Quantum Black Hole Entropy, Localization and the Stringy Exclusion Principle

Supersymmetric localization has lead to remarkable progress in computing quantum corrections to BPS black hole entropy. The program has been successful especially for computing perturbative corrections to the Bekenstein-Hawking area formula. In this work, we consider non-perturbative corrections related to polar states in the Rademacher expansion, which describes the entropy in the microcanonical ensemble. We propose that these non-perturbative effects can be identified with a new family of saddles in the localization of the quantum entropy path integral. We argue that these saddles, which are euclidean $AdS_2\times S^1\times S^2$ geometries, arise after turning on singular fluxes in M-theory on a Calabi-Yau. They cease to exist after a certain amount of flux, resulting in a finite number of geometries; the bound on that number is in precise agreement with the stringy exclusion principle. Localization of supergravity on these backgrounds gives rise to a finite tail of Bessel functions in agreement with the Rademacher expansion. As a check of our proposal, we test our results against well-known microscopic formulas for one-eighth and one-quarter BPS black holes in $\mathcal{N}=8$ and $\mathcal{N}=4$ string theory respectively, finding agreement. Our method breaks down precisely when mock-modular effects are expected in the entropy of one-quarter BPS dyons and we comment upon this. Furthermore, we mention possible applications of these results, including an exact formula for the entropy of four dimensional $\mathcal{N}=2$ black holes.Comment: 66 page

Exact holography and black hole entropy in N=8 and N=4 string theory

We compute the exact entropy of one-eighth and one-quarter BPS black holes in N=8 and N=4 string theory respectively. This includes all the N=4 CHL models in both K3 and T^4 compactifications. The main result is a measure for the finite dimensional integral that one obtains after localization of supergravity on AdS_2xS^2. This measure is determined entirely by an anomaly in supersymmetric Chern-Simons theory on local AdS_3 and takes into account the contribution from all the supergravity multiplets. In Chern-Simons theory on compact manifolds this is the anomaly that computes a certain one-loop dependence on the volume of the manifold. For one-eighth BPS black holes our results are a first principles derivation of a measure proposed in arXiv:1111.1161, while in the case of one-quarter BPS black holes our result computes exactly all the perturbative or area corrections. Moreover, we argue that instantonic contributions can be incorporated and give evidence by computing the measure which matches precisely the microscopics. Along with this, we find an unitary condition that truncates the answer to a finite sum of instantons in perfect agreement with a microscopic formula. Our results solve a number of puzzles related to localization in supergravity and constitute a larger number of examples where holography can be shown to hold exactly.Comment: 49 pages; clarifications in the discussion of section 4.3, results unchanged; other minor typos correcte

Nonperturbative black hole entropy and Kloosterman sums

Non-perturbative quantum corrections to supersymmetric black hole entropy often involve nontrivial number-theoretic phases called Kloosterman sums. We show how these sums can be obtained naturally from the functional integral of supergravity in asymptotically AdS_2 space for a class of black holes. They are essentially topological in origin and correspond to charge-dependent phases arising from the various gauge and gravitational Chern-Simons terms and boundary Wilson lines evaluated on Dehn-filled solid 2-torus. These corrections are essential to obtain an integer from supergravity in agreement with the quantum degeneracies, and reveal an intriguing connection between topology, number theory, and quantum gravity. We give an assessment of the current understanding of quantum entropy of black holes.Comment: 35 pages; minor changes, JHEP versio

Localization & Exact Holography

We consider the AdS_2/CFT_1 holographic correspondence near the horizon of big four-dimensional black holes preserving four supersymmetries in toroidally compactified Type-II string theory. The boundary partition function of CFT_1 is given by the known quantum degeneracies of these black holes. The bulk partition function is given by a functional integral over string fields in AdS_2. Using recent results on localization we reduce the infinite-dimensional functional integral to a finite number of ordinary integrals over a space of localizing instantons. Under reasonable assumptions about the relevant terms in the effective action, these integrals can be evaluated exactly to obtain a bulk partition function. It precisely reproduces all terms in the exact Rademacher expansion of the boundary partition function as nontrivial functions of charges except for the Kloosterman sum which can in principle follow from an analysis of phases in the background of orbifolded instantons. Our results can be regarded as a step towards proving `exact holography' in that the bulk and boundary partition functions computed independently agree for finite charges. Since the bulk partition function defines the quantum entropy of the black hole, our results enable the evaluation of perturbative as well as nonperturbative quantum corrections to the Bekenstein-Hawking-Wald entropy of these black holes

Localization in Supergravity and Quantum AdS4/CFT3AdS_4/CFT_3 Holography

We compute the quantum gravity partition function of M-theory on $AdS_4 \times X_7$ by using localization techniques in four-dimensional gauged supergravity obtained by a consistent truncation on the Sasaki-Einstein manifold $X_{7}$. The supergravity path integral reduces to a finite dimensional integral over two collective coordinates that parametrize the localizing instanton solutions. The renormalized action of the off-shell instanton solutions depends linearly and holomorphically on the "square root" prepotential evaluated at the center of $AdS_{4}$. The partition function resembles the Laplace transform of the wave function of a topological string and with an assumption about the measure for the localization integral yields an Airy function in precise agreement with the computation from the boundary ABJM theory on a 3-sphere. Our bulk quantum gravity computation is nonperturbatively exact in four-dimensional Planck length but ignores corrections due to brane-instantons.Comment: 32 pages; v2: minor changes, JHEP versio

Quantum entropy of supersymmetric black holes

We review recent progress concerning the quantum entropy of a large class of supersymmetric black holes in string theory both from the microscopic and macroscopic sides. On the microscopic field theory side, we present new results concerning the counting of black hole microstates for charge vectors with nontrivial arithmetic duality invariants. On the macroscopic gravitational side, we present a novel application of localization techniques to a supergravity functional integral to compute the quantum entropy of these black holes. Localization leads to an enormous simplification of a path integral of string theory in AdS_2 by reducing it to a finite dimensional integral. The localizing solutions are labeled by n_v+1 parameters, with n_v the number of vector multiplets in the theory of N=2 supergravity. As an example we show, for four dimensional large black holes which preserve four supersymmetries in toroidally compactified IIB string theory, that the macroscopic degeneracy precisely agrees with all the terms in an exact Rademacher expansion of the microscopic answer except for Kloosterman sums which in principle can be computed. Generalizing previous work, these finite charge contributions to the leading Bekenstein-Hawking entropy can also be viewed as an instance of "exact holography" in the context of AdS_2/CFT_1 correspondence.Comment: arXiv admin note: substantial text overlap with arXiv:1111.1161, arXiv:1012.026