Ricci-flat string algebroids

Abstract

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.

52 pages. Updated introduction and references; new proposition about coupled instantons; new examples; typos corrected; enlarged discussions. Comments welcome