Liouville Quantum Gravity on the complex tori

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

1 Institut de Physique Theorique
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 DMA - Département de Mathématiques et Applications - ENS Paris

In this paper, we construct Liouville Quantum Field Theory (LQFT) on the toroidal topology in the spirit of the 1981 seminal work by Polyakov. Our approach follows the construction carried out by the authors together with A. Kupiainen in the case of the Riemann sphere. The difference is here that the moduli space for complex tori is non trivial. Modular properties of LQFT are thus investigated. This allows us to sum up the LQFT on complex tori over the moduli space, to compute the law of the random Liouville modulus, therefore recovering (and extending) formulae obtained by physicists, and make conjectures about the relationship with random planar maps of genus one, eventually weighted by a conformal field theory and conformally embedded onto the torus.

Mots clés: Liouville Quantum Gravity; complex tori; modular covariance; Gaussian multiplicative chaos; KPZ formula; Polyakov formula

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]

François David, Rémi Rhodes, Vincent Vargas. Liouville Quantum Gravity on the complex tori. Journal of Mathematical Physics, 2016, 57 (2 ). ⟨hal-01161637⟩