Liouville quantum gravity on the annulus

TL;DR

This paper rigorously constructs Liouville quantum gravity on an annulus, addressing boundary and moduli space complexities, and connects it to conformal field theory, KPZ formula, and random planar maps.

Contribution

It extends Liouville quantum gravity construction to annuli, handling boundary and moduli space issues, and links the theory to physical and probabilistic models.

Findings

Recovered the Weyl anomaly formula for the annulus

Proved the finiteness of the partition function over moduli space

Established the joint law of Liouville measures and the random modulus

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 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.