arXiv 0812.42302008-12-23math.DGmath.SG

辛旋量上曲率张量的结构

Structure of the curvature tensor on symplectic spinors

Svatopluk Krýsl

备注：: 17 pages, 1 figure
发表信息：: Journal of Geometry and Physics, 2010, Vol. 60, Issue 9, 1251--1261
DOI：: 10.1016/j.geomphys.2010.04.004

摘要：

我们研究辛流形$(M^{2l},ω)$配备了一个辛挠自由仿射（也称为Fedosov）连接$\nabla$，并承认亚格结构。令$\mathcal{S}$是所谓的辛旋量丛，令$R^S$是与Fedosov联络$\nabla.$相关联的辛旋量协变导数$\nabla^S$的曲率张量场。已知辛旋量或值外微分2-形式的空间$Γ(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$关于结构群分解为三个不变空间，在这种情况下，该结构群是元群$Mp(2l,\mathbb{R})$。对于一个辛旋量场$ϕ\in Γ(M,\mathcal{S}),$，我们用联络$\nabla.$的辛Ricci和辛Weyl曲率张量场，显式地计算了$R^Sϕ\in Γ(M,\bigwedge^2T^*M \otimes \mathcal{S})$在上述三个不变空间上的投影。

英文摘要：

We study symplectic manifolds $(M^{2l},ω)$ 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, $Γ(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$ 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 $ϕ\in Γ(M,\mathcal{S}),$ we compute explicitly the projections of $R^Sϕ\in Γ(M,\bigwedge^2T^*M \otimes \mathcal{S})$ 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.

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