Quantum exponentials for the modular double and applications in gravity models
Thomas G. Mertens
TL;DR
The paper advances a Gauss–Euler-like decomposition of the modular double SL$_q^+(2,\mathbb{R})$ via $R(g)=g_{b}(\gamma f)\,e^{2\phi H}\,g_b^*(\beta e)$, with $g_b$ the quantum dilogarithm, to connect the modular double to quantum group representations and gravity amplitudes. It verifies this structure by deriving Whittaker and hyperbolic representation matrices, reproducing known results (e.g., Ip) and establishing dual Casimir difference equations that act as a quantum Laplacian on the group manifold; it further constructs the regular representation and discusses one-sided gravitational wavefunctions, giving a gravity interpretation in terms of edge states and horizons. The findings enable a $q$-BF formulation of amplitudes for 2d Liouville gravity and 3d gravity, tying lower-dimensional quantum gravity to the rich representation theory of the modular double. The work also sketches Hopf-duality interpretations and points toward possible supersymmetric extensions.
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.