By making use of current-algebra Ward identities we study renormalization of general anisotropic current-current interactions in 2D. We obtain a set of algebraic conditions that ensure the renormalizability of the theory to all orders. In a certain minimal prescription we compute the βeta function to all orders.Left-right current-current interactions in 2D arise in many physical systems, for example in the KosterlitzThouless transitions and in the study of free electrons in random potentials. The chiral Gross-Neveu model is the simplest example which is symmetry preserving (isotropic) [1]. Because such interactions are marginal in 2D, the couplings are dimensionless and there is usually no small parameter to expand in order to explore strong coupling.In this work we determine the βeta function to all orders in a certain minimal prescription. This should be sufficient for the study of fixed points. The primary tool that allows us to isolate the log divergences at arbitrary order are the Ward identities for the currents. Kutasov performed a similar computation in the simpler isotropic case (i.e. non-abelian Thirring model, which is equivalent to the chiral Gross-Neveu model) but argued his result was the leading order in 1/k where k is the level of the current algebra [4]. We believe our result to be exact.We do not expect that there is anything dramatically new to learn for the isotropic case. The behavior of models like the Gross-Neveu model is very well understood. This is in contrast to the anisotropic, i.e. symmetry breaking perturbations where one can expect richer phenomena. In the anisotropic case not all perturbations are in fact renormalizable. We find three renormalizability conditions that ensure the theory is renormalizable to all orders. Given these conditions are satisfied, we find a compact expression for the summation of all orders in perturbation theory.We know of no reason to expect additional nonperturbative corrections to the βeta function due for instance to instantons. Our result should be a useful tool for exploring strong coupling physics for the many models in this class. In this Letter we only report the main result and defer applications to future publications.Consider a conformal field theory with Lie algebra symmetry realized in the standard way [2,3]. It possesses left and right conserved currents, J a (z),J a (z), where z = x + iy,z = x − iy, satisfying the operator product expansion (OPE)and similarly forJ a (z). η ab in the above equation is a metric (Killing form) and k is the level. We include the case of Lie superalgebras with applications to disordered systems in mind. Here each current J a has a grade [a] = 0 or 1, and the tensors satisfy:For superalgebras, η ab is generally not diagonalizable, but we have η ab η bc = δ c a . The conformal field theory can be perturbed by marginal operators which are bilinear in the currents. The most general action iswhere S cft is the conformal field theory with the currentalgebra symmetry, and d A aā are certain tensors that define the mod...

We study the noncommutative generalization of (euclidean) integrable models in two-dimensions, specifically the sine-and sinh-Gordon and the U (N ) principal chiral models. By looking at tree-level amplitudes for the sinh-Gordon model we show that its naïve noncommutative generalization is not integrable. On the other hand, the addition of extra constraints, obtained through the generalization of the zero-curvature method, renders the model integrable. We construct explicit nonlocal non-trivial conserved charges for the U (N ) principal chiral model using the Brezin-Itzykson-Zinn-Justin-Zuber method. 1 email:cabrera@ift.unesp.br 2 email:marco@if.ufrj.br 4 (e βφ ⋆ ⋆∂(e −βφ ⋆ ) − e −βφ ⋆ ⋆∂(e βφ ⋆ )) σ 3 (3.34) we obtain the same equation of motion 2 as in [17] ∂(e βφ ⋆ ⋆∂(e −βφ ⋆ ) − e −βφ ⋆ ⋆∂(e βφ ⋆ )) = 2m 2 sinh ⋆ (βφ) (3.35) 2 After replacing β by iβ. 8 but different additional constraints ∂(e −βφ ⋆ ) − 1 4 {e −βφ ⋆ , e βφ ⋆ ⋆∂(e −βφ ⋆ ) − e −βφ ⋆ ⋆∂(e βφ ⋆ )} ⋆ = 0 (3.36) ∂(e βφ ⋆ ) − 1 4 {e βφ ⋆ , e −βφ ⋆ ⋆∂(e βφ ⋆ ) − e βφ ⋆ ⋆∂(e −βφ ⋆ )} ⋆ = 0 . (3.37)It is straightforward to show that these constraints can be written as total derivatives, and that they vanish in the limit θ → 0.

We study a series of N = 1 supersymmetric integrable particle theories in d = 1 + 1 dimensions. These theories are represented as integrable perturbations of specific N = 1 superconformal field theories. Starting from the conjectured S-matrices for these theories, we develop the Thermodynamic Bethe Ansatz (TBA), where we use that the 2-particle S-matrices satisfy a free fermion condition. Our analysis proves a conjecture by E. Melzer, who proposed that these N = 1 supersymmetric TBA systems are "folded" versions of N = 2 supersymmetric TBA systems that were first studied by P. Fendley and K. Intriligator.

We find new integrable boundary conditions, depending on a free parameter g, for the O(N ) nonlinear σ model, which are of nondiagonal type, that is, particles can change their "flavor" through scattering off the boundary. These boundary conditions are derived from a microscopic boundary lagrangian, which is used to establish their integrability, and exhibit integrable flows between diagonal boundary conditions investigated previously. We solve the boundary Yang-Baxter equation, connect these solutions to the boundary conditions, and examine the corresponding integrable flows.1