Homogeneous G2 and Sasakian instantons on the Stiefel 7-manifold

Andrés J. Moreno; Luis E. Portilla

paper

Homogeneous G2 and Sasakian instantons on the Stiefel 7-manifold

arXiv:2406.06753

Abstract

We study homogeneous instantons on the seven dimensional Stiefel manifold V in the context of $G_{2}$ and Sasakian geometry. According to the reductive decomposition of V we provide an explicit description of all invariant $G_{2}$ and Sasakian structures. In particular, we characterise the invariant $G_{2}$ - structures inducing a Sasakian metric, among which the well known nearly parallel $G_{2}$ -structure (Sasaki- Einstein) is included. As a consequence, we classify the invariant connections on homogeneous principal bundles over V with gauge group U(1) and SO(3), satisfying either the $G_{2}$ or the Sasakian instanton condition. In addition, we study infinitesimal deformations of $G_{2}$ -instantons on coclosed $G_{2}$ -manifolds using a spinorial approach. By means of a Weitzenböck-type formula with torsion, we obtain curvature obstructions to the existence of non-trivial infinitesimal deformations and prove rigidity results for certain homogeneous $G_{2}$ -instantons.

Section 4.3 removed. Section 5 added, 23 pages, 1 figure. Comments are welcome