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# Advances in Theoretical and Mathematical Physics

## Volume 26 (2022)

### Number 6

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

Pages: 1669 – 1704

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n6.a3

#### Authors

#### 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.

Published 30 June 2023