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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 $text{U}_q(mathfrak{osp}(1|2, mathbb{R}))$.
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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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In this note, we consider the question of classicality for the theory which is known to be the effective description of two-dimensional black holes - the Morse quantum mechanics. We calculate the Wigner function and the Fisher information characterizing classicality/quantumness of single-particle systems and briefly discuss further directions to study.
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