From quantum groups to Liouville and dilaton quantum gravity

A bstract 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 N N = 1 supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of U q ( sl sl (2 , ℝ)) and review its applications to Liouville gravity. We then derive the corresponding matrix element for U q ( osp osp (1 | 2 , ℝ)) and apply it to explain structural features of N N = 1 Liouville supergravity. We show that this matrix element has the following properties: (1) its q → 1 limit is the classical OSp + (1 | 2 , ℝ) Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in N N = 1 Liouville supergravity, and (3) it leads to 3 j -symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for N 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 U q ( sl sl (2 , ℝ)) or the quantum supergroup U q ( osp osp (1 | 2 , ℝ)).