Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation $E\sim k^{z}$ and dynamical exponent $z>1$. The asymptotic growth of the number of states of these theories can be described by an extension of Cardy formula that depends on $z$. We show that this result can be recovered by counting the partitions of an integer into $z$-th powers, as proposed by Hardy and Ramanujan a century ago. This gives a novel relationship between the characteristic energy of the dispersion relation with the cylinder radius and the ground state energy. For free bosons with Lifshitz scaling, this relationship is shown to be identically fulfilled by virtue of the reflection property of the Riemann $ζ$-function. The quantum Benjamin-Ono$_{2}$ (BO$_{2}$) integrable system, relevant in the AGT correspondence, is also analyzed. As a holographic realization, we provide a special set of boundary conditions for which the reduced phase space of Einstein gravity with a couple of $U(1)$ fields on AdS$_3$ is described by the BO$_{2}$ equations. This suggests that the phase space can be quantized in terms of quantum BO$_{2}$ states. Indeed, in the semiclassical limit, the ground state energy of BO$_{2}$ coincides with the energy of global AdS$_{3}$, and the Bekenstein-Hawking entropy for BTZ black holes is recovered from the anisotropic extension of Cardy formula.
The Bekenstein-Hawking entropy of an extremal vacuum black ring is derived from a microscopic counting of states. The entropy of extremal Kaluza-Klein black holes with ergospheres is also derived.
The entropy of a black hole can be obtained by counting states in loop quantum gravity. The dominant term depends on the Immirzi parameter involved in the quantization and is proportional to the area of the horizon, while there is a logarithmic correction with coefficient -1/2.
We show that an extremal Kerr black hole, appropriately lifted to M-theory, can be transformed to a Kaluza-Klein black hole in M-theory, or a D0-D6 charged black hole in string theory. Since all the microstates of the latter have recently been identified, one can exactly reproduce the entropy of an extremal Kerr black hole. We also show that the topology of the event horizon is not well defined in M-theory.
We consider four-dimensional dyonic single-center BPS black holes in the $N=2$ STU model of Sen and Vafa. By working in a region of moduli space where the real part of two of the three complex scalars $S, T, U$ are taken to be large, we evaluate the quantum entropy function for these BPS black holes. In this regime, the subleading corrections point to a microstate counting formula partly based on a Siegel modular form of weight two. This is supplemented by another modular object that takes into account the dependence on $Y^0$, a complex scalar field belonging to one of the four off-shell vecto…
In a recent letter, Cadoni and Mignemi proposed a formulation for the statistical computation of the 2D black holes entropy. We present a criticism about their formulation.
We show that the arguments proposed by Park and Yee against our recent derivation of the statistical entropy of 2D black holes do not apply to the case under consideration
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/CFT_1 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.
We present a microscopical derivation of the entropy of the black hole solutions of the Jackiw-Teitelboim theory. We show that the asymptotic symmetry of two-dimensional (2D) Anti-de Sitter space is generated by a central extension of the Virasoro algebra. Using a canonical realization of this symmetry and Cardy's formula we calculate the statistical entropy of 2D black holes, which turns out to agree, up to a factor $\sqrt 2$, with the thermodynamical result.
We discuss a previous attempt at a microscopic counting of the entropy of asymptotically flat non-extremal black-holes. This method used string dualities to relate 4 and 5 dimensional black holes to the BTZ black hole. We show how the dualities can be justified in a certain limit, equivalent to a near horizon limit, but the resulting spacetime is no longer asymptotically flat.
The exact entropy of two-charge supersymmetric black holes in N=4 string theories is computed to all orders using Wald's formula and the supersymmetric attractor equations with an effective action that includes the relevant higher curvature terms. Classically, these black holes have zero area but the attractor equations are still applicable at the quantum level. The quantum corrected macroscopic entropy agrees precisely with the microscopic counting for an infinite tower of fundamental string states to all orders in an asymptotic expansion.
We quantize the D1-D5-P microstate geometries known as superstrata directly in supergravity. We use Rychkov's consistency condition [hep-th/0512053] which was derived for the D1-D5 system; for superstrata, this condition turns out to be strong enough to fix the symplectic form uniquely. For the $(1,0,n)$ superstrata, we further confirm this quantization by a bona-fide explicit computation of the symplectic form using the semi-classical covariant quantization method in supergravity. We use the resulting quantizations to count the known supergravity superstrata states, finding agreement with pre…
We quantize the moduli space of regular D1-D5 microstates, directly from Type IIB SUGRA. The moduli space is parametrized by a smooth closed non-selfintersecting curve in four dimensions, and we derive that the components of the curve satisfy chiral boson commutation relations, with the correct value of the effective Planck constant previously conjectured using U-duality. We use the Crnkovic-Witten-Zuckerman covariant quantization method, previously used to quantize the `bubbling AdS' geometries, combined with a certain new `consistency condition' which allows us to reduce the computation to q…
We compute the topologically twisted index for general $\mathcal{N} = 2$ supersymmetric field theories on $\mathbb{H}_2\times S^1$. We also discuss asymptotically $AdS_4$ magnetically charged black holes with hyperbolic horizon, in four-dimensional $\mathcal{N} = 2$ gauged supergravity. With certain assumptions, put forward by Benini, Hristov and Zaffaroni, we find precise agreement between the black hole entropy and the topologically twisted index, for $ABJM$ theories.
We examine certain two-charge supersymmetric states with spin in five-dimensional string theories which can be viewed as small black rings when the gravitational coupling is large. Using the 4D-5D connection, these small black rings correspond to four-dimensional non-spinning small black holes. Using this correspondence, we compute the degeneracy of the microstates of the small black rings exactly and show that it is in precise agreement with the macroscopic degeneracy to all orders in an asymptotic expansion. Furthermore, we analyze the five-dimensional small black ring geometry and show qual…
Many three-dimensional $\mathcal{N}=2$ SCFTs admit a universal partial topological twist when placed on hyperbolic Riemann surfaces. We exploit this fact to derive a universal formula which relates the planar limit of the topologically twisted index of these SCFTs and their three-sphere partition function. We then utilize this to account for the entropy of a large class of supersymmetric asymptotically AdS$_4$ magnetically charged black holes in M-theory and massive type IIA string theory. In this context we also discuss novel AdS$_2$ solutions of eleven-dimensional supergravity which describe…
We derive the Bekenstein-Hawking entropy for a class of BPS black holes in the massive type IIA supergravity background AdS$_4 \times S^6$ from a microscopic counting of supersymmetric ground states in a holographically dual field theory. The counting is performed by evaluating the topologically twisted index of three-dimensional $\mathcal{N}=2$ Chern-Simons-matter gauge theories in the large $N$ limit. The $\mathcal{I}$-extremization principle is shown to match the attractor mechanism for the near-horizon geometries constructed in the four-dimensional dyonic $\mathcal{N}=2$ gauged supergravit…
The topologically twisted index for ABJM theory with gauge group $U(N)_k \times U(N)_{-k}$ has recently been shown, in the large-$N$ limit, to reproduce the Bekenstein-Hawking entropy of certain magnetically charged asymptotically AdS$_4$ black holes. We numerically study the index beyond the large-$N$ limit and provide evidence that it contains a subleading logarithmic term of the form $-1/2\log N$. On the holographic side, this term naturally arises from a one-loop computation. However, we find that the contribution coming from the near horizon states does not reproduce the field theory answ…
We use recently established AdS$_6$/CFT$_5$ dualities to count the microstates of magnetically charged AdS$_6 \times S^2 \times Σ$ black holes in Type IIB. The near-horizon limit is described by solutions with AdS$_2 \times Σ_{\mathfrak{g}_1} \times Σ_{\mathfrak{g}_2} \times S^2 \times Σ$ geometry, where $Σ_{\mathfrak{g}_i}$ are Riemann surfaces of constant curvature and $Σ$ is a further Riemann surface over which the geometry is warped. Our results show that the topologically twisted indices of the proposed dual superconformal field theories precisely reproduce the Bekenstein-Hawking entropy …
We analyse the power counting renormalizability of scalar theory in Lifshitz spacetime in D+2 dimensions. We show the spectral dimension becomes 2+(D/z) (z is the critical exponent) after integrating out the radion field. We comment on the AdS/CFT correspondence, and on how to avoid the Lifshitz singularity by flowing into AdS spacetime in the infrared. We also comment on the quantum gravity in Lifshitz spacetime.
We study the renormalizability in theories of a self-interacting Lifshitz scalar field. We show that although the statement of power-counting is true at one-loop order, in generic cases where the scalar field is dimensionless, an infinite number of counter terms are involved in the renormalization procedure. This problem can be avoided by imposing symmetries, the shift symmetry in the present paper, which allow only a finite number of counter terms to appear. The symmetry requirements might have important implications for the construction of matter field sectors in the Horava-Lifshitz gravity.
Reproducing the integer count of black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In this paper, we show that, by using supersymmetric localization, the gravitational path integral for $\frac{1}8$-BPS black holes in $\mathcal{N}=8$ supergravity reproduces the index obtained in the string theory construction of such black holes, including all non-perturbatively suppressed geometries. A more refined argument then shows that, not only the black hole index, but also the total number of black hole microstates within an energy window above extr…
In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N=4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centered black holes as well.
In hep-th/0412322, Gaiotto, Strominger and Yin proposed a novel way of counting black hole microstates by counting the quantum mechanical ground states of probe branes placed in the near-horizon black hole background. We discuss the generalization of this proposal to the case of two-charge D0-D4 `small' black holes in type IIA. We also describe the construction of BPS D-brane probes in the near-horizon region of the 2-charge D1-D5 system in type IIB. Based on hep-th/0505176, hep-th/0607150.
We provide a universal microscopic counting for the microstates of the asymptotically AdS black holes and black strings that arise as solutions of the half-maximal gauged supergravity in four and five dimensions. These solutions can be embedded in all M-theory and type II string backgrounds with an AdS vacuum and sixteen supercharges and provide an infinite set of examples dual to $\mathcal{N}=2$ and $\mathcal{N}=4$ conformal field theories in four and three dimensions, respectively. The counting is universal and it is performed by either studying the large $N$ limit of the relevant supersymme…
We numerically study the topologically twisted index of several three-dimensional supersymmetric field theories on a genus $g$ Riemann surface times a circle, $Σ_g\times S^1$. We show that for a large class of theories with leading term of the order $N^{3/2}$, where $N$ is generically the rank of the gauge group, there is a universal logarithmic correction of the form $\frac{g-1}{2} \log N$. We explain how this logarithmic subleading correction can be obtained as a one-loop effect on the dual supergravity theory for magnetically charged, asymptotically AdS$_4\times M^7$ black holes for a large…
Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Department of Theoretical Physics, University of Geneva（日内瓦大学理论物理系）；Laboratory for Theoretical Fundamental Physics, Institute of Physics, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院物理研究所基础物理实验室）
We derive the leading g_s perturbation of the SUGRA fields generated by a supersymmetric configuration of respectively 1, 2 or 4 D3-branes intersecting at an arbitrary angle via the computation of the string theory disk scattering amplitude of one massless NSNS field interacting with open strings stretched between the branes. The configuration with four branes is expected to be relevant for black hole microstate counting in four dimensions.
We study the superconfomal index of four-dimensional toric quiver gauge theories using a Bethe Ansatz approach recently applied by Benini and Milan. Relying on a particular set of solutions to the corresponding Bethe Ansatz equations we evaluate the superconformal index in the large $N$ limit, thus avoiding to take any Cardy-like limit. We present explicit results for theories arising as a stack of $N$ D3 branes at the tip of toric Calabi-Yau cones: the conifold theory, the suspended pinch point gauge theory, the first del Pezzo theory and $Y^{p,q}$ quiver gauge theories. For a suitable choice…
We study the topologically twisted index of a certain Chern-Simons matter theory with $SU(N)$ level $k$ gauge group on a genus $g$ Riemann surface times a circle. For this theory it is known that the logarithm of the topologically twisted index grows as $N^{5/3}$ and that it matches the Bekenstein-Hawking entropy of certain magnetically charged asymptotically $AdS_4\times S^6$ black holes in massive type IIA supergravity. Through a combination of numerical and analytical techniques we study the subleading in $N$ structure. We demonstrate precise analytic cancellation of terms of orders $N\log\…
We give a geometric interpretation of the entropy of the supertubes with fixed conserved charges and angular momenta in two different approaches using the DBI action and the supermembrane theory. By counting the geometrically allowed microstates, it is shown that both the methods give consistent result on the entropy. In doing so, we make the connection to the gravity microstates clear.
We analyse the power counting renormalizability of the quantum field theory of Einstein or Einstein-Gauss-Bonnet gravity in D+2 dimensional Lifshitz spacetime. We show the spectral dimension becomes 2+(D/z) at the UV region where z is the critical exponent. Since it is larger than two, the quantum theory of Einstein gravity is not power counting renormalizable. For the pure Einstein-Gauss-Bonnet gravity, where Lifshitz spacetime is allowed only when the parameters are fine tuned, it happens that the graviton modes do not propagate and the quantum field theory is accidentally renormalizable whe…
Loop Gravity provides a microscopic derivation of Black Hole entropy. In this paper, I show that the microstates counted admit a semiclassical description in terms of shapes of a tessellated horizon. The counting of microstates and the computation of the entropy can be done via a mapping to an equivalent statistical mechanical problem: the counting of conformations of a closed polymer chain. This correspondence suggests a number of intriguing relations between the thermodynamics of Black Holes and the physics of polymers.
We present a necessary and sufficient condition to falsify whether a Hawking radiation spectrum indicates unitary emission process or not from the perspective of information theory. With this condition, we show the precise values of Bekenstein-Hawking entropies for Schwarzschild black holes and Reissner-Nordström black holes can be calculated by counting the microstates of their Hawking radiations. In particular, for the extremal Reissner-Nordström black hole, its number of microstate and the corresponding entropy we obtain are found to be consistent with the string theory results. Our finding…
We study the large $N$ expansion of twisted partition functions of 3d $\mathcal{N}=2$ superconformal field theories arising from $N$ M5-branes wrapped on a hyperbolic 3-manifold, $M_3$. Via the 3d-3d correspondence, the partition functions of these 3d ${\cal N}=2$ superconformal field theories are related to simple topological invariants on the 3-manifold. The partition functions can be expressed using only classical and one-loop perturbative invariants of $PSL(N,\mathbb{C})$ Chern-Simons theory around irreducible flat connections on $M_3$. Using mathematical results on the asymptotics of the …
This paper addresses a long standing problem, the counting of the microstates of supersymmetric asymptotically AdS black holes in terms of a holographically dual field theory. We focus on a class of asymptotically AdS$_4$ static black holes preserving two real supercharges which are dual to a topologically twisted deformation of the ABJM theory. We evaluate in the large $N$ limit the topologically twisted index of the ABJM theory and we show that it correctly reproduces the entropy of the AdS$_4$ black holes. An extremization of the index with respect to a set of chemical potentials is require…
In this note I derive the flat space limit of the modified Cardy formula associated with inner horizons and show that it reproduces the correct Galilean conformal field theory counting of flat space cosmology microstates. l also determine the entropy of flat space cosmologies in flat space chiral gravity in this way. In addition, I derive a Cardy-like expression for flat space cosmologies with spin-3 charges and thus give a prediction for the corresponding Galilean conformal field theory counting of flat space cosmology microstates with spin-3 charges.
In [1] the horizon fluffs proposal is put forward to identify microstates of generic non-extremal three-dimensional Banados-Teitelboim-Zanelli (BTZ) black holes. The proposal is that black hole microstates, which were dubbed as horizon fluffs, are states labelled by the conserved charges associated with non-trivial diffeomorphisms on the near horizon geometry and these states are not in the coadjoint orbits of the asymptotic Virasoro algebra at the Brown-Henneaux central charge, associated with the BTZ black holes. It is also known that AdS3 Einstein gravity has more general black hole solutio…
The Bekenstein-Hawking entropy of certain black holes can be computed microscopically in string theory by mapping the elusive problem of counting microstates of a strongly gravitating black hole to the tractable problem of counting microstates of a weakly coupled D-brane system, which has no event horizon, and indeed comfortably fits on the head of a pin. We show here that, contrary to widely held beliefs, the entropy of spherically symmetric black holes can easily be dwarfed by that of stationary multi-black-hole ``molecules'' of the same total charge and energy. Thus, the corresponding pin-s…
We present a one-function family of solutions to 4D vacuum Einstein equations. While all diffeomorphic to the same extremal Kerr black hole, they are labeled by well-defined conserved charges and are hence distinct geometries. We show that this family of solutions forms a phase space the symplectic structure of which is invariant under a $U(1)$ Kac-Moody algebra generated by currents $\mathbb{J}_n$ and Virasoro generators $\mathbb{L}_n$ with central charge six times angular momentum of the black hole. This symmetry algebra is well-defined everywhere in the spacetime, near the horizon or in the…
We analyze the structure of heavy multitrace BPS operators in N = 1 superconformal quiver gauge theories that arise on the worldvolume of D3-branes on an affine toric cone. We exhibit a geometric procedure for counting heavy mesonic operators with given U(1) charges. We show that for any fixed linear combination of the U(1) charges, the entropy is maximized when the charges are in certain ratios. This selects preferred directions in the charge space that can be determined with the help of a piece of string. We show that almost all heavy mesonic operators of fixed U(1) charges share a universal…
We consider general black holes in D=5, N=2 supergravity coupled to vector multiplets, and discuss the issue of microstate counting from various viewpoints. The statistical entropy is computed for the near-extremal case using the central charge of the $AdS_2$ factor appearing in the near-horizon geometry. Furthermore, we explicitly construct the duality transformation connecting electrically charged black holes to magnetically charged black strings, under which the $AdS_2 \times S^3$ near horizon geometry becomes $AdS_3 \times S^2$. For $AdS_3$ the counting of microstates correctly reproduces …
The Bekenstein-Hawking entropy for certain BPS-saturated black holes in string theory has recently been derived by counting internal black hole microstates at weak coupling. We argue that the black hole microstate can be measured by interference experiments even in the strong coupling region where there is clearly an event horizon. Extracting information which is naively behind the event horizon is possible due to the existence of statistical quantum hair carried by the black hole. This quantum hair arises from the arbitrarily large number of discrete gauge symmetries present in string theory.
At the present work, it is studied the extension of F (R) gravities to the new recently proposed theory of gravity, the so-called Horava-Lifshitz gravity, which provides a way to make the theory power counting renormalizable by breaking Lorentz invariance. It is showed that dark energy can be well explained in the frame of this extension, just in terms of gravity. It is also explored the possibility to unify inflation and late-time acceleration under the same mechanism, providing a natural explanation the accelerated expansion.
We study unitarity and renormalizability in the Lifshitz scalar field theory, which is characterized by an anisotropic scaling between the space and time directions. Without the Lorentz symmetry, both the unitarity and the renormalizability conditions are modified from those in relativistic theories. We show that for renormalizability, an extended version of the power counting condition is required in addition to the conventional one. The unitarity bound for S-matrix elements also gives stronger constraints on interaction terms because of the reference frame dependence of scattering amplitudes…
We study the renormalization of an N = 1 supersymmetric Lifshitz sigma model in three dimensions. The sigma model exhibits worldvolume anisotropy in space and time around the high-energy z = 2 Lifshitz point, such that the worldvolume is endowed with a foliation structure along a preferred time direction. In curved backgrounds, the target-space geometry is equipped with two distinct metrics, and the interacting sigma model is power-counting renormalizable. At low energies, the theory naturally flows toward the relativistic sigma model where Lorentz symmetry emerges. In the superspace formalism…
We consider the relationship between renormalizability and unitarity at a Lifshitz point in d dimensions. We test tree unitarity for theories containing only scalars and fermions, and for pure gauge theory. In both cases, we find the requirement of weighted power-counting renormalizability is equivalent to that of tree unitarity.
