We analyse the spectrum of perturbations of the de Sitter space on the one hand, while on the other hand we compute the location of the poles in the Conformal Field Theory (CFT) propagator at the border.The coincidence is striking, supporting a dS/CFT correspondence. We show that the spectrum of thermal excitations of the CFT at the past boundary I − together with that spectrum at the future boundary I + is contained in the quasi-normal mode spectrum of the de Sitter space in the bulk.(1)eabdalla@fma.if.usp.br (2)karlucio@fma.if.usp.br (3)dals@df.ufscar.br

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We discuss the relation between bulk de Sitter three-dimensional spacetime and the corresponding conformal field theory at the boundary, in the framework of the exact quasinormal mode spectrum. We show that the quasinormal mode spectrum corresponds exactly to the spectrum of thermal excitations of Conformal Field Theory at the past boundary I − , together with the spectrum of a Conformal Field Theory at the future boundary I + .

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or A(1) 1 model, Izergin-Korepin or A(2) 2 model, sl(2|1) model and the osp(2|1) model. We find that there is a general solution for the A (1) 1 and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the A (2) 2 and osp(2|1) models we find three general solutions, being two complete reflection K-matrices solutions and one incomplete reflection K-matrix solution with some null entries. In both models these solutions have two free parameters. Integrable spin-1 Hamiltonians with general boundary interactions are also presented. Several reduced solutions from these general solutions are presented in the appendices.(1.7)and K + (u) is the reflection matrix which satisfy an left RE. Given a solution K − (u) of (1.7), one can show that the corresponding quantitysatisfy the left RE. Here ρ is a crossing parameter and U is a crossing matrix both being specific to each model [6,8]. t i stands for the transposition taken in the i-th space and tr 0 is the trace taken in the auxiliary space.

2n−1 , and D(2)

We consider a formulation of the algebraic Bethe ansatz for the six vertex model with non-diagonal open boundaries. Specifically, we study the case where both left and right K-matrices have an upper triangular form. We show that the main difficulty entailed by those form of the K-matrices is the construction of the excited states. However, it is possible to treat this problem with aid of an auxiliary transfer matrix and by means of a generalized creation operator.

We have considered the Zamolodchikov-Fateev and the Izergin-Korepin models with diagonal reflection boundaries. In each case the eigenspectrum of the transfer matrix is determined by application of the algebraic Bethe Ansatz .

The nineteen-vertex models of Zalomodchikov-Fateev, Izergin-Korepin and the supersymmetric osp(1|2) with periodic boundary conditions are studied. We find the spectrum of these quantum spin chain using the Coordinate Bethe Ansatz. The approach is a suitable parametrization of their wavefunctions. We also applied the Algebraic Bethe Ansatz in order to obtain the eigenvalues and eigenvectors of the corresponding transfer matrices.