Two-dimensional critical loop models such as the dense and dilute O(n) models are expected to be exactly solvable through a conformal field theory whose primary fields V(r,s)V_{(r,s)} insert 2r2r open loop segments ("legs") of momentum ss, or modify the weight of enclosing loops when r=0r=0. Jacobsen, Nivesvivat, Ribault and Roux (2025) conjectured a closed-form expression — a product over the eight sign choices of inverse Barnes double-gamma functions — for the three-point structure constants, and verified it by transfer-matrix numerics for 43 label triples; the 2026 joint synthesis of Ang, Cai, Jacobsen, Nivesvivat, Roux, Sun and Wu proposed the same formula as exact for all allowed labels, with bootstrap and lattice evidence plus rigorous probabilistic derivations that however remain restricted to three diagonal fields, three spinless two-leg fields, and Potts spin clusters (the last established under the scaling-limit conjecture for Potts spin clusters). The open problem is to establish the formula rigorously — prove or disprove it — for the full class of allowed labels (rir_i half-integers with risir_i s_i integer and r1+r2+r3r_1+r_2+r_3 integer), for example by conformal-loop-ensemble and Liouville-quantum-gravity methods, by Bethe-ansatz computation of the excited-state form factors governing the lattice scaling limit, or by algebraic constructions in the periodic Temperley-Lieb algebra, including a controlled treatment of the indecomposable s=1s=1 lattice sectors.

Background

Loop models and their conformal field theory. A critical two-dimensional loop model, such as the O(n) model on the square or hexagonal lattice, is a statistical ensemble of non-intersecting closed loops in which every loop carries a fugacity nn (each loop configuration is weighted by nn raised to the number of loops). At criticality the continuum limit is a conformal field theory (CFT — a field theory invariant under local conformal maps) with central charge c=13−6β2−6β−2c=13-6\beta^2-6\beta^{-2} (the additive anomaly parameter labelling the universality class) and loop fugacity n=−2cos⁡(πβ2)n=-2\cos(\pi\beta^2); the dilute phase corresponds to β2∈(1, 3/2]\beta^2\in(1,\,3/2] and the dense phase to β2∈[1/2, 1)\beta^2\in[1/2,\,1), together covering 0≤n≤20\le n\le 2.

The primary fields are labelled by Kac labels (r,s)(r,s) with r∈12Nr\in\frac{1}{2}\mathbb{N} and s∈Cs\in\mathbb{C} (the standard parametrization of Virasoro representation labels), with conformal spin rs∈Zrs\in\mathbb{Z}. A field V(r,s)V_{(r,s)} with r>0r>0 inserts 2r2r open loop segments — legs — emanating from the insertion point, each leg acquiring a phase eiπse^{i\pi s} when it winds around that point; for r=0r=0 the field is called diagonal and changes the weight of loops enclosing it from nn to w=2cos⁡(πs)w=2\cos(\pi s). The left and right conformal dimensions are Δ(r,s)=((rβ−sβ−1)2−(β−β−1)2)/4\Delta_{(r,s)}=\left((r\beta-s\beta^{-1})^2-(\beta-\beta^{-1})^2\right)/4 and Δ(r,−s)=((rβ+sβ−1)2−(β−β−1)2)/4\Delta_{(r,-s)}=\left((r\beta+s\beta^{-1})^2-(\beta-\beta^{-1})^2\right)/4.

Three-point functions. Global conformal invariance fixes the coordinate dependence of a three-point function of primary fields up to a single number, the three-point structure constant C123C_{123}. In ordinary CFTs the operator product expansion (OPE — the axiom that a product of two local fields expands into a sum over fields with computable coefficients) reduces all correlation functions to these constants. For loop models this reduction fails: loops are non-local objects, so the existence of OPEs does not follow from local conformal symmetry, and the spectrum at generic nn involves indecomposable (non-diagonalizable) representations of the Virasoro algebra (the infinite-dimensional algebra of local conformal symmetries in two dimensions). Determining all three-point structure constants is nevertheless the central missing step of the program "critical loop models are exactly solvable".

History. In the diagonal sector, Delfino and Viti (2010) predicted the three-point connectivity of critical percolation clusters in terms of the imaginary DOZZ formula — the analytic continuation to imaginary charges of the DOZZ formula giving the exact three-point structure constants of Liouville field theory (itself proved rigorously by Kupiainen, Rhodes and Vargas in 2020) — and Ikhlef, Jacobsen and Saleur (2016) treated the analogous nesting-loop statistics. The bootstrap program for four-point functions of loop models (Grans-Samuelsson, Jacobsen, Nivesvivat, Ribault and Saleur 2023; Nivesvivat, Ribault and Jacobsen 2024) produced an analytic ansatz — a reference three-point structure constant built from the Barnes double gamma function Γβ\Gamma_\beta, the meromorphic function defined by the shift relations Γβ(x+β)/Γβ(x)=2π ββx−1/2/Γ(βx)\Gamma_\beta(x+\beta)/\Gamma_\beta(x)=\sqrt{2\pi}\,\beta^{\beta x-1/2}/\Gamma(\beta x) and its β↔β−1\beta\leftrightarrow\beta^{-1} counterpart.

The conjecture. Jacobsen, Nivesvivat, Ribault and Roux (2025) conjectured that this ansatz is the exact three-point structure constant for all fields,

C(r1,s1)(r2,s2)(r3,s3)  =  ∏ϵ1,ϵ2,ϵ3=±Γβ−1 ⁣(β+β−12+β2∣∑i=13ϵiri∣+β−12∑i=13ϵisi),C_{(r_1,s_1)(r_2,s_2)(r_3,s_3)} \;=\; \prod_{\epsilon_1,\epsilon_2,\epsilon_3=\pm} \Gamma_\beta^{-1}\!\left(\frac{\beta+\beta^{-1}}{2}+\frac{\beta}{2}\left|\sum_{i=1}^{3}\epsilon_i r_i\right|+\frac{\beta^{-1}}{2}\sum_{i=1}^{3}\epsilon_i s_i\right),

together with the field-normalization-invariant ratio

ω123  =  C123C000C011C022C033,\omega_{123} \;=\; C_{123}\sqrt{\frac{C_{000}}{C_{011}C_{022}C_{033}}},

where the subscript 0 denotes the degenerate identity field V(0,1−β2)V_{(0,1-\beta^2)}, C0iiC_{0ii} are two-point structure constants given explicitly by the same Barnes double-gamma normalization, C(0,1−β2)(r,s)(r,s)=∏±Γβ−2(β±1)∏±,±Γβ−1(β±1+rβ±sβ−1)C_{(0,1-\beta^2)(r,s)(r,s)}=\prod_\pm\Gamma_\beta^{-2}(\beta^{\pm1})\prod_{\pm,\pm}\Gamma_\beta^{-1}(\beta^{\pm1}+r\beta\pm s\beta^{-1}) (so that C011,C022,C033C_{011},C_{022},C_{033} are this expression at (r,s)=(ri,si)(r,s)=(r_i,s_i)), and C000C_{000} is the partition function, i.e. the three-point constant of three identity fields V(0,1−β2)V_{(0,1-\beta^2)}. They verified ω\omega by transfer-matrix computations (building the cylindrical lattice column by column and diagonalizing the resulting operator, an action of the Temperley-Lieb algebra — the diagram algebra generated by local arc-reconnection moves) for 43 label triples, successfully in almost all cases; the discrepancies are confined to the s=1s=1 sectors, where the relevant modules of the Jones-Temperley-Lieb algebra (the unoriented variant of the Temperley-Lieb diagram algebra) are indecomposable and the transfer matrix has parity-degenerate ground states, so the finite-size limit is uncontrolled. On the probability side, the continuum limit of these models is the conformal loop ensemble CLEκ_\kappa (a conformally invariant random countable collection of loops, each locally a Schramm-Loewner evolution curve — the canonical conformally invariant random curve — with κ=4/β2\kappa=4/\beta^2), whose coupling to Liouville quantum gravity (random surfaces built from the Gaussian free field) has yielded rigorous three-point computations: Ang, Cai, Sun and Wu proved the diagonal case and the spinless two-leg case ω(1,0)(1,0)(1,0)\omega_{(1,0)(1,0)(1,0)} ("three points on the same loop"), and Cai, Liu, Wu and Zhuang (2025) proved the Potts spin-cluster three-point connectivity constant, as a conformal-loop-ensemble observable that coincides with the lattice spin-cluster probability under the scaling-limit conjecture for Potts spin clusters. The 2026 joint letter of Ang, Cai, Jacobsen, Nivesvivat, Roux, Sun and Wu proposed the same formula as exact for all allowed labels, with evidence from the transfer matrix, the bootstrap, and these special probabilistic derivations. What does not exist, for legged fields with nonzero momentum ss and for mixed legged/diagonal triples ("enclosures"), is any rigorous derivation: the lattice route is still obstructed at s=1s=1, and the probabilistic route is, by its authors' own description, case-by-case.

Problem Statement

Prove or disprove the following conjecture. For critical loop models at generic loop fugacity (equivalently generic β2\beta^2, in particular in the dense and dilute phases β2∈[1/2, 3/2]\beta^2\in[1/2,\,3/2]), the normalization-invariant three-point structure constants of the primary fields V(r,s)V_{(r,s)} are exactly ω123=C123C000/(C011C022C033)\omega_{123}=C_{123}\sqrt{C_{000}/(C_{011}C_{022}C_{033})} with C123C_{123} given by the product of inverse Barnes double-gamma functions displayed in the Background (equations (1.7) and (1.9) of Jacobsen et al. 2025; equations (3) and (6) of Ang et al. 2026), for every triple of allowed labels — ri∈12Nr_i\in\frac{1}{2}\mathbb{N}, si∈Cs_i\in\mathbb{C} with risi∈Zr_i s_i\in\mathbb{Z} and r1+r2+r3∈Zr_1+r_2+r_3\in\mathbb{Z} — where the three-point function is that of the continuum limit of the critical loop model, defined as the scaling limit of the lattice loop-model three-point functions and equivalently described by the corresponding conformal-loop-ensemble observable.

The statement must be established beyond the cases that are already rigorous: three diagonal fields (ri=0r_i=0), three spinless two-leg fields (V(1,0)V_{(1,0)} in all three slots), and the Potts spin-cluster three-point connectivity (established, as a conformal-loop-ensemble observable, under the scaling-limit conjecture for Potts spin clusters). In particular, a complete answer must cover legged fields with nonzero momentum (r>0r>0, s≠0s\neq 0) and mixed legged/diagonal triples (enclosures). Any sound route completes the problem, for instance: (a) a probabilistic proof via the coupling of CLEκ_\kappa with Liouville quantum gravity, generalizing the gluing constructions used for the solved cases; (b) an exact derivation from the lattice by quantum integrability — computing the excited-state form-factor ratio and overlap factors of the finite-size expression C123(L)=⟨V3∣O2∣V1⟩/⟨V0∣O2∣V2⟩C_{123}(L)=\langle V_3|\mathcal{O}_2|V_1\rangle/\langle V_0|\mathcal{O}_2|V_2\rangle by Bethe-ansatz techniques (the exact-solution method for integrable lattice models) for the 19-vertex (Izergin-Korepin) chain, the integrable vertex model associated with the square-lattice O(n) loop model, together with a controlled prescription for the s=1s=1 indecomposable sectors that disentangles the parity-degenerate leading eigenstates so that the lattice limit exists; or (c) an algebraic construction relating the lattice state-operator morphism O(r,s)\mathcal{O}_{(r,s)} (link patterns with labelled defects) to the fusion product of periodic Temperley-Lieb modules. A disproof must exhibit a specific allowed label triple together with a rigorous argument, or a controlled computation with explicit error bounds, showing that the continuum three-point constant differs from the formula; a finite-size discrepancy without convergence control does not suffice.

References

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  3. Morris Ang, Gefei Cai, Xin Sun, Baojun Wu, "Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond", Journal of the American Mathematical Society (2026), doi:10.1090/jams/1086, arXiv:2107.01788, https://arxiv.org/abs/2107.01788
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