$T$-dual solutions and infinitesimal moduli of the $G_2$-Strominger system
Andrew Clarke, Mario García-Fernández, Carl Tipler
Abstract
We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a\ncompact $7$-dimensional manifold. The coupling is via an equation for $4$-forms\nwhich appears in supergravity and generalized geometry, known as the Bianchi\nidentity. First studied by Friedrich and Ivanov, the resulting system of\npartial differential equations describes compactifications of the heterotic\nstring to three dimensions, and is often referred to as the $G_2$-Strominger\nsystem. We study the moduli space of solutions and prove that the space of\ninfinitesimal deformations, modulo automorphisms, is finite dimensional. We\nalso provide a new family of solutions to this system, on $T^3$-bundles over\n$K3$ surfaces and for infinitely many different instanton bundles, adapting a\nconstruction of Fu-Yau and the second named author. In particular, we exhibit\nthe first examples of $T$-dual solutions for this system of equations.\n