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

Neeb, Karl-Hermann

doi:10.5802/jolt.1423

Neeb, Karl-Hermann ¹

¹Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany

Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 139-199

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 Makarevič 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, including de Sitter and anti-de Sitter space, in some detail.

Article information
Cite this article

Received: 2025-02-17
Accepted: 2025-05-05
Published online: 2026-06-17

DOI: 10.5802/jolt.1423

Classification: 22E46, 81R05, 53C50
Keywords: causal manifolds, causal symmetric space, flag manifold, Jordan algebra, Jordan spacetime, $\mathrm{AdS}/\mathrm{CFT}$-correspondence

Author's affiliations:

Neeb, Karl-Hermann ¹

¹ Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany

Neeb, Karl-Hermann. Open orbits in causal flag manifolds, modular flows and wedge regions. Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 139-199. doi: 10.5802/jolt.1423

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