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Quantization of Teichmüller spaces and the quantum dilogarithm. (English) Zbl 0897.57014
From the introduction: “Despite the progress in understanding and solving the quantum Chern-Simons field theory with a compact gauge group, much remains unclear in the case of noncompact groups. The major mathematical motivation is provided by the possibility of constructing new topological three-manifold invariants.”
“The purpose of the present letter is to quantize the Teichmüller space of punctured surfaces with the Weil-Peterson symplectic structure. This Teichmüller space 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. We start from the Penner parametrization of the decorated Teichmüller space where the mapping class group is realized explicitly through rational transformations generated by composition of the elementary Ptolemy transformation. The latter transformation is canonical and the quantum dilogarithm implements this transformation on the quantum level. Our approach is similar to the combinatorial quantization of the Chern-Simons theory with compact gauge groups”.

57M99 General low-dimensional topology
37D99 Dynamical systems with hyperbolic behavior
57M50 General geometric structures on low-dimensional manifolds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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