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Microscopic and Macroscopic Entropy of Extremal Black Holes in String Theory

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Abstract

This is a short review summarizing the current status of the comparison between microscopic and macroscopic entropy of extremal BPS black holes in string theory.

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... Indeed, for extremal black holes, not only the microscopic explanation of the entropy S is absent, but even the value S itself of the entropy is uncertain. Although some suggestions have been worked out that yield S = 0 [16][17][18], the entropy of an extremal black hole is still an open problem, as string theory claims that it is in fact given by the Bekenstein-Hawking entropy S = A + /4 [19,20], see also [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] on this discussion. ...
... Case 2. For R → r + and r + → r − , i.e., for δ = ε λ , with λ kept fixed according to Eq. (20), and ε → 0 we get from Eqs. (23)- (22) M (r + , ε, δ) = r + , Q(r + , ε, δ) = r + . ...
... For R → r + and r + → r − , i.e., for δ = ε λ , with λ kept fixed according to Eq. (20), and ε → 0 we get from Eq. (28) Put back intermediate step ...
Preprint
Using a unified approach we study the entropy of extremal black holes through the entropy of an electrically charged thin shell. We encounter three cases in which a shell can be taken to its own gravitational or horizon radius and become an extremal spacetime. In case 1, we use a non extremal shell, calculate all the thermodynamics quantities including the entropy, take it to the horizon radius, and then take the extremal limit. In case 2, we take the extremal limit and the horizon radius limit simultaneously, i.e., as the shell approaches its horizon radius it also approaches extremality. In case 3, we build first an extremal shell, and then take its horizon radius. We find that the thermodynamic quantities in general have different expressions in the three different cases. The entropy is the Bekenstein-Hawking entropy S=A+/4 S=A_+/4 (where A+A_+ is the horizon area) in cases 1 and 2, and in case 3 it can be any well-behaved function of A+A_+. The contributions from the various thermodynamic quantities for the entropy in all three cases are distinct. Indeed, in cases 1 and 2 the limits agree in what concerns the entropy but they disagree in the behavior of all other thermodynamic quantities. Cases 2 and 3 disagree in what concerns the entropy but agree in the behavior of the local temperature and electric potential. Case 2 is in a sense intermediate between cases 1 and 3. Our approach sheds light on the extremal black hole entropy issue.
... In the absence of a microscopic explanation, a bottom-up approach would to be study the quantum corrections to (2) in the low energy effective theory of gravity plus matter. These corrections provide unambiguous data testing candidate models, in similar spirit of that in the context of black hole microstate counting [2][3][4][5][6] and Higher-spin/CFT dualities [7][8][9][10]. As an illustration, let us say one asserts that the dS horizon entropy S macro for 3D pure gravity (computed in [11]) counts the number ( ) of partitions of . ...
... Note that either (6) or (32) are UV-divergent and need to be regulated. The equality (33) means that the schemeindependent part (e.g. the coefficient of the logarithmic divergence) of both sides agree. ...
... Uniquely fixing a reference system requires an extra physical input. In the current case, such an input is provided by demanding (47) to equal the Euclidean path integral (6). As it turns out, such a As emphasized in [15], this choice of boundary condition turns out to be irrelevant. ...
Preprint
In this short note, we review some recent progress in understanding the 1-loop corrections to the Gibbons-Hawking entropy, which amounts to studying free fields on the de Sitter static patch and the round sphere. After briefly surveying the unitary irreducible representations of the de Sitter group SO(1,d+1) and their Harish-Chandra characters, we discuss the Lorentzian interpretation for the 1-loop sphere path integral for a scalar. After that we comment on how the results are modified by edge contributions for spinning fields.
... -2 -entropy. However, there are often huge technical challenges to overcome in evaluating them [7,8]. In this paper, we will explain how to compute logarithmic corrections for all rotating (and non-rotating) as well as charged (and uncharged) black holes in the low-energy model of Einstein-Maxwell-dilaton theory by structuring a common and efficient setup. ...
... Similarly, it is possible to embed the Schwarzschild-AdS, Reissner-Nordström-AdS and Kerr-AdS black holes in a consistent EMD theory with a negative cosmological constant (abbreviated to EMD-AdS theory) and their intersecting sectors in gauged supergravity [53,56,58,59]. These embedding choices of black hole backgrounds effectively intensify the prospect of microscopic consistency of calculated quantum correction results inside string theory [3][4][5][6][7][8][9][10][11][12][13]. To date, pioneered by Ashoke Sen and collaborators and then followed by many other groups, the logarithmic corrections are mostly reported for the full Kerr-Newman family of black holes in EM theory [25][26][27]38] and all N ≥ 1 ungauged supergravity [22-24, 28, 30-32, 34-37, 39]. ...
... Inside the framework of Einstein's general relativity, the entropy of a black hole is described by the seminal Bekenstein-Hawking area law (BHAL) [1,2], which is equal to one-quarter of the area of the event horizon. String theory has already gained remarkable success by providing a counting of microstates underlying the entropy of various classes of flat and AdS black holes to establish the BHAL (e.g., see [3][4][5][6][7][8][9][10][11][12][13]). For a non-trivial consistency check, there has been trendy progress on the macroscopic front (the low energy or IR limit), i.e., in the description of Einstein gravity by incorporating possible quantum gravitational correction to BHAL describing black hole entropy (semi-)classically and approximately at tree level. ...
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We calculate the logarithmic correction to the entropy of asymptotically flat and AdS black holes (rotating, non-rotating, charged, and uncharged) embedded in Einstein-Maxwell-dilaton (EMD) theories with U(1)-charged. The leading quantum gravitational corrections are achieved in both extremal and non-extremal limits of black hole temperature by designing a common Euclidean gravity setup that evaluates the "logarithmic term'' from one-loop effective actions via heat kernel method-based calculations. EMD theories are universal building blocks of compactified string theory or supergravity models in 4D. For a concrete example, we generalize the entire setup and calculate logarithmic corrections for black holes in U(1)2U(1)^2-charged EMD models intersecting with N=4\mathcal{N}=4 ungauged and gauged bosonic supergravity. In contrast to flat backgrounds, all the AdS4_4 results are found to be \textit{non-topological}, providing a wider "infrared window'' into the microscopic degrees of freedom of black holes in string theory.
... Euclidean gravity methods [1] have been incredibly successful as an IR window into black hole microstates, even beyond the leading order in G N . For example, as demonstrated in [2][3][4][5][6], 1-loop Euclidean path integrals compute logarithmic corrections to black hole entropy that are in perfect agreement with the microscopic results in string theory or holographic CFT. At 1-loop, the path integral receives corrections from quadratic fluctuations of matter fields and the graviton around the black hole, and reduces to functional determinants of differential operators. ...
... This Schrödinger equation is same as that of the spacelike Liouville quantum mechanics. 6 Notice that the information about the black hole geometry (except for its temperature T H ) and the scalar (its mass and angular momentum) becomes completely invisible. A near-horizon observer studying (3.21) would not be able to distinguish the black hole spacetime (2.1) and the Rindler-like wedge (3.17) (see figure 1); they would obtain an S-matrix (see appendix A for details) ...
... In figure 4, with an example of a scalar on static BTZ, we show a comparison of ∆ρ l (ω) on the complex ω-plane for two different choices of reference. 6 We thank Daniel Kapec for pointing this out. 7 We resolve the t −2 pole in the factors multiplying χ QNM (t) by t −2 → 1 While we have focused on the case of scalars, it is straightforward to generalize our arguments to a Dirac spinor. ...
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A bstract When computing the ideal gas thermal canonical partition function for a scalar outside a black hole horizon, one encounters the divergent single-particle density of states (DOS) due to the continuous nature of the normal mode spectrum. Recasting the Lorentzian field equation into an effective 1D scattering problem, we argue that the scattering phases encode non-trivial information about the DOS and can be extracted by “renormalizing” the DOS with respect to a reference. This defines a renormalized free energy up to an arbitrary additive constant. Interestingly, we discover that the 1-loop Euclidean path integral, as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free energy to be that on a Rindler-like region, and the renormalized DOS captures the quasinormal modes for the scalar. We support these claims with the examples of scalars on static BTZ, Nariai black holes and the de Sitter static patch. For black holes in asymptotically flat space, the renormalized DOS is captured by the phase of the transmission coefficient whose magnitude squared is the greybody factor. We comment on possible connections with recent works from an algebraic point of view.
... Euclidean gravity methods [1] have been incredibly successful as an IR window into black hole microstates, even beyond the leading order in G N . For example, as demonstrated in [2,3,4,5,6], 1-loop Euclidean path integrals compute logarithmic corrections to black hole entropy that are in perfect agreement with the microscopic results in string theory or holographic CFT. At 1-loop, the path integral receives corrections from quadratic fluctuations of matter fields and the graviton around the black hole, and reduces to functional determinants of differential operators. ...
... While in principle there could be a holomorphic part contributing to ∆ρ(ω), in all the explicit examples we have checked, ∆ρ(ω) does not receive such a contribution and ∆ρ QNM (ω) gives the complete answer. In such case, substituting (3.23) into (3.16) and doing the ω-integral, 6 we have ...
... While we have focused on the case of scalars, it is straightforward to generalize our arguments 6 We resolve the t −2 pole in the factors multiplying χQNM(t) by t −2 → 1 ...
Preprint
When computing the ideal gas thermal canonical partition function for a scalar outside a black hole horizon, one encounters the divergent single-particle density of states (DOS) due to the continuous nature of the normal mode spectrum. Recasting the Lorentzian field equation into an effective 1D scattering problem, we argue that the scattering phases encode non-trivial information about the DOS and can be extracted by "renormalizing" the DOS with respect to a reference. This defines a renormalized free energy up to an arbitrary additive constant. Interestingly, the 1-loop Euclidean path integral, as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free energy to be that on a Rindler space, and the renormalized DOS captures the quasinormal modes for the scalar. We support these claims with the examples of scalars on static BTZ, Nariai black holes and the de Sitter static patch. For black holes in asymptotically flat space, the renormalized DOS is captured by the phase of the transmission coefficient whose magnitude squared is the greybody factor. We comment on possible connections with recent works from an algebraic point of view.
... These counting formulas, also often known as the black hole partition functions, give an integer for the indices corresponding to the BPS microstates underlying the given supersymmetric black hole. On the gravity side, Sen's quantum entropy function formalism posits that a path integral computation in string theory on AdS 2 ×K near-horizon geometry should give the black hole indices in the full quantum theory [7][8][9], where K is a compact manifold. Since this computation only refers to the near-horizon geometry, the answer is expected not to be sensitive to the nature of the solution far away from the horizon. ...
... Having obtained the hair modes as solutions to non-linear supergravity equations for both 4d and 5d black holes, we now turn to the discussion of hair removed partition functions. The hair removed 4d and 5d partition functions themselves are interesting quantities, as they are expected to be obtainable on the gravity side from the quantum entropy function formalism [7][8][9]. In section 7.1, we review the microscopic considerations relevant for our discussion. ...
... The horizon partition functions perfectly match. 9 This is not a coincidence. Using information from table 4 and appropriate periodicity of the modes we see that the hair removed partition functions match in all cases. ...
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A bstract Although BMPV black holes in flat space and in Taub-NUT space have identical near-horizon geometries, they have different indices from the microscopic analysis. For K3 compactification of type IIB theory, Sen et al. in a series of papers identified that the key to resolving this puzzle is the black hole hair modes: smooth, normalisable, bosonic and fermionic degrees of freedom living outside the horizon. In this paper, we extend their study to N = 4 CHL orbifold models. For these models, the puzzle is more challenging due to the presence of the twisted sectors. We identify hair modes in the untwisted as well as twisted sectors. We show that after removing the contributions of the hair modes from the microscopic partition functions, the 4d and 5d horizon partition functions agree. Special care is taken to present details on the smoothness analysis of hair modes for rotating black holes, thereby filling an essential gap in the literature.
... Based on the AdS/CFT correspondence the proposed quantum entropy function of [19,20] captures both kinds of corrections and accounts for exponentially suppressed contributions 2 as demanded by the microscopic index formula [9,14,25]. See [10,26,27] for reviews and [28-33] for more recent studies of the (quarter-BPS) quantum entropy that rely on localization of the supergravity path integral. ...
... It has been argued [56] that S-and T-transformations (i.e., those inherited from the parent theory compatible with the orbifolding procedure) do not mix dyon charges with twisted and untwisted sector charges in the winding number sense. 10 However, the definition of twisted curve classes in DT theory a priori only concerns the E 8 (− 1 2 )-part of the (co-)homology lattice of (K3 × T 2 )/Z 2 , which is a sublattice of the electric charge lattice Λ e , while the twisted and untwisted charge sectors in physics (independently of the 10 Physically this distinction does not apply for the unorbifolded case, for which the electric and magnetic charge lattices are isomorphic and the U-duality group acts transitively on the unit-torsion dyon charges. Due to duality invariance of the BPS index we hence expect only one quarter-BPS partition function. ...
... It has been argued [56] that S-and T-transformations (i.e., those inherited from the parent theory compatible with the orbifolding procedure) do not mix dyon charges with twisted and untwisted sector charges in the winding number sense. 10 However, the definition of twisted curve classes in DT theory a priori only concerns the E 8 (− 1 2 )-part of the (co-)homology lattice of (K3 × T 2 )/Z 2 , which is a sublattice of the electric charge lattice Λ e , while the twisted and untwisted charge sectors in physics (independently of the 10 Physically this distinction does not apply for the unorbifolded case, for which the electric and magnetic charge lattices are isomorphic and the U-duality group acts transitively on the unit-torsion dyon charges. Due to duality invariance of the BPS index we hence expect only one quarter-BPS partition function. ...
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A bstract Motivated by recent advances in Donaldson-Thomas theory, four-dimensional N \mathcal{N} N = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ 2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B (2) ⊂ Sp 4 (ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T ² ) / ℤ 2 , as suggested by string duality with type IIA theory on X .
... (1). The other approach, based on string theory calculations, yields that the entropy of extremal black holes is given by the Bekenstein-Hawking entropy, Eq. (2) [27,28], see also [29][30][31][32][33][34][35][36][37][38]. On the other hand, for (3+1)-dimensional nonextremal black holes, the entropy is the original unambiguous Bekenstein-Hawking entropy, S = A+ 4G of Eq. (2) [39][40][41]. ...
... This reasoning is purely classical, and inclusion of the backreaction due to quantum fields can destroy this picture. On the other hand, the proposal put forward by string theory leads to S = A+ 4G , Eq. (2), i.e., to a Bekenstein-Hawking entropy for extremal black hole [27,28], see [2] for the BTZ black hole. Our conclusion that 0 ≤ S(A + ) ≤ A+ 4G , see Eq. (54), incorporates both the S = 0 and the S = A+ 4G results. ...
Preprint
In a (2+1)-dimensional spacetime with a negative cosmological constant, the thermodynamics and the entropy of an extremal rotating thin shell, i.e., an extremal rotating ring, are investigated. The outer and inner regions are taken to be the Ba\~{n}ados-Teitelbom-Zanelli (BTZ) spacetime and the vacuum ground state anti-de Sitter (AdS) spacetime, respectively. By applying the first law of thermodynamics to the extremal shell one shows that its entropy is an arbitrary function of the gravitational area A+A_+ alone, S=S(A+)S=S(A_+). When the shell approaches its own gravitational radius r+r_+ and turns into an extremal rotating BTZ black hole, it is found that the entropy of the spacetime remains such a function of A+A_+. It is thus vindicated, that extremal black holes, here extremal BTZ black holes, have different properties from the corresponding nonextremal black holes, which have the Bekenstein-Hawking entropy S(A+)=A+4GS(A_+)= \frac{A_+}{4G}, where G is the gravitational constant. It is argued that for the extremal case 0S(A+)A+4G0\leq S(A_+)\leq \frac{A_+}{4G}. Thus, rather than having just two entropies for extremal black holes, as previous results debated, 0 and A+4G\frac{A_+}{4G}, it is shown that extremal black holes may have a continuous range of entropies, limited by precisely those two entropies. Surely, the entropy that a particular extremal black hole picks must depend on past processes, notably on how it was formed. It is also found a remarkable relation between the third law of thermodynamics and the impossibility for a massive body to reach the velocity of light. In the procedure, it becomes clear that there are two distinct angular velocities for the shell, the mechanical and thermodynamic angular velocities. In passing, we clarify, for a static spacetime with a thermal shell, the meaning of the Tolman temperature formula at a generic radius and at the shell. (Abridged version).
... Logarithmic corrections to black hole entropy are very powerful: they are governed by low energy data that probe non-trivially any theory of quantum gravity that attempts to account for the black hole microstates. As such, they are a successful and robust test in several situations [9,10,11,12]. For CHL models both d(Q) and its 4D(5D) supersymmetric black hole counterpart are known explicitly, and the agreement is remarkable (see Appendix C for a quick review of this class of black holes). ...
... while the remaining n V vectormultiplet gauge fields, that couple to the D2 charges, have field strength F a = −φ a dr ∧ dt + p a sin(θ)dθ ∧ dφ , (C. 12) with p a the magnetic charges. The scalar fields, on the other hand, are determined in terms of the electric fields and the magnetic charges, that is, LX 0 = φ 0 , LX a = φ a + ip a , (C. 13) with p 0 = 0. ...
Preprint
We discuss the application of Siegel Modular Forms to Black Hole entropy counting. The role of the Igusa cusp form χ10\chi_{10} in the D1D5P system is well-known, and its transformation properties are what allows precision microstate counting in this case. We apply a similar method to extract the Fourier coefficients of other Siegel modular and paramodular forms, and we show that they could serve as candidates for other types of black holes. We investigate the growth of their coefficients, identifying the dominant contributions and the leading logarithmic corrections in various regimes. We also discuss similarities and differences to the behavior of χ10\chi_{10}, and possible physical interpretations of such forms both from a microscopic and gravitational point of view.
... Automorphic forms associated to the symplectic group have played an important role in the understanding of the microscopic structure of the entropy of certain types of supersymmetric black holes (refs. [30,31] contain reviews). Such forms can be obtained by lifting classical modular forms to Siegel forms and for the models considered so far the origin can in fact be traced to elliptic modular forms [32]. ...
... where Y > 0 means that Y leads to a positve definite quadratic form. The symplectic group, with elements M written as in (30), acts on Z ∈ H n as ...
Preprint
Automorphic inflation is an application of the framework of automorphic scalar field theory, based on the theory of automorphic forms and representations. In this paper the general framework of automorphic and modular inflation is described in some detail, with emphasis on the resulting stratification of the space of scalar field theories in terms of the group theoretic data associated to the shift symmetry, as well as the automorphic data that specifies the potential. The class of theories based on Eisenstein series provides a natural generalization of the model of j-inflation considered previously.
... Of course if the classical picture were to be correct, the four laws of black hole mechanics are not true thermodynamical equations but just a mere analogy without much deep implication. However, there is mounting evidence from various considerations (such as the microscopic degree of freedom counting for extremal black holes in string theory in terms of the D-brane charges [8,9] and in 2+1 dimensional gravity in terms of the asymptotic symmetries etc.) that this is not the case and black holes do have temperature and entropy with perhaps important implications for quantum gravity. ...
... For higher dimensions, the simple form (9) does not lead to a unitary theory, therefore one needs to add at least quadratic terms in curvature inside the determinant. After a rather tedious computation laid out in detail in [10,11] the following theory was found: ...
Preprint
There is a class of higher derivative gravity theories that are in some sense natural extensions of cosmological Einstein's gravity with a unique maximally symmetric classical vacuum and only a massless spin-2 excitation about the vacuum and no other perturbative modes. These theories are of the Born-Infeld determinantal form. We show that the macroscopic dynamical entropy as defined by Wald for bifurcate Killing horizons in these theories are equivalent to the geometric Bekenstein-Hawking entropy (or more properly Gibbons-Hawking entropy for the case of de Sitter spacetime) but given with an effective gravitational constant which encodes all the information about the background spacetime and the underlying theory. We also show that the higher curvature terms increase the entropy. We carry out the computations in generic n-dimensions including the particularly interesting limits of three, four and infinite number of dimensions. We also give a preliminary discussion about the black hole entropy in generic dimensions for the BI theories.
... The claim that logarithmic corrections computed from the IR theory agree with results for the UV completion has been successfully tested in many cases where string theory provides a microscopic counting formula for black hole microstates. We refer to [10,11] for a broad overview and [12][13][14][15] for more recent developments in AdS 4 /CFT 3 . Logarithmic corrections have also been evaluated for a plethora of other black holes [16,17] where a microscopic account still awaits. ...
... To date, there is no known microstate counting formula that, when compared to the black hole entropy, accounts for terms that involve c = 0. For example, in all cases considered in [10,11,39], the object of interest is an index, or a closely related avatar, and the resulting logarithmic terms nicely accommodate quantum corrections when C local is controlled by a alone. The challenge of reproducing the logarithmic correction when c is non-vanishing comes from the intricate dependence on the black hole parameters that the Weyl tensor gives to C local . ...
Preprint
We compute the leading logarithmic correction to black hole entropy on the non-BPS branch of 4D N2{\cal N}\geq 2 supergravity theories. This branch corresponds to finite temperature black holes whose extremal limit does not preserve supersymmetry, such as the D0D6D0-D6 system in string theory. Starting from a black hole in minimal Kaluza-Klein theory, we discuss in detail its embedding into N=8,6,4,2{\cal N}=8, 6, 4, 2 supergravity, its spectrum of quadratic fluctuations in all these environments, and the resulting quantum corrections. We find that the c-anomaly vanishes only when N6{\cal N}\geq 6, in contrast to the BPS branch where c vanishes for all N2{\cal N}\geq 2. We briefly discuss potential repercussions this feature could have in a microscopic description of these black holes.
... Unfortunately it is precisely the case when d = 1 that (1.2) is most subtle. A strict zero temperature version of (1.2) was used by Sen and collaborators to calculate logarithmic corrections to extremal black hole entropy [13][14][15][16][17][18][19]. Zero temperature partition functions count ground states, so these authors really calculated a particular correction to the quantity Z throat (β = ∞, Q, J) = Tr groundstates ∼ S c log 0 e S0 . ...
... The spectral density calculations highlight another reason that the DHS approach seems likely to make contact with the near-horizon analyses. Although the factor of T 3/2 arises from discrete zero modes in AdS 2 /NHEK, most of the spectrum in these geometries is actually continuous (and contributes known logarithmic corrections to the extremal entropy [19]). Indeed, it is somewhat atypical to have so many discrete eigenvalues on a noncompact infinite-volume geometry. ...
Preprint
Recent work on the quantum mechanics of near-extremal non-supersymmetric black holes has identified a characteristic T3/2T^{3/2} scaling of the low temperature black hole partition function. This result has only been derived using the path integral in the near-horizon region and relies on many assumptions. We discuss how to derive the T3/2T^{3/2} scaling for the near-extremal rotating BTZ black hole from a calculation in the full black hole background using the Denef-Hartnoll-Sachdev (DHS) formula, which expresses the 1-loop determinant of a thermal geometry in terms of a product over the quasinormal mode spectrum. We also derive the spectral measure for fields of any spin in Euclidean BTZ and use it to provide a new proof of the DHS formula and a new, direct derivation of the BTZ heat kernel. The computations suggest a path to proving the T3/2T^{3/2} scaling for the asymptotically flat 4d Kerr black hole.
... It is natural to expect topological logarithmic corrections in the UV given the known examples of microscopic counting of black hole entropy [11,[29][30][31][32][77][78][79]. This is also automatic if the 4d theory comes from an odd-dimensional theory by Kaluza-Klein reduction because C local = 0 in odd dimensions. ...
... We might hope to use the topological nature of logarithmic corrections as a criterion for a low-energy theory to admit a UV completion. In the available examples of microscopic counting, the logarithmic correction is indeed topological [11,[29][30][31][32][77][78][79]. Such a criteria would greatly constrain effective supergravity theories as it gives rather stringent conditions similar to anomaly cancellation. ...
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A bstract We compute the logarithmic correction to the entropy of asymptotically AdS 4 black holes in minimal N \mathcal{N} N = 2 gauged supergravity. We show that for extremal black holes the logarithmic correction computed in the near horizon geometry agrees with the result in the full geometry up to zero mode contributions, thus clarifying where the quantum degrees of freedom lie in AdS spacetimes. In contrast to flat space, we observe that the logarithmic correction for supersymmetric black holes can be non-topological in AdS as it is controlled by additional four-derivative terms other than the Euler density. The available microscopic data and results in 11d supergravity indicate that the full logarithmic correction is topological, which suggests that the topological nature of logarithmic corrections could serve as a diagnosis of whether a low-energy gravity theory admits an ultraviolet completion.
... As a leading candidate for a theory of quantum gravity, string theory has already provided, in the work of Strominger and Vafa, a microstate counting of the entropy for certain asymptotically flat black holes [6]. It has also provided, through work by Sen and collaborators, an explanation for the logarithmic corrections to the entropy [7]. ...
... Further discussion about the entropy of black hole in different ensembles can be found in[7,31]. ...
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A bstract We investigate logarithmic corrections to the entropy of supersymmetric, rotating, asymptotically AdS 5 black holes and black strings. Within the framework of the AdS/CFT correspondence, the entropy of these black objects is determined, on the field theory side, by the superconformal index and the refined topologically twisted index of N \mathcal{N} N = 4 supersymmetric Yang-Mills theory, respectively. We read off the logarithmic correction from those field-theoretic partition functions. On the gravity side, we take the near-horizon limit and apply the Kerr/CFT correspondence whose associated charged Cardy formula describes the degeneracy of states at subleading order and determines the logarithmic correction to the entropy. We find perfect agreement between these two approaches. Our results provide a window into precision microstate counting and demonstrate the efficacy of low-energy, symmetry-based approaches such as the Kerr/CFT correspondence for asymptotically AdS black objects under certain conditions.
... These various features were known for a long time, but they were greatly developed and refined about a decade ago, especially by A. Sen [16,[39][40][41]. A highlight of this advance was the case of supersymmetric black holes in string theory where the coefficients from loops of massless particles computed in effective field theory were shown to agree precisely with the corresponding expansion of microscopic counting formulae [42,43]. This result gives confidence that the various contributions to the logarithmic corrections have been correctly understood. ...
... e The dependence of the zero-mode contribution on the theory and the ensemble is far from trivial. It was discussed by Sen in many interesting situations [17,[43][44][45]. The results were summarized in an Appendix of [41]. ...
Preprint
We review and extend recent progress on the quantum description of near-extremal black holes in the language of effective quantum field theory. With black holes in Einstein-Maxwell theory as the main example, we derive the Schwarzian low energy description of the AdS2_2 region from a spacetime point of view. We also give a concise formula for the symmetry breaking scale, we relate rotation to supersymmetry, and we discuss quantum corrections to black hole entropy in detail.
... Supergravity theories are particular class of vacuums that can be realized as the lowenergy truncation of string theories compactified down to four space-time dimensions. Logarithmic entropy corrections have been extensively investigated in supergravity theories, which already have a well-established microscopic counterpart within string theory [60,61]. But, most of the popularly known examples [4-10] of logarithmic entropy corrections in supergravities are for extremal black holes with non-rotating geometry. ...
... Moreover, the complex nature of the calculated logarithmic correction results will enhance the technical difficulties a lot more in their microscopic reproduction. The calculated macroscopic logarithmic correction to the Bekenstein-Hawking entropy of extremal Reissner-Nordström black holes in N = 4 and N = 8 EMSGTs (originally reported in [5]) already have an exact microscopic agreement in string theory (see [60,61] and references therein). No such concrete microscopic answers (particularly in string theory) for any other calculated non-extremal and remaining extremal corrections are reported to date. ...
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A bstract We calculate the first three Seeley-DeWitt coefficients for fluctuation of the massless fields of a N \mathcal{N} N = 2 Einstein-Maxwell supergravity theory (EMSGT) distributed into different multiplets in d = 4 space-time dimensions. By utilizing the Seeley-DeWitt data in the quantum entropy function formalism, we then obtain the logarithmic correction contribution of individual multiplets to the entropy of extremal Kerr-Newman family of black holes. Our results allow us to find the logarithmic entropy corrections for the extremal black holes in a fully matter coupled N \mathcal{N} N = 2 , d = 4 EMSGT, in a particular class of N \mathcal{N} N = 1, d = 4 EMSGT as consistent decomposition of N \mathcal{N} N = 2 multiplets ( N \mathcal{N} N = 2 → N \mathcal{N} N = 1) and in N \mathcal{N} N ≥ 3 , d = 4 EMSGTs by decomposing them into N \mathcal{N} N = 2 multiplets ( N \mathcal{N} N ≥ 3 → N \mathcal{N} N = 2). For completeness, we also obtain logarithmic entropy correction results for the non-extremal Kerr-Newman black holes in the matter coupled N \mathcal{N} N ≥ 1 , d = 4 EMSGTs by employing the same Seeley-DeWitt data into a different Euclidean gravity approach developed in [17].
... The near-horizon geometry of extremal black holes universally contain an AdS 2 throat. This feature plays a crucial role in the microscopic understanding of black hole entropy in string theory [1][2][3]. In the recent years, there has been an upheaval of interest in the AdS 2 /CFT 1 correspondence that has revealed the chaotic nature of black holes [4,5] and the unitarity of black hole evaporation [6,7]. ...
... where, K is a constant depending only on N . 3 We have used the S-modular transformation of the Dedekind-eta function η(−1/τ ) = √ −iτ η(τ ). Combining the above two equations, the final result for the left moving character (at low temperatures) is ...
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A bstract Two-dimensional conformal field theories with Virasoro symmetry generically contain a Schwarzian sector. This sector is related to the near-horizon region of the near-extremal BTZ black hole in the holographic dual. In this work we generalize this picture to CFTs with higher spin conserved currents. It is shown that the partition function in the near-extremal limit agrees with that of BF higher spin gravity in AdS 2 which is described by a generalized Schwarzian theory. We also provide a spectral decomposition of Schwarzian partition functions via the WN {\mathcal{W}}_N W N fusion kernel and consider supersymmetric generalizations.
... The near-horizon geometry of extremal black holes universally contain an AdS 2 throat. This feature plays a crucial role in the microscopic understanding of black hole entropy in string theory [1][2][3]. In the recent years, there has been an upheaval of interest in the AdS 2 /CFT 1 correspondence that has revealed the chaotic nature of black holes [4,5] and the unitarity of black hole evaporation [6,7]. ...
... where, K is a constant depending only on N . 3 We have used the S-modular transformation of the Dedekind-eta function η(−1/τ ) = √ −iτ η(τ ). Combining the above two equations, the final result for the left moving character (at low temperatures) is ...
Preprint
Two-dimensional conformal field theories with Virasoro symmetry generically contain a Schwarzian sector. This sector is related to the near-horizon region of the near-extremal BTZ black hole in the holographic dual. In this work we generalize this picture to CFTs with higher spin conserved currents. It is shown that the partition function in the near-extremal limit agrees with that of BF higher spin gravity in AdS2_2 which is described by a generalized Schwarzian theory. We also provide a spectral decomposition of Schwarzian partition functions via the WN\mathcal{W}_N fusion kernel and consider supersymmetric generalizations.
... give an integer for the indices corresponding to the BPS microstates underlying the given supersymmetric black hole. On the gravity side, Sen's quantum entropy function formalism posits that a path integral computation in string theory on AdS 2 × K near-horizon geometry should give the black hole indices in the full quantum theory [7][8][9], where K is a compact manifold. Since this computation only refers to the near-horizon geometry, the answer is expected not to be sensitive to the nature of the solution far away from the horizon. ...
... Having obtained the hair modes as solutions to non-linear supergravity equations for both 4d and 5d black holes, we now turn to the discussion of hair removed partition functions. The hair removed 4d and 5d partition functions themselves are interesting quantities, as they are expected to be obtainable on the gravity side from the quantum entropy function formalism [7][8][9]. In section 7.1, we review the microscopic considerations relevant for our discussion. ...
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Although BMPV black holes in flat space and in Taub-NUT space have identical near-horizon geometries, they have different indices from the microscopic analysis. For K3 compactification of type IIB theory, Sen et al in a series of papers identified that the key to resolving this puzzle is the black hole hair modes: smooth, normalisable, bosonic and fermionic degrees of freedom living outside the horizon. In this paper, we extend their study to N = 4 CHL orbifold models. For these models, the puzzle is more challenging due to the presence of the twisted sectors. We identify hair modes in the untwisted as well as twisted sectors. We show that after removing the contributions of the hair modes from the microscopic partition functions, the 4d and 5d horizon partition functions agree. Special care is taken to present details on the smoothness analysis of hair modes for rotating black holes, thereby filling an essential gap in the literature.
... A prototypical case is that of the Strominger-Vafa [1] black hole which admits an AdS 3 × S 3 decoupling limit and a dual description in terms of a 2-dimensional SCFT [2] often dubbed D1D5 CFT. The key breakthrough obtained in this approach is a precise account of the Bekenstein-Hawking entropy formula and its generalizations in terms of a microscopic counting for several BPS configurations, see [3] for a recent review. It is very interesting to go beyond the counting problem and ask if the detailed understanding of the microstates of supersymmetric black holes can be used to shed any light on the conceptual puzzles that arise when formulating quantum mechanics in a black hole background. ...
Preprint
We compute correlators of two heavy and two light operators in the strong coupling and large c limit of the D1D5 CFT which is dual to weakly coupled AdS3_3 gravity. The light operators have dimension two and are scalar descendants of the chiral primaries considered in arXiv:1705.09250, while the heavy operators belong to an ensemble of Ramond-Ramond ground states. We derive a general expression for these correlators when the heavy states in the ensemble are close to the maximally spinning ground state. For a particular family of heavy states we also provide a result valid for any value of the spin. In all cases we find that the correlators depend non-trivially on the CFT moduli and are not determined by the symmetries of the theory, however they have the properties expected for correlators among pure states in a unitary theory, in particular they do not decay at large Lorentzian times.
... It has been argued in [11], using the su(1, 1|1) superconformal algebra of the AdS 2 × Σ g near-horizon region to the black hole, that the states associated to the pure single-center BPS black holes 12 have (−1) F = 1, and thus they are precisely counted by the index. This argument is essentially the same as the one given in [22] (and nicely summarized e.g. in [52]) for BPS black holes in flat space. This is not true for the states coming from multi-center black holes and hair, whose number however we might expect to be subleading. ...
Preprint
We study the entropy of static dyonic BPS black holes in AdS4_4 in 4d N=2\mathcal{N}=2 gauged supergravities with vector and hyper multiplets, and how the entropy can be reproduced with a microscopic counting of states in the AdS/CFT dual field theory. We focus on the particular example of BPS black holes in AdS4×S6_4 \times S^6 in massive Type IIA, whose dual three-dimensional boundary description is known and simple. To count the states in field theory we employ a supersymmetric topologically twisted index, which can be computed exactly with localization techniques. We find perfect match at leading order.
... whereᾱ γ = κα γ and g(n) is effective number of micro states giving rise to the macroscopic black hole configuration, as we consider the black hole as a system in a micro-canonical set up. Many quantum gravity theories attempt to visualize this number of available microstates through their counting schemes and have been successful in doing so [12,59,60]. Therefore, recovery of this relation from microscopic counting does not really serve as a very strong discriminator between the available quantum gravity models. ...
Preprint
We analyze the emission spectrum of a (fundamentally quantum) black hole in the Kerr-Newman family by assuming a discretization of black hole geometry and the holographic entropy-area relation. We demonstrate that, given the above structure of black hole entropy, a macroscopic black hole always has non-continuously separated mass states and therefore they descend down in discrete manner. We evaluate the step size of the discrete spectrum, which vanishes in the extremal limit, leading to a continuum spectrum as expected from thermal nature of black holes. This further reveals an interesting relation, in each class, between the dynamic and kinematic length scales for all black holes belonging to the Kerr-Newman family, pointing towards a possible universal character across the class, dependent only on black hole mass. Further, we have presented the computation of maximum number of emitted quanta from the black hole as well as an estimation of its lifetime. We also argue the independence of these features from the presence of additional spacetime dimensions.
... is indeed found through a Euclidean path integral approach to extremal black hole entropy, both in BTZ black holes [17] and in Reissner-Nordström black holes [18], whereas in contradiction, the Bekenstein-Hawking upper limit of Eq. (3), S = A+ 4G , see also Eq. (1), is found through string theory techniques in extremal cases, namely, in (2+1)-dimensional extremal rotating BTZ black holes [19], and in (3+1)-dimensional extremal Reissner-Nordström black holes [20], following the breakthrough worked out in (4+1) dimensions [21,22], see also [23][24][25][26][27][28][29][30][31][32] for further studies on thermodynamics and entropy of extremal black holes. In a sense, Eq. (3) fills the gap between Euclidean path integral approaches and string theory techniques for the entropy of extremal black holes. ...
Preprint
Using a thin shell, the first law of thermodynamics, and a unified approach, we study the thermodymanics and find the entropy of a (2+1)-dimensional extremal rotating Ba\~{n}ados-Teitelbom-Zanelli (BTZ) black hole. The shell in (2+1) dimensions, i.e., a ring, is taken to be circularly symmetric and rotating, with the inner region being a ground state of the anti-de Sitter (AdS) spacetime and the outer region being the rotating BTZ spacetime. The extremal BTZ rotating black hole can be obtained in three different ways depending on the way the shell approaches its own gravitational or horizon radius. These ways are explicitly worked out. The resulting three cases give that the BTZ black hole entropy is either the Bekenstein-Hawking entropy, S=A+4GS=\frac{A_+}{4G}, or it is an arbitrary function of A+A_+, S=S(A+)S=S(A_+), where A+=2πr+A_+=2\pi r_+ is the area, i.e., the perimeter, of the event horizon in (2+1) dimensions. We speculate that the entropy of an extremal black hole should obey 0S(A+)A+4G0\leq S(A_+)\leq\frac{A_+}{4G}. We also show that the contributions from the various thermodynamic quantities, namely, the mass, the circular velocity, and the temperature, for the entropy in all three cases are distinct. This study complements the previous studies in thin shell thermodynamics and entropy for BTZ black holes. It also corroborates the results found for a (3+1)-dimensional extremal electrically charged Reissner-Nordstr\"om black hole.
... Logarithmic corrections to extremal black hole entropy are not universally positive. See, e.g., [52,53]. ...
Preprint
We analyze infrared consistency conditions of 3D and 4D effective field theories with massive scalars or fermions charged under multiple U(1) gauge fields. At low energies, one can integrate out the massive particles and thus obtain a one-loop effective action for the gauge fields. In the regime where charge-independent contributions to higher-derivative terms in the action are sufficiently small, it is then possible to derive constraints on the charge-to-mass ratios of the massive particles from requiring that photons propagate causally and have an analytic S-matrix. We thus find that the theories need to contain bifundamentals and satisfy a version of the weak gravity conjecture known as the convex-hull condition. Demanding self-consistency of the constraints under Kaluza-Klein compactification, we furthermore show that, for scalars, they imply a stronger version of the weak gravity conjecture in which the charge-to-mass ratios of an infinite tower of particles are bounded from below. We find that the tower must again include bifundamentals but does not necessarily have to occupy a charge (sub-)lattice.
... The macroscopically computed logarithms serve as a litmus test for any proposed enumeration of quantum black hole microstates which is more refined than the test provided by the area law. Sen extensively analyzed a number of stringy examples all of which passed the test with flying colors [7], and further noted that the macroscopic computation does not match the loop gravity result. He also posed a matching of the logarithms as a challenge for Kerr/CFT [8][9][10][11]. ...
Preprint
It has been shown by A. Sen that logarithmic corrections to the black hole area-entropy law are entirely determined macroscopically from the massless particle spectrum. They therefore serve as powerful consistency checks on any proposed enumeration of quantum black hole microstates. Sen's results include a macroscopic computation of the logarithmic corrections for a five-dimensional near extremal Kerr-Newman black hole. Here we compute these corrections microscopically using a stringy embedding of the Kerr/CFT correspondence and find perfect agreement.
... These results can be used to describe the one-loop Euclidean gravitational partition function and complement the analysis of logarithmic corrections to Kerr and Kerr-AdS BH thermodynamics recently studied in [25,26]. We remark that one-loop effective actions in supergravity were used to compute logarithmic corrections to the entropy of supersymmetric BHs [27,28]. By complementing the analysis performed in this paper with the study of the zero-modes of the one-loop operators one should be able to compute logarithmics corrections to the entropy of Kerr-(A)dS BHs. ...
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We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024
... When this rescaling is implemented in the quantum entropy function, it shows that there is a correction to the black hole entropy of the form S BH (q i ) → · · · + a grav log Λ 2 + · · · . (5.2) 16 For a review of the original work see [41]. ...
Article
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A bstract We quantify how constraints on light states affect the asymptotic growth of heavy states in weak Jacobi forms. The constraints we consider are sparseness conditions on the Fourier coefficients of these forms, which are necessary to interpret them as gravitational path integrals. Using crossing kernels, we extract the leading and subleading behavior of these coefficients and show that the leading Cardy-like growth is robust in a wide regime of validity. On the other hand, we find that subleading corrections are sensitive to the constraints placed on the light states, and we quantify their imprint on the asymptotic growth of states. Our approach is tested against the generating function of symmetric product orbifolds, where we provide new insights into the factors contributing to the asymptotic growth of their Fourier coefficients. Finally, we use our methods to revisit the UV/IR connection that relates black hole microstate counting to modular forms. We provide a microscopic interpretation of the logarithmic corrections to the entropy of BPS black holes in N \mathcal{N} N = 2, 4 ungauged supergravity in four and five dimensions, and tie it to consistency conditions in AdS 3 /CFT 2 .
... For the future, the vision ultimately is that all the various contributions to the partition function, in gravity and in CFT, whether boundary conditions correspond to an index or not, can be disentangled. Significant strides have been taken towards this goal in the most favorable circumstances, such as asymptotically flat spacetimes with at least 1 8 of the supersymmetries [66][67][68][69][70]. For asymptotically AdS spacetimes with maximal supersymmetry, the setting we have studied, the current research frontier is at a lower level of understanding, but recent years have witnessed much progress, using a variety of techniques [71][72][73]. ...
Article
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A bstract Supersymmetric AdS black hole solutions exist only when their angular momenta and charges satisfy a certain constraint that depends on the dimension. We show that these nonlinear relations on the conserved charges agree with a computation in the dual supersymmetric CFT in its free limit, with interactions entering only through a uniform rescaling of all charges. Our computations apply to the highly non-trivial charge constraints for AdS 4 , AdS 5 and AdS 7 black holes, and generalize an earlier one for the analogous constraint in AdS 3 . Our results suggest a microscopic understanding of AdS black holes beyond the scope of supersymmetric indices.
... In this limit the far region decouples [9] and one expects that the relevant part of the black hole Hilbert space can be equivalently captured by gravitational dynamics in the throat according to the Kerr/CFT correspondence. The analogous formalism, when applied to spherically symmetric black holes, has led to precise matches of bulk gravitational calculations and microscopic counts [10,11]. ...
Article
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Recent work has shown that loop corrections from massless particles generate 3 2 log T Hawking corrections to black hole entropy which dominate the thermodynamics of cold near-extreme charged black holes. Here we adapt this analysis to near-extreme Kerr black holes. Like AdS 2 × S 2 , the near-horizon extreme Kerr (NHEK) metric has a family of normalizable zero modes corresponding to reparametrizations of boundary time. The path integral over these zero modes leads to an infrared divergence in the one-loop approximation to the Euclidean NHEK partition function. We regulate this divergence by retaining the leading finite temperature correction in the NHEK scaling limit. This “not-NHEK” geometry lifts the eigenvalues of the zero modes, rendering the path integral infrared finite. The quantum-corrected near-extremal entropy exhibits 3 2 log T Hawking behavior characteristic of the Schwarzian model and predicts a lifting of the ground state degeneracy for the extremal Kerr black hole. Published by the American Physical Society 2024
... For the future, the vision ultimately is that all the various contributions to the partition function, in gravity and in CFT, whether boundary conditions correspond to an index or not, can be disentangled. Significant strides have been taken towards this goal in the most favorable circumstances, such as asymptotically flat spacetimes with at least 1 8 of the supersymmetries [59][60][61][62][63]. For asymptotically AdS spacetimes with maximal supersymmetry, the setting we have studied, the current research frontier is at a lower level of understanding, but recent years have witnessed much progress, using a variety of techniques [64][65][66]. ...
Preprint
Supersymmetric AdS black hole solutions exist only when their angular momenta and charges satisfy a certain constraint that depends on the dimension. We show that these nonlinear relations on the conserved charges agree with a computation in the dual supersymmetric CFT in its free limit, with interactions entering only through a uniform rescaling of all charges. Our computations apply to the highly non-trivial charge constraints for AdS4_4, AdS5_5 and AdS7_7 black holes, and generalize an earlier one for the analogous constraint in AdS3_3. Our results suggest a microscopic understanding of AdS black holes beyond the scope of supersymmetric indices.
... These results can be used to describe the one-loop Euclidean gravitational partition function and complement the analysis of logarithmic corrections to Kerr BH thermodynamics recently studied in [25]. We remark that one-loop effective actions in supergravity were used to compute logarithmic corrections to the entropy of supersymmetric BHs [26,27]. By complementing the analysis performed in this paper with the study of the zero-modes of the one-loop operators one should be able to compute logarithmics corrections to the entropy of Kerr-(A)dS BHs. ...
Preprint
Full-text available
We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ODEs with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasi-normal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulae for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss exact properties of the thermal version of the BH effective actions. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulae can be given for higher spin matter and radiation fields in more general gravitational backgrounds.
... The holographic principle of gravity provides profound insights into both macroscopic and microscopic properties of black holes in general relativity. Stemming from the Anti de-Sitter (AdS)/conformal field theory (CFT) correspondence, the entropy of BPS black holes in string theory admits microscopic interpretations by state counting [1,2]. The crucial mechanism that enables these descriptions is the conformal symmetry, which is applied from the AdS factor in the black hole near horizon geometry. ...
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A bstract Recent studies on the holographic descriptions of Kerr black holes indicate that the conformal or the warped conformal symmetries are responsible for the Kerr black hole physics at both background and perturbation levels. In the present paper, we extend the validity of these studies to the case of accelerating Kerr black hole. By invoking a set of non-trivial diffeomorphisms near the horizon bifurcation surface of the accelerating Kerr black hole, the Dirac brackets among charges of the diffeomorphisms form the symmetry algebra of a warped CFT which consists of one Virasoro and one Kac-Moody algebra with central extensions. This provides the evidence for warped CFTs being possible holographic dual to accelerating Kerr black holes. The thermal entropy formula of the warped CFT fixed by modular parameters and vacuum charges reproduces the entropy of the rotating black hole with acceleration.
... Ever since the seminal work [2], the Euclidean gravitational path integral has been a prominent tool that has led to tremendous progress in thermodynamic and entanglement aspects of quantum black holes. In some cases, one finds exact agreement with microscopic calculations in string theory or holographic CFTs, even beyond the leading order in G N [3][4][5][6][7]. Operationally, one starts with a formal path integral integrating over all metrics and matter fields, and then expands g = g * + δg, Φ = Φ * + δφ around the saddle points (g * , Φ * ) ...
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A bstract We extend a formula for 1-loop black hole determinants by Denef, Hartnoll, and Sachdev (DHS) to spinning fields on any ( d + 1)-dimensional static spherically symmetric black hole. By carefully analyzing the regularity condition imposed on the Euclidean eigenfunctions, we reveal an unambiguous bulk-edge split in the 1-loop Euclidean partition function for tensor fields of arbitrary integer spin: the bulk part captures the “renormalized” thermal canonical partition function recently discussed in [1]; the edge part is related to quasinormal modes (QNMs) that fail to analytically continue to a subset of Euclidean modes with enhanced fall-offs near the origin. Since the edge part takes the form of a path integral on S d− 1 , this suggests that these are associated with degrees of freedom living on the bifurcation surface in the Lorentzian two-sided black hole geometry. For massive higher spin on static BTZ and massive vector on Nariai black holes, we find that the edge partition function is related to the QNMs with lowest overtone numbers.
... Some supersymmetric black holes (such as the extreme Reissner-Nordström black hole) do have extra symmetry that non-extreme black holes don't have: they admit Killing spinor fields leading to the extreme black hole to be invariant under supersymmetric transformations. Extreme black holes take a special place in counting micro states related to entropy in black holes: large class of extremal supersymmetric black holes in string theory have been studied in the counting of mircrostates [3,4]. ...
Article
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We study neutral massless scalar field perturbations around an extreme dilaton black hole in 2 + 1 dimensions: the wave equations of the massless scalar field is shown to be exactly solvable in terms of Whittaker functions. Thus, the quasinormal modes are computed exactly and shown to be purely imaginary: we show the existence of stable and unstable modes. Interestingly, the quasinormal modes do not depend on the black holes parameters and the fundamental mode is always unstable and depends only on the parameters of the test field. Also, we determine the quasinormal frequencies via the improved asymptotic iteration method which shows a good agreement with the analytical results.
... Some supersymmetric black holes (such as the extreme Reissner-Nordstrom black hole) do have extra symmetry that non-extreme black holes don't have: they admit Killing spinor fields leading to the extreme black hole to be invariant under supersymmetric transformations. Extreme black holes take a special place in counting micro states related to entropy in black holes: large class of extremal supersymmetric black holes in string theory have been studied in the counting of mircrostates [3,4]. ...
Preprint
We study neutral massless scalar field perturbations around an extreme dilaton black hole in 2 +1 dimensions: the wave equations of the massless scalar field is shown to be exactly solvable in terms of Whittaker functions. Thus, the quasinormal modes are computed exactly and shown to be purely imaginary: we show the existence of stable and unstable modes. Interestingly, the quasinormal modes do not depend on the black holes parameters and the fundamental mode is always unstable and depends only on the parameters of the test field. Also, we determine the quasinormal frequencies via the improved asymptotic iteration method which shows a good agreement with the analytical results.
... Hence computations of these corrections to the entropy of black holes in various supergravity theories have become a rich arena to explore in both microscopic and macroscopic sectors. References [20,21] present an excellent comparative study of macroscopic and microscopic entropy corrections of various black holes in different supergravity theories. By successfully comparing the logarithmic corrections to the entropy of a black hole between both sectors (microscopic and macroscopic), one can test a consistent theory of quantum gravity. ...
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A bstract We reviewed the field redefinition approach of Seeley-DeWitt expansion for the determination of Seeley-DeWitt coefficients from arXiv:1505.01156 . We apply this approach to compute the first three Seeley-DeWitt coefficients for “non-minimal” N \mathcal{N} N = 1 Einstein-Maxwell supergravity in four dimensions. Finally, we use the third coefficient for the computation of the logarithmic corrections to the Bekenstein-Hawking entropy of non-extremal black holes following arXiv:1205.0971 . We determine the logarithmic corrections for non-extremal Kerr-Newman, Kerr, Reissner-Nordström and Schwarzschild black holes in “non-minimal” N \mathcal{N} N = 1, d = 4 Einstein-Maxwell supergravity.
... To do so, recall the microstate counting of asymptotically flat black holes which has a long history and is understood incomparably better. Many precision agreements were established, not just at the leading order but also for higher derivative corrections, quantum corrections, and far beyond [9][10][11]. Moreover, in most cases it has been understood why these agreements hold with the precision they do. ...
Article
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A bstract We develop the thermodynamics of black holes in AdS 4 and AdS 7 near their BPS limit. In each setting we study the two distinct deformations orthogonal to the BPS surface as well as their nontrivial interplay with each other and with BPS properties. Our results illuminate recent microscopic calculations of the BPS entropy. We show that these microscopic computations can be leveraged to also describe the near BPS regime, by generalizing the boundary conditions imposed on states.
... As a leading candidate for a theory of quantum gravity, string theory has alredy provided, in the work of Strominger and Vafa, a microstate counting of the entropy for certain asymptotically flat black holes [6]. It has also provided, through work by Sen and collaborators, an explanation for the logarithmic corrections to the entropy [7]. ...
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We investigate logarithmic corrections to the entropy of supersymmetric, rotating, asymptotically AdS5_5 black holes and black strings. Within the framework of the AdS/CFT correspondence, the entropy of these black objects is determined, on the field theory side, by the superconformal index and the refined topologically twisted index of N=4\mathcal{N}=4 supersymmetric Yang-Mills theory, respectively. We read off the logarithmic correction from those field-theoretic partition functions. On the gravity side, we take the near-horizon limit and apply the Kerr/CFT correspondence whose associated charged Cardy formula describes the degeneracy of states at subleading order and determines the logarithmic correction to the entropy. We find perfect agreement between these two approaches. Our results provide a window into precision microstate counting and demonstrate the efficacy of low-energy, symmetry-based approaches such as the Kerr/CFT correspondence for asymptotically AdS black objects.
... Supergravity theories can be realized as the low energy gravity end of compactified string theories (mainly on Calabi-Yau manifold), and this makes logarithmic correction results in supergravities a more significant IR testing grounds for string theories. 2 Following the developments discussed in the previous paragraph, the analysis of the effective action in N = 1, d = 4 EMSGT is highly inquisitive for the determination of logarithmic correction to the entropy 2 An excellent comparative review of logarithmic corrections and their microscopic consistency in string theory can be found in [44,45]. ...
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A bstract We study one-loop covariant effective action of “non-minimally coupled” N \mathcal{N} N = 1, d = 4 Einstein-Maxwell supergravity theory by heat kernel tool. By fluctuating the fields around the classical background, we study the functional determinant of Laplacian differential operator following Seeley-DeWitt technique of heat kernel expansion in proper time. We then compute the Seeley-DeWitt coefficients obtained through the expansion. A particular Seeley-DeWitt coefficient is used for determining the logarithmic correction to Bekenstein-Hawking entropy of extremal black holes using quantum entropy function formalism. We thus determine the logarithmic correction to the entropy of Kerr-Newman, Kerr and Reissner-Nordström black holes in “non-minimally coupled” N \mathcal{N} N = 1, d = 4 Einstein-Maxwell supergravity theory.
... To do so, recall the microstate counting of asymptotically flat black holes which has a long history and is understood incomparably better. Many precision agreements were established, not just at the leading order but also for higher derivative corrections, quantum corrections, and far beyond [9][10][11]. Moreover, in most cases it has been understood why these agreements hold with the precision they do. ...
Preprint
We develop the thermodynamics of black holes in AdS4_4 and AdS7_7 near their BPS limit. In each setting we study the two distinct deformations orthogonal to the BPS surface as well as their nontrivial interplay with each other and with BPS properties. Our results illuminate recent microscopic calculations of the BPS entropy. We show that these microscopic computations can be leveraged to also describe the near BPS regime, by generalizing the boundary conditions imposed on states.
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In gravitational collapses, the horizon and singularity’s realisation in the finite future of the proper time used co-moving observer happens in the future of infinitely far away future of the normal time used outside probe. To the latter the horizon and singularity defined in the singularity theorem are physical realities only in the sense of uncertainty principle and ensemble interpretation. We provide two exact time dependent solution families to the Einstein equation and show that they form a pair of complementary description for the microscopic state of black holes by showing that the Bekenstein–Hawking entropy formula follows properly from their canonical wave function’s degeneracy. We also develop an eXact One Body method for general relativity two-body dynamics whose conservative part calls no post newtonian approximation as input and applies to the full three stages of black hole binary merger events. By this method, we analytically calculate the gravitational wave forms following from such merger processes. In the case black holes carry exact and apriori horizon and singularity our wave forms agree with those following from conventional effective one body method but exhibit more consistent late time behaviour. In the case black holes carry only asymptotic horizon and extended inner structure thus experiencing banana shape deformation as the merger occurs, our wave forms exhibit all features especially the late time quasi-normal mode type oscillation seen in real observations.
Chapter
This last chapter contains some topics in non-perturbative superstring theory and basic applications of the duality web between SUSY string theories:
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We consider various aspects of Sen's classical entropy function formalism for asymptotically AdS4 black holes with emphasis on its efficacy to capture higher derivative corrections to the Bekenstein-Hawking entropy. The formalism has the important advantage of being based on near-horizon symmetries and does not require knowledge of the full interpolating supergravity solution, nor of its AdS4 asymptotics. For the static case, we focus on applying the entropy function formalism in the presence of various higher derivative terms motivated in conformal supergravities; we find agreement with recently reported results utilizing the full black hole solutions and Wald's entropy formula. For the rotating case, we demonstrate that a modified version of the formalism generates a background that coincides precisely with the Bardeen-Horowitz limit of known rotating, electrically charged AdS4 black holes and provides a swift approach to the black hole entropy, including higher derivatives corrections. We conclude that Sen's classical entropy function formalism is a viable and highly efficient approach to capturing higher-derivative corrections to the entropy of asymptotically AdS4 black holes albeit naturally missing certain relations arising from global aspects of the full black hole solution.
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A bstract We calculate the logarithmic correction to the entropy of asymptotically flat and AdS black holes (rotating, non-rotating, charged, and uncharged) embedded in Einstein-Maxwell-dilaton (EMD) theories with U (1)-charged. The leading quantum gravitational corrections are achieved in both extremal and non-extremal limits of black hole temperature by designing a common Euclidean gravity setup that evaluates the “logarithmic term” from one-loop effective actions via heat kernel method-based calculations. EMD theories are universal building blocks of compactified string theory or supergravity models in 4D. For a concrete example, we generalize the entire setup and calculate logarithmic corrections for black holes in U (1) ² -charged EMD models intersecting with N \mathcal{N} N = 4 ungauged and gauged bosonic supergravity. In contrast to flat backgrounds, all the AdS 4 results are found to be non-topological, providing a wider “infrared window” into the microscopic degrees of freedom of black holes in string theory.
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In this work we derive the Bekenstein-Hawking entropy formula, S=A4lp2, from the following minimal assumptions: (i) there is a minimum area, Amin, proportional to lp2; (ii) the event horizon area, A, is tessellated by N=A/Amin distinguishable units; and (iii) the internal structure of these units is that of an infinite tower of internal levels. Although our results are model independent, this internal structure can be realized as the excitations of more fundamental entities such as, for instance, strings or loop quantum gravity spin networks. Even more, once the microstates of the black hole are taken to be singlets formed within the infinite tower of states describing the whole event horizon, the correction term −32logA emerges from our model. Finally, some comments regarding the applicability of the present model to extremal black holes, as well as possible relationships with spectral geometry and other approaches are pointed out. Our results are independent of the dimension of the black hole and whether it is rotating or not.
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A bstract Motivated by the prospect of constraining microscopic models, we calculate the exact one-loop corrected de Sitter entropy (the logarithm of the sphere partition function) for every effective field theory of quantum gravity, with particles in arbitrary spin representations. In doing so, we universally relate the sphere partition function to the quotient of a quasi-canonical bulk and a Euclidean edge partition function, given by integrals of characters encoding the bulk and edge spectrum of the observable universe. Expanding the bulk character splits the bulk (entanglement) entropy into quasinormal mode (quasiqubit) contributions. For 3D higher-spin gravity formulated as an sl( n ) Chern-Simons theory, we obtain all-loop exact results. Further to this, we show that the theory has an exponentially large landscape of de Sitter vacua with quantum entropy given by the absolute value squared of a topological string partition function. For generic higher-spin gravity, the formalism succinctly relates dS, AdS ± and conformal results. Holography is exhibited in quasi-exact bulk-edge cancelation.
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We report on an investigation of Ruppeiner curvature RU for (d+1)-dimensional hyperbolic black holes in anti-de Sitter spacetimes and its connection with Renormalization Group (RG) flows in the dual d-dimensional conformal field theories (CFTs) in Minkowski spacetimes. A repulsive type interaction among microstructures is found, which is weaker for positive mass black holes and grows stronger for negative mass black holes at low temperatures. In particular, we show that RU evaluated along a zero mass curve is a universal constant, depending only on the dimension of space-time. The extremal black holes are pointed out to have a positive Ruppeiner curvature irrespective of their horizon topology.
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We argue that String Theory and Loop Quantum Gravity canVaid, Deepak be thought of as describing different regimes of a single unified theory of quantum gravity. LQG can be thought of as providing the pre-geometric exoskeleton out of which macroscopic geometry emerges and String Theory then becomes the effective theory which describes the dynamics of that exoskeleton. The core of the argument rests on the claim that the Nambu-Goto action of String Theory can be viewed as the expectation value of the LQG area operator evaluated on the string worldsheet. A concrete result is that the string tension of String Theory and the Barbero-Immirzi parameter of LQG turn out to be proportional to each other.
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We compute the macroscopic entropy of the supersymmetric rotating dyonic strings carrying linear momentum in 6D (1, 0) supergravity with curvature squared corrections. Our calculation is based on Sen’s entropy function formalism applied to the near-horizon geometry of the string solution taking the form of an extremal Bañados-Teitelboim-Zanelli ×S3. The final entropy formula states that the two independent supersymmetric completions of Riemann tensor squared contribute equally to the entropy. A further S3 compactification of the 6D theory results in a matter coupled 3D supergravity model in which the quantization condition of the SU(2)R Chern-Simons level implies the horizon value of the dilaton is not modified by higher derivative interactions beyond supersymmetric curvature squared terms.
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We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in N = 4 and N = 8 supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in N = 4 supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the N = 8 supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS 2 /CF T 1 correspondence. We also extend our analysis to N = 2 supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.
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We review and extend recent attempts to find a precise relation between extremal black hole entropy and degeneracy of microstates using AdS2/CFT1 correspondence. Our analysis leads to a specific relation between degeneracy of black hole microstates and an appropriately defined partition function of string theory on the near horizon geometry — named the quantum entropy function. In the classical limit this reduces to the usual relation between statistical entropy and Wald entropy.
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We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in N=4 \mathcal{N} = {4} and N=8 \mathcal{N} = {8} supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in N=4 \mathcal{N} = {4} supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the N=8 \mathcal{N} = {8} supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS 2/CF T 1 correspondence. We also extend our analysis to N=2 \mathcal{N} = {2} supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.
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For dyons in heterotic string theory compactified on a six-torus, with electric charge vector Q and magnetic charge vector P, the positive integer I ≡ gcd(Q ∧ P) is an invariant of the U-duality group. We propose the microscopic theory for computing the spectrum of all dyons for all values of I, generalizing earlier results that exist only for the simplest case of I = 1. Our derivation uses a combination of arguments from duality, 4d-5d lift, and a careful analysis of fermionic zero modes. The resulting degeneracy agrees with the black hole degeneracy for large charges and with the degeneracy of field-theory dyons for small charges. It naturally satisfies several physical requirements including integrality and duality invariance. As a byproduct, we also derive the microscopic (0, 4) superconformal field theory relevant for computing the spectrum of five-dimensional Strominger-Vafa black holes in ALE backgrounds and count the resulting degeneracies.
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N=4 supersymmetric string theories contain negative discriminant states whose numbers are known precisely from microscopic counting formulae. On the macroscopic side, these results can be reproduced by regarding these states as multi-centered black hole configurations provided we make certain identification of apparently distinct multi-centered black hole configurations according to a precise set of rules. In this paper we provide a physical explanation of such identifications, thereby establishing that multi-centered black hole configurations reproduce correctly the microscopic results for the number of negative discriminant states without any ad hoc assumption.
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In these notes we describe recent progress in understanding finite size corrections to the black hole entropy. Much of the earlier work concerning quantum black holes has been in the limit of large charges when the area of the even horizon is also large. In recent years there has been substantial progress in understanding the entropy of supersymmetric black holes within string theory going well beyond the large charge limit. It has now become possible to begin exploring finite size effects in perturbation theory in inverse size and even nonperturbatively, with highly nontrivial agreements between thermodynamics and statistical mechanics. Unlike the leading Bekenstein-Hawking entropy which follows from the two-derivative Einstein-Hilbert action, these finite size corrections depend sensitively on the phase under consideration and contain a wealth of information about the details of compactification as well as the spectrum of nonperturbative states in the theory. Finite-size corrections are therefore very interesting as a valuable window into the microscopic degrees of freedom of the quantum theory.
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Certain helicity trace indices of charged states in N=4 and N=8 superstring theory have been computed exactly using their explicit weakly coupled microscopic description. These indices are expected to count the exact quantum degeneracies of black holes carrying the same charges. In order for this interpretation to be consistent, these indices should be positive integers. We prove this positivity property for a class of four/five dimensional black holes in type II string theory compactified on T^6/T^5 and on K3 \times T^2/S^1. The proof relies on the mock modular properties of the corresponding generating functions.
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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.
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We use localization to evaluate the functional integral of string field theory on AdS_2xS^2 background corresponding to the near horizon geometry of supersymmetric black holes in 4d compactifications with N=2 supersymmetry. In particular, for a theory containing n_v + 1 vector multiplets, we show that the functional integral localizes exactly onto an ordinary integral over a finite-dimensional submanifold in the field space labeling a continuous family of instanton solutions in which auxiliary fields in the vector multiplets are excited with nontrivial dependence on AdS_2 coordinates. These localizing solutions are universal in that they follow from the off-shell supersymmetry transformations and do not depend on the choice of the action. They are parametrized by n_v +1 real parameters C^I, ; I= 0,..., n_v that correspond to the values of the auxiliary fields at the center of AdS_2. In the Type-IIA frame, assuming D-terms evaluate to zero on the solutions for reasons of supersymmetry, the classical part of the integrand equals the absolute square of the partition function of the topological string as conjectured by Ooguri, Strominger, and Vafa; however evaluated at the off-shell values of scalar fields at the center of AdS_2. In addition, there are contributions from one-loop determinants, brane-instantons, and nonperturbative orbifolds that are in principle computable. These results thus provide a concrete method to compute exact quantum entropy of these black holes including all perturbative and nonperturbative corrections and can be used to establish a precise relation between the quantum degeneracies of black holes and the topological string.
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For BPS black holes with at least four unbroken supercharges, we describe how the macroscopic entropy can be used to compute an appropriate index, which can be then compared with the same index computed in the microscopic description. We obtain exact results incorporating all higher order quantum corrections in the limit when only one of the charges, representing momentum along an internal direction, approaches infinity keeping all other charges fixed at arbitrary finite values. In this limit, we find that the microscopic index is controlled by certain anomaly coefficients whereas the macroscopic index is controlled by the coefficients of certain Chern-Simons terms in the effective action. The equality between the macroscopic and the microscopic index then follows as a consequence of anomaly inflow. In contrast, the absolute degeneracy does not have any such simple expression in terms of the anomaly coefficients or coefficients of Chern-Simons terms. We apply our analysis to several examples of spinning black holes in five dimensions and non-spinning black holes in four dimensions to compute the index exactly in the limit when only one of the charges becomes large, and find perfect agreement with the result of exact microscopic counting. Our analysis resolves a puzzle involving M5-branes wrapped on a 5-cycle in K3 × T 3.
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We survey recent results on the exact dyon spectrum in a class of N = 4 supersymmetric string theories, and discuss how the results can be understood from the macroscopic viewpoint using AdS(2)/CFT1 correspondence. The comparison between the microscopic and the macroscopic results includes power suppressed corrections to the entropy, the sign of the index, logarithmic corrections and also the twisted index measuring the distribution of discrete quantum numbers among the microstates. (Based on lectures given by A.S. at the 12th Marcel Grossmann Meeting On General Relativity, 12-18 Jul 2009, Paris, France; CERN Winter School on Supergravity, Strings, and Gauge Theory, 25-29 January 2010; String Theory: Formal Developments And Applications, 21 Jun - 3 Jul 2010, Cargese, France, and notes taken by I.M. at the Cargese school.)
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We evaluate the one loop determinant of matter multiplet fields of 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.
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AdS 2/CFT 1 correspondence leads to a prescription for computing the degeneracy of black hole states in terms of path integral over string fields living on the near horizon geometry of the black hole. In this paper we make use of the enhanced supersymmetries of the near horizon geometry and localization techniques to argue that the path integral receives contribution only from a special class of string field configurations which are invariant under a subgroup of the supersymmetry transformations. We identify saddle points which are invariant under this subgroup. We also use our analysis to show that the integration over infinite number of zero modes generated by the asymptotic symmetries of AdS 2 generate a finite contribution to the path integral.
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Macroscopic entropy of an extremal black hole is expected to be determined completely by its near horizon geometry. Thus two black holes with identical near horizon geometries should have identical macroscopic entropy, and the expected equality between macroscopic and microscopic entropies will then imply that they have identical degeneracies of microstates. An apparent counterexample is provided by the 4D-5D lift relating BMPV black hole to a four dimensional black hole. The two black holes have identical near horizon geometries but different microscopic spectrum. We suggest that this discrepancy can be accounted for by black hole hair, -- degrees of freedom living outside the horizon and contributing to the degeneracies. We identify these degrees of freedom for both the four and the five dimensional black holes and show that after their contributions are removed from the microscopic degeneracies of the respective systems, the result for the four and five dimensional black holes match exactly. Comment: LaTeX file, 30 pages; v2: minor changes; v3: minor changes; v4: note added
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Attention is paid to the fact that temperature of a classical black hole can be derived from the extremality condition of its free energy with respect to variation of the mass of a hole. For a quantum Schwarzschild black hole evaporating massless particles the same condition is shown to result in the following one-loop temperature T=(8πM)1(1+σ(8πM2)1)T=(8\pi M)^{-1} (1+\sigma (8\pi M^2)^{-1}) and entropy S=4πM2σlogMS = 4\pi M^2 - \sigma\log M expressed in terms of the effective mass M of a hole together with its radiation and the integral of the conformal anomaly σ\sigma that depends on the field species. Thus, in the given case quantum corrections to T and S turn out to be completely provided by the anomaly. When it is absent (σ=0\sigma=0), which happens in a number of supersymmetric models, the one-loop expressions of T and S preserve the classical form. On the other hand, if the anomaly is negative (σ<0\sigma<0) an evaporating quantum hole seems to cease to heat up when its mass reaches the Planck scales. Comment: 10 pages, Latex file
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A recently discovered relation between 4D and 5D black holes is used to derive exact (weighted) BPS black hole degeneracies for 4D N = 8 string theory from the exactly known 5D degeneracies. A direct 4D microscopic derivation in terms of weighted 4D D-brane bound state degeneracies is sketched and found to agree.
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We find the exact spectrum of a class of quarter BPS dyons in a generic N = 4 supersymmetric Z(N) orbifold of type IIA string theory on K3 x T-2 or T-6. We also find the asymptotic expansion of the statistical entropy to first non-leading order in inverse power of charges and show that it agrees with the entropy of a black hole carrying same set of charges after taking into account the effect of the four derivative Gauss-Bonnet term in the effective action of the theory.
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Attention is paid to the fact that the temperature of a classical black hole can be derived from the extremality condition of its free energy with respect to the variation of the mass of a hole. For a quantum Schwarzschild black hole evaporating massless particles the same condition is shown to result in the one-loop temperature T=(8\ensuremath{\pi}M{)}^{\mathrm{\ensuremath{-}}1}[1+\ensuremath{\sigma}(8\ensuremath{\pi}M2{\mathit{M}}^{2}{)}^{\mathrm{\ensuremath{-}}1}] and entropy S=4\ensuremath{\pi}M2{\mathit{M}}^{2}-\ensuremath{\sigma}lnM expressed in terms of the effective mass M of a hole together with its radiation and the integral of the conformal anomaly \ensuremath{\sigma} that depends on the field species. Thus, in the given case quantum corrections to T and S turn out to be completely provided by the anomaly. When it is absent (\ensuremath{\sigma}=0), which happens in a number of supersymmetric models, the one-loop expressions of T and S preserve the classical form. On the other hand, if the anomaly is negative (\ensuremath{\sigma}0) an evaporating quantum hole seems to cease to heat up when its mass reaches the Planck scales.
Article
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.
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The conditions for fully supersymmetric backgrounds of general N=2 locally supersymmetric theories are derived based on the off-shell superconformal multiplet calculus. This enables the derivation of a non-renormalization theorem for a large class of supersymmetric invariants with higher-derivative couplings. The theorem implies that the invariant and its first order variation must vanish in a fully supersymmetric background. The conjectured relation of one particular higher-derivative invariant with a specific five-dimensional invariant containing the mixed gauge-gravitational Chern-Simons term is confirmed.
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Over the past few years the understanding of the microscopic theory of black hole entropy has made important conceptual progress by recognizing that the degeneracies are encoded in partition functions which are determined by higher rank automorphic representations, in particular in the context of Siegel modular forms of genus two. In this brief review some of the elements of this framework are highlighted. One of the surprising aspects is that the Siegel forms that have appeared in the entropic framework are geometric in origin, arising from weight two cusp forms, hence from elliptic curves.
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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 near-horizon 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.
Article
A recently discovered relation between 4D and 5D black holes is used to derive exact (weighted) BPS black hole degeneracies for 4D Script N = 8 string theory from the exactly known 5D degeneracies. A direct 4D microscopic derivation in terms of weighted 4D D-brane bound state degeneracies is sketched and found to agree.
Article
For dyons in heterotic string theory compactified on a six-torus, with electric charge vector Q and magnetic charge vector P, the positive integer I ≡ gcd(Q ∧ P) is an invariant of the U-duality group. We propose the microscopic theory for computing the spectrum of all dyons for all values of I, generalizing earlier results that exist only for the simplest case of I = 1. Our derivation uses a combination of arguments from duality, 4d-5d lift, and a careful analysis of fermionic zero modes. The resulting degeneracy agrees with the black hole degeneracy for large charges and with the degeneracy of field-theory dyons for small charges. It naturally satisfies several physical requirements including integrality and duality invariance. As a byproduct, we also derive the microscopic (0, 4) superconformal field theory relevant for computing the spectrum of five-dimensional Strominger-Vafa black holes in ALE backgrounds and count the resulting degeneracies.
Article
The microscopic formula for the degeneracies of 1/8 BPS black holes in type II string theory compactified on a six dimensional torus can be expressed as a sum of several terms. One of the terms is a function of the Cremmer-Julia invariant and gives the leading contribution to the entropy in the large charge limit. The other terms, which give exponentially subleading contribution, depend not only on the Cremmer-Julia invariant, but also on the arithmetic properties of the charges, and in fact exist only when the charges satisfy special arithmetic properties. We identify the origin of these terms in the macroscopic formula for the black hole entropy, based on quantum entropy function, as the contribution from non-trivial saddle point(s) in the path integral of string theory over the near horizon geometry. These saddle points exist only when the charge vectors satisfy the arithmetic properties required for the corresponding term in the microscopic formula to exist. Furthermore the leading contribution from these saddle points in the large charge limit agrees with the leading asymptotic behaviour of the corresponding term in the degeneracy formula.
Article
The microscopic formula for the degeneracies of 1/8 BPS black holes in type II string theory compactified on a six dimensional torus can be expressed as a sum of several terms. One of the terms is a function of the Cremmer-Julia invariant and gives the leading contribution to the entropy in the large charge limit. The other terms, which give exponentially subleading contribution, depend not only on the Cremmer-Julia invariant, but also on the arithmetic properties of the charges, and in fact exist only when the charges satisfy special arithmetic properties. We identify the origin of these terms in the macroscopic formula for the black hole entropy, based on quantum entropy function, as the contribution from non-trivial saddle point(s) in the path integral of string theory over the near horizon geometry. These saddle points exist only when the charge vectors satisfy the arithmetic properties required for the corresponding term in the microscopic formula to exist. Furthermore the leading contribution from these saddle points in the large charge limit agrees with the leading asymptotic behaviour of the corresponding term in the degeneracy formula.
Article
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.
Article
We find the exact spectrum of a class of quarter BPS dyons in a generic Script N = 4 supersymmetric Bbb ZN orbifold of type IIA string theory on K3 × T2 or T6. We also find the asymptotic expansion of the statistical entropy to first non-leading order in inverse power of charges and show that it agrees with the entropy of a black hole carrying same set of charges after taking into account the effect of the four derivative Gauss-Bonnet term in the effective action of the theory.
Article
We find the most general bosonic solution to the localization equations describing the contributions to the quantum entropy of supersymmetric black holes in four-dimensional N=2 supergravity coupled to n_v vector multiplets. This requires the analysis of the BPS equations of the corresponding off-shell supergravity (including fluctuations of the auxiliary fields) with AdS2 \times S2 attractor boundary conditions. Our work completes and extends the results of arXiv:1012.0265 that were obtained for the vector multiplet sector, to include the fluctuations of all the fields of the off-shell supergravity. We find that, when the auxiliary SU(2) gauge field strength vanishes, the most general supersymmetric configuration preserving four supercharges is labelled by n_v+1 real parameters corresponding to the excitations of the conformal mode of the graviton and the scalars of the n_v vector multiplets. In the general case, the localization manifold is labelled by an additional SU(2) triplet of one-forms and a scalar function.
Article
Euclidean gravity method has been successful in computing logarithmic corrections to extremal black hole entropy in terms of low energy data, and gives results in perfect agreement with the microscopic results in string theory. Motivated by this success we apply Euclidean gravity to compute logarithmic corrections to the entropy of various non-extremal black holes in different dimensions, taking special care of integration over the zero modes and keeping track of the ensemble in which the computation is done. These results provide strong constraint on any ultraviolet completion of the theory if the latter is able to give an independent computation of the entropy of non-extremal black holes from microscopic description. For Schwarzschild black holes in four space-time dimensions the macroscopic result seems to disagree with the existing result in loop quantum gravity.
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We compute the second Seely-DeWitt coefficient of the kinetic operator of the metric and gauge fields in Einstein-Maxwell theory in an arbitrary background field configuration. We then use this result to compute the logarithmic correction to the entropy of an extremal Kerr-Newmann black hole.
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The Bekenstein-Hawking area-entropy relation is derived for a class of five-dimensional extremal black holes in string theory by counting the degeneracy of BPS solition bound states.
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We present a microscopic index formula for the degeneracy of dyons in four-dimensional N = 4 string theory. This counting formula is manifestly symmetric under the duality group, and its asymptotic growth reproduces the macroscopic Bekenstein-Hawking entropy. We give a derivation of this result in terms of the type 11 five-brane compactified on K3, by assuming that its fluctuations are described by a closed string theory on its world-volume. We find that the degeneracies are given in terms of the denominator of a generalized super Kac-Moody algebra. We also discuss the correspondence of this result with the counting of D-brane states.
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We find general relations between the on-shell gravitational trace anomaly A_N, and the logarithmic correction Delta S_N to the entropy of "large" BPS extremal black holes in N>1 supergravity theories in D=4 space-time dimensions (recently computed by Sen [arXiv:1108.3842]). For (generalized) self-mirror theories (all having A_N = 0), we obtain the result DeltaS_N = - Delta S_(8-N) = 2 - N/2, whereas for generic theories the trace anomaly tildeA_N of the fully dualized theory turns out to coincide with 2Delta S_N, up to a model-independent shift: tildeA_N = 2Delta S_N - 1. We also speculate on N=1 theories displaying "large" extremal black hole solutions.
Article
We compute logarithmic corrections to the entropy of rotating extremal black holes using quantum entropy function i.e. Euclidean quantum gravity approach. Our analysis includes five dimensional supersymmetric BMPV black holes in type IIB string theory on T^5 and K3 x S^1 as well as in the five dimensional CHL models, and also non-supersymmetric extremal Kerr black hole and slowly rotating extremal Kerr-Newmann black holes in four dimensions. For BMPV black holes our results are in perfect agreement with the microscopic results derived from string theory. In particular we reproduce correctly the dependence of the logarithmic corrections on the number of U(1) gauge fields in the theory, and on the angular momentum carried by the black hole in different scaling limits. We also explain the shortcomings of the Cardy limit in explaining the logarithmic corrections in the limit in which the (super)gravity description of these black holes becomes a valid approximation. For non-supersymmetric extremal black holes, e.g. for the extremal Kerr black hole in four dimensions, our result provides a stringent testing ground for any microscopic explanation of the black hole entropy, e.g. Kerr/CFT correspondence.
Article
Single centered BPS black hole solutions exist only when the charge carried by the black hole has positive discriminant. On the other hand the exact dyon spectrum in heterotic string theory compactified on T 6 is known to contain states with negative discriminant. We show that all of these negative discriminant states can be accounted for as two centered black holes. Thus after the contribution to the index from the two centered black holes is subtracted from the total microscopic index, the index for states with negative discriminant vanishes even for finite values of charges, in agreement with the results from the black hole side. Bound state metamorphosis — which requires us to identify certain apparently different two centered configurations according to a specific set of rules — plays a crucial role in this analysis. We also generalize these results to a class of CHL string theories.
Article
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.
Article
Single centered supersymmetric black holes in four dimensions have spherically symmetric horizon and hence carry zero angular momentum. This leads to a specific sign of the helicity trace index associated with these black holes. Since the latter are given by the Fourier expansion coefficients of appropriate meromorphic modular forms of Sp(2,Z) or its subgroup, we are led to a specific prediction for the signs of a subset of these Fourier coefficients which represent contributions from single centered black holes only. We explicitly test these predictions for the modular forms which compute the index of quarter BPS black holes in heterotic string theory on T^6, as well as in Z_N CHL models for N=2,3,5,7.
Article
AdS 2/CFT 1 correspondence predicts that the logarithm of a ZN {\mathbb{Z}_N} twisted index over states carrying a fixed set of charges grows as 1/N times the entropy of the black hole carrying the same set of charges. In this paper we verify this explicitly by calculating the microscopic ZN {\mathbb{Z}_N} twisted index for a class of states in the CHL models. This demonstrates that black holes carry more information about the microstates than just the total degeneracy.
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In four dimensional string theories with N=4 \mathcal{N} = 4 and N=8 \mathcal{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 × 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 \mathcal{N} = 4 supersymmetric string theories. In particular the partition function is a modular form of a subgroup of Sp(2;Z) {\text{Sp}}\left( {2;\mathbb{Z}} \right) 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 ZN {\mathbb{Z}_N} orbifold of the attractor geometry produces the desired contribution.
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BMPV black holes in flat transverse space and in Taub-NUT space have identical near horizon geometries but different microscopic degeneracies. It has been proposed that this difference can be accounted for by different contribution to the degeneracies of these black holes from hair modes, -- degrees of freedom living outside the horizon. In this paper we explicitly construct the hair modes of these two black holes as finite bosonic and fermionic deformations of the black hole solution satisfying the full non-linear equations of motion of supergravity and preserving the supersymmetry of the original solutions. Special care is taken to ensure that these solutions do not have any curvature singularity at the future horizon when viewed as the full ten dimensional geometry. We show that after removing the contribution due to the hair degrees of freedom from the microscopic partition function, the partition functions of the two black holes agree. Comment: 40 pages, LaTeX
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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. Comment: LaTeX file, 27 pages; v2: minor corrections
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We consider a general, classical theory of gravity in n dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, ξa\xi^a, on spacetime one can associate a local symmetry and, hence, a Noether current (n1)(n-1)-form, j{\bf j}, and (for solutions to the field equations) a Noether charge (n2)(n-2)-form, Q{\bf Q}. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole m