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Comptes Rendus. Physique
A probabilistic approach of Liouville field theory
Comptes Rendus. Physique, Volume 21 (2020) no. 6, pp. 561-569.

Part of the special issue: Prizes of the French Academy of Sciences 2019 (continued)

In this article, we present the Liouville field theory, which was introduced in the eighties in physics by Polyakov as a model for fluctuating metrics in 2D quantum gravity, and outline recent mathematical progress in its study. In particular, we explain the probabilistic construction of this theory carried out by David–Kupiainen–Rhodes–Vargas in [] and how this construction connects to the modern and general approach of Conformal Field Theories in physics, called conformal bootstrap and based on representation theory.

Dans cet article, nous présentons la théorie des champs de Liouville, qui fut introduite en physique dans les années 80 par Polyakov comme modèle de métriques aléatoires dans le cadre de la gravité quantique 2D, et donnons un aperçu de la construction probabiliste de cette théorie proposée par David–Kupiainen–Rhodes–Vargas dans []. Nous expliquons comment cette construction se relie à l’approche moderne des théories conformes de champs en physique appelée conformal bootstrap et basée sur la théorie des représentations.

Published online:
DOI: 10.5802/crphys.43
Classification: 60D05, 81T40, 81T20
Keywords: $2D$ quantum gravity, path integral, Liouville field theory, conformal bootstrap
Rémi Rhodes 1, 2; Vincent Vargas 3

1 Aix-Marseille University (AMU), Institut de Mathématiques (I2M), 39 rue F. Joliot Curie, Marseille, France.
2 Institut Universitaire de France (IUF)
3 Institut Galilée, Université Paris 13, 99 avenue Jean-Baptiste Clément, Villetaneuse, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {R\'emi Rhodes and Vincent Vargas},
     title = {A probabilistic approach of {Liouville} field theory},
     journal = {Comptes Rendus. Physique},
     pages = {561--569},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {21},
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     year = {2020},
     doi = {10.5802/crphys.43},
     language = {en},
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VL  - 21
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PB  - Académie des sciences, Paris
DO  - 10.5802/crphys.43
LA  - en
ID  - CRPHYS_2020__21_6_561_0
ER  - 
%0 Journal Article
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%A Vincent Vargas
%T A probabilistic approach of Liouville field theory
%J Comptes Rendus. Physique
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Rémi Rhodes; Vincent Vargas. A probabilistic approach of Liouville field theory. Comptes Rendus. Physique, Volume 21 (2020) no. 6, pp. 561-569. doi : 10.5802/crphys.43. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.43/

[1] François David; Antti Kupiainen; Ré Rhodes; Vincent Vargas Liouville quantum gravity on the Riemann sphere, Commun. Math. Phys., Volume 342 (2016) no. 3, pp. 869-907 | Zbl

[2] A. M. Polyakov Quantum geometry of bosonic strings, Phys. Lett., B, Volume 103 (1981) no. 3, pp. 207-210 | DOI

[3] Luis F. Alday; Davide Gaiotto; Yuji Tachikawa Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys., Volume 91 (2010) no. 2, pp. 167-197 | Zbl

[4] Colin Guillarmou; Rémi Rhodes; Vincen Vargas Polyakov’s formulation of 2d bosonic string theory, Publ. Math., Inst. Hautes Étud. Sci., Volume 130 (2019), pp. 111-185 | Zbl

[5] V. G. Knizhnik; A. M. Polyakov; A. B. Zamolodchikov Fractal structure of 2D-quantum gravity, Modern Phys. Lett. A, Volume 3 (1988) no. 8, pp. 819-826 | DOI

[6] Christophe Garban Quantum gravity and the KPZ formula [after Duplantier-Sheffield], Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058 (Astérisque), Volume 352 (2013), pp. 315-354 | Zbl

[7] Bertrand Duplantier; Jason Miller; Scott Sheffield Liouville quantum gravity as a mating of trees (2014) (https://arxiv.org/abs/1409.7055)

[8] Jean-Pierre Kahane Sur le chaos multiplicatif, Ann. Sci. Math. Québec, Volume 9 (1985) no. 2, pp. 105-150 | Zbl

[9] Rémi Rhodes; Vincent Vargas Gaussian multiplicative chaos and applications: a review, Probab. Surv., Volume 11 (2014), pp. 315-392 | Zbl

[10] Yan V. Fyodorov; Jean-Philippe Bouchaud Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential, J. Phys., Volume 41 (2008) no. 37, 372001, 12 pages | Zbl

[11] Alexander A. Belavin; Alexander M. Polyakov; Alexander B. Zamolodchikov Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys., Volume 241 (1984) no. 2, pp. 333-380 | Zbl

[12] Harald Dorn; Hans-Jörg Otto Two- and three-point functions in Liouville theory, Nuclear Phys., Volume 429 (1994) no. 2, pp. 375-388 | Zbl

[13] Alexander Borisovich Zamolodchikov; Alexeĭ Borisovich Zamolodchikov Conformal bootstrap in Liouville field theory, Nuclear Phys., Volume 477 (1996) no. 2, pp. 577-605 | Zbl

[14] Antti Kupiainen; Rémi Rhodes; Vincent Vargas Integrability of Liouville theory: proof of the DOZZ formula, Ann. Math., Volume 191 (2020) no. 1, pp. 81-166 | Zbl

[15] Antti Kupiainen; Rémi Rhodes; Vincent Vargas The DOZZ formula from the path integral, J. High Energy Phys., Volume 2018 (2018) no. 5, 94, 24 pages | Zbl

[16] Colin Guillarmou; Aantti Kupiainen; R’emi Rhodes; Vincent Vargas Conformal bootstrap in Liouville Theory (2005) (https://arxiv.org/abs/2005.11530)

[17] Davesh Maulik; Andrei Okounkov Quantum groups and quantum cohomology, Astérisque, 408, Société Mathématique de France, 2019 | Zbl

[18] Olivier Schiffmann; Eric Vasserot Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A 2 , Publ. Math., Inst. Hautes Étud. Sci., Volume 118 (2013), pp. 213-342 | Zbl

[19] Promit Ghosal; Guillaume Remy; Xin Sun; Yi Sun Probabilistic conformal blocks for Liouville CFT on the torus (2020) (https://arxiv.org/abs/2003.03802)

[20] Julien Dubédat; Hao Shen Stochastic Ricci Flow on Compact Surfaces (2019) (https://arxiv.org/abs/1904.10909)

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