[en] I discuss in this paper the continuum limit of integrable spin chains based on the superalgebras sl(N/K) . The general conclusion is that, with the full 'supersymmetry', none of these models is relativistic. When the supersymmetry is broken by the generator of the sub u(1) , Gross-Neveu models of various types are obtained. For instance, in the case of sl(N/K) with a typical fermionic representation on every site, the continuum limit is the GN model with N colors and K flavors. In the case of sl(N/1) and atypical representations of spin j , a close cousin of the GN model with N colors and j flavors with flavor anisotropy is obtained. The Dynkin parameter associated with the fermionic root, while providing solutions of the Yang-Baxter equation with a continuous parameter, thus does not give rise to any new physics in the field theory limit. These features generalize to the case where an impurity is embedded in the system

[en] I discuss and extend the recent proposal of Leclair and Mussardo for finite temperature correlation functions in integrable QFTs. I give further justification for its validity in the case of one-point functions of conserved quantities. I also argue that the proposal is not correct for two- (and higher-) point functions, and give some counterexamples to justify that claim

[en] The relations between integrable systems and conformal field theories have first been worked out in the continuum limit. Using Coulomb gas and Bethe ansatz techniques, critical exponents and partition functions were obtained for various lattice models giving exact results of practical interest as well as a precise correspondence with conformal theories. It was later realized that the two subjects exhibit numerous similarities, and although this is maybe unexpected a priori, that some prints of the conformal invariant structure are present on the lattice before the continuum limit. We review here recent progress on this question

[en] The obtention of the supersymmetric 19-vertex model from an inhomogeneous 6-vertex model, is discussed. The 19-vertex model is considered the underlying theory and applied to calculate the partition function of the torus. A similar scheme to that for the critical models obtention, concerning the restriction procedure at special points of the gaussian line (of the 6-vertex model) is used. The critical line, in the supersymmetric model, is described by a free superfield and related to the XY model. The ''fused'' lattice models are outlined. It is shown that the results can be generalized to the models obtained by SU(2) coset construction

[en] Two dimensional critical systems are studied using transformation to free fields and conformal invariance methods. The relations between the two approaches are also studied. The analytical results obtained generally depend on universality hypotheses or on renormalization group trajectories which are not established rigorously, so numerical verifications, mainly using the transfer matrix approach, are presented. The exact determination of critical exponents; the partition functions of critical models on toruses; and results as the critical point is approached are discussed

[en] Theoretical studies concerning the connection between critical physical systems and the conformal theories are reviewed. The conformal theory associated to a critical (integrable) lattice model is derived. The obtention of the central charge, critical exponents and torus partition function, using renormalization group arguments, is shown. The quantum group structure, in the integrable lattice models, and the theory of Visaro algebra representations are discussed. The relations between off-critical integrable models and conformal theories, in finite geometries, are studied

[en] Functional relations play a key role in the study of integrable models. We argue in this paper that for massless field theories at zero temperature, these relations can in fact be interpreted as monodromy relations. Combined with a recently discovered duality, this gives a way to bypass the Bethe ansatz, and compute directly physical quantities as solutions of a linear differential equation, or as integrals over a hyperelliptic curve. We illustrate these ideas in details in the case of the c=1 theory, and the associated boundary sine-Gordon model

[en] We explain in this paper how a meaningful irrelevant perturbation theory around the infra-red (strong coupling) fixed point can be carried out for integrable quantum impurity problems. This is illustrated in details for the spin-((1)/(2)) Kondo model, where our approach gives rise to the complete low temperature expansion of the resistivity, beyond the well known T² Fermi liquid behaviour. We also consider the edge states tunneling problem, and demonstrate by Keldysh techniques that the DC current satisfies an exact duality between the UV and IR regimes. This corresponds physically to a duality between the tunneling of Laughlin quasi particles and electrons, and, more formally, to the existence of an exact instantons expansion. The duality is deeply connected with integrability, and could not have been expected a priori

[en] These lectures aim at introducing some basic algebraic concepts on lattice integrable models, in particular quantum groups, and to discuss some connections with knot theory and conformal field theories. The list of contents is: Vertex models and Yang-Baxter equation; Quantum sl(2) algebra and the Yang-Baxter equation; U_q sl(2) as a symmetry of statistical mechanical models; Face models; Face models attached to graphs; Yang-Baxter equation, braid group and link polynomials

[en] After the general developments of conformal invariance in two dimensions, it was realized that the study of critical models in finite geometries, in addition to the practical information it could provide through finite size scaling, was also of great conceptual interest. The simplest example is the case of the torus, a genus 1 surface which is thus not conformally equivalent to the plane. This geometry appears quite frequently in lattice calculations for systems with periodic boundary conditions, and is also very natural from the point of view of string theory. We will discuss briefly in these notes the main results obtained so far in this simple case