We study the reflection of magnons from a D5-brane in the framework of the AdS/CFT correspondence. We consider two possible orientations of the D5-brane with respect to the reference vacuum state, namely vacuum states aligned along "vertical" and "horizontal" directions. We show that the reflections are of the achiral type. We also show that the reflection matrices satisfy the boundary Yang-Baxter equations for both orientations. In the horizontal case the reflection matrix can be interpreted in terms of a bulk S-matrix, S(p, −p), and factorizability of boundary scattering therefore follows from that of bulk scattering. Finally, we solve the nested coordinate Bethe ansatz for the system in the vertical case to find the Bethe equations. In the horizontal case, the Bethe equations are of the same form as those for the closed string.

In this work we use the q-oscillator formalism to construct the atypical (short) supersymmetric representations of the centrally extended U q (su(2|2)) algebra. We then determine the S-matrix describing the scattering of arbitrary bound states. The crucial ingredient in this derivation is the affine extension of the aforementioned algebra. IntroductionIntegrable systems constitute a special class of models in mathematics and physics. Their properties allow them to be solved exactly and thus they appear to be a very useful playground for studying various systems. One common feature shared by these models is that they are closely related to some underlying algebraic structures. Thus for most of the quantum integrable systems there is some sort of large and powerful symmetry hidden in the origins of it, for example a Yangian or a quantum affine algebra. A particularly interesting example is the Hubbard model.-1 -The Hubbard model, which was named after John Hubbard, is the simplest model of interacting particles on a lattice, with only two terms in the Hamiltonian, the hopping term (kinetic energy) and the Coulomb potential [1]. The model describes an ensemble of particles in a periodic potential at sufficiently low temperatures such that all the particles may be considered to be in the lowest Bloch band and also any long-range interactions between the particles are considered to be weak enough and thus are ignored. It is based on the tight-binding approximation of the superconducting systems and the motion of electrons between the atoms of a crystalline solid. Despite its apparent simplicity, it is very rich applications and generalizations describing including phase shifts and a plethora of interesting phenomena. In the case when interactions between particles on different sites of the lattice can not be neglected and are included, the model is often referred to as the Extended Hubbard model. The particles can either be fermions, as in Hubbard's original work, or bosons, and the model is then referred as either the BoseHubbard model or the boson Hubbard model that can be used to study systems such as bosonic atoms on an optical lattice (for a decent overview of various generalizations see reprint volumes [2][3][4] and also a more recent book [5]).A very specific class of models are those that share features of the one-dimensional Hubbard model and the supersymmetric t-J model [6]. The very interesting case being the Alcaraz and Bariev model [7] having an extra spin-spin interaction term in the Hamiltonian and showing some characteristics of superconductivity. This model can be viewed as a quantum deformation of the Hubbard model in much the same way as the Heisenberg XXZ model is a quantum deformation of the XXX model. This model has a specific R-matrix which can not be written as a function of the difference of two associated spectral parameters. This paradigm is related to the very interesting but at the same time complicated algebraic properties of the model.In recent years there has been renewed in...

We study the remnants of Yangian symmetry of AdS/CFT magnons reflecting from boundaries with no degrees of freedom. We present the generalized twisted boundary Yangian of open strings ending on boundaries which preserve only a subalgebra h of the bulk algebra g, where (g,h) is a symmetric pair. This is realized by open strings ending on the D3 brane known as the Y=0 maximal giant graviton in AdS_5 x S^5. We also consider the Yangian symmetry of the boundary which preserves an su(1|2) subalgebra only