Deformations of homogeneous associative submanifolds in nearly parallel $G_2$-manifolds

Abstract

A nearly parallel $G_2$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y)={\mathbb{R}}_{>0}\times Y$ has the holonomy group contained in $\mathrm{Spin}(7)$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$. An associative submanifold in $Y$ is a minimal 3-submanifold whose cone is Cayley. We study its deformations, namely, Cayley cone deformations, explicitly when it is homogeneous in the 7-sphere $S^7$.

36 pages, v3: minor corrections, published version

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