Conformal blocks and cohomology in genus 0

Belkale, Prakash; Mukhopadhyay, Swarnava

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

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 $G_2$.

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 $G_2$.

Classification : 17B67, 14H60, 32G34, 81T40
Keywords: conformal blocks, logarithmic forms, singular cohomology
Mot clés : blocs conformes, formes logarithmiques, cohomologie singulière

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

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

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

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

[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 | Zbl

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

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

[8] Looijenga, Eduard Unitarity of $\mathrm{SL}\left(2\right)$-conformal blocks in genus zero, J. Geom. Phys., Volume 59 (2009) no. 5, pp. 654-662 | DOI | MR | Zbl

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

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

[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, Volume 100 (1995) no. 1-3, pp. 93-102 | DOI | MR | Zbl

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

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

[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 | Zbl

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

[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 (Adv. Stud. Pure Math.), Volume 19, Academic Press, Boston, MA, 1989, pp. 459-566 | MR | Zbl

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

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