Supersymmetric R\'enyi entropy and charged hyperbolic black holes

The supersymmetric R'enyi entropy across a spherical entangling surface in a $d$-dimensional SCFT with flavor defects is equivalent to a supersymmetric partition function on ${\mathbb{H}}^{d-1}\times {\mathbb{S}}^1$, which can be computed exactly using localization. We consider the holographically dual BPS solutions in $(d+1)$-dimensional matter coupled supergravity $(d=3,5)$, which are charged hyperbolically sliced AdS black holes. We compute the renormalized on-shell action and the holographic supersymmetric R'enyi entropy and show a perfect match with the field theory side. Our setup allows a direct map between the chemical potentials for the global symmetries of the field theories and those of the gravity solutions. We also discuss a simple case where angular momentum is added.