Projective Flat Connections on Moduli Spaces of Riemann Surfaces and the Knizhnik--Zamolodchikov Equations

O. K. Sheinman

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 307-319.

Résumé

A global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus with marked points is developed. Here, the globality means that one uses the Krichever–Novikov algebras of gauge and conformal symmetries (i.e., of global symmetries) instead of the loop and Virasoro algebras, which are local in this context. A thorough account of the global approach with all necessary details from the theory of Krichever–Novikov algebras and their representations was given by the author earlier (Usp. Mat. Nauk, 1999, vol. 54, no. 1; 2004, vol. 59, no. 4). This paper focuses on the geometric ideas that underlie our construction of conformal blocks. We prove the invariance of these blocks with respect to the (generalized) Knizhnik–Zamolodchikov connection and the projective flatness of this connection.

Bibliographie

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

[2] Beauville A., Laszlo Y., “Conformal blocks and generalized theta functions”, Commun. Math. Phys., 164 (1994), 385–419 | DOI | MR | Zbl

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

[4] Enriquez B., Felder G., Solutions of the KZB equations in genus $\geqslant1$, , 1999 arXiv: /math.QA/9912198 | MR

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

[6] Grinevich P. G., Orlov A. Yu., Flag spaces in KP theory, Virasoro action on $\det D$ and Segal–Wilson $\tau$-function, Preprint CLNS 945/89, Cornell Univ., Ithaca, NY, 1989 | MR

[7] Harris J., Morrison I., Moduli of curves, Springer, New York, Berlin, Heidelberg, 1998 | MR

[8] Hitchin N., “Flat connections and geometric quantization”, Commun. Math. Phys., 131 (1990), 347–380 | DOI | MR | Zbl

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

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

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

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

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

[15] Schlichenmaier M., Verallgemeinerte Krichever–Novikov Algebren und deren Darstellungen, PhD Thes. Univ. Mannheim, 1990

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

[17] Schlichenmaier M., “Differential operator algebras on compact Riemann surfaces”, Generalized symmetries in physics (Clausthal, 1993), eds. H.-D. Doebner, V. K. Dobrev, A. G. Ushveridze, World Sci. Publ., River Edge, NJ, 1994, 425–434 | MR

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

[19] Schlichenmaier M., “Higher genus affine algebras of Krichever–Novikov type”, Moscow Math. J., 3:4 (2003), 1395–1427 ; arXiv: math.QA/0210360 | MR | Zbl

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

[21] Shlikhenmaier M., Sheinman O. K., “Teoriya Vessa–Zumino–Vittena–Novikova, uravneniya Knizhnika–Zamolodchikova i algebry Krichevera–Novikova”, UMN, 54:1 (1999), 213–250 ; arXiv: math.QA/9812083 | MR

[22] Shlikhenmaier M., Sheinman O. K., “Uravneniya Knizhnika–Zamolodchikova dlya polozhitelnogo roda i algebry Krichevera–Novikova”, UMN, 59:4 (2004), 147–180 ; arXiv: math.AG/0312040 | MR

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

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

[25] Sheinman O. K., “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

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

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

[28] Sheinman O. K., “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

[29] Sheinman O. K., “Affine Krichever–Novikov algebras, their representations and applications”, Geometry, topology, and mathematical physics, Sel. pap. S. P. Novikov's seminar (Moscow, 2002–2003), AMS Transl. Ser. 2, 212, eds. V. M. Buchstaber, I. M. Krichever, Amer. Math. Soc., Providence, RI, 2004, 297–316 ; arXiv: math.RT/0304020 | MR | Zbl

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

[31] Tyurin A. N., Quantization, classical and quantum field theory and theta functions, CRM Monogr. Ser., 21, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[32] Ueno K., “Introduction to conformal field theory with gauge symmetries”, Geometry and physics, Proc. Conf. Aarhus Univ., 1995, Lect. Notes Pure and Appl. Math., 184, eds. J. E. Andersen et al., M. Dekker, New York, 1997, 603–745 | MR | Zbl