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.
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Kashaev, R.M. Quantization of Teichmüller Spaces and the Quantum Dilogarithm. Letters in Mathematical Physics 43, 105–115 (1998). https://doi.org/10.1023/A:1007460128279
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DOI: https://doi.org/10.1023/A:1007460128279