Conformal blocks  and cohomology in genus 0

Belkale, Prakash; Mukhopadhyay, Swarnava

doi:10.5802/aif.2893

Conformal blocks and cohomology in genus 0

Belkale, Prakash; Mukhopadhyay, Swarnava

Annales de l'Institut Fourier, Volume 64 (2014) no. 4, p. 1669-1719

Abstract
Résumé

We give a characterization of conformal blocks in terms of the singular cohomology of suitable smooth projective varieties, in genus $0$ for classical Lie algebras and $G_{2}$ .

Nous donnons une caractérisation des blocs conformes en termes de cohomologie singulière des variétés projectives lisses appropriées, dans le genre $0$ pour les algèbres de Lie classiques et $G_{2}$ .

MR 3329676 | Zbl 06387320

DOI : https://doi.org/10.5802/aif.2893
Classification: 17B67, 14H60, 32G34, 81T40
Keywords: conformal blocks, logarithmic forms, singular cohomology

@article{AIF_2014__64_4_1669_0,
     author = {Belkale, Prakash and Mukhopadhyay, Swarnava},
     title = {Conformal blocks  and cohomology in genus 0},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     pages = {1669-1719},
     doi = {10.5802/aif.2893},
     mrnumber = {3329676},
     zbl = {06387320},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_4_1669_0}
}

Belkale, Prakash; Mukhopadhyay, Swarnava. Conformal blocks  and cohomology in genus 0. Annales de l'Institut Fourier, Volume 64 (2014) no. 4, pp. 1669-1719. doi : 10.5802/aif.2893. http://www.numdam.org/item/AIF_2014__64_4_1669_0/

References

[1] Awata, Hidetoshi; Tsuchiya, Akihiro; Yamada, Yasuhiko Integral formulas for the WZNW correlation functions, Nuclear Phys. B, Tome 365 (1991) no. 3, pp. 680-696 | Article | MR 1136712

[2] Beauville, Arnaud Conformal blocks, fusion rules and the Verlinde formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Bar-Ilan Univ., Ramat Gan (Israel Math. Conf. Proc.) Tome 9 (1996), pp. 75-96 | MR 1360497 | Zbl 0848.17024

[3] Belkale, Prakash Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 for arbitrary Lie algebras, J. Math. Pures Appl. (9), Tome 98 (2012) no. 4, pp. 367-389 | Article | MR 2968161 | Zbl 1277.14035

[4] Bezrukavnikov, Roman; Finkelberg, Michael; Schechtman, Vadim Factorizable sheaves and quantum groups, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1691 (1998), pp. x+287 | MR 1641131 | Zbl 0938.17016

[5] Brieskorn, Egbert Sur les groupes de tresses [d’après V. I. Arnolʼd], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin (1973), p. 21-44. Lecture Notes in Math., Vol. 317 | Numdam | MR 422674 | Zbl 0277.55003

[6] Deligne, Pierre Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971) no. 40, pp. 5-57 | Article | Numdam | MR 498551 | Zbl 0219.14007

[7] Feigin, Boris; Schechtman, Vadim; Varchenko, Alexander On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys., Tome 170 (1995) no. 1, pp. 219-247 http://projecteuclid.org/euclid.cmp/1104272957 | Article | MR 1331699 | Zbl 0842.17043

[8] Looijenga, Eduard Unitarity of $SL (2)$ -conformal blocks in genus zero, J. Geom. Phys., Tome 59 (2009) no. 5, pp. 654-662 | Article | MR 2518993 | Zbl 1165.32304

[9] Looijenga, Eduard The KZ system via polydifferentials, Arrangements of hyperplanes—Sapporo 2009, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 62 (2012), pp. 189-231 | MR 2933798 | Zbl 1260.32004

[10] Ramadas, T. R. The “Harder-Narasimhan trace” and unitarity of the KZ/Hitchin connection: genus 0, Ann. of Math. (2), Tome 169 (2009) no. 1, pp. 1-39 | Article | MR 2480600 | Zbl 1167.32011

[11] Schechtman, Vadim V.; Terao, Hiroaki; Varchenko, Alexander N. Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra, Tome 100 (1995) no. 1-3, pp. 93-102 | Article | MR 1344845 | Zbl 0849.32025

[12] Schechtman, Vadim V.; Varchenko, Alexander N. Arrangements of hyperplanes and Lie algebra homology, Invent. Math., Tome 106 (1991) no. 1, pp. 139-194 | Article | MR 1123378 | Zbl 0754.17024

[13] Serre, Jean-Pierre Complex semisimple Lie algebras, Springer-Verlag, Berlin, Springer Monographs in Mathematics (2001), pp. x+74 (Translated from the French by G. A. Jones, Reprint of the 1987 edition) | Article | MR 1808366 | Zbl 1058.17005

[14] Sorger, Christoph La formule de Verlinde, Astérisque (1996) no. 237, pp. Exp. No. 794, 3, 87-114 (Séminaire Bourbaki, Vol. 1994/95) | Numdam | MR 1423621 | Zbl 0878.17024

[15] Tsuchiya, Akihiro; Kanie, Yukihiro Vertex operators in conformal field theory on $P^{1}$ and monodromy representations of braid group, Conformal field theory and solvable lattice models (Kyoto, 1986), Academic Press, Boston, MA (Adv. Stud. Pure Math.) Tome 16 (1988), pp. 297-372 | MR 972998 | Zbl 0661.17021

[16] Tsuchiya, Akihiro; Ueno, Kenji; Yamada, Yasuhiko Conformal field theory on universal family of stable curves with gauge symmetries, Integrable systems in quantum field theory and statistical mechanics, Academic Press, Boston, MA (Adv. Stud. Pure Math.) Tome 19 (1989), pp. 459-566 | MR 1048605 | Zbl 0696.17010

[17] Ueno, Kenji Conformal field theory with gauge symmetry, American Mathematical Society, Providence, RI; Fields Institute for Research in Mathematical Sciences, Toronto, ON, Fields Institute Monographs, Tome 24 (2008), pp. viii+168 | MR 2433154 | Zbl 1188.81004

[18] Varchenko, A. Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, World Scientific Publishing Co., Inc., River Edge, NJ, Advanced Series in Mathematical Physics, Tome 21 (1995), pp. x+371 | Article | MR 1384760 | Zbl 0951.33001