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

66 pages, v4: fixed further typos

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• $T\bar{T}$ in JT Gravity and BF Gauge Theory
• Supergroup Structure of Jackiw-Teitelboim Supergravity
• The dilaton gravity hologram of double-scaled SYK
• Wormholes, branes and finite matrices in sine dilaton gravity
• Quantum exponentials for the modular double and applications in gravity models
• Supersymmetric Virasoro Minimal Strings
• 3d Gravity as a random ensemble
• Phase transitions for deformations of JT supergravity and matrix models
• Branes in JT (super)gravity from group theory
• Equivalences between 2D dilaton gravities, their asymptotic symmetries, and their holographic duals
• Quantum group origins of edge states in double-scaled SYK
• Gravitational wavefunctions in JT supergravity
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• Asymptotics of Weil-Petersson volumes and two-dimensional quantum gravities
• Fiducial observers and the thermal atmosphere in the black hole quantum throat
• Probing sine dilaton gravity with flow central charge