Symplectic spinors and Hodge theory

TL;DR

This paper explores the representation theory and cohomology of symplectic spinors, revealing dualities, decompositions, and elliptic properties that extend Hodge theory to complex symplectic structures and operators.

Contribution

It introduces new decompositions of symplectic spinor modules, characterizes elliptic complexes, and applies advanced index theorems to symplectic geometry.

Findings

Decomposition of exterior forms into irreducible modules with hidden symmetries.

Identification of elliptic complexes related to symplectic spinors.

Application of index theory to symplectic Hodge structures.

Abstract

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.