Nearly parallel $G_2$-manifolds: formality and associative submanifolds

TL;DR

This paper constructs new examples of non-formal and formal 7-manifolds with special geometric structures, analyzes their minimal models, and identifies associative submanifolds with deformations, advancing understanding of $G_2$-geometry.

Contribution

It provides new examples of non-formal Sasaki-Einstein 7-manifolds, determines minimal models for certain fiber bundles, and constructs associative submanifolds with deformation families.

Findings

Aloff-Wallach spaces are formal.

Constructed non-formal simply connected compact Sasaki-Einstein 7-manifolds.

Found associative minimal submanifolds with non-trivial deformations.

Abstract

We construct new examples of non-formal simply connected compact Sasaki-Einstein 7-manifolds. We determine the minimal model of the total space of any fibre bundle over $CP^2$ with fibre $S^1\times S^2$ or $S^3/Z_p$ ($p>0$), and we apply this to conclude that the Aloff-Wallach spaces are formal. We also find examples of formal manifolds and non-formal manifolds, which are locally conformal parallel $Spin(7)$-manifolds. On the other hand, we construct associative minimal submanifolds in the Aloff-Wallach spaces and in any regular Sasaki-Einstein 7-manifold; in particular, in the space $Q(1,1,1)=(SU(2) \times SU(2) \times SU(2))/ (U(1) \times U(1))$ with the natural $S^1$-family of nearly parallel $G_2$-structures induced by the Sasaki-Einstein structure. In each of those cases, we obtain a family of non-trivial associative deformations.