On the quantum entropy function in 4d gauged supergravity

We analyze BPS black hole attractors in the conformal 4d gauged supergravity formalism and apply the technique known as supergravity localization in order to evaluate Sen's quantum entropy function in the $\mathrm{AdS}_2\!\times\!\mathrm{S}^2$ near-horizon geometry. Under certain assumptions, we reduce the exact expression of the functional integral to a finite-dimensional integral for a number of supersymmetric black holes in gauged supergravity with AdS asymptotics subject to a holographic description via a dual field theory. Examples include the asymptotically $\mathrm{AdS}_4\!\times\!\mathrm{S}^7$ Cacciatori-Klemm black holes in M-theory and the asymptotically $\mathrm{AdS}_5\!\times\!\mathrm{S}^5$ generalizations of Gutowski-Reall black holes and Benini-Bobev black strings in type IIB, as well as the recently constructed asymptotically $\mathrm{AdS}_4\!\times\!\mathrm{S}^6$ solutions in massive type IIA. Our results provide an important first step towards a gravitational counterpart to the exact evaluation of supersymmetric partition functions at finite $N$ for the holographically dual field theories in these examples.