Symplectic spinors and Hodge theory
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 (SchurWeylHowe type duality) given by a representation of the Lie superalgebra $\mathfrak{osp}(12)$ 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 RaritaSchwinger and the symplectic Dirac operator of Habermann. Further, we use the FomenkoMishchenko generalization of the AtiyahSinger 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.02026
 arXiv:
 arXiv:1708.02026
 Bibcode:
 2017arXiv170802026K
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Algebraic Topology;
 Mathematics  Representation Theory;
 15A66;
 53C07;
 53C10;
 53D05;
 53D35;
 58A14;
 81S10;
 11F27
 EPrint:
 41 pages