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

TL;DR

This paper explores the construction of abelian varieties linked to spin manifolds and Hodge structures, inspired by superstring theory, connecting geometric, algebraic, and topological aspects.

Contribution

It broadens the framework for associating abelian varieties to spin manifolds and Hodge structures, integrating ideas from Weil Jacobians, algebraic groups, and Tannakian categories.

Findings

Constructs principal abelian varieties from spin manifolds.

Relates index theory to polarization of Jacobians.

Connects moduli spaces to Teichmüller theory concepts.

Abstract

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.