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Motivated by construction in Algebraic Quantum Field Theory we introducewedge domains in compactly causal symmetric spaces M=G/H, which includes inparticular anti de Sitter space in all dimensions and its coverings. Our wedgedomains generalize Rindler wedges in Minkowski space. The key geometricstructure we use is the modular flow on M defined by an Euler element in theLie algebra of G. Our main geometric result asserts that three seeminglydifferent characterizations of these domains coincide: the positivity domain ofthe modular vector field; the domain specified by a KMS like analytic extensioncondition for the modular flow; and the domain specified by a polardecomposition in terms of certain cones.  In the second half of the article we show that our wedge domains shareimportant properties with wedge domains in Minkowski space. If G is semisimple,there exist unitary representations of G and isotone covariant nets of realsubspaces defined for any open subset of M, which assign to connectedcomponents of the wedge domains a standard subspace whose modular groupcorresponds to the modular flow on M. This corresponds to theBisognano--Wichmann property in Quantum Field Theory. We also show that the setof G-translates of the connected components of the wedge domain provides ageometric realization of the abstract wedge space introduced by the firstauthor and V. Morinelli.

Wedge domains in compactly causal symmetric spaces