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arXiv 1010.2632hep-th
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六维超引力的旋量几何和Killing旋量方程

Spinorial geometry and Killing spinor equations of 6-D supergravity

Mehmet Akyol, George Papadopoulos

备注：
34 pages, Minor changes
发表信息：
Class.Quant.Grav.28:105001,2011
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摘要：

我们求解了任意张量、矢量和标量多重态耦合的6维（1，0）超引力的Killing旋量方程。Killing旋量的各向同性群是$Sp(1)\cdot Sp(1)\ltimes \bH (1)$、$U(1)\cdot Sp(1)\ltimes \bH (2)$、$Sp(1)\ltimes \bH (3,4)$、$Sp(1) (2)$、$U(1) (4)$和$\{1\} (8)$，其中括号中是每种情况下保持的超对称数。如果各向同性群是非紧的，时空允许一个平行的零1-形式相对于一个扭转的引力多重态的3-形式场强度的联系。相关联的向量场是Killing，3-形式是根据时空几何确定的。$Sp(1)\ltimes \bH$的情况下，允许一个后代的解决方案，保持3出4超对称性，由于hyperini Killing旋量方程。如果各向同性群是紧的，时空允许一个由1-形式旋量双线构成的自然框架。在$Sp(1)$和U（1）情形中，时空分别允许3个和4个关于挠率联络的平行1-形式。相关的向量场是Killing，在某些附加的限制下，时空是一个带有纤维的主丛，一个洛伦兹李群。Killing旋量方程对所有其他领域施加的条件也被确定。

英文摘要：

We solve the Killing spinor equations of 6-dimensional (1,0)-supergravity coupled to any number of tensor, vector and scalar multiplets in all cases. The isotropy groups of Killing spinors are $Sp(1)\cdot Sp(1)\ltimes \bH (1)$, $U(1)\cdot Sp(1)\ltimes \bH (2)$, $Sp(1)\ltimes \bH (3,4)$, $Sp(1) (2)$, $U(1) (4)$ and $\{1\} (8)$, where in parenthesis is the number of supersymmetries preserved in each case. If the isotropy group is non-compact, the spacetime admits a parallel null 1-form with respect to a connection with torsion the 3-form field strength of the gravitational multiplet. The associated vector field is Killing and the 3-form is determined in terms of the geometry of spacetime. The $Sp(1)\ltimes \bH$ case admits a descendant solution preserving 3 out of 4 supersymmetries due to the hyperini Killing spinor equation. If the isotropy group is compact, the spacetime admits a natural frame constructed from 1-form spinor bi-linears. In the $Sp(1)$ and U(1) cases, the spacetime admits 3 and 4 parallel 1-forms with respect to the connection with torsion, respectively. The associated vector fields are Killing and under some additional restrictions the spacetime is a principal bundle with fibre a Lorentzian Lie group. The conditions imposed by the Killing spinor equations on all other fields are also determined.

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