Equivariant localization for $D=5$ gauged supergravity

We consider supersymmetric solutions of $D=5$ Euclidean gauged supergravity coupled to an arbitrary number of vector multiplets. We consider solutions that admit both the R-symmetry Killing vector, $\mathcal{K}$, constructed as a bilinear in the Killing spinor, as well as an additional Killing vector $\ell$. Using $\ell$ to perform a dimensional reduction to $D=4$, $\mathcal{N}=2$ gauged supergravity, we show how the $D=5$ on-shell action can be computed using equivariant localization. We illustrate the formalism with some examples, computing the supersymmetric Casimir energy and the supersymmetric index of the dual SCFT without using the explicit supergravity solutions.