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تسجيل مستخدم جديدFrom Quantum Groups to Liouville and Dilaton Quantum Gravity
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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 $qto 1$ 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 $text{U}_q(mathfrak{osp}(1|2, mathbb{R}))$.
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