Euclidean Freedman-Schwarz model

The N=4 gauged SU(2)$\times$SU(1,1) supergravity in four-dimensional Euclidean space is obtained via a consistent dimensional reduction of the N=1, D=10 supergravity on $S^3\times AdS_3$. The dilaton potential in the theory is proportional to the difference of the two gauge coupling constants, which is due to the opposite signs of the curvatures of $S^3$ and $AdS_3$. As a result, the potential can be positive, negative, or zero-depending on the values of the constants. A consistent reduction of the fermion supersymmetry transformations is performed at the linearized level, and special attention is paid to the Euclidean Majorana condition. A further reduction of the D=4 theory is considered to the static, purely magnetic sector, where the vacuum solutions are studied. The Bogomol'nyi equations are derived and their essentially non-Abelian monopole-type and sphaleron-type solutions are presented. Any solution in the theory can be uplifted to become a vacuum of string or M-theory.