Exotic Holonomy on Moduli Spaces of Rational Curves

TL;DR

This paper explores the existence of exotic holonomy connections on moduli spaces of rational curves, constructing new examples and analyzing their torsion properties, revealing novel holonomy representations.

Contribution

It constructs a torsion-free connection on the moduli space of rational contact curves and introduces a new exotic holonomy related to a six-dimensional representation of SL(2)×SL(2).

Findings

Existence of torsion-free connections with exotic holonomy on moduli spaces.

Construction of a connection on the space of all rational curves with specific torsion properties.

Identification of a new exotic holonomy as a six-dimensional representation of SL(2)×SL(2).

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 $\Z$ of all rational curves in $\W$. We construct a connection on $\Z$ whose restriction to $\Y$ is torsion free. However, the connection on $\Z$ has torsion unless both $\Y$ and $\Z$ 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.