Coupling solutions of BGG-equations in conformal spin geometry

TL;DR

This paper develops explicit formulas and conditions for coupling solutions of BGG-equations in conformal spin geometry, enabling the construction of new solutions and analyzing their geometric properties.

Contribution

It introduces a simple integrability condition for normal solutions and a procedure to couple known solutions to generate new ones in conformal spin geometry.

Findings

Explicit coupling formulas for conformal spin structures

Conditions linking almost Einstein scales, conformal Killing forms, and twistor spinors

Invariant decomposition of conformal Killing fields using generic twistor spinors

Abstract

BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of BGG-equations to produce new ones. Employing a suitable calculus for conformal spin structures this yields explicit coupling formulas and conditions between almost Einstein scales, conformal Killing forms and twistor spinors. Finally we discuss a class of generic twistor spinors that provides an invariant decomposition of conformal Killing fields.