Parallel tractor extension and ambient metrics of holonomy split g₂

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 G₂ of the exceptional Lie group, and is equal to G₂ for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinitedimensional family of metrics of holonomy equal to split G₂. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G₂ 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.