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Homogeneous G2 and Sasakian instantons on the Stiefel 7-manifold

Andrés J. Moreno, Luis E. Portilla

TL;DR

This work classifies and analyzes invariant ${ m G}_2$- and Sasakian-structures on the 7D Stiefel manifold $V^{5,2}$ and studies homogeneous instantons on associated bundles. By exploiting a reductive decomposition, the authors parametrize invariant ${ m G}_2$-structures and Sasakian metrics, identify coclosed and nearly parallel cases, and connect ${ m G}_2$-instantons with self-dual contact instantons within the Sasakian framework. They construct and classify invariant connections on homogeneous bundles with gauge groups ${ m U}(1)$ and ${ m SO}(3)$, showing that the invariant ${ m SO}(3)$-connection on the nontrivial bundle is simultaneously a ${ m G}_2$- and SDCI, and they establish Yang–Mills properties in this setting. Finally, they extend spinorial deformation theory to coclosed ${ m G}_2$-structures on $V^{5,2}$, deriving explicit obstruction formulas via a Weitzenböck-type identity with torsion and proving rigidity for a nonempty range of the metric parameter $y_2$. These results illuminate the moduli and rigidity phenomena for homogeneous ${ m G}_2$-instantons in a rich Sasakian context and link to transversely holomorphic/Hermitian structures on associated bundles.

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.

Homogeneous G2 and Sasakian instantons on the Stiefel 7-manifold

TL;DR

This work classifies and analyzes invariant - and Sasakian-structures on the 7D Stiefel manifold and studies homogeneous instantons on associated bundles. By exploiting a reductive decomposition, the authors parametrize invariant -structures and Sasakian metrics, identify coclosed and nearly parallel cases, and connect -instantons with self-dual contact instantons within the Sasakian framework. They construct and classify invariant connections on homogeneous bundles with gauge groups and , showing that the invariant -connection on the nontrivial bundle is simultaneously a - and SDCI, and they establish Yang–Mills properties in this setting. Finally, they extend spinorial deformation theory to coclosed -structures on , deriving explicit obstruction formulas via a Weitzenböck-type identity with torsion and proving rigidity for a nonempty range of the metric parameter . These results illuminate the moduli and rigidity phenomena for homogeneous -instantons in a rich Sasakian context and link to transversely holomorphic/Hermitian structures on associated bundles.

Abstract

We study homogeneous instantons on the seven dimensional Stiefel manifold V in the context of and Sasakian geometry. According to the reductive decomposition of V we provide an explicit description of all invariant and Sasakian structures. In particular, we characterise the invariant - structures inducing a Sasakian metric, among which the well known nearly parallel -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 or the Sasakian instanton condition. In addition, we study infinitesimal deformations of -instantons on coclosed -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 -instantons.
Paper Structure (12 sections, 23 theorems, 117 equations, 1 figure, 1 table)

This paper contains 12 sections, 23 theorems, 117 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Homogeneous $\mathrm{G}_2$ and contact metric structures on $V^{5,2}$
  3. $\mathrm{G}_2$-structures on $V^{5,2}$
  4. Homogeneous Sasakian metrics on $V^{5,2}$
  5. $\mathrm{G}_2$ and self dual contact instantons on $V^{5,2}$
  6. Homogeneous Principal Bundles on $V^{5,2}$
  7. Homogeneous ${\rm SO}(3)$-principal bundles
  8. Homogeneous $\mathrm{G}_2$ and self dual contact instantons (SDCI)
  9. Yang--Mills connections
  10. Infinitesimal deformations and rigidity of $\mathrm{G}_2$-instantons
  11. Infinitesimal deformation of $\mathrm{G}_2$-instantons
  12. Weitzenböck inequality and curvature obstruction

Key Result

Lemma 2.1

The forms in eq: su3-structure are $\mathrm{ad}(\mathfrak{so}(2)\oplus\mathfrak{so}(3))$-invariant, therefore, they induce a ${\rm SO}(5)$-invariant ${\rm SU}(3)$-structure $(\omega,\mathop{\mathrm{Re}}\Omega)$ on $G_{2}(\mathbb{R}^5)$. Furthermore, we have that $d\omega=0,$$d(\mathop{\mathrm{Re}}\O

Figures (1)

  • Figure 1: Plot of the right--hand side of the Weitzenböck inequality as a function of the metric parameter $y_2$. This shows that the expression becomes strictly negative for a non-empty interval of positive values of $y_2$.

Theorems & Definitions (50)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Theorem 2.1
  • proof
  • Remark 2.5
  • ...and 40 more