Coupled $\operatorname{G}_2$-instantons

TL;DR

This paper introduces coupled instanton equations on spin manifolds, explores their special solutions in dimensions 6 and 7, and provides new methods for constructing instantons and analyzing their geometric relations, with implications in physics and mathematics.

Contribution

It formulates coupled instanton equations involving multiple geometric structures, offers solutions in specific dimensions, and connects these to generalized geometry and Ricci-flat metrics.

Findings

Complete solution to a key open problem in coupled instanton construction.

Existence of approximate coupled G2-instantons on contact Calabi-Yau 7-manifolds.

New methods linking coupled instantons to generalized Ricci-flat metrics.

Abstract

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 the heterotic $\operatorname{G}_2$ 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 Gonz\'alez 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 $\operatorname{G}_2$-structures with torsion coupled to $\operatorname{G}_2$-instantons, in dimension $7$, we also establish results around the second problem. The last part of the present work carefully studies the approximate solutions to the heterotic $\operatorname{G}_2$-system constructed by the third and fourth authors on contact Calabi--Yau $7$-manifolds, for which we prove the existence of approximate coupled $\operatorname{G}_2$-instantons and generalized Ricci-flat metrics.