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

TL;DR

This work constructs a unified geometric framework for generalized spacetimes defined by cubic forms from Euclidean Jordan algebras of degree three and their Freudenthal triple systems, organizing their symmetries through quasiconformal actions. It provides explicit geometric realizations of the conformal and quasiconformal groups, including $\mathrm{SO}(d+2,4)$ for generic Jordan families and the exceptional groups $\mathrm{F}_{4(4)}$, $\mathrm{E}_{6(2)}$, $\mathrm{E}_{7(-5)}$, $\mathrm{E}_{8(-24)}$ for simple Jordan algebras, with detailed 5-graded decompositions and differential operator realizations. The paper connects these geometric constructions to $N=2$ Maxwell-Einstein supergravity theories, outlining how U-duality groups emerge in $d=5,4,3$ dimensions, and shows how dilatonic and spinorial extensions of Minkowski spacetime arise from Jordan-algebra data. Finally, it derives minimal unitary realizations by quantizing the quasiconformal actions, giving explicit eigenvalues of the quadratic Casimir and completing the representation theory picture for these spacetime symmetries in a way relevant to U-duality, black hole charge spaces, and higher-dimensional quantum gravity frameworks.

Abstract

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) XSO(1,1) and SO(d,2)XSO(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. 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 F_4(4), E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our earlier work hep-th/0409272.