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 ${?}_{\mathrm{id},\mathrm{id},\mathrm{adj}}$ 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.