On Connections of Conformal Field Theory and Stochastic Lœwner 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 $\frac{\kappa }{2}{L}_{-1}^2-2L_{-2}$ 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.

Added references for sections 4&5, updated 2 references. 9 figures, text will be modified / expanded

References in corpus (6)

• The dimension of the SLE curves
• Conformal Field Theories of Stochastic Loewner Evolutions
• SLE martingales and the Virasoro algebra
• Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk
• On Conformal Field Theory and Stochastic Loewner Evolution
• The Brownian loop soup

Cited by in corpus (10)

• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• Conformal Random Geometry
• SLE and Virasoro representations: Fusion
• Random loops and conformal field theory
• The Global Geometry of Stochastic Lœwner Evolutions
• Local martingales associated with Schramm-Loewner evolutions with internal symmetry
• Schramm--Loewner-evolution-type growth processes corresponding to Wess--Zumino--Witten theories
• A Renormalisation Group approach to Stochastic Lœwner Evolutions and the Doob h-transform
• CFT Interpretation of Merging Multiple SLE Traces
• SLE and Virasoro representations: localization