Hubert Saleur

Common structures between finite systems and conformal field theories through quantum groups

Abstract We discuss in this paper algebraic structures that are common to finite integrable lattice systems and conformal field theories. The concept of quantum group plays a major role in our study, and a detailed theory of representations of U q [SU( n +1)] for q a root of unity is given. We obtain in particular a discrete analog of the Feigin-Fuchs construction, with corresponding concepts of null vectors or unitarity. The modular transformation S -matrix is also obtained from finite lattice considerations.

Quantum field theory for the multi-variable Alexander-Conway polynomial

Abstract We investigate in this paper the quantum field theory description of the multi-variable Alexander polynomial (Δ). We first study the WZW model on the GL(1, 1). It presents a number of interesting features including non-compactness, non-simplicity, 1/ k 2 quantum corrections, and logarithms in the current blocks. We compute the four-point functions, determine the scalar product of current blocks and make contact with the U q gl(1, 1) quantum group. We then discuss the gl(1,1) Chern-Simons model and recover the eight relations that uniquely determine Δ, as shown recently by Murakami. We discuss also how Δ can be computed by link surgery. The analysis runs into some difficulties due to the facts that Δ is zero in the braid group approach unless a strand is open, and that Δ vanishes for a split link. We propose some formal solutions to these. In appendices we give the full set of U q gl(1, 1) 6 j coefficients, we deal with details of the free field representations of gl(1, 1) (1) , and we explain how the Burau matrix can be computed via monodromy and contour representation of a link, establishing the connection between quantum field theory and a recent paper by Moody.

Discrete scale invariance, complex fractal dimensions, and log-periodic fluctuations in seismicity

We discuss in detail the concept of discrete scale invariance and show how it leads to complex critical exponents and hence to the log-periodic corrections to scaling exhibited by various measures of seismic activity close to a large earthquake singularity. Discrete scale invariance is first illustrated on a geometrical fractal, the Sierpinsky gasket, which is shown to be fully described by a complex fractal dimension whose imaginary part is a simple function (inverse of the logarithm) of the discrete scaling factor. Then, a set of simple physical systems (spins and percolation) on hierarchical lattices is analyzed to exemplify the origin of the different terms in the discrete renormalization group formalism introduced to tackle this problem. As a more specific example of rupture relevant for earthquakes, we propose a solution of the hierarchical time-dependent fiber bundle of Newman et al. [1994] which exhibits explicitly a discrete renormalization group from which log-periodic corrections follow. We end by pointing out that discrete scale invariance does not necessarily require an underlying geometrical hierarchical structure. A hierarchy may appear “spontaneously” from the physics and/or the dynamics in a Euclidean (nonhierarchical) heterogeneous system. We briefly discuss a simple dynamical model of such mechanism, in terms of a random walk (or diffusion) of the seismic energy in a random heterogeneous system.

Precursors, aftershocks, criticality and self-organized criticality

We present a simple model of earthquakes on a pre-existing hierarchical fault structure. The system self-organizes at large times in a stationary state with a power law Gutenberg-Richter distribution of earthquake sizes. The largest fault carries irregular great earthquakes preceded by precursors developing over long time scales and followed by aftershocks obeying an Omoris law. The cumulative energy released by precursors follows a time-to-failure power law with log-periodic structures, qualifying a large event as an effective dynamical (depinning) critical point. Down the hierarchy, smaller earthquakes exhibit the same phenomenology, albeit with increasing irregularities.

Boundary energy and boundary states in integrable quantum field theories

Abstract We study the ground-state energy of integrable 1 + 1 quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new “R-channel TBA”, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S -matrix of O ( n ) and minimal models.

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n+m|n)/[U(1)×U(n+m−1|n)]≅CPn+m−1|n (projective superspaces) and OSp(2n+m|2n)/OSp(2n+m−1|2n)≅S2n+m−1|2n (superspheres), n, m integers, −2⩽m⩽2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within Landau–Ginzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in non-interacting fermions with quenched disorder.

THE BLOB ALGEBRA AND THE PERIODIC TEMPERLEY-LIEB ALGEBRA

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the ‘blob’ algebra. We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. We complete the analysis using results from the study of the blob algebra.

Polymers and percolation in two dimensions and twisted N = 2 supersymmetry

We show how a large class of geometrical critical systems including dilute polymers, polymers at the theta point, percolation and to some extent brownian motion, are described by a twisted N=2 supersymmetric theory with k = 1 (it is broken in the dense polymer phase that is described simply by a η, ξ system). This allows us to give for the first time a consistent conformal field theory description of these problems. The fields that were described so far by formally allowing half integer labels in the Kac table are built and their four point functions studied. Geometrical operators are organized in a few representations of the twisted N = 2 algebra. A noticeable feature is that in addition to Neveu Schwartz and Ramond, a sector with quarter twists has sometimes to be introduced. The algebra of geometrical operators is determined. Fermions boundary conditions are geometrically interpreted, and the partition functions that were so far defined formally as generating functions for the critical exponents are naturally understood, sector by sector. Twisted N = 2 provides moreover a very unified description of all these geometrical models, explaining for instance why the exponents of polymers and percolation coincide. It must be stressed that the physical states in these geometrical problems are not the physical states for string theory, which are usually extracted by the BRS cohomology. In polymers for instance, QBRS is precisely the operator that creates polymers out of the vacuum, such that the topological sector is the sector without any polymers. It seems that twisted N = 2 is the correct continuum limit (in two dimensions) for models with Parisi Sourlas supersymmetry. Some possible explanation of this fact is advanced. As an example of application of N = 2 supersymmetry we discuss in the first appendix the still unsolved problem of backbone of percolation. We conjecture in particular the value D = 25/16 for the fractal dimension of the backbone, in good agreement with numerical computations. Finally some of these ideas are extended to the off critical case in the second appendix where it is shown how to give a meaning to the n → 0 limit of the O(n) model S matrix recently introduced by Zamolodchikov by introducing fermions.

Exact conductance through point contacts in the nu =1/3 fractional quantum Hall Effect.

The conductance for tunneling through an impurity in a Luttinger liquid is described by a universal scaling function. We compute this scaling function exactly, by using the thermodynamic Bethe ansatz and a kinetic (Boltzmann) equation. This model has been proposed to describe resonant tunneling through a point contact between two {nu}=1/3 quantum Hall edges. Recent experiments on quantum Hall devices agree well with our exact results. We also derive the exact conductance and {ital I}({ital V}) curve, out of equilibrium, in this fully interacting system.

Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models

Partition functions of critical 2D models on a torus can be derived from their microscopic formulation and their free field representation in the continuum limit. This is worked out explicitly for theO(n) andQ-state Potts model. Forn orQ integer we recover results obtained from conformal invariance, but our procedure also extends to nonintegral values. In the latter case the expansion on characters of the Virasoro algebra involves real coefficients of either sign. The operator content of both models is discussed in detail.