic. The paper leaves open two linked questions: (1) a perturbed-conformal-field-theory reinterpretation of the discrete integral conditions away from the critical line, identifying the perturbing operators - subtle near η=0\eta=0 where the η\eta-perturbation is marginal (Kosterlitz-Thouless); (2) whether the quantum-group current construction can be extended to produce, in at least some model of this class, fully discretely-holomorphic lattice operators satisfying both discrete Cauchy-Riemann halves - the ingredient that underpinned rigorous scaling-limit conformal-invariance proofs for dimers, Ising, and percolation.

Background

Two-dimensional exactly solved lattice models at criticality are expected to be described, in the scaling limit, by conformal field theories (CFTs), but for most models this correspondence rests on heuristic matches of critical exponents and finite-size spectra rather than on proof. A central device for making such correspondences rigorous is discrete holomorphicity. In a two-dimensional field theory written in complex coordinates z=x+itz=x+it, zˉ=x−it\bar z=x-it, a conserved current obeys ∂zˉJ−∂zJˉ=0\partial_{\bar z}J-\partial_z\bar J=0; if it also obeys ∂zˉJ+∂zJˉ=0\partial_{\bar z}J+\partial_z\bar J=0, then ∂zˉJ=0\partial_{\bar z}J=0 and JJ is holomorphic, the hallmark of a CFT. A lattice operator satisfying discrete versions of both equations is called discretely holomorphic (or preholomorphic); such operators feed into discrete complex-analysis arguments, and this is precisely how conformal invariance was proved for dimers, the Ising model, and critical percolation. As Ikhlef and Weston note, these - together with a partial result for hexagonal-lattice self-avoiding walks, whose connective-constant proof rests on a parafermionic observable - remain essentially the only models with rigorous critical scaling limits.

The six-vertex model places arrows on the edges of a square lattice with the ice rule (two arrows in and two out at each vertex), with weights a,b,ca,b,c for the three vertex types and anisotropy Δ=(a2+b2−c2)/(2ab)\Delta=(a^2+b^2-c^2)/(2ab). For −1<Δ<1-1<\Delta<1 the model is critical (massless, algebraically decaying correlations) and its scaling limit is the c=1c=1 compactified free boson CFT; Δ=1\Delta=1 is the Kosterlitz-Thouless transition to the ferroelectric phase, where the perturbation driving the transition is marginal. In the parametrization used by Ikhlef and Weston the weights are built from sinh⁡(λ+η)\sinh(\lambda+\eta), sinh⁡λ\sinh\lambda, sinh⁡η\sinh\eta, so that Δ=cosh⁡η\Delta=\cosh\eta: the critical line is purely imaginary η=iθ\eta=i\theta with 0<θ<π0<\theta<\pi, and η=0\eta=0 is the Δ=1\Delta=1 Kosterlitz-Thouless point. Through the vertex-face correspondence (Baxter's intertwiners) the six-vertex model maps to a trigonometric solid-on-solid (SOS) model, a model of heights on one sublattice of the square lattice; with the cyclic restriction of heights at η=iπ(p−p′)/p\eta=i\pi(p-p')/p (p′<pp'<p coprime, the cyclic SOS or CSOS model) the CFT limit has central charge c=1−6(p−p′)2/(pp′)c=1-6(p-p')^2/(pp').

Bernard and Felder (1991) showed how to construct conserved currents on the lattice for models whose weights carry a quantum-group Uq(sl2)U_q(\mathrm{sl}_2) symmetry (a deformation of the Lie algebra symmetry, with deformation parameter q=eηq=e^{\eta}): the currents are quasi-local, meaning that they sit at a lattice insertion point together with a path (a tail) running to the boundary, and they satisfy a discrete analogue of current conservation. Ikhlef, Weston, Wheeler and Zinn-Justin (2013) and Ikhlef and Weston (2015) developed this construction for dense and dilute loop models and for the chiral Potts model; in the chiral Potts case a careful analysis of the critical limit of the discrete conservation condition led to an identification of the model as a perturbed CFT that agreed with and extended a prediction of Cardy. Ikhlef and Weston (2017) carry the construction through for the six-vertex model: the currents yield parafermionic lattice operators of spin 1+iη/π1+i\eta/\pi obeying discrete-integral (current-conservation) conditions around lattice plaquettes embedded in the complex plane, which are identified with primary fields of the c=1c=1 free boson; under the vertex-face correspondence these map to SOS operators with spins 11 and 1+2iη/π1+2i\eta/\pi, identified in the CSOS case with screening charges and the Φ1,3\Phi_{1,3} primary of the c<1c<1 theory.

The construction stops short in two ways, both acknowledged by the authors. First, the discrete integral conditions are established on the critical line; away from it the lattice model is massive, and no perturbed-CFT interpretation of the conditions - of the kind carried out for the chiral Potts model - is known, the obstacle near η=0\eta=0 being the marginal (Kosterlitz-Thouless) nature of the perturbation. Second, the currents satisfy only one discrete half of the Cauchy-Riemann relations: they are half-holomorphic parafermions of the type introduced by Riva and Cardy, not discretely holomorphic, so they do not by themselves support rigorous scaling-limit proofs. Fendley (2021) later generalized the current construction using braided tensor categories (a topological structure from knot invariants and anyon theory) and sharpened the second obstruction: the conservation condition supplies only half as many constraints as there are degrees of freedom on the square lattice, and in the three-state Potts chain a lattice parafermion satisfying the conservation condition is known to scale to a mixture of holomorphic and anti-holomorphic operators. Neither question has been resolved in the subsequent literature: an operator-formalism extension (Rigas 2023) produces discretely holomorphic parafermions in several further Ising-descended models (two-color Ashkin-Teller, loop O(1)O(1), staggered and odd eight-vertex, and abelian sandpile models), but by an operator formalism distinct from the quantum-group current approach, and the height function of the full-plane six-vertex model with −1≤Δ≤−1/2-1\le\Delta\le-1/2 has been proved to converge to a scaled Gaussian free field by transfer-matrix spectral methods (Duminil-Copin, Kozlowski, Lammers and Manolescu 2026), a rigorous scaling-limit result obtained without discretely-holomorphic observables and without treating the off-critical current relations. Whether the framework can nevertheless be extended to produce fully discretely-holomorphic operators in some model, and what the discrete integral conditions mean off criticality, are the two questions below.

Problem Statement

The problem consists of the two linked open questions posed at the end of Ikhlef and Weston (2017) about the quasi-local conserved currents and parafermionic operators constructed there from the quantum-group (Bernard-Felder) approach in the six-vertex model with anisotropy η\eta and in its vertex-face image, the trigonometric SOS model. The common objective is to convert the paper's heuristic, critical-line identifications of these lattice operators with conformal-field-theory fields into controlled statements: massive and off-critical for the first question, rigorous and critical for the second.

  1. Off-critical (perturbed-CFT) reinterpretation of the discrete integral conditions. The six-vertex and SOS parafermions obey discrete-integral (current-conservation) conditions around lattice plaquettes embedded in the complex plane, valid on the critical line η=iθ\eta=i\theta, 0<θ<π0<\theta<\pi (equivalently Δ=cos⁡θ∈(−1,1)\Delta=\cos\theta\in(-1,1)), where the model is massless and described by the c=1c=1 compactified free boson. Away from this line the lattice model is massive. Give a lattice-level reinterpretation of the discrete integral conditions via perturbed conformal field theory: determine how the quasi-local currents and their discrete integral relations deform under perturbations of the weights off the critical line, and identify the perturbing CFT operator or operators that control these deformations. The model to follow is the chiral Potts analysis of Ikhlef and Weston (2015), where such an analysis identified the perturbing field and agreed with and extended a prediction of Cardy. For the six-vertex model the analysis must in particular handle the neighbourhood of η=0\eta=0 (equivalently Δ=1\Delta=1), where the perturbation associated with η\eta is marginal - the Kosterlitz-Thouless point of the c=1c=1 Gaussian theory - so that a first-order perturbed-CFT treatment is insufficient.

  2. Full discrete holomorphicity. Determine whether the quantum-group current construction, or a suitable extension of it, can be adapted or supplemented so as to produce, in at least some model of this class (six-vertex, trigonometric or cyclic SOS, or a closely related integrable lattice model), fully discretely-holomorphic lattice operators: parafermionic operators satisfying discrete versions of both Cauchy-Riemann equations, ∂zˉJ−∂zJˉ=0\partial_{\bar z}J-\partial_z\bar J=0 and ∂zˉJ+∂zJˉ=0\partial_{\bar z}J+\partial_z\bar J=0, so that ∂zˉJ=0\partial_{\bar z}J=0 holds in the discrete sense, rather than only the current-conservation half. The question admits a definitive answer in either direction: an explicit construction in a genuinely interacting (non-free-fermion) model of the class, with a proof of both discrete relations for the stated weights and lattice, or a principled no-go result - an argument covering the natural extensions of the framework (modified tails or twists, lattice embeddings, alternative quantum-group or braided-category data) showing that no such operators can be produced. The stakes are explicit in the source: fully discretely-holomorphic operators are the ingredient that underpinned the rigorous scaling-limit conformal-invariance proofs for dimers, Ising, and percolation, and none is known for any genuinely interacting model of this class.

A complete resolution of the stated problem addresses both questions, each either affirmatively or negatively. The off-critical question (1) is not answered merely by asserting the marginal subtlety near η=0\eta=0, and the holomorphicity question (2) is not answered merely by re-deriving the already known conservation half.

References

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