Exotic Holonomy on Moduli Spaces of Rational Curves

Bryant \cite{Br} proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger \cite{Ber}. These connections occur on moduli spaces $\text{Y}$ of rational contact curves in a contact threefold $\text{W}$. Therefore, they are naturally contained in the moduli space $\mathbb{Z}$ of all rational curves in $\text{W}$. We construct a connection on $\mathbb{Z}$ whose restriction to $\text{Y}$ is torsion free. However, the connection on $\mathbb{Z}$ has torsion unless both $\text{Y}$ and $\mathbb{Z}$ are flat. We also show the existence of a new exotic holonomy which is a certain sixdimensional representation of $\text{Sl}\times \text{Sl}$. We show that every regular $H_3$-connection (cf. \cite{Br}) is the restriction of a unique connection with this holonomy.