Dressing Transformations and the Algebraic--Geometrical Solutions in the Conformal Affine sl(2)sl(2) Toda Model

It is shown that the algebraic--geometrical (or quasiperiodic) solutions of the Conformal Affine $sl(2)$ Toda model are generated from the vacuum via dressing transformations. This result generalizes the result of Babelon and Bernard which states that the $N$--soliton solutions are generated from the vacuum by dressing transformations.Comment: 12 pages, latex, no figure

Equations in dual variables for Whittaker functions

It is known that the Whittaker functions $w(q,\lambda)$ associated to the group SL(N) are eigenfunctions of the Hamiltonians of the open Toda chain, hence satisfy a set of differential equations in the Toda variables $q_i$. Using the expression of the $q_i$ for the closed Toda chain in terms of Sklyanin variables $\lambda_i$, and the known relations between the open and the closed Toda chains, we show that Whittaker functions also satisfy a set of new difference equations in $\lambda_i$.Comment: 13 page

A Note on the Symplectic Structure on the Dressing Group in the sinh--Gordon Model

We analyze the symplectic structure on the dressing group in the \shG\, model by calculating explicitly the Poisson bracket \{g\x g\} where $g$ is the \dg\, element which creates a generic one soliton solution from the vacuum. Our result is that this bracket does not coincide with the Semenov--Tian--Shansky one. The last induces a Lie--Poisson structure on the \dg . To get the bracket obtained by us from the Semenov--Tian--Shansky bracket we apply the formalism of the constrained Hamiltonian systems. The constraints on the \dg\, appear since the element which generates one solitons from the vacuum has a specific form.Comment: 11 pages, latex, no figures, some statements corrected, the end of Sec. 3 and the whole Sec. 4 totally revise

The symplectic structure of rational Lax pair systems

We consider dynamical systems associated to Lax pairs depending rationnally on a spectral parameter. We show that we can express the symplectic form in terms of algebro--geometric data provided that the symplectic structure on L is of Kirillov type. In particular, in this case the dynamical system is integrable.Comment: 8 pages, no figure, Late

Quantum Group Generators in Conformal Field Theory

These are notes of a seminar given at the 30th International Symposium on the Theory of Elementary Particles, Berlin-Buckow, August 1996. The material is derived from collaborations with E. Cremmer and J.-L. Gervais, and C. Klimcik, and is partially new. Within the general framework of Poisson-Lie symmetry, we discuss two approaches to the problem of constructing moment maps, or q-deformed Noether charges, that generate the quantum group symmetry which appears in many conformal field theories. Concretely, we consider the case of $U_q(sl(2))$ and the operator algebra that describes Liouville theory and other models built from integer powers of screenings in the Coulomb gas picture.Comment: 21 pages, LaTe

Continuum limit of the Volterra model, separation of variables and non standard realizations of the Virasoro Poisson bracket

The classical Volterra model, equipped with the Faddeev-Takhtadjan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including $L\_0$. The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.Comment: Latex, 43 pages Synchronized with the to be published versio

Higher index focus-focus singularities in the Jayne-Cummings-Gaudin model : symplectic invariants and monodromy

We study the symplectic geometry of the Jaynes-Cummings-Gaudin model with $n=2m-1$ spins. We show that there are focus-focus singularities of maximal Williamson type $(0,0,m)$. We construct the linearized normal flows in the vicinity of such a point and show that soliton type solutions extend them globally on the critical torus. This allows us to compute the leading term in the Taylor expansion of the symplectic invariants and the monodromy associated to this singularity.Comment: 39 page

Dressing Symmetries

We study Lie-Poisson actions on symplectic manifolds. We show that they are generated by non-Abelian Hamiltonians. We apply this result to the group of dressing transformations in soliton theories; we find that the non-Abelian Hamiltonian is just the monodromy matrix. This provides a new proof of their Lie-Poisson property. We show that the dressing transformations are the classical precursors of the non-local and quantum group symmetries of these theories. We treat in detail the examples of the Toda field theories and the Heisenberg model.Comment: (29 pages

Multi-mode solitons in the classical Dicke-Jaynes-Cummings-Gaudin Model

We present a detailed analysis of the classical Dicke-Jaynes-Cummings-Gaudin integrable model, which describes a system of $n$ spins coupled to a single harmonic oscillator. We focus on the singularities of the vector-valued moment map whose components are the $n+1$ mutually commuting conserved Hamiltonians. The level sets of the moment map corresponding to singular values may be viewed as degenerate and often singular Arnold-Liouville torii. A particularly interesting example of singularity corresponds to unstable equilibrium points where the rank of the moment map is zero, or singular lines where the rank is one. The corresponding level sets can be described as a reunion of smooth strata of various dimensions. Using the Lax representation, the associated spectral curve and the separated variables, we show how to construct explicitely these level sets. A main difficulty in this task is to select, among possible complex solutions, the physically admissible family for which all the spin components are real. We obtain explicit solutions to this problem in the rank zero and one cases. Remarkably this corresponds exactly to solutions obtained previously by Yuzbashyan and whose geometrical meaning is therefore revealed. These solutions can be described as multi-mode solitons which can live on strata of arbitrary large dimension. In these solitons, the energy initially stored in some excited spins (or atoms) is transferred at finite times to the oscillator mode (photon) and eventually comes back into the spin subsystem. But their multi-mode character is reflected by a large diversity in their shape, which is controlled by the choice of the initial condition on the stratum

Quantum sl_n Toda field theories

We quantize $sl_n$ Toda field theories in a periodic lattice. We find the quantum exchange algebra in the diagonal monodromy (Bloch wave) basis in the case of the defining representation. In the $sl_3$ case we extend the analysis also to the second fundamental representation. We clarify, in particular, the relation of Jimbo and Rosso's quantum $R$ matrix with the quantum $R$ matrix in the Bloch wave basis.Comment: 19 pages, SISSA-ISAS 110/92/E