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## Conformal blocks and generalized theta functions (1994)

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Venue: | Comm. Math. Phys |

Citations: | 141 - 8 self |

### Citations

1057 |
Infinite dimensional Lie algebras,
- Kac
- 1990
(Show Context)
Citation Context ... central extension and the Tate residue (4.9) We want to show that at the level of Lie algebras, the extension (E) is the universal central extension which appears in the theory of Kac-Moody algebras =-=[K]-=-. This is essentially known (see e.g. [A-D-K], where very similar computations appear). We have included the computation because it is extremely simple and gives a nice generalization of the residue d... |

918 |
Quantum field theory and the Jones polynomial,
- Witten
- 1989
(Show Context)
Citation Context ...mension in a purely combinatorial way, giving the famous Verlinde formula ([V], see cor. 8.6). The isomorphism B c (r) �� \Gamma! H 0 (SU X (r); L c ) is certainly known to the physicists -- see e=-=.g. [W]-=-. Our point is that this can be proved in a purely mathematical way. In fact we hope to convince the reader that even in an infinite-dimensional context, the methods of algebraic geometry provide a fl... |

506 | The irreducibility of the space of curves of given genus,
- Deligne, Mumford
- 1969
(Show Context)
Citation Context ...osition of the Kac-Moody groups theory [Ku, Sl]. 3. The stack SL r (O)nSL r (K)=SL r (AX ) Stacks We will need a few properties of stacks. Rather than giving formal definitions (for which we refer to =-=[D-M]-=- and especially [L-MB]), we will try here to give a rough idea of what stacks are and what they are good for. Many geometric objects (like vector bundles on a fixed variety, or varieties of a given ty... |

291 |
Fusion Rules And Modular Transformations In 2d Conformal Field Theory,”
- Verlinde
- 1988
(Show Context)
Citation Context ...y [T-U-Y] this implies that the space H 0 (SU X (r); L c ) satisfies the socalledsfusion rules, which allow to compute its dimension in a purely combinatorial way, giving the famous Verlinde formula (=-=[V], se-=-e cor. 8.6). The isomorphism B c (r) �� \Gamma! H 0 (SU X (r); L c ) is certainly known to the physicists -- see e.g. [W]. Our point is that this can be proved in a purely mathematical way. In fac... |

272 |
Loop groups and equations of the KdV type,
- Segal, Wilson
- 1985
(Show Context)
Citation Context ...on the quotient Q = SL r (K)=SL r (O), which will turn out to be a much nicer object. Q as a Grassmannian The quotient space Q is related to the infinite Grassmannian used by the Japanese school (see =-=[S-W]-=-) in the following way. For any k-algebra R, define a lattice in R((z)) r as a sub-R[[z]]-module W of R((z)) r which is projective of rank r, and such that S z \Gamman W = R((z)) r . It is an exercise... |

189 |
Conformal field theory on universal family of stable curves with gauge symmetries, Integrable systems in quantum field theory and statistical mechanics,
- Tsuchiya, Ueno, et al.
- 1989
(Show Context)
Citation Context ...sion of the space of c th -order theta functions on the Jacobian of X, and is sometimes called the space of generalized theta functions. We will prove that it is canonically isomorphic to B c (r). By =-=[T-U-Y]-=- this implies that the space H 0 (SU X (r); L c ) satisfies the socalledsfusion rules, which allow to compute its dimension in a purely combinatorial way, giving the famous Verlinde formula ([V], see ... |

178 |
Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques,
- Drezet, Narashimhan
- 1989
(Show Context)
Citation Context ...� (see [M], prop. 15). It then follows from lemma 7.8 that the Picard group of SLX (r) is generated by the determinant bundle L (the corresponding statement for the moduli space SU X (r) is proved i=-=n [D-N]-=-). Appendix to x7: Integration of integrable highest weight modules (according to Faltings) In this appendix we want to show that integrable highest weight representations of the Lie algebra b sl r (K... |

144 | Théorie des intersections et théorème de Riemann-Roch - Berthelot, A - 1971 |

137 |
Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures
- Peskine, Szpiro
- 1973
(Show Context)
Citation Context ...0 ! O r\Gamma1 P 1 A[i] \Gamma\Gamma\Gamma! OP 1(N) r (M i1 ;:::;M ir ) \Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma! OP 1(rN) ! 0 ; where A[i] is obtained by deleting the i-th column of A(t) (see e.g. =-=[P-S]-=-, lemme 3.1). Taking cohomology we see that the map (P j ) 7! P k P k M ik of S r N into S rN is surjective, which implies the surjectivity of T A(t) (det). Therefore \Gamma (N) 0 is smooth, with the ... |

91 |
A proof for the Verlinde formula
- Faltings
- 1994
(Show Context)
Citation Context ...July 1992 we heard of G. Faltings beautiful ideas, which should prove at the same time both our result and that of [T-U-Y] (in the more general case of principal bundles). These ideas are sketched in =-=[F]-=-, but (certainly due to our own incompetence) we were unable to understand some of the key points in the proof. We have therefore decided after some time to write a complete version of our proof, if o... |

68 | Infinite Grassmannians and moduli spaces of G-bundles, Vector Bundles on Curves—New Directions
- Kumar
- 1997
(Show Context)
Citation Context ...ur proof, if only to provide an introduction to Faltings' ideas. Part of our results have been obtained independently (also in the context of principal bundles) by S. Kumar, Narasimhan and Ramanathan =-=[K-N-R]-=-. We would like to thank G. Faltings for useful letters, and V. Drinfeld for pointing out an inaccuracy in an earlier version of this paper. 1. The ind-groups GL r (K) and SL r (K). k-spaces and ind-s... |

61 |
Fusion rings and geometry.
- Gepner
- 1991
(Show Context)
Citation Context ...terms of the representation theory of SL r (k) -- more precisely in terms of the fusion algebra associated to this group. In the case (of interest here) of SL r (k), this computation has been done in =-=[G]-=-; the reader will find a treatment valid for all classical groups (and possibly more accessible to mathematicians) in the Appendix of [F]. The outcome is the following formula 1 : COROLLARY 8.6 (Verli... |

51 | Demazure character formula in arbitrary Kac-Moody setting, - Kumar - 1987 |

44 |
Formules de caracteres pour les algebres de Kac-Moody generales,
- Mathieu
- 1988
(Show Context)
Citation Context ...M 7.7 (S. Kumar, Mathieu).-- The space H 0 (Q; L c �� ) is isomorphic (as a b sl r (K)-module) to the dual of the basic representation V c of level c of b sl r (K). This theorem is proved in [Ku] =-=and [M]-=-, with one important difference. Both S. Kumar and Mathieu define the structure of ind-variety on SL r (K)=SL r (O) in an ad hoc way, using representation theory of Kac-Moody algebras; we must show th... |

36 |
Un analogue global du cône nilpotent
- Laumon
- 1988
(Show Context)
Citation Context ...a trivialization of V r E. Forgetting F gives a morphism of stacks of SL s;d X (r) to SLX (r); the (reduced) substack SLX (r) SLX (r) ss is the union of these images (for variable s; d). According to =-=[L]-=-, cor. 2.10, the dimension of SL s;d X (r) is (g \Gamma 1)(r 2 \Gamma 1 + s 2 \Gamma rs) \Gamma rd, so the codimension of its image is at least rd, which iss2. Since HN is a torsor over SLX (r) N the ... |

36 |
Semistable bundles over an elliptic curve,
- Tu
- 1993
(Show Context)
Citation Context ... We have a forgetful morphism ' : SLX (r) ss \Gamma! SU X (r). It is known that the determinant bundle L is the pull back of a line bundle (that we will still denote by L) on SU X (r) (see [D-N], and =-=[Tu]-=- for the case g = 1). PROPOSITION 8.4.-- Let c 2 N. The map ' : H 0 (SU X (r); L c ) \Gamma! H 0 (SLX (r) ss ; L c ) is an isomorphism. Let us choose an integer Ns2g. We claim that both spaces can be ... |

11 |
On the geometry of Schubert varieties attached to Kac-Moody Lie algebras, Can.Math.Soc.Conf.Proc. on ‘Algebraic geometry
- Slodowy
- 1986
(Show Context)
Citation Context ...-variety on SL r (K)=SL r (O) in an ad hoc way, using representation theory of Kac-Moody algebras; we must show that it coincides with our functorial definition. For instance Kumar, following Slodowy =-=[Sl]-=-, considers the representation V c for a fixed c, and a highest weight vector v c . The subgroup SL r (O) is the stabilizer of the line kv c in P(V c ), so the map g 7! gv c induces an injection i c :... |

8 |
Moret-Bailly: Champs algebriques, prepublication de Université de
- Laumon
- 1992
(Show Context)
Citation Context ... we'll work over an algebraically closed field k of characteristic 0. A k-algebra will always be assumed to be associative, commutative and unitary. Our basic objects will be k-spaces in the sense of =-=[L-MB]-=-: by definition, a k-space (resp. a k-group) is a functor F from the category of k-algebras to the category of sets (resp. of groups) which is a sheaf for the faithfully flat topology. Recall 1 that t... |

4 |
Residues of differentials on curves, Ann. scient
- Tate
- 1968
(Show Context)
Citation Context ...lly known (see e.g. [A-D-K], where very similar computations appear). We have included the computation because it is extremely simple and gives a nice generalization of the residue defined by Tate in =-=[T]-=-. Let us start from the central extension (F). Since F(V) is the group of invertible elements of the associative algebra End(V)= End f (V), its Lie algebra is simply the quotient of the Lie algebra En... |

2 |
Algèbre Commutative. Ch 5 à 7
- Bourbaki
- 2006
(Show Context)
Citation Context ...L r (K)=GL r (O) and the set of isomorphism classes of rank r vector bundles on X which are trivial on X . But a projective module over a Dedekind ring is free if and only if its determinant is free (=-=[B]-=-, ch. 7, x4, prop. 24), hence our assertion for GL r . The same proof applies for SL r . (1.9) Our first goal in the following sections will be to show that the bijection defined in cor. 1.8 comes act... |

1 |
KAC: The infinite wedge representation and the reprocity law for algebraic curves
- ARBARELLO, CONCINI, et al.
- 1989
(Show Context)
Citation Context ...4.9) We want to show that at the level of Lie algebras, the extension (E) is the universal central extension which appears in the theory of Kac-Moody algebras [K]. This is essentially known (see e.g. =-=[A-D-K]-=-, where very similar computations appear). We have included the computation because it is extremely simple and gives a nice generalization of the residue defined by Tate in [T]. Let us start from the ... |