Denis Bernard

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA1(1) introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakovs 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.

Quantum group symmetries and non-local currents in 2D QFT

We construct and study the implications of some new non-local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions, including the sine-Gordon theory and its generalization to affine Toda theory. These non-local currents provide a non-perturbative formulation of the theories. The symmetry algebras correspond to the quantum affine Kac-Moody algebras. TheS-matrices are completely characterized by these symmetries. FormalS-matrices for the imaginary-coupling affine Toda theories are thereby derived. The application of theseS-matrices to perturbed coset conformal field theory is studied. Non-local charges generating the finite dimensional Quantum Group in the Liouville theory are briefly presented. The formalism based on non-local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d.

Fractional Supersymmetries in Perturbed Coset Cfts and Integrable Soliton Theory

Abstract We study integrable perturbations of the coset CFTs. The models are characterized by two fractional supersymmetries that are dual to each other. Generally, these models can be considered as restrictions of new integrable field theories we call fractional super soliton field theories. We study the connections with other models such as perturbations of WZW models, super sine-Gordon theory, Gross-Neveu models, and principal chiral models.

Yang-Baxter equation in long-range interacting systems

We consider the su(p) spin chains with long-range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities of the dynamical models and we diagonalise them. In the spin chain case, we establish the connection between the degeneracies of the spectrum and the representation theory of the Yangians. We use a correspondence with the dynamical models to diagonalise the Hamiltonian. Finally, we extend the previous results to the case of a trigonometric R-matrix.

On the Wess-Zumino-Witten models on the torus☆

Abstract We discuss the Ward identities of the Wess-Zumino-Witten models on Riemann surfaces and point out some ambiguities in the description of the zero modes of the currents. In the case of the torus, we show how to describe them and we write the Ward identities in such a way that they become complete. We examine in detail how the Ward identities are related to the Kubo-Martin-Schwinger condition. As an illustration of this formulation, we present a new proof of the Weyl-Kac character formula. The proof essentially relies on the mixed Virasoro × Kac-Moody Ward identities and explains the relation of the heat equation on the group manifold to the Weyl-Kac character formula.

Residual Quantum Symmetries of the Restricted {Sine-Gordon} Theories

Various aspects of the Restricted sine-Gordon (RSG) theories for the coupling β28π = pp+1 are examined. The particle spectrum is interpreted in terms of generalized kinks for the RSG potential, which is shown to effectively contain p−1 degenerate vacua. Simple arguments based on a study of the topological charge show that in the massless limit the RSG theories reduce to the Feigin-Fuchs construction. We find that the RSG S-matrices are characterized by a symmetry which generalizes supersymmetry to a fractional supersymmetry, and that the particles have fractional spin and statistics. In accordance with these results, we find some non-local chiral algebras in the minimal models that generalize the superconformal algebra, and whose currents are conserved in perturbation theory.

Hidden Yangians in

We define non-local conserved currents in massive current algebras in two dimensions. Our approach is algebraic and non-perturbative. The non-local currents give a quantum field realization of the Yangians. We show how the noncocommutativity of the Yangians is related to the non-locality of the currents. We discuss the implications of the existence of non-local conserved charges on theS-matrices.

2

Stochastic Loewner evolutions (SLEκ) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLEκ evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLEκ processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLEκ zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems for generalized crossing probabilities whose coefficients are multipoint CFT correlation functions. It provides a direct link between conformal correlation functions and probabilities of stopping time events in SLEκ evolutions. We point out a relation between SLEκ processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from SLEκ data.

D massive current algebras

Abstract This review provides an introduction to two dimensional growth processes. Although it covers a variety of processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm–Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramms argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the remarkable properties of SLE are explained, together with the tools for computing with it. This review has been written with the aim of filling the gap between the mathematical and the physical literature on the subject.

Conformal field theories of Stochastic Loewner evolutions

The mechanistic bases for gene essentiality and for cell mutational resistance have long been disputed. The recent availability of large protein interaction databases has fuelled the analysis of protein interaction networks and several authors have proposed that gene dispensability could be strongly related to some topological parameters of these networks. However, many results were based on protein interaction data whose biases were not taken into account. In this article, we show that the essentiality of a gene in yeast is poorly related to the number of interactants (or degree) of the corresponding protein and that the physiological consequences of gene deletions are unrelated to several other properties of proteins in the interaction networks, such as the average degrees of their nearest neighbours, their clustering coefficients or their relative distances. We also found that yeast protein interaction networks lack degree correlation, i.e. a propensity for their vertices to associate according to their degrees. Gene essentiality and more generally cell resistance against mutations thus seem largely unrelated to many parameters of protein network topology.