Conjectured inversion relations for thermalized interaction-round-a-face lattice models built from rational conformal field theories

Belavin, Gepner and Ramos Cabezas (2022) conjectured two inversion relations for the 'thermalized' (elliptic, off-critical) interaction-round-a-face (IRF) lattice models constructed from the data of an arbitrary rational conformal field theory (RCFT), and used them — via Baxter's corner transfer matrix method — to compute local state probabilities in regime III; in the parent work of Gepner (2021) the same relations determine the free energies in all four regimes. The relations are established only for the WZW-based elliptic face models (the Jimbo–Miwa–Okado and Kuniba affine-algebra models, the Andrews–Baxter–Forrester SOS model, and hard hexagon); for a general RCFT the thermalized Boltzmann weights themselves are not constructed, so the inversion relations remain conjectures about a hypothetical elliptic solution of the Yang–Baxter equation. The problem is to prove both relations for the general class from the underlying RCFT data, to determine the precise conditions on the theory and the field under which they hold, or to refute them by an explicit counterexample. No proof, delimitation, or refutation of the conjecture is known.

Background

Two-dimensional lattice models are called solvable when their Boltzmann weights satisfy the Yang–Baxter equation (YBE), an algebraic identity in a complex spectral parameter $u$ that guarantees that the row-to-row transfer matrices commute for different values of $u$, making exact computations possible. A large class of solvable models is furnished by conformal field theory: in an interaction-round-a-face (IRF, or face) model the microscopic variables sit on the vertices of the lattice, each vertex variable takes values in the finite set of primary fields of a rational conformal field theory (RCFT), and the Boltzmann weight of a face vanishes unless the four surrounding fields obey the fusion rules of that RCFT. Here an RCFT is a two-dimensional conformal field theory with finitely many primary fields; each primary field $a$ carries a conformal dimension $\Delta_a$, fields combine according to integer fusion coefficients, and the exchange (braiding) of two identical primary fields inside a four-point function acts by a braiding matrix whose eigenvalues $\lambda_a=\epsilon_a\exp(\pi i(2\Delta_h-\Delta_a))$, with $\epsilon_a=\pm1$ a statistics sign, are fixed by the conformal dimensions alone (Moore–Seiberg).

Gepner proposed in 1992 that for any RCFT $\mathcal O$ and any primary field $h$ whose fusion products with itself and with its conjugate field $\bar h$ are multiplicity-free, the face model IRF$(\mathcal O,h,h)$ is solvable. Writing the fusion product as $h\cdot h=\sum_{r=0}^{n-1}\psi_r$ (the fields $\psi_r$ are the $n$ blocks of the braiding, and $\Delta_r$ denotes the conformal dimension of $\psi_r$), the R-matrix is obtained by Baxterization of the braiding matrix,

$R_i^{h,h}(u)=\sum_{a=0}^{n-1} f_a(u)P_i^a,\qquad f_a(u)=\frac{\prod_{j=1}^{a}\sin(\zeta_{j-1}-u)\prod_{j=a+1}^{n-1}\sin(\zeta_{j-1}+u)}{\prod_{j=1}^{n-1}\sin(\zeta_{j-1})},$

where $P_i^a$ are the projectors onto the eigenspaces of the braiding operator at site $i$ and the crossing parameters $\zeta_r=\frac{\pi}{2}(\Delta_{r+1}-\Delta_r)$ are built from the dimensions of the fields in the fusion product. For the Wess–Zumino–Witten (WZW) models — the RCFTs whose primary fields are the integrable highest-weight representations of an affine Lie algebra — this construction reproduces the classical exactly solved elliptic face models of Jimbo, Miwa and Okado (for the algebras $A,B,C,D$) and Kuniba (for $G_2$).

The trigonometric solution above is critical ($q=0$). Away from criticality one 'thermalizes' the model by replacing the sine by the odd Jacobi theta function $\theta_1(u,q)$, where $-1<q<1$ is the elliptic modulus; the resulting elliptic model has four standard regimes of the spectral parameter and modulus (regime III is $0<q<1$ and $0<u<l$, with the crossing parameter $l$ defined in the next sentence). Baxter's inversion relation method is the main tool for solving such models: if the R-matrices obey the two inversion relations stated below, the partition function per site $\kappa(u)$ obeys two functional equations, $\kappa(u)\kappa(-u)=\rho(u)\rho(-u)$ and $\kappa(l+u)\kappa(l-u)=\tilde\rho(u)\tilde\rho(-u)$, which determine the free energy in each regime; here $l=\pi\Delta_{\rm adjoint}/2$ is the crossing parameter and $\Delta_{\rm adjoint}$ is the conformal dimension of the first non-unit field in the ordered fusion product $h\cdot\bar h$ (for a WZW model this field is the adjoint representation, with $\Delta_{\rm adjoint}=g/(k+g)$ in terms of the dual Coxeter number $g$ and the level $k$).

The two inversion relations at issue are conjectures about the thermalized model. They read

$R_i^{h,h}(u)\,R_i^{h,h}(-u)=\rho(u)\rho(-u)\,\mathbf 1_i,\qquad \rho(u)=\prod_{r=0}^{n-2}\frac{\theta_1(\zeta_r-u,q)}{\theta_1(\zeta_r,q)},$

for the R-matrix braiding $h$ with $h$, and

$R_i^{h,\bar h}(u)\,R_i^{h,\bar h}(-u)=\tilde\rho(u)\tilde\rho(-u)\,\mathbf 1_i,\qquad \tilde\rho(u)=\prod_{r=0}^{n-2}\frac{\theta_1(\tilde\zeta_r-u,q)}{\theta_1(\zeta_r,q)},$

for the R-matrix braiding $h$ with $\bar h$, where the tilded quantities are built in the same way from the fusion product $h\cdot\bar h=\sum_{r=0}^{n-1}\tilde\psi_r$ (the untilded denominator in $\tilde\rho$ reflects a normalization convention for $R^{h,\bar h}$). For the trigonometric solution the first relation is an algebraic identity of the Baxterization ansatz — the product $f_a(u)f_a(-u)$ is independent of $a$ — and the second follows from the first together with the crossing relation that maps $R^{h,h}$ to $R^{h,\bar h}$. The elliptic case is genuinely open: for a general RCFT the thermalized Boltzmann weights are not known at all — Gepner states explicitly that 'We do not know how to thermalize the Boltzmann weights. So instead we will thermalize the inversion relations' — so the relations are conjectures about a hypothetical elliptic solution of the YBE that reduces to the trigonometric one as $q\to0$. They are verified whenever explicit elliptic weights exist: the WZW-based face models of Jimbo–Miwa–Okado and Kuniba, the Andrews–Baxter–Forrester solid-on-solid model, and hard hexagon (the case $\mathcal O=SU(2)_3$, where the free energy obtained from the relations agrees with Baxter's exact solution in all four regimes). Belavin, Gepner and Ramos Cabezas adopted these relations (their eqs. (3.2) and (3.10)) as the foundation for a computation of the local state probabilities by Baxter's corner transfer matrix method: the corner transfer matrix is the transfer matrix of one quadrant of the lattice, and its diagonalization reduces the probability of finding a given primary field at the lattice origin in regime III to a one-dimensional configuration sum, which is conjectured to be expressible through coset branching functions (the graded multiplicities in the character decomposition of a coset conformal field theory $G/\mathcal O$). The same program also conjectures that the crossing multipliers, the face-dependent prefactors in the crossing relation, are given by the characters of the underlying RCFT — proved so far only for WZW models — but the present problem concerns the inversion relations on which the corner-transfer-matrix analysis rests.

Problem Statement

Establish, delimit, or refute the two conjectured inversion relations for the thermalized IRF$(\mathcal O,h,h)$ lattice models built from an arbitrary rational conformal field theory. In the notation of the background, for every RCFT $\mathcal O$ and every primary field $h$ whose fusion products $h\cdot h=\sum_{r=0}^{n-1}\psi_r$ and $h\cdot\bar h=\sum_{r=0}^{n-1}\tilde\psi_r$ are multiplicity-free, with $\zeta_r=\frac{\pi}{2}(\Delta_{r+1}-\Delta_r)$, $\tilde\zeta_r=\frac{\pi}{2}(\tilde\Delta_{r+1}-\tilde\Delta_r)$ and elliptic modulus $q$, the conjecture of Belavin-Gepner-Ramos Cabezas (eqs. (3.2) and (3.10) there, following Gepner 2021) asserts that the thermalized R-matrices satisfy

$R_i^{h,h}(u)\,R_i^{h,h}(-u)=\rho(u)\rho(-u)\,\mathbf 1_i,\qquad \rho(u)=\prod_{r=0}^{n-2}\frac{\theta_1(\zeta_r-u,q)}{\theta_1(\zeta_r,q)},$

and

$R_i^{h,\bar h}(u)\,R_i^{h,\bar h}(-u)=\tilde\rho(u)\tilde\rho(-u)\,\mathbf 1_i,\qquad \tilde\rho(u)=\prod_{r=0}^{n-2}\frac{\theta_1(\tilde\zeta_r-u,q)}{\theta_1(\zeta_r,q)},$

up to the normalization convention of $R^{h,\bar h}$, where $R^{h,h}$ and $R^{h,\bar h}$ are the elliptic (thermalized) solutions of the Yang-Baxter equations that reduce to the trigonometric Baxterizations of the braiding matrices of $h$ with $h$ and with $\bar h$ in the limit $q\to0$. A complete answer must do one of the following:

1. Prove both relations rigorously for the general class from the RCFT data (fusion rules, conformal dimensions, braiding eigenvalues) — which necessarily includes establishing the existence of the thermalized solution itself, since for a general RCFT the elliptic Boltzmann weights are currently not constructed; or
2. Determine the precise conditions on the RCFT $\mathcal O$ and the field $h$ under which the relations hold, thereby delimiting exactly the class of models for which the corner-transfer-matrix free-energy and local-state-probability computations are valid; or
3. Refute the conjecture by exhibiting an explicit model — an RCFT and a field $h$, together with genuine thermalized weights solving the Yang-Baxter equation and reducing to the RCFT braiding at $q\to0$ — for which one of the two relations fails.

The two relations are the joint input to Baxter's corner-transfer-matrix computation of the local state probabilities in regime III and to the free-energy functional equations in all four regimes, so the common objective of all three resolution modes is to place the RCFT-based thermalized-IRF program on a rigorous footing.

References

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