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

We study the symmetries of generalized spacetimes and corresponding phasespaces defined by Jordan algebras of degree three. The generic Jordan family offormally real Jordan algebras of degree three describe extensions of theMinkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation,Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) andSO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simpleJordan algebras of degree three correspond to extensions of Minkowskianspacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra(2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthaltriple systems defined over these Jordan algebras describe conformallycovariant phase spaces. Following hep-th/0008063, we give a unified geometricrealization of the quasiconformal groups that act on their conformal phasespaces extended by an extra "cocycle" coordinate. For the generic Jordan familythe quasiconformal groups are SO(d+2,4), whose minimal unitary realizations aregiven. The minimal unitary representations of the quasiconformal groups F_4(4),E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in ourearlier work hep-th/0409272.Comment: A typo in equation (37) corrected and missing titles of some references added. Version to be published in JHEP. 38 pages, latex fil