  We continue the effort of defining and evaluating the quantum entropy
function for supersymmetric black holes in 4d ${\cal N} = 2$ gauged
supergravity, initiated in [1803.05920]. The emphasis here is on the missing
steps in the previous localization analysis, mainly dealing with one-loop
determinants for abelian vector multiplets and hypermultiplets on the
non-compact space $\mathbb{H}_2 \times \Sigma_{\rm g}$ with particular boundary
conditions. We use several different techniques to arrive at consistent
results, which have a most direct bearing on the logarithmic correction terms
to the Bekenstein-Hawking entropy of said black holes.

High Energy Physics - Theory