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${\rm SL}_2$ quantum
trace in quantum Teichmüller theory via writhe
Hyun Kyu Kim, Thang T Q Lê and Miri Son |
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Algebraic & Geometric Topology 23 (2023) 339–418
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Abstract |
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Quantization of the Teichmüller space of a punctured Riemann surface is an approach to –dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop in gives rise to a natural trace-of-monodromy function on the Teichmüller space. For any ideal triangulation of , this function is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of . An important problem was to construct a quantization of this function, , namely to replace it by a noncommutative Laurent polynomial in the quantum variables. This problem, which is closely related to the framed protected spin characters in physics, has been solved by Allegretti and Kim using Bonahon and Wong’s quantum trace for skein algebras, and by Gabella using Gaiotto, Moore and Neitzke’s Seiberg–Witten curves, spectral networks, and writhe of links. We show that these two solutions to the quantization problem coincide. We enhance Gabella’s solution and show that it is a twist of the Bonahon–Wong quantum trace. |
Keywords
quantum Teichmüller spaces, Bonahon–Wong quantum trace,
skein algebra of surfaces, Seiberg–Witten curves and
spectral networks, quantum cluster varieties
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Mathematical Subject Classification
Primary: 13F60, 46L85, 53D55, 81R60
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References |
Publication
Received: 2 April 2021
Revised: 20 August 2021
Accepted: 30 September 2021
Published: 27 March 2023
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Authors |
| Hyun Kyu Kim | |
| School of Mathematics Korea Institute for Advanced Study Seoul South Korea |
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| Thang T Q Lê | |
| School of Mathematics Georgia Institute of Technology Atlanta, GA United States |
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| Miri Son | |
| Department of Mathematics Rice University Houston, TX United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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