Moduli spaces in AdS 4 supergravity

We study the structure of the supersymmetric moduli spaces of = 1 and = 2 supergravity theories in AdS ₄ backgrounds. In the = 1 case, the moduli space cannot be a complex submanifold of the Kähler field space, but is instead real with respect to the inherited complex structure. In = 2 supergravity the same result holds for the vector multiplet moduli space, while the hypermultiplet moduli space is a Kähler submanifold of the quaternionic-Kähler field space. These findings are in agreement with AdS/CFT considerations.