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The entropy of a class of asymptotically AdS$_4$ magnetically charged BPSblack holes can be obtained by extremizing the topologically twisted index ofthe 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 twistedcompactifications of M2-branes living at a Calabi-Yau four-fold has beenrecently proposed. In this paper we investigate the relation between the twoextremization principles. We show that the two extremization procedures areequivalent for theories without baryonic symmetries, which include ABJM and thetheory dual to the non-toric Sasaki-Einstein manifold $V^{5,2}$. We thenconsider a class of quivers dual to M2-branes at toric Calabi-Yau four-foldsfor 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 thisconstruction, that we call mesonic twist, and we show that the gravitationalextremization problem and $\mathcal{I}$-extremization are equivalent. Wecomment on more general cases.

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