Toric gravitational instantons in gauged supergravity

We introduce a general class of toric gravitational instantons in $D=4$, $\mathcal{N}=2$ gauged supergravity, namely Euclidean supersymmetric solutions with $U(1)^2$ isometry. Such solutions are specified by a "supergravity labelled polytope", where the labels encode the 4-manifold topology, the choice of magnetic fluxes, and certain signs associated to the Killing spinor. Equivariant localization then allows us to write down the gravitational free energy for such a solution, assuming it exists, and study its properties. These results open the way for a systematic study of holography in this setting, where the dual large $N$ field theories are defined on the boundary 3-manifolds, which are (squashed) lens spaces $L(p,q)$ or generalizations with orbifold singularities.