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

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 $C{P}^2$ 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.

35 pages, no figures; v2. added references; v3. title changed, clarifications in the Introduction and in the proof of Theorem 4.2, rewritten some parts to improve presentation