From Quantum Groups to Liouville and Dilaton Quantum Gravity

TL;DR

This paper explores the quantum group symmetries underlying 2D Liouville and dilaton supergravity models, extending known results to supersymmetric cases and connecting them to quantum groups, Whittaker functions, and boundary operator couplings.

Contribution

It derives new matrix elements for quantum supergroups and links Liouville supergravity to dilaton supergravity via quantization of Poisson structures.

Findings

Derived the $ ext{U}_q( ext{osp}(1|2, ext{R}))$ Whittaker function.

Connected quantum group matrix elements to black hole state densities.

Matched boundary operator couplings with quantum group representations.

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