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arXiv:1807.05603 · Nuclear Physics B

Three blocks solvable lattice models and Birman–Murakami–Wenzl algebra

Alexander A. Belavin, Doron Gepner

Published Nov 22, 2018• 4 citations
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Abstract

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.

DOI
10.1016/j.nuclphysb.2018.11.009

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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.
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.