The almost Einstein operator for $(2, 3, 5)$ distributions

TL;DR

This paper develops tractor calculus for $(2, 3, 5)$ distributions, explicitly computes the first BGG operator, and reveals its solutions as almost Einstein scales, connecting geometric structures with conformal holonomy and providing new computational tools.

Contribution

It introduces explicit formulas for the tractor connection and the first BGG operator in $(2, 3, 5)$ geometry, linking solutions to almost Einstein scales and holonomy.

Findings

Solutions of the operator are automatically normal.

The kernel of the operator corresponds to almost Einstein scales.

The conformal holonomy is shown to be $ ext{G}_2$ for certain structures.

Abstract

For the geometry of oriented $(2, 3, 5)$ distributions $(M, {\mathbf D})$, which correspond to regular, normal parabolic geometries of type $(\mathrm{G}_2, P)$ for a particular parabolic subgroup $P < \mathrm{G}_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator $\Theta_0$ associated to the $7$-dimensional irreducible representation of $\mathrm{G}_2$. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator are automatically normal, yielding a geometric interpretation of $\ker \Theta_0$: For any $(M, {\mathbf D})$, this kernel consists precisely of the almost Einstein scales of the Nurowski conformal structure on $M$ that ${\mathbf D}$ determines. We apply our formula for $\Theta_0$ (1) to recover efficiently some known solutions, (2) to construct a distribution with root type $[3, 1]$ with a nonzero solution, and (3) to show efficiently that the conformal holonomy of a particular $(2, 3, 5)$ conformal structure is equal to $\mathrm{G}_2$.