Holographic duals of refined partition functions

Recent years have witnessed lots of progress in the computation of supersymmetric partition functions of SCFTs on curved manifolds via localization. The twisted partition function on product manifolds of the form $S^1 \times \Sigma_g$, where $\Sigma_g$ is a two-dimensional Riemann surface, is of particular relevance due to its role in the microstate counting for magnetic static AdS$_4$ black holes realizing the topological twist. We review here supergravity solutions having as conformal boundary more general 3d manifolds. We first focus on solutions (AdS-Taub-NUT and AdS-Taub-Bolt) having as boundary a circle bundle over $\Sigma_g$, showing the matching of their on-shell action with the large $N$ limit of the partition function of the dual CFT. We then discuss some recent results for a challenging example, which involves the refinement by angular momentum. The gravitational backgrounds in this case are rotating supersymmetric AdS$_4$ black holes. We review the salient features of two different classes of such solutions in theories of supergravity with uplift in M-theory, and comment on the current status of their entropy counting in the dual CFT.