Ruijsenaars wavefunctions as modular group matrix coefficients
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911918882553856 |
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| author | Di Francesco, Philippe Kedem, Rinat Khoroshkin, Sergey Schrader, Gus Shapiro, Alexander |
| author_facet | Di Francesco, Philippe Kedem, Rinat Khoroshkin, Sergey Schrader, Gus Shapiro, Alexander |
| contents | We give a description of the Hallnäs--Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $S\in SL(2,\mathbb{Z})$ acting on the Hilbert space of $GL(2)$ quantum Teichmüller theory on the punctured torus. The $GL(2)$ Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framed $GL(2)$-local systems on the punctured torus, and an $SL(2,\mathbb{Z})$-equivariant embedding of the $GL(2)$ spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ruijsenaars wavefunctions as modular group matrix coefficients Di Francesco, Philippe Kedem, Rinat Khoroshkin, Sergey Schrader, Gus Shapiro, Alexander Mathematical Physics Quantum Algebra Representation Theory Exactly Solvable and Integrable Systems We give a description of the Hallnäs--Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $S\in SL(2,\mathbb{Z})$ acting on the Hilbert space of $GL(2)$ quantum Teichmüller theory on the punctured torus. The $GL(2)$ Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framed $GL(2)$-local systems on the punctured torus, and an $SL(2,\mathbb{Z})$-equivariant embedding of the $GL(2)$ spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety. |
| title | Ruijsenaars wavefunctions as modular group matrix coefficients |
| topic | Mathematical Physics Quantum Algebra Representation Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2402.14214 |