Unified Maxwell-Einstein and Yang-Mills-Einstein supergravity theories in four dimensions

We study unified N ≤ 2 Maxwell-Einstein supergravity theories (MESGTs) and unified Yang-Mills Einstein supergravity theories (YMESGTs) in four dimensions. As their defining property, these theories admit the action of a global or local symmetry group that is (i) simple, and (ii) acts irreducibly on all the vector fields of the theory, including the ''graviphoton''. Restricting ourselves to the theories that originate from five dimensions via dimensional reduction, we find that the generic Jordan family of MESGTs with the scalar manifolds M ≤ [SU(1,1)/U(1)]×[SO(2,n)/SO(2)×SO(n)] are all unified in four dimensions with the unifying global symmetry group SO(2,n). Of these theories only one can be gauged so as to obtain a unified YMESGT with the gauge group SO(2,1). Three of the four magical supergravity theories defined by simple euclidean Jordan algebras J₃ ^{double-struck A sign}(double-struck A sign ≤ ℂ,ℍ,double- struck O sign) of degree 3 are unified MESGTs in four dimensions. The MESGTs defined by J₃^ℂ and J₃ ^{double-struck O sign} can furthermore be gauged so as to obtain a 4D unified YMESGT with gauge groups SO(3,2) and SO(6,2), respectively. The generic non-Jordan family and the theories whose scalar manifolds are homogeneous but not symmetric do not lead to unified MESGTs in four dimensions. The three infinite families of unified five-dimensional MESGTs defined by simple lorentzian Jordan algebras of degree p≠4, whose scalar manifolds are non-homogeneous, do not lead directly to unified MESGTs in four dimensions under dimensional reduction. However, since their manifolds are non-homogeneous we are not able to completely rule out the existence of symplectic sections in which these theories become unified in four dimensions.