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

Abstract: The exact quantum entropy of BPS black holes can be evaluated using localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the $Q\mathcal{V}$ operator). We compute the 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 the Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in $\mathcal{N}=2$ supergravity. We explain cancellations concerning $\frac18$-BPS black holes in $\mathcal{N}=8$ supergravity that were observed in arXiv:1111.1161. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory.