3/10/2008

Scott Sheffield
Courant Institute

Quantum zippers

I will discuss some recent work on the random two dimensional geometries of Liouville quantum gravity. Roughly speaking, the geometries in question have the form e^{h(z)} d2 z where h is a multiple of the Gaussian free field on a planar domain. Some care is needed to make this precise, since the Gaussian free field is defined only as a random distribution, not as a function. However, once the basic objects are defined, we can prove that

1. The dimensions of random fractals defined in Euclidean and quantum geometries are related by the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula (joint work with Bertrand Duplantier). 2. When two independent free boundary random geometries are conformally welded ("zipped together") at their boundaries, the law of the resulting interface is a form of the Schramm-Loewner evolution (SLE). 3. The continuum geometries discussed above are the scaling limits (at least in one rather weak sense) of certain discrete random planar maps (a.k.a. "discrete quantum gravity" models).