Quantum exponentials for the modular double and applications in gravity models

Abstract

In this note, we propose a decomposition of the quantum matrix group SL${}_q^{+}(2,\mathbb{R})$ as (deformed) exponentiation of the quantum algebra generators of Faddeev's modular double of ${\text{U}}_q(\mathfrak{sl}(2,\mathbb{R}))$. The formula is checked by relating hyperbolic representation matrices with the Whittaker function. We interpret (or derive) it in terms of Hopf duality, and use it to explicitly construct the regular representation of the modular double, leading to the Casimir and its modular dual as the analogue of the Laplacian on the quantum group manifold. This description is important for both 2d Liouville gravity, and 3d pure gravity, since both are governed by this algebraic structure. This result builds towards a $q$-BF formulation of the amplitudes of both of these gravitational models.

32 pages, v3: added material on regular representation and Hopf duality, typos fixed and references added, matches published version

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• Branes in JT (super)gravity from group theory
• Les Houches lectures on two-dimensional gravity and holography
• Quantum group origins of edge states in double-scaled SYK
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes