High Energy Theory Seminar - Bertrand Duplantier, Institute for Theoretical Physics, Saclay - Schramm-Loewner Evolution and Liouville Quantum Gravity

Wed, Feb 27, 2013, 3:30 pm to 4:30 pm
PCTS Seminar Room
Liouville quantum gravity in two dimensions is described by the ``random Riemannian manifold'' obtained by changing the Lebesgue measure in the plane by a random conformal factor, the exponential of the Gaussian free field. This ``random surface" is believed to be the continuum scaling limit of certain discretized random surfaces that can be studied with combinatorics and random matrix theory. When boundary arcs of a Liouville quantum gravity random surface are conformally welded to each other (in a boundary quantum-length-preserving way) the resulting interface is a random curve described by the Schramm-Loewner evolution (SLE). This allows to develop a theory of quantum fractal measures, consistent with the Knizhnik-Polyakov-Zamolochikov (KPZ) relation, and to analyze their evolution under conformal welding maps related to SLE. As an application, one can construct quantum length and boundary intersection measures on the SLE curve itself. (Joint work with Scott Sheffield, MIT- Math)