Supersymmetry, localization and quantum entropy function

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.

Section: Jhep07(2017)022mentioning

confidence: 99%

Section: Jhep07(2017)022mentioning

confidence: 99%

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

Gomes

2017

“…We refer the reader to [27] for more details about this solution. Here we merely mention that the parameters u, v, R, τ and V are determined completely in terms of the electric and magnetic charges (Q, P ) of the black hole, and u = v. R and τ do not scale with the AdS radius u.…”

Section: Jhep03(2014)043mentioning

confidence: 99%

“…Additionally, it can also be shown [17] that these geometries obey the appropriate fall-off conditions for them to be included in the string path integral in (1.1). We refer the reader to [15,17,27] for more details regarding this orbifold. In this paper, we will perform a further test of this proposal.…”

Section: Jhep03(2014)043mentioning

confidence: 99%

Heat kernels on the AdS2 cone and logarithmic corrections to extremal black hole entropy

Gupta

Lal

Thakur

2014

We develop new techniques to efficiently evaluate heat kernel coefficients for the Laplacian in the short-time expansion on spheres and hyperboloids with conical singularities. We then apply these techniques to explicitly compute the logarithmic contribution to black hole entropy from an N = 4 vector multiplet about a Z N orbifold of the nearhorizon geometry of quarter-BPS black holes in N = 4 supergravity. We find that this vanishes, matching perfectly with the prediction from the microstate counting. We also discuss possible generalisations of our heat kernel results to higher-spin fields over Z N orbifolds of higher-dimensional spheres and hyperboloids.

“…In particular, we can define the exact quantum entropy of a black hole as a functional integral in the gravitational theory with boundary conditions set by the attractor mechanism [1]. The exact computation of such functional integrals has been made possible due to the powerful technique of localization [2] applied to supergravity [3][4][5]. Although some important hurdles remain to be crossed in this program, we can -with some explicitly-stated assumptions -begin to write formulas for the perturbatively exact quantum entropy of a supersymmetric black hole in string theories with 8 supercharges in four dimensions [6].…”

Section: Introductionmentioning

confidence: 99%

Single-centered black hole microstate degeneracies from instantons in supergravity

Murthy

Reys

2016

We obtain holographic constraints on the microscopic degeneracies of black holes by computing the exact macroscopic quantum entropy using localization, including the effects of string worldsheet instantons in the supergravity effective action. For 1 4 -BPS black holes in type II string theory on K3 × T 2 , the constraints can be explicitly checked against expressions for the microscopic BPS counting functions that are known in terms of certain mock modular forms. We find that the effect of including the infinite sum over instantons in the holomorphic prepotential of the supergravity leads to a sum over Bessel functions with successively sub-leading arguments as in the Rademacher expansion of Jacobi forms -but begins to disagree with such a structure near an order where the mock modular nature becomes relevant. This leads to a systematic method to recover the polar terms of the microscopic degeneracies from the degeneracy of instantons (the Gromov-Witten invariants). We check explicitly that our formula agrees with the known microscopic answer for the first seven values of the magnetic charge invariant.