Symplectic Killing spinors

Let $(M,\omega)$ be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection $\nabla.$ Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial differential equation and they are the main topic of this paper. We derive a necessary condition satisfied by a symplectic Killing spinor field. The advantage of this condition consists in the fact that it is expressed by a zeroth order operator. This condition helps us substantionally to compute the symplectic Killing spinor fields for the standard symplectic vector spaces and the round sphere $S^2$ equipped with the volume form of the round metric.Comment: 17 pages, proofs and examples extended conference tal

Symplectic spinors and Hodge theory

Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a representation of the Lie superalgebra $\mathfrak{osp}(1|2)$ in this case. We also determine ranges of the induced exterior symplectic spinor derivatives when they are restricted to bundles induced by the irreducible submodules mentioned. This duality is used to decompose curvature tensors of covariant derivatives induced by a Fedosov connection to a symplectic spinor bundle, for characterizing a subcomplex of the de Rham complex twisted by the spinors and for proving that the complex is of elliptic type. Part of the dual is used to characterize Fedosov manifolds admitting symplectic Killing spinors and for relating the spectra of the symplectic Rarita--Schwinger and the symplectic Dirac operator of Habermann. Further, we use the Fomenko--Mishchenko generalization of the Atiyah--Singer index theorem to prove a kind of Hodge theory is valid for elliptic complexes with differentials on projective and finitely generated bundles over the algebra of compact operators and certain further ones.Comment: 41 pages; habilitation thesis (short version) for the process of association to Charles University in Pragu

Classification of 1st order symplectic . . .

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