Exotic Holonomy on Moduli Spaces of Rational Curves

Abstract

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 $\Y$ of rational contact curves in a contact threefold $\W$. Therefore, they are naturally contained in the moduli space $\mathbb{Z}$ of all rational curves in $\W$. We construct a connection on $\mathbb{Z}$ whose restriction to $\Y$ is torsion free. However, the connection on $\mathbb{Z}$ has torsion unless both $\Y$ and $\mathbb{Z}$ are flat. We also show the existence of a new exotic holonomy which is a certain sixdimensional representation of $\Sl \times \Sl$. We show that every regular $H_3$-connection (cf. \cite{Br}) is the restriction of a unique connection with this holonomy.

30 pages, AMS-TeX, uses pictex

Cited by in corpus (2)

• On the Incompleteness of Berger's List of Holonomy Representations
• Exotic holonomies $\E_7^{(a)}$