Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps

TL;DR

This paper explores the geometry of G2-manifolds by constructing intermediate Jacobians, defining Yukawa couplings, and establishing G2-analogues of Abel-Jacobi maps that relate moduli spaces of special structures.

Contribution

It introduces a new framework linking G2-structures, intermediate Jacobians, and Abel-Jacobi maps, extending classical geometric concepts to G2-geometry.

Findings

Defined the universal intermediate Jacobian J for G2-structures

Related Yukawa coupling to a pseudo-Kahler structure on J

Constructed G2-analogues of Abel-Jacobi maps for moduli spaces

Abstract

We study the moduli space of torsion-free G2-structures on a fixed compact manifold, and define its associated universal intermediate Jacobian J. We define the Yukawa coupling and relate it to a natural pseudo-Kahler structure on J. We consider natural Chern-Simons type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang-Mills connections), and also deformed Donaldson-Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel-Jacobi maps.