On-shell action and the Bekenstein-Hawking entropy of supersymmetric black holes in $AdS_6$

TL;DR

This work demonstrates, within Euclidean $F(4)$ gauged supergravity, that the renormalized on-shell action matches the Bekenstein-Hawking entropy for supersymmetric $AdS_6$ black holes, thereby supporting the AdS/CFT correspondence through holographic renormalization. It extends the analysis to supersymmetric $AdS_6$ black strings and black branes, showing their on-shell actions reproduce the central charge of 2d theories and the $S^3$ free energy of 3d theories, respectively. A central technical achievement is the careful construction of counterterms (including $S_{ ext{R}^2}$, $S_{F^2}$, and $S_ ho$) that cancel divergences against the precise asymptotics dictated by supersymmetry, with the resulting finite action equaling boundary-theory quantities. The results reveal a uniform structure in the on-shell actions of black objects arising from topological twisting in $F(4)$ gauged supergravity, supporting a coherent holographic picture across dimensions and linking microscopic twisted-index counts to gravity-side entropy via the bulk action.

Abstract

Recently, entropy of static supersymmetric black holes in $AdS_6$ was microscopically counted by the topologically twisted index of five-dimensional superconformal field theories. However, the AdS/CFT dictionary states that the partition function of the boundary field theory corresponds to the on-shell action of the bulk quantum gravity. In this paper, we aim to explain the microscopic counting of black hole entropy by the topologically twisted index. We calculate the renormalized on-shell action and show that the on-shell action equals the Bekenstein-Hawking entropy of the supersymmetric black holes in $AdS_6$.