Classification of 1st order symplectic spinor operators over contact projective geometries

TL;DR

This paper classifies first-order invariant differential operators between specific bundles in contact projective geometries, extending classical operators like Dirac and twistor operators to this geometric setting.

Contribution

It provides a complete classification of first-order invariant operators in contact projective geometries, including existence conditions and examples related to classical operators.

Findings

At most one invariant operator per bundle pair up to scalar

Derived explicit conditions for operator existence

Connected to classical Dirac, twistor, Rarita-Schwinger operators

Abstract

We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic or generalized Kostant spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one $1^{st}$ order invariant differential operator up to a complex multiple and give an equivalence condition for the existence of such an operator. Contact projective analogues of the well known Dirac, twistor and Rarita-Schwinger operators appearing in Riemannian geometry are special examples of these operators.