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 $Ad{S}_2\times S^2$ 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 $\frac{1}{8}$-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.

47 pages; v3 accepted for publication in JHEP

References in corpus (14)

• Quantum Entropy Function from AdS(2)/CFT(1) Correspondence
• Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function
• Higher Derivative Corrections to Non-supersymmetric Extremal Black Holes in N=2 Supergravity
• Nonperturbative black hole entropy and Kloosterman sums
• Supersymmetric Index from Black Hole Entropy
• Arithmetic of Quantum Entropy Function
• Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
• Holographic Description of AdS_2 Black Holes
• A Farey tale for N=4 dyons
• Localization on Three-Manifolds
• Extended Supersymmetry on Curved Spaces
• Asymptotic Expansion of the N=4 Dyon Degeneracy
• Quantum black hole entropy and the holomorphic prepotential of N=2 supergravity
• Deformations of special geometry: in search of the topological string

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• Construction of all N=4 conformal supergravities
• Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds
• Dualities in CHL-Models
• An approach to BPS black hole microstate counting in an N=2 STU model
• On Exponentially Suppressed Corrections to BMPV Black Hole Entropy