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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

Algebraic & Geometric Topology 23 (2023) 339–418

Quantization of the Teichmüller space of a punctured Riemann surface S is an approach to 3–dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop γ in S gives rise to a natural trace-of-monodromy function 𝕀(γ) on the Teichmüller space. For any ideal triangulation Δ of S, this function 𝕀(γ) is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of Δ. An important problem was to construct a quantization of this function, 𝕀(γ), 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 SL 2 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.

quantum Teichmüller spaces, Bonahon–Wong quantum trace, skein algebra of surfaces, Seiberg–Witten curves and spectral networks, quantum cluster varieties
Mathematical Subject Classification
Primary: 13F60, 46L85, 53D55, 81R60
Received: 2 April 2021
Revised: 20 August 2021
Accepted: 30 September 2021
Published: 27 March 2023
Hyun Kyu Kim
School of Mathematics
Korea Institute for Advanced Study
South Korea
Thang T Q Lê
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States
Miri Son
Department of Mathematics
Rice University
Houston, TX
United States

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