0. Introduction 1. Borel decomposition and the 2-Toda lattice 2. Two-Toda τ -functions and Pfaffian τ̃ -functions 3. The Pfaffian Toda lattice and skew-orthogonal polynomials 4. The (s = −t)-reduction of the Virasoro vector fields 5. A representation of the Pfaffian τ̃ -function as a symmetric matrix integral 6. String equations and Virasoro constraints 7. Virasoro constraints with boundary terms...

ABSTRACT Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the SU(∞) Nahm equations and the continuous Toda theory, we construct the quantum/Moyal deformations of the self dual membrane in terms of the q-Moyal star product . The q deformations of the SU(∞) Nahm equations are studied and explicit solutions are given. The continu...

One fruitful motivating principle of much research on the family of integrable systems known as “Toda lattices” has been the heuristic assumption that the periodic Toda lattice in an affine Lie algebra is directly analogous to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper shows that the analogy is not perfect. A discrepancy arises because the natural generalizatio...

The Toda lattice defined by the Hamiltonian H = 12 ∑n i=1 p 2 i + ∑n−1 i=1 νie qi−qi+1 with νi ∈ {±1}, which exhibits singular (blowing up) solutions if some of the νi = −1, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group G = SL(n,RI ). The subsystem is T Ge, where Ge = N+AN− consists of the determinant one matrices...

Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with integrable Z-gradations is discussed.

A new class of integrable two-dimensional dilaton gravity theories, in which scalar matter fields satisfy the Toda equations, is proposed. The simplest case of the Toda system is considered in some detail, and on this example we outline how the general solution can be obtained. We also demonstrate how the wave-like solutions of the general Toda systems can be simply derived. In the dilaton grav...

We explore the connection between an infinite system of particles in R2 described by a bi–dimensional version of the Toda equations with the theory of orthogonal polynomials in two variables. We define a 2D Toda lattice in the sense that we consider only one time variable and two space variables describing a mesh of interacting particles over the plane. We show that this 2D Toda lattice is rela...

We consider Baxter Q-operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local L-functions compatible with this identification. This gives a con...

The general properties of a class of two-dimensional dilaton gravity (DG) theories with multiexponential potentials are studied and a subclass of these theories, in which the equations of motion reduce to Toda and Liouville equations, is treated in detail. A combination of parameters of the equations should satisfy a certain constraint that is identified and solved for the general multi-exponen...

Locality is analyzed for Toda field theories by noting novel chiral description in the conventional nonchiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operato...