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Two Perspectives of the 2D Unit Area Quantum Sphere and Their Equivalence

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Abstract

2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantum gravity and is the theory of world-sheet in string theory. Recently there has been growing interest in studying LQG in the realm of probability theory: David et al. (Liouville quantum gravity on the Riemann sphere. Commun Math Phys 342(3):869–907, 2016) and Duplantier et al. (Liouville quantum gravity as a mating of trees. ArXiv e-prints: arXiv:1409.7055, 2014) both provide a probabilistic perspective of the LQG on the 2D sphere. In particular, in each of them one may find a definition of the so-called unit area quantum sphere. We examine these two perspectives and prove their equivalence by showing that the respective unit area quantum spheres are the same. This is done by considering a unified limiting procedure for defining both objects.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Rämistr. 101, 8092, Zurich, Switzerland

    Juhan Aru

  2. Sorbonne Universities, UPMC Univ Paris 06, Paris, France

    Yichao Huang

  3. Department of Mathematics, ENS Paris, CNRS, PSL Research University, 45 Rue d’Ulm, 75005, Paris, France

    Yichao Huang

  4. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Xin Sun

Authors
  1. Juhan Aru
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  2. Yichao Huang
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  3. Xin Sun
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Correspondence to Xin Sun.

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Communicated by H.-T. Yau

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Aru, J., Huang, Y. & Sun, X. Two Perspectives of the 2D Unit Area Quantum Sphere and Their Equivalence. Commun. Math. Phys. 356, 261–283 (2017). https://doi.org/10.1007/s00220-017-2979-6

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  • DOI: https://doi.org/10.1007/s00220-017-2979-6

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