Wick rotations and real GIT

Abstract

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.

27 pages

References in corpus (4)

• How to Wick rotate generic curved spacetime
• Pseudo-Riemannian VSI spaces II
• On holomorphic Riemannian geometry and submanifolds of Wick-related spaces
• Pseudo-Riemannian VSI spaces

Cited by in corpus (4)

• Wicked metrics
• Wick-rotations of pseudo-Riemannian Lie groups
• Real GIT with applications to compatible representations and Wick-rotations
• Universal Walker metrics