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The exact quantum entropy of BPS black holes can be evaluated usinglocalization in supergravity. An important ingredient in this program, that hasbeen lacking so far, is the one-loop effect arising from the quadraticfluctuations of the exact deformation (the $Q\mathcal{V}$ operator). We computethe fluctuation determinant for vector multiplets and hyper multiplets around$Q$-invariant off-shell configurations in four-dimensional $\mathcal{N}=2$supergravity with $AdS_{2} \times S^{2}$ boundary conditions, using theAtiyah-Bott fixed-point index theorem and a subsequent zeta functionregularization. Our results extend the large-charge on-shell entropycomputations in the literature to a regime of finite charges. Based on ourresults, we present an exact formula for the quantum entropy of BPS black holesin $\mathcal{N}=2$ supergravity. We explain cancellations concerning$\frac18$-BPS black holes in $\mathcal{N}=8$ supergravity that were observed inarXiv:1111.1161. We also make comments about the interpretation of alogarithmic term in the topological string partition function in the low energysupergravity theory.

Functional determinants, index theorems, and exact quantum black hole entropy