Agnaldo A. da Silva Jr.

arXiv2026

Ricci-flat string algebroids

We investigate geometric structures defined by 4-forms, encompassing important classes of geometric structures with torsion including almost Hermitian, $\mathrm{G}_2$, $\mathrm{Spin}(7)$, and almost quaternion/hyper Hermitian. Within this unified framework, the underlying 4-form naturally determines geometric data that define a string algebroid over the manifold. In particular, it gives rise to a canonical notion of instanton, thereby linking these geometries to gauge-theoretic structures in generalized geometry and heterotic string theory. We further prove that these data induce generalized Ricci-flat metrics. Finally, we examine the main examples in the literature and construct several explicit ones.

math.DG↗

arXiv2024

Coupled $\operatorname{G}_2$-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 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 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 $\operatorname{G}_2$-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.

math.DG↗