Coarse-Graining the Lin-Maldacena Geometries
aa r X i v : . [ h e p - t h ] A ug arXiv:0705.4308 Coarse-Graining theLin-Maldacena Geometries
Hsien-Hang Shieh, Greg van Anders, and Mark Van RaamsdonkDepartment of Physics and Astronomy, University of British Columbia6224 Agricultural Road, Vancouver, B.C., V6T 1Z1, Canada
Abstract
The Lin-Maldacena geometries are nonsingular gravity duals to degeneratevacuum states of a family of field theories with SU (2 |
4) supersymmetry. In thisnote, we show that at large N , where the number of vacuum states is large,there is a natural ‘macroscopic’ description of typical states, giving rise to a setof coarse-grained geometries. For a given coarse-grained state, we can associatean entropy related to the number of underlying microstates. We find a simpleformula for this entropy in terms of the data that specify the geometry. Wesee that this entropy function is zero for the original microstate geometries andmaximized for a certain “typical state” geometry, which we argue is the gravitydual to the zero-temperature limit of the thermal state of the corresponding fieldtheory. Finally, we note that the coarse-grained geometries are singular if andonly if the entropy function is non-zero. Introduction
Recently, several fascinating new examples of gauge-theory / gravity duality haveemerged [1] for which the field theory has a discrete highly degenerate basis of vacuumstates yet we have an explicit non-singular geometry corresponding to each element ofthe basis.The field theories include the Plane-Wave Matrix Model (a one-parameter defor-mation of the low-energy theory of D0-branes [3]), a maximally supersymmetric 2+1dimensional gauge theory on S [4], N = 4 SUSY Yang-Mills theory on S /Z k , andtype IIA Little String Theory on S [1, 4, 6]. Each of these theories has SU (2 |
4) su-persymmetry, which may be used to argue that the numerous classical vacuum states(reviewed in section 2) remain degenerate in the quantum theory, and in particular,must be present at strong coupling. In [1] (following [8]), Lin and Maldacena searchedfor supergravity solutions with the same SU (2 |
4) symmetry, and found nonsingularsolutions in one-to-one correspondence with each element of a natural basis of vacuumstates for each of the field theories. In the following discussion and sections 2 to 5of this paper, we focus on the example of the Plane-Wave Matrix Model (reviewed insection 2), but we discuss the other theories in detail in section 6.While the geometries corresponding to basis vacuum states in each case are the sameasymptotically, they differ even in their topology in the infrared. Since the genericvacuum state in the field theory is a linear superposition of basis elements, such astate cannot be dual to a single non-singular supergravity solution with fixed topology(assuming there are observables that can detect topology), but must simply be dual toa quantum superposition of the topologically different geometries. Similarly, genericmixed states in the field theory, such as the zero-temperature limit of the thermal state,involve microstates corresponding to many different topologies so we might expect thata gravitational dual description in terms of a single geometry is impossible. On theother hand, there are many examples of geometries believed to be dual to thermal statesof field theories, and these thermal states involve enormous numbers of microstates thatcan be very different in the infrared. Mathur and collaborators have advocated (see [10]for a review) that we should interpret the thermal state geometry as a coarse-graineddescription of the underlying microstates, just as the homogeneous configuration thatwe use to describe the thermal state of a gas in a box is a coarse-graining of the truemicrostates of the atoms. Specifically, the macroscopic description of any almost anystate in the underlying ensemble of microstates is extremely close to one particularcoarse-grained configuration, the thermal equilibrium state. We will see that in ourcase also, there is a natural way to coarse-grain (i.e. give a macroscopic descriptionof) geometries corresponding to typical microstates, and that most of the microstateshave a coarse-grained description that is very close to a particular geometry, whichwe propose is the correct dual to the zero temperature limit of the thermal state. The construction is completely analogous to the construction of gravity duals to half BPS statesof N=4 SUSY Yang-Mills theory [8]. As in that case, the smooth supergravity solutions correspond-ing particular states can have large curvatures, and thus are only approximations to the true dualgeometries which should minimize the α ′ -corrected low-energy effective action.
1n this geometry, the complicated topological features that distinguish the individualmicrostate geometries are replaced by a singularity. The details of the coarse graining procedure are described in section 3 below, butwe give the essential idea here. The supergravity fields in the Lin-Maldacena geome-tries are determined in terms of the potential for an axially symmetric electrostaticsproblem involving a certain number of parallel coaxial charged conducting disks ina background electric field. The number, locations and charges of the disks are de-termined by the data specifying the field theory vacuum. We will find that typicalfield theory vacua correspond to electrostatics configurations with a large number ofclosely spaced disks whose radii are very small compared with the separation betweenthe disks. At large N , such a configuration has a natural coarse-grained descriptionas a smoothly varying charge distribution on the axis. Inserting the potential arisingfrom this coarse-grained configuration into the Lin-Maldacena supergravity solution,one finds a singular geometry. Since all of the nontrivial topological features are as-sociated with the regions between the disks in the electrostatics configurations (theseregions map to topologically non-trivial throats in the supergravity solutions), we seethat the complicated topologies that characterize individual microstates are replacedby a singularity in the coarse-grained description. A completely analogous coarse-graining has been discussed [5, 14, 15, 16] for thehalf-BPS sector of N = 4 SUSY Yang-Mills theory. There, the microstate geome-tries are the type IIB LLM geometries [8], constructed in terms of droplets of a two-dimensional incompressible fluid, and the coarse-grained description allows for configu-rations with arbitrary density of the fluid between zero and the maximal density. Onesignificant difference is that all of the states we consider are ground states for the fieldtheory, whereas the LLM discussion relates to a special class of excited states withenergy equal to an R-charge.As emphasized in [5], a given coarse-grained configuration provides an approxima-tion to a very large number of microstates, just as in the thermodynamic description ofordinary physical systems. Further, there is one preferred coarse-grained configuration,analogous to the thermal equilibrium state, which is very close to the coarse-grained de-scription of almost any randomly chosen microstate. For the type IIB LLM geometries,the geometry corresponding to this preferred state was determined in [5] and dubbedthe “hyperstar” geometry. In section 3 of this paper, we determine the correspondinggeometry for the Plane-Wave Matrix Model. In our case, the ensemble of microstateswe consider is just the set of vacuum states, or alternately the set of states that con-tribute (each with equal weight) to the T → T → For a general discussion of conditions under which field theory states can be associated withsemiclassical geometries, see [27] in the LLM context and [28] in the D1-D5 context. The radii of the disks are determined by the other information via a constraint. We should note however, that for the case of closely spaced disks, the supergravity approximationis not valid for the region between the disks, so the classical topological features that we are discussingshould be understood to be replaced by some stringy analogue.
2o the preferred states in other restricted ensembles, analogous to the type IIB super-star [11], and discuss thermal geometries for the remaining SU (2 |
4) symmetric theoriesin section 6.As for an ordinary thermodynamic system, the thermal states we derive shouldmaximize an entropy functional that measures the number of microstates nearby anarbitrary coarse-grained configuration. In section 4, we derive such an entropy func-tional, and find that it may be written simply in terms of the data that specify thegeometry. We find that this functional is indeed maximized by the thermal state ge-ometry of section 4. Further, we note that for all the coarse-grained configurations,those for which the entropy functional vanishes are the ones that coincide the originalnon-singular microstate geometries. On the other hand, configurations with non-zeroentropy are necessarily singular.In the general proposal by Mathur and collaborators, black hole geometries withhorizons are to be understood as coarse-grained descriptions of underlying horizon-freemicrostate geometries. In the present setup, the coarse graining leads to geometrieswith naked singularities uncloaked by horizons, but this is to be expected since thenumber of microstates in our case is not large enough to give a classical finite-areahorizon in the supergravity limit. It may be that a horizon develops as we move fromthe supergravity approximation to solutions minimizing the full low-energy effectiveaction, but, as we will see, realizing this would necessarily involve understanding both α ′ and string loop corrections. SU (2 | symmetric matrix quantum mechan-ics and the dual Lin-Maldacena Geometries In this section, we review the Plane-Wave matrix model, its vacua, and the dual geome-tries constructed by Lin and Maldacena. The other SU (2 |
4) symmetric field theoriesare discussed in section 6. We will see that each of these theories has a large degener-acy of vacuum states at the classical level. This degeneracy remains at the quantumlevel, since the representation theory of SU (2 |
4) does not allow for states with arbi-trarily small non-zero energies, and therefore does not allow the zero-energy states inthe classical limit of the theory to receive corrections to their energy [17, 18].
The Plane-Wave Matrix Model [3] is a massive deformation of the supersymmetricmatrix quantum mechanics describing decoupled low-energy D0-branes in flat space. It is described by a dimensionless Hamiltonian H = Tr (cid:18) P A + 12 ( X i / + 12 ( X a / + i ⊤ γ Ψ This is similar to the Polchinski-Strassler deformation of N = 4 SUSY Yang-Mills theory [2], butin this case, we preserve all 32 supersymmetries. i gǫ ijk X i X j X k − g ⊤ γ A [ X A , Ψ] − g X A , X B ] (cid:19) , (1)where A = 1 , . . . , i = 1 , . . . ,
3, and a = 4 , . . . ,
9. Here, the scalars X A and 16-component fermions Ψ are hermitian N × N matrices, and P A is the matrix of canon-ically conjugate momenta. Apart from N , the size of the matrices, the theory has onedimensionless parameter g , such that the theory is weakly coupled for small enough g . For this theory, the classical vacua, each with zero energy, are described by X a = 0 a = 4 , . . . , X i = 13 g J i i = 1 , , , where J i give any reducible representation of the SU (2) algebra. These vacua arein one-to-one correspondence with partitions of N , since we may have in general n k copies of the k -dimensional irreducible representation such that P k kn k = N . Below,it will be convenient to represent such a partition by a Young diagram with N boxes,containing n k columns of length k .In the D0-brane picture, a block-diagonal configuration with n k copies of the k -dimensional irreducible representation is associated classically with concentric D2-brane fuzzy spheres, with n k spheres at radius proportional to k . On the other hand,it was argued in [4] that at sufficiently strong coupling, such a configuration is betterdescribed as a collection of concentric fivebranes, with multiplicities and radii given interms of the numbers and lengths of columns in the dual Young diagram. For generalvalues of parameters, we can interpret the solution as a fuzzy configuration with bothD2-brane and NS5-brane characteristics. This will be apparent from the dual gravi-tational solutions, which include throats carrying D2-brane flux and throats carryingNS5-brane flux in the infrared part of the geometry.
The vacua of the matrix model each preserve SU (2 |
4) symmetry. In [1], Lin and Malda-cena searched for type IIA supergravity solutions preserving the same SU (2 |
4) symme-try (more precisely, with isometries given by the bosonic subgroup SO (6) × SO (3) × U (1)of SU (2 | V ( r, z )) feeds The model was introduced originally as a matrix model for M-theory on the maximally super-symmetric eleven-dimensional plane-wave. For this we are required to take a limit N → ∞ with g N ∼ N . In the present work, we will mainly be concerned with the usual ’t Hooft large N limitwith λ fixed. In [4], the matrix model was discussed in the context of its conjectured description of M-theoryon a plane-wave background. There, the fivebranes were M5-branes, while here we are considering alimit with fixed λ , dual to a IIA background, so the fivebranes are NS5 branes. SU (2 |
4) symmetric theories described in section 6, the constructiondiffers only by a choice of boundary conditions (background potential or the pres-ence/absence of infinite-sized conducting plates). The solution to these electrostaticsproblems has been discussed in [20].We now describe the electrostatics problem in detail and then review some generalfeatures of the dual supergravity solutions. Common to all vacua, we have in theelectrostatics problem an infinite conducting plate at z = 0 (on which we may assumethat the potential vanishes), and a background potential V ∞ = V ( r z − z ) . (2)In addition, corresponding to a matrix model vacuum with Q i copies of the d i -dimensionalirreducible representation, we have conducting disks with charge Q i parallel to the in-finite plate and centred at r = 0, z = d i . In order that the supergravity solution isnon-singular, the radii R i of the disks must be chosen so that the charge density at theedge vanishes.The parameters of the matrix model are related to the parameters in the electro-statics problem as N = P Q i d i and g ∝ /V . The coordinates r and z in the electrostatics problem form two of the nine spatialcoordinates in the geometry. In addition, for each value of r and z , we have an S andan S with radii that depend on ( r, z ). The S shrinks to zero size on the r = 0 axis,while the S shrinks to zero size at the locations of the conducting plates, so we havevarious non-contractible S s and S s corresponding to paths that terminate on differentplates or on different segments of the vertical axis respectively. This is illustrated infigure 1. As shown in [1], through an S corresponding to a path surrounding plateswith a total charge of Q , we have N = Q units of flux from the dual of the Ramond-Ramond four-form, suggesting the presence of N D2-branes. Similarly, through an S corresponding to a path between plates separated by a distance d , we have N = d unitsof H-flux, suggesting that this part of the geometry between the plates is describingthe degrees of freedom of N NS5-branes.Since the matrix model is a massive deformation of the maximally supersymmetricquantum mechanics describing low-energy D0-branes in flat-space, we should expectthat the dual supergravity solutions correspond to infrared modifications of the near-horizon D0-brane geometry. Indeed, the solutions are asymptotically the same as thenear-horizon D0-brane solution, with the strong-coupling region in the infrared replacedby smooth topological features that depend on the choice of vacuum. Our conventions here are slightly different from the ones in [1], as we describe in appendix A. Σ Σ rz X =
GravityElectrostaticsGauge Theory Σ Σ k ... d J k ... d J ... d J J QkQk d d k R RkQ1Q d
Figure 1: Mapping between matrix model vacua, electrostatics configurations, andgeometries. For illustrative purposes, we have replaced the S × S s associated toeach point ( r, z ) with S × S . In the full geometry, the dotted segment maps toa submanifold Σ that is topologically S × S (simply connected) rather than the S × S shown here. Similarly, the dashed segment maps to a submanifold Σ that istopologically S × S rather than the S × S here.6 Coarse-Graining the Lin-Maldacena Geometries
For large N , the plane-wave matrix model has of order exp( √ N /π ) independent vacualabelled by reducible dimension N representations of SU (2). In this section, we willargue that as for standard thermodynamic systems (e.g. particles in a box), if weuse coarse-grained, macroscopic variables to describe the states, then despite the largenumber of possible microscopic states, the description of a randomly chosen microstatewill, with very high probability, be extremely close to the average or “thermal equilib-rium” state. We will see explicitly what the coarse-grained description of this averagestate is in our case, and see that there is a natural way to associate a geometry tothis (and more general) coarse-grained configurations. We will interpret the resultinggeometry as the zero-temperature limit of the thermal state, since this state has adensity matrix with equal contributions from each basis vacuum state. Much of thediscussion in this section follows ideas in [5] for the LLM geometries. We begin by understanding the macroscopic variables appropriate in our case. Aswe will see, typical gauge theory states for large N will correspond to electrostaticsconfigurations with large numbers of charged disks at unit separation. The microstateconfigurations are specified by giving the (integer) charge at each discrete location onthe vertical axis. Since the extent of the disk configurations on this axis will be muchlarger than the disk separations (typically by a factor of √ N as we will see), it issensible to characterize configurations by a macroscopic charge density Q ( z ). This, wecan define by averaging the microscopic charge over a distance much larger than thedisk separations, but much smaller than the vertical extent of the disk configuration.Thus, in the coarse-grained description of states, Q ( z ) should be a smooth function.We still need to understand how the charge Q ( z ) should be arranged in the direc-tions perpendicular to z (recall that for the microstates it spreads out dynamically onthe charged conducting disks), but first it will be helpful to see what Q ( z ) looks likefor typical states. In the microscopic description, the charges Q n at position z = n label how many timesthe irreducible representation of dimension n appears, and are subject to the constraint ∞ X n =1 nQ n = N . (3)We would now like to ask what a typical randomly chosen representation looks like.To do this, we first note that the independent vacuum states of the matrix modelare in one-to-one correspondence with the quantum states of a free massless boson onan interval (a.k.a. a quantum guitar string) with energy E − E = ~ ωN , where ω is7he frequency of the lowest mode. In this analogy, Q n give the number of particles offrequency nω . For large N , where the energy and number of particles are large, we knowthat a thermodynamic description is appropriate, and that any macroscopic quantitiesevaluated for a randomly chosen microstate are extremely likely to be extremely closeto the average values.For our discussion, we will be interested in the average coarse-grained charge dis-tribution defined above, so we start by computing the expected value of Q n for each n . This is equivalent to calculating the expected particle numbers for our gas of freebosons in the microcanonical ensemble at energy E = N (setting ~ = ω = 1). Forlarge N , this should agree up to tiny corrections with the result as computed in thecanonical ensemble, so long as we choose the temperature such that the expected valueof the energy is N . The calculation is much simpler in the canonical ensemble, sincenow we can sum over all states without a constraint.To study the canonical ensemble, we write a partition function [21] Z = X Q n e − β P nQ n = Y n X Q n e − βnQ n = Y n − e − βn . (4)From this, the expectation value of Q n is found (for example by changing the β in frontof Q n to α , differentiating ln( Z ) with respect to − αn , and setting α = β ) to be h Q n i = 1 e βn − . (5)The expected value of energy is h N i = − ∂ β ln( Z ) = X n ne βn − ≈ π β , where the last line assumes that the sum can be approximated by an integral (validfor large N ). Solving for β in terms of N and plugging in to (5), we find h Q n i = 1 e πn √ N − . (6)Thus, the coarse-grained approximation to a typical microstate will have a linearcharge density very close to h Q ( z ) i = 1 e πz √ N − . (7)Or, defining x = z/ √ N and √ N q ( x ) to be the charge density in terms of x , we have h q ( x ) i = 1 e πx √ − . (8)8 .3 Supergravity solution for the average state We would now like to understand the supergravity solution corresponding to the aver-age coarse-grained configuration we have found. To do this, we first need to understandprecisely how the charge Q ( z ) should be distributed in the horizontal directions. Forthe microstates, the actual distribution of charge is determined dynamically, since thecharges are free to move on conducting disks whose radii are determined by the con-straint that the charge density at the edge vanishes. However, we will now see that thetypical configurations for large N with fixed λ have disks whose radii are much smallerthan the separation between the disks. Thus, in the coarse-grained picture for typicalstates, we can take the charge distribution to sit on the vertical axis.To understand how large the disks should be, we note that for the microstates,having conducting disks with the correct radii is necessary in order to avoid singularitiesin the supergravity solution. If we simply place all the charge on the axis, singularitiesshould appear (wherever ∂ r V = 0). These cannot be at radii much larger than theoriginal radii of the disks, since at these large radii, the electrostatics potential shouldbe modified only slightly when we move all the charge to the axis. Thus, the distancescale defined by the sizes of the disks should be the same as the typical coordinatedistance from the axis where singularities appear in the modified configuration. Wewill now use this to estimate the radii of the disks for the typical configurations.For a charge distribution Q ( z ) on the vertical axis, the corresponding potential willbe given by [21] V ( r, z ) = V ( r z − z ) + Z ∞ dz ′ Q ( z ′ ) ( p r + ( z − z ′ ) − p r + ( z + z ′ ) ) , (9)where the second term arises from the image charges below the infinite conductingplate. It is straightforward to check that such a potential for smooth Q ( z ) always givesrise to a singular supergravity solution [21]. The singularity appears at the locus ofpoints where the radial component of the electric field vanishes [1]. To estimate thisradius, we note that for slowly varying Q ( z ), the radial electric field near the axis isgiven by E r ( r ) = − rzV + 2 Q ( z ) r , so the singularity is located at r = s Q ( z ) zV . (10)From (7), we see that for z of order √ N , the typical value of the charge on each diskis of order one, while for z of order one, the typical charge is of order √ N . Recallingthat V ∼ /g , we estimate that the typical radii of the disks will be r ∼ s λN z = O ( √ N ) , This should be a good approximation so long as r is small compared with Q/Q ′ . Sfrag replacements r rz z
Figure 2: Coarse-graining for large disks. The shaded region represents a solid con-ductor that conducts only in the horizontal directions. r ∼ s λN z = O (1) . In either case, for large N and fixed λ the typical radii go to zero. Thus, in the coarse-grained description of typical states in the ’t Hooft limit, we can take all the chargeto be located on the z -axis. This leads us to the following conclusion: the geometrydual to the T = 0 thermal state of the plane-wave matrix model at large N is given bythe Lin-Maldacena solution (31), with potential (9) determined in terms of the chargedistribution (7). It may be that for some coordinate choice, the solution takes a simpler,more explicit form, but we have not investigated this.
For large N and fixed λ , we have seen that the typical states have electrostatics config-urations for which the disks are small relative to their separations, so that the chargecan simply be taken to lie on the vertical axis in the coarse-grained description. How-ever, it is also useful to have a coarse-grained description of states in cases where theradii of the disks is larger than their separations. This is relevant, for example, if weallow λ to scale as a power of N , or for fixed λ in restricted ensembles for which werestrict the number of fivebranes (as in section 5).In such cases, the coarse-grained picture will have the closely spaced disks replacedby a uniform material that conducts only in the directions perpendicular to the z -axis.This material will have some smooth profile described by a radius function R ( z ) and10arry charges such that total charge on the conductor between heights z and z + dz is Q ( z ). Just as the radii of the disks in the original setup are determined by thecharges, we should expect that R ( z ) in the coarse-grained situation will be determinedby Q ( z ). Specifically, it turns out that the shape R ( z ) of the conductor must be chosensuch that the surface charge density vanishes. This R ( z ) gives the coordinate locationof the singularity in the supergravity solution corresponding to a given coarse-grained Q ( z ). The details of this coarse graining procedure and the mathematical procedurethat determines R ( z ) in terms of Q ( z ) are described in appendix B. In thermodynamic systems, we can often associate an entropy with coarse-grainedconfigurations that are more general than the state of thermal equilibrium for thewhole system. In this section, we give a functional that associates an entropy to ageneral coarse-grained Lin-Maldacena geometry and discuss its properties. A similarentropy functional has been derived recently for the LLM geometries in [26, 27].
As a familiar example, consider an ideal monatomic gas in a box. For a given energy E , we can find the entropy of the whole system, but we could also talk about theentropy of a state where all the particles are in one half of the box (but are otherwisein a typical configuration). More generally, we can associate an entropy to an arbitraryconfiguration for which we specify the particle density and energy density (the macro-scopic variables) as a function of position, as long as these vary only over macroscopicscales.For illustrative purposes, we will work out this example, starting with quantities ascalculated in the canonical ensemble. Up to an additive constant, the entropy for N particles in thermal equilibrium at temperature T in volume V is given by S = N k (ln(
V /N ) + 32 ln( T )) . On the other hand, the average energy is E = 3 / N kT .
Defining the particle density ρ and the energy density ρ E , we can then write an ex-pression for an entropy density in terms of ρ and ρ E as s = S/V = − ρ ln( ρ ) + 32 ρ ln( 23 ρ E /k ) . Finally, the entropy associated with some general coarse-grained state is S [ ρ, ρ E ] = Z dV (cid:26) − ρ ln( ρ ) + 32 ρ ln( 23 ρ E /k ) (cid:27) , Z dV ρ = N , and Z dV ρ E = E .
We can check that the entropy functional is maximized subject to the constraints forconstant ρ and ρ E .Thus, to define the entropy functional, we split the system up into macroscopic parts(the volume elements), determine the entropy for each of these parts as a function of thecoarse-grained variables of the part, and then write the entropy of the whole system asa sum of the individual entropies, with the constraint that the coarse-grained variablesare consistent with any specified global quantities (such as energy). Now we move on to the plane-wave matrix model vacua. In this case, the variable thatwe use to describe our coarse-grained configurations is the charge density q ( x ) (recallthat we defined x = z/ √ N . Let us now consider the interval [ x, x + dx ) as a subsystemof our analog thermodynamic system. The charge in this interval, q ( x ) dx is given as asum of independent microscopic variables q ( x ) dx = Q n + · · · + Q n + l , which are also independent of the variables that determine Q outside the interval. Here n = x √ N and l = dx √ N . We assume that the coarse graining is over macroscopicdistances, in other words that the number l of individual degrees of freedom contribut-ing to Q ( x ) dx is large. Thus, we should have 1 ≪ l ≪ n . Now, for the subsystem, wehave the partition function Z = n + l Y k = n − e − βk . This gives free energy F ≈ lT ln(1 − e − nβ ) , and energy ¯ E = h nQ n + · · · + ( n + l ) Q n + l i≈ nle nβ − . The entropy is then S = ( E − F ) /T = l (cid:20) νβe nβ − − ln(1 − e − nβ ) (cid:21) . s ( z ) = S/l or equivalently s ( x ) = √ N S/l . We would like to expressthis in terms of the average charge density Q ( x ) in the interval, given by Q = h Q n + · · · + Q n + l i /dx ≈ √ N ¯ E/ ( nl )= √ N e nβ − . Solving for β in terms of Q , and substituting into the formula for s , we find s ( x ) = √ N (( q + 1) ln( q + 1) − q ln( q )) , where we have defined q = Q/ √ N .Thus, we can associate to a coarse-grained configuration described by a chargedensity q ( x ) an entropy S [ q ( x )] = √ N Z dx [( q + 1) ln( q + 1) − q ln( q )] . (11)Allowed vacua of the matrix model are subject to the constraint Z dxxq ( x ) = 1 . (12)We can now check that maximizing (11) subject to the constraint (12) gives the cor-rect result for the charge density. Introducing a Lagrange multiplier for the constraintand varying with respect to q , we findln( q + 1) − ln( q ) + Λ x = 0 . This gives q ( x ) = 1 e Λ x − , and enforcing the constraint yields Λ = π/ √ . Thus, we reproduce (8).For more general coarse-grained configurations, it is clear from (11) that the entropywill be nonzero if there is any interval ( x , x ) for which q ( x ) is continuous and nonzero.Thus, the only way to have a vanishing entropy functional with a nonzero net chargeis to have the charge located at discrete points on the axis such that q ( x ) is a sum ofdelta functions, as we have in the microstate configurations. In this case, the entropyvanishes since for large q , we have( q + 1) ln( q + 1) − q ln( q ) ∼ ln( q ) (large q ) Technically, such a q ( x ) can only appear as a coarse-grained configuration in the limit where wetake the coarse-graining scale to zero. Thus, for any non-zero coarse-graining scale, the entropy willbe non-zero for all configurations. Z ln( δ ( x − a )) dx = 0 . Recalling that the D2- and NS5-brane fluxes are quantized properly in the supergravitysolutions if and only if the charges are quantized and located at integer values of z ,we conclude that the entropy function is zero if and only if q ( x ) corresponds to amicrostate geometry. Consequently, all coarse-grained configurations with non-zeroentropy correspond to singular supergravity solutions.Our formula (11) gives the entropy as a simple expression in terms of q ( x ), whichin turn directly determines the geometry. In this sense, it is a geometrical formula forthe entropy. We might also ask whether there is any direct relation to a horizon area(or Wald’s generalization [24]) in this case. However, as is typical in examples with alarge amount of supersymmetry, the singular coarse-grained geometries that we obtainhave no horizons. On the other hand, both the curvature and the dilaton diverge atthe singularities, so the supergravity solution should receive both α ′ and string loopcorrections. It is possible that the fully corrected solutions have horizons.Following [25], we might hope that an appropriate definition of a stretched horizonaround the singularity would have area that reproduces the entropy (perhaps upto numerical factors). In fact, our setup should provide a very stringent test of anyproposed definition of a stretched horizon, if we demand that it correctly reproduces thefunctional dependence of the entropy on q ( x ). Unfortunately, as we show in appendixC, the necessary location of a stretched horizon whose area would reproduce our entropyis parametrically closer to the singularity than either the radius where the curvaturebecomes large or the radius where the dilaton becomes large. At this scale, it isprobably naive to expect that a simple area would reproduce the entropy. The T = 0 thermal solution we have found is analogous to the ‘hyperstar’ geometryof [5], dual to the coarse-grained typical state of N = 4 SUSY Yang-Mills theory on S with a U (1) ∈ SO (6) R-charge equal to energy. For that theory, there is a relatedgeometry known as the ‘superstar’ that has been understood as the geometry dual tothe equilibrium state in a more restricted ensemble for which the number of D-branesin the spacetime is fixed. There are similar restricted ensembles that are natural toconsider in our case.To understand these, we recall that the microstate geometries contain various non-contractible S cycles carrying NS5-brane flux and non-contractible S cycles carrying It was shown in [1] that the metric components in a general LM geometry 31 will be continuousand nonzero (except for points on the conducting disks ) for all potential V satisfying the threedimensional Laplace equation. From this it is straightforward to see that the region outside of thecoarse-grained conducting disks is causally connected. Possible definitions considered in the literature include the locus of points where the curvaturebecomes strong, where the dilaton becomes strong, where the local temperature equals the Hagedorntemperature, or where microstates begin to differ significantly from each other. Sfrag replacements S S rz Figure 3: Example electrostatics configuration showing the non-contractible cycles S and S carrying the largest amount of NS5-brane and D2-brane flux respectively.D2-brane flux. For a given microstate, there will be some 3-cycle in the geometrycarrying a maximal number of units N of NS5-brane flux and some 6-cycle carrying amaximal number of units N of D2-brane flux, as shown in figure 3. We loosely referto N and N as the number of NS5-branes and D2-branes in the geometry. Just as weunderstood the typical states in general, we can also ask about the form of the typicalstates in ensembles where either N or N or both are fixed.To do this, we note that the total number of units of NS5-brane flux is given bythe largest j for which Q j = 0, while the number of units of D2-brane flux is givenby the total charge P j Q j . If we consider a Young diagram with Q j rows of length j , then N and N are the total number of rows and columns in the Young diagramrespectively. The problem of studying typical Young diagrams with a fixed numberof rows (or equivalently a fixed number of columns) is precisely the one studied in[5] to understand typical states in the hyperstar ensemble of LLM geometries, whilethe problem of studying typical Young diagrams with a fixed number of rows andcolumns is precisely the one studied in [5] to determine the typical configurations inthe (generalized) superstar ensemble. Thus, we can directly carry over those results tofind the q ( x ). N For fixed N , we simply restrict the partition function (4) to n ≤ N . The expectedvalue of Q n is given by the same formula, h Q n i = 1 e βn − , (13)but now the expected value of N is h N i = N X n =1 ne βn − ≈ N f ( βN ) , f ( x ) ≡ x Li (1 − e − x ) . Q ( z ) = 1 e zN f − ( N/N ) − , z ≤ N . Note that in the unrestricted ensemble, the typical extent of the charge distribution wasof order √ N , so we only have a significant difference from the unrestricted ensemblewhen N is of order √ N or smaller. One interesting case is that where we fix N tosome large but finite value in the large N limit. In this case, we find β = N N , and Q ( z ) ≈ NN z . In this case, our estimate (10) for the size of the disks gives r ∼ p λ/N , so the disksare large compared to their separations for λ ≫ N . In this case, we need to use themethods of appendix B to determine the appropriate coarse-grained geometry. N or fixed N and N For fixed N (with either fixed or unrestricted N ), it is simplest to work in a grandcanonical ensemble where we introduce a chemical potential for N and tune it to getthe correct value. We will therefore consider the partition function Z ( β, µ ) = X Q j e − P ( βj + µ ) Q j . (14)From this, we obtain a charge distribution h Q ( z ) i = 1 e βz + µ − , (15)where β and µ are fixed by demanding N = *X j jQ j + = N X j =1 je βj + µ − , (16)as before, and *X j Q j + = N X j =1 e βj + µ − . (17)In general, β and µ are complicated functions of N and N , but as pointed out in [5],there is a simple special case where we take β → µ . This gives the solutionin the case where we restrict N N = 2 N .
16n this case, the charge density is constant Q ( z ) = N N , ≤ z ≤ N , and the supergravity solution may be written very explicitly in terms of ordinaryfunctions. This case corresponds to a triangular Young diagram, which in the LLMcase gives rise to the original superstar geometry.We also get a simple expression for the charge distribution in the case where N islarge but fixed in the large N limit with N unrestricted. In this case, a straightforwardcalculation gives Q ( z ) = N N e − N N z . SU (2 | symmetric theories So far, we have discussed the Plane-Wave Matrix Model. However, Lin and Malda-cena [1] also identified supergravity duals to the vacua of other, higher dimensional,field theories with SU (2 |
4) supersymmetry. These are the aforementioned maximallysupersymmetric Yang-Mills theory on R × S [4], N = 4 SYM theory on S /Z k , andtype IIA Little String Theory on S [4, 1, 6]. Aspects of the relations among thesetheories and the Plane-Wave Matrix Model have been discussed in [22, 23].In this section we will analyze these theories in the same way as we have for thePlane-Wave Matrix Model. For the higher-dimensional theories, the construction ofdual supergravity solutions differs only in the boundary conditions for the electro-statics problem. The individual microstates are still distinguished by the locationsand charges of finite-sized conducting disks, so the coarse-graining procedure and theentropy functional are exactly the same as in the Plane-Wave Matrix Model. S × R We will first consider maximally supersymmetric field theory on S × R . This theorycan be derived as a limit of the Plane-Wave Matrix Model [4], or of N = 4 SYM on S /Z k in the limit k → ∞ [1].The field content of this theory is the same as the usual low-energy D2-brane gaugetheory, with an SU ( N ) gauge field together with fermions and seven scalar fields. Sixof the scalar fields are associated with the SO (6) R-symmetry of the theory. Theremaining one comes from the dimensional reduction when the k → ∞ limit is takenin N = 4 SYM on S /Z k . We will refer to this scalar as Φ. The vacua of this fieldtheory are given by Φ = − diag( n , n , . . . , n N ), and F = dA = Φ sin θdθdφ , where the n i are integers, and θ and φ are the usual coordinates on S .The different vacua of the theory are labelled by the multiplicities of the integersin the vacuum configurations of Φ and F .17 .1.2 Supergravity The supergravity dual to this theory shares many similarities with the dual to Plane-Wave Matrix Model. As in the Plane-Wave Matrix Model case, the disks are parallel,circular, and centred at r = 0, z = d i . In this case, however, the auxiliary electrostaticsproblem has no infinite disks, and the background potential is given by V ∞ = W ( r − z ) . (18)As before, non-singular solutions will have disks with radii R i chosen so that the chargedensity vanishes at the edge of each disk.Corresponding to a vacuum with N i copies of the integer n i will be an electrostaticsconfiguration with disks at positions d i = πn i / Q i = π N i /
8. Thegauge theory parameters are related to the electrostatics ones as g ∝ /W , and N = P N i .In similar fashion to the Plane-Wave Matrix Model case, we can find the potentialfor the system with coarse-grained charge density Q to be V ( r, z ) = W ( r − z ) + Z ∞−∞ dz ′ Q ( z ′ ) p r + ( z − z ′ ) . (19) As we have described above, the vacua of this theory are labelled by a set of integersand their multiplicities. Since the integers specifying the vacuum can be arbitrarilylarge (the only restriction is that the sum of multiplicities is N ), we have an infinitenumber of vacua in this case. In the electrostatics picture, this corresponds to the factthat the plates are allowed to sit anywhere on the z -axis, with the only restriction thatthe total charge is N . As a result, quantities such as the charge at any location willaverage to zero, and we cannot see any natural way to define a typical configurationin this case for the unrestricted ensemble.On the other hand, we do get a well defined thermal configuration in an ensemblewhere we fix the number of NS5-branes, as in section 5. This corresponds to fixingthe separation between the highest and lowest disk. For the SU ( N ) theory, we shoulddemand also that the sum of integers times their multiplicities is zero, so we end upwith a finite set of vacuum states. For coarse-grained typical states, the total charge N will be evenly distributed between the N plates, so the coarse-grained charge densitywill be Q ( z ) = NN , − N ≤ z ≤ N . Another way to obtain a non-trivial electrostatics configuration is to recall thedefinition of this theory as a k → ∞ limit of N = 4 SYM on S /Z k . If we instead takea limit in which N → ∞ and k → ∞ with N/k = ξ fixed then the resulting theorywill have a T = 0 thermal state arising from the electrostatics potential V ( r, z ) =18 ( r − z ) − ( πξ ) / (2) ln( r ). The corresponding geometry will have a string likesingularity with entropy density s = (1 + ξ ) ln (1 + ξ ) − ξ ln ξ. (20) N = 4 Yang-Mills theory on S /Z k This theory and its vacua can be obtained from N = 4 SYM on S in the followingmanner, as outlined in [7]. We can coordinatize the S using the metric ds S = 14 [(2 dψ + cos θdφ ) + dθ + sin θdφ ] (21)where θ and φ are the usual coordinates on S , and ψ is an angular variable withperiod 2 π . The orbifold is obtained by identifying ψ ∼ ψ + 2 π/k . The vacua of thefield theory are given by the space of flat connections, modulo gauge transformations,on S /Z k . The orbifold allows for vacua of the form A = − diag( n , n , . . . , n N ) dψ , sothat e πn i /k are k th roots of unity. This ensures that A has unit holonomy around thefull angular direction ψ , which is topologically trivial. To label the vacua uniquely, wewill restrict the integers n i to be on the interval [0 , k ). In the supergravity picture, the background potential for N = 4 Yang-Mills theoryon S /Z k is the same as in (18), but the electrostatics configuration is required to beperiodic in z with period πk/
2. Even though the background potential is not periodicin z , the part of the potential that determines the charge densities on the disks is. Sothe electrostatics solution will have a periodic part that arises from the charged disksin addition to the background piece.The periodic arrays of conducting disks are, in turn, related to the vacua of thefield theory. For a vacuum that has N i repetitions of the integer n i , the correspondingelectrostatics configuration will have a set of charged conducting disks at positions z = πn i / , π ( n i ± k ) / , π ( n i ± k ) / , . . . , each carrying charge π N i /
8. The gaugetheory parameters are given in terms of the electrostatics parameters by g k ∝ /W and N = P N i .Here the potential for the system with coarse-grained charge density Q is V ( r, z ) = W ( r − z ) + Z ∞−∞ dz ′ Q ( z ′ ) p r + ( z − z ′ ) , (22)where Q has a of period πk/
2. 19 .2.3 Typical states
Having described the field theory vacua and the corresponding auxiliary electrostaticsconfigurations, we would like to consider the typical state.To find the typical configuration in this case, we can use the partition function (14)with β = 0. We can fix µ by imposing N = k e µ − , (23)which means e − µ = NN + k , (24)and the typical vacuum will have q = N/k .Up to an overall constant, the electrostatic potential can be found outside the chargedistribution to be V ( r, z ) = W ( r − z ) − ( πN ) / (2 k ) ln( r ). It is singular, and has anentropy of S = k (cid:18)(cid:18) Nk (cid:19) ln (cid:18) Nk (cid:19) − Nk ln (cid:18) Nk (cid:19)(cid:19) . (25) S Type IIA Little String Theory on S was defined originally by its supergravity dual,found in [1] and described below. Using this supergravity dual, it has been arguedthat this theory can be defined by particular double-scaling limits of either the Plane-Wave Matrix Model [6], the maximally supersymmetric Yang-Mills theory on S × R or N = 4 Yang-Mills theory on S /Z k [7]. In this case, for the theory associated with k fivebranes we have two infinite conductingplates separated by a distance k . As shown by Lin and Maldacena [1], we can have anon-trivial potential V ( r, z ) = 1 g I (cid:16) rk (cid:17) sin (cid:16) zk (cid:17) (26)between the plates for which the corresponding geometry has an infinitely long throatcarrying NS5-brane flux. The parameter g is related to the size of the sphere on whichthe NS5-branes sit, as measured in units of α ′ (the dimensionful coupling of the LittleString Theory).We can consider adding additional charged conducting disks to this system whilekeeping the number of units of NS5-brane flux fixed. In the electrostatics picture, thiscorresponds to adding some number of finite charged conducting disks in the regionbetween the two infinite disks. The disks can sit at positions d i = πn i /
2, where theintegers n i are in the interval [1 , k ), and carry finite charges N i .20 .3.3 Typical states As for the 2+1 dimensional case, the number of vacua here is infinite if we allowarbitrary configurations finite disks in between the infinite conducting plates. However,it is interesting to consider some restricted ensembles.First, we add some fixed number N of units of D0-brane flux. This requires that X i iN i = N .
In this case, the counting problem is identical to that is section 5.1, so we obtain thesame typical charge distribution. Of course, the supergravity solution will be differenthere, since the background potential is now (26).Alternatively, we could consider an ensemble of geometries in which the number ofunits of D2-brane charge is fixed. In that case it is again convenient to use (14) with β = 0. Fixing the asymptotic charge we find that N = ( N −
1) 1 e µ − , (27)which can be inverted to give e − µ = N N + N − . (28)The typical state will have h Q j i = 1 e µ − N N − , (29)and the entropy of this configuration is, for N ≫ S = N (cid:18)(cid:18) N N (cid:19) ln (cid:18) N N (cid:19) − N N ln (cid:18) N N (cid:19)(cid:19) . (30) Acknowledgements
We are grateful to Henry Ling, and especially Vijay Balasubramanian for many use-ful discussions. This work has been supported in part by the Natural Sciences andEngineering Research Council of Canada, the Killam Trusts, the Alfred P. Sloan Foun-dation, and the Canada Research Chairs programme.
A The Lin-Maldacena solutions
The general Lin-Maldacena SU (2 | α ′ in the metric) is given by [1] ds = ¨ V − V − V ′′ ! / ( − V ¨ V − V dt + − V ′′ ˙ V ( dρ + dη ) + 4 d Ω + 2 V ′′ ˙ V ∆ d Ω ) , = 4( ¨ V − V ) − V ′′ ˙ V ∆ ,C = − V ′ ˙ V ¨ V − V dt , (31) F = dC , C = − V V ′′ ∆ dt ∧ d Ω ,H = dB , B = 2 ˙ V ˙ V ′ ∆ + η ! d Ω , ∆ ≡ ( ¨ V − V ) V ′′ − ( ˙ V ′ ) . In these equations, the potential V uses slightly different conventions from the onewe discussed. The potential V lm here is related to our potential V by V lm ( r, z ) = π V ( 2 π r, π z ) . B Coarse-graining for large disks
For certain parameter values, or in restricted ensembles, the typical states are suchthat the radii of the disks are large compared to their separations. As we noted above,in this case, the macroscopic description will replace the closely spaced disks with asolid material that conducts only in the horizontal directions.Such a conductor has the following properties. Since the charges are free to rear-range themselves in the directions perpendicular to z , they will do so in such a waythat the final potential inside the conductor is a function only of z , ensuring that theelectric field in the r and θ directions is zero. There will generally be some chargedistribution inside the conductor, given by ρ ( z ) = − π V ′′ ( z ) , (32)so ρ is also a function only of z . The remaining charge will build up at the surfaceof the conductor. In general, the shape R ( z ) for the conductor, and the linear chargedistribution Q ( z ) on the conductor, together with some fixed background potential willdetermine the charge density ρ ( z ) inside the conductor and the surface charge density σ ( z ), determined from ρ ( z ) via Q ( z ) = πR ( z ) ρ ( z ) + 2 πR ( z ) σ ( z ) p R ′ ( z )) . (33)On the other hand, for some special choice of R ( z ), the surface charge density willvanish. This is the coarse-grained analogue of the constraint that the charge densityshould vanish at the tip of the disks. 22 .1 The variational problem We will now set up the mathematical problem that determines R ( z ) and ρ ( z ) from Q ( z ). We start by assuming some fixed R ( z ) and Q ( z ).Outside the conductor, the potential will be given by V + ( r, z ) = V ( r, z ) + ˜ V ( r, z ) , where ˜ V is the potential due to the charges in the conductor, which should vanish atlarge r and z . Since ˜ V is an axially symmetric solution of Laplace’s equation, we canexpand it in terms of Bessel functions,˜ V ( z ) = Z ∞ duu A ( u ) e − zu J ( ru ) . Inside the conductor, the potential will be some function V − ( z ). The unknown functions A ( u ) and V − ( z ), together with the charge density ρ ( z ) inside the conductor and thecharge density σ ( z ) on the surface of the conductor will be determined by the twoequations (32) and (33), and the boundary condition ~E + ( R ( z ) , z ) − ~E − ( z ) = 4 πσ ( z )ˆ n . (34)In our case, we wish to fix R ( z ) by the constraint that the surface charge densityvanishes. Then the electric field must be continuous across the boundary of the con-ductor, and since the electric field is vertical inside, we must have ∂ r V ( R ( z ) , z ) = 0.Explicitly, we have ∂ r V ( R ( z ) , z ) − Z ∞ due − zu A ( u ) J ( R ( z ) u ) = 0 . (35)This determines R ( z ) in terms of A ( u ). Given this, the potential inside the conductor isdetermined by the z component of the boundary condition (34), or simply by continuityof the potential across the boundary, so V − ( z ) = V ( R ( z ) , z ) + Z ∞ duu A ( u ) e − zu J ( R ( z ) u ) . Finally, we can use (32) and (33) to write an equation relating A ( u ) and Q ( z ), Q ( z ) = − R ( z )( ∂ z V ( z ) + Z ∞ duuA ( u ) e − zu J ( R ( z ) u )) . (36)To summarize, A ( u ) is determined by the integral equation (36) where R ( z ) isdetermined in terms of A via (35).In practice, it is far simpler to determine R ( z ) and Q ( z ) given some A ( u ), or moregenerally some solution to the Laplace equation that arises from any set of axiallysymmetric localized charges. We could also parametrize our solution to the Laplaceequation via the multipole data rather than the function A ( u ). As an example of23his approach, we can come up with an explicit coarse-grained supergravity solutionstarting with the simplest non-trivial solution ˜ V , namely the potential from a dipolelocalized at the origin (the infinite conducting plane at z = 0 forces the potential tobe an odd function of z .). In this case, we have˜ V ( r, z ) = p z ( r + z ) . The radial electric field for the full potential is then E r ( r, z ) = − ∂ r V + ( r, z ) = − V rz + 3 p rz ( r + z ) . Requiring that this is zero gives r = 0 or z = 0 or z + r = x , where we define x = (cid:18) p V (cid:19) . Thus, in this case, the profile of the conductor is spherical. From (36), we can nowdetermine the corresponding charge density Q ( z ). We find Q ( z ) = 52 V z ( x − z ) . As a check, we find that the total dipole moment for this configuration is Z ∞ dz zQ ( z ) = p . So we have at least one example where we know both the geometry and the Youngdiagram explicitly. Note that for this case, the typical height for the plates and thetypical size are the same, of order x . In terms of the field theory parameters, we have V ∼ /g and p = 2 N , so x ∼ λ . Thus, our coarse-grained description should bevalid as long as λ is large. The typical charge on one of the plates in the correspondingmicrostate geometries is Q ∼ V x ∼ N/λ . In section 3, we saw that this charge is oforder one for typical distributions, so it is only for λ ∼ N that the geometry we haveconstructed has an entropy of the same order of magnitude as the thermal state. (Itis important to note that for a fixed configuration of disks (i.e. fixed p/V ∼ λ ), thecorresponding charge distribution changes as a function of N .) C Stretched horizons
In this appendix we investigate the possibility that the area (or some generalizationof area) of a suitably defined stretched horizon might reproduce the entropy formula2411). We focus on a particularly simple specific example of a coarse-grained geome-try, and find that a stretched horizon whose area would reproduce the entropy wouldnecessarily be parametrically closer to the singularity than both the scale x s where thestring coupling becomes of order one, and the scale x c where the curvature becomesstring scale.The geometry we focus on is the thermal state geometry of the super Yang Millstheory on S /Z k . In this case the potential is simply − N π k log ρ + V ( ρ − η ) (37)where V ∼ g ym k as identified in [7]. The potential is singular at ρ = 0, which violatesthe regularity condition on the LM geometry. The boundary of the coarse-grainedconducting disks is at ρ = r q πN kV , and the supergravity solution is ds = (cid:18) N V kπ (cid:19) / (cid:26) − V kρ ) N π dt + 8 V (2 V ρ − N π/ k ) ( dρ + dη ) + 4 d Ω + 2 k (2 V ρ − N π/ k ) N π d Ω (cid:27) e = ( N π/k )16 V (2 V ρ − N π/ k ) , C = − k (2 V ρ − N π/ k ) πN dt ∧ d Ω , B = 2 ηd Ω(38)We see explicitly that the geometry is singular at ρ = r ∼ p g N , which is exactlythe edge of the disks, but there is no horizon in this geometry. This solution has beenconsidered in [8], where it was pointed out that the singularity is related to the Z k orbifold singularity in the IIB language. We will assume the stretched horizon to be aconstant ρ surface respecting the translational symmetry along the η direction. Using ρ = r + x , we find the string coupling becomes of order one at x s = 18 √ πV ∼ ( g ym k ) . (39)The Ricci scalar can be calculated noticing the fibred structure of the metric, R string = 3 r V kN π V kρ − N π V kρ − N π . (40)We see that it diverges at exactly the boundary of the coarse-grained conducting disks.The curvature becomes of string scale at x c ∼ . (41)In the above we have assumed g ym N ≫ g ym N ≫ ≫ g ym k , Recently, these ideas have been explored in the context of coarse-grained LLM microstates [12],though a prescription for defining a stretched horizon that generally reproduces the entropy of coarse-grained states has not emerged.
N/k ≫
1. The area of an 8-surface at constant t and ρ = R ( η ) canbe calculated to be (in the Einstein frame) A = 2 / ω ω p R ′ ( z ) q ( ¨ V − V ) ˙ V / , (42)where ω , ω are the volume elements of the two-sphere and five-sphere respectively.Specializing to R ( z ) = r + x and to the metric (38), we get A = 16 ω ω √ N π (4 V kρ − N π ) / k . (43)We note that it is a monotonically increasing function with the distance from r . Usingthis and evaluating at x c , x s we find A c ∼ N / g / ym k ,A s ∼ ( g ym k ) A c . (44)The Bekenstein-Hawking entropy formula S = AG N gives ( G N = g s = ( g ym k ) ) S c ∼ g / ym k / (cid:18) Nk (cid:19) / ,S s = ( g ym k ) S c . (45)As expected S c ≫ S s . According to the entropy functional (11), the entropy associatedwith the geometry (38) is S = − k ln( N/k ) + ( N + k ) ln( N/k + 1) . (46)In the large N/k limit it becomes S ∼ k (ln( N/k ) + 1) , (47)which is much smaller than both S s , S c . Here, both α ′ and string loop corrections arevery important. 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