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



Andrew Clarke (Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ, Brazil)

Mario Garcia-Fernandez (Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Cantoblanco, Madrid, Spain)

Carl Tipler (Laboratoire de Mathématiques de Bretagne Atlantique, Université Bretagne Occidentale, Brest, France)


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