Liouville Quantum Gravity on the Riemann sphere

François David (1) , Antti Kupiainen (2) , Rémi Rhodes (3) , Vincent Vargas (4, 5)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 Department of Mathematics and Statistics [Helsinki]
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
4 ENS-PSL - École normale supérieure - Paris
5 DMA - Département de Mathématiques et Applications - ENS Paris

In this paper, we rigorously construct $2d$ Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov "Quantum Geometry of bosonic strings". We also establish some of its fundamental properties like conformal covariance under PSL$_2(\mathbb{C})$-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quantum Gravity.

Keywords: Liouville Quantum Gravity; quantum field theory; Gaussian multiplicative chaos; KPZ formula; KPZ scaling laws; Polyakov formula

Domains: Probability [math.PR]; Mathematical Physics [math-ph]

François David, Antti Kupiainen, Rémi Rhodes, Vincent Vargas. Liouville Quantum Gravity on the Riemann sphere. Communications in Mathematical Physics, 2016, 342, pp.869 - 907. ⟨10.1007/s00220-016-2572-4⟩. ⟨cea-01509870⟩