$\mathcal{I}$-Extremization for AdS$_4$ Black Holes: Master Volume, Free Energy, and Baryonic Charges

TL;DR

This paper demonstrates that the entropy of certain AdS$_4$ black holes can be derived through an $ ext{I}$-extremization approach based on toric geometry, connecting geometric and holographic methods.

Contribution

It proves the reformulation of black hole entropy as an $ ext{I}$-extremization problem for toric geometries and simplifies related extremization proofs.

Findings

Black hole entropy can be obtained via $ ext{I}$-extremization.

Established the link between master volume and entropy in toric geometries.

Provided examples illustrating the $ ext{I}$-extremization framework.

Abstract

In a previous paper, we proposed an entropy function for AdS$_4$ BPS black holes in M-theory with general magnetic charges, resolving a long-standing puzzle about baryonic charges in three-dimensional holography and offering a prediction for the large-$N$ limit of several partition functions whose saddle points have yet to be found. The entropy function is constructed from the master volume of the internal manifold. In this paper, we prove that the entropy of a general class of black holes based on toric geometry can indeed be reformulated as an $\mathcal{I}$-extremization problem, and we provide a set of examples. As an aside, we also simplify existing proofs of the equivalence between $a$-, $c$-, and $F$-extremizations and their gravitational duals.