Contents

# Contents

## 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.

## References

Surveys include

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

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 July 10, 2019 at 07:32:20. See the history of this page for a list of all contributions to it.