Current Algebra of Classical Non-Linear Sigma Models

The current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is analyzed. It is found that introducing, in addition to the Noether current $j_\mu$ associated with the global symmetry of the theory, a composite scalar field $j$, the algebra closes under Poisson brackets.Comment: 6 page

On determinant representations of scalar products and form factors in the SoV approach: the XXX case

In the present article we study the form factors of quantum integrable lattice models solvable by the separation of variables (SoV) method. It was recently shown that these models admit universal determinant representations for the scalar products of the so-called separate states (a class which includes in particular all the eigenstates of the transfer matrix). These results permit to obtain simple expressions for the matrix elements of local operators (form factors). However, these representations have been obtained up to now only for the completely inhomogeneous versions of the lattice models considered. In this article we give a simple algebraic procedure to rewrite the scalar products (and hence the form factors) for the SoV related models as Izergin or Slavnov type determinants. This new form leads to simple expressions for the form factors in the homogeneous and thermodynamic limits. To make the presentation of our method clear, we have chosen to explain it first for the simple case of the $XXX$ Heisenberg chain with anti-periodic boundary conditions. We would nevertheless like to stress that the approach presented in this article applies as well to a wide range of models solved in the SoV framework.Comment: 46 page

Factorizing FF-matrices and the XXZ spin-1/2 chain: A diagrammatic perspective

Using notation inherited from the six-vertex model, we construct diagrams that represent the action of the factorizing $F$-matrices associated to the finite length XXZ spin-1/2 chain. We prove that these $F$-matrices factorize the tensor $R^{\sigma}_{1... n}$ corresponding with elements of the permutation group. We consider in particular the diagram for the tensor $R^{\sigma_c}_{1... n}$, which cyclically permutes the spin chain. This leads us to a diagrammatic construction of the local spin operators $S_i^{\pm}$ and $S_i^{z}$ in terms of the monodromy matrix operators.Comment: 26 pages, extra references added, typographical errors correcte

Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV

We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variable (SOV) representation hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial character of the Q-function allows us then to show that a finite system of equations of generalized Bethe type can be similarly used to describe the complete transfer matrix spectrum.Comment: 28 page

Irreducibility of fusion modules over twisted Yangians at generic point

With any skew Young diagram one can associate a one parameter family of "elementary" modules over the Yangian \Yg(\g\l_N). Consider the twisted Yangian \Yg(\g_N)\subset \Yg(\g\l_N) associated with a classical matrix Lie algebra \g_N\subset\g\l_N. Regard the tensor product of elementary Yangian modules as a module over \Yg(\g_N) by restriction. We prove its irreducibility for generic values of the parameters.Comment: Replaced with journal version, 18 page

The universal R-matrix and its associated quantum algebra as functionals of the classical r-matrix: the sl2sl_{2} case

Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ and its universal quantum $R$-matrix are explicitely constructed as functionals of the associated classical $r$-matrix. In this framework, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ is naturally imbedded in the universal envelopping algebra of the $sl_{2}$ current algebra