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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.
Surveys include
Wikipedia, Liouville field theory
Luis F. 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)
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 $SL(2,\mathbb{R})$-WZW-model and analytically continued Wess-Zumino-Witten theory is discussed in
Nobuyuki 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)
Last revised on November 5, 2018 at 12:16:08. See the history of this page for a list of all contributions to it.