Construction of conformally invariant higher spin operators using transvector algebras

Abstract

This paper deals with a systematic construction of higher spin operators, defined as conformally invariant differential operators acting on functions on flat space \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^m$\end{document}Rm with values in an arbitrary half-integer irreducible representation for the spin group. To be more precise, the higher spin version of the Dirac operator and associated twistor operators will be constructed as generators of a transvector algebra, hereby generalising the well-known fact that the classical Dirac operator on \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^m$\end{document}Rm and its symbol generate the orthosymplectic Lie superalgebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(1,2)$\end{document}osp(1,2). To do so, we will use the extremal projection operator and its relation to transvector algebras. In the second part of the article, the conformal invariance of the constructed higher spin operators will be proven explicitly.