Exactly Marginal Deformations and Their Supergravity Duals

We study the space of supersymmetric ${\mathrm{AdS}}_5$ solutions of type IIB supergravity corresponding to the conformal manifold of the dual $\mathcal{N}=1$ conformal field theory. We show that the background geometry naturally encodes a generalized holomorphic structure, dual to the superpotential of the field theory, with the existence of the full solution following from a continuity argument. In particular, this work allows us to address the long-standing problem of finding the gravity dual of the generic $\mathcal{N}=1$ deformations of $\mathcal{N}=4$ conformal field theory: even if we are not able to give it in a fully explicit form, we provide a proof-of-existence of the supergravity solution. Using this formalism, we derive a new result for the Hilbert series of the deformed field theories.