Open orbits in causal flag manifolds, modular flows and wedge regions
Karl-Hermann Neeb
TL;DR
The paper classifies causal flag manifolds as conformal completions of simple Euclidean Jordan algebras, showing that such manifolds exist precisely for hermitian groups of tube type and can be realized as Shilov boundaries of symmetric tube domains. Open orbits under symmetric subgroups, termed causal Makarevič spaces, are parameterized by involutions of the Jordan algebra and split into compactly/non-compactly causal types, with modular flows governed by Euler elements. It provides a detailed taxonomy of these spaces, including Cayley-type and Pierce-type cases, and analyzes positivity (wedge) regions, their (non-)global hyperbolicity, and their Lorentzian realizations, especially in the Minkowski, de Sitter, and anti-de Sitter settings. The work leverages Jordan-algebraic structure and Cayley transforms to obtain uniform geometric descriptions of group-type spaces and explores implications for Algebraic Quantum Field Theory and holography. Overall, the results illuminate how symmetry, causality, and modular dynamics intertwine in highly structured spacetime analogs and their boundaries.
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.