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Vertex algebras and algebraic curves
 Mathematical Surveys and Monographs 88 (2001), Amer. Math.Soc. MR1849359 (2003f:17036
"... Vertex operators appeared in the early days of string theory as local operators describing propagation of string states. Mathematical analogues of these operators were discovered in representation theory of affine KacMoody algebras in the works of Lepowsky–Wilson [LW] and I. Frenkel–Kac [FK]. In or ..."
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Cited by 93 (9 self)
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Vertex operators appeared in the early days of string theory as local operators describing propagation of string states. Mathematical analogues of these operators were discovered in representation theory of affine KacMoody algebras in the works of Lepowsky–Wilson [LW] and I. Frenkel–Kac [FK]. In order to formalize the emerging structure and motivated
The line bundles on the moduli of parabolic Gbundles over curves and their sections
, 1997
"... : Let X be a complex, smooth, complete and connected curve and G be a complex simple and simply connected algebraic group. We compute the Picard group of the stack of quasiparabolic Gbundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the correspond ..."
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Cited by 69 (4 self)
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: Let X be a complex, smooth, complete and connected curve and G be a complex simple and simply connected algebraic group. We compute the Picard group of the stack of quasiparabolic Gbundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the corresponding spaces of global sections with the vacua spaces of Tsuchiya, Ueno and Yamada. The method uses the uniformization theorem which describes these stacks as double quotients of certain infinite dimensional algebraic groups. We describe also the dualizing bundle of the stack of Gbundles and show that it admits a unique square root, which we construct explicitly. If G is not simply connected, the square root depends on the choice of a thetacharacteristic. These results about stacks allow to recover the DrezetNarasimhan theorem (for the coarse moduli space) and to show an analogous statement when G = Sp 2r . We prove also that the coarse moduli spaces of semistable SOr bundles are not loca...
Infinite Grassmannians and moduli spaces of Gbundles
 Math. Annalen
, 1994
"... These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in th ..."
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Cited by 57 (2 self)
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These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in that we have
Vector bundles on curves and generalized theta functions: recent results and open problems
 Cambridge University Press
, 1995
"... Abstract. The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a nonabelian generalization of the classical theta functions. New ideas coming from mathematical physics have ..."
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Cited by 49 (3 self)
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Abstract. The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a nonabelian generalization of the classical theta functions. New ideas coming from mathematical physics have shed a new light on these spaces of sections—allowing notably to compute their dimension (Verlinde’s formula). This survey paper is devoted to giving an overview of these ideas and of the most important recent results on the subject.
Vilonen Perverse Sheaves on affine Grassmannians and Langlands Duality, electronic preprint math.AG/9911050
"... In this paper we outline a proof of a geometric version of the Satake isomorphism. Namely, given a connected, complex algebraic reductive group G we show that the tensor category of representations of the dual group L G is naturally equivalent to ..."
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Cited by 43 (3 self)
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In this paper we outline a proof of a geometric version of the Satake isomorphism. Namely, given a connected, complex algebraic reductive group G we show that the tensor category of representations of the dual group L G is naturally equivalent to
Conformal blocks, fusion rules and the Verlinde formula
 BarIlan Univ
, 1993
"... The Verlinde formula computes the dimension of certain vector spaces, the spaces of conformal blocks, which are the basic objects of a particular kind of quantum field theories, the socalled Rational Conformal Field Theories (RCFT). These spaces appear as spaces of global multiform sections of some ..."
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Cited by 36 (0 self)
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The Verlinde formula computes the dimension of certain vector spaces, the spaces of conformal blocks, which are the basic objects of a particular kind of quantum field theories, the socalled Rational Conformal Field Theories (RCFT). These spaces appear as spaces of global multiform sections of some flat vector
Principal Gbundles over elliptic curves
, 1998
"... 1. Introduction. Let E be an elliptic curve with origin p0, and let G be a complex simple algebraic group. For simplicity, we shall only consider the case where G is simply connected, although all of the methods discussed below can be extended to the case of a general group G. The goal of this note ..."
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Cited by 35 (2 self)
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1. Introduction. Let E be an elliptic curve with origin p0, and let G be a complex simple algebraic group. For simplicity, we shall only consider the case where G is simply connected, although all of the methods discussed below can be extended to the case of a general group G. The goal of this note is to announce some results concerning the
Hitchin's and WZW connections are the same
, 1998
"... Introduction :\Gamma Let X be an algebraic curve over the field C of complex numbers which is assumed to be smooth, connected and projective. For simplicity, we assume that the genus of X is ? 2 . Let G be a simple simply connected group and MG (X) the coarse moduli scheme of semistable Gbundles on ..."
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Cited by 27 (2 self)
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Introduction :\Gamma Let X be an algebraic curve over the field C of complex numbers which is assumed to be smooth, connected and projective. For simplicity, we assume that the genus of X is ? 2 . Let G be a simple simply connected group and MG (X) the coarse moduli scheme of semistable Gbundles on X . Any linear representation determines a line bundle \Theta on M and some non negative integer l (the Dynkin index of the representation, cf [KNR], [LS]). Its is known that the choice of a (closed) point x 2 X(C) (and, a priori, of a formal coordinate near x ) of X determines an isomorphism (see (5.4)) between the projective space of conformal blocks PB l (X) (for G ) of level l and the space PH 0 (MG (X); \Theta) of generalized theta
Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups in the singular case
 In preparation
"... Abstract. We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)Verlinde bundles over Teichmüller space, is asymptotically faithful, that is the intersection over all levels of the kerne ..."
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Cited by 23 (7 self)
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Abstract. We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)Verlinde bundles over Teichmüller space, is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least 3. For the genus 2 case, this intersection is exactly the order two subgroup, generated by the hyperelliptic involution, in the case of even degree and n = 2. Otherwise the intersection is also trivial in the genus 2 case. 1.