References & Citations
Mathematics > Differential Geometry
Title: Coupled $\operatorname{G}_2$-instantons
(Submitted on 19 Apr 2024 (v1), last revised 6 May 2024 (this version, v2))
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.
Submission history
From: Mario Garcia-Fernandez [view email][v1] Fri, 19 Apr 2024 15:07:51 GMT (170kb)
[v2] Mon, 6 May 2024 23:07:45 GMT (173kb)
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