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

TL;DR

This paper explores the symmetries of generalized spacetimes derived from Jordan algebras, detailing their conformal groups, phase spaces, and minimal unitary realizations, extending previous work on quasiconformal groups.

Contribution

It provides a unified geometric framework for quasiconformal groups acting on conformal phase spaces of Jordan algebra-based spacetimes, including explicit minimal unitary representations.

Findings

Describes extensions of Minkowski spacetime via Jordan algebras in critical dimensions.

Identifies quasiconformal groups acting on these phase spaces as SO(d+2,4).

Provides minimal unitary realizations of these quasiconformal groups.

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.