From Quantum Groups to Liouville and Dilaton Quantum Gravity

TL;DR

This work reveals a unifying quantum-group perspective on 2d Liouville and dilaton gravity by constructing and analyzing mixed-parabolic Whittaker functions for the q-deformed algebras U_q(sl(2,R)) and U_q(osp(1|2,R)). It demonstrates how these representation-theoretic objects encode Liouville gravity amplitudes, boundary vertex functions, and the Plancherel measure, and it provides a Lagrangian footing via graded Poisson-sigma models that naturally yield the same quantum-group symmetry upon quantization. The results extend to N=1 supersymmetry and connect Liouville gravity to sinh-type dilaton supergravity, offering a cohesive framework that links gravity amplitudes to quantum-group data and opens avenues toward integrable structures and higher-N generalizations.

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

Figures (4)

• Figure 1: Left: wavefunction $\psi_{R, \nu\mu}(g)$ in a mixed parabolic basis as a two-boundary state. Right: two-boundary slicing of the disk.
• Figure 2: The contour $\mathcal{C}$ used to define $\phi_{\alpha}^{\hbox{$+$}}(t)$.
• Figure 3: Deformed JT gravity disk amplitude with two boundary operators, interpretable as undeformed JT gravity with a gas of defects localized in the interior of the disk (blue blob).
• Figure 4: Left: contour $\mathcal{C}$ used to define $I_\alpha(x)$. Right: contour $\mathcal{C}$ used to define the $q$-deformed version $\mathcal{I}^{\epsilon}_{\alpha}(x)$.