Coupled G₂-instantons

We introduce the coupled instanton equations for a metric, a spinor, a three-form, and a connection on a bundle, over a spin manifold. Special solutions in dimensions 6 and 7 arise, respectively, from the Hull--Strominger and heterotic G₂ systems. 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. In this respect, we present two open questions regarding how these different geometric conditions are intertwined. A positive answer is expected from recent developments in the physics literature in work by De la Ossa, Larfors and Svanes, and in the mathematics literature, for the case of Calabi--Yau manifolds, in recent work by the second author jointly with Gonzalez Molina. We give a complete solution to the first of these two problems, for G₂-structures with torsion coupled to G₂-instantons, in the seven-dimensional case, and also establish some partial results for the second problem. The last part of the present work carefully studies the approximate solutions to the heterotic G₂-system constructed by the third and fourth authors on contact Calabi--Yau 7-manifolds, for which we prove the existence of approximate coupled G₂-instantons and generalized Ricci-flat metrics.