Moduli of $G_2$ structures and the Strominger system in dimension 7

TL;DR

This paper studies the moduli space of $G_2$ structures with torsion coupled with $G_2$-instantons on 7-manifolds, using elliptic operator theory and relating to generalized geometry, as an analogue of the Strominger system.

Contribution

It introduces a new coupled system of PDEs in 7-dimensions inspired by supergravity, and proves the finiteness of the moduli space of solutions.

Findings

Moduli space of solutions is finite dimensional.

Established relation to generalized geometry.

Formulated an analogue of the Strominger system in 7D.

Abstract

We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. The resulting system of partial differential equations can be regarded as an analogue of the Strominger system in $7$-dimensions. We initiate the study of the moduli space of solutions and show that it is finite dimensional using elliptic operator theory. We also relate the associated geometric structures to generalized geometry.