Elliptic de Sitter space: $dS/Z_2$

We propose that, for every event in de Sitter space, there is a CPT-conjugate event at its antipode. Such an “elliptic” $Z_2$ identification of de Sitter space provides a concrete realization of observer complementarity: every observer has complete information. It is possible to define the analogue of an S matrix for quantum gravity in elliptic de Sitter space that is measurable by all observers. In a holographic description, S-matrix elements may be represented by correlation functions of a dual (conformal field) theory that lives on the single boundary sphere. S-matrix elements are de Sitter invariant, but have different interpretations for different observers. We argue that Hilbert states do not necessarily form representations of the full de Sitter group, but just of the subgroup of rotations. As a result, the Hilbert space can be finite dimensional and still have a positive norm. We also discuss the elliptic interpretation of de Sitter space in the context of type ${\mathrm{IIB}}^{*}$ string theory.