Renormalizability of Liouville Quantum Gravity at the Seiberg bound

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

1 Institut de Physique Theorique
2 Department of Mathematics and Statistics [Helsinki]
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
4 DMA - Département de Mathématiques et Applications - ENS Paris

Liouville Quantum Field Theory can be seen as a probabilistic theory of 2d Riemannian metrics e φ(z) dz 2 , conjecturally describing scaling limits of discrete 2d-random surfaces. The law of the random field φ in LQFT depends on weights α ∈ R that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [2] in the case when the weights are below the so called Seiberg bound: α < Q where Q parametrizes the random surface model in question. These correspond to conical singularities in the classical setup. In this paper, we construct LQFT in the case when the Seiberg bound is saturated which can be seen as the probabilistic version of Riemann surfaces with cusp singularities. Their construction involves methods from Gaussian Multiplicative Chaos theory at criticality.

Mots clés: Liouville Quantum Gravity; quantum field theory; Gaussian multiplicative chaos; KPZ formula; KPZ scaling laws; Polyakov formula; punctures; cusp singularity; uniformization theorem

Domaines: Probabilités [math.PR]

François David, Antti Kupiainen, Rémi Rhodes, Vincent Vargas. Renormalizability of Liouville Quantum Gravity at the Seiberg bound. 2015. ⟨hal-01161638⟩