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

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.