Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory

We first investigate a construction of principally polarized abelian varieties attached to certain spin manifolds, due to Witten and Moore and Witten around 2000. The index theorem for the Dirac operator associated to the spin structure implies integrality of a natural skew pairing on the topological K-group. The latter serves as a principal polarization. We place the construction in a broader framework, relate it to Weil's intermediate Jacobian and point out how it leads to abelian varieties canonically associated to polarized even weight Hodge structures. The latter construction can also be explained in terms of algebraic groups which might be useful from the point of view of Tannakian categories. The constructions depend on moduli much as in Teichmüller theory although the period maps in general are only real analytic.