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International Conference on High Energy Physics (ICHEP)

A large family of IRF solvable lattice models based on WZW models

Juan Ramos, Vladimir Belavin, Doron Gepner

Published Nov 8, 2022• 2 citations
Publisher

Abstract

We consider the lattice models based on affine algebras described by Jimbo et al., for the algebras 𝐴𝐵𝐶𝐷 and by Kuniba et al. for 𝐺2. We find a general formula for the crossing multipliers of these models, with the result that these crossing multipliers are given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in a large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We also elaborate on some details of the computation of the Local state probability of these models, conjecturing an expression for it.

DOI
10.22323/1.414.0432

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We consider the lattice models based on affine algebras described by Jimbo et al., for the algebras 𝐴𝐵𝐶𝐷 and by Kuniba et al. for 𝐺2. We find a general formula for the crossing multipliers of these models, with the result that these crossing multipliers are given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in a large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We also elaborate on some details of the computation of the Local state probability of these models, conjecturing an expression for it.
Abstract We study the large class of solvable lattice models, based on the data of conformal field theory. These models are constructed from any conformal field theory. We consider the lattice models based on affine algebras described by Jimbo et al, for the algebras ABCD and by Kuniba et al for G 2. We find a general formula for the crossing multipliers of these models. It is shown that these crossing multipliers are also given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in this large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We use this result to study the local state probabilities of these models and show that they are given by the branching rule, in regime III.
Local State Probabilities of Solvable Lattice Models: Relatives of An(1)A_{n}^{(1)}An(1)​ Family
We present the results for the local state probabilities (LSP) of the solvable lattice models, constructed around rational conformal field theory given by WZW model on SO(3)4R=SU(2)4R/Z2SO(3)_{4 R}=SU(2)_{4 R} / Z_{2}SO(3)4R​=SU(2)4R​/Z2​ together with primary field ϕ1ϕ_{1}ϕ1​(symmetric tensor of degree 2). Some conjectures for the LSP for some higher rank relatives of An(1)A_{n}^{(1)}An(1)​ face models are also presented.
We conjecture the inversion relations for thermalized solvable interaction round the face (IRF) two dimensional lattice models. We base ourselves on an ansatz for the Baxterization described in the 90's. We solve these inversion relations in the four main regimes of the models, to give the free energy of the models, in these regimes. We use the method of Baxter in the calculation of the free energy of the hard hexagon model. We believe these results to be quite general, shared by most of the known IRF models. Our results apply equally well to solvable vertex models. Using the expression for the free energy we calculate the critical exponent α, and from it the dimension of the perturbing (thermal) operator in the fixed point conformal field theory (CFT). We show that it matches either the coset O/G or G/O, where O is the original CFT used to define the model and G is some unknown CFT, depending on the regime. This agrees with known examples of such models by Huse and Jimbo et al.
Birman–Murakami–Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the SU(2) 2×2 fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.
A bstract This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the su(3)k \mathfrak{su}{(3)}_k su(3)k​ su 3 k affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum R matrix of the Uq(sl(3)) {U}_q\left(\mathfrak{sl}(3)\right) Uq​(sl(3)) U q sl 3 quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of R matrix elements.
This paper represents a continuation of our previous work, where the Bolzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically the unrestricted face model, based on the su(3)k\mathfrak{su}(3)_ksu(3)k​ affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum RRR matrix of the Uq(sl(3))U_q(\mathfrak{sl} (3))Uq​(sl(3)) quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of RRR matrix elements.