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We define a three-dimensional quantum theory of gravity as the holographic dual of the Liouville conformal field theory. The theory is consistent and unitary by definition. The corresponding theory of gravity with negative cosmological constant has peculiar properties. The quantum theory has no normalisable AdS3 vacuum. The model contains primary black holes with zero spin. All states can be interpreted as black holes dressed with boundary gravitons. There is a unique universal interaction between these states consistent with unitarity and the conformal symmetry of the model. This theory of gravity, though conceptually isolated from other models of quantum gravity, is worth scrutinising.
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We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with $mathcal{N} = 1$ supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of $text{U}_q(mathfrak{sl}(2, mathbb{R}))$ and review its applications to Liouville gravity. We then derive the corresponding matrix element for $text{U}_q(mathfrak{osp}(1|2, mathbb{R}))$ and apply it to explain structural features of $mathcal{N} = 1$ Liouville supergravity. We show that this matrix element has the following properties: (1) its $qto 1$ limit is the classical $text{OSp}^+(1|2, mathbb{R})$ Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in $mathcal{N} = 1$ Liouville supergravity, and (3) it leads to $3j$-symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for $mathcal{N} = 1$ Liouville supergravity. This object should likewise be of interest in the context of integrability of supersymmetric relativistic Toda chains. We furthermore relate Liouville (super)gravity to dilaton (super)gravity with a hyperbolic sine (pre)potential. We do so by showing that the quantization of the target space Poisson structure in the (graded) Poisson sigma model description leads directly to the quantum group $text{U}_q(mathfrak{sl}(2, mathbb{R}))$ or the quantum supergroup $text{U}_q(mathfrak{osp}(1|2, mathbb{R}))$.
We consider a gravitational perturbation of the Jackiw-Teitelboim (JT) gravity with an arbitrary dilaton potential and study the condition under which the quadratic action can be seen as a $Tbar{T}$-deformation of the matter action. As a special case, the flat-space JT gravity discussed by Dubovsky et al[arXiv:1706.06604 ] is included. Another interesting example is a hyperbolic dilaton potential. This case is equivalent to a classical Liouville gravity with a negative cosmological constant and then a finite $Tbar{T}$-deformation of the matter action is realized as a gravitational perturbation on AdS$_2$.
General properties of a class of two-dimensional dilaton gravity (DG) theories with multi-exponential potentials are studied and a subclass of these theories, in which the equations of motion reduce to Toda and Liouville equations, is treated in detail. A combination of parameters of the equations should satisfy a certain constraint that is identified and solved for the general multi-exponential model. From the constraint it follows that in DG theories the integrable Toda equations, generally, cannot appear without accompanying Liouville equations. We also show how the wave-like solutions of the general Toda-Liouville systems can be simply derived. In the dilaton gravity theory, these solutions describe nonlinear waves coupled to gravity as well as static states and cosmologies. A special attention is paid to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible, with the aim to gain a better understanding of realistic theories reduced to dimensions 1+1 and 1+0 or 0+1.
There has been a proposal that infrared quantum effects of massless interacting field theories in de-Sitter space may provide time-dependent screening of the cosmological constant. As a concrete model of the proposal, we study the three loop corrections to the energy-momentum tensor of massless $lambda phi^4$ theory in the background of classical Liouville gravity in $D=2$ dimensional de-Sitter space. We find that the cosmological constant is screened in sharp contrast to the massless $lambda phi^4$ theory in $D=4$ dimensions due to the sign difference between the cosmological constant of the Liouville gravity and that of the Einstein gravity. To argue for the robustness of our prediction, we introduce the concept of time-dependent infrared counter-terms and examine if they recover the de-Sitter invariance in the $lambda phi^4$ theory in comparison with the Sine-Gordon model where it was possible.
In this paper we study a connection between Jackiw-Teitelboim (JT) gravity on two-dimensional anti de-Sitter spaces and a semiclassical limit of $c<1$ two-dimensional string theory. The world-sheet theory of the latter consists of a space-like Liouville CFT coupled to a non-rational CFT defined by a time-like Liouville CFT. We show that their actions, disk partition functions and annulus amplitudes perfectly agree with each other, where the presence of boundary terms plays a crucial role. We also reproduce the boundary Schwarzian theory from the Liouville theory description. Then, we identify a matrix model dual of our two-dimensional string theory with a specific time-dependent background in $c=1$ matrix quantum mechanics. Finally, we also explain the corresponding relation for the two-dimensional de-Sitter JT gravity.
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