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Quantization of the universal Teichmüller space. (English) Zbl 1230.37057

Author’s abstract: “In the first part of this article we describe the complex geometry of the universal Teichmüller space, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The universal Teichmüller space contains classical Teichmüller spaces \(T(G)\), where \(G\) is a Fuchsian group, as complex submanifolds. The quotient Diff\(_{+}(S^{1})\)/Möb\((S^{1})\) of the diffeomorphism group of the unit circle modulo Möbius transformations can be considered as a ’smooth’ part of the universal Teichmüller space. The second part is devoted to the quantization of the universal Teichmüller space. The smooth part Diff\(_{+}(S^{1})\)/Möb\((S^{1})\) may be quantized, using its embedding into the Hilbert-Schmidt Grassmannian. However, this quantization method does not apply to the whole universal Teichmüller space, to which the ’quantized calculus’ of Connes and Sullivan may be applied.”

MSC:

37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
53D50 Geometric quantization
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References:

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