Publication : t15/160

Renormalizability of Liouville Quantum Gravity at the Seiberg bound

David F. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)
Kupiainen A. (Department of Mathematics, University of Helsinki P.O. Box 68, FIN-00014 Helsinki, FINLAND)
Rhodes R. (LAMA, Université Paris-Est Marne-la-Vallée, Champs sur Marne, France)
Vargas V. (DMA, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France)
Liouville Quantum Field Theory can be seen as a probabilistic theory of 2d Riemannian metrics $e^{phi(z)}dz^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $phi$ in LQFT depends on weights $alphain mathbb{R}$ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in cite{DKRV} in the case when the weights are below the so called Seiberg bound: $alpha<Q$ 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.
Année de publication : 2017
DOI : doi:10.1214/17-EJP113
Preprint : arXiv:1506.01968
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Langue : Anglais

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