On the On-Shell: The Action of AdS$_4$ Black Holes

TL;DR

This paper calculates the on-shell action of AdS4 black holes in gauged supergravity, showing it equals the black hole entropy and clarifies the holographic relation with the dual field theory.

Contribution

It demonstrates the equivalence between the on-shell action and entropy for BPS black holes in AdS4 and clarifies the holographic correspondence with the twisted ABJM theory.

Findings

On-shell action equals minus the black hole entropy.

Divergences are canceled with appropriate boundary conditions.

Entropy is localized on the horizon geometry, indicating no hair.

Abstract

We compute the on-shell action of static, BPS black holes in AdS$_4$ from ${\cal N}=2$ gauged supergravity coupled to vector multiplets and show that it is equal to minus the entropy of the black hole. Holographic renormalization is used to demonstrate that with appropriate boundary conditions on the scalar fields, the divergent and finite contributions from the asymptotic boundary vanish. The entropy arises from the extrinsic curvature on $\Sigma_g\times S^1$ evaluated at the horizon, where $\Sigma_g$ may have any genus $g\geq 0$. This provides a clarification of the equivalence between the partition function of the twisted ABJM theory on $\Sigma_g\times S^1$ and the entropy of the dual black hole solutions. It also demonstrates that the complete entropy resides on the AdS$_2\times \Sigma_g$ horizon geometry, implying the absence of hair for these gravity solutions.