Abstract
LetSU X r be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofL k with the space of conformal blocks defined in terms of representations of the Lie algebrasl r (C((z))). It follows in particular that the dimension ofH 0(SU X r,L k) is given by the Verlinde formula.
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[A-D-K] Arbarello, E., De Concini, C., Kac, V.: The infinite wedge representation and the reciprocity law for algebraic curves. Proc. of Symp. in Pure Math.49, 171–190 (1989)
[B] Bourbaki, N.: Algèbre commutative, ch. 5 to 7. Paris: Masson 1985
[D-M] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. IHES36, 75–110 (1969)
[D-N] Drezet, J.M., Narasimhan, M.S.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math.97, 53–94 (1989)
[F] Faltings, G.: A proof for the Verlinde formula. J. Alg. Geom., to appear
[G] Gepner, D.: Fusion rings and geometry. Commun. Math. Phys.141, 381–411 (1991)
[K] Kac, V.: Infinite dimensional Lie algebras. Progress. in Math.44, Boston: Birkhäuser 1983
[Ku] Kumar, S.: Demazure character formula in arbitrary Kac-Moody setting. Invent. Math.89, 395–423 (1987)
[K-N-R] Kumar, S., Narasimhan, M.S., Ramanathan, A.: Infinite Grassmannian and moduli space ofG-bundles. Preprint (1993)
[L] Laumon, G.: Un analogue global du cône nilpotent. Duke Math. J.57, 647–671 (1988)
[L-MB] Laumon, G., Moret-Bailly, L.: Champs algébriques. Prépublication Université Paris-Sud (1992)
[M] Mathieu, O.: Formules de caractères pour les algèbres de Kac-Moody générales. Astérisque159–160 (1988)
[P-S] Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. Publ. Math. IHES42, 47–119 (1973)
[SGA4 1/2] Cohomologie étale. Séminaire de Géométrie algébrique SGA4 1/2, par Deligne, P. Lecture Notes569. Berlin, Heidelberg, New York: Springer 1977
[SGA6] Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie algébrique SGA 6, dirigé par Berthelot, P., Grothendieck, A., Illusie, L. Lecture Notes225. Berlin, Heidelberg, New York: Springer 1971
[S1] Slodowy, P.: On the geometry of Schubert varieties attached to Kac-Moody Lie algebras. Can. Math. Soc. Conf. Proc.6, 405–442 (1984)
[S-W] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES61, 5–65 (1985)
[T] Tate, J.: Residues of differentials on curves. Ann. Scient. Éc. Norm. Sup.1 (4ème série). 149–159 (1968)
[Tu] Tu, L.: Semistable bundles over an elliptic curve. Adv. in Math.98, 1–26 (1993)
[T-U-Y] Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Studies in Pure Math.19, 459–566 (1989)
[V] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360–376 (1988)
[W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1986)
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Communicated by G. Felder
Both authors were partially supported by the European Science Project “Geometry of Algebraic Varieties,” Contract no. SCI-0398-C(A)
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Beauville, A., Laszlo, Y. Conformal blocks and generalized theta functions. Commun.Math. Phys. 164, 385–419 (1994). https://doi.org/10.1007/BF02101707
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DOI: https://doi.org/10.1007/BF02101707