Recent Advances in Quantum Integrable Systems

Titles, abstracts and transparencies of talks

L. Amico:    Integrable spin-boson models

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 Uq(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

V. Caudrelier:    N.C.

A. Chakrabarti:    Memories of Daniel
Text (pdf file)

E. Corrigan:    N.C.

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 sl2 loop algebra symmetry if the q parameter is given by a root of unity, (q0)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 sl2 loop algebra. Moreover, we prove that every finite-dimensional highest weight representation of the sl2 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.

M. Jimbo:    N.C.

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 sl2 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 Uq(sln) 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 Uq(osp(1/2)) symmetry

We consider a Gaudin model related to the q-deformed superalgebra Uq(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. Shiraishi:    N.C.

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

Transparencies (pdf file)

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

Titles and abstracts of posters

Y. Ishimoto:    Logarithmic correlation functions and moduli integrals in minimal string theory

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[xj,d xk/dt}]=δjk+imθjkf. 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 divθB=ηθ and ∂Bs / ∂t + εkjs ∂Ej/∂xk =A1d2f / dt2 +A2 df / dt +A3, where ηθ, A1, A2 and A3 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 divθB=ηθ 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.

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