$T$-Dual solutions and infinitesimal moduli of the $G_2$-Strominger system

TL;DR

This paper investigates the moduli space of solutions to the $G_2$-Strominger system, establishing finiteness of infinitesimal deformations and constructing new solutions including the first examples of $T$-dual solutions.

Contribution

It proves the finite dimensionality of the infinitesimal moduli space and constructs new explicit solutions, including $T$-dual solutions, for the $G_2$-Strominger system.

Findings

The space of infinitesimal deformations is finite dimensional.

New solutions on $T^3$-bundles over $K3$ surfaces are constructed.

First examples of $T$-dual solutions for this system are provided.

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. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes compactifications of the heterotic string to three dimensions, and is often referred to as the $G_2$-Strominger system. We study the moduli space of solutions and prove that the space of infinitesimal deformations, modulo automorphisms, is finite dimensional. We also provide a new family of solutions to this system, on $T^3$-bundles over $K3$ surfaces and for infinitely many different instanton bundles, adapting a construction of Fu-Yau and the second named author. In particular, we exhibit the first examples of $T$-dual solutions for this system of equations.