Real GIT with applications to compatible representations and Wick-rotations

TL;DR

This paper explores real geometric invariant theory and compatible representations to understand Wick-rotations of pseudo-Riemannian manifolds, providing new conditions for when such manifolds can be Wick-rotated to different signatures.

Contribution

It extends previous results by establishing necessary and sufficient conditions for Wick-rotations between pseudo-Riemannian manifolds of arbitrary signatures.

Findings

Derived existence conditions for Wick-rotations between different signatures.

Proved an invariance theorem for Wick-rotations of arbitrary signatures.

Generalized previous results to broader classes of pseudo-Riemannian manifolds.

Abstract

Motivated by Wick-rotations of pseudo-Riemannian manifolds, we study real geometric invariant theory (GIT) and compatible representations. We extend some of the results from earlier works \cite{W2,W1}, in particular, we give some sufficient as well as necessary conditions for when pseudo-Riemannian manifolds are Wick-rotatable to other signatures. For arbitrary signatures, we consider a Wick-rotatable pseudo-Riemannian manifold with closed $O(p,q)$-orbits, and thus generalise the existence condition found in \cite{W1}. Using these existence conditions we also derive an invariance theorem for Wick-rotations of arbitrary signatures.