Structure of the curvature tensor on symplectic spinors
| Autor: | Svatopluk Krýsl |
|---|---|
| Rok vydání: | 2010 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Symplectic group Mathematical analysis General Physics and Astronomy Metaplectic structure Symplectic representation 53D05 53C07 58J10 Symplectic vector space Differential Geometry (math.DG) Mathematics - Symplectic Geometry FOS: Mathematics Symplectic Geometry (math.SG) Geometry and Topology Symplectomorphism Mathematics::Symplectic Geometry Moment map Mathematical Physics Mathematics Symplectic geometry Symplectic manifold |
| Zdroj: | Journal of Geometry and Physics. 60:1251-1261 |
| ISSN: | 0393-0440 |
| DOI: | 10.1016/j.geomphys.2010.04.004 |
| Popis: | We study symplectic manifolds $(M^{2l},\omega)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative $\nabla^S$ associated to the Fedosov connection $\nabla.$ It is known that the space of symplectic spinor valued exterior differential 2-forms, $\Gamma(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$ decomposes into three invariant spaces with respect to the structure group, which is the metaplectic group $Mp(2l,\mathbb{R})$ in this case. For a symplectic spinor field $\phi \in \Gamma(M,\mathcal{S}),$ we compute explicitly the projections of $R^S\phi \in \Gamma(M,\bigwedge^2T^*M \otimes \mathcal{S})$ onto the three mentioned invariant spaces in terms of the symplectic Ricci and symplectic Weyl curvature tensor fields of the connection $\nabla.$ Using this decomposition, we derive a complex of first order differential operators provided the Weyl tensor of the Fedosov connection is trivial. Comment: 17 pages, 1 figure |
| Databáze: | OpenAIRE |
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