Talk page
Title:
Conformal restriction, conformal geometry and vertex algebras
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Abstract:
Conformal loop ensembles (CLE) provides a measure-theoretic description of the scaling or "continuum" limits of critical statistical models. It describes the scaling limit of cluster boundaries through non-intersecting random loops. One can extract its expected properties through the more general "conformal restriction systems". The scaling limits of critical models are also believed to be described by conformal field theory (CFT), which rather uses algebraic ideas, based on vertex algebras. Relating these two approaches is an important problem, with potential for a deep understanding of criticality. I will overview my work on this. It is based on relating conformal-symmetry fields to a geometry of conformal maps, in order to construct them in the context of conformal restriction systems. These CFT fields contain the stress-energy tensor, and give rise to the Virasoro vertex operator algebra, the most basic algebraic structure in CFT. I will give a brief introduction on vertex algebras, CLE and conformal restriction systems, then explain my results.
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