Entropy Function Formalism Research Articles

Attractor horizons in six-dimensional type IIB supergravity

We consider near horizon geometries of extremal black holes in six-dimensional type IIB supergravity. In particular, we use the entropy function formalism to compute the charges and thermodynamic entropy of these solutions. We also comment on the role of attractor mechanism in understanding the entropy of the Hopf T-dual solutions in type IIA supergravity.

Logarithmic corrections to $${\mathcal{N} = 2}$$ black hole entropy: an infrared window into the microstates

Logarithmic corrections to the extremal black hole entropy can be computed purely in terms of the low energy data -- the spectrum of massless fields and their interaction. The demand of reproducing these corrections provides a strong constraint on any microscopic theory of quantum gravity that attempts to explain the black hole entropy. Using quantum entropy function formalism we compute logarithmic corrections to the entropy of half BPS black holes in N=2 supersymmetric string theories. Our results allow us to test various proposals for the measure in the OSV formula, and we find agreement with the measure proposed by Denef and Moore if we assume their result to be valid at weak topological string coupling. Our analysis also gives the logarithmic corrections to the entropy of extremal Reissner-Nordstrom black holes in ordinary Einstein-Maxwell theory.

Near horizon data and physical charges of extremal AdS black holes

We compute the physical charges and discuss the properties of a large class of five-dimensional extremal AdS black holes by using the near horizon data. Our examples include baryonic and electromagnetic black branes, as well as supersymmetric spinning black holes. In the presence of the gauge Chern–Simons term, the five-dimensional physical charges are the Page charges. We carry out the near horizon analysis and compute the four-dimensional charges of the corresponding black holes by using the entropy function formalism and show that they match the Page charges.

Logarithmic corrections to extremal black hole entropy from quantum entropy function

We evaluate the one loop determinant of matter multiplet fields of $ \mathcal{N} = 4 $ supergravity in the near horizon geometry of quarter BPS black holes, and use it to calculate logarithmic corrections to the entropy of these black holes using the quantum entropy function formalism. We show that even though individual fields give non-vanishing logarithmic contribution to the entropy, the net contribution from all the fields in the matter multiplet vanishes. Thus logarithmic corrections to the entropy of quarter BPS black holes, if present, must be independent of the number of matter multiplet fields in the theory. This is consistent with the microscopic results. During our analysis we also determine the complete spectrum of small fluctuations of matter multiplet fields in the near horizon geometry.

Rotating black holes with equal-magnitude angular momenta in d = 5 Einstein-Gauss-Bonnet theory

We construct rotating black hole solutions in Einstein-Gauss-Bonnet theory in five spacetime dimensions. These black holes are asymptotically flat, and possess a regular horizon of spherical topology and two equal-magnitude angular momenta associated with two distinct planes of rotation. The action and global charges of the solutions are obtained by using the quasilocal formalism with boundary counterterms generalized for the case of Einstein-Gauss-Bonnet theory. We discuss the general properties of these black holes and study their dependence on the Gauss-Bonnet coupling constant $\alpha$. We argue that most of the properties of the configurations are not affected by the higher derivative terms. For fixed $\alpha$ the set of black hole solutions terminates at an extremal black hole with a regular horizon, where the Hawking temperature vanishes and the angular momenta attain their extremal values. The domain of existence of regular black hole solutions is studied. The near horizon geometry of the extremal solutions is determined by employing the entropy function formalism.

A twist in the dyon partition function

In four dimensional string theories with N=4 and N=8 supersymmetries one can often define twisted index in a subspace of the moduli space which captures additional information on the partition function than the ones contained in the usual helicity trace index. We compute several such indices in type IIB string theory on K3 x T^2 and T^6, and find that they share many properties with the usual helicity trace index that captures the spectrum of quarter BPS states in N=4 supersymmetric string theories. In particular the partition function is a modular form of a subgroup of Sp(2,Z) and the jumps across the walls of marginal stability are controlled by the residues at the poles of the partition function. However for large charges the logarithm of this index grows as 1/n times the entropy of a black hole carrying the same charges where n is the order of the symmetry generator that is used to define the twisted index. We provide a macroscopic explanation of this phenomenon using quantum entropy function formalism. The leading saddle point corresponding to the attractor geometry fails to contribute to the twisted index, but a Z_n orbifold of the attractor geometry produces the desired contribution.

The entropy function for the extremal Kerr-(anti-)de Sitter black holes

Based on Sen's entropy function formalism, we consider the Bekenstein–Hawking entropy of the extremal Kerr-(anti-)de Sitter black holes in 4-dimensions. Unlike the extremal Kerr black hole case with flat asymptotic geometry, where the Bekenstein–Hawking entropy S is proportional to the angular momentum J, we get a quartic algebraic relation between S and J by using the known solution to the Einstein equation. We recover the same relation in the entropy function formalism. Instead of full geometry, we write down an ansatz for the near horizon geometry only. The exact form of the unknown functions and parameters in the ansatz are obtained by solving the differential equations which extremize the entropy function. The results agree with the nontrivial relation between S and J. We also study the Gauss–Bonnet correction to the entropy exploiting the entropy function formalism. We show that the term, though being topological thus does not affect the solution, contributes a constant addition to the entropy because the term shifts the Hamiltonian by that amount.

Black hole entropy, flat directions and higher derivatives

Higher order derivative corrections to the Einstein-Maxwell action are considered and an explicit form is found for the corrections to the entropy of extremal black holes. We speculate on the properties of these corrections from the point of view of small black holes and in the case when the classical black hole potential exhibits flat directions. A particular attention is paid to the issue of stability of several solutions, including large and small black holes by using properties of the Hessian matrix of the effective black hole potential. This is done by using a model independent expression for such matrix derived within the entropy function formalism.

Contributions of Riemann invariants to the entropy of extremal black holes

We use the entropy function formalism introduced by A. Sen to obtain the entropy of AdS2 × Sd−2 extremal and static black holes in four and five dimensions, with higher derivative terms of a general type. Starting from a generalized Einstein-Maxwell action with nonzero cosmological constant, we examine all possible scalar invariants that can be formed from the complete set of Riemann invariants (up to order 10 in derivatives). The resulting entropies show the deviation from the well known Bekenstein-Hawking area law S = A/4G for Einstein's gravity up to second order derivatives.

Arithmetic of quantum entropy function

Quantum entropy function is a proposal for computing the entropy associated with the horizon of a black hole in the extremal limit, and is related via AdS/CFT correspondence to the dimension of the Hilbert space in a dual quantum mechanics. We show that in N=4 supersymmetric string theories, quantum entropy function formalism naturally explains the origin of the subtle differences between the microscopic degeneracies of quarter BPS dyons carrying different torsion, i.e. different arithmetical properties. These arise from additional saddle points in the path integral -- whose existence depends on the arithmetical properties of the black hole charges -- constructed as freely acting orbifolds of the original AdS_2\times S^2 near horizon geometry. During this analysis we demonstrate that the quantum entropy function is insensitive to the details of the infrared cutoff used in the computation, and the details of the boundary terms added to the action. We also discuss the role of the asymptotic symmetries of AdS_2 in carrying out the path integral in the definition of quantum entropy function. Finally we show that even though quantum entropy function is expected to compute the absolute degeneracy in a given charge and angular momentum sector, it can also be used to compute the index. This can then be compared with the microscopic computation of the index.

A comment on Kerr–CFT and Wald entropy

We point out that the entropies of black holes in general diffeomorphism invariant theories, computed using the Kerr–CFT correspondence and the Wald formula (as implemented in the entropy function formalism), need not always agree. A simple way to illustrate this is to consider Einstein–Gauss–Bonnet gravity in four dimensions, where the Gauss–Bonnet term is topological. This means that the central charge of Kerr–CFT computed in the Barnich–Brandt–Compere formalism remains the same as in Einstein gravity, while the entropy computed using the entropy function gives a universal correction proportional to the Gauss–Bonnet coupling. We argue that at least in this example, the Kerr–CFT result is the physically reasonable one. The resolution to this discrepancy might lie in a better understanding of boundary terms.

Asymptotic expansion of the 𝒩 = 4 dyon degeneracy

We study various aspects of power suppressed as well as exponentially suppressed corrections in the asymptotic expansion of the degeneracy of quarter BPS dyons in N=4 supersymmetric string theories. In particular we explicitly calculate the power suppressed corrections up to second order and the first exponentially suppressed corrections. We also propose a macroscopic origin of the exponentially suppressed corrections using the quantum entropy function formalism. This suggests a universal pattern of exponentially suppressed corrections to all extremal black hole entropies in string theory.

More on entropy function formalism for non-extremal branes

We find R4 correction to the non-extremal D1D5P solution of the supergravity by exactly solving the differential equations of motion and by using the entropy function formalism. In both cases, we find the same correction to the supergravity solution. We then calculate the correction to the entropy using the free energy method and the entropy function formalism. The results are the same.

2D gravity onAdS2with Chern-Simons corrections

We study 2D Maxwell-dilaton gravity with higher order corrections given by the Chern-Simons term. The model admits three distinctive $AdS_2$ vacuum solutions. By making use of the entropy function formalism we find the entropy of the solutions which is corrected due to the presence of the Chern-Simons term. We observe that the form of the correction depends not only on the coefficient of the Chern-Simons term, but also on the sign of the electric charge; pointing toward the chiral nature of the dual CFT. Using the asymptotic symmetry of the theory as well as requiring a consistent picture we can find the central charge and the level of U(1) current. Upon uplifting the solutions to three dimensions we get purely geometric solutions which will be either $AdS_3$ or warped $AdS_3$ with an identification.

Entropy function andAdS2/CFT1correspondence

Wald's formula for black hole entropy, applied to extremal black holes, leads to the entropy function formalism. We manipulate the entropy computed this way to express it as the logarithm of the ground state degeneracy of a dual quantum mechanical system. This provides a natural definition of the extremal black hole entropy in the full quantum theory. Our analysis also clarifies the relationship between the entropy function formalism and the Euclidean action formalism.

(Un)attractor black holes in higher derivative AdS gravity

We investigate five-dimensional static (non-)extremal black hole solutions in higher derivative Anti-de Sitter gravity theories with neutral scalars non-minimally coupled to gauge fields. We explicitly identify the boundary counterterms to regularize the gravitational action and the stress tensor. We illustrate these results by applying the method of holographic renormalization to computing thermodynamical properties in several concrete examples. We also construct numerical extremal black hole solutions and discuss the attractor mechanism by using the entropy function formalism.

Thermodynamics of Born–Infeld black holes

We discuss the horizon structure for Born–Infeld black holes in the context of Einstein–Born–Infeld gravity. We show that the entropy function formalism agrees with a direct calculation of the entropy. With the entropy function formalism we also obtain the entropy when an axion–dilaton system as well as gravitational derivative corrections are included.

Higher derivative correction to Kaluza–Klein black hole solution

We investigate the attractor mechanism in a Kaluza–Klein black hole solution in the presence of higher derivative terms. In particular, we discuss the attractor behavior of static black holes by using the effective potential approach as well as the entropy function formalism. We consider different higher derivative terms with a general coupling to the moduli field. For the R 2 theory, we use an effective potential approach, looking for solutions which are analytic near the horizon and showing that they exist and enjoy attractor behavior. The attractor point is determined by extremization of the modified effective potential at the horizon. We study the effect of the general higher derivative corrections of R n terms. Using the entropy function we define the modified effective potential and we find the conditions to have the attractor solution. In particular for a single charged Kaluza–Klein black hole solution we show that a higher derivative correction dresses the singularity for an appropriate coupling, and we can find the attractor solution.

Entropy function and higher derivative corrections to entropies in (anti-)de Sitter space

We first briefly discuss the relation between black hole thermodynamics and the entropy function formalism. We find that an equation which governs the relationship between Sen's entropy function and black hole entropy, can quickly give higher order corrections to entropy of pure (anti-) de Sitter space without knowing the corrected metric. We also show that near horizon geometry and the entropy function extremization is no longer required for pure (anti-)de Sitter space. The entropy of (anti-)de Sitter space and Schwarzschild-(anti-) de Sitter black holes together with Gauss-Bonnet terms, $R^2$ terms and $R^4$ terms are calculated as concrete examples.

Euclidean methods and the entropy function

AbstractThe attractor mechanism implies that the supersymmetric black hole near horizon solution is defined only in terms of the conserved charges and is therefore independent of asymptotic moduli. Starting only with the near horizon geometry, Sen's entropy functional formalism computes the entropy of an extreme black hole by means of a Legendre transformation where the electric fields are defined as conjugated variables to the electric charges. However, traditional Euclidean methods require the knowledge of the full geometry to compute the black hole thermodynamic quantities. We establish the connection between the entropy functional formalism and the standard Euclidean formalism taken at zero temperature. We find that Sen's entropy function f (on‐shell) matches the zero temperature limit of the Euclidean action. Moreover, Sen's near horizon angular and electric fields agree with the chemical potentials that are defined from the zero‐temperature limit of the Euclidean formalism.