arXiv math-ph/04100292004-10math-phmath.MP

论共形场论与随机LÜwner演化的联系

On Connections of Conformal Field Theory and Stochastic Lœwner Evolution

Roland Friedrich

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

摘要：

这篇手稿探讨了一类称为“随机Loewner演化”（SLE）和共形场论（CFT）的随机过程之间的联系。首先回顾了一些重要的结果，我们利用的续集，特别是概念的共形限制和“restrcition鞅”，最初介绍了共形限制（G. F。Lawler等人）。然后给出了SLE与维拉索罗代数的表示论之间的一个联系的显式构造。特别地，我们解释的沃德恒等式的限制性质和中心荷的布朗气泡的密度。然后，我们表明，这种解释允许有关的$κ$的随机过程的中心荷$c$的共形场论。这是通过在维拉索罗代数的第二级退化的最高权重表示来实现的。然后，我们从理论物理学的观点出发，推导出同样的关系式。特别是，我们探讨SLE和几何的基础模空间之间的关系。最后，我们概述了一个一般的建设，它允许构建任意黎曼曲面上的随机曲线。其关键是考虑正则算子$\fracκ{2}L^2_{-1}-2L_{-2}$与边界场的结合，边界场是退化的最高权场$ψ$，作为适当模空间上扩散的生成元。

英文摘要：

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 $κ$ 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 $\fracκ{2}L^2_{-1}-2L_{-2}$ in conjunction with a boundary field that is a degenerate highest-weight field $ψ$ as the generator of a diffusion on an appropriate moduli space.

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