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SLE and the free field: Partition functions and couplings

Author: Julien Dubédat
Journal: J. Amer. Math. Soc. 22 (2009), 995-1054
MSC (2000): Primary 60G17, 60K35
Published electronically: April 29, 2009
MathSciNet review: 2525778
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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 $\kappa >0$. This identifies SLE curves as local observables of the free field.

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Additional Information

Julien Dubédat
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027
MR Author ID: 710651

Keywords: Schramm-Loewner Evolutions, free field
Received by editor(s): February 27, 2008
Published electronically: April 29, 2009
Additional Notes: The author was partially supported by NSF grant DMS0804314
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.