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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.
Kim, Hyun Kyu 1 ; Lê, Thang T Q 2 ; Son, Miri 3
@article{10_2140_agt_2023_23_339,
author = {Kim, Hyun Kyu and L\^e, Thang T Q and Son, Miri},
title = {SL2 quantum trace in quantum {Teichm\"uller} theory via writhe},
journal = {Algebraic and Geometric Topology},
pages = {339--418},
publisher = {mathdoc},
volume = {23},
number = {1},
year = {2023},
doi = {10.2140/agt.2023.23.339},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.339/}
}
TY - JOUR AU - Kim, Hyun Kyu AU - Lê, Thang T Q AU - Son, Miri TI - SL2 quantum trace in quantum Teichmüller theory via writhe JO - Algebraic and Geometric Topology PY - 2023 SP - 339 EP - 418 VL - 23 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.339/ DO - 10.2140/agt.2023.23.339 ID - 10_2140_agt_2023_23_339 ER -
%0 Journal Article %A Kim, Hyun Kyu %A Lê, Thang T Q %A Son, Miri %T SL2 quantum trace in quantum Teichmüller theory via writhe %J Algebraic and Geometric Topology %D 2023 %P 339-418 %V 23 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.339/ %R 10.2140/agt.2023.23.339 %F 10_2140_agt_2023_23_339
Kim, Hyun Kyu; Lê, Thang T Q; Son, Miri. SL2 quantum trace in quantum Teichmüller theory via writhe. Algebraic and Geometric Topology, Tome 23 (2023) no. 1, pp. 339-418. doi: 10.2140/agt.2023.23.339
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