Aims of this article

Date: 2015-10-07; view: 483.

The original papers on SLE are mostly both long and difficult, using, moreover, concepts and methods foreign to most theoretical physicists. There are reviews, in particular those by Werner [12] and by Lawler [13] which cover much of the important material in the original papers. These are however written for mathematicians. A more recent review by Kager and Nienhuis [14] describes some of the mathematics in those papers in way more accessible to theoretical physicists, and should be essential reading for any reader who wants then to tackle the mathematical literature. A complete bibliography up to 2003 appears in [15].

However, the aims of the present article are more modest. First, it does not claim to be a thorough review, but rather a semi-pedagogical introduction. In fact some of the material, presenting some of the existing results from a slightly different, and hopefully clearer, point of view, has not appeared before in print. The article is directed at the theoretical physicist familiar with the basic concepts of quantum field theory and critical behaviour at the level of a standard graduate textbook, and with a theoretical physicist's knowledge of conformal mappings and stochastic processes. It is not the purpose to prove anything, but rather to describe the concepts and methods of SLE, to relate them to other ideas in theoretical physics, in particular CFT, and to illustrate them with a few simple computations, which, however, will be presented in a thoroughly non-rigorous manner. Thus, this review is most definitely not for mathematicians interested in learning about SLE, who will no doubt cringe at the lack of preciseness in some of the arguments and perhaps be puzzled by the particular choice of material. The notation used will be that of theoretical physics, for example for expectation value, and so will the terminology. The word ‘martingale' has just made its only appearance. Perhaps the largest omission is any account of the central arguments of LSW [6] which relate SLE to various aspects of Brownian motion and thus allow for the direct computation of many critical exponents. These methods are in fact related to two-dimensional quantum gravity, whose role in this is already the subject of a recent long article by Duplantier [16].