Sigma model for Ernst equation: Lax representation,Bäcklund transformation and divergence-free currents

A sigma model associated with the Ernst equation is derived. This sigma model is described by the Belinsky-Zakharov-type completely integrable equation and is formally equivalent to the usual sigma model in curved two-dimensional space. The corresponding Lax representation, BÄcklund transformation, and divergence-free currents are obtained.     

KAC-moody algebras,Yang-Baxter-Zamolodchikov-Faddeev algebras and integrable field theories

A large class of integrable two-dimensional field theories exhibit Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and Kac-Moody (KM) algebras. Examples of them are chiral fermionic models, sigma models and Wess-Zumino-Witten sigma models. With their help an explicit link is found between representations of YBZF and KM algebras.     

On the geometry of classically integrable two-dimensional non-linear sigma models

A master equation expressing the zero curvature representation of the equations of motion of a two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. Special attention is paid to those representations possessing a spectral parameter. Furthermore, a closer connection between integrability and T-duality transformations is emphasised. Finally, new integrable non-linear sigma models are found and all their corresponding Lax pairs depend on a spectral parameter.     

Integrable mappings for noncommutative objects

The method of integrable mappings is generalized to the noncommutative case. Hierarchies of integrable systems corresponding to the noncommutative Darboux-Toda substitution in the two-dimensional spaces and superspaces are constructed.     

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.     

The sigma model (dual) representation for a two-parameter family of integrable quantum field theories

We introduce and study two-parameter families of integrable field theories. The perturbative and non-perturbative methods are used to justify their factorized scattering theory in the form of the direct products of two S-matrices of the sine-Gordon model. The Bethe ansatz technique is applied for the calculation of the observables in the strong coupling region. The results are in the exact agreement with ones following from the sigma model action which is a two-parameter U(1) ⊗ (1) symmetrical deformation of the O(4) non-finear sigma model. The application of the sigma model representation to related perturbed conformal field theories is discussed.     

Classical and Quantum Integrable Sigma Models. Ricci Flow, “Nice Duality” and Perturbed Rational Conformal Field Theories

Journal of Experimental and Theoretical Physics - We consider classical and quantum integrable sigma models and their relations with the solutions of renormalization group equations. We say that an...     

LETTER: String Theory and an Integrable Model in Two-Dimensional Gravity with Dynamical Torsion

The integrability character of an integrable model of two-dimensional gravity with bosonic string coupling in Riemann-Cartan space is studied in the present letter. The equations of motion in the model are reduced to a nonlinear integrable equation. The general numerical solutions of this equation are found. In addition, the exact solution of scalar curvature is obtained.     

The Lie-Poisson structure of integrable classical non-linear sigma models

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into ther-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matricesr ands are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrixc. It is proposed that all these Poisson brackets taken together are, representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.Suported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/1-1Supported by the Studienstiftung des Deutschen Volkes     

Extending integrable hamiltonian systems from 2 to N dimensions

We describe a method of constructing N-dimensional integrable hamiltonian systems starting from two-dimensional ones. Several models are examined. Included are the two candidates for integrability discovered by Lakshmanan and Sahadevan for which we find the integrals of motion. Results for other N-dimensional integrable hamiltonian systems are also presented.     

On integrable two-dimensional generalizations of nonlinear Schrödinger type equations

New Bäcklund transformations of integrable 2 + 1 dimensional generalisations of nonlinear Schrödinger type equations are found. The corresponding Miura transformations and modified equations are constructed. The Bäcklund transformations being treated as two-dimensional chain equations provide examples of new integrable difference-differential equations.     

<Emphasis Type="Italic">N</Emphasis> = 1 Supersymmetric Sigma Model with Boundaries,I

 We study an N=1 two-dimensional non-linear sigma model with boundaries representing, e.g., a gauge fixed open string. We describe the full set of boundary conditions compatible with N=1 superconformal symmetry. The problem is analyzed in two different ways: by studying requirements for invariance of the action, and by studying the conserved supercurrent. We present the target space interpretation of these results, and identify the appearance of partially integrable almost product structures. Received: 27 November 2001 / Accepted: 16 August 2002 Published online: 19 December 2002 Acknowledgements. We are grateful to Ingemar Bengtsson and Andrea Cappelli for discussions and comments. MZ would like to thank the ITP, Stockholm University, where part of this work was carried out. UL acknowledges support in part by EU contract HPNR-CT-2000-0122 and by NFR grant 650-1998368. Communicated by R.H. Dijkgraaf     

Lie algebroids,Lie groupoids and TFT

We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.     

Static elliptic minimal surfaces in $$\hbox {AdS}_4$$

The Ryu–Takayanagi conjecture connects the entanglement entropy in the boundary CFT to the area of open co-dimension two minimal surfaces in the bulk. Especially in \(\hbox {AdS}_4\), the latter are two-dimensional surfaces, and, thus, solutions of a Euclidean non-linear sigma model on a symmetric target space that can be reduced to an integrable system via Pohlmeyer reduction. In this work, we construct static minimal surfaces in \(\hbox {AdS}_4\) that correspond to elliptic solutions of the reduced system, namely the cosh-Gordon equation, via the inversion of Pohlmeyer reduction. The constructed minimal surfaces comprise a two-parameter family of surfaces that include helicoids and catenoids in H\(^3\) as special limits. Minimal surfaces that correspond to identical boundary conditions are discovered within the constructed family of surfaces and the relevant geometric phase transitions are studied.     

Integrable Systems and Two-Dimensional Gravity from Conformally Reduced WZNW Model

Imposing constraints with an integer ordering on WZNW model a large series of conformal invariant integrable systems will result. In this letter, a general approach for imposing the first and the second class constraints based on an arbitrary grading scheme of the Lie algebras of the WZNW groups is presented. The first order constraints correspond to integrable systems containing super Toda and conformal affine Toda systems as examples and are related to two-dimensional induced gravity, whilst the second order constraints correspond to supersymmetric-like integrable systems containing super Toda and conformal affine super Toda systems (for super WZNW groups) and are conjectured to be related to twodimensional induced supergravity.     

Solving topological defects via fusion

Integrable defects in two-dimensional integrable models are purely transmitting thus topological. By fusing them to integrable boundaries new integrable boundary conditions can be generated, and, from the comparison of the two solved boundary theories, explicit solutions of defect models can be extracted. This idea is used to determine the transmission factors and defect energies of topological defects in sinh-Gordon and Lee–Yang models. The transmission factors are checked in Lagrangian perturbation theory in the sinh-Gordon case, while the defect energies are checked against defect thermodynamic Bethe ansatz equations derived to describe the ground-state energy of diagonal defect systems on a cylinder. Defect bootstrap equations are also analyzed and are closed by determining the spectrum of defect bound-states in the Lee–Yang model.     

Bäcklund-calogero group and general form of integrable equations for the two-dimensional Gelfand-Dikij-Zakharov-Shabat problem bilocal approach

The general form of the integrable equations and their Bäcklund transformations connected with the general two-dimensional Gelfand-Dikij-Zakharov-Shabat spectral problem is found within the framework of the generalized AKNS method. The bilocal tensor product of the solutions of the spectral problem is used successively, which essentially simplifies the calculations of recursion operators. The transformation properties of the integrable equations and Bäcklund transformations under the gauge group are discussed.     

Two-dimensional nonlinear string-type equations and their exact integration

On the basis of a group-theoretical formulation for exactly integrable two-dimensional nonlinear dynamical systems associated with the local part of an arbitrary graded Lie algebra, we study a string-type subclass of the equations. Explicit expressions are obtained for their general solutions.     

The supersymmetric soliton correlation matrix

Supersymmetric systems in (2/2) dimensions integrable by the supersymmetric generalization of the Zakharov-Shabat ?dressing? method are studied. The supersymmetric version of the ?soliton correlation matrix? is used to obtain multi- soliton solutions to generic supersymmetric systems of Zakharov-Mikhailov- Shabat type, together with their reductions under finite automorphism groups. The sypersymmetric S² sigma model is worked out as an explicit application of the method.