Annecy
A construction of universal dynamical R-matrices is presented. It is based on solutions F of a shifted cocycle relation. F provides a twist from the usual universal R-matrix to a dynamical one, solution of the Gervais--Neveu--Felder equation.
Saclay
We shall discuss the so-called "passive scalar problem" which consists in describing the statistical properties of a scalar field (eg. the temperature) advected by turbulent flows. We shall describe the origins of anomalous scalings present in this problem and relate them to a few properties of Lagrangian trajectories in turbulent flows.
Saclay
We study the extension of matrix string theory from a flat background to a curved background. After recalling the ingredients of matrix string theory in a flat background we shall present the one loop calculation of the effective action in a curved background. We shall compare our results with the strings equations of motion. Some comments on D0 branes will be given.
Potsdam
The investigation of lower-dimensional gravity is presently receiving a lot of attention because of its connection with string theory and black hole physics. In this context many papers have been devoted to the discussion of two-dimensional models of dilaton gravity coupled to various matter fields [1]. Two-dimensional pure dilaton gravity is completely integrable for any choice of the dilatonic potential. The model can be quantized exactly, thanks to its integrability properties, and the resulting quantum theory exhibits at the quantum level the Birkhoff theorem [2].
ENS Lyon
We summarize the CFT description of the Luttinger liquid theory a la Haldane and show how this can be used to provide a simple description of edge excitations of 1/(2p+1) quantum Hall sequences. Luttinger CFTs corresponding to the nu =1/(2p+1) filling fractions appear to be rational CFTs (RCFT). Edge excitations above a given bulk excitation are described by a twisted version of the Luttinger effective theory. Generators of the extended symmetry algebra are identified as edge fermions creators and annihilators, thus giving a physical meaning to the RCFT point of view on edge excitations of these sequences.
Montpellier
The determination of universal quantities in statistical mechanics from integrable field theory is illustrated through the example of the q-state Potts model and percolation.
King's London
We explain how to obtain the classical soliton solutions of affine Toda theory on a half line by a simple method of mirror images. We notice that the integrable boundary conditions found by the Durham group impose that a soliton is converted into an antisoliton when it reflects off the boundary. This suggests the existence of new integrable boundary conditions which are soliton preserving. We present these and show how integrability can easily be demonstrated by generalizing Slyanin's formalism. We explore the implications which these classical considerations have for the study of quantum reflection matrices.
Durham
Recent microscopic derivations of black hole radiation based on string theory have essentially relied on Maldacena's conjecture of an AdS/CFT correspondence. In the simpler case of pure 2+1 gravity (with negative cosmological constant), this correspondence is known to be a well established fact. By working in this framework, we are able to quantize gravity in the presence of an external scalar field. We find that the coupling between the scalar field and gravity is equivalently described by a perturbed conformal field theory at the boundary of AdS_3. This allows us to perform a microscopic computation of the transition rates between black hole states due to interaction with the scalar field. Thus, Hawking radiation is derived from a microscopic theory, and we find agreement with the semiclassical result, including greybody factors. This result also has application to four and five dimensional black holes in supergravity.
Cambridge
The classical bosonic principal chiral model in 1+1 dimensions contains infinitely many local conserved quantities which correspond to symmetric invariant tensors of the underlying group $G$. These local charges always commute with the non-local (Yangian) charges, but their algebra amongst themselves is more subtle. One infinite commuting subset can be found for any $G$ by taking powers of the energy-momentum tensor; another commuting subset can be found by taking traces of products in the defining representation for $G=U(N), SO(N), Sp(N)$ (but not for $SU(N)$). A formula is given interpolating these two subsets when $G$ is either orthogonal or symplectic and it is conjectured that the local charges so defined will commute for any fixed value of the interpolating parameter $\alpha$. For $G=SO(2n)$ a special value $\alpha = 1/n$ is picked out by requiring in addition commutation with the `Pfaffian' charge. Analogous results hold for the supersymmetric version of the chiral model but there are also novel features---in particular an `extra' finite set of commuting conserved quantities associated with antisymmetric invariants of $G$ (including $SU(N)$). Survival of some or all of these local charges at the quantum level has interesting implications for the $S$-matrix structures in both the bosonic and supersymmetric models.
SISSA
Some basic issues of D-brane dynamics will be discussed, using as a concrete example the D-particle or D0-brane. These issues include the probing of space-time at a very small scale and a possible non perturbative formulation of M-theory known as Matrix theory.
Durham
This talk is the report of a still-in-progress joint project with Marian Stanishkov. We propose an alternative description of the spectrum of local fields in the classical limit of the integrable quantum field theories. Our approach is based on symmetries and provides a systematic way of deriving the null-vectors that appear in this construction. We present explicit results for the case of the $A_1^{(1)}$-(m)KdV and the $A_2^{(2)}$-(m)KdV hierarchies, different classical limits of perturbed 2D CFT's. The construction of a new Virasoro-like symmetry algebra (completly different from the conformal one) will be briefly discussed. Some hints about quantization and off-critical treatment are also given specifically in relation to the deformation of the Virasoro-like symmetry after these procedures.
King's London
We review our solution of Seiberg-Witten models with gauge groups A_n or D_n expressing the scalar modes in terms of conformal blocks of the rational logarithmic conformal field theory with c=-2. We discuss first steps in generalizing this approach towards situations where the moduli space is a non-hyperelliptic curve, e.g. in the case of product groups.
ENS Paris
It has been known for some time that 10 D super Yang-Mills equations are equivalent to flatness conditions on superhyperplanes. Although they are thus integrable in a weak sense, no explicit solutions are known yet, since the Lax pair gives complicated non linear $\sigma$-model type consistency conditions. The basic novel point is to observe that there exists an on-shell gauge where half of the fields vanish, such that these complicated conditions become linear. Then they reduce to (super)selfduality conditions in eight variables whose general solution is obtained in closed form. The remaining (non linear) equations reduce to Yang type equations in eight (super) variables which seem to be tractable.
Madrid
Discussion from the point of view of holography of string representation of Wilson loops and zig-zag invariance, and also on the holographic interpretation and proof of the c-theorem.
SISSA
One-dimensional quantum spin systems exhibit plateaux in their magnetization curves when subjected to external magnetic fields. Such plateaux generalize spin-gaps in zero field: For spin-S Heisenberg chains a famous conjecture by Haldane predicts a spin-gap for integer S = 1, 2, ... while none otherwise (S=1/2, 3/2, ...). This talk summarizes results on magnetization plateaux with particular emphasis on N coupled Heisenberg chains, so-called `spin ladders'. The appearance of plateaux in magnetization curves can be studied using numerical methods, a strong-coupling approach and Abelian bosonization. A first important issue is the quantization condition on the magnetization to permit the appearance of a plateau, a question which in zero magnetic field is addressed by Haldane's conjecture. A second item are the universality classes associated to the transitions at the boundaries of magnetization plateaux.
Madrid
We will discuss the world volume actions of the Type IIB five-brane and the Kaluza-Klein monopole solutions. This will be done, making use of the T-duality between Type IIA and Type IIB supergravity and between the five-brane and monopole solutions. Together they form a closed web of dualities that enables us to write down the Type IIB actions.
Potsdam
Topological charges of single solitons in affine Toda theory are expected to fill out the weights of the fundamental representations, but the known classical solutions do not do so. It shall be described how to construct new single soliton solutions with new charges which fill some of the missing weights, but unfortunately not all. The inverse scattering method shall be used for this, but for those who are not au fait with the method, it shall also be described how to get the solutions in terms of restrictions of multi-soliton versions of the soliton solutions already known for these theories.
Durham
We calculate the exact N-dependence of the effective 16-fermion vertex at the 1-instanton level, and find precise agreement in the large-N limit with the prediction of the type IIB superstring on AdS_5 x S^5. This suggests that the string theory prediction for the 1-instanton amplitude considered here is not corrected by higher-order terms in the alpha' expansion.
Santiago
Recently, a new method for generalizing the zero-curvature method for two-dimensional integrable systems to higher dimensions has been proposed, as described in the talk by J. Sanchez Guillen. I discuss the application of this method in two important examples, that of the principal chiral model, and that of the CP(1) model, both in 2+1 dimensions. In the two cases I define an integrable sub-model (a restricted model with infinitely many conserved currents), and I show the explicit calculation of the conserved currents.
ENS Lyon
Representation theory of Drinfel'd twist for quantum affine sl2 algebra is used to compute scalar products of Bethe states (leading to Gaudin formula) and to solve the Quantum Inverse Problem for the XXZ Heisenberg spin-1/2 finite chain. Form factors of local spin operators are then calculated in terms of determinants of usual functions of the parameters of the model. The spontaneous magnetization is computed in the thermodymamic limit (Baxter formula) and finite size corrections are evaluated.
Santiago
We summarize recent work aimed to the construction of new integrable quantum field theories whose equations of motion are related to the non-abelian version of the affine Toda equations. The resulting theories, referred to as the Homogeneous sine-Gordon (HSG) and Symmetric Space sine-Gordon (SSSG) theories, admit a natural description as integrable perturbations of CFT's associated with cosets of the form $G_k/U(1)^{\times p}$, where $G$ is a semi-simple Lie group and $p$ is an integer number $\geq0$ and $\leq {\rm rank\/}(G)$. Among their main features, they are not parity invariant and their spectrum exhibits unstable particles that manifest as resonance poles in the S-matrix.
ICTP
We analyze the boundary conditions that preserve the quantum integrability of the nonlinear sigma model and of the Gross-Neveu model on the half-line. We show that for the nonlinear sigma model Neumann, Dirichlet and a mixed boundary condition are quantum integrable, and that for the Gross-Neveu model the condition \psi_\pm^a = \pm \psi_\mp^a is quantum integrable. We also comment on the boundary condition found by Corrigan and Sheng for the $O(3)$ nonlinear sigma model.
SISSA
The lowest representatives of the Form Factors relative to the trace operators of $N=1$ Super Sinh-Gordon Model are exactly calculated. The novelty of their determination consists in solving a coupled set of unitarity and crossing equations. Analytic continuations of the Form Factors as functions of the coupling constant allow the study of interesting models in a uniform way, among these the latest model of the Roaming Series and the minimal supersymmetric models as investigated by Schoutens. A fermionic version of the $c$-theorem is also proved and the corresponding sum-rule derived.
SISSA
The Matrix String Theory, i.e. the two dimensional U(N) SYM with N=(8,8) supersymmetry, has classical BPS instantons built of Affine Toda solitons. The instantons are described by a bordered Riemann surface interpolating between different string configurations. Matrix String Theory amplitudes around such a classical BPS background, in the strong Yang-Mills coupling, are thus candidates to represent the perturbative string amplitudes. I will describe recent work showing that the leading contribution is proportional to the right factor g_s^{-\chi}, where \chi is the Euler characteristic of the interpolating Riemann surface and g_s is the string coupling.
Potsdam
The canonical formulation of d=2,N=16 supergravity is presented. We work out the supersymmetry generators (including all higher order spinor terms) and the N=16 superconformal constraint algebra. We then describe the construction of the conserved non-local charges associated with the affine E9 symmetry of the classical equations of motion. These charges are shown to commute weakly with the supersymmetry constraints, and hence with all other constraints. Under commutation, they close into a quadratic algebra of Yangian type, which is formally the same as that of the bosonic theory. The Lie-Poisson action of E9 on the classical solutions is exhibited explicitly. Further implications of our results are discussed.
Potsdam
A new class of integrable systems is obtained by deforming quantum field theories with a factorized scattering operator in a way hat preserves the bootstrap S-matrix. The deformation parameter \beta plays the role of an inverse temperature for the thermal equilibrium states associated with the Rindler wedge \beta =2\pi being the AQFT value. The form factor approach unambiguously defines these novel systems and provides an explicit computational scheme just as for \beta =2 \pi. For \beta \neq 2 \pi the kinematical arena is deformed but is by construction compatible with the full non-perturbative dynamics of the systems.
Swansea
The mathematical structure of electromagnetic duality still has to be found but it certainly includes the theory of modular functions, that is, functions transforming simply under the action of the modular group and its generalisations. The partition function of free Maxwell theory on a closed, connected four-manifold $M_4$ with otherwise arbitrary topology provides a non trivial example. This can be expressed as a generalised sort of theta functions associated with the lattice of allowed $U(1)$ magnetic charges. Properties of such theta functions are given, including the special features that arise when $M_4$ does not admit uncharged Dirac spinors.
Annecy
I present the results published in the paper hep-th/9803243, where we construct an algebra homomorphism between the Yangian Y(sl(n)) and the finite W-algebras W(sl(np),n.sl(p)) for any p. We show how this result can be applied to determine properties of the finite dimensional representations of such W-algebras.
Potsdam
Dimensional reduction of various gravity models leads to effectively two-dimensional field theories described by gravity coupled nonlinear G/H coset space sigma-models. They are integrable in the sense of an underlying linear system. The coordinate dependence of the associated spectral parameter however gives rise to several new features. The classical infinite-dimensional symmetry group (the Geroch group) is generated by the Lie-Poisson action of a complete set of nonlocal conserved charges. Canonical quantization of this structure leads to a twisted Yangian double with fixed central extension at a critical level.
Santiago
The zero curvature methods for two dimensional integrable models are generalized to d+1 dimensions. The new local flatness conditions are used to represent the equations of motion of some well known relativistic theories in 2+1 and 3+1 dimensions like BF, Chern-Simons and self-dual Yang-Mills and to discover new integrable models with infinite conserved currents, which are easily obtained. The construction is done both in spacetime and in the loopspace.
Amsterdam
In recent years there have been a number of successful checks of S-duality in N=4 supersymmetric Yang-Mills theory in the case when the gauge symmetry is broken to an abelian group. When the unbroken group is non-abelian however, the implementation of S-duality is much more intricate. An important prerequisite is the classification of the electric, magnetic and dyonic charge sectors of the theory. In my talk I will explain how this classification can be achieved, and propose a formulation of S-duality in the non-abelian setting.
ENS Paris
When a Lagrangian theory is invariant under a local (gauge) symmetry, the usual Noether construction for the conserved charges has to be completed. The basic object associated to this gauge symmetry is a (D-2)-form (instead of the usual (D-1)-form Noether current) to be integrated at spatial infinity. This (D-2)-form is called a superpotential and can be computed locally as a functional of the fields in a straightforward way. The examples of gravity (in the new GL(D,R) first order formalism), Yang-Mills and Chern-Simons theories will be presented.
Annecy
A two-loop calculation in N=4 super Yang-Mills is proposed which can serve as a test of Maldacena's conjecture. The theory is formulated in terms of N=2 harmonic superspace, where manifestly supersymmetric Feynman rules exist. Then we compute the correlator of four gauge invariant composite operators made out of hypermultiplets. The result is given by scalar two-loop integrals having logarithmic short distance singularities. The hypermultiplet correlator is related by $SU(4)$ and conformal supersymmetry to the correlator $Tr F^2 Tr F \tilde{F} Tr F^2 Tr F \tilde{F}$. The AdS supergravity counterpart of the latter has recently been computed and its asymptotic behaviour is in agreement with our results.
Bologna
We analyse the extension of the Destri-de Vega nonlinear integral equation (NLIE) to describe excited state scaling functions in sine-Gordon (sG)/massive Thirring (mTh) theory. We discuss the results obtained in the UV and IR regimes, and perform numerical studies in the interpolating range of scales, to compare to data coming from an extension of the Truncated Conformal Space (TCS) method to perturbed c=1 Conformal Field Theory (CFT). Both the numerical and analytic results strongly support the validity of the NLIE as a description of excited states of sG/mTh theory in finite spatial volume. We present the extension of the NLIE to states with odd value of the topological charge and discuss the relation between Bethe quantisation rules, locality and boundary conditions on the cylinder.
Saclay
This talk is based on results from a still-in-progress project in collaboration with Patrick Dorey (Durham), Paolo Provero (Turin) and Stefano Vinti (M\"unster). The phase diagram of the Z_6 spin model in two dimensions is partially reconsidered in view of new exact results from perturbed conformal field theory. Recently-obtained Montecarlo data give support to the conjectured scenario.
Swansea
After a brief review of the duality conjectured by Maldacena between p+1 dimensional conformal field theories and superstring (M) theory on AdS_{p+2}xS_{D-p-2} (D=10(11)), some recent achievements in the construction of a superconformal invariant action on the world volume of a p-brane in AdS_{p+2}xS_{D-p-2} will be discussed.
Tokyo
The sine-Gordon model defined on the half-line $[0, \infty)$ with the boundary potential $V(\varphi(0)) = -M \cos \beta/2 (\varphi(0) - \varphi_0)$ has been shown to be integrable by Sklyanin and Ghoshal-Zamolodchikov. It is well-known that the sine-Gordon theory on a line can be obtained by the continuum limit of the XYZ spin-1/2 chain. The general form of the integrable boundary interaction for the open XYZ chain has been derived by Inami and Konno by constructing the reflection matrix explicitly. Hence, it is expected that the continuum limit of the integrable open XYZ chain gives rise to the boundary sine-Gordon theory . We show this connection using the perturbation theory of the XYZ chain at the $U(1)$ symmetric point and the bosonization method.
Bonn
In the solution of the superintegrable chiral Potts model (which is integrable both due to the Onsager algebra and from Yang-Baxter relations),special $Z_N$-symmetrical polynomials introduced by Albertini, McCoy, Perk and Baxter play a central role. As B.Davies has shown, these polynomials appear in the decomposition of finite-dimensional representations of the Onsager algebra into $sl(2,C)$-representations. The energy levels of the model are obtained in terms of the zeros of these polynomials. All zeros appear on the negative real axis. We give simple analytic expressions which determine all zeros, except for the largest and smallest ones, to exponential precision in the chain size $L$. The leading finite size dependence of the energy eigenvalues is shown to be determined just by the extreme zeros. We derive a special formula for these zeros. The finite size corrections to the known thermodynamic limit expressions for the lowest eigenvalues follow from Euler-McLaurin techniques.
King's London
We consider the perturbation of boundary conditions of integrable field theories in the UV and IR limits, using conformal field theory and TBA techniques respectively. We find the exact relation of the parametrisations of the perturbations in the two regimes, and find evidence for an intriguing new form of boundary conditions.
King's London
After a review of branes and their dynamics we consider the possible intersection of the fivebranes. We show how preserving supersymmetry leads to special spacetime geometries. Finally we derive low energy effective actions from the the intersecting brane configurations.
Heriot-Watt
We consider the analogue of the 6-vertex model constructed from alternating spin $n/2$ and spin $m/2$ lines, where $1\leq n \leq m-1$. We identify the transfer matrix and the space on which it acts in terms of the representation theory of $U_q(\widehat{sl}_2)$. We diagonalise the transfer matrix and compute the S-matrix.
Montpellier
A relation between the familiy of commuting continuous transfer-matrices and the thermodynamic Bethe ansatz (TBA) equations was established recently in the case of (massless) sin-Gordon type structures [V.Bazhanov, S.Lukyanov and A.Zamolodchikov, hep-th/9412229, hep0th/9604044, hep-th/9805008]. In the talk this problem is reconsidered for the sinh-Gordon type situation which appears to be quite different from the previous relation. A formal construction of the Q-operators and there relation to the TBA equation is also discussed.