Coupled G2-instantons

In this paper, we introduce the coupled instanton equations for a metric, a spinor, a 3-form, and a connection on a bundle, over a spin manifold. Special solutions in dimensions [Formula: see text] and [Formula: see text] arise, respectively, from the Hull–Strominger and the heterotic [Formula: see text] system. The equations are motivated by recent developments in theoretical physics and can be recast using generalized geometry; we investigate how coupled instantons relate to generalized Ricci-flat metrics and also to Killing spinors on a Courant algebroid. We present two open questions regarding how these different geometric conditions are intertwined, for which a positive answer is expected from recent developments in the physics literature by De la Ossa, Larfors and Svanes, and in the mathematics literature on Calabi–Yau manifolds, in recent work by the second-named author with González Molina. We give a complete solution to the first of these problems, providing a new method for the construction of instantons in arbitrary dimensions. For [Formula: see text]-structures with torsion coupled to [Formula: see text]-instantons, in dimension [Formula: see text], we also establish results around the second problem. The last part of this work carefully studies the approximate solutions to the heterotic [Formula: see text]-system constructed by the third and fourth authors on contact Calabi–Yau [Formula: see text]-manifolds, for which we prove the existence of approximate coupled [Formula: see text]-instantons and generalized Ricci-flat metrics.