Introduction

It has been known essentially since Riemann that one can associate to any compact Riemann surface X an abelian variety, the Jacobian JX, together with a divisor Θ (well-defined up to translation) that can be defined both in a geometric way and as the zero locus of an explicit function, the Riemann theta function. The geometry of the pair (JX, Θ) is intricately (and beautifully) related to the geometry of X.

The idea that higher-rank vector bundles should provide a non-abelian analogue of the Jacobian appears already in the influential paper [We] of A. Weil (though the notion of vector bundle does not appear as such in that paper!). The construction of the moduli spaces was achieved in the 1960's, mainly by D. Mumford and the mathematicians of the Tata Institute. However it is only recently that the study of the determinant line bundles on these moduli spaces and of their spaces of sections has made clear the analogy with the Jacobian. This is largely due to the intrusion of Conformal Field Theory, where these spaces have appeared (quite surprisingly for us!) as fundamental objects.