Geodesic
Quantization of the Universal Teichm\"uller Space
Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 173-200.

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In the first part of the paper, we describe the Kähler geometry of the universal Teichmüller space, which can 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 $\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ of the diffeomorphism group of the unit circle modulo Möbius transformations can be considered as a “smooth” part of the universal Teichmüller space. In the second part we describe how to quantize $\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ by embedding it in an infinite-dimensional Siegel disc. This quantization method does not apply to the whole universal Teichmüller space. However, this space can be quantized using the “quantized calculus” of A. Connes and D. Sullivan. 
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     title = {Quantization of the {Universal} {Teichm\"uller} {Space}},
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     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a12/}
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A. G. Sergeev. Quantization of the Universal Teichm\"uller Space. Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 173-200. http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a12/

[1] Alfors L., Lektsii po kvazikonformnym otobrazheniyam, Mir, M., 1969 | MR

[2] Ahlfors L. V., “Some remarks on Teichmüller's space of Riemann surfaces”, Ann. Math. Ser. 2, 74 (1961), 171–191 | DOI | MR | Zbl

[3] Bowen R., “Hausdorff dimension of quasi-circles”, Publ. Math. IHES, 50 (1979), 11–25 | MR | Zbl

[4] Connes A., Géométrie non commutative, Intereditions, Paris, 1990 | MR | Zbl

[5] Lehto O., Univalent functions and Teichmüller spaces, Springer, Berlin, 1987 | MR | Zbl

[6] Nag S., The complex analytic theory of Teichmüller spaces, J. Wiley and Sons, New York, 1988 | MR | Zbl

[7] Nag S., “A period mapping in universal Teichmüller space”, Bull. Amer. Math. Soc., 26 (1992), 280–287 | DOI | MR | Zbl

[8] Nag S., Sullivan D., “Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle”, Osaka J. Math., 32 (1995), 1–34 | MR | Zbl

[9] Nag S., Verjovsky A., “$\operatorname{Diff}(S^1)$ and the Teichmüller spaces”, Commun. Math. Phys., 130 (1990), 123–138 | DOI | MR | Zbl

[10] Pressli E., Sigal G., Gruppy petel, Mir, M., 1990 | MR

[11] Power S. C., Hankel operators on Hilbert space, Res. Notes Math., 64, Pitman, Boston, 1982 | MR | Zbl

[12] Sniatycki J., Geometric quantization and quantum mechanics, Springer, New York, 1980 | MR | Zbl

[13] Segal G., “Unitary representations of some infinite-dimensional groups”, Commun. Math. Phys., 80 (1981), 301–342 | DOI | MR | Zbl

[14] Shale D., “Linear symmetries of free boson fields”, Trans. Amer. Math. Soc., 103 (1962), 149–167 | DOI | MR | Zbl

[15] Tuynman G. M., Mathematical structures in field theories, Proc. sem. 1983–1985, V. 1, Geometric quantization, CWI Syllabus, 8, Math. Centr., Amsterdam, 1985 | MR | Zbl

[16] Woodhouse N., Geometric quantization, Clarendon Press, Oxford, 1980 | MR | Zbl