Baxter 2000 Workshop Program

The Baxter Revolution in Mathematical Physics

The Australian National University

13-19 February 2000

Abstracts

Changrim Ahn (Korea)
Reflection amplitudes and thermodynamic Bethe Ansatz for simply and non-simply laced affine Toda theories
We study the ultraviolet asymptotics in affine Toda theories. These models are considered as perturbed non-affine Toda theories. We calculate the reflection amplitudes, which relate different exponential fields with the same quantum numbers. Using these amplitudes we derive the quantization condition for the vacuum wave function, describing zero-mode dynamics, and calculate the UV asymptotics of the effective central charge of simply and non-simply laced affine Today theories. These asymptotics are in a good agreement with thermodynamic Bethe ansatz results. Using this formalism, one can confirm the free energies of the affine Today theories derived recently using the reflection amplitudes.
F.C. Alcaraz (Brasil)
Exact solution of asymmetric diffusion with second-class particles of arbitrary size
The exact solution of the asymmetric exclusion problem with first- and scond-class particles is presented. In this model the particles (size 1) of both classes are located at lattice points, and diffuse with equal asymmetric rates, but particles in the first class do not distinguish those in the second class from holes (empty sites). We generalize and solve exactly this model by considering molecules in the first and second class with sizes $s_1$ and $s_2$ ($s_1,s_2 = 0,1,2,\ldots$), in units of lattice spacing, respectively. The solution is derived by a Bethe ansatz of nested type.
Helen Au-Yang (Oklahoma)
Studies of Baxter's solution for the free energy of the Chiral Potts model
Using the functional relations derived for the chiral Potts model, Baxter has obtained several results for its free energy. In this work many of the subtleties are reviewed to show his incredible accomplishments. His calculations are then extended to include other regions.
Murray Batchelor (ANU)
Ladders
I will discuss a general family of solvable ladder models which can be constructed from the su(N) lattice models. These include models of higher spin. Baxter's early contribution to the su(3) model will also be mentioned.
(Work in collaboration with Jan de Gier and Mark Maslen).
Vladimir Bazhanov (ANU)
The mysterious Q
Abstract not available
Peter Bouwknegt (Adelaide)
The universal chiral partition function in a fractional quantum Hall basis
In this talk I will explain new formulas for the characters of affine Lie algebras inspired by the study of non-Abelian fractional quantum Hall states.
Geoff Campbell (ANU)
Transforming q series into new Dirichlet series involving Riemann zeta functions
Abstract not available
Alan Carey (Adelaide)
Calogero-Sutherland systems
I will explain how to use the representation theory of loop groups to construct anyon fields. These anyons may be used to construct a W-algebra. One of the generators of this W-algebra may be considered as a second quantised Calogero-Sutherland Hamiltonian and used to construct the eigenvectors and eigenvalues of the Calogero-Sutherland model. By using anyons at non-zero temperature the so-called elliptic CS model may also be solved.
Ian Enting (CSIRO, Melbourne)
The computational complexity of series expansions for the Potts model using the finite lattice method
The finite lattice method of series expansion has been a powerful way of obtaining series expansions for lattice statistics problems, particularly in two dimensions. In order to assess the potential for further development, the computational complexity of the method is analysed for various series expansions of Potts model. Several new types of expansion are discussed.
Omar Foda (Melbourne)
Combinatorial aspects of Baxter-type models
Abstract not available
Peter Forrester
Random matrices in a bigger picture
I'll talk on the relationship of random matrices to a quantum many body problem. Then I'll discuss recent analytic results for the ground state structure function.
Vlad Fridkin (RIMS, Kyoto)
$\eta$-dependence of the ground state energy of the finite XXZ spin chain
With a view to establishing an exact representation of the ground state energy of the finite size XXZ spin chain in terms of elementary functions, we concentrate on the crossing-parameter $\eta$- dependence around $\eta=\pi/3$ for which the ground state energy is known. The approach taken involves the use of a physical solution $Q$ of Baxter's t-Q equation, corresponding to the ground state, as well as a non-physical solution $P$ of the same equation. The calculation of $P$ and then of the first ground state derivative is covered. A possible application to the theory of percolation is mentioned briefly.
Vlad Fridkin, Don Zagier and Yuri Stroganov (RIMS, Kyoto)
Ground state of the quantum symmetric finite size XXZ spin chain with anisotropy parameter $\Delta = \frac{1}{2}$
We find an analytic solution of the Bethe Ansatz equations for the special case of a finite XXZ spin chain with free boundary conditions and with a complex surface field which provides for $U_q(sl(2))$ symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter $\Delta = \frac{1}{2}$ corresponding to $q^3 = -1$.
Jan de Gier (ANU)
Fractional magnetization plateaus in exactly solved spin ladders
The su(n) generalisations of the XXX spin chain can be interpreted in a natural way as spin ladders. Using the integrability of these models the critical behaviour of ladder systems can be studied in great detail. Particularly interesting is the case when ladders are placed in a magnetic field where they exhibit magnetisation plateaus at fractional values of the total magnetisation.
Mo-Lin Ge (Tianjin)
Yangian symmetry in Yang-Baxter systems: applications
For a given rational solution of the Yang-Baxter equation the trT(u) involved RTT relation is set up that leads to the Yangian algebra and Hamiltonian simultaneously. For SL(2) and T0(n) we show that the Haldane-Shastry model is related to RTT relation in this way. It turns out that [T0(2), J_2]=0 and [T0(3), J_2]=0. If T0(2) and I,J commute with J_2, where the generators I and J form Y(SL(2)) as given by Drinfeld. Therefore, the J_2 may be viewed as a new quantum number for some many-body system. As the applications we take the following examples:

(I) Hydrogen atom
Based on the point of view of the quantum tensor space, we rederive the spectrum of Hydrogen atom in terms of Y(SL(2)) representation theory of Chari-pressley. Furthermore, we show that the monopole harmonic Yqlm of Wu-Yang may be regarded as the simplest realization of Y(SL(2)).
(II) Haldane-Shastry model
The connection between such a model and J_2 is established for H2 in the Hamiltonian family with long-ranged interaction.
(III) The new quantum numbers of J_2 for three and four spin-1/2 systems are found. With J_2 the degeneracies for I_2 and I_3, where I=?, can be specified, namely, the quantum numbers are {I_2, I_3, J_2}. The transition operators for different states of such few-body spin systems are explicitly shown.
Mark Gould (Brisbane)
Type II quantum superalgebras and integrable models
Quantum (affine) superalgebras have recently played a prominent role in the construction and analysis of one-dimensional integrable models of strongly correlated fermions. In this talk the structure of the Type II quantum superalgebra Uq[osp(m/n)] is briefly reviewed. In particular Cartan-Weyl generators are obtained in terms of which the q-Serre (and higher order) relations, as well as Casimir invariants, are simply expressible. Two new families of spectral-dependent R-matrices R(x) with Uq[osp(m/n)]-invariance are obtained for general m and n, arising from the vector irrep. These are associated with the twisted quantum affine superalgebra Uq[gl(m/n)(2)] and the untwisted quantum affine superalgebra Uq[osp(m/n)(1)]. They give rise to two distinct classes of integrable "fermionic" models with a grade-selfadjoint local Hamiltonian.
Miklos Gulacsi (ANU)
The exact Schrieffer-Wolff transformation
Abstract not available
Tony Guttmann (Melbourne)
Progress in the study of the susceptibility of 2d Ising model
In collaborative work with Will Orrick, Bernie Nickel and Larry Glasser, some progress has been made in the study of the structure of the susceptibility of the two-dimensional Ising model. In particular, expressing the susceptibility as the sum of (2k+1) particle excitations, and transforming each of these to the variable proposed by Baxter for the magnetisation, certain intruiging elliptic function parameterisations are observed. Similar results hold for the low temperature susceptibility too. These and other observations will be discussed.
Chris Hamer (UNSW, Sydney)
Finite-size scaling and effective Lagrangians at quantum phase transitions
A brief review is given of finite-size scaling theory at quantum phase transitions, and its connections with the 'effective Lagrangian' approach in field theory. In one space dimension, the theory of conformal invariance makes universal predictions for the finite-size scaling corrections. In higher dimensions, Leutwyler, Hasenfratz and others have shown how similar predictions can be made in cases where spontaneous breakdown of a continuous symmetry gives rise to Goldstone bosons. Conversely, measurements of the finite-size scaling behaviour can give estimates of the parameters of the underlying effective Lagrangian. These points are illustrated for somne quantum spin systems.
Iwan Jensen (Melbourne)
Enumerations of lattice animals and trees
Using the finite lattice method we have counted the number of lattice animals up to size 46. Series has also been calculated for the radius of gyration and perimeter-weighted moments. The method has also been used to generate series for lattice trees.
Vaughan Jones (UC, Berkeley)
The quantum dodecahedron
By building a statistical mechanical model on any planar graph, with spin set the vertices of the E_8 Coxeter graph we present a new proof of the existence of E_8 type subfactors and argue that they "are" the quantum dodechahedral groups. The method uses a result of Graham and Lehrer on the affine or annular Temperley-Lieb algebra.
Rinat Kashaev (St Petersburg)
Quantum dilogarithm and strongly coupled quantum discrete Liouville model
Some remarkable properties of the ``non-compact'' quantum dilogarithm are analysed. It is used in formulating the quantum discrete Liouville model in the strongly coupled regime, 1
Andreas Klumper (Dortmund)
Thermodynamics of integrable quantum chains
There exist two methods for the calculation of thermodynamic quantities of integrable quantum systems. The thermodynamic Bethe ansatz (TBA) is the oldest method and conceptually simple, yielding however infinitely many integral equations for the isotropic spin-1/2 Heisenberg chain. In contrast, the newer quantum transfer matrix (QTM) approach yields just two non-linear integral equations, but is mathematically much more involved.

In this talk I will show how to apply the TBA construction to the Heisenberg chain and derive the exact nonlinear integral equations of the QTM approach for just {\em two} functions describing the elementary excitations. Using these equations the magnetic susceptibility $\chi$ and specific heat $C$ versus temperature $T$ of the spin $S = 1/2$ antiferromagnetic uniform Heisenberg chain are calculated to high accuracy for $5\times10^{-25}\leq T/J\leq 5$. The $\chi(T)$ data agree very well at low $T$ with the asymptotically exact theoretical low-$T$ prediction of S. Lukyanov (1998). The unknown coefficients of the second and third lowest-order logarithmic correction terms in this theory for $C(T)$ are estimated from the $C(T)$ data.
Vladimir Korepin (Stony Brook)
Solution of inverse scattering problem for eight-vertex model
Abstract not available
Atsuo Kuniba (Tokyo)
Soliton cellular automata from vertex models
I report on a curious interplay between classical integrable systems and quantum integrable systems. The former example is a soliton cellular automaton connected to the ultradiscrete limit of the KP equation, and the latter is a family of fusion vertex models at $q=0$. The two systems are identified and the combinatorial R matrix in the crystal base theory is found to emerge as the S matrix of the ultradiscrete solitons. (Joint work with G.Hatayama, K.Hikami, R.Inoue, T.Takagi and T.Tokihiro.)
Jean-Marie Maillard (Paris)
Let's Baxterise
We recall the concept of baxterisation of an R-matrix, or of a monodromy matrix, and show that the baxterisation approach is a ``win-win'' strategy. Three situations can be distinguished. Firstly the baxterisation procedure works and one gets that way the key ingredient to analyze a model of lattice statistical mechanics, namely the ``canonical'' parameterization of the model, compatible with all the symmetries of the model (discrete or continuous) : this parameterization is most of the time associated with abelian varieties. Secondly the baxterisation procedure does not work, because the orbits of the discrete symmetry group associated with the baxterisation procedure are finite, and therefore getting a parameterization becomes quite hopeless (algebraic varieties of the so-called ``general type'', higher genus curves in the simplest cases ...). However, writing this very finiteness condition enables to actually get the equations of these algebraic varieties. Thirdly, the baxterisation procedure does not work, because the orbits of the group associated with the baxterisation procedure are chaotic (though corresponding most of the time to measure-preserving dynamical systems) : the model has to be ruled out as far as ``reasonable'' analytical studies are concerned. The Yang-Baxter integrable models, or, more generally, the models which can be analytically studied, correspond to the first two situations for which the baxterisation procedure works or yields finite order orbits. In fact, even in the third generically chaotic case, the baxterisation technics enable to find efficiently where the subvarieties, corresponding to the first two analytically interesting situations, are actually located. This paper is illustrated by many examples of baxterisation, in particular the heuristic example of the baxterisation of the Baxter model, with a detailed analysis of the finite order situations. One also gives a generalization of these results, namely the baxterisation of the sixteen vertex model which is, generically, non Yang-Baxter integrable. One finally shows that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying Yang-Baxter structure is not clear, and not used. For instance, we actually give several examples of baxterisation of differential operators, starting with the simple example of the baxterisation of the Toda L-operator, and other simple local quantum Lax matrices.
Vladimir Mangazeev (ANU)
Some exact results for the Zamolodchikov model
Using $Q$-matrix method we obtain the functional equations for the spectrum of the transfer-matrices for the three-layer Zamolodchikov model. We analyze a distribution of Bethe-roots for the ground state and obtain a set of integral equations for the density functions in the thermodynamic limit. This set of equations can be solved exactly by Fourier integrals.
J. Rodrigo R. Martinez (Koln)
The spin-1/2 XXZ quantum chain at finite magnetic fields: Crossover phenomena driven by temperature
We investigate the asymptotic behaviour of spin-spin correlation functions for the integrable Heisenberg chain. To this end we use the Quantum Transfer Matrix (QTM) technique developed in [1] which results in a set of non-linear integral equations (NLIE) whose solution gives for the largest eigenvalue the free energy and by modifying the paths of integration the next-leading eigenvalues and hence the correlation lengths. At finite field $h>0$ and sufficiently high temperature the next-leading eigenvalue is unique and given by a 2-string solution to the QTM taking real values thus resulting into exponentially decaying correlations with antiferromagnetic oscillations. At sufficiently low temperature a different behaviour is expected on grounds of predictions by conformal field theory (CFT). In particular, correlations with incommensurate $2k_F$ oscillations are expected. As a consequence of this, the QTM has to develop complex conjugate pairs of eigenvalues at sufficiently low temperatures.
We present analytical and numerical investigations of the QTM confirming the above expectation and establishing a well defined crossover temperature $T_c(h)$ at which the 1-string eigenvalue to the QTM gets degenerate with the 2-string solution. Among other things we find a simple particle-hole picture for the excitations of the QTM and we make contact with the dressed charge formulation of CFT. For general temperatures the correlation lengths can be obtained by numerically solving the NLIE.
[1] A. Kl"umper, Z. Phys. B 91 (1993) 507.
(Joint work in collaboration with A. Kluemper, C. Scheeren and M. Shiroishi.)
Barry McCoy (Stony Brook)
The Baxter Revolution
I review the revolutionary impact Rodney Baxter has had on statistical mechanics from his solution of the 8-vertex model in 1971, the invention of corner transfer matrices in 1976, to the creation of the RSOS models in 1984 and his continuing current work on the chiral Potts model.
Barry McCoy (Stony Brook)
The $sl_2$ loop algebra symmetry of the six-vertex model at roots of unity
It will be shown that the six vertex model (XXY spin chain) with $\Delta=(q+q^{-1})/2 and $q^{2N}=1$ has an invariance under the loop algebra of $sl_2$ which produces sets of degenerate eigenvalues. For $\Delta=0$ the multiplicity of these degeneracies will be computed. Generalizations to the XYZ model will also be presented. (Joint work with Tetsuo Deguchi and Klaus Fabricius.)
James McGuire and Charlotte Dirk (Florida)
An extension of the Bethe ansatz: the quantum three-particle ring
The quantum problem of three impenetrable particles of arbitrary mass confined to a ring is solved by the Bethe ansatz. The solution of this problem is intimately related to the solution of Laplace's equation in the interior of an arbitrary acute triangle; a problem thought insoluble by Bethe ansatz methods.
Tetsuji Miwa (RIMS, Kyoto)
Combinatorics of coinvariants
I will report on our recent work with Feigin, Loktev, Kedem and Mukhin on $\widehat{sl}_2$ coinvariants. We establish recursion relations which give a bijection between a set of monomial basis of the coinvariant space and the set of Verlinde paths.
Tomoki Nakanishi (Nagoya)
Bethe equation at $q=0$ and weight multiplicities
The $U_q(X_n^{(1)})$ Bethe equation is studied at $q=0$. A linear congruence equation is proposed related to the string solutions. The number of its off-diagonal solutions is expressed in terms of an explicit combinatorial formula and gives the weight multiplicities of the quantum space.
Bernard Nienhuis (Amsterdam)
Packing order: tiles and colors
A number of exactly solved two-dimensional statistical models can be formulated as stackings of geometrical objects. Recently three such models have been solved, which admit a phase with non-crystallographic rotational symmetry. These models do not satisfy the Yang-Baxter equation. One of them however turns out to be a particular limit of a Yang-Baxter solvable model. In this talk I will discuss possibilities to bring also the other models into the happy Yang-Baxter fold.
Jae Dong Noh (Seoul)
Symmetry property and incompleteness of the Bethe ansatz wave functions for the Heisenberg XXZ chain
We investigate the incompleteness of the Bethe ansatz solutions with normalizable wave functions for the Heisenberg $XXZ$ chain using their symmetry properties. The eigenstates of the Heisenberg Hamiltonian are simultaneous eigenstates of the translation operator $T$ and the lattice inversion operator $V$ in the subspace of $\Omega=\pm 1$ with $\Omega$ the eigenvalue of $T$. We show that the Bethe ansatz cannot produce the non-degenerate eigenstates which have quantum numbers $(\Omega,\Upsilon)=(\pm 1,\mp 1)$ with $\Upsilon$ the eigenvalue of $V$. It is also shown that such states exist in any nontrivial magnetization sector, which proves the incompleteness of the Bethe ansatz solutions. Surprisingly, the missing states include low-lying states in the zero magnetization sector. The number of missing states is estimated to diverge exponentially with the chain length.
Jaan Oitmaa (UNSW, Sydney)
Ladders, combs, bilayers and brushes
We discuss the physics of a number of low-dimensional s=1/2 quantum antiferromagnets. These systems show a number of, at first sight, surprising features which are absent in the corresponding classical systems. Our group has used both numerical and analytic methods to obtain new results which illustrate the important interplay between magnetic order and singlet formation leading to "spin liquid" ground states.
Aleks Owczarek (Melbourne)
The combinatorics of the free fermion condition in vertex models
The free fermion condition of the six-vertex model provides a 5 parameter sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple, hence allowing explicit solutions. Such conditions arose originally in early field-theoretic S-matrix approaches. Here we provide a combinatorial explanation for the condition in terms of a generalised Gessel-Viennot involution. By doing so we extend the use of the Gessel-Viennot theorem, originally devised for non-intersecting walks only, to a special weighted type of intersecting walk, and hence express the partition function of N such walks starting and finishing at fixed endpoints in terms of the single walk partition functions.
Paul A. Pearce (Melbourne)
Integrable and conformal boundary conditions
General integrable boundary conditions are presented for the critical A-D-E lattice models of Pasquier. These models are associated with the minimal sl(2) conformal field theories. In the continuum scaling limit the integrable lattice boundary conditions give rise to a complete set of conformal boundary conditions for these theories.
Jacques H.H. Perk (Oklahoma)
Wavevector-dependent susceptibilities in quasiperiodic Ising models
Using the various functional relations for correlation functions in planar Ising models, new results are obtained for the q-dependent susceptibility for Ising models with quasiperiodic coupling constants. The effects are clearest if the interactions are both attractive and repulsive according to a quasiperiodic pattern.
Yaroslav Pugai (RIMS, Kyoto)

Free field construction for ABF model in regime II
The Wakimoto construction for the quantum affine algebra $U_q(\slthten)$ admits a reduction to the $q$-deformed parafermion algebras. We interpret the latter theory as a free field realization of the Andrews-Baxter-Forrester models in regime II. We give multi-particle form factors of some local operators on the lattice and compute their scaling limit, where the models are described by a massive field theory with $\Z_k$ symmetric minimal scattering matrices. (Joint work with M. Jimbo, T. Konno, S. Odake, J. Shiraishi).
Reinout Quispel (La Trobe, Melbourne)

Piecewise-linear integrable systems
Abstract not available
Chaiho Rim (Chonju, Korea)

Reflection amplitudes and one point function of (affine) Toda field theories
We calculate the reflection amplitudes for Toda field theories, which relate different exponential fields with the same quantum number. And we find one-point function of the exponential field for the corresponding non-affine field theories with the help of the reflection amplitudes and the symmetries of the theory.
Katherine Seaton (Latrobe, Melbourne)
The inversion relation and the dilute $A_{3,4,6}$ eigenspectrum
On the basis of the result obtained by applying Baxter's exact perturbative approach to the dilute $A_3$ model to give the $E_8$ mass spectrum, the dilute $A_L$ inversion relation was used to predict the eigenspectra in the $L=4$ and $L=6$ cases. (The approach has now been applied to confirm the $L=4$/$E_7$ expression.) In calculating the next-to-leading term in the correlation lengths (or equivalently masses), the inversion relation condition gives a surprisingly simple result in all three cases, and for all masses.
Fedor Smirnov
Dual Baxter's equations and quantum algebraic geometry
We consider two integrable quantum mechanical problems which are related by duality $\hbar\to1/\hbar$. These models have the common set of eigen-functions and the spectrum is described by pair of dual Baxter's equations. We explain the relation of this duality to the duality between differential forms and cycles for certain "quantized" algebraic variety.
Junji Suzuki (Shizuoka)
Functional relations in Stokes multipliers (tentative)
I consider a class of linear differential equations possessing an irregular singular point at infinity. Particularly, I will discuss functional relations among Stokes multipliers.
Craig Tracy (UC, Davis)
Applications of random matrix theory to combinatorics and growth processes
In this talk we review recent applications of random matrix theory to combinatorical problems of the Robinson-Schensted-Knuth type. The appearance of RMT distributions in growth processes will be briefly described at the end of the talk.
Miki Wadati (Tokyo)
Symmetric and non-symmetric bases of quantum integrable particle systems with long-range interactions
We review our recent work on quantum integrable particle systems with long-range interactions. In particular, we discuss the symmetric orthogonal basis of the Calogero model and the non-symmetric orthogonal basis of the corresponding Calogero model with distinguishable particles. The Rodrigues formulas for the polynomial part of them, i.e., Hi-Jack and the nonsymmetric multivariable Hermite polynomials, are constructed. Some related topics are also discussed. (in collaboration with H. Ujino, Y. Komori and A. Nishino)
S. Ole Warnaar (Amsterdam)
Bailey's lemma and Kostka polynomials
In the theory of symmetric functions an important role is played by the Kostka polynomials. Seemingly unrelated to these polynomials is Bailey's lemma, which is a highly efficient lemma for proving q-series identities. In this talk I will show how Kostka polynomials may be used to find extensions of the Bailey lemma.
Paul Wiegmann (Chicago)
Arithmetics and Cantor spectra of integrable models: Hofstadter problem
I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571 ,1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation is known to be a pure singular continuum with a rich hierarchical structure. Few years ago it has been found that the almost Mathieu operator is integrable. An asymptotic solution of this operator became possible due analysis the Bethe Ansatz equations.
Fred Wu (Boston)
Dimer statistics and spanning trees
The problem of enumerating spanning trees and dimers on lattices and nonorientable surfaces is considered. Closed-form expressions are obtained for the generating functions for finite as well as infinite lattices in two and higher dimensions. Results are also obtained for a simple quartic net embedded on the Moebius strip and the Klein bottle. Generalizing an observation of Temperley, it is also shown that there exists a bijection between spanning tree configurations on any planar graph G and dimer configurations on a related planar graph.
Fred Wu (Boston)
Density of Fisher zeroes of the Ising model
Closed-form expressions are obtained for the density of the Fisher zeroes, or zeroes of the partition function in the complex temperature plane, for the Ising model in zero field as well as in a pure imaginary field $i\pi/2$. Results are given for the simple-quartic, triangular, honeycomb, and the kagome lattices. It is found that the density diverges logarithmically at points on its loci.
Ruihong Yue (Xian)
Elliptic dynamical Gaudin model and their Bethe Ansatz
We propose a new method of constructing elliptic dynamical Gaudin models related to an arbitrary simple Lie algebra and diagonalizing their Hamiltonian in the frame of the Wakimoto modules over affine algebras at critical level. The eigen-functions of the hamiltonian are obtained by restricting certain invariant correlation functions on tensor product of Wakimoto modules, which is a direct generalization of Feigin, Frenkel and Reshetikhin's approach from complex plane to elliptic curves. The Bethe ansatz equations (for $sl_n$ case) are determined by reqirement of the existance of singular vectors in Wakimoto modules.
Alexey Zamolodchikov (Montpellier)
Yang-Baxter equations in integrable relativistic field theory
I review the applications of Yang-Baxter relations to find out explicitely the S-matrices of 2D integrable field theories. The sin-Gordon model and the O(N) sigma model are taken as examples. Also I comment on some applications of these exact scattering theories to recover different off-mass-shell characteristics, in particular the vacuum energy.