Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 037, 67 pages      arXiv:2405.14727      https://doi.org/10.3842/SIGMA.2026.037

Quantized Geodesic Lengths for Teichmüller Spaces: Algebraic Aspects

Hyun Kyu Kim
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, South Korea

Received June 09, 2025, in final form March 13, 2026; Published online April 15, 2026

Abstract
In 1980's H. Verlinde suggested to construct and use a quantization of Teichmüller spaces to construct spaces of conformal blocks for the Liouville conformal field theory. This suggestion led to a mathematical formulation by Fock in 1990's and later by Fock, Goncharov and Shen, called the modular functor conjecture, based on the Chekhov-Fock quantum Teichmüller theory. In 2000's, Teschner combined the Chekhov-Fock version and the Kashaev version of quantum Teichmüller theory to construct a solution to a modified form of the conjecture. We embark on a direct approach to the conjecture based on the Chekhov-Fock(-Goncharov) theory. We construct quantized trace-of-monodromy along simple loops via Bonahon and Wong's quantum trace maps developed in 2010's, and investigate algebraic structures of them, which will eventually lead to construction and properties of quantized geodesic length operators. We show that a special recursion relation used by Teschner is satisfied by the quantized trace-of-monodromy, and that the quantized trace-of-monodromy for disjoint loops commute in a certain strong sense.

Key words: quantized geodesic length; quantum Teichmüller space; quantized trace-of-monodromy; Bonahon-Wong quantum trace; modular functor conjecture; quantum cluster variety.

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