Parallel tractor extension and ambient metrics of holonomy split $G_2$

Abstract

The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form $G_2$ of the exceptional Lie group, and is equal to $G_2$ for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split $G_2$. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in $G_2$ is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.