For the critical two-dimensional Ising model there is now an exact bosonization theorem: in any finitely connected planar domain with wired, free, or plus/minus boundary arcs, the square of a scaling-limit correlation of the primary fields (spin, disorder, energy, fermions), subject to two parity conditions, equals the correlation of an explicit Wick-ordered vertex-operator expression in a compactified Gaussian free field (Bayraktaroglu, Izyurov, Virtanen, and Webb, 2023/2025). The theorem is currently restricted to planar domains, and its proof couples the model to general boundary conditions through a Fourier–Walsh expansion that the authors do not expect to admit a closed form. This problem asks to complete the correspondence in its natural geometric and perturbative generality: (i) extend the bosonization identity to critical Ising correlations on compact Riemann surfaces of genus at least two, where no lattice-to-continuum result of this type is known and named obstructions stand in the way (discretizations with conical singularities, failure of the degeneration and non-vanishing lemmas used in lower genus); and (ii) extend it to the off-critical (massive) model in the same boundary-condition generality, where only simply connected domains with plus boundary conditions have been treated, via the sine-Gordon model at the free fermion point. Genus one was resolved in 2025, confirming long-standing physics predictions, which sharpens the contrast with the still-open higher-genus case.

Background

The Ising model is a lattice model of ±1±1 spins with ferromagnetic nearest-neighbor interactions; in two dimensions it is exactly solvable and has a critical point at which its large-scale fluctuations are described by a conformal field theory (CFT) — a quantum field theory invariant under conformal maps — with central charge c=1/2c=1/2 (a numerical parameter of a CFT counting its effective number of degrees of freedom; c=1/2c=1/2 is the value for a single real free fermion). The primary fields of this CFT are the spin field σσ (the local magnetization), the disorder field μμ (dual to σσ, measuring frustration of boundary conditions), the energy density εε, and the two chiral Majorana fermions ψ,ψ∗ψ,ψ^* (the holomorphic and anti-holomorphic fermion fields; a Majorana fermion is a fermion that is its own antiparticle). Chelkak, Hongler, and Izyurov proved that all multi-point correlations of these fields have universal scaling limits in planar domains with wired (fixed-spin), free, plus, or minus boundary conditions on designated boundary arcs.

Bosonization is the identification of such fermionic fields with vertex operators — exponentials of a free bosonic field. The bosonic object here is the compactified Gaussian free field (GFF): a massless Gaussian field ΦΦ on the domain ΩΩ, taken modulo √2⋅π⋅Z√2·π·ℤ so that it is circle-valued, with boundary data given by a harmonic function ξξ that is constant on each boundary arc and jumps at the finitely many transition points between arcs; the associated probability measure is proportional to exp⁡(−⟨∇ξ,∇ξ⟩Ω,reg/4π)\exp(−⟨∇ξ,∇ξ⟩_{Ω,\mathrm{reg}}/4π), where the Dirichlet energy is regularized because it diverges logarithmically at the jump points. Correlations are defined by Wick ordering (denoted :⋅::·:), i.e. by subtracting the singular self-contractions encoded in the Green's function of ΩΩ and in harmonic measures.

The source theorem states that whenever the number of insertions from {σ,ψ,ψ∗}\{σ,ψ,ψ^*\} and the number from {μ,ψ,ψ∗}\{μ,ψ,ψ^*\} are both even, the square of the Ising correlation equals a compactified-GFF correlation, ⟨Oz1…OzN⟩Ω2=⟨O^z1…O^zN⟩Ω,⟨O_{z_1}…O_{z_N}⟩^2_Ω=⟨Ô_{z_1}…Ô_{z_N}⟩_Ω, with dictionary σzi↦√2:cos⁡(√(1/2)Φ(zi)):σ_{z_i}↦√2:\cos(√(1/2)Φ(z_i)):, μzi↦√2:sin⁡(√(1/2)Φ(zi)):μ_{z_i}↦√2:\sin(√(1/2)Φ(z_i)):, εzi↦−(1/2):∣∇Φ(zi)∣2:ε_{z_i}↦−(1/2):|∇Φ(z_i)|^2:, and the fermions mapping to constant multiples of the derivatives of ΦΦ. The right-hand side is a concrete formula in the Green's function and harmonic measures of ΩΩ. If a parity condition fails, the Ising correlation vanishes identically while the bosonic side need not; it then corresponds to Ising correlations with different boundary conditions.

The proof is analytic-geometric. Passing to the Schottky double of ΩΩ (the compact Riemann surface obtained by gluing ΩΩ to a mirror copy along the boundary), fermion correlations become Pfaffians of the Szegő kernel (the projection kernel onto holomorphic sections of a spin bundle over the double; a spin structure — a choice of such a bundle, equivalently a square root of the line bundle of holomorphic one-forms — fixes the signs with which fermion fields traverse the surface's cycles), and the identity for squared correlations follows from the Hejhal–Fay identity, which expresses products of Szegő kernels through Abelian differentials (holomorphic one-forms) and Riemann theta functions (special functions encoding the period matrix of the surface). The general multiply connected case is obtained by degeneration: n+kn+k small handles are attached to the double, one per interior insertion and one per free arc, and then pinched. The coupling to general boundary conditions enters through Fourier–Walsh coefficients — the coefficients of the boundary-condition indicator functions in the Walsh basis — and it is this step the authors do not expect to become more concrete. A parallel combinatorial route identifies the squared Ising correlation with the correlation of two independent Ising copies, i.e. with a dimer model (a random perfect matching of a bipartite graph), whose height function (an integer-valued function on the faces encoding the local geometry of the matching) converges to the compactified GFF (Dubédat; Duminil-Copin–Lis; Basok), although the known dimer results do not fully cover the boundary conditions of the analytic theorem.

This rigorous, lattice-to-continuum bosonization is distinct from the classical formal bosonization of the c=1/2c=1/2 CFT on Riemann surfaces in the physics literature of the 1980s–90s, which postulates operator identifications but proves nothing about scaling limits of lattice correlations.

Problem Statement

Complete the rigorous bosonization of the critical Ising model — the identity between squared Ising correlations and compactified Gaussian free field correlations established for multiply connected planar domains — in its natural geometric and perturbative generality, by resolving the following interlocked extensions left open by the source theorem.

(1) Higher genus. Prove, or disprove, an analogous bosonization identity for scaling-limit correlations of the critical Ising model on compact Riemann surfaces of genus g≥2g≥2, for arbitrary spin structures. A positive resolution must specify the lattice discretization (which necessarily carries conical singularities by the Gauss–Bonnet theorem), fix spin-structure conventions, state the identity including any parity or topological conditions, and identify its right-hand side explicitly — for instance in terms of theta functions, the Szegő kernel, or the compactified GFF on the surface; it should rigorously justify the formal conformal-field-theory predictions for such correlations.

(2) Off-critical regime. Extend the identity to the near-critical (massive) Ising model in the boundary-condition and connectivity generality of the source theorem — finitely connected domains with wired, free, and plus/minus boundary arcs — identifying the right-hand side with the massive field theory: the sine-Gordon model (a scalar field with a cosine self-interaction) at the free fermion point, i.e. the coupling β=4πβ=4π at which it is equivalent to a free fermion, or an equivalent construction. The currently known case covers only simply connected domains with plus boundary conditions.

A rigorous resolution of either component constitutes an answer; since component (1) is posed as a prove-or-disprove task, a rigorous disproof for component (1) equally does. Partial results that substantially extend the known torus (genus one) or near-critical simply connected cases — for example, higher-spin or disorder fields on the torus, or the massive regime in multiply connected domains — are within the scope of this problem as partial progress, but they do not by themselves constitute a resolution of either component; purely formal conformal-field-theory identifications without lattice-to-continuum convergence content are out of scope.

References

  1. Baran Bayraktaroglu, Konstantin Izyurov, Tuomas Virtanen, Christian Webb. Bosonization of Primary Fields for the Critical Ising Model on Multiply Connected Planar Domains. Communications in Mathematical Physics 406, 222 (2025). DOI: 10.1007/s00220-025-05392-5, https://doi.org/10.1007/s00220-025-05392-5. Preprint: arXiv:2312.02960, https://arxiv.org/abs/2312.02960
  2. Baran Bayraktaroglu, Konstantin Izyurov. Critical Ising correlations on a torus. Preprint: arXiv:2506.11324, https://arxiv.org/abs/2506.11324
  3. S. C. Park, Tuomas Virtanen, Christian Webb. Near-critical Ising, sine-Gordon at the free fermion point, and bosonization. Preprint: arXiv:2512.11304, https://arxiv.org/abs/2512.11304
  4. Mikhail Basok. Dimers on Riemann surfaces and compactified free field. Preprint: arXiv:2309.14522, https://arxiv.org/abs/2309.14522
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  7. Hugo Duminil-Copin, Marcin Lis. On the double random current nesting field. Probability Theory and Related Fields 175, 937–955 (2019). DOI: 10.1007/s00440-019-00899-0, https://doi.org/10.1007/s00440-019-00899-0. Preprint: arXiv:1712.02305, https://arxiv.org/abs/1712.02305
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  9. Alex Karrila, Tuomas Virtanen, Christian Webb. Operator Product Expansions of Derivative Fields in the Sine-Gordon Model. Journal of Statistical Physics 193(3), 38 (2026). DOI: 10.1007/s10955-026-03585-3, https://doi.org/10.1007/s10955-026-03585-3. Preprint: arXiv:2503.21326, https://arxiv.org/abs/2503.21326
  10. Roland Bauerschmidt, Scott Mason, Christian Webb. Twisted Dirac operators and fractional correlations of the massless sine-Gordon model at the free fermion point. Preprint: arXiv:2508.14806, https://arxiv.org/abs/2508.14806
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