Spin-boson models find applications in a variety of different contexts,
such as atomic physics, semiconducting heterostructures,
dissipative quantum-systems, and more recently quantum computation.
We construct models describing spin-boson interaction using a quantum
inverse scattering procedure with open boundary conditions;
particular care is devoted to the so called counter-rotating terms
(important for many applications).
The exact solution of the models we find is achieved by first
diagonalizing the transfer matrix of an auxiliary XYZ model with mixed
representations, j and s; then contracting su(2)_{j}, transforming spins
into bosons.

Transparencies (pdf file)

Powerpoint version

**A. Belavin:**
*Four-Point function in minimal Liouville gravity.
*

The problem of correlation functions in minimal 2D gravity is considered. We evaluate the four-point correlation with one degenerate matter field and three general primary matter fields (with generic non-degenerate dimensions). The integral over moduli space is reduced to the boundary terms with the use of higher equations of motion of the Liouville field theory and evaluted explicitely .

**H. Boos:**
*The correlation functions of the XYZ chain
*

We discuss our recent results on the correlation functions for the
XYZ model. This is a generalization of our previous results for
the XXX and XXZ models based on the direct solving the reduced
qKZ equation. The corner stone of our construction is the so-called
trace function of any element of the Sklyanin algebra.
In some particular cases we reproduce known result by Lashkevich
and Pugai.

Transparencies (pdf file)

**E. Buffenoir:**
*Universal approach to Quantum Dynamical Coboundary Problem
*

Along the study of Quantum Dynamical Yang-Baxter Equation, motivated
originally by its occurence in the study of Toda Field theories and IRF
integrable models, the picture has emerged that solutions of these
equations for any quantum envelopping algebra U_{q}(G) are
associated to
solutions of Quantum Yang-Baxter equation through quantum dynamical
cocycles. It appeared that a universal expression of these objects can
be obtained explicitely by using an auxiliary linear equation called
ABRR equation. However, in spite of numerous works on these structures,
there was no answer to the problem, except in the first rank case,
adressed since the beginning of this story, about the general solutions
of the dynamical coboundary equation, also known as Vertex-IRF
transform, associated to these dynamical cocycles.
In this talk, we give a completely new universal approach to the dynamical
coboundary problem allowing to build a universal expression of the
solutions of dynamical coboundary problem in the higher rank case.

**O. Castro Alvaredo:**
*Form factors from algebraic Bethe ansatz for higher spin representations*

In this talk I will present new formulae for the form factors of spin operators in the XXX and XXZ quantum spin chains. These formulae hold for generic spin representations of the operators considered, as well as for generic spin representations at the remaining sites of the chain (which can be different at different sites). Therefore our results apply to any 'mixed' spin chains, such as alternating chains or spin chains with impurities. Remarkably, the expressions found admit a very compact form, thanks to certain complicated identities involving higher spin eigenvalues of the transfer matrix.

Transparencies (pdf file) , second file

**A. Chakrabarti:**
*Memories of Daniel
*

Text (pdf file)

**N. Crampe:**
*Y(so(N)) or Y(sp(N)) and integrable models
*

I will speak of my first work with Daniel, who was my PhD supervisor. We introduce briefly the FRT formalism for the half loop algebra based on the Lie algebra so(N) or sp(N) and for one of its deformation, the Yangian. These algebras appear in different integrable models: spin chains, Gaudin model, Calogero model and Principal Chiral Models.

**T. Deguchi:**
* Regular Bethe states as highest weight vectors of the sl(2) loop algebra
of the XXZ spin chain at roots of unity: Proofs of conjectures by
Fabricius and McCoy
*

The Hamiltonian of the XXZ spin chain has the sl_{2} loop algebra
symmetry if the q parameter is given by a root of unity,
(q_{0})^{2N}=1,
for an integer N. Recently, for the XXZ spin chain at roots of unity,
Fabricius and McCoy have proposed important conjectures such as
`Drinfeld polynomials of XXZ Bethe states'. Motivated by them, we show
in some sectors that regular Bethe ansatz eigenvectors are highest
weight vectors and generate irreducible representations of the sl_{2}
loop algebra. Moreover, we prove that every finite-dimensional highest
weight representation of the sl_{2} loop algebra is irreducible. We thus
derive the dimensions of the highest weight representation generated by
a given regular Bethe state through the Drinfeld polynomial, which is
expressed explicitly in terms of the Bethe roots.

**V.K. Dobrev:**
*Characters of D=4 conformal supersymmetry*

We give character formulae for the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2|N). Using these we derive decompositions of the long superfields as they descend to the unitarity threshold. These results may be applied to the problem of operators with protected scaling dimensions in quantum field or string theory framework. This is very important for the applications since the existence of such operators imply "non-renormalization theorems" at the quantum level. Our character formulae results are also applicable to irreps of the complex Lie superalgebras sl(4|N).

**A. Doikou:**
*Analytical Bethe ansatz for integrable open spin chains
*

I will talk about our recent work with Daniel, completed
during my stay in Annecy. I will review some of the algebraic and
physical aspects of integrable open spin chains associated to
(super) Yangians. In particular, I will briefly discuss the
corresponding algebraic setting, and I will also present the
spectrum and Bethe ansatz equations computed via the analytical
Bethe ansatz formulation.

Transparencies (pdf file)

**B. Doyon:**
*Finite-temperature form factors in the free
Majorana theory*

I will present a study of the large distance expansion of correlation
functions in the free massive Majorana theory at finite temperature, alias
the Ising field theory at zero magnetic field on a cylinder. I will
develop a method that mimics the spectral decomposition, or form factor
expansion, of zero-temperature correlation functions, introducing the
concept of ``finite-temperature form factors''. My techniques are
different from those of previous attempts in this subject, and my results
seem to be in disagreement with those previously obtained. I will explain
how an appropriate analytical continuation of finite-temperature form
factors gives form factors in the quantization scheme on the circle. I
will show that finite-temperature form factor expansions are able to
reproduce expansions in form factors on the circle. I will calculate
finite-temperature form factors of non-interacting fields (fields that are
local with respect to the fundamental
fermion field), and observe that they are given by a mixing of
their zero-temperature form factors and of those of other fields
of lower scaling dimension. I will then calculate finite-temperature form
factors of order and disorder fields. For this purpose, I will derive the
Riemann-Hilbert problem that completely specifies the set of
finite-temperature form factors of general twist fields (order and
disorder fields and their descendants). This Riemann-Hilbert problem is
different from the zero-temperature one, and so are its solutions. My
results agree with the known form factors on the circle of order and
disorder fields.

Transparencies (pdf file)

**L. Feher:**
*Spin Calogero models obtained from dynamical r-matrices
and geodesic motion*

In the talk we report our recent work with B.G. Pusztai in which
we study classical integrable systems based on dynamical r-matrices
corresponding to automorphisms of self-dual Lie algebras, *g*.
We apply a construction due to Li and Xu to associate spin Calogero type
models with the r-matrices and show that their equation of motion is a
projection of the natural geodesic equation on a Lie group G with Lie
algebra *g*. The phase space of the model is interpreted as a
Hamiltonian reduction of an open submanifold of the cotangent bundle
T^{*}G, using the symmetry arising from the adjoint action of G twisted
by the underlying automorphism. This shows the integrability of the
resulting systems and gives an algorithm to solve them. As examples we
present new integrable models built on the involutive diagram automorphisms
of the real split and compact simple Lie algebras. We also apply the
dynamical r-matrix construction to the cyclic permutation automorphism of
a semisimple Lie algebra composed of identical factors and find that it
yields certain models investigated earlier by different methods.

Transparencies (pdf file)

**D. Fioravanti:**
*Gemetrical loci of CFTs from a Lie algebra*

We will describe the appearance of specific algebraic KdV potentials as a consequence of a requirement on integro-differential expressions. These expressions are generated by means of vector fields acting on the KdV field and forming a Virasoro algebra, as a consequence of a previous paper by Stanishkov and the author (1998). The ``almost'' rational KdV fields are described in terms of a geometrical loci of complex points. A class of solutions of these constraints has recently appeared as a description of any conformal Verma module without degeneration, thanks to a work by Bazhanov, Lukyanov and A. Zamolodchikov. We will also give hints on the meaning of these contraints and similar ones in the rational Calogero-Moser systems by exploiting the KP into KdV reduction. In this respect, we will attept to motivate this work in the perspective of an off-critical Deformed Virasoro symmetry (cf. Rossi's talk in the afternoon).

**F. Göhmann:**
*Integral formulae for finite temperature
correlation functions of the XXZ chain*

I shall present integral formulae for finite temperature
density matrix elements and correlation functions of the
antiferromagnetic XXZ chain in the thermodynamic limit. I explain how
these formulae are derived by combinig the quantum transfer
matrix formalism to thermodynamics with algebraic Bethe ansatz
techniques for the calculation of matrix elements. I further
discuss applications like the direct numerical evaluation of the
integrals and high order high temperature expansions.

Transparencies (pdf file)

**M. Hallnas:**
*A classification of exactly solvable 1D many-body
systems with local interactions*

As is well known, a general local interaction in 1D is characterized by four real parameters. We propose a novel physical interpretation of these parameters and classify as well as solve all cases for which the coordinate Bethe ansatz is consistent. This yields exact solutions of two families of systems, one with two independent coupling constants deforming the well known delta interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta- and so called delta-prime interaction. If time permits I will also briefly discuss some results on extensions of these models by classical root systems. A method which in particular can be used to describe systems with boundaries. The talk is based on joint work with Edwin Langmann and Cornelius Paufler.

**H. Konno:**
*Correlation Functions of the Spin 1 XYZ Model
*

**G. Korchemsky:**
*Noncompact Heisenberg spin chains induced by
Yang-Mills theories*

Transparencies (pdf file)

**P. Kulish:**
*Quantization of super - Korteweg - de Vries
equations*

The algebraic structures related with integrability properties of superconformal field theory (SCFT) are introduced with underlying example of super - KdV equations. The SCFT counterparts of Baxter's Q-operator are constructed. The fusion relations for the transfer-matrices in different representsations are obtained to apply Bazhanov-Lukyanov-Zamolodchikov method to study their spectra.

**M. Lassalle:**
*Macdonald polynomials of type B, C, D
*

We recall the definition and properties of Macdonald polynomials associated
with reduced root systems. For type B, C, D, we give some results and conjectures,
including explicit expansions for systems of rank 2.

Transparencies (pdf file)

**J.M. Maillet:**
*On corrrelation functions of the XXZ chain*

Transparencies (pdf file) ,
experimental figure (pdf file),
theoretical figure (pdf file)

**B. McCoy:**
*Root of unity symmetries in the 6 and 8 vertex model*

I will discuss the phenomena that when the crossing parameter η
satisfies a
root of unity condition the eigenvalues of the 6 and 8 vertex model decompose
into degenerate subspaces whose dimension is a power of two. For the 6
vertex model this degeneracy is explained by the existence of an
sl_{2}
loop algebra which can be exactly formulated in some cases and
in others inferred from numerical computations. For both the 6 and 8
vertex model these degeneracies follow from a deep analogy between the
8 vertex and the chiral Potts model and a new
conjectured functional equation for the 8 vertex model. For the 8
vertex model the TQ equation will be discussed and it will be shown
that while three different definitions of Q have been made that there
is no definition of an auxillary matrix Q for which a TQ equation
has been proven for chains of odd length which is valid for all values
of η.

**M. Mintchev:**
*Reflection-transmission algebras and their
applications to QFT with impurities*

Inspired by factorized scattering from reflecting and transmitting impurities in integrable systems, we propose and analyse a generalization of the Zamolodchikov-Faddeev algebra. Distinguished elements of the new algebra, called reflection and transmission generators, encode the particle-impurity interactions. We will present several applications of this framework to QFT with defects.

**S. Pakuliak:**
*Universal weight function and nested Bethe ansatz*

The projections of the product of the Drinfeld's currents onto
intersections of the different Borel subalgebras in the current realization of
the quantum affine algebra U_{q}(sl_{n}) are calculated.
It was shown that this
projections has the structure of the off-shell nested Bethe ansatz vectors.

Transparencies (pdf file)

**F. Pan:**
*Exactly solvable gl(m/n) Bose-Fermi systems
*

A simple SUSY Bose-Fermi Hamiltonian and a class of hard-core Bose-Fermi
Hamiltonians with high order terms constructed by using the gl(m|n)
generators is shown to be exactly solvable. Excitation energies and
corresponding wavefunctions are obtained by using a simple algebraic Bethe
ansatz.

Transparencies (pdf file)

Powerpoint version

**Y.H. Quano:**
*Form factors of the eight-vertex model at
reflectionless points*

Form factors in the eight-vertex model are considered on the basis of
the bosonization scheme for the eight-vertex SOS model and vertex-face
correspondence. At so-called `reflectionless points', the S-matrix of the eight-vertex
model becomes (anti-)diagonal, so that the matter will become simpler than
usual. We wish to show that the form factors of some local operators of the
eight-vertex model at reflectionless points can be expressed in terms of
the sum of theta functions without any integrals.

Transparencies (pdf file)

**O. Ragnisco:**
*Curvature from quantum deformations (provisional) *

It is shown how non-standard quantum deformations of sl(2) provide in a
natural way integrable systems living on spaces with (possibly non
constant) curvature.

Transparencies (pdf file)

**F. Ravanini:**
*Excited states and finite size spectrum of super
Sine Gordon model*

Inspired by the NLIE on the lattice homogeneous XXZ chain recently
obtained by J. Suzuki, both for vacuum and excited states, we present a
derivation from Bethe Ansatz of a continuum limit NLIE describing the
excited states of the Super Sine-Gordon model. It generalizes the NLIE
proposed by C. Dunning for the vacuum sometimes ago. The infrared and
ultraviolet limits are explored making contact with the factorized scattering
formulation and the conformal limit (c=3/2) of the theory.

Transparencies (pdf file)

**V. Rittenberg:**
*Spin chains of different kinds - common work with Daniel
*

**V. Rivasseau:**
*Recherches avec Daniel en theorie des cordes
*

**M. Rossi:**
*Scattering factors of the XYZ model and their relation with
elliptic algebras and deformed Virasoro algebra.*

By using the technique of the nonlinear integral equation, scattering factors for the first excitations of the XYZ model are obtained. They appear to be elliptic deformations of the corresponding scattering factors of the sine-Gordon model. In particular, Baxter's elliptic R-matrix and the structure function of the deformed Virasoro algebra are (re)found in this framework as describing the scattering respectively of the elliptic deformation of the soliton-antisoliton system and of the first breathers.

**G. Satta:**
*Gaudin models with U _{q}(osp(1/2)) symmetry
*

We consider a Gaudin model related to the q-deformed superalgebra
U_{q}(osp(1|2)). We present an exact solution to that system diagonalizing a
complete set of commuting observables, and providing the corresponding
eigenvectors and eigenvalues. The approach used is based on the coalgebra
supersymmetry of the model.

Transparencies (pdf file)

**T. Sedrakyan:**
*Integrable structure behind non-Gaussian unitary ensembles
of random matrices
*

I will discuss the relations between random matrix theories (RMT) and theories of classical integrable systems . Importance of the analyze of integrable structures behind RMT is connected with a possibility to develop exact nonperturbative techniques allowing to exactly calculate correlation functions, which govern the spectrum in RMT. An important development in this context is the link between nonlinear replica σ model for U(N) invariant Gaussian ensemble of random matrices and Painlevé hierarchy of exactly solvable Toda lattice equations (TLE). Extension of this link to non-Gaussian RMT leads to the Graded (or q-deformed) Toda lattice equations. compact form.

**J. Suzuki:**
*The finite size property of the boundary XXZ model
in attractive regime
*

**Y. Yamada:**
*On elliptic K-matrices
*

We give some explicit form of elliptic K-matrices with several parameters.

Transparencies (pdf file)

**Al. Zamolodchikov:**
*Liouville field theory coupled to perturbed CFT
*

We study the simplest examples of minimal string theory, which are the unitary (p,q) minimal models coupled to two-dimensional gravity (Liouville field theory). We show that four-point Liouville correlation functions of `tachyons' exhibit logarithmic singularities, and that the theory turns out to be logarithmic. Their moduli integrations and the relation with Zamolodchikov's logarithmic degenerate fields will be discussed.

**Z. Nagy:**
*Spin chains from dynamical quadratic algebras
*

We present a construction of integrable quantum spin chains where local spin-spin interactions are weighted by ``position''-dependent potential containing abelian non-local spin dependance. This construction applies to the previously defined three general quadratic reflection-type algebras: respectively non-dynamical, semidynamical, fully dynamical.

**S. Posta:**
*On three particle matrix Calogero models
*

We use a method developed in C. Burdik and O. Navratil, A method for construction of the matrix solvable models, J. Phys. Math. Gen. 38 (2005) 1533--1542 to construct new matrix solvable models. We apply this method to the models of the A2 type and obtain solvable models of the system with the matrix NxN potential.

**M.B. Sedra:**
*Noncommutative geometry framework and the Feynman's
proof of Maxwell equations*

The main focus of the present work is to study the Feynman's proof of the
Maxwell equations using the NC geometry framework. To accomplish this
task,we consider two kinds of noncommutativity formulations going along
the same lines as Feynman's approach. This allows us to go beyond the
standard case and discover non-trivial results. In fact, while the first
formulation gives rise to the static Maxwell equations, the second
formulation is based on the following assumption
m[x_{j},d x_{k}/dt}]=δ_{jk}+imθ_{jk}f.
The results extracted
from the second formulation are more significant since they are associated
to a non trivial θ-extension of the Bianchi-set of Maxwell
equations. We find div_{θ}B=η_{θ} and
∂B_{s} / ∂t +
ε_{kjs} ∂E_{j}/∂x_{k}
=A_{1}d^{2}f / dt^{2}
+A_{2} df / dt +A_{3},
where
η_{θ}, A_{1}, A_{2} and A_{3}
are local functions
depending on
the NC θ-parameter. The novelty of this proof in the NC space is
revealed notably at the level of the corrections brought to the previous
Maxwell equations. These corrections correspond essentially to the
possibility of existence of magnetic charges sources that we can associate
to the magnetic monopole since div_{θ}B=η_{θ}
is not vanishing in general.

**A. Zuevskiy:**
*Vertex operator solutions to Ricci flow
equation
*

In the group-theoretical approach, we construct classes of special solutions to the Ricci flow equations.