# Wick rotations and real GIT

## TL;DR

This paper introduces a framework for Wick rotations using real slices of holomorphic Riemannian manifolds and applies real geometric invariant theory to study their existence, exemplified by rotating a G2-holonomy manifold.

## Contribution

It develops a new approach to Wick rotations via real forms of complex Lie groups and uses real GIT to analyze their existence and properties.

## Key findings

- Established criteria for the existence of Wick rotations between pseudo-Riemannian spaces.
- Applied real GIT to determine non-existence in certain cases.
- Explicitly Wick rotated a G2-holonomy manifold to a split-G2 holonomy manifold.

## 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.