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SLE and Virasoro Representations: Fusion

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Abstract

We continue the study of null-vector equations in relation with partition functions of (systems of) Schramm–Loewner Evolutions (SLEs) by considering the question of fusion. Starting from n commuting SLEs seeded at distinct points, the partition function satisfies n null-vector equations (at level 2). We show how to obtain higher level null-vector equations by coalescing the seeds one by one. As an example, we extend Schramm’s formula (for the position of a marked bulk point relatively to a chordal SLE trace) to an arbitrary number of SLE strands.

The argument combines input from representation theory—the study of Verma modules for the Virasoro algebra—with regularity estimates, themselves based on hypoellipticity and stochastic flow arguments.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Julien Dubédat

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  1. Julien Dubédat
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Correspondence to Julien Dubédat.

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Communicated by M. Salmhofer

J. Dubédat was Partially supported by NSF Grant DMS-1005749 and the Alfred P. Sloan Foundation.

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Dubédat, J. SLE and Virasoro Representations: Fusion. Commun. Math. Phys. 336, 761–809 (2015). https://doi.org/10.1007/s00220-014-2283-7

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