algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In the context of quantum field theory, Liouville theory is the name of a certain type of 2d CFT.
An argument (via Chern-Simons gravity, but see the caveats there) that 3d quantum gravity with negative cosmological constant has as boundary field theory 2d Liouville theory is due to (Coussaert-Henneaux-vanDriel 95). See also at AdS3-CFT2 and CS-WZW correspondence.
On the non-critical bosonic string via Liouville theory (cf. Polyakov action):
Alexander Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207-210 [doi:10.1016/0370-2693(81)90743-7, pdf]
Jean-Loup Gervais, André Neveu, The dual string spectrum in Polyakov’s quantization (I), Nuclear Physics B 199 1 (1982) 59-76 [doi:10.1016/0550-3213(82)90566-1]
Jean-Loup Gervais, André Neveu, Dual string spectrum in Polyakov’s quantization (II). Mode separation, Nuclear Physics B 209 1 (1982) 125-145 [doi:10.1016/0550-3213(82)90105-5]
Thomas L. Curtright, Charles B. Thorn, Conformally Invariant Quantization of the Liouville Theory, Phys. Rev. Lett. 48 (1982) 1309; Erratum Phys. Rev. Lett. 48, (1982) 1768 [doi:10.1103/PhysRevLett.48.1309]
Review and survey:
Luis Alday, Davide Gaiotto, Yuji Tachikawa, appendix A.1 of Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett.Math.Phys.91:167-197, 2010 (arXiv:0906.3219)
Harold Erbin, Notes on 2d quantum gravity and Liouville theory (2015) [pdf, pdf]
See also:
Rigorous construction of the path integral, the DOZZ formula, and the conformal bootstrap for Liouville theory:
François David, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Liouville Quantum Gravity on the Riemann sphere, Communications in Mathematical Physics volume 342, pages869–907 (2016) (arxiv:1410.7318)
Colin Guillarmou, Rémi Rhodes, Vincent Vargas, Polyakov’s formulation of 2d bosonic string theory, Publ. Math. IHES 130 (2019) 111–185 [arXiv:1607.08467, doi:10.1007/s10240-019-00109-6]
Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Integrability of Liouville theory: proof of the DOZZ Formula, Annals of Mathematics, 191 1 (2020) 81-166 (arxiv:1707.08785, doi:10.4007/annals.2020.191.1.2)
Antti Kupiainen, Rémi Rhodes, Vincent Vargas, The DOZZ formula from the path integral, Journal of High Energy Physics volume 2018, Article number: 94 (2018) (arXiv:1803.05418 doi:10.1007/JHEP05(2018)094)
Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Conformal bootstrap in Liouville Theory [arXiv:2005.11530]
Review:
Construction as a functorial field theory following Segal 1988
Review in:
Vincent Vargas, Lecture notes on Liouville theory and the DOZZ formula (arXiv:1712.00829)
Rémi Rhodes, Vincent Vargas, A probabilistic approach of Liouville field theory, Comptes Rendus. Physique, Tome 21 (2020) no. 6, pp. 561-569. (doi:10.5802/crphys.43)
An argument (via Chern-Simons gravity, but see the caveats there) that 3d quantum gravity with negative cosmological constant has as boundary field theory 2d Liouville theory is due to
O. Coussaert, Marc Henneaux, P. van Driel, The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961-2966 (arXiv:gr-qc/9506019)
Leon A. Takhtajan, Lee-Peng Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Commun.Math.Phys. 239 (2003) 183-240 (arXiv:math/0204318)
Abstract: We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen’s quasi-Fuchsian reciprocity. We prove that the classical action is a Kahler potential of the Weil-Petersson metric. We also prove that Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups and their free combinations.
The relation to quantum Teichmüller theory is discussed/reviewed in:
Jörg Teschner, On the relation between quantum Liouville theory and the quantized Teichmüller spaces, Int. J. Mod. Phys. A 19S2:459-477,2004 (arxiv:hep-th/0303149)
Dylan Allegretti, Notes on Quantum Teichmüller theory (pdf)
Jörg Teschner, Quantization of moduli spaces of flat connections and Liouville theory, proceedings of the International Congress of Mathematics 2014 (arXiv:1405.0359)
Relation to the -gauged WZW-model and analytically continued Wess-Zumino-Witten theory:
Noboyuki Ishibashi, Extra Observables in Gauged WZW Models, Nucl.Phys. B379 (1992) 199-219 (arXiv:hep-th/9110071)
Jian-Feng Wu, Yang Zhou, From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality (arXiv:0911.1922)
Further discussion:
On conformal blocks for Liouville theory:
Babak Haghighat, Yihua Liu, Nicolai Reshetikhin, Flat Connections from Irregular Conformal Blocks [arXiv:2311.07960]
Xia Gu, Babak Haghighat, Kevin Loo, Irregular Fibonacci Conformal Blocks [arXiv:2311.13358]
Babak Haghighat: Flat Connections from Irregular Conformal Blocks, talk at CQTS (Feb 2024) [video: zm, kt]
Relation ot Liouville theory to the Nambu-Goto action:
Via celestial holography:
Last revised on September 11, 2024 at 06:23:40. See the history of this page for a list of all contributions to it.