Bethe vectors of gl(3)-invariant integrable models, their scalar products and form factors

This short note corresponds to a talk given at "Lie Theory and Its Applications in Physics", (Varna, Bulgaria, June 2013) and is based on joint works with S. Belliard, S. Pakuliak and N. Slavnov, see arXiv:1206.4931, arXiv:1207.0956, arXiv:1210.0768, arXiv:1211.3968 and arXiv:1312.1488.Comment: 15 page

Vertex operators for boundary algebras

We construct embeddings of boundary algebras B into ZF algebras A. Since it is known that these algebras are the relevant ones for the study of quantum integrable systems (with boundaries for B and without for A), this connection allows to make the link between different approaches of the systems with boundaries. The construction uses the well-bred vertex operators built recently, and is classified by reflection matrices. It relies only on the existence of an R-matrix obeying a unitarity condition, and as such can be applied to any infinite dimensional quantum group.Comment: 11 pages, no figure, Latex2

Integrable systems with impurity

After reviewing some basic properties of RT algebras, which appear to be the natural framework to deal with integrable systems in presence of an impurity, we show how any integrable system (including these possessing translation invariance) can be promoted to an integrable system with an impurity which can reflect and transmit particles. The technics allows bulk translation invariant $S$-matrices while avoiding the no-go theorem stated recently about these laters. Presented at the Vth International workshop on Lie theory and its applications in physics, Varna (Bulgaria), June 16-22, 2003Comment: 14 pages; Misprint in eq. (3.6) and (3.7) correcte

Yangian realisations from finite W algebras

We construct an algebra homomorphism between the Yangian Y(sl(n)) and the finite W-algebras W(sl(np),n.sl(p)) for any p. We show how this result can be applied to determine properties of the finite dimensional representations of such W-algebras.Comment: 26 pages, Latex2

Direct computation of scattering matrices for general quantum graphs

We present a direct and simple method for the computation of the total scattering matrix of an arbitrary finite noncompact connected quantum graph given its metric structure and local scattering data at each vertex. The method is inspired by the formalism of Reflection-Transmission algebras and quantum field theory on graphs though the results hold independently of this formalism. It yields a simple and direct algebraic derivation of the formula for the total scattering and has a number of advantages compared to existing recursive methods. The case of loops (or tadpoles) is easily incorporated in our method. This provides an extension of recent similar results obtained in a completely different way in the context of abstract graph theory. It also allows us to discuss briefly the inverse scattering problem in the presence of loops using an explicit example to show that the solution is not unique in general. On top of being conceptually very easy, the computational advantage of the method is illustrated on two examples of "three-dimensional" graphs (tetrahedron and cube) for which other methods are rather heavy or even impractical.Comment: 20 pages, 4 figure

Generalized coordinate Bethe ansatz for non diagonal boundaries

We compute the spectrum and the eigenstates of the open XXX model with non-diagonal (triangular) boundary matrices. Since the boundary matrices are not diagonal, the usual coordinate Bethe ansatz does not work anymore, and we use a generalization of it to solve the problem.Comment: 11 pages; References added and misprints correcte

Algebraic approach to multiple defects on the line and application to Casimir force

An algebraic framework for quantization in presence of arbitrary number of point-like defects on the line is developed. We consider a scalar field which interacts with the defects and freely propagates away of them. As an application we compute the Casimir force both at zero and finite temperature. We derive also the charge density in the Gibbs state of a complex scalar field with defects. The example of two delta-defects is treated in detail.Comment: 24 pages, 10 figure