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

TL;DR

This paper proves that for static, BPS AdS$_4$ black holes in ${\cal N}=2$ FI-gauged supergravity, the holographically renormalized on-shell action equals the negative of the black hole entropy, $\widetilde{{\cal S}}_{\rm on-shell}|_{\rm BPS}=-\frac{A_h}{4G_N}$. The authors use holographic renormalization, a Legendre transform for Neumann boundary conditions on scalar fields, and a reduction to a one-dimensional BPS action to show that divergences cancel and the finite boundary contributions vanish, leaving only the horizon term. The entropy arises from the extrinsic curvature at the horizon, and the analysis clarifies the equivalence between the twisted ABJM partition function on $Σ_g\times S^1$ and black hole entropy, implying no hair for these solutions. The method relies on a universal structure of the BPS equations and boundary terms and is robust against explicit knowledge of the full bulk solution, suggesting broad applicability to similar AdS/CFT contexts and potential generalizations to other dimensions and matter content.

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 $Σ_g\times S^1$ evaluated at the horizon, where $Σ_g$ may have any genus $g\geq 0$. This provides a clarification of the equivalence between the partition function of the twisted ABJM theory on $Σ_g\times S^1$ and the entropy of the dual black hole solutions. It also demonstrates that the complete entropy resides on the AdS$_2\times Σ_g$ horizon geometry, implying the absence of hair for these gravity solutions.