Formally integrable complex structures and parallel vector cross products

The notion of a multilinear vector cross product (VCP) has been introduced by Gray as a natural generalization of the notion of an almost complex structure. In my lecture I shall present a correspondence between parallel VCPs on a Riemannian manifold $M$ and parallel almost complex structures on a higher dimensional knot space over $M$ endowed with a $L^2$-metric. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively. Using VCPs, I shall also show similarities between integrable complex structures on one hand and torsion-free $G_2$-and $\Spin(7)$-structures on the other hand. My talk is based on my joint works with D. Fiorenza, K. Kawai, L. Schwachhöfer, J. Vanzura, and L. Vitagliano.