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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 4(358), Pages 147–180 (Mi umn760)  

This article is cited in 16 scientific papers (total in 16 papers)

Knizhnik–Zamolodchikov equations for positive genus and Krichever–Novikov algebras

M. Schlichenmaiera, O. K. Sheinmanbc

a University of Luxembourg
b Steklov Mathematical Institute, Russian Academy of Sciences
c Independent University of Moscow

Abstract: In this paper a global operator approach to the Wess–Zumino–Witten–Novikov theory for compact Riemann surfaces of arbitrary genus with marked points is developed. The term ‘global’ here means that Krichever–Novikov algebras of gauge and conformal symmetries (that is, algebras of global symmetries) are used instead of loop algebras and Virasoro algebras (which are local in this context). The basic elements of this global approach are described in a previous paper of the authors (Russ. Math. Surveys 54:1 (1999)). The present paper gives a construction of the conformal blocks and of a projectively flat connection on the bundle formed by them.

DOI: https://doi.org/10.4213/rm760

Full text: PDF file (476 kB)
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English version:
Russian Mathematical Surveys, 2004, 59:4, 737–770

Bibliographic databases:

UDC: 517.774
MSC: Primary 17B66, 17B67, 81R10; Secondary 14H15, 14H55, 30F30
Received: 15.03.2004

Citation: M. Schlichenmaier, O. K. Sheinman, “Knizhnik–Zamolodchikov equations for positive genus and Krichever–Novikov algebras”, Uspekhi Mat. Nauk, 59:4(358) (2004), 147–180; Russian Math. Surveys, 59:4 (2004), 737–770

Citation in format AMSBIB
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\paper Knizhnik--Zamolodchikov equations for positive genus and Krichever--Novikov algebras
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\pages 147--180
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    This publication is cited in the following articles:
    1. Fialowski A., Schlichenmaier M., “Global geometric deformations of current algebras as Krichever-Novikov type algebras”, Comm. Math. Phys., 260:3 (2005), 579–612  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. O. K. Sheinman, “Projective Flat Connections on Moduli Spaces of Riemann Surfaces and the Knizhnik–Zamolodchikov Equations”, Proc. Steklov Inst. Math., 251 (2005), 293–304  mathnet  mathscinet  zmath
    3. Sheinman O.K., “Krichever-Novikov algebras and their representations”, Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemporary Mathematics Series, 391, 2005, 313–321  crossref  mathscinet  zmath  isi
    4. Fialowski A., Schlichenmaier M., “Global geometric deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebras”, Internat. J. Theoret. Phys., 46:11 (2007), 2708–2724  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. O. K. Sheinman, “Krichever–Novikov Algebras, their Representations and Applications in Geometry and Mathematical Physics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S85–S161  mathnet  crossref  crossref  zmath
    6. Schlichenmaier M., “Higher Genus Affine Lie Algebras of Krichever - Novikov Type”, Difference Equations, Special Functions and Orthogonal Polynomials, 2007, 589–599  crossref  mathscinet  zmath  isi
    7. Schlichenmaier M., “A global operator approach to Wess-Zumino-Novikov-Witten models”, XXVI Workshop on Geometrical Methods in Physics, AIP Conference Proceedings, 956, 2007, 107–119  crossref  mathscinet  zmath  adsnasa  isi
    8. M. Schlichenmaier, O. K. Sheinman, “Central extensions of Lax operator algebras”, Russian Math. Surveys, 63:4 (2008), 727–766  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Wagemann F., “Deformations of Lie algebras of vector fields arising from families of schemes”, J. Geom. Phys., 58:2 (2008), 165–178  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Schlichenmaier M., “Classification of central extensions of Lax operator algebras”, Geometric Methods in Physics, AIP Conference Proceedings, 1079, 2008, 227–234  crossref  mathscinet  zmath  adsnasa  isi
    11. Schlichenmaier M., “Deformations of the Witt, Virasoro, and Current Algebra”, Generalized Lie Theory in Mathematics, Physics and Beyond, 2009, 219–234  crossref  mathscinet  zmath  isi
    12. MARTIN SCHLICHENMAIER, “KRICHEVER-NOVIKOV TYPE ALGEBRAS — PERSONAL RECOLLECTIONS OF JULIUS WESS”, Int. J. Mod. Phys. Conf. Ser, 13:01 (2012), 158  crossref
    13. Cox B. Guo X. Lu R. Zhao K., “N-Point Virasoro Algebras and Their Modules of Densities”, Commun. Contemp. Math., 16:3 (2014), 1350047  crossref  mathscinet  zmath  isi
    14. O. K. Sheinman, “Lax operator algebras and integrable systems”, Russian Math. Surveys, 71:1 (2016), 109–156  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. Schlichenmaier M., “N -point Virasoro algebras are multipoint Krichever–Novikov-type algebras”, Commun. Algebr., 45:2 (2017), 776–821  crossref  mathscinet  zmath  isi  elib  scopus
    16. Cox B., Guo X., Lu R., Zhao K., “Simple Superelliptic Lie Algebras”, Commun. Contemp. Math., 19:3 (2017), 1650032  crossref  mathscinet  zmath  isi
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