The semiclassical limit of Liouville conformal field theory

Lacoin, Hubert; Rhodes, Rémi; Vargas, Vincent

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 4, pp. 1031-1083.

La théorie conforme des champs de Liouville sur la sphère de Riemann (LCFT) a récemment été construite via la théorie des probabilités par David–Kupiainen et les deux derniers auteurs. Dans ce papier, on étudie la relation entre LCFT et la théorie classique de Liouville via la limite semi-classique. LCFT dépend d’un paramètre $\gamma \in (0,2)$ et la limite $\gamma \to 0$ correspond à la limite semi-classique. Dans le régime semi-classique, on détermine la limite des fonctions de corrélation et du champ de Liouville associé sous une condition de courbure négative (pour la métrique limite). On établit également un résultat de grandes déviations pour le champ de Liouville : comme attendu, la fonctionnelle de grandes déviations est l’action de Liouville classique. Comme corollaire, on obtient une preuve probabiliste du théorème de Takhtajan–Zograf qui relie l’action de Liouville classique (pris en son minimum) aux paramètres accessoires de Poincaré. Enfin, on énonce des conjectures dans le cas de la courbure positive (incluant l’étude des quantum spheres introduits par Duplantier–Miller–Sheffield).

A rigorous probabilistic construction of Liouville conformal field theory (LCFT) on the Riemann sphere was recently given by David–Kupiainen and the last two authors. In this paper, we focus on the connection between LCFT and the classical Liouville field theory via the semiclassical approach. LCFT depends on a parameter $\gamma \in (0,2)$ and the limit $\gamma \to 0$ corresponds to the semiclassical limit of the theory. Within this asymptotic and under a negative curvature condition (on the limiting metric of the theory), we determine the limit of the correlation functions and of the associated Liouville field. We also establish a large deviation result for the Liouville field: as expected, the large deviation functional is the classical Liouville action. As a corollary, we give a new (probabilistic) proof of the Takhtajan–Zograf theorem which relates the classical Liouville action (taken at its minimum) to Poincaré’s accessory parameters. Finally, we gather conjectures in the positive curvature case (including the study of the so-called quantum spheres introduced by Duplantier–Miller–Sheffield).

Classification : 81T40, 81T20, 60D05
Mots clés : Liouville Quantum Theory, Gaussian multiplicative chaos, Polyakov formula, uniformization, accessory parameters, semiclassical analysis.

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