Infinite grassmannians and moduli spaces of G-bundles

Abstract

Let C be a smooth projective irreducible algebraic curve over C of any genus and G a connected simply-connected simple affine algebraic group over C. In this paper we elucidate the relationship between (1) the space of vacua (conformal blocks) defined in conformal field theory, using an integrable highest weight representation of the affine Kac-Moody algebra associated to G and (2) the space of regular sections (generalised theta functions) of a line bundle on the moduli space TA of semistable principal G-bundles on C. Fix a point p in C and let ~p (resp. kp) be the completion of the local ring #p of C at p (resp. the quotient field of C~p). Let ;~ := G(kp) (the kp-rational points of the algebraic group G) be the loop group of G and let ;~ := G(#~p) be the standard maximal parahoric subgroup of ~:. Then the generalised flag variety X := N / . ~ is an inductive limit of projective varieties, in fact of generalised Schubert varieties. One has a natural .~r line bundle s on X (cf. Sect. 2.2), and the Picard group Pic(X) is isomorphic to Z which is generated by s (Proposition 2.3), where ,~f is the universal central extension of .%: by the multiplicative group C* (cf. Sect. 2.2). By an analogue of the Borel-Weil theorem proved in the Kac-Moody setting by Kumar (and also by Mathieu), the space H~162 s of the regular sections of the line bundle Y-(dxo) := Y-(Xo) | (for any d = 0) is canonically isomorphic with the full vector space dual V(dXo)* of the (integrable) highest weight (irreducible) module V(dXo) of the affine KacMoody group .~ (or of the associated affine Kac-Moody algebra, which is a certain one dimensional central extension of the loop algebra (Lie G) | with highest weight dXo (cf. Sect. 6.1).