Yvan Saint-Aubin

Conformal invariance in two-dimensional percolation

The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the seventeenth century. It was introduced more recently into mathematics by S. R. Broadbent and J. M. Hammersley ([BH]) and is a branch of probability theory that is especially close to statistical mechanics. Broadbent and Hammersley distinguish between two types of spreading of a fluid through a medium, or between two aspects of the probabilistic models of such processes: diffusion processes, in which the random mechanism is ascribed to the fluid; and percolation processes, in which it is ascribed to the medium. A percolation process typically depends on one or more probabilistic parameters. For example, if molecules of a gas are absorbed at the surface of a porous solid (as in a gas mask) then their ability to penetrate the solid depends on the sizes of the pores in it and their positions, both conceived to be distributed in some random manner. A simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner (that could be determined by a periodic graph), and to be open (thus very large) or closed (thus smaller than the molecules) with probabilities p and 1 − p. As p increases the probability of deeper penetration of the gas into the interior of the solid grows.

Degenerate conformal field theories and explicit expressions for some null vectors

Abstract Degenerate conformal field theories are characterized by the finite number of primary fields involved in their description and by the existence of null operators that lead to differential equations for the correlation functions. These null operators are related to null vectors in the unitary representations of the discrete series of the Virasoro algebra. Each such representation possesses two leading null vectors. In the representation with the highest weight h p,q =[((m+1)p−mq) 2 −1]/4m(m+1) , these null vectors are labeled ψp,q and ψm−p,m+1−q. We give the explicit expression for the null vectors ψp,1 and ψ1,m+1−q, i.e. for q=1 or p=m−1 respectively and show how they can be used to construct correlation functions of degenerate theories.

Bäcklund transformations and soliton-type solutions for σ models with values in real Grassmannian spaces

A Bäcklund transformation for the two-dimensional σ model with values in oriented real Grassmannian spaces is constructed using the known Bäcklund transformation for the SO(n) principal models. The construction provides a natural way to linearize the differential system of the Bäcklund transformation. In the case of the Sn≈SO(n+1/SOn) model, the Backlund transformation reduces to that of Pohlmeyer. The one-soliton solution for S2∼SO(3)/SO(2) is obtained analytically and plots of one-soliton and two-soliton solutions are displayed.

Algebro-geometric aspects of the Bethe equations

Although the Ansatz introduced by Bethe in 1931 ([B]) has been exploited repeatedly by physicists, who have adapted it successfully to a variety of problems, it has never been given a careful mathematical treatment. As a result there is often a disquieting imprecision in its formulation that discourages a resolute pursuit of its analytical consequences; moreover, and more to the point here, its algebraic charm has been little appreciated. Two years ago, the present authors undertook a study of the equations with standard techniques from algebraic geometry. The enterprise, rewarding as it has been, has taken more time and energy than expected. Complete proofs, even adequate understanding, have cost a great deal of effort and patience, and there are still gaps, but the project is nearing completion, and in this paper we describe, albeit in a somewhat provisional form, the principal features of the treatment. Details will appear in [BL]. There is no need here to recall the physical origins of the eigenvalue problem treated by Bethe. The mathematical problem is that of finding the eigenvalues and eigenvectors of an operator on a space of dimension 2 N . This space is

Infinite Dimensional Lie Algebras Acting on the Solution Space of Various

Infinite‐dimensional Lie algebras of infinitesimal transformations acting on the solution space of various two‐dimensional σ models are investigated. The main tools are (i) Takasaki’s interpretation [Commun. Math. Phys. 94, 35 (1984)] of the solutions of the associated linear system in terms of points in an infinite‐dimensional Grassmann manifold and (ii) Mikhailov’s reduction procedure [Physica D 3, 73 (1981)] for linear systems. Takasaki’s approach leads, for the σ models with values in a Lie group G, to a set of transformations that has the structure of the loop algebra g ⊗R[t,t−1], where g is the Lie algebra of G. (This algebra has already been encountered by Dolan [Phys. Rev. Lett. 47, 1371 (1981)] and by Wu [Nucl. Phys. B 211, 160 (1983)] among others.) The σ models with a Wess–Zumino term are also considered; the algebraic structure is found to be the same. Finally, Mikhailov’s procedure is used to study the σ models with values in a Riemannian symmetric space (RSS) G/H which is not a Lie group. Th...

\sigma

Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, TN, is an element of the periodic Temperley?Lieb algebra , where N is the number of sites on a section of the cylinder, and ? = ?q ? q?1 = 2cos?? and ? the weights of contractible and non-contractible loops. The thermodynamic limit of TN is believed to describe a conformal field theory of central charge c = 1 ? 6?2/(?(? ? ?)). The abstract element TN acts naturally on (a sum of) spaces , similar to those upon which the standard modules of the (classical) Temperley?Lieb algebra act. These spaces known as sectors are labeled by the numbers of defects d and depend on a twist parameter v that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of TN both between sectors with distinct defect numbers and within a given sector.

Models

Bauer, di Francesco, Itzykson and Zuber recently proposed an algorithm to construct all singular vectors of the Virasoro algebra. It is based on the decoupling of (already known) singular fields in the fusion process. We show the extension of their algorithm to the Neveu-Schwarz superalgebra.

Jordan cells of periodic loop models

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of (the central charge) and of the highest weight hpq = [(p (m + 2) − qm)2 − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψp,q (of weight hpq + pq/2) and ψm−p, m+2−q (of weight hpq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.

FUSION AND THE NEVEU-SCHWARZ SINGULAR VECTORS

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley–Lieb algebra introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley–Lieb algebra. Besides the defining parameters β = u2 + u−2 with u = eiλ/2 (weight of contractible loops) and α (weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix TN(λ, ν) depends on the anisotropy ν and the spectral parameter λ that fixes the model. (The thermodynamic limit of TN is believed to describe conformal field theory of central charge c = 1 − 6λ2/(π(λ − π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v, and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner between these link representations and the eigenspaces of Sz of the XXZ models. This map is shown to be an isomorphism for generic values of u and v, and the critical curves in the plane of these parameters for which fails to be an isomorphism are given.

AN EXPLICIT FORMULA FOR SOME SINGULAR VECTORS OF THE NEVEU–SCHWARZ ALGEBRA

A Fortuin-Kasteleyn cluster on a torus is said to be of type {a,b},a,b in Z , if it is possible to draw a curve belonging to the cluster that winds a times around the first cycle of the torus as it winds -b times around the second. Even though the Q -Potts models make sense only for Q integers, they can be included into a family of models parametrized by beta = square root of Q for which the Fortuin-Kasteleyn clusters can be defined for any real beta(0,2] . For this family, we study the probability pi({a,b}) of a given type of clusters as a function of the torus modular parameter tau=tau(r)+itau(i). We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. For example, the behavior of pi({1,0}) is studied for tau(i) --> infinity . Exponents describing these behaviors are defined and related to weights h(r,s) of the extended Kac table for r and s integers, but also half-integers. Numerical simulations are also presented. Possible relationship with recent works and conformal loop ensembles is discussed.