I. M. Krichever

Integrability and Seiberg-Witten exact solution

The exact Seiberg-Witten (SW) description of the light sector in the

The Dispersionless Lax Equations and Topological Minimal Models

N=2

Integrable Systems.I

SUSY

Quantum Integrable Models and Discrete Classical Hirota Equations

4d

Algebras of virasoro type, riemann surfaces and structures of the theory of solitons

Yang-Mills theory is reformulated in terms of integrable systems and appears to be a Gurevich-Pitaevsky (GP) solution to the elliptic Whitham equations. We consider this as an implication that dynamical mechanism behind the SW solution is related to integrable systems on the moduli space of instantons. We emphasize the role of the Whitham theory as a possible substitute of the renormalization-group approach to the construction of low-energy effective actions.Abstract The exact Seiberg-Witten (SW) description of the light sector in the N = 2 SUSY 4 d Yang-Mills theory [N. Seiberg and E. Witten, Nucl. Phys. B 430 (1994) 485 (E); B 446 (1994) 19] is reformulated in terms of integrable systems and appears to be a Gurevich-Pitaevsky (GP) [A. Gurevich and L. Pitaevsky, JETP 65 (1973) 65; see also, S. Novikov, S. Manakov, L. Pitaevsky and V. Zakharov, Theory of solitons] solution to the elliptic Whitham equations. We consider this as an implication that the dynamical mechanism behind the SW solution is related to integrable systems on the moduli space of instantons. We emphasize the role of the Whitham theory as a possible substitute of the renormalization-group approach to the construction of low-energy effective actions.

Vector Bundles and Lax Equations on Algebraic Curves

It is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless Lax equations. The exact formulae for the tree level partition functions ofAn topological minimal models are found. The Virasoro constraints for the analogue of the τ-function of the dispersionless Lax equation corresponding to these models are proved.

THE EFFECTIVE PREPOTENTIAL OF N=2 SUPERSYMMETRIC SU(NC) GAUGE THEORIES

Integrable systems which do not have an “obvious“ group symmetry, beginning with the results of Poincare and Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960’s. Although a number of fundamental methods of mathematical physics were based essentially on the perturbation-theory analysis of the simplest integrable examples, ideas about the structure of nontrivial integrable systems did not exert any real influence on the development of physics.

Spin generalization of the Ruijsenaars-Schneider model, the non-Abelian 2D Toda chain, and representations of the Sklyanin algebra

Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota’s bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota’s equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter’s Q-operator are solutions to the auxiliary linear problems for classical Hirota’s equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for Ak−1-type models appear as discrete time equations of motions for zeros of classical τ -functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota’s equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained. Department of Mathematics of Columbia University and Landau Institute for Theoretical Physics Kosygina str. 2, 117940 Moscow, Russia James Franck Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA James Franck Institute and and Enrico Fermi Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA and Landau Institute for Theoretical Physics Joint Institute of Chemical Physics, Kosygina str. 4, 117334, Moscow, Russia and ITEP, 117259, Moscow, Russia 1

Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

In particular, the vacuum vector in the Fok representation is an example of a vector Φ0, although in quantum theory there arises a quite complicated algebraic aggregate composed of different Verma modules (cf. [1, 2]). The geometric approach of Polyakov et al., to the introduction of interactions in the theory of a string necessarily leads to complicated problems of the algebraic geometry of Riemann surfaces [3, 4]. However, the role of the Virasoro algebra in this approach is absolutely not apparent. The goal of the present paper is

Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

Abstract: The Hamiltonian theory of zero-curvature equations with spectral parameter on an arbitrary compact Riemann surface is constructed. It is shown that the equations can be seen as commuting flows of an infinite-dimensional field generalization of the Hitchin system. The field analog of the elliptic Calogero-Moser system is proposed. An explicit parameterization of Hitchin system based on the Tyurin parameters for stable holomorphic vector bundles on algebraic curves is obtained.