New R-matrices from representations of braid-monoid algebras based on superalgebras

In this paper we discuss representations of the Birman–Wenzl–Murakami algebra as well as of its dilute extension containing several free parameters. These representations are based on superalgebras and their baxterizations permit us to derive novel trigonometric solutions of the graded Yang–Baxter equation. In this way we obtain the multiparametric R-matrices associated to the $U_q[\mathit{sl}{(r|2m)}^{(2)}]$, $U_q[\mathit{osp}{(r|2m)}^{(1)}]$ and $U_q[\mathit{osp}{(r=2n|2m)}^{(2)}]$ quantum symmetries. Two other families of multiparametric R-matrices not predicted before within the context of quantum superalgebras are also presented. The latter systems are indeed non-trivial generalizations of the $U_q[D_{n+1}^{(2)}]$ vertex model when both distinct edge variables statistics and extra free-parameters are admissible.     

Nonlinear integral equations for the finite size effects of RSOS and vertex-models and related quantum field theories

Starting from critical RSOS lattice models with appropriate inhomogeneities, we derive two component nonlinear integral equations to describe the finite volume ground state energy of the massive ${\varphi }_{\mathrm{id},\mathrm{id},\mathrm{adj}}$ perturbation of the $\mathit{SU}{(2)}_k\times \mathit{SU}{(2)}_{k^{\prime }}/\mathit{SU}{(2)}_{k+k^{\prime }}$ coset models. When $k^{\prime }\to \infty$ while the value of k is fixed, the equations correspond to the current–current perturbation of the $\mathit{SU}{(2)}_k$ WZW model. Then modifying one of the kernel functions of these equations, we propose two component nonlinear integral equations for the fractional supersymmetric sine-Gordon models. The lattice versions of our equations describe the finite size effects in the corresponding lattice models, namely in the critical $\mathrm{RSOS}(k,q)$ models, in the isotropic higher-spin vertex models, and in the anisotropic higher-spin vertex models. Numerical and analytical checks are also performed to confirm the correctness of our equations. These type of equations make it easier to treat the excited state problem.