The exact quantum integrability problem of the membrane is investigated. It is found that the spherical membrane moving in flat target spacetime backgrounds is an exact quantum integrable system for a particular class of solutions of the light-cone gauge equations of motion : a dimensionally-reduced SU(∞) Yang-Mills theory to one temporal dimension . Crucial ingredients are the exact integrabil...

For each one of the Lie algebras gln and g̃ln, we constructed a family of integrable generalizations of the Toda chains characterized by two integers m+ and m−. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m+ = m−, we find a symmetric reduction for each general...

0. Introduction 1. Operators Λ and ε with [Λ, ε] = 1 and the δ-function 2. The two-Toda lattice 3. Bilinear Fay identities and a new identity for the two-Toda τ -functions 4. Higher Fay identities for the two-Toda lattice 5. Eigenfunction expansions and vertex operators 6. A remarkable trace formula 7. Two-Toda symmetries and the ASV-correspondence 8. Fredholm determinants of Christoffel-Darbou...

The solution of the classical open-chain n-body Toda problem is derived from an ansatz and is found to have a highly symmetric form. The proof requires an unusual identity involving Vandermonde determinants. The explicit transformation to action-angle variables is exhibited. [email protected] Permanent address 1 The Toda chain is one of the paradigmatic examples of an integrable many-body s...

A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequenci...

We propose integral representations for wave functions of Bn, Cn, and Dn open Toda chains at zero eigenvalues of the Hamiltonian operators thus generalizing Givental representation for An. We also construct Baxter Q-operators for closed Toda chains corresponding to Lie algebras B∞, C∞, D∞, affine Lie algebras B (1) n , C (1) n , D (1) n and twisted affine Lie algebras A (2) 2n−1 and A (2) 2n . ...

We study the ultraviolet asymptotics in non-simply laced affine Toda theories considering them as perturbed non-affine Toda theories, which possess the extended conformal symmetry. We calculate the reflection amplitudes, in non-affine Toda theories and use them to derive the quantization condition for the vacuum wave function, describing zero-mode dynamics. The solution of this quantization con...

Several inverse spectral problems are solved by a method which is based on exact solutions of the semi-infinite Toda lattice. In fact, starting with a well-known and appropriate probability measure μ, the solution αn(t), bn(t) of the Toda lattice is exactly determined and by taking t = 0, the solution αn(0), bn(0) of the inverse spectral problem is obtained. The solutions of the Toda lattice wh...

Sufficient conditions for constructing a set of solutions of the Toda lattice are analyzed. First, under certain conditions the invariance of the spectrum of J t is established in the complex case. Second, given the tri-diagonal matrix J t defining a Toda lattice solution, the dynamic behavior of zeros of polynomials associated to J t is analyzed. Finally, it is shown by means of an example how...

Ordinary and gl(n,R) generalized Toda systems as well as a related hierarchy are probed with respect to certain quantization characteristics. “Quantum” canonical and Poisson transformations are used to study quantizations of transformed Toda systems. With a Lax pair setting, a hierarchy of related systems are shown and their quantizations discussed. Finally, comments are added about quantum asp...