Quantization of Teichmüller spaces and the quantum dilogarithm

TL;DR

This work shows that the Teichmüller space of punctured surfaces with the Weil–Petersson structure can be obtained as a Hamiltonian reduction of a finite‑dimensional phase space associated with decorated triangulations. Upon quantization, the mapping class group actions are realized by quantum dilogarithm transformations, giving rise to (projective) representations. The paper develops both non‑compact (infinite‑dimensional) and compact (root‑of‑unity, finite‑dimensional) quantizations, connected through a common combinatorial framework based on Ptolemy moves and pentagon relations. This ties quantum Teichmüller theory to quantum hyperbolic invariants and the broader program of combinatorial Chern–Simons quantization. The results illuminate how quantum dilogarithms encode modular transformations in the quantum geometry of surfaces.

Abstract

The Teichmüller space of punctured surfaces with the Weil-Petersson symplectic structure and the action of the mapping class group is realized as the Hamiltonian reduction of a finite dimensional symplectic space where the mapping class group acts by symplectic rational transformations. Upon quantization the corresponding (projective) representation of the mapping class group is generated by the quantum dilogarithms.

Figures (4)

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Theorems & Definitions (14)

• Definition 1
• Theorem 1: Penner 1987a, 1987b
• Theorem 2: Penner 1987a
• Definition 2
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6