Coulomb-gas identification of the critical $U_{\mathfrak{q}}(\mathfrak{sl}_3)$ web models: exact central charges, electromagnetic conformal weights, and the fractal dimension of critical webs

The $U_q(\mathfrak{sl}_3)$ web model (the $n=3$ Kuperberg $A_2$ spider model) on the hexagonal lattice generalises the $O(N)$ loop model to trivalent webs with a quantum-group symmetry. Numerical diagonalisation of its transfer matrix shows that, for each sampled $q=e^{i\gamma}$ on the unit circle, the model has a dense and a dilute critical phase, and its electromagnetic (defect) partition functions $Z_K^{\bm e,\bm m}$ are finite-size precursors of pairs of continuum electromagnetic operators. The exact identification of the continuum limit — a Coulomb-gas derivation of the central charges of both phases and closed-form expressions for the electromagnetic conformal weights $h_{\bm e,\bm m}$, $\bar h_{\bm e,\bm m}$ — was explicitly deferred by the founding paper to a companion publication that has still not appeared, and with it the confirmation of the conjectured fractal dimension $D_f=2-(h_{\bm e_0,\bm\rho}+\bar h_{\bm e_0,\bm\rho})$ of one-connected-component critical webs at $q=e^{i\pi/3}$. The problem is to establish this Coulomb-gas/conformal-field-theory identification, including its sector content and its geometrical corollaries. Follow-up work has produced integrable manifolds for the rank-two web models, an exact central-charge formula quoted from unpublished work, and a rigorous construction of compactified imaginary Toda theory expected to describe the scaling limit — but the published lattice-to-continuum identification, the electromagnetic conformal weights, and the fractal-dimension confirmation remain open.

Background

Two-dimensional geometrical lattice models are a meeting point of statistical mechanics, integrability and conformal field theory (CFT). The prototype is the $O(N)$ loop model on the hexagonal lattice: configurations are families of closed, non-crossing curves, and every closed loop receives a fugacity $N$. For $-2\le N\le 2$ one writes $N=q+q^{-1}$ with $|q|=1$, and the model is critical in two phases: a dense phase, in which loops fill the lattice, and a dilute phase, attained at a special critical point. Both continuum limits are described by the Coulomb gas — the theory of a free compactified boson with a background electric charge, reached on the lattice by mapping configurations to a locally conserved, integer-valued height variable whose coarse-grained fluctuations become the boson. In that description the coupling fixes the central charge $c$ (the number characterising the CFT; it controls the finite-size scaling $f_L=f_\infty-\pi c_{\rm eff}/(6L^2)$ of the free energy per site on a cylinder of circumference $L$), while electric charges (implemented on the lattice by modified seam, or twist, operators — the Boltzmann weights along a distinguished column of the cylinder, the seam, are altered so as to insert a charge) and magnetic charges (implemented by defect lines running along the cylinder) label the scaling operators, whose conformal weights $(h,\bar h)$ govern the finite-size scaling of the corresponding modified partition functions. Magnetic defects also have a direct geometric meaning: their two-point functions are connectivity (indicator) functions, and in special limits they yield the fractal dimension of the critical curves — e.g. at $N=0$ the dilute model describes self-avoiding walks, i.e. non-intersecting random paths ($c=0$), and the dense phase gives dense polymers ($c=-2$).

The algebraic origin of this structure is the Temperley–Lieb algebra — the diagram algebra of non-crossing pairings generated by the local lattice moves — which is in Schur–Weyl duality with the quantum group $U_q(\mathfrak{sl}_2)$, a $q$-deformation of the universal enveloping algebra of $\mathfrak{sl}_2$ (Schur–Weyl duality means that the diagram algebra and the quantum group act on the same local configuration spaces and their representation theories determine each other); the local Boltzmann weights are intertwiners (maps commuting with the quantum-group action) of $U_{-q}(\mathfrak{sl}_2)$. The web models of Lafay, Gainutdinov and Jacobsen generalise this to higher rank: for $n=3$ the configurations are oriented trivalent webs built from the Kuperberg $A_2$ spider — the diagrammatic category of intertwiners of $U_q(\mathfrak{sl}_3)$ — in which every vertex is a $U_q(\mathfrak{sl}_3)$ intertwiner and each closed connected component is reduced, via Kuperberg's local relations (elimination of digons, squares and loops), to loops whose total weight is the quantum integer $[3]_q=q^{2}+1+q^{-2}$ (the $q$-analogue of the number 3; it vanishes at $q=e^{i\pi/3}$). The weight of a web is therefore a non-local object, and one achievement of the founding paper (Nucl. Phys. B 979 (2022) 115789) was a local reformulation as a vertex model with seven states per link of the hexagonal lattice, making the $U_{-q}(\mathfrak{sl}_3)$ symmetry of the local transfer matrices explicit and enabling a transfer-matrix treatment on the strip and on the cylinder.

On the strip, the row-to-row transfer matrix is a full $U_{-q}(\mathfrak{sl}_3)$ intertwiner. On the cylinder — the geometry relevant for extracting conformal data — it is not; it retains only invariance under the Cartan subalgebra, i.e. the two commuting diagonal generators $H_1,H_2$. Consequently the Hilbert space decomposes into sectors labelled by weights $\bm m=n_1\bm w_1+n_2\bm w_2$ of the full weight lattice $\mathcal R_3^*$ (a weight is the pair of simultaneous eigenvalues of $H_1,H_2$ on a state; $\bm w_1,\bm w_2$ are the fundamental weights, the generators of the lattice dual to the simple roots), not only by dominant weights, and two weights in the same Weyl orbit (related by reflections in the simple-root hyperplanes) can define inequivalent sectors, because a defect line of charge $\bm m$ picks up the phase $q^{2(\bm\rho,\bm m)}$ when winding around the cylinder ($\bm\rho$ is the Weyl vector, the half-sum of the positive roots). Electric charges $\bm e$ are introduced by modifying the seam operator. The resulting electromagnetic partition functions $Z_K^{\bm e,\bm m}$ are the finite-size precursors of pairs of electromagnetic operators. The paper solves the associated combinatorial problem — the classification of irreducible open webs on the strip and on the cylinder (any irreducible cylinder web with nonzero magnetic charge is a concatenation of elementary blocks) — and thereby fixes the sector structure of the transfer matrix.

Numerically (exact diagonalisation for circumferences $L=5$ and $6$; the zero-magnetisation sector already has dimension 5881 at $L=6$) the model has, for each sampled $q=e^{i\gamma}$, $\gamma\in[0,\pi]$, a dense and a dilute critical phase, in close analogy with the loop model. At $q=e^{i\pi/5}$ the effective central charges read $c_{\rm eff}\approx 6/5$ (dilute) and $\approx 4/5$ (dense) — values the authors announced to be exact; at $q=e^{i\pi/4}$ the critical point in the dilute phase can be identified with spin interfaces of the critical three-state Potts model; at $q=e^{i\pi/3}$, where $[3]_q=0$, one finds $c_{\rm eff}=0$ (dilute) and $c_{\rm eff}=-2$ (dense), the analogues of the self-avoiding-walk and dense-polymer points. The dense-phase central charges coincide with those of an integrable fifteen-vertex model (a solvable square-lattice vertex model with fifteen allowed local arrow configurations).

What the founding paper explicitly leaves open, deferring it to a companion publication, is the analytic continuum description: a Coulomb-gas derivation of the exact central charges as functions of $q$ — equivalently, a $W_3$-symmetric CFT, since two bosons with an $\mathfrak{sl}_3$-type charge lattice carry the extended symmetry algebra generated by a spin-3 current in addition to the stress tensor ($W_3$) — and closed-form expressions for the electromagnetic conformal weights $h_{\bm e,\bm m}$, $\bar h_{\bm e,\bm m}$. A stated corollary concerns the fractal dimension of critical webs: at $q=e^{i\pi/3}$, after renormalising the partition function so that only one-connected-component webs contribute (the $n=3$ analogue of the polymer limit), the paper conjectures that the two-point indicator function of two points belonging to the same connected web is governed by the weights $(h,\bar h)=(h_{\bm e_0,\bm\rho},\bar h_{\bm e_0,\bm\rho})$ with $\bm e_0=\bm\rho/3$, so that the fractal dimension is $D_f=2-(h_{\bm e_0,\bm\rho}+\bar h_{\bm e_0,\bm\rho})$.

Problem Statement

For the critical $U_q(\mathfrak{sl}_3)$ ($A_2$ Kuperberg) web model on the hexagonal lattice, with $q=e^{i\gamma}$ on the unit circle, establish the Coulomb-gas / conformal-field-theory identification of its dense and dilute critical phases. The single objective is the analytic continuum-limit identification of this lattice model; its joint deliverables are: (1) a derivation, from the lattice model (for example via a two-component height-variable mapping or an equivalent construction), of the exact central charges of the dense and dilute critical phases as functions of the lattice parameter — reproducing the numerically observed values, including $c_{\rm eff}=6/5$ (dilute) and $c_{\rm eff}=4/5$ (dense) at $q=e^{i\pi/5}$ and the $c_{\rm eff}=0/-2$ pair at $q=e^{i\pi/3}$; and (2) closed-form expressions for the electromagnetic conformal weights $h_{\bm e,\bm m}$, $\bar h_{\bm e,\bm m}$ that govern the scaling of the electromagnetic partition functions $Z_K^{\bm e,\bm m}/Z_K$, for magnetic charges $\bm m$ in the $\mathfrak{sl}_3$ weight lattice and electric charges $\bm e$, consistent with the lattice sector structure (Cartan-labelled sectors, winding phases $q^{2(\bm\rho,\bm m)}$) and with the finite-size spectra; and (3) as a direct corollary of (2), the confirmation or refutation of the conjectured fractal dimension of one-connected-component critical webs at $q=e^{i\pi/3}$ (where $[3]_q=0$ and the model renormalises to single-component webs), namely

$D_f=2-\left(h_{\bm e_0,\bm\rho}+\bar h_{\bm e_0,\bm\rho}\right),\qquad \bm e_0=\tfrac{1}{3}\bm\rho.$

The three deliverables are scientifically linked: (3) is the geometrical consequence of the operator content determined in (2), and both test the continuum theory whose central charge is fixed in (1).

All three deliverables are required outputs: a complete resolution reports the exact central charges (1), the closed-form electromagnetic weights (2), and the explicit value of $D_f$ at $q=e^{i\pi/3}$ that they yield — confirming or refuting the conjectured formula — the latter following from (2) by setting $\bm e_0=\bm\rho/3$; the central charges and the weights may be obtained from a single Coulomb-gas construction.

References

1. Augustin Lafay, Azat M. Gainutdinov, Jesper Lykke Jacobsen, "$U_{\mathfrak{q}}(\mathfrak{sl}_3)$ web models: Locality, phase diagram and geometrical defects", Nuclear Physics B 979 (2022) 115789, DOI: 10.1016/j.nuclphysb.2022.115789, arXiv:2107.10106, https://arxiv.org/abs/2107.10106
2. Augustin Lafay, Azat M. Gainutdinov, Jesper Lykke Jacobsen, "Integrability of rank-two web models", Nuclear Physics B 1002 (2024) 116530, DOI: 10.1016/j.nuclphysb.2024.116530, arXiv:2311.14805, https://arxiv.org/abs/2311.14805
3. Yi-An Yao, "Compactified imaginary Toda theory", arXiv:2605.25494 (2026), https://arxiv.org/abs/2605.25494
4. Augustin Lafay, "Geometrical lattice models, algebraic spiders and applications to random geometry", PhD thesis, Sorbonne Université, defended 12 September 2022, HAL: tel-03966966, https://theses.hal.science/tel-03966966
5. Augustin Lafay, Azat M. Gainutdinov, Jesper Lykke Jacobsen, "$U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces", Journal of Statistical Mechanics: Theory and Experiment 2021 (2021) 053104, DOI: 10.1088/1742-5468/abfcb3, arXiv:2101.00282, https://arxiv.org/abs/2101.00282
6. Greg Kuperberg, "Spiders for rank 2 Lie algebras", Communications in Mathematical Physics 180 (1996) 109–151, DOI: 10.1007/BF02101184, arXiv:q-alg/9712003, https://arxiv.org/abs/q-alg/9712003
7. Janko Böhm, Jesper Lykke Jacobsen, Yunfeng Jiang, Yang Zhang, "Geometric algebra and algebraic geometry of loop and Potts models", Journal of High Energy Physics 2022, article 68 (2022), DOI: 10.1007/jhep05(2022)068, arXiv:2202.02986, https://arxiv.org/abs/2202.02986
8. Augustin Lafay, Ian Le, Julien Roussillon, "Degenerate conformal blocks for the $W_3$ algebra at c=2 and connection probabilities in the triple dimer model", arXiv:2402.12013 (2024), https://arxiv.org/abs/2402.12013