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

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 3 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.

Purely coclosed G$_{\mathbf2}$-structures on 2-step nilpotent Lie groups

“…In theoretical physics, purely coclosed G 2 -structures are closely related to the G 2 -Strominger system of equations that arises considering the Killing spinor equations of 10-dimensional string theory [23] on a compact 7-manifold (see e.g. [4,17] for more details and for the complete description of this system). Indeed, by [12,13], the gravitino and dilatino Killing spinor equations with dilaton function f on a compact 7-manifold M are equivalent to the following system of equations for a G 2 -structure ϕ on M :…”

“…Solutions to the G 2 -Strominger system have been recently obtained in [4] on T 3 -bundles over K3 surfaces. Previously, two examples of solutions with constant dilaton function were described in [10].…”

“…(i) If dim(n ′ ) = 1, there exists a purely coclosed G 2 -structure on n inducing the metric g if and only if tr 2 (j(z) 2 ) = 4 tr(j(z) 4 ) for every z ∈ n ′ . (ii) If dim(n ′ ) = 2, there exists a purely coclosed G 2 -structure on n inducing the metric g if and only if there exists an oriented 4-dimensional subspace r ⊂ r with d(n ′ ) * ⊂ Λ 2 r * such that for every orthonormal basis [6,22], allows us to give the explicit description, up to automorphisms, of the metrics induced by purely coclosed G 2structures on every 2-step nilpotent Lie algebra admitting such structures and with derived algebra of dimension at most two.…”

“…It is clear that for any other vector z ′ = λz in n ′ , the corresponding matrix A ′ = −j(z ′ ) verifies A ′ = λA, so the same equation holds by homogeneity. Conversely, suppose that g is an inner product on n for which tr 2 (j(z) 2 ) = 4 tr(j(z) 4 ) holds for every z in n ′ . Consider again a unit vector z ∈ n ′ and a g-orthonormal basis {e 1 , .…”

“…One can prove that the condition tr 2 (j(z) 2 ) = 4 tr(j(z) 4 ) cannot hold whenever j(z) has rank 2, so we can state the following.…”