Wedge Domains in Compactly Causal Symmetric Spaces

Abstract

AbstractMotivated by constructions in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces $M=G/H$, which includes in particular anti-de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. The key geometric structure we use is the modular flow on $M$ defined by an Euler element in the Lie algebra of $G$. Our main geometric result asserts that three seemingly different characterizations of these domains coincide: the positivity domain of the modular vector field, the domain specified by a KMS-like analytic extension condition for the modular flow, and the domain specified by a polar decomposition in terms of certain cones. In the second half of the article we show that our wedge domains share important properties with wedge domains in Minkowski space. If $G$ is semisimple, there exist unitary representations $(U,{\mathcal {H}})$ of $G$ and isotone covariant nets of real subspaces $\textsf {H}({\mathcal {O}}) \subseteq {\mathcal {H}}$, defined for any open subset ${\mathcal {O}} \subseteq M$, which assign to connected components of the wedge domains a standard subspace whose modular group corresponds to the modular flow on $M$. This corresponds to the Bisognano–Wichmann property in Quantum Field Theory. We also show that the set of $G$-translates of the connected components of the wedge domain provides a geometric realization of the abstract wedge space introduced by the first author and V. Morinelli.