Exotic holonomies $\E_7^{(a)}$

Abstract

It is proved that the Lie groups $\E_7^{(5)}$ and $\E^{(7)}_7$ represented in $\R^{56}$ and the Lie group $\E_7^{\C}$ represented in $\R^{112}$ occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.