We extend the discussion of the "Kerr/CFT correspondence" and its recent developments to the more general gauge/gravity correspondence in the full extremal black hole space-time of the bulk by using a technique of the holographic renormalization group (RG) flow. It is conjectured that the extremal black hole space-time is holographically dual to the chiral two dimensional field theory. Our example is a typical four dimensional Reissner-Nordstrom black hole, a system in which the M5-brane is wrapped on four cycles of Calabi-Yau threefold. In five dimensional supergravity view point this near horizon geometry is $AdS_3\times S^2$, and three dimensional gravity coupled to moduli fields is effectively obtained after a dimensional reduction on $S^2$. Constructing the Hamilton-Jacobi equation, we define the holographic RG flow from the three dimensional gravity. The central charge of the Virasoro algebra is calculable from the conformal anomaly at the point where the beta function defined from gravity side becomes zero. In general, we can also identify the c-function of the dual two dimensional field theory. We show that these flow equations are completely equivalent to not only BPS but also non-BPS attractor flow equations of the muduli fields. The attractor mechanism by which the values of the moduli fields are fixed at the event horizon of the extremal black hole can be understood equivalently to the fact that the RG flows are fixed at the critical points in the dual field theory.
We present a simple and yet rigorous derivation of the flow equations for the supersymmetric black-hole solutions of all 4-dimensional supergravities based on the recently found general form of all those solutions.
We examine in details the entropy function formalism for non-extremal D3, M2, and M5-branes that their throat approximation is given by Schwarzschild black hole in AdS_{p+2} \times S^{D-(p+2)}. We show that even though there is no attractor mechanism in the non-extremal black holes/branes, the entropy function formalism does work and the entropy is given by the entropy function at its saddle point.
We review the construction of multi-centered black hole solutions through dimensional reduction over time. This method does not rely on Killing spinor equations or gradient flow equations, but on solving the second order field equations in terms of harmonic functions. The black hole attractor equations are obtained directly from the field equations.
We report on recent advances in the study of critical points of the ``black hole effective potential'' V_{BH} (usually named \textit{attractors}) of N=2, d=4 supergravity coupled to n_{V} Abelian vector multiplets, in an asymptotically flat extremal black hole background described by 2n_{V}+2 dyonic charges and (complex) scalar fields which are coordinates of an n_{V}-dimensional Special Kahler manifold.
Motivated by the new higher D-supergravity solutions on intersecting attractors obtained by Ferrara et al. in [Phys.Rev.D79:065031-2009], we focus in this paper on 8D maximal supergravity with moduli space [SL(3,R)/SO(3)]x[SL(2,R)/SO(2)] and study explicitly the attractor mechanism for various configurations of extremal black p- branes (anti-branes) with the typical near horizon geometries AdS_{p+2}xS^{m}xT^{6-p-m} and p=0,1,2,3,4; 2<=m<=6. Interpretations in terms of wrapped M2 and M5 branes of the 11D M-theory on 3-torus are also given. Keywords: 8D supergravity, black p-branes, attractor me…
We present a method which allows to deform extremal black hole solutions into non-extremal solutions, for a large class of supersymmetric and non-supersymmetric Einstein-Vector-Scalar type theories. The deformation is shown to be largely independent of the details of the matter sector. While the line element is dressed with an additional harmonic function, the attractor equations for the scalars remain unmodified in suitable coordinates, and the values of the scalar fields on the outer and inner horizon are obtained from their fixed point values by making specific substitutions for the charges…
We establish an attractor mechanism for the horizon metric of asymptotically locally AdS4 supersymmetric black holes. The Kähler moduli of the metric are not fixed in the asymptotic region, however supersymmetry dictates that the near horizon metric must be the constant curvature one. We show how this mechanism is realized in detail for four-dimensional N=2 gauged supergravity coupled to vector multiplets by focusing on the STU model. A similar analysis is performed for gauged supergravity theories in five, six, and seven dimensions where we establish the same mechanism by extending previous r…
We provide a field theory interpretation of the attractor mechanism for asymptotically AdS$_4$ dyonic BPS black holes whose entropy is captured by the supersymmetric index of the twisted ABJM theory at Chern-Simons level one. We holographically compute the renormalized off-shell quantum effective action in the twisted ABJM theory as a function of the supersymmetric fermion masses and the arbitrary vacuum expectation values of the dimension one scalar bilinear operators and show that extremizing the effective action with respect to the vacuum expectation values of the dimension one scalar bilin…
Reconsidering the analysis of the moduli space of N=2 eight dimensional supergravity coupled to seven scalars, we propose a new scalar manifold factorization given by \frac{\textsc {SO(2,2)}}{\textsc{SO(2)}\times {\textsc{SO(2)}}}\times \frac{\textsc{SO(2,1)}}{\textsc{SO(2)}}\times \textsc {SO(1,1)}. This factorization is supported by the appearance of three solutions of Type IIA extremal black p-branes (p=0,1,2) with AdS_{p+2}\times S^{6-p} near-horizon geometries in eight dimensions. We analyze the corresponding attractor mechanism. In particular, we give an interplay between the scalar mani…
In this paper we elaborate on the relation between the entropy formula of Wald and the "entropy function" method proposed by A. Sen. For spherically symmetric extremal black holes, it is shown that the expression of extremal black hole entropy given by A. Sen can be derived from the general entropy definition of Wald, without help of the treatment of rescaling the AdS_2 part of near horizon geometry of extremal black holes. In our procedure, we only require that the surface gravity approaches to zero, and it is easy to understand the Legendre transformation of f, the integration of Lagrangian …
We have learnt that the Bekenstein-Hawking entropy of supersymmetric black holes in five-dimensional minimal supergravity can be reproduced from microstate counting via the AdS/CFT and Kerr/CFT correspondence. In this note, we demostrate the coincidence of both formalisms for extremal and near-extremal black holes in five-dimensional minimal supergravity. Connection to AdS_2 gravity is also mentioned.
These lectures provide a pedagogical, introductory review of the so-called Attractor Mechanism (AM) at work in two different 4-dimensional frameworks: extremal black holes in N=2 supergravity and N=1 flux compactifications. In the first case, AM determines the stabilization of scalars at the black hole event horizon purely in terms of the electric and magnetic charges, whereas in the second context the AM is responsible for the stabilization of the universal axion-dilaton and of the (complex structure) moduli purely in terms of the RR and NSNS fluxes. Two equivalent approaches to AM, namely th…
We study non-supersymmetric extremal black hole excitations of 4d N=2 supersymmetric string vacua arising from compactification on Calabi-Yau threefolds. The values of the (vector multiplet) moduli at the black hole horizon are governed by the attractor mechanism. This raises natural questions, such as "what is the distribution of attractor points on moduli space?" and "how many attractor black holes are there with horizon area up to a certain size?" We employ tools developed by Denef and Douglas to answer these questions.
We study certain properties of the low energy regime of a theory which resembles four dimensional YM theory in the framework of a non-critical holographic gravity dual. We use for the latter the near extremal $AdS_6$ non-critical SUGRA. We extract the glueball spectra that associates with the fluctuations of the dilaton, one form and the graviton and compare the results to those of the critical near extremal $D4$ model and lattice simulations. We show an area law behavior for the Wilson loop and screening for the 't Hooft loop. The Luscher term is found to be $-{3/24}\frac π{L}$. We derive the…
We perform a systematic search for static solutions in different sectors of 5d $N=8$ supergravities with compact and non-compact gauged R-symmetry groups, finding new and listing already known backgrounds. Due to the variety of possible gauge groups and resulting scalar potentials, the maximally symmetric vacua we encounter in these theories can be Minkowski, de Sitter, or anti-de Sitter. There exist BPS and non-BPS near-horizon geometries and full solutions with all these three types of asymptotics, corresponding to black holes, branes, strings, rings, and other black objects with more exotic…
We consider the $\mathrm{AdS}_3/\mathrm{CFT}_2$ description of Reissner-Nordstrøm black holes by studying their uplifted counterparts in five dimensions. Assuming a natural size of the extra dimension, the near horizon geometries for the extremal limit are exactly $\mathrm{AdS}_3 \times \mathrm{S}^2$. We compute the scattering amplitude of a scalar field, with a mode near threshold of frequency and extra dimensional momentum, by a near extremal uplifted black hole. The absorption cross section agrees with the two point function of the CFT dual to the scalar field.
We study flows of non-supersymmetric attractor black holes in the context of gauge/gravity correspondence. As our bulk theory, we consider the Einstein-Maxwell-Dilaton system with a single dilaton field coupled to two Maxwell fields and make a relevant deformation by adding a bare potential to the dilaton field. We find two types of extremal black hole solutions with attractor mechanism: The one smooth at the horizon and the other non-smooth. We show from both bulk and boundary theory perspective that the former is thermodynamically unstable, while the latter is stable.
We study extremal black hole solutions of D=5 Gauss-Bonnet gravity coupled to a system of gauge and scalar fields. As in Einstein gravity, we find that the values of the scalar fields on the horizon must extremize a certain effective potential that depends on the black hole charges. If the matrix of second derivatives of the effective potential at this extremum has positive eigenvalues, we give evidence, based on a near horizon perturbative expansion, that the attractor mechanism continues to hold in this general class of theories. We numerically construct solutions to a particular simple sing…
We argue that extremal black holes of N=4 Poincare supergravity coupled to conformal matter have a quantum-exact dual 5-d description in the maximally supersymmetric extension of the Randall-Sundrum theory. This dual is a the classical, static supergravity solution describing a string with both Neveu-Schwarz and Ramond charge with respect to different antisymmetric tensor doublets. We also discuss the issue of the singularity present in a class of such supergravity solutions found by Cvetic, Lu and Pope.
We study the entropy of non-supersymmetric extremal black holes which exhibit attractor mechanism by making use of the entropy function. This method, being simple, can be used to calculate corrections to the entropy due to higher order corrections to the action. In particular we apply this method for five dimensional non-supersymmetric extremal black hole which carries two magnetic charges and find the R^2 corrections to the entropy. Using the behavior of the action evaluated for the extremal black hole near the horizon, we also present a simple expression for C-function corrected by higher or…
We study the attractor mechanism for extremal non-BPS black holes with an infinite throat near horizon geometry, developing, as we do so, a physical argument as to why such a mechanism does not exist in non-extremal cases. We present a detailed derivation of the non-supersymmetric attractor equation. This equation defines the stabilization of moduli near the black hole horizon: the fixed moduli take values specified by electric and magnetic charges corresponding to the fluxes in a Calabi Yau compactification of string theory. They also define the so-called double-extremal solutions. In some ex…
Due to the attractor mechanism, the entropy of an extremal black hole does not vary continuously as we vary the asymptotic values of various moduli fields. Using this fact we argue that the entropy of an extremal black hole in string theory, calculated for a range of values of the asymptotic moduli for which the microscopic theory is strongly coupled, should match the statistical entropy of the same system calculated for a range of values of the asymptotic moduli for which the microscopic theory is weakly coupled. This argument does not rely on supersymmetry and applies equally well to nonsupe…
Using the low energy limit of type IIB superstring theory, we obtain the non-extremal limit of deformed conifold geometry which is dual to the IR limit of large N thermal QCD.At low temperatures, the extremal geometry without black hole is favored while at high temperatures, the field theory is described by non-extremal black hole geometry. We compute the ten dimensional on shell action for extremal and non-extremal geometries and demonstrate that at a critical temperature $T_c$ there is a first order confinement to deconfinement phase transition. We compute $T_c$ as a function of 'tHooft coup…
We analyze the fermionic wigging of 1/2-BPS (electric) extremal black hole attractors in N=2, D=5 ungauged Maxwell-Einstein supergravity theories, by exploiting anti-Killing spinors supersymmetry transformations. Regardless of the specific data of the real special geometry of the manifold defining the scalars of the vector multiplets, and differently from the D=4 case, we find that there are no corrections for the near--horizon attractor value of the scalar fields; an analogous result also holds for 1/2-BPS (magnetic) extremal black string. Thus, the attractor mechanism receives no fermionic c…
From rotating attractors to extremal black holes with axionic hair
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro（阿威罗大学数学与数学应用研究与发展中心（CIDMA）数学系）
We introduce a new attractor mechanism to find the entropy for spherically symmetric extremal black holes. The key ingredient is to find a two-dimensional (2D) dilaton gravity with the dilaton potential $V(ϕ)$. The condition of an attractor is given by $\nabla^2ϕ=V(ϕ_0)$ and $\bar{R}_2=-V^{\prime}(ϕ_0)$ and for a constant dilaton $ ϕ=ϕ_0$, these are also used to find the location of the degenerate horizon $r=r_{e}$ of an extremal black hole. As a nontrivial example, we consider an extremal regular black hole obtained from the coupled system of Einstein gravity and nonlinear electrodynamics. Th…
In this note, we explore holographic attributes of four-dimensional near-extremal Reissner-Nordstrom black hole solutions in ungauged ${\cal N}=2$ supergravity theories at the two-derivative level by recasting them as a specific first-order deformation in solution space, associated with an infinitesimal Harrison transformation, of black holes in an $AdS_2$ space-time. Specifically, we use this link to exhibit how bulk properties, such as mass and entropy, of four-dimensional near-extremal black holes are holographically encoded in the one-dimensional boundary theory dual to gravity in an infin…
We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N=2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N=2 MESGTs are the quasiconformal groups QConf(J) ass…
In this note, we use the first order formalism for attractor flows in N=2 SUGRA extremal black hole backgrounds to establish a formal correspondence between the RG flow of moduli in the underlying N=(2,2) SCFT and bulk attractor flows. Starting from a study of moduli flow trajectories in the CFT, we derive a potential which generates the aforesaid flow. This potential is shown to be a symplectic invariant with the same form as the black hole potential which drives the attractor flows in the bulk. We use these results to make comments on the non-renormalization of extremal black hole entropy an…
We report on recent results in the study of extremal black hole attractors in N=2, d=4 ungauged Maxwell-Einstein supergravities. For homogeneous symmetric scalar manifolds, the three general classes of attractor solutions with non-vanishing Bekenstein-Hawking entropy are discussed. They correspond to three (inequivalent) classes of orbits of the charge vector, which sits in the relevant symplectic representation R_{V} of the U-duality group. Other than the 1/2-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge. The complete classif…
We present a c-function for spherically symmetric, static and asymptotically flat solutions in theories of four-dimensional gravity coupled to gauge fields and moduli. The c-function is valid for both extremal and non-extremal black holes. It monotonically decreases from infinity and in the static region acquires its minimum value at the horizon, where it equals the entropy of the black hole. Higher dimensional cases, involving $p$-form gauge fields, and other generalisations are also discussed.
We examine the attractor mechanism for extremal black holes in the context of five dimensional N = 2 supergravity and show that attractor points are unique in the extended vector multiplet moduli space. Implications for black hole entropy are discussed.
In this paper we present a first order formulation for non-extremal Anti-de Sitter black hole solutions in four dimensional $\mathcal{N}=2$ U(1)-gauged Supergravity. The dynamics is determined in terms of a quantity $\mathcal{W}$ which plays the role of a superpotential for the gauging potential in the action. We show how the first order flow arises from writing the action as a sum of squares and we identify the superpotential driving the first order flow for two classes of solutions (electric and magnetic) of the $t^3$ model. After identifying $\mathcal{W}$, we study the Hamilton-Jacobi holog…
We substantiate a covariant proposal for the holographic reflected entropy in $CFT$s dual to non-static $AdS$ geometries from the bulk extremal entanglement wedge cross section in the literature with explicit computations in the $AdS_3/CFT_2$ scenario. In this context we obtain the reflected entropy for zero and finite temperature time dependent bipartite mixed states in $CFT_{1+1}$s with a conserved charge dual to bulk rotating extremal and non-extremal BTZ black holes through a replica technique. Our results match exactly with the corresponding extremal entanglement wedge cross section for t…
We show that strongly-coupled, translation-invariant holographic IR phases at finite density can be classified according to the scaling behaviour of the metric, the electric potential and the electric flux introducing four critical exponents, independently of the details of the setup. Solutions fall into two classes, depending on whether they break relativistic symmetry or not. The critical exponents determine key properties of these phases, like thermodynamic stability, the (ir)relevant deformations around them, the low-frequency scaling of the optical conductivity and the nature of the spect…
The attractor mechanism for the four-dimensional ${\cal N}=2$ supergravity black hole solution is analyzed in the case of the D0-D4 system. Our analyses are based on newly derived exact solutions, which exhibit explicitly the attractor mechanism for extremal non-BPS black holes. Our solutions account for the moduli as general complex fields, while in almost all non-BPS solutions obtained previously, the moduli fields are restricted to be purely imaginary. It is also pointed out that our moduli solutions contain an extra parameter that is not contained in solutions obtained by replacing the cha…
As a test for the non-supersymmetric attractor mechanism, we consider extremal Reissner-Nordström-(anti-)de Sitter black holes. Based on the simple observation that the near-horizon geometry of a generic extremal black hole contains two-dimensional anti-de Sitter factor even in the presence of the positive cosmological constant, we apply Ashoke Sen's entropy function method to compute the entropy of these black holes. We find the results which exactly agree with the Bekenstein-Hawking entropy. We also obtain the constant higher-order correction to the entropy due to the Gauss-Bonnet term.
We study holographic Wilsonian RG in a general class of asymptotically AdS backgrounds with a U(1) gauge field. We consider free charged Dirac fermions in such a background, and integrate them up to an intermediate radial distance, yielding an equivalent low energy dual field theory. The new ingredient, compared to scalars, involves a `generalized' basis of coherent states which labels a particular half of the fermion components as coordinates or momenta, depending on the choice of quantization (standard or alternative). We apply this technology to explicitly compute RG flows of charged fermio…
Non-extremal black holes are endowed with geometric invariants related to their horizon areas. We extend earlier work on hot attractor black holes to higher dimensions and add a scalar potential. In addition to the event and Cauchy horizons, when we complexify the radial coordinate, non-extremal black holes will generically have other horizons as well. We prove that the product of all of the horizon areas is independent of variations of the asymptotic moduli further generalizing the attractor mechanism for extremal black holes. In the presence of a scalar potential, as typically appears in gau…
We study nearly extreme black holes with nearly AdS$_2$ horizon geometry in various settings inspired by string theory. Our focus is on the scales of the nAdS$_2$ region and their relation to microscopic theory. These scales are determined by a generalization of the attractor mechanism for extremal black holes and realized geometrically as the normal derivatives along the extremal attractor flow. In some cases the scales are equivalently determined by the charge dependence of the extremal attractor by itself. Our examples include near extreme black holes in $D\geq 4$ dimensions, AdS boundary c…
We discuss some of the basic features of extremal black holes in four-dimensional extended supergravities. Firstly, all regular solutions display an attractor behavior for the scalar field evolution towards the black hole horizon. Secondly, they can be obtained by solving first order flow equations even when they are not supersymmetric, provided one identifies a suitable superpotential W which also gives the black hole entropy at the horizon and its ADM mass at spatial infinity. We focus on N=8 supergravity and we review the basic role played by U-duality of the underlying supergravity in dete…
Freudenthal duality (F-duality), an anti-involution of charge vectors keep the entropy and attractor solutions invariant for an extremal supersymmetric black hole. In this paper, we analyze the effect of F-duality on the entropy of a near-extremal $STU$ black hole in $\mathcal{N}=2$ ungauged, four-dimensional supergravity. We consider double-extremal black holes, whose attractor solutions are fixed in terms of the black hole charges throughout the moduli space. It is well known that JT gravity governs the dynamics of the near-horizon regions of higher dimensional, near-extremal black holes. Ow…
These lectures give an elementary introduction to the subject of four dimensional black holes (BHs) in supergravity and the Attractor Mechanism in the extremal case. Some thermodynamical properties are discussed and some relevant formulae for the critical points of the BH effective potential are given. The case of Maxwell-Einstein-axion-dilaton (super)gravity is discussed in detail. Analogies among BH entropy and multipartite entanglement of qubits in quantum information theory, as well moduli spaces of extremal BH attractors, are also discussed.
We study the application of our recent holographic entanglement negativity conjecture for mixed states of adjacent subsystems in conformal field theories with a conserved charge. In this context we obtain the holographic entanglement negativity for zero and finite temperature mixed state configurations in $d$-dimensional conformal field theories dual to bulk extremal and non extremal charged $AdS_{d+1}$ black holes. Our results conform to quantum information theory expectations and constitute significant consistency checks for our conjecture.
We present the main features of the physics of extremal black holes embedded in supersymmetric theories of gravitation, with a detailed analysis of the attractor mechanism for BPS and non-BPS black-hole solutions in four dimensions.
Non-extremal black holes in $\mathcal{N}=2$ supergravity have two horizons, the geometric mean of whose areas recovers the horizon area of the extremal black hole obtained from taking a smooth zero temperature limit. In prior work (arxiv:1410.3478), using the attractor mechanism, we deduced the existence of several moduli independent invariant quantities obtained from averaging over a decoupled inter-horizon region. We establish that non-extremal geometries at the Reissner--Nordström point, where the scalar moduli are held fixed, can be lifted to solutions in supergravity with a near-horizon A…
By making use of the entropy function formalism we study the generalized attractor equations in the four dimensional N=2 supergravity in presence of higher order corrections. This result might be used to understand a possible ensemble one could associate to an extremal black hole.
We propose that holographic entanglement entropy can be calculated at arbitrary orders in the bulk Planck constant using the concept of a "quantum extremal surface": a surface which extremizes the generalized entropy, i.e. the sum of area and bulk entanglement entropy. At leading order in bulk quantum corrections, our proposal agrees with the formula of Faulkner, Lewkowycz, and Maldacena, which was derived only at this order; beyond leading order corrections, the two conjectures diverge. Quantum extremal surfaces lie outside the causal domain of influence of the boundary region as well as its …
We use the relation between extremal black hole solutions in five- and in four-dimensional N=2 supergravity theories with cubic prepotentials to define the entropy function for extremal black holes with one angular momentum in five dimensions. We construct two types of solutions to the associated attractor equations.
We study attractor mechanism in extremal black holes of Einstein-Born-Infeld theories in four dimensions. We look for solutions which are regular near the horizon and show that they exist and enjoy the attractor behavior. The attractor point is determined by extremization of the effective potential at the horizon. This analysis includes the backreaction and supports the validity of non-supersymmetric attractors in the presence of higher derivative interactions in the gauge field part.
Motivated by the recently proposed Kerr/CFT correspondence, we investigate the holographic dual of the extremal and non-extremal rotating linear dilaton black hole in Einstein-Maxwell-Dilaton-Axion Gravity. For the case of extremal black hole, by imposing the appropriate boundary condition at spatial infinity of the near horizon extremal geometry, the Virasoro algebra of conserved charges associated with the asymptotic symmetry group is obtained. It is shown that the microscopic entropy of the dual conformal field given by Cardy formula exactly agrees with Bekenstein-Hawking entropy of extrema…
The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscop…
We propose defect extremal surface as the holographic counterpart of boundary quantum extremal surface. The defect extremal surface is defined by minimizing the Ryu-Takayanagi surface corrected by the defect theory. This is particularly interesting when the RT surface crosses or terminates on the defect. In a simple set up of AdS/BCFT, we find that the defect extremal surface formula gives precisely the same results of the boundary quantum extremal surface. We provide a decomposition procedure of an AdS bulk with a defect brane to see clearly how Island formula emerges from a brane world syste…
We consider near horizon geometries of extremal black holes in six-dimensional type IIB supergravity. In particular, we use the entropy function formalism to compute the charges and thermodynamic entropy of these solutions. We also comment on the role of attractor mechanism in understanding the entropy of the Hopf T-dual solutions in type IIA supergravity.
Extremal black branes are of interest because they correspond to the ground states of field theories at finite charge density in gauge/gravity duality. The geometry of such a brane need not be translationally invariant in the spatial directions along which it extends. A less restrictive requirement is that of homogeneity, which still allows points along the spatial directions to be related to each other by symmetries. In this paper, we find large new classes of homogeneous but anisotropic extremal black brane horizons, which could naturally arise in gauge/gravity dual pairs. In 4+1 dimensional…
