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

• dilaton

• AGT correspondence

• Jackiw-Teitelboim gravity

• list of functorial field theories

References

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:

• 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)

• 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:

• Sourav Chatterjee, Edward Witten, Liouville Theory: An Introduction to Rigorous Approaches [arXiv:2404.02001]

Construction as a functorial field theory following Segal 1988

• Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Segal’s axioms and bootstrap for Liouville Theory [arXiv:2112.14859]

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)

• Martin Hairer: Probabilistic interpretation of quantum field theories [arXiv:2501.14361]

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.

more in relation to $AdS_3$/$CFT_2$-duality:

• Lin Chen, Ling-Yan Hung, Yikun Jiang, Bing-Xin Lao: Quantum 2D Liouville Path-Integral Is a Sum over Geometries in $AdS_3$ Einstein Gravity [arXiv:2403.03179]

• Ning Bao, Ling-Yan Hung, Yikun Jiang, Zhihan Liu: QG from SymQRG: $AdS_3/CFT_2$ Correspondence as Topological Symmetry-Preserving Quantum RG Flow [arXiv:2412.12045]

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)

Further discussion:

• Pavel Haman, Alfredo Iorio, Classical gravitational anomalies of Liouville theory [arXiv:2306.09825]

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:

• Yuri Makeenko, The Nambu-Goto string as higher-derivative Liouville theory [arXiv:2407.01136]

Via celestial holography:

• Igor Mol: Partial Differential Equations for MHV Celestial Amplitudes in Liouville Theory [arXiv:2409.05936]

In the context of D=2 quantum gravity:

• Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez: The complex Liouville string [arXiv:2409.17246]

• Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez: The complex Liouville string: the worldsheet [arXiv:2409.18759]

• Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez: The complex Liouville string: the matrix integral [arXiv:2410.07345]

• Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez: The complex Liouville string: worldsheet boundaries and non-perturbative effects [arXiv:2410.09179]

• Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann: The complex Liouville string: the gravitational path integral [arXiv:2501.10265]

On timelike Liouville theory:

• Sourav Chatterjee. Rigorous results for timelike Liouville field theory (2025). (arXiv:2504.02348).

category: physics