  We look into a construction of principal abelian varieties attached to
certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to
place it in a broader framework. This is related to Weil intermediate Jacobians
but it also suggests to associate abelian varieties 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 Teichm\"uller theory
although the period maps in general are only real analytic. One of the nice
features is how the index for certain differential operators canonically
associated to the geometry of the situation (spin structure, complex structure
etc.) leads to integrality of skew pairings on the topological K-group (coming
from the Index Theorem) which then serves as a polarization for the jacobian.
