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 Uspekhi Mat. Nauk, 2009, Volume 64, Issue 6(390), Pages 117–168 (Mi umn9331)

Orbifold Riemann surfaces: Teichmüller spaces and algebras of geodesic functions

M. Mazzoccoa, L. O. Chekhovbcd

a Loughborough University, UK
b Alikhanov Institute for Theoretical and~Experimental Physics, Moscow
c Steklov Mathematical Institute, Moscow
d Laboratoire Poncelet Franco--Russie, Moscow

Abstract: A fat graph description is given for Teichmüller spaces of\linebreak Riemann surfaces with holes and with ${\mathbb Z}_2$- and ${\mathbb Z}_3$-orbifold points (conical singularities) in the Poincaré uniformization. The corresponding mapping class group transformations are presented, geodesic functions are constructed, and the Poisson structure is introduced. The resulting Poisson algebras are then quantized. In the particular cases of surfaces with $n$ ${\mathbb Z}_2$-orbifold points and with one and two holes, the respective algebras $A_n$ and $D_n$ of geodesic functions (classical and quantum) are obtained. The infinite-dimensional Poisson algebra ${\mathfrak D}_n$, which is the semiclassical limit of the twisted $q$-Yangian algebra $Y'_q(\mathfrak{o}_n)$ for the orthogonal Lie algebra $\mathfrak{o}_n$, is associated with the algebra of geodesic functions on an annulus with $n$ ${\mathbb Z}_2$-orbifold points, and the braid group action on this algebra is found. From this result the braid group actions are constructed on the finite-dimensional reductions of this algebra: the $p$-level reduction and the algebra $D_n$. The central elements for these reductions are found. Also, the algebra ${\mathfrak D}_n$ is interpreted as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple point.
Bibliography: 36 titles.

Keywords: conical singularities, moduli space, geodesic algebra, quantization.

DOI: https://doi.org/10.4213/rm9331

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English version:
Russian Mathematical Surveys, 2009, 64:6, 1079–1130

Bibliographic databases:

Document Type: Article
UDC: 515.165.7+517.545
MSC: Primary 30F60, 32G15; Secondary 53D17

Citation: M. Mazzocco, L. O. Chekhov, “Orbifold Riemann surfaces: Teichmüller spaces and algebras of geodesic functions”, Uspekhi Mat. Nauk, 64:6(390) (2009), 117–168; Russian Math. Surveys, 64:6 (2009), 1079–1130

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn9331
• https://doi.org/10.4213/rm9331
• http://mi.mathnet.ru/eng/umn/v64/i6/p117

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This publication is cited in the following articles:
1. E. M. Chirka, “Prostranstva Teikhmyullera”, Lekts. kursy NOTs, 15, MIAN, M., 2010, 3–150
2. Chekhov L., Mazzocco M., “Shear coordinate description of the quantized versal unfolding of a $D_4$ singularity”, J. Phys. A, 43:44 (2010), 442002, 13 pp.
3. Felikson A., Shapiro M., Tumarkin P., “Cluster algebras and triangulated orbifolds”, Adv. Math., 231:5 (2012), 2953–3002
4. Felikson A., Tumarkin P., “Bases For Cluster Algebras From Orbifolds”, Adv. Math., 318 (2017), 191–232
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