On Moduli Spaces in $\mathbf{A}\mathbf{d}{\mathbf{S}}_{\mathbf{4}}$ Supergravity

We study the structure of the supersymmetric moduli spaces of $\mathcal{N}=1$ and $\mathcal{N}=2$ supergravity theories in $Ad{S}_4$ backgrounds. In the ${\cal N}=1$ case, the moduli space cannot be a complex submanifold of the K"ahler field space, but is instead real with respect to the inherited complex structure. In $\mathcal{N}=2$ supergravity the same result holds for the vector multiplet moduli space, while the hypermultiplet moduli space is a K"ahler submanifold of the quaternionic-K"ahler field space. These findings are in agreement with AdS/CFT considerations.