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A bstract 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 $$ \mathcal{N} $$ N = 1 supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of U q ( $$ \mathfrak{sl} $$ sl (2 , ℝ)) and review its applications to Liouville gravity. We then derive the corresponding matrix element for U q ( $$ \mathfrak{osp} $$ osp (1 | 2 , ℝ)) and apply it to explain structural features of $$ \mathcal{N} $$ N = 1 Liouville supergravity. We show that this matrix element has the following properties: (1) its q → 1 limit is the classical OSp + (1 | 2 , ℝ) Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in $$ \mathcal{N} $$ N = 1 Liouville supergravity, and (3) it leads to 3 j -symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for $$ \mathcal{N} $$ 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 U q ( $$ \mathfrak{sl} $$ sl (2 , ℝ)) or the quantum supergroup U q ( $$ \mathfrak{osp} $$ osp (1 | 2 , ℝ)).
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Macdonald Operators and Quantum Q-Systems for Classical Types
We propose solutions of the quantum Q-systems of types BN,CN,DN in terms of q-difference operators, generalizing our previous construction for the Q- system of type A. The difference operators are interpreted as q-Whittaker limits of discrete time evolutions of Macdonald-van Diejen type operators. We conjecture that these new operators act as raising and lowering operators for q-Whittaker functions, which are special cases of graded characters of fusion products of KR- modules.
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- Award ID(s):
- 1802044
- PAR ID:
- 10326431
- Editor(s):
- Alexeev, A.; Frenkel, E.; Rosso, M.; Webster, B.; Yakimov, M.
- Date Published:
- Journal Name:
- Representation Theory, Mathematical Physics, and Integrable Systems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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