Cohomology in algebra, geometry, physicsand statistics

Twistor complexes in symplectic geometry

For a manifold with a vanishing second Stiefel--Whitney class and equipped with a symplectic form, it is possible to define the so-called symplectic spinor bundle (B. Kostant) that is a parallel notion to the spinor bundle on a Riemannian manifold. The fibre of the bundle is an infinite dimensional complex vector space which is called the space of symplectic spinors. It is a direct sum of two irreducible representations of the connected double cover of the symplectic group.

 The tensor product of exterior forms on the manifold with the symplectic spinor bundle ("twisted" deRham complex) splits into subbundles and the symplectic twistor operators are defined with help of them. We describe their construction. Computing a representation-theoretic characteristic (Schur--Weyl--Howe type duality), we can determine the symbols of the symplectic twistor operators and prove the ellipticity of the cohomological complexes formed by these operators.
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