Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the minkowskian spacetimes by an extra ``dilatonic'' coordinate, whose rotation, Lorentz and conformal groups are SO(d−1), SO(d−1,1) × SO(1,1) and SO(d,2) × SO(2,1), respectively. The generalized spacetimes described by simple Jordan algebras of degree three correspond to extensions of minkowskian spacetimes in the critical dimensions (d = 3,4,6,10) by a dilatonic and extra (2,4,8,16) commuting spinorial coordinates, respectively. Their rotation, Lorentz and conformal groups are those that occur in the first three rows of the Magic Square. The Freudenthal triple systems defined over these Jordan algebras describe conformally covariant phase spaces. Following hep-th/0008063, we give a unified geometric realization of the quasiconformal groups that act on their conformal phase spaces extended by an extra ``cocycle'' coordinate. For the generic Jordan family the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are given. The minimal unitary representations of the quasiconformal groups F4(4), E6(2), E7(−5) and E8(−24) of the simple Jordan family were given in our earlier work [10].