nLab
Liouville theory

Idea

In the context of quantum field theory, Liouville theory is the name of a certain type of 2d CFT.

Properties

Relation to 3d quantum gravity

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.

Related concepts

• AGT correspondence

• Jackiw-Teitelboim gravity

References

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)

See also:

• Wikipedia, Liouville field theory

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)

• 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 in:

• Vincent Vargas, Lecture notes on Liouville theory and the DOZZ formula (arXiv:1712.00829)

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 $SL(2,\mathbb{R})$-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)