SLE for theoretical physicists

This article provides an introduction to Schramm(stochastic)-Loewner evolution (SLE) and to its connection with conformal field theory, from the point of view of its application to two-dimensional critical behaviour. The emphasis is on the conceptual ideas rather than rigorous proofs.Comment: 43 pages, to appear in Annals of Physics; v.2: published version with minor correction

Non periodic Ishibashi states: the su(2) and su(3) affine theories

We consider the su(2) and su(3) affine theories on a cylinder, from the point of view of their discrete internal symmetries. To this end, we adapt the usual treatment of boundary conditions leading to the Cardy equation to take the symmetry group into account. In this context, the role of the Ishibashi states from all (non periodic) bulk sectors is emphasized. This formalism is then applied to the su(2) and su(3) models, for which we determine the action of the symmetry group on the boundary conditions, and we compute the twisted partition functions. Most if not all data relevant to the symmetry properties of a specific model are hidden in the graphs associated with its partition function, and their subgraphs. A synoptic table is provided that summarizes the many connections between the graphs and the symmetry data that are to be expected in general.Comment: 19 pages, 3 figure

The TT‾T\overline T deformation of quantum field theory as random geometry

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant $\det T$ of the stress tensor, commonly referred to as $T\overline T$. Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk and a more general simply connected domain. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space. We also discuss possible generalizations to higher dimensions.Comment: 32 pages. Final published version! Solution for t>0 clarifie

Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications

We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n->0 of the O(n) model, and percolation as the limit Q->1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c->0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O(-2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter.Comment: 37 pages. v2: minor corrections and additions. Submitted to Special Issue of J. Phys. A on Logarithmic CF

Quantum Network Models and Classical Localization Problems

A review is given of quantum network models in class C which, on a suitable 2d lattice, describe the spin quantum Hall plateau transition. On a general class of graphs, however, many observables of such models can be mapped to those of a classical walk in a random environment, thus relating questions of quantum and classical localization. In many cases it is possible to make rigorous statements about the latter through the relation to associated percolation problems, in both two and three dimensions.Comment: 23 pages. To appear in '50 years of Anderson Localization', E Abrahams, ed. (World Scientific)

Linking numbers for self-avoiding walks and percolation: application to the spin quantum Hall transition

Non-local twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which, in the limits n -> 0 (resp. Q -> 1) count the numbers of self-avoiding loops (resp. percolation clusters) surrounding a given point. This yields many results, for example the distribution of the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. These twist operators correspond to (1,2) in the Kac classification of conformal field theory, so that their higher-point correlations, which describe linking numbers around multiple points, may be computed exactly. As an application we compute the exact value \sqrt 3/2 for the dimensionless conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.Comment: 4 pages, 3 figures; final version as will appear in PR

Crossing Formulae for Critical Percolation in an Annulus

An exact formula is given for the probability that there exists a spanning cluster between opposite boundaries of an annulus, in the scaling limit of critical percolation. The entire distribution function for the number of distinct spanning clusters is also given. These results are found using Coulomb gas methods. Their forms are compared with the expectations of conformal field theory.Comment: 7 pages, 1 figure; v.2,3: minor corrections; v.4: published versio

The Legacy of Ken Wilson

This is a brief account of the legacy of Ken Wilson in statistical physics, high energy physics, computing and education.Comment: Written version of a talk given at the Ken Wilson Memorial Session, StatPhys 25, Seoul, July 2013. To appear in the conference proceedings in J. Stat. Mec