Wick旋转与实GIT
Wick rotations and real GIT
摘要：
我们通过把伪黎曼流形看作全纯黎曼流形的实切片来定义威克旋转。从标架丛观点出发，不同伪黎曼空间之间的威克旋转可以通过它们的结构群来研究，这些结构群是相应复化李群的实形式（复李群$O(n,\mathbb{C})$的不同实形式$O(p,q)$）。通过这种方式，我们可以使用真正的GIT（几何不变理论）来推导出几个新的结果，关于这种威克旋转的存在和不存在。作为一个显式例子，我们Wick将一个已知的$G_2$ -完整流形旋转到一个具有分裂- $G_2$完整流形的伪黎曼流形.
英文摘要：
We define Wick-rotations by considering pseudo-Riemannian manifolds as real slices of a holomorphic Riemannian manifold. From a frame bundle viewpoint Wick-rotations between different pseudo-Riemannian spaces can then be studied through their structure groups which are real forms of the corresponding complexified Lie group (different real forms $O(p,q)$ of the complex Lie group $O(n,\mathbb{C})$). In this way, we can use real GIT (geometric invariant theory) to derive several new results regarding the existence, and non-existence, of such Wick-rotations. As an explicit example, we Wick rotate a known $G_2$-holonomy manifold to a pseudo-Riemannian manifold with split-$G_2$ holonomy.
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