From Quantum Groups to Liouville and Dilaton Quantum Gravity

Abstract

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 $q\to 1$ 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 $\text{U}_q(\mathfrak{osp}(1|2, \mathbb{R}))$.