On Conformal Field Theory and Stochastic Loewner Evolution

Abstract

We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with boundary using Conformal Field Theory methods. We propose in particular a CFT construction for a probability measure on (clouded) paths, and check it against known restriction properties. The probability measure can be thought of as a section of the determinant bundle over moduli spaces of Riemann surfaces. Loewner evolutions have a natural description in terms of random walk in the moduli space, and the stochastic diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle.

24 pages, 4 figures, LaTeX; v2: added section 4.1, references and minor clarifications, version to appear in NPB

References in corpus (4)

• The dimension of the SLE curves
• Conformal Field Theories of Stochastic Loewner Evolutions
• SLE_k growth processes and conformal field theories
• SLE martingales and the Virasoro algebra

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• Towards a conformal field theory for Schramm-Loewner evolutions
• Quantum Hall transitions: An exact theory based on conformal restriction
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• Logarithmic Conformal Null Vectors and SLE
• Large deviations of multichordal SLE${}_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians
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• Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties
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• The Global Geometry of Stochastic Lœwner Evolutions
• Schramm-Loewner evolution with Lie superalgebra symmetry
• Local martingales associated with Schramm-Loewner evolutions with internal symmetry
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• The Correlator Toolbox, Metrics and Moduli
• SLE($\kappa ,\rho$)and Boundary Coulomb Gas
• From the Conformal Anomaly to the Virasoro Algebra
• Note on Schramm-Loewner evolution for superconformal algebras
• Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian
• A Renormalisation Group approach to Stochastic Lœwner Evolutions and the Doob h-transform
• Coset construction of Virasoro minimal models and coupling of Wess-Zumino-Witten theory with Schramm-Loewner evolution
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