Quantum Teichmüller space
L. Chekhov, V. V. Fock
TL;DR
The paper tackles the problem of quantizing the Teichmüller space of open surfaces while preserving the mapping class group symmetry. It develops a deformation-quantized framework using a graph-based coordinate system and quantum dilogarithm flips, proving a pentagon-like consistency result (Theorem 4) that ensures a well-defined action of the modular groupoid. Key contributions include an explicit family of algebras $\mathcal{T}^{\hbar}(S)$, a central subalgebra generated by face sums, and unitary representations parametrized by hole lengths, all encapsulated in a symmetry between $\hbar$ and $1/\hbar$ and linked to Liouville conformal blocks. This work provides a rigorous algebraic scaffold for quantum Teichmüller geometry with potential implications for 2+1D gravity and conformal field theory.
Abstract
We describe explicitly a noncommutative deformation of the *-algebra of functions on the Teichmüller space of Riemann surfaces with holes equivariant w.r.t. the mapping class group action.