Maximally Supersymmetric AdS Solutions and their Moduli Spaces

We study maximally supersymmetric AdS$_D$ solutions of gauged supergravities in dimensions $D \geq 4$. We show that such solutions can only exist if the gauge group after spontaneous symmetry breaking is a product of two reductive groups $H_R \times H_\mathrm{mat}$, where $H_R$ is uniquely determined by the dimension D and the number of supersymmetries N while $H_\mathrm{mat}$ is unconstrained. This resembles the structure of the global symmetry groups of the holographically dual SCFTs, where $H_R$ is interpreted as the R-symmetry and $H_\mathrm{mat}$ as the flavor symmetry. Moreover, we discuss possible supersymmetry preserving continuous deformations, which correspond to the conformal manifolds of the dual SCFTs. Under the assumption that the scalar manifold of the supergravity is a symmetric space we derive general group theoretical conditions on these moduli. Using these results we determine the AdS solutions of all gauged supergravities with more than 16 real supercharges. We find that almost all of them do not have supersymmetry preserving deformations with the only exception being the maximal supergravity in five dimensions with a moduli space given by $SU(1,1)/U(1)$. Furthermore, we determine the AdS solutions of four-dimensional N=3 supergravities and show that they similarly do not admit supersymmetric moduli.