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

TL;DR

This work derives an exact quantum entropy formula for BPS black holes in four-dimensional $\\mathcal{N}=2$ supergravity by applying supersymmetric localization to the quantum entropy functional. The authors compute the one-loop determinants from vector and hyper multiplet fluctuations around the $AdS_2\\times S^2$ near-horizon geometry using the Atiyah-Bott fixed-point index theorem and zeta-function regularization, yielding a perturbatively exact expression $W^{\\text{pert}}(q,p)=\\int_{\\mathcal{M}_{Q}}\\prod_I d\\phi^I \; e^{-\\pi q_I\\phi^I + 4\\pi \\text{Im}F((\\phi^I + i p^I)/2)} Z_{\\text{det}}(\\phi)$. The determinant factors decompose into an induced measure and a one-loop piece, with explicit results $Z_{\\text{1-loop}}^{\\text{vec}} = e^{\\mathcal{K}/12}$ and $Z_{\\text{1-loop}}^{\\text{hyp}} = e^{-\\mathcal{K}/12}$, leading to an $a_0$-dependent logarithmic correction that matches known large-charge results and clarifies cancellations in extended supergravities. The paper also discusses how the exact gravitational entropy connects to topological string theory via the prepotential $F$ and OSV-type conjectures, while highlighting the importance of the induced measure and potential wall-crossing invariance of a black-hole index. Overall, the results provide a perturbatively exact framework for quantum black hole entropy in $\\mathcal{N}=2$ theories and illuminate deep links to topological strings and microscopic degeneracies.

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.