We discuss the relation between matrix models and the Seiberg–Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitean one-matrix model in the planar (large N) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham τ-function. The corresponding Whi...

We present a hermitian matrix chain representation of the general solution of the Hirota bilinear difference equation of three variables. In the large N limit this matrix model provides some explicit particular solutions of continuous differential Hirota equation of three variables. A relation of this representation to the eigenvalues of transfer matrices of 2D quantum integrable models is disc...

In this contribution we review the theory of integrability of quantum systems in one spatial dimension. We introduce the basic concepts such as the Yang-Baxter equation, commuting currents, and the algebraic Bethe ansatz. Quite extensively we present the treatment of integrable quantum systems at finite temperature on the basis of a lattice path integral formulation and a suitable transfer matr...

This paper gives a new integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: ut = ∂ 5 xu − 2 3 , ut = ∂ 5 x (u 1 3 )xx − 2(u 1 6 )x u ; uxxt + 3uxxux + uxxxu = 0. The first two are in the positive order hierarchy while the 3rd one is in the negative order hierarchy. The whole hierarchy is shown integrable through solving a key...

Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group structures keep non-deformed in the course of quantizing the integrable system although their treatment is to be changed. Manifest examples of the KP/Toda hiera...

in the fundamental representation. Conspicuously absent from this list is A (1) n for n > 1. This is because in order to demonstrate integrability, we assume that the corresponding R matrix has crossing symmetry, which is not true in the case A (1) n for n > 1. Sklyanin has demonstrated that an integrable open spin chain can be constructed with an R matrix that satisfies nothing more than the Y...

As another evidence for the matrix Discrete Light Cone formulation of M theory, we show how general integrable Hamiltonian systems emerge from BPS bound states of k longitudinal fivebranes. Such configurations preserve eight supercharges and by chain of dualities can be related to the solution of N = 2 four-dimensional gauge theories. Underlying Hitchin systems on the bare spectral curve with k...

This is the second part of the paper devoted to the general proof of canon-icity of Bäcklund transformation (BT) for a Hamiltonian integrable system governed by SL(2)-invariant r-matrix. Introducing an extended phase space from which the original one is obtained by imposing a 1st kind constraint, we are able to prove the canonicity of BT in a new way. The new proof allows to explain naturally t...

Two-particle scattering amplitudes for integrable relativistic quantum field theory in 1+1 dimensions can normally have at most singularities of poles and zeros along the imaginary axis in the complex rapidity plane. It has been supposed that single particle amplitudes of the exact boundary reflection matrix exhibit the same structure. In this paper, single particle amplitudes of the exact boun...

We discuss the relation between matrix models and the Seiberg–Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitean one-matrix model in the planar (large N) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham τ-function. The corresponding Whi...