From Fake Supergravity to Superstars

The fake supergravity method is applied to 5-dimensional asymptotically AdS spacetimes containing gravity coupled to a real scalar and an abelian gauge field. The motivation is to obtain bulk solutions with R x S^3 symmetry in order to explore the AdS/CFT correspondence when the boundary gauge theory is on R x S^3. A fake supergravity action, invariant under local supersymmetry through linear order in fermion fields, is obtained. The gauge field makes things more restrictive than in previous applications of fake supergravity which allowed quite general scalar potentials. Here the superpotential must take the form W(\phi) ~ exp(-k\phi) + c exp(2\phi/(3k)), and the only freedom is the choice of the constant k. The fermion transformation rules of fake supergravity lead to fake Killing spinor equations. From their integrability conditions, we obtain first order differential equations which we solve analytically to find singular electrically charged solutions of the Lagrangian field equations. A Schwarzschild mass term can be added to produce a horizon which shields the singularity. The solutions, which include "superstars", turn out to be known in the literature. We compute their holographic parameters.