Geodesic
Knizhnik–Zamolodchikov equations for positive genus and Krichever–Novikov algebras
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 4, pp. 737-770 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{RM_2004_59_4_a3,
     author = {M. Schlichenmaier and O. K. Sheinman},
     title = {Knizhnik{\textendash}Zamolodchikov equations for positive genus and {Krichever{\textendash}Novikov} algebras},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {737--770},
     year = {2004},
     volume = {59},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2004_59_4_a3/}
}
TY  - JOUR
AU  - M. Schlichenmaier
AU  - O. K. Sheinman
TI  - Knizhnik–Zamolodchikov equations for positive genus and Krichever–Novikov algebras
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2004
SP  - 737
EP  - 770
VL  - 59
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/RM_2004_59_4_a3/
LA  - en
ID  - RM_2004_59_4_a3
ER  - 
%0 Journal Article
%A M. Schlichenmaier
%A O. K. Sheinman
%T Knizhnik–Zamolodchikov equations for positive genus and Krichever–Novikov algebras
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2004
%P 737-770
%V 59
%N 4
%U http://geodesic.mathdoc.fr/item/RM_2004_59_4_a3/
%G en
%F RM_2004_59_4_a3
M. Schlichenmaier; O. K. Sheinman. Knizhnik–Zamolodchikov equations for positive genus and Krichever–Novikov algebras. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 4, pp. 737-770. http://geodesic.mathdoc.fr/item/RM_2004_59_4_a3/

[1] D. Bernard, “On the Wess–Zumino–Witten models on Riemann surfaces”, Nuclear Phys. B, 309:1 (1988), 145–174 | DOI | MR

[2] L. Bonora, M. Rinaldi, J. Russo, K. Wu, “The Sugawara construction on genus-$g$ Riemann surfaces”, Phys. Lett. B, 208:3–4 (1988), 440–446 | DOI | MR

[3] B. Enriquez, G. Felder, Solutions of the KZB equations in genus $\ge1$, arXiv: math/9912198

[4] G. Felder, Ch. Wieczerkowski, “Conformal blocks on elliptic curves and the Knizhnik–Zamolodchikov–Bernard equations”, Comm. Math. Phys., 176:1 (1996), 133–161 | DOI | MR | Zbl

[5] J. Harris, I. Morrison, Moduli of Curves, Springer-Verlag, New York, 1998 | MR

[6] N. J. Hitchin, “Flat connections and geometric quantization”, Comm. Math. Phys., 131:2 (1990), 347–380 | DOI | MR | Zbl

[7] V. Kats, Beskonechnomernye algebry Li, Mir, M., 1993 | MR | Zbl

[8] V. G. Kac, A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras, World Scientific, Teaneck, 1987 | MR

[9] V. G. Knizhnik, A. B. Zamolodchikov, “Current algebra and Wess–Zumino model in two dimensions”, Nuclear Phys. B, 247:1 (1984), 83–103 | DOI | MR | Zbl

[10] I. M. Krichever, S. P. Novikov, “Algebry tipa Virasoro, rimanovy poverkhnosti i struktury teorii solitonov”, Funkts. analiz i ego pril., 21:2 (1987), 46–63 | MR | Zbl

[11] I. M. Krichever, S. P. Novikov, “Algebry tipa Virasoro, rimanovy poverkhnosti i struny v prostranstve Minkovskogo”, Funkts. analiz i ego pril., 21:4 (1987), 47–61 | MR | Zbl

[12] I. M. Krichever, S. P. Novikov, “Algebry tipa Virasoro, tenzor energii-impulsa i operatornye razlozheniya na rimanovykh poverkhnostyakh”, Funkts. analiz i ego pril., 23:1 (1989), 24–40 | MR | Zbl

[13] I. M. Krichever, S. P. Novikov, “Golomorfnye rassloeniya i kommutiruyuschie raznostnye operatory. Dvukhtochechnye konstruktsii”, UMN, 55:3 (2000), 181–182 | MR | Zbl

[14] A. Ruffing, T. Deck, M. Schlichenmaier, “String branchings on complex tori and algebraic representations of generalized Krichever–Novikov algebras”, Lett. Math. Phys., 26:1 (1992), 23–32 | DOI | MR | Zbl

[15] V. A. Sadov, “Bases on multipunctured Riemann surfaces and interacting strings amplitudes”, Comm. Math. Phys., 136:3 (1991), 585–597 | DOI | MR | Zbl

[16] M. Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces, Lecture Notes in Phys., 322, Springer-Verlag, Berlin, 1989 | MR | Zbl

[17] M. Schlichenmaier, “Krichever–Novikov algebras for more than two points”, Lett. Math. Phys., 19:2 (1990), 151–165 | DOI | MR | Zbl

[18] M. Schlichenmaier, “Krichever–Novikov algebras for more than two points: explicit generators”, Lett. Math. Phys., 19:4 (1990), 327–336 | DOI | MR | Zbl

[19] M. Schlichenmaier, “Central extensions and semi-infinite wedge representations of Krichever–Novikov algebras for more than two points”, Lett. Math. Phys., 20:1 (1990), 33–46 | DOI | MR | Zbl

[20] M. Schlichenmaier, Verallgemeinerte Krichever–Novikov Algebren und deren Darstellungen, Ph.D. Thesis, Universität Mannheim, Mannheim, 1990

[21] M. Schlichenmaier, “Degenerations of generalized Krichever–Novikov algebras on tori”, J. Math. Phys., 34:8 (1993), 3809–3824 | DOI | MR | Zbl

[22] M. Schlichenmaier, “Differential operator algebras on compact Riemann surfaces”, Generalized Symmetries in Physics (Clausthal, Germany, 1993), eds. H.-D. Doebner, V. K. Dobrev, A. G. Ushveridze, World Scientific, River Edge, 1994, 425–434 | MR

[23] M. Schlichenmaier, “Local cocycles and central extensions for multipoint algebras of Krichever–Novikov type”, J. Reine Angew. Math., 559 (2003), 53–94 | MR | Zbl

[24] M. Schlichenmaier, “Higher genus affine algebras of Krichever–Novikov type”, Mosc. Math. J. (to appear) | MR

[25] M. Schlichenmaier, O. K. Sheinman, “The Sugawara construction and Casimir operators for Krichever–Novikov algebras”, J. Math. Sci. (New York), 92:2 (1998), 3807–3834 ; arXiv: q-alg/9512016 | DOI | MR | Zbl

[26] M. Shlikhenmaier, O. K. Sheinman, “Teoriya Vessa–Zumino–Vittena–Novikova, uravneniya Knizhnika–Zamolodchikova i algebry Krichevera–Novikova”, UMN, 54:1 (1999), 213–249 | MR

[27] O. K. Sheinman, “Ellipticheskie affinnye algebry Li”, Funkts. analiz i ego pril., 24:3 (1990), 51–61 | MR

[28] O. K. Sheinman, “Affinnye algebry Li na rimanovykh poverkhnostyakh”, Funkts. analiz i ego pril., 27:4 (1993), 54–62 | MR | Zbl

[29] O. K. Sheinman, “Representations of Krichever–Novikov algebras”, Topics in Topology and Mathematical Physics, ed. S. P. Novikov, Amer. Math. Soc., Providence, RI, 1995, 185–197 | MR | Zbl

[30] O. K. Sheinman, “Fermionnaya model predstavlenii affinnykh algebr Krichevera–Novikova”, Funkts. analiz i ego pril., 35:3 (2001), 60–72 | MR | Zbl

[31] O. K. Sheinman, “Second order Casimirs for the affine Krichever–Novikov algebras $\widehat{\mathfrak{gl}}_{g,2}$ and $\widehat{\mathfrak{sl}}_{g,2}$”, Mosc. Math. J., 1:4 (2001), 605–628 ; arXiv: math/0109001 | MR | Zbl

[32] O. K. Sheinman, “Kazimiry vtorogo poryadka dlya affinnykh algebr Krichevera–Novikova $\widehat{\mathfrak{{{gl}}}_{g,2}}$ i $\widehat{\mathfrak{{{sl}}}_{g,2}}$”, UMN, 56:5 (2001), 189–190 | MR | Zbl

[33] O. K. Sheinman, Krichever–Novikov algebras, their representations and applications, arXiv: math/0304020 | MR

[34] A. Tsuchiya, K. Ueno, Y. Yamada, “Conformal field theory on universal family of stable curves with gauge symmetries”, Adv. Stud. Pure Math., 19 (1989), 459–566 | MR | Zbl

[35] K. Ueno, “Introduction to conformal field theory with gauge symmetries”, Geometry and Physics (Aarhus, 1995), eds. J. E. Andersen et al., Dekker, New York, 1997, 603–745 | MR | Zbl