Geometry of $\mathcal{I}$-extremization and black holes microstates

The entropy of a class of asymptotically AdS$_4$ magnetically charged BPS black holes can be obtained by extremizing the topologically twisted index of the dual three-dimensional field theory. This principle is known as $\mathcal{I}$-extremization. A gravitational dual of $\mathcal{I}$-extremization for a class of theories obtained by twisted compactifications of M2-branes living at a Calabi-Yau four-fold has been recently proposed. In this paper we investigate the relation between the two extremization principles. We show that the two extremization procedures are equivalent for theories without baryonic symmetries, which include ABJM and the theory dual to the non-toric Sasaki-Einstein manifold $V^{5,2}$. We then consider a class of quivers dual to M2-branes at toric Calabi-Yau four-folds for which the $\mathcal{I}$-functional can be computed in the large $N$ limit, and depends on three mesonic fluxes. We propose a gravitational dual for this construction, that we call mesonic twist, and we show that the gravitational extremization problem and $\mathcal{I}$-extremization are equivalent. We comment on more general cases.