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 Kac-Moody algebras in the works of Lepowsky–Wilson [LW] and I. Frenkel–Kac [FK]. In order to formalize the emerging structure and motivated

: 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 quasi-parabolic G-bundles 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 G-bundles 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 theta-characteristic. These results about stacks allow to recover the Drezet-Narasimhan 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 semi-stable SOr -bundles are not loca...

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

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 non-abelian 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.

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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

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

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 so-called Rational Conformal Field Theories (RCFT). These spaces appear as spaces of global multiform sections of some flat vector

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 G-bundles 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

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 hyper-elliptic involution, in the case of even degree and n = 2. Otherwise the intersection is also trivial in the genus 2 case. 1.