SLE and the free field: Partition functions and couplings

Abstract

Schramm-Loewner Evolutions ($\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of $\SLE$ and the free field with appropriate boundary conditions; this involves $\zeta$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of $\SLE$ with the free field, showing that, in a precise sense, chordal $\SLE$ is the solution of a stochastic "differential" equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general $\kappa >0$.

55 pages, 4 figures. v2: additional material

References in corpus (6)

• Duality of Chordal SLE
• Boundary Partitions in Trees and Dimers
• Euler integrals for commuting SLEs
• The Full Scaling Limit of Two-Dimensional Critical Percolation
• Duality of Schramm-Loewner Evolutions
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Cited by in corpus (6)

• General beta Jacobi corners process and the Gaussian Free Field
• Extreme boundary conditions and random tilings
• Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model
• Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE
• Radial SLE martingale-observables
• Hadamard's formula and couplings of SLEs with free field