Verlinde spaces and Prym theta functions

1 Introduction The Verlinde formula is a remarkable—and potentially very useful—new tool in the geometry of algebraic curves which is borrowed from conformal field theory. In the first instance it is a trigonometric expression which assigns a natural number N l (G, g) to data consisting of a semisimple algebraic group G, a nonnegative integer g and an auxiliary integer l ∈ Z. In physics N l (G, g) is interpreted as the dimension of the space of conformal blocks at level l in the Wess-Zumino-Witten model of conformal field theory on a compact Riemann surface of genus g. In algebraic geometry this 'space of conformal blocks' is identified with a vector space of the form H 0 (M C (G), L l), where M C (G) is the moduli scheme (or stack) of semistable principal G-bundles over C, and L is an ample line bundle on this moduli space generating the Picard group. The feature of the Verlinde formula which motivates this paper is its 'nu-merology'. Namely, when one computes the numbers N l (G, g) for the classical simple groups one finds that they obey interesting identities which lead one to certain conjectures about the geometry of M C (G). (See [OW].) In this article we are concerned with identities linking the complex spin groups G = Spin m with the configuration of principally polarised Prym varieties associated to the curve C via its unramified double covers. This connection was first observed in [O2] for the odd spin groups Spin 2n+1 ; here we shall give a systematic account of both the odd and even cases. 1 The moduli space M C (SO m) has, for m ≥ 3, two connected components labelled by the second Stiefel-Whitney class w 2 of the bundles. Each of these components has a J 2 (C)-Galois cover which is a moduli space for Clifford bundles with fixed spinor norm line bundle ξ ∈ Pic(C), and whose isomorphism class depends only on deg ξ mod 2. When ξ = O C the Galois cover is precisely M C (Spin m) since by definition Spin m is the kernel of the spinor norm; the 'sister' space which arises when deg ξ is odd, we denote by M − C (Spin m). In each case the fibre J 2 (C) over M C (SO m) parametrises liftings of a given SO m-bundle to a Clifford …