Prym Varieties and the Moduli of Spin Bundles

We show that the Verlinde formula for moduli spaces of spin bundles on an algebraic curve gives dimensions of direct sums of spaces of theta functions over the finite set of Prym varieties of unramified double covers of the curve. We then construct a natural duality (though we do not show it is nondegenerate) between the latter spaces and the spaces of sections of the theta bundles on the moduli spaces of spin bundles corresponding to the orthogonal representation. Finally, it is remarked that this should be viewed as a special case of a conjectured `strange duality' for spin bundles.