Exact Holography and Black Hole Entropy in N = 8 $$ \mathcal{N}=8 $$ and N = 4 $$ \mathcal{N}=4 $$ String Theory

Abstract

Abstract: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 2 × S 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 a unitary condition that truncates the answer to a finite sum of instantons in perfect agreement with a microscopic formula. Our results therefore 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.

Single-centered black hole microstate degeneracies from instantons in supergravity

“…Further it was argued that the symmetries fix the form of the graviton multiplet determinant up to a constant that can be matched to an on-shell computation [59]. The latter has been addressed in various papers [36,60,61] although we think it is fair to say that a full satisfactory…”

“…While this paper was being prepared for publication, we received [36] on the arXiv. As we use some of the technical analysis presented in that paper, we present a brief review of the relevant parts in section 4 and a refinement of some of the details in appendix B.…”

“…It has been argued recently in [36] that the prefactor P 1 = 2m + 4. The function (3.14) then agrees precisely with the leading Bessel function in the Rademacher expansion of the microscopic theory (2.9) with the right weight, argument, and prefactor.…”

“…We find, however, that the arguments of the Bessel functions here decrease (as we expect) up to a point, but then increase indefinitely, thus showing that this sum is not convergent! A solution to this puzzle was presented recently in [36] by making a choice of contour γ in (4.1) and analyzing the contributions to the degeneracies from each term in the Fourier expansion. With this choice of contour, almost all of the infinite number of Bessel functions turn out to be highly suppressed, and one is left with a finite number of I-Bessel functions, consistent with the structure of the leading c = 1 term of the Rademacher expansion (2.9).…”

“…With this choice of contour, almost all of the infinite number of Bessel functions turn out to be highly suppressed, and one is left with a finite number of I-Bessel functions, consistent with the structure of the leading c = 1 term of the Rademacher expansion (2.9). We now review this analysis, and use the contour prescription of [36] to make a detailed comparison between the expansion of the integral (4.7) and the c = 1 term of the Rademacher expansion (2.9). In making this comparison, we provide some clarifications about the details of the analysis of [36], which we present in appendix B.…”