Abelian varieties and theta functions as invariants for compact Riemannian manifolds; constructions inspired by superstring theory

In some forms of superstring theory particular theta-functions come up as partition functions. The associated Abelian varieties come either from certain cohomology groups of the underlying universe or, in more recent theories (e.g. [Witten] [Mo-Wi]), are linked to their K-groups. Expressed in mathematical terms, one canonically associates to the cohomology or the K-theory of an even dimensional compact spin manifold a principally polarized Abelian variety. Moreover, if the dimension is 2 mod 8 a particular line bundle is singled out whose first Chern class is the principal polarization. This bundle thus has a non-zero section represented by a theta function which, after suitable normalization, is indeed the partition function of the underlying theory. Let me give some further detail on the physical motivation. There are several types of superstring theories, e.g. type I which is self-dual and types IIA and IIB wich are related via T -duality. The theories start from a spacetime Y which in a first approximation can be taken to be Y = X × T where T is the time-“axis” 1 and X is some compact Riemannian manifold. In Type IIA theory the Ramond-Ramond field is a closed differential form G = G0 + G2 + · · · on X with components of all even degrees while in type IIB G is an odd degree closed differential form on X. Moreover, these forms are integral (that is they have integral periods over integral homology