Stable de Sitter Vacua in 4 Dimensional Supergravity Originating from 5   Dimensions
  The five dimensional stable de Sitter ground states in N=2 supergravity obtained by gauging SO(1,1) symmetry of the real symmetric scalar manifold (in particular a generic Jordan family manifold of the vector multiplets) simultaneously with a subgroup R_s of the R-symmetry group descend to four dimensional de Sitter ground states under certain conditions. First, the holomorphic section in four dimensions has to be chosen carefully by using the symplectic freedom in four dimensions; and second, a group contraction is necessary to bring the potential into a desired form. Under these conditions, stable de Sitter vacua can be obtained in dimensionally reduced theories (from 5D to 4D) if the semi-direct product of SO(1,1) with R^(1,1) together with a simultaneous R_s is gauged. We review the stable de Sitter vacua in four dimensions found in earlier literature for N=2 Yang-Mills Einstein supergravity with SO(2,1) x R_s gauge group in a symplectic basis that comes naturally after dimensional reduction. Although this particular gauge group does not descend directly from five dimensions, we show that, its contraction does. Hence, two different theories overlap in certain limits. Examples of stable de Sitter vacua are given for the cases: (i) R_s=U(1)_R, (ii) R_s=SU(2)_R, (iii) N=2 Yang-Mills/Einstein Supergravity theory coupled to a universal hypermultiplet. We conclude with a discussion regarding the extension of our results to supergravity theories with more general homogeneous scalar manifolds. 
O. Ogetbil