hepth/0509011
Instabilities of NearExtremal Smeared Branes
and the Correlated Stability Conjecture
Troels Harmark, Vasilis Niarchos, Niels A. Obers
The Niels Bohr Institute
Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
, ,
Abstract
We consider the classical and local thermodynamic stability of non and nearextremal Dbranes smeared on a transverse direction. These two types of stability are connected through the correlated stability conjecture for which we give a proof in this specific class of branes. The proof is analogous to that of Reall for unsmeared branes, and includes the construction of an appropriate twoparameter offshell family of smeared Dbrane backgrounds. We use the boost/Uduality map from neutral black strings to smeared black branes to explicitly demonstrate that nonand nearextremal smeared branes are classically unstable, confirming the validity of the conjecture. For nearextremal smeared branes in particular, we show that a natural definition of the grand canonical ensemble exists in which these branes are thermodynamically unstable, in accord with the conjecture. Moreover, we examine the connection between the unstable GregoryLaflamme mode of charged branes and the marginal modes of extremal branes. Some features of Tduality and implications for the finite temperature dual gauge theories are also discussed.
Contents
 1 Introduction
 2 Preliminaries
 3 The correlated stability conjecture for charged branes
 4 The GregoryLaflamme mode for smeared branes
 5 Connection to marginal modes of extremal smeared branes
 6 The CSC for nearextremal branes
 7 Conclusions
 A The GregoryLaflamme mode
 B An offshell twoparameter family of black branes
1 Introduction
The gauge/gravity correspondence [1] implies a deep connection between the thermodynamics of nearhorizon brane backgrounds and that of the dual nongravitational theories. In particular, this suggests that the gauge theory dual of a classically unstable brane background must have a corresponding phase transition. This hints at an interesting connection between the classical stability of brane backgrounds and the thermodynamic stability of the dual nongravitational theories, and thereby to the thermodynamic stability of the brane backgrounds themselves.
Gregory and Laflamme [2, 3] were the first to study the classical stability of neutral black strings, as well as certain charged branes. They showed that neutral black strings are unstable under perturbations that oscillate in the direction in which the string extends, provided that the wavelength of the perturbation is larger than the order of the horizon radius. When the extended direction is compactified on a circle, this means in particular that for masses below/above a critical mass the black string is unstable/stable. Already in this original work, a global thermodynamic argument for the instability was given, namely that for small masses the entropy of a localized black hole is higher than that of the black string with the same mass, suggesting a possible endpoint for the decay of the black string in that case.^{1}^{1}1The viewpoint that an unstable uniform black string decays to a localized black hole has been challenged in [4] (in this connection see also the discussion in the reviews [5, 6]).
This deep connection between classical and thermodynamic stability of brane solutions was recently made more precise with a conjecture formulated by Gubser and Mitra [7, 8]. The conjecture, referred to as the Correlated Stability Conjecture (CSC), states that for systems with translational symmetry and infinite extent, a GregoryLaflamme (GL) instability arises precisely when the system has a local thermodynamic instability. In further detail, local thermodynamic stability is defined here as positivedefiniteness of the Hessian matrix of second derivatives of the mass with respect to the entropy and any charges that can be redistributed over the direction in which the GL instability is supposed to occur. Thus, the CSC relates a local thermodynamic instability to a perturbative dynamical instability. The latter is the classical gravitational instability arising from perturbations of the brane background.
For magnetically charged Dbrane solutions in String/Mtheory, a semiclassical proof of the CSC was given by Reall [9] using the Euclidean path integral formulation of gravity.^{2}^{2}2See also [10]. See furthermore [11, 12, 13, 14] for work on the classical stability of charged branes. A key ingredient in this proof is the relation between the threshold mode of the classical instability and a Euclidean negative mode in the semiclassical path integral. In this way, for this class of branes it was shown that a classical instability appears precisely when the specific heat becomes negative. The CSC has also been considered and confirmed recently in more complicated settings involving bound states of branes [15, 16, 17]. A class of counterexamples in a different setting appeared very recently in [18]. Further comments related to these examples can be found in Section 3.
Another interesting class of branes is that of smeared Dbranes. Here, we call an electrically charged brane smeared if it has a direction with translational symmetry along which the brane is not charged. For magnetically charged branes we follow the opposite convention. When compactified on the isometric direction, a smeared Dbrane is related by Tduality to a Dbrane wrapping the Tdual circle. In particular, the nearextremal limit of the latter has a dual description in terms of a dimensional supersymmetric YangMills (SYM) theory compactified on a circle. Thus, the issue of stability of smeared black branes has nontrivial implications for the stability of the dual SYM theories on , where the torus consists of the compact spatial world volume direction and the Euclidean time direction.
Moreover, there is an interesting connection between neutral black strings and smeared Dbranes. Building on the original observation of Ref. [19], it was shown in [20] (see also [21, 22, 23]) that the phases of KaluzaKlein black holes (see the reviews [5, 6, 24]) can be mapped onto phases of non and nearextremal Dbranes with a circle in their transverse space. The map is a sequence of boost and Uduality transformations, and includes as a special case the map of neutral black strings to non and nearextremal Dbranes that are uniformly smeared on a transverse circle.
For neutral black strings wrapped on a circle a new static phase of nonuniform strings [25, 26, 27, 28] is known to emerge at the GL transition point, where the unstable mode is at threshold. In Refs. [21, 22, 20] it was observed that the GL point is mapped onto a critical mass/energy for non/nearextremal branes, where a new phase of non and nearextremal branes, nonuniformly distributed on the circle, emerges. This picture strongly suggests that non and nearextremal smeared branes exhibit classical instabilities. In particular, the threshold mode of the black string is mapped directly onto the threshold mode of non and nearextremal smeared black Dbranes. More generally, it is natural to expect that the (timedependent) unstable mode can also be obtained from the neutral GL unstable mode by a suitable generalization of the boost/Uduality map [22].
The aim of this paper is to study the classical stability and CSC for non and nearextremal smeared branes in detail.^{3}^{3}3Our brane solutions do not include KaluzaKlein bubbles. The classical stability of another class of smeared branes, involving bubbles, has been considered in Ref. [29]. For solutions involving branes and bubbles, see also [30]. Our analysis will show in a quantitative way how the presence of charge affects the GL instability of neutral black strings, and what happens when we take the nearextremal limit. An important tool that we use is the boost/Uduality map mentioned above. In this work we restrict ourselves to smeared Dbranes of type II string theory, but the other 1/2BPS branes can be treated similarly.
The main points of this paper can be summarized as follows:

We argue that when applying the CSC to smeared branes one should consider thermodynamic stability in the grand canonical ensemble since the charge can redistribute itself in the direction in which the brane is smeared. Nonextremal smeared branes are thermodynamically unstable in this ensemble and according to the CSC these branes should have a classical instability. The choice of ensemble is independent of whether the brane is electrically or magnetically charged.

We give a proof of the CSC for smeared branes. One of the essential elements is the explicit construction of an appropriate offshell twoparameter family of Euclidean backgrounds connected to these branes.

Following Ref. [22], we give an explicit construction of the GL unstable mode for nonextremal smeared branes. More specifically, we map the timedependent unstable GL mode of neutral black strings to a timedependent unstable mode of nonextremal smeared branes. This confirms the validity of the CSC for nonextremal smeared branes.

We present a detailed analysis of the nearextremal limit and the validity of the CSC in this case. We show explicitly that the nearextremal limit of the unstable GL mode of nonextremal smeared branes is welldefined and hence that nearextremal smeared branes are classically unstable. According to the CSC this means that nearextremal smeared branes are thermodynamically unstable in the grand canonical ensemble. Nevertheless, it seems that in the nearextremal limit the charge cannot vary anymore, and it is not a priori obvious how to define a grand canonical ensemble for these branes. We show, however, that a natural definition of such an ensemble exists. This implies a new version of the first law of thermodynamics for nearextremal smeared branes, involving a rescaled chemical potential. In this nearextremal grand canonical ensemble, the nearextremal smeared branes are thermodynamically unstable. In this way, we resolve an apparent puzzle, showing that the CSC is also correct for nearextremal branes.^{4}^{4}4Note that this also resolves the contradiction found in the numerical investigations of Ref. [31].

We examine the connection between the GL mode for charged branes and marginal modes for extremal smeared branes. We show that in the extremal limit the GL mode of nearextremal smeared branes precisely becomes the marginal mode of extremal smeared branes. As a consequence of this fact the extremal smeared branes are in a sense arbitrarily close to being unstable.

We discuss various issues related to Tduality on smeared Dbranes and the properties of the dual gauge theories.
The outline of this paper is as follows. We start in Section 2 by reviewing how non and nearextremal smeared Dbranes of type II string theory can be obtained from the neutral black string solutions using the boost/Uduality map. We also review the main results on the thermodynamics of these branes, which we will need when considering the CSC.
In Section 3, we will review the CSC and discuss the criteria that determine the choice of thermodynamic ensemble. Then following the arguments of Reall [9], we present a proof of the CSC for smeared branes. Some of the subtle issues that are not fully resolved will be discussed as well. We also comment on the general proof of the CSC and the validity of the CSC in the presence of compact directions.
We continue in Section 4 by showing that, with a suitable generalization of the boost/Uduality transformation reviewed in Section 2, we can map the GL unstable mode of the neutral black string to a timedependent unstable mode for nonextremal smeared branes. This map relies on an argument first noticed in Ref. [22]. We then take the nearextremal limit of this and show that it gives a welldefined unstable mode for nearextremal smeared branes. As a consequence we conclude that the latter are also classically unstable.
In Section 5 we consider the extremal case. We show that the extremal branes have marginal modes of any wavelength and that these modes can be recovered by taking the extremal limit of the GL mode of charged smeared branes. This means that when an extremal smeared brane is perturbed in such a way that it becomes nonBPS the resulting brane background becomes unstable. Furthermore, for any marginal perturbation of the extremal smeared branes one can find a corresponding nonBPS continuation.
We return to the CSC in Section 6, where we give the appropriate definition of the grand canonical ensemble in the nearextremal case. We then show that, in this ensemble, nearextremal branes are thermodynamically unstable. Together with the explicit construction of the unstable mode for nearextremal branes in Section 4, this confirms the validity of the CSC in this case as well.
We conclude with a short summary and some open questions in Section 7. Two appendices are also included. In Appendix A we present the differential equations determining the neutral GL mode. In Appendix B we provide the details of the offshell twoparameter family of black brane backgrounds that is essential for the proof of the CSC for smeared branes in Section 3.
2 Preliminaries
In this section we review how the non and nearextremal smeared Dbranes can be obtained from the neutral black string by using the boost/Uduality map, and the nearextremal limit. For later use throughout the paper, we also review the main results on the thermodynamics of these branes.
2.1 Nonand near extremal smeared branes from boost/Uduality
We begin by recalling how one obtains the nonextremal smeared Dbrane solution of type II string theory from the uniform black string solution in pure gravity, by applying a sequence of transformations including a boost and Udualities.^{5}^{5}5This method of “charging up” neutral solutions was originally conceived in [32] where it was used to obtain black branes from neutral black holes.
The starting point is the metric of the uniform black string in dimensions:
(2.1) 
By adding flat directions, this solution is trivially uplifted to a solution of elevendimensional (super)gravity
(2.2) 
Here, the coordinates are used for flat directions and for the one that will be taken to be the eleventh direction when reducing from Mtheory to type IIA string theory. The metric (2.2) is thus a vacuum solution of Mtheory. Performing a Lorentzboost in the direction
(2.3) 
and dropping the label ‘new’ gives the boosted metric
(2.4) 
Since we have an isometry in the direction we can now make an Sduality in the direction to obtain a solution of type IIA string theory. This gives a nonextremal D0brane solution of type IIA string theory which is uniformly smeared along the space parameterized by and . With a Tduality transformation on each of the directions , we deduce the solution for a nonextremal Dbrane smeared along the direction,
(2.5) 
written in the string frame. The function is given in (2.1). One can also use further Udualities to obtain the backgrounds for smeared F1strings and NS5branes, but we choose to focus on Dbranes in this paper.
By one further Tduality in the transverse direction, we can relate the Dbrane smeared on a circle to that of a Dbrane wrapped on the Tdual circle. Since this Tduality is important for defining the nearextremal limit and computing quantities in the dual gauge theories of nearextremal Dbranes smeared on a transverse circle, we give the relevant Tduality relations here. Assuming that the coordinate in (2.5) is compactified on a circle of circumference , , and denoting the string coupling by , we find that the circumference of the Tdual circle (on which the Dbrane is wrapped) and the Tdual string coupling are related by
(2.6) 
where is the string length. Introducing the gauge theory quantities and for the dimensional supersymmetric YangMills compactified on a spatial circle, it also follows that we have the relation
(2.7) 
Another useful definition is the following
(2.8) 
where we used the Tduality and gauge theory relations given above along with the definition .
Nearextremal limit
We define the nearextremal limit as
(2.9) 
where are the coordinates appearing in the nonextremal background (2.5). Using this limit, the resulting solution for nearextremal smeared Dbranes is
(2.10) 
where is defined in (2.7).
The classical and local thermodynamic stability of the non and nearextremal Dbrane backgrounds given above is the main object of our investigation in this paper. The boost/Uduality transformation and the nearextremal limit described above will prove very useful in this context.
We recall here that the sequence of boost and Uduality transformations reviewed above can be equally well applied to other solutions of pure gravity with a compact circle direction. In particular, in Ref. [20] this map is discussed in great detail, and applied to both black holes on cylinders and nonuniform black strings, generating non and nearextremal black branes that are either localized on a transverse circle or nonuniformly distributed on the circle.
2.2 Thermodynamics of non and nearextremal smeared branes
We now review some important results on the thermodynamics of the non and nearextremal smeared Dbranes obtained above.
The thermodynamics of the nonextremal Dbranes (2.5) smeared on a transverse circle of circumference , is given by
(2.11) 
where is the worldvolume of the brane. Here the extensive quantities, , and correspond respectively to the mass density, charge density and entropy density along the transverse direction. is the temperature and the chemical potential. These quantities satisfy the usual first law of thermodynamics .
Correspondingly, the thermodynamic quantities of nearextremal Dbranes (2.10) are given by
(2.12) 
Here, is the energy above extremality defined by in the nearextremal limit (2.9), while and , defined in (2.7), (2.8) are finite in the limit. The quantities in (2.12) satisfy the usual first law of thermodynamics .
Canonical and grand canonical ensemble
In what follows, we will make extensive use of both the canonical as well as grand canonical ensemble, so we briefly review the definition of these ensembles here.
In the canonical ensemble, the temperature and charge are kept fixed and the appropriate thermodynamic potential is the Helmholtz free energy
(2.13) 
The condition for thermodynamic stability in the canonical ensemble is
(2.14) 
positivity of the specific heat .
In the grand canonical ensemble, the temperature and chemical potential are kept fixed, so the appropriate thermodynamic potential is the Gibbs free energy
(2.15) 
The condition for thermodynamic stability in the grand canonical ensemble is
(2.16) 
where is the specific heat and is the inverse isothermal electric permittivity. The condition (2.16) follows from demanding that the Hessian of the Gibbs free energy is negative definite. In fact, because of the matrix relation , where is the Hessian of the mass , this is equivalent to demanding that is positive definite, as required by local thermodynamic equilibrium.
Using the thermodynamic quantities for nonextremal branes in (2.11) and the definitions in (2.16) one computes in this case^{6}^{6}6These expressions are most easily computed using the identity that
(2.17) 
With these results and the conditions for thermodynamic stability reviewed above, it is not difficult to determine the conditions for thermodynamic stability of nonextremal smeared Dbranes. For , positivity of the specific heat becomes
(2.18) 
showing that there is a lower bound on the charge of the branes in order to be thermodynamically stable in the canonical ensemble. On the other hand, we have
(2.19) 
which is incommensurate with the condition in (2.18). For we have that and . Hence for any it is impossible to satisfy the requirement (2.16) of thermodynamic stability in the grand canonical ensemble.
We now turn to the case of nearextremal branes. Since the charge has been sent to infinity, there seems to be for this case only one relevant ensemble, namely the canonical ensemble. The corresponding Helmholtz potential is given by
(2.20) 
and thermodynamic stability requires positivity of the specific heat . From the thermodynamics (2.12) one easily obtains the simple relation
(2.21) 
As a check, the same result is obtained by taking the nearextremal limit () of the expression (2.17) for . For any nonzero temperature, the specific heat is thus positive for nearextremal smeared Dbranes when , which is a wellknown result.
Since in the nearextremal limit the charge goes to infinity and the chemical potential goes to one, it seems that these quantities cannot be varied anymore. This suggests that it does not make sense to consider the grand canonical ensemble for nearextremal branes. Moreover, one could consider what happens to the quantity in (2.17) when taking the nearextremal limit. Since in the nearextremal limit, we clearly have that in the limit. This would seem to imply that, infinitesimally close to the nearextremal limit, nonextremal branes are marginally thermodynamically stable in the grand canonical ensemble. As we will see below when we consider the correlated stability conjecture, which relates classical and thermodynamic stability, this fact implies a puzzle in view of the fact that nearextremal branes are classically unstable. However, in Section 6 we will demonstrate that there is a natural resolution of this apparent violation of the CSC.
Tduality
Finally we have a few remarks on the effect of the Tduality along the direction (see Section 2.1) on the thermodynamics. Under this transformation the thermodynamic quantities (2.11) and (2.12) for non and nearextremal branes are invariant. Thus, the thermodynamics of a Dbrane smeared on a circle is identical to that of a Dbrane wrapped on the Tdual circle. Despite this fact, in the next section we will see that when applying the CSC there is a qualitative distinction between the two cases.
3 The correlated stability conjecture for charged branes
3.1 Formulation of the conjecture
In [7, 8] Gubser and Mitra put forward an intriguing conjecture that relates classical and thermodynamic instabilities in gravitational systems. The precise form of this conjecture states that a gravitational system with a noncompact symmetry ( a black brane with noncompact worldvolume) is classically stable, if and only if, it is locally thermodynamically stable. For a system with conserved charges this criterion of local thermodynamic stability translates (after the appropriate choice of ensemble) into a positivity criterion for the Hessian matrix , which involves the second derivatives of the mass with respect to the entropy and the charges . A partial proof of this conjecture has been given by Reall in [9], who considered the case of magnetically charged black branes. However, a general proof of the conjecture is yet to be found.
Before even applying the CSC one should first make the appropriate choice of thermodynamic ensemble. This choice is important, because it affects crucially the discussion of local thermodynamic stability and ultimately the validity of the conjecture as such. The usual practice is to consider gravity in the canonical ensemble, where the temperature is kept fixed and the partition function of the system is expressed as a function of . In general situations with an arbitrary number of charges , however, the choice of the ensemble is not always obvious and part of the conjecture should involve a clear statement that singles out the right choice. In what follows, we attempt to illuminate this point for a large class of smeared or unsmeared, magnetically or electrically charged black branes.
To be more concrete, consider a generic (nonextremal) black Dbrane solution in type II string theory. By let us denote the set of noncompact directions, along which the brane exhibits translational symmetry. This solution may be charged electrically or magnetically under a form gauge field, with a corresponding charge . An electrically charged brane will be called smeared along , whenever it is not charged along this direction. For magnetically charged branes, we follow the opposite convention and call the brane smeared along , whenever it is charged along .^{7}^{7}7This definition is also very natural for D3branes which are selfdual under electricmagnetic duality. In principle, the choice of thermodynamic ensemble depends crucially on whether we treat as a parameter that has been fixed or as a parameter that can vary freely.
In the first case, we consider the system in the canonical ensemble, where the partition function depends on the temperature and the fixed charge. As explained in Section 2.2, in that case local thermodynamic stability requires the positivity of the specific heat (2.14). The situation is slightly different, when the charge is allowed to vary freely. Then, we keep the corresponding chemical potential fixed and discuss local thermodynamic stability in the grand canonical ensemble. In this ensemble the partition function depends on and the temperature and the Hessian of the mass is positive definite precisely when (2.16) is satisfied.
The proposal is that thermodynamic computations should be done in the grand canonical ensemble with respect to the charge , when the branes are smeared along at least one of the directions . The opposite should be true when the brane is not smeared in any of the directions . Note that this statement applies equally well to electrically or magnetically charged branes. Indeed, any sensible formulation of the CSC should be invariant under the electricmagnetic duality.
As a simple illustration, consider a nonextremal D0brane solution smeared on a transverse direction in the type IIA theory. The corresponding supergravity solution in the string frame takes the form
(3.1) 
The D0brane charge is smeared along the direction in this example and as a result it can be redistributed there freely. Hence, it is appropriate to consider local thermodynamic stability in the grand canonical ensemble, where is allowed to vary.
With a Tduality transformation along the background (3.1) turns into the D1brane solution
(3.2)  
The D1brane is now charged along the direction and the corresponding charge is a fixed quantity. Accordingly, we should now consider local thermodynamic stability in the canonical ensemble.
As another example consider the D0D2 bound state [15, 16, 17]. In this system, the D0brane is fully embedded inside the D2brane and smeared along its two transverse directions inside the worldvolume of the D2brane. Then, according to the previous discussion, the CSC should be applied in the grandcanonical ensemble with respect to the charge of the D0brane, but in the canonical ensemble with respect to the charge of the D2brane.
The above choice of thermodynamics fits very nicely with the classical stability properties of these solutions. As we see explicitly in later sections, the smeared D0branes exhibit a GL instability, which persists even in the nearextremal limit. The thermodynamic instability arises naturally in the grand canonical ensemble, but is absent in the canonical ensemble above some critical value of the charge where the specific heat is strictly positive (see eq. (2.18)). The D1branes, on the other hand, are known to be stable in supergravity for large enough charges. This feature is captured correctly by the local thermodynamic stability analysis in the canonical ensemble.
More generally, it is wellknown in thermodynamics that one can move back and forth between the canonical and grand canonical ensembles with the appropriate Legendre transform. Here we see that within the context of the CSC it is natural to associate a Legendre transform with a Tduality transformation in supergravity. The appearance of the Legendre transform is an essential feature of supergravity that treats momentum and winding modes asymmetrically. In the full string theory, where momentum and winding are exchanged by Tduality the Legendre transform would be unnecessary. A momentum instability for a smeared brane would transform under Tduality into a winding instability for a wrapping brane.^{8}^{8}8Similar statements on this point appeared in [16].
3.2 The CSC for smeared branes
It is interesting to consider the proof of the CSC for smeared black branes in the grand canonical ensemble. This is a first step towards the extension of the proof of [9] in more general situations, where besides the mass one can also vary an additional set of charges.
More specifically, consider the case of a nonextremal smeared Dbrane on a transverse direction , given in Eqs. (2.5). In order to address the relation between classical stability and thermodynamics we repeat the basic elements of the argument that appears in [9] (we refer the reader to that paper for a more detailed description). In a nutshell, one has to show the following facts (see also the recent discussion in [17]):

Demonstrate the existence of an appropriate family of Euclidean backgrounds for which the Euclidean action (relative to flat space) takes the form
(3.3) The generic point in this family is an offshell background, which does not satisfy the Einstein equations of motion, but satisfies the appropriate Hamiltonian constraints. In the present case, we consider a twoparameter family of backgrounds parameterized by two variables, and . This should be contrasted to the unsmeared case analyzed in [9], where the corresponding family is onedimensional. In (3.3) is the inverse temperature and the chemical potential. For some value of the background becomes an exact solution of the equations of motion. At this point the parameters , and are precisely the energy, entropy and charge of the corresponding black brane. For other values of the interpretation of the functions , and is not important. An explicit construction of this twoparameter family will be discussed in a moment.

Verify that the norm of the onshell perturbations on the space of theories is positive. This ensures that the analysis is restricted to normalizable onshell perturbations excluding any nonphysical negativenorm conformal perturbations. The precise form of the norm is related to the Lichnerowicz operator and can be defined as follows. For compactness, let us denote the field perturbations collectively by . is in general a multiindex label and the fields may include scalar, metric and gauge field perturbations. The ansatz for a static GL mode is and the norm on the space of perturbations is defined through the metric . This metric appears in the linearized equations of motion in the following way
(3.4) One must check that is positive definite. This will ensure that the action decreases, and therefore an instability exists, precisely when the Hessian fails to be positive definite. Indeed, for quadratic perturbations
(3.5) 
A final point, which was emphasized also in [17], is the following. One should demonstrate that there is sufficient overlap between the offshell deformations of point and the actual onshell perturbations . In [9], it was pointed out that a path in the family of offshell geometries is not related directly to an eigenfunction of the Lichnerowicz operator, but rather some linear combination of eigenfunctions. This suggests that when the action decreases along this path, at least one of the eigenvalues of the Lichnerowicz operator must be negative and therefore an actual onshell instability should exist. It is not immediately obvious, however, that the converse is also true. To complete the proof one should demonstrate that the offshell deformations and the actual onshell perturbations cover the same linear space.
Before addressing each of the above points in the case of the smeared Dbranes (2.5) it will be useful to recall how the above points facilitate the connection between the thermodynamics and the classical stability analysis. First of all, since the black branes (2.5) are actual solutions of the Einstein equations of motion they extremize the action at special points of the twoparameter family. This implies the vanishing of the first derivatives
(3.6) 
or equivalently the standard equations
(3.7) 
The quadratic perturbation of the action along the surface parameterized by involves the Hessian matrix
(3.8) 
which can be rewritten more compactly as a matrix equation
(3.9) 
In this expression is the Jacobian of the transformation from the variables to and is the Hessian of the Gibbs free energy. Assuming to be an invertible matrix, equation (3.9) allows for a direct connection between classical stability and local thermodynamic stability. Indeed, given the validity of points () and above, classical stability requires to have positive eigenvalues and through (3.9) this requirement translates to being negative definite. We have already argued that this is equivalent to the conditions (2.16).
We now proceed to demonstrate points  above. The first point requires the construction of a twoparameter family of Euclidean backgrounds for smeared Dbranes that satisfies (3.3). The detailed construction appears in Appendix B and boils down to the following facts. One can show that there is at least one twoparameter family of Euclidean backgrounds satisfying all the Hamiltonian constraints. These constraints include the vanishing of the Hamiltonian, the vanishing of ten momenta and the Gauss constraint. The requirement (3.3) is an automatic consequence of the validity of these constraints [33, 34]. The explicit form of the family arises from the general spherically symmetric ansatz
(3.10) 
in the following way. The functions and have to be chosen so that the Hamiltonian constraints are satisfied and so that for specific points in the family we get the onshell solutions (2.5) in the Einstein frame. In addition, certain boundary conditions must be imposed. As usual for gravity in the canonical ensemble, boundary conditions are imposed at an asymptotic boundary at , where the Euclidean time direction is compactified at a radius . The functions and are both vanishing at the horizon and the relation between and follows from the condition of regularity at , which in general takes the form
(3.11) 
The twoparameter family of Appendix B is based on the special ansatz
(3.12) 
where is some integration constant that can be fixed by imposing the appropriate boundary conditions. With this ansatz one can satisfy the Hamiltonian constraints by suitably expressing the functions and in terms of and , which remain free. This gives rise to a twoparameter family of black brane backgrounds controlled by the free functions and . In this manner, the family appearing in Appendix B is a generalization of the oneparameter family of Euclidean geometries presented in [9], following previous work in Refs. [35, 36]. In [9] one could satisfy the Hamiltonian constraints with a set of backgrounds depending on one free function, here we can satisfy them with two free functions.
The choice of the functions and is arbitrary up to certain boundary conditions. For the metric they are as follows. First, we keep the boundary conditions at the boundary invariant. This gives a single temperature for the whole family. Also, to keep the topology invariant the functions and continue to vanish at and by regularity at we demand that their derivatives satisfy (3.11). For , should remain positive. Similarly for the gauge field, in order to obtain a family with a single chemical potential , we want to keep the value of the gauge field at fixed. Since we can satisfy all of these conditions on the actual onshell solutions (2.5), we can also satisfy them for the above offshell family, at least infinitesimally close to the onshell points. This is enough for the purpose of proving the CSC.^{9}^{9}9Similar statements apply to smeared black branes in the nearextremal limit.
Note that we can deduce a completely different offshell family of Euclidean black branes by applying the boost/Uduality procedure of Section 2.1 on an elevendimensional neutral offshell family of the same form as in [9]. There are several problems with such a family. First of all, it would be a oneparameter family, since we start with a oneparameter family in eleven dimensions. Secondly, it would not satisfy all the Hamiltonian constraints of type II supergravity, but only part of them  more precisely, a linear combination of the Gauss constraint and the Hamiltonian. This fact alone would be enough to show that this family has the property (3.3),^{10}^{10}10To demonstrate (3.3) we need to show that the Hamiltonian of the system receives contributions from boundary terms only. This is true for the offshell family coming from the boost/Uduality procedure, because the Hamiltonian receives only boundary contributions in eleven dimensions. This continues to hold through the boost/Uduality procedure. Indeed, the specific family is timeindependent and hence the Hamiltonian is proportional to the Lagrangian, which is Uduality invariant. but this would not be fully satisfactory for the purposes of the CSC proof.
The boost/Uduality procedure is much more useful in demonstrating the second point . The actual form of the onshell perturbations will be discussed in the next section, but we are already in position to argue in favor of the validity of . From the GL analysis it is clear that the norm of the onshell perturbations in eleven dimensions is positive. The norm is clearly invariant under rotations and boosts and should remain positive under Uduality, which is of course a symmetry of the theory. Thus, the positivity of the norm is guaranteed.
The final point is slightly more subtle. Even in the unsmeared case of [9] this is a point that has not been shown rigorously. We do not have anything new to add on this point for the present case of smeared Dbranes, but in general it is natural to expect that there will be sufficient overlap between the eigenfunctions of the Lichnerowicz operator and the family of offshell deformations for systems that are specified uniquely by the full set of conserved charges. For example, systems with scalar hair do not fall into this category. This crucial point was put forward in a very recent paper by Friess, Gubser and Mitra [18], who found explicit counterexamples to the CSC. In the context of the proof outlined in the present paper, it is clear that in these examples the validity of point breaks down.
Comments on the general proof of the CSC
The correlated stability conjecture is expected to hold for more general gravitational systems with chemical potentials and conjugate charges ( is an arbitrary positive integer). The basic elements of Reall’s proof in this general case have been sketched recently in [17] and are a straightforward generalization of the points (), () and () appearing above. More specifically, in the grand canonical ensemble one has to show the existence of an parameter family of offshell deformations with Euclidean action
(3.13) 
and then verify the validity of the points () and () above.
The CSC in the presence of compact directions
We would like to conclude this section with a few words about the validity of the CSC in the presence of compact directions. Clearly, this is a situation where the CSC is expected to break down. This can be seen explicitly, for example, in neutral uniform black strings on a transverse circle (2.1). The specific heat of the uniform branch is always negative and local thermodynamic stability would suggest the presence of a classical instability for black strings of any mass. Instead, the explicit classical stability analysis by Gregory and Laflamme (see below) exhibits a marginally stable threshold mode at a specific critical GL mass and a classical instability below the critical mass. The uniform solution is stable above . It is not difficult to see what goes wrong with the CSC proof of [9] in this case. The arguments based on thermodynamics predict the existence of a negative eigenvalue mode for the dimensionally reduced Lichnerowicz operator, but the actual static GL mode is . For a transverse circle parameterized by the momentum is quantized in integer multiples of ( being the circumference of the ) and the static mode can exist only at specific values of the ratio (see (2.1) for the definition of ), or equivalently at specific values of the mass. The moral of this example is the following. In general, one has to fit the GL static mode into the compact directions and this is not automatic. Despite this complication, there is still some practical value in the CSC in the presence of compact directions. Although it does not fix the precise value of the critical point, it predicts that a static mode exists whenever it can fit into the compact directions. It is in this spirit that we use the CSC later on in the context of nonextremal and nearextremal branes smeared on a transverse circle.
4 The GregoryLaflamme mode for smeared branes
In this section we show that the boost/Uduality transformation reviewed in Section 2.1 can be used to transform the timedependent unstable mode of a neutral black string, found by Gregory and Laflamme, to a timedependent unstable mode of non and nearextremal smeared Dbranes. As explained below, this section relies crucially on an argument of [22].
4.1 Neutral black strings
As discovered by Gregory and Laflamme [2, 3], the black string in dimensional Minkowskispace (2.1) is classically unstable under linear perturbations of the form
(4.1) 
where , , and are functions for which the perturbation solves the linearized Einstein equations. Note that we define , , and to be functions of the parameter , which is well defined under the nearextremal and extremal limits in the charged case below. The gauge conditions on are the tracelessness condition (A.3) and the transversality conditions (A.4). The Einstein equations reduce to the four independent equations (A.5)(A.8) that appear in Appendix A.
Combining the gauge conditions with the linearized Einstein equations of motion one can derive a single second order differential equation on
(4.2) 
The explicit form of the dependent rational functions and appears in Appendix A. This equation depends only on , and the variable . Then one gets a curve of the possible values of and for which this equation has a solution. We plot a sketch of this curve found numerically in [2, 3] in Figure 1. Notice that the wellknown threshold GL mode appears at the critical value for which .
4.2 Nonextremal smeared Dbranes
In Section 2.1 we reviewed the boost/Uduality transformation of [20] that takes a neutral black string to a Dbrane solution smeared on a transverse direction. In this section we describe the transformation that takes the unstable GL mode of the neutral black string to an unstable mode of the nonextremal Dbrane solution smeared on a transverse circle.
The transformation should convert the neutral black string solution (2.2) to the smeared Dbrane solution (2.5). If we try naively to apply the boost/Uduality transformation reviewed in Section 2.1 to the GL mode (4.1) we find a nonnormalizable exponential dependence on (the critical case is an exception). As was pointed out in [22], to avoid this problem we may consider a complex metric perturbation, and try to find a complex transformation that i) Gets rid of the exponential dependence, ii) Still has the same effect on the zeroth order part of the metric, on the neutral black string metric. At the end one can take the real part of the transformed perturbation. Note that this last step works only because of the linearity of the perturbed Einstein equations of motion.
The precise form of the transformation we look is as follows. First define the coordinates and by the following complex rotation of the and coordinates
(4.3) 
Notice that the neutral black string metric (2.2) embedded in eleven dimensional gravity is invariant under (4.3) since . The transformation (4.3) is then supplemented by a boost that takes the coordinates to the coordinates given by
(4.4) 
These rotations are such that
(4.5) 
Then we can write the real part of the boosted perturbation of the eleven dimensional metric (2.4) as
(4.6) 
Using the relation we find
(4.7) 
where we have made the relabellings and . Notice that the complex rotation (4.3) precisely ensures that the exponential factor does not have a dependence. We can therefore apply the same Uduality transformations on (4.6)(4.7) as on the boosted neutral black string. After these Uduality transformations, consisting of one Sduality and Tdualities, we conclude that the perturbed nonextremal Dbrane smeared on a transverse direction can be written as^{11}^{11}11As usual, in the special case of D3branes the gauge field strength is self dual, so that we have .
(4.8) 
The perturbed Dbrane background appears here in one compact expression. From this one can easily deduce the explicit form of the perturbation by expanding to first order. The unstable mode appearing in [22] is equivalent to (4.8) for , though written in a different gauge.
Note that the functions , , and are still solutions to Eqs. (A.5)(A.8) with given by (A.2). In particular is a solution of Eq. (4.2). All of these functions depend on the variable just as in the case of the neutral black string perturbation.
Furthermore, we see from (4.5) that and . Hence, we can obtain as function of by using the functional dependence for the neutral black string (as sketched for in Figure 1). We note that the critical point is mapped to the critical point , which corresponds to the marginal mode of the Dbrane with a transverse direction. This marginal mode appears at the origin of the nonuniform phase of nonextremal Dbranes with a transverse direction [20]. From Figure 1 we also deduce that for any and the function never exhibits timedependent modes with wavelength smaller than the critical one.
The existence of the perturbation (4.8) proves explicitly that a nonextremal Dbrane with a transverse flat direction is always unstable as a gravitational background. This result meshes nicely with the unstable thermodynamics in (2.18), (2.19) and confirms the validity of the CSC if we consider it in the grand canonical ensemble. In contrast to this, a Dbrane wrapped on the Tdual circle is known to be classically stable in supergravity close to extremality. This fact is in agreement with the CSC if we consider it in the canonical ensemble (see (2.18)).
4.3 Nearextremal limit
We now consider the nearextremal limit (2.9) of the unstable mode (4.8) for nonextremal Dbranes with a transverse direction. The nearextremal limit of the unperturbed nonextremal solutions (2.5) has been discussed already in Section 2, giving the nearextremal Dbrane background (2.10).
Applying the limit (2.9) on the background (4.8) gives the following perturbation of the nearextremal Dbrane (2.10)
(4.9) 
We note that the nearextremal limit (2.9) keeps and fixed, and moreover keeps , , and fixed as functions of the variable which is now given by
(4.10) 
Therefore, the functions , , and are still solutions of the Eqs. (A.5)(A.8) with given by (A.2). In particular is still a solution of Eq. (4.2).
We have thus shown that the nearextremal limit of the timedependent mode of smeared branes is welldefined and hence that nearextremal smeared Dbranes are classically unstable. Then according to the CSC this means that nearextremal Dbranes should be thermodynamically unstable in the grand canonical ensemble. However, as noted in Section 2.2, we do not have a priori a definition of this ensemble for nearextremal branes, since it seems that the charge cannot vary anymore. We will resolve this puzzle in Section 6, where we will show that there does exist an appropriate definition of this ensemble in the nearextremal limit. We will see that in this ensemble nearextremal smeared Dbranes are indeed thermodynamically unstable, in accordance with the above result and the CSC.
5 Connection to marginal modes of extremal smeared branes
In this section we find a connection between the GL mode of charged smeared branes described in Section 4 and the marginal modes of extremal smeared branes.
5.1 Marginal modes for extremal smeared branes
In general, the solutions for extremal Dbranes distributed along a single flat direction is given as
(5.1) 
where the harmonic function obeys
(5.2) 
away from the source distribution. Given the appropriate boundary conditions the general solution is
(5.3) 
where the charge distribution function is arbitrary. Now, the extremal smeared Dbrane case corresponds obviously to being constant. But, it is clearly possible to add to the constant charge distribution a single mode of any wavenumber , in which case the harmonic function takes the form
(5.4) 
where is defined in (2.7) and the function solves
(5.5) 
The solutions of Eq. (5.5) are of the form , being the modified Bessel function of the second kind. The modes given by (5.4) are marginal in the sense that the modified solution solves the supergravity equations. In this sense the extremal smeared branes have marginal modes of any wavelength.
It is interesting to consider what happens to such marginal modes if one perturbs the extremal brane so that it gets a temperature. This is one of the things which we address in Section 5.2. Clearly from the study of nonextremal Dbranes distributed on transverse directions, one does not expect the existence of static solutions for any charge distribution .^{12}^{12}12In fact one only knows the case of the uniform distribution where is constant and the case of the localized distribution where with a constant and a length. These corresponds to the uniform and localized phase of nonextremal branes on a circle [20]. This is one example where gravity behaves vastly different from, say, electromagnetism where solutions with arbitrary charge distributions exist, although most of them are not stable when interactions are taken into account.
Since we shall explore in the following a connection with nearextremal branes, we need to consider briefly the nearhorizon limit (2.9) of the extremal Dbrane background given by (5.1) and (5.4). This gives
(5.6) 
Note that the function has been rescaled appropriately, and that still obeys Eq. (5.5).
5.2 Extremal limit of the GregoryLaflamme perturbation
In this section we consider the extremal limit of the GL mode for nearextremal smeared Dbranes, as found in Section 4.3. As we see in a moment, this gives a connection between the marginal modes of extremal smeared branes considered in Section 5.1 and the GL modes of nearextremal smeared branes.
Let us take the extremal limit of the perturbation of nearextremal Dbranes smeared on a transverse direction, as given by (4.9). The extremal limit corresponds to with and kept fixed. We first remark that since we want the argument of the cosine factor to remain finite, we need to stay finite in the limit. Hence, we need and to remain fixed in the limit . In terms of the dispersion diagram of Figure 1 this means that we move closer and closer to the point on the left part of the curve. We see that the variable defined above stays finite in the limit. Note also that this limit necessarily has the consequence that the timedependent exponential factor in in (4.9) disappears. One can now see from in (4.9) that we need to stay finite. By using this in Eqs. (A.3)(A.8) and demanding consistency of this system of equations, we get similar conditions on , and , so that in total the extremal limit is given as
(5.7) 
Applying the extremal limit (5.7) to the perturbed nearextremal solution given in (4.9), we get the following background