First BGG operators on homogeneous conformal geometries

TL;DR

This paper develops an algebraic method to explicitly find solutions to first BGG operators on homogeneous conformal geometries, with applications to holonomy and conserved quantities, especially in general relativity.

Contribution

It introduces an invariant calculus for solving first BGG operators explicitly on homogeneous conformal geometries, including conformal Killing tensors and twistor spinors.

Findings

Explicit solutions for conformal Killing tensors and twistor spinors.

Applications to holonomy reductions and conserved quantities.

Demonstrations on geometries from general relativity.

Abstract

We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing--Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.