On Connections of Conformal Field Theory and Stochastic L{\oe}wner Evolution

Abstract

This manuscript explores the connections between a class of stochastic processes called "Stochastic Loewner Evolution" (SLE) and conformal field theory (CFT). First some important results are recalled which we utilise in the sequel, in particular the notion of conformal restriction and of the "restrcition martingale", originally introduced in Conformal restriction (G.F. Lawler, et al). Then an explicit construction of a link between SLE and the representation theory of the Virasoro algebra is given. In particular, we interpret the Ward identities in terms of the restriction property and the central charge in terms of the density of Brownian bubbles. We then show that this interpretation permits to relate the $\kappa$ of the stochastic process with the central charge $c$ of the conformal field theory. This is achieved by a highest-weight representation which is degenerate at level two, of the Virasoro algebra. Then we proceed by giving a derivation of the same relations, but from the theoretical physics point of view. In particular, we explore the relation between SLE and the geometry of the underlying moduli spaces. Finally we outline a general construction which allows to construct random curves on arbitrary Riemann surfaces. The key to this is to consider the canonical operator $\frac{\kappa}{2}L^2_{-1}-2L_{-2}$ in conjunction with a boundary field that is a degenerate highest-weight field $\psi$ as the generator of a diffusion on an appropriate moduli space.