Open orbits in causal flag manifolds, modular flows and wedge regions

TL;DR

This paper explores the geometry of open orbits in causal flag manifolds related to Lie groups, focusing on their structures, modular flows, and applications in algebraic quantum field theory, especially in Lorentzian geometries.

Contribution

It characterizes invariant causal structures on flag manifolds, analyzes open orbits under symmetric subgroups, and studies modular flows and positivity regions in the context of AQFT.

Findings

Invariant causal structures exist only for hermitian groups of tube type.

Determined Euler elements and positivity regions for non-compactly causal spaces.

Identified conditions for the existence of modular flows and described globally hyperbolic subsets.

Abstract

We study open orbits of symmetric subgroups of a simple connected Lie group G on a causal flag manifold. First we show that a flag manifold M of G carries an invariant causal structure if and only if G is hermitian of tube type and M is the conformal completion of the corresponding simple euclidean Jordan algebra, resp., the Shilov boundary of the associated symmetric tube domain. We then study open orbits in M under symmetric subgroups, also called causal Makarevic spaces, from the perspective of applications in Algebraic Quantum Field Theory (AQFT). A key motivation is the geometry of corresponding modular flows. The open orbits are reductive causal symmetric spaces, which arise in two flavors: compactly causal and non-compactly causal ones. In the non-compactly causal case we determine the corresponding Euler elements and their positivity regions. For compactly causal spaces, modular flows do not always exist and we determine when this is the case. Then the positivity regions of the modular flows are not globally hyperbolic, but these spaces contain other interesting globally hyperbolic subsets that can be described in terms of the conformally flat Jordan coordinates via Cayley charts. We discuss the Lorentzian case, involving de Sitter and anti-de Sitter space in some detail.