Speaker: Guillaume Baverez
Title:
Schramm-Loewner Evolution, quantum Liouville theory and Teichmüller theory
Abstract:
SLE is a stochastic version of the Loewner equation and conjecturally
describes the scaling limit of interfaces of critical statistical mechanics
models. In the years following its introduction, it was understood that SLE
should also arise as the random curve produced by the isometric conformal
welding of planar domains equipped with a measure given by the exponential
of the Gaussian Free Field. On the one hand, the Loewner equation gives a
convenient framework for computations but on the other hand it offers only
limited geometric perspectives since it is based on the Riemann mapping
theorem rather than Poincaré uniformisation.
In this talk, I will outline an ongoing (or rather beginning) programme
aiming at a generalisation of SLE to arbitrary Riemann surfaces, taking the
conformal welding as the starting point. In this setting, SLE is naturally
and intrinsically defined as a probability measure on the homotopy class of
a prime geodesic. I will start by reviewing the deterministic theories at
stake (Liouville equation, conformal welding, Teichmüller theory...), then
build on this material to introduce the probabilistic ones and explain the
links between them. If time permits, I will also explain how the conformal
welding approach fits into Segal's axiomatisation of conformal field theory.