Wedge domains in non-compactly causal symmetric spaces

Abstract

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory, modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric spaces G/H. This class contains de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of $${{\mathfrak {g}}}$$ . We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of these seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of these spaces has a natural extension to a non-compactly causal symmetric space of the form $$G_{{\mathbb {C}}}/G^c$$ where $$G^c$$ is a certain real form of the complexification $$G_{{\mathbb {C}}}$$ of G. As $$G_{{\mathbb {C}}}/G^c$$ is non-compactly causal, it also contains three types of wedge domains. Our results says that the intersection of these domains with G/H agree with the wedge domains in G/H.