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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta"> <mml:semantics> <mml:mi> ζ </mml:mi> <mml:annotation encoding="application/x-tex">\zeta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> κ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\kappa &gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This identifies SLE curves as local observables of the free field.