Hairy Black Holes and Duality in an Extended Supergravity Model

We consider a $D=4$, $\mathcal{N}=2$ gauged supergravity with an electromagentic Fayet-Iliopoulos term. We restrict to the uncharged, single dilaton consistent truncation and point out that the bulk Lagrangian is self-dual under electromagnetic duality. Within this truncation, we construct two families of exact hairy black hole solutions, which are asymptotically $Ad{S}_4$. When a duality transformation is applied on these solutions, they are mapped to two other inequivalent families of hairy black hole solutions. The mixed boundary conditions of the scalar field correspond to adding a triple-trace operator to the dual field theory action. We also show that the this truncation contains all the consistent single dilaton truncations of gauged $\mathcal{N}=8$ supergravity with a possible $\omega$-deformation.