Liouville quantum gravity on the annulus

Abstract

In this work, we construct Liouville quantum gravity on an annulus in the complex plane. This construction is aimed at providing a rigorous mathematical framework to the work of theoretical physicists initiated by Polyakov in 1981. It is also a very important example of a conformal field theory (CFT). Results have already been obtained on the Riemann sphere and on the unit disk, so this paper will follow the same approach. The case of the annulus contains two difficulties: it is a surface with two boundaries and it has a non-trivial moduli space. We recover the Weyl anomaly—a formula verified by all CFT—and deduce from it the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula. We also show that the full partition function of Liouville quantum gravity integrated over the moduli space is finite. This allows us to give the joint law of the Liouville measures and of the random modulus and to write the conjectured link with random planar maps.