RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 4 scientific papers (total in 4 papers)

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

Full text: PDF file (1171 kB)
References: PDF file   HTML file

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
Received: 10.11.2009

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
\Bibitem{MazChe09}
\by M.~Mazzocco, L.~O.~Chekhov
\paper Orbifold Riemann surfaces: Teichm\"uller~spaces and algebras of geodesic functions
\jour Uspekhi Mat. Nauk
\yr 2009
\vol 64
\issue 6(390)
\pages 117--168
\mathnet{http://mi.mathnet.ru/umn9331}
\crossref{https://doi.org/10.4213/rm9331}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2640967}
\zmath{https://zbmath.org/?q=an:05711123}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2009RuMaS..64.1079M}
\elib{http://elibrary.ru/item.asp?id=20425328}
\transl
\jour Russian Math. Surveys
\yr 2009
\vol 64
\issue 6
\pages 1079--1130
\crossref{https://doi.org/10.1070/RM2009v064n06ABEH004653}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000278425000002}
\elib{http://elibrary.ru/item.asp?id=15299784}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77951282463}


Linking options:
  • http://mi.mathnet.ru/eng/umn9331
  • https://doi.org/10.4213/rm9331
  • http://mi.mathnet.ru/eng/umn/v64/i6/p117

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. E. M. Chirka, “Prostranstva Teikhmyullera”, Lekts. kursy NOTs, 15, MIAN, M., 2010, 3–150  mathnet  crossref  zmath  elib
    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.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Felikson A., Shapiro M., Tumarkin P., “Cluster algebras and triangulated orbifolds”, Adv. Math., 231:5 (2012), 2953–3002  crossref  mathscinet  zmath  isi  elib  scopus
    4. Felikson A., Tumarkin P., “Bases For Cluster Algebras From Orbifolds”, Adv. Math., 318 (2017), 191–232  crossref  mathscinet  zmath  isi  scopus
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:560
    Full text:138
    References:48
    First page:17

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019