We study simple space-time symmetry groups G which act on a space-time manifold Å = G H which admits a G-invariant global causal structure. We classify pairs (G, Å) which share the following additional properties of conformal field theory: 1) The stability subgroup H of o ∈ Å is the identity component of a parabolic subgroup of G, implying factorization H = MAN − , where M generalizes Lorentz transformations, A dilatations, and N − special conformal transformations. 2) special conformal transformations ξ ∈ N − act trivially on tangent vectors v ∈ T o Å. The allowed simple Lie groups G are the universal coverings of SU (m, m), SO(2, D), Sp(l, Ê), SO * (4n) and E 7(−25) andH are particular maximal parabolic subgroups. They coincide with the groups of fractional linear transformations of Euklidean Jordan algebras whose use as generalizations of Minkowski space time was advocated by Günaydin. All these groups G admit positive energy representations. It will also be shown that the classical conformal groups SO(2, D) are the only allowed groups which possess a time reflection automorphism T ; in all other cases space-time has an intrinsic chiral structure.

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