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 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.

31 pages

Cited by in corpus (5)

• The Fyodorov-Bouchaud formula and Liouville conformal field theory
• Integrability of boundary Liouville conformal field theory
• Liouville quantum gravity with matter central charge in $(1,25)$: a probabilistic approach
• Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes
• Weak Liouville quantum gravity metrics with matter central charge $\mathbf{c}\in (-\infty ,25)$