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

Understand

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.