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Liouville Quantum Gravity on the Riemann Sphere

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Abstract

In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov. We 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 formula. We also make precise conjectures about the relationship of the theory to scaling limits of random planar maps conformally embedded onto the sphere.

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Authors and Affiliations

  1. Institut de Physique Théorique, CNRS, URA 2306, CEA, IPhT, Gif-sur-Yvette, France

    François David

  2. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Antti Kupiainen

  3. Université Paris-Est Marne la Vallée, LAMA, Champs sur Marne, France

    Rémi Rhodes

  4. ENS Ulm, DMA, 45 rue d’Ulm, 75005, Paris, France

    Vincent Vargas

Authors
  1. François David
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  2. Antti Kupiainen
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  4. Vincent Vargas
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Corresponding author

Correspondence to Rémi Rhodes.

Additional information

Communicated by F. Toninelli

A. Kupiainen: Supported by the Academy of Finland.

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David, F., Kupiainen, A., Rhodes, R. et al. Liouville Quantum Gravity on the Riemann Sphere. Commun. Math. Phys. 342, 869–907 (2016). https://doi.org/10.1007/s00220-016-2572-4

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