hep-th/9808079 NSF-ITP-98-076 McGill/98-13

The AdS/CFT Correspondence and a New Positive Energy Conjecture for General Relativity

Gary T. Horowitz^{†}^{†} E-mail:

Institute for Theoretical Physics, UCSB, Santa Barbara, CA 93106

Physics Department, University of California, Santa Barbara, CA 93106

and

Robert C. Myers^{†}^{†} E-mail:

Institute for Theoretical Physics, UCSB, Santa Barbara, CA 93106

Department of Physics, McGill University, Montréal, PQ H3A 2T8

We examine the AdS/CFT correspondence when the gauge theory is considered on a compactified space with supersymmetry breaking boundary conditions. We find that the corresponding supergravity solution has a negative energy, in agreement with the expected negative Casimir energy in the field theory. Stability of the gauge theory would imply that this supergravity solution has minimum energy among all solutions with the same boundary conditions. Hence we are lead to conjecture a new positive energy theorem for asymptotically locally Anti-de Sitter spacetimes. We show that the candidate minimum energy solution is stable against all quadratic fluctuations of the metric.

1. Introduction

There is growing evidence for a remarkable correspondence between string theory in Anti-de Sitter (AdS) spacetime and a conformal field theory (CFT) [1,2,3]. In particular, type IIB superstring theory on is believed to be completely equivalent to super Yang-Mills theory in four dimensions [1]. For many applications, it suffices to consider just the low energy limit of the superstring theory, namely, supergravity. There is a well defined total energy for any spacetime which asymptotically approaches AdS [4], and part of the correspondence is that this energy agrees with energy in the gauge theory. For solutions which approach AdS globally, there are positive energy theorems which ensure that this energy cannot be negative [5], in agreement with the stability of the gauge theory vacuum.

Witten [6] has suggested that one can describe ordinary (i.e., nonsupersymmetric) Yang-Mills gauge theory by compactifying one direction on a
circle and requiring antiperiodic boundary conditions for the fermions around
the circle. In this case, the additional fermions and scalars would
acquire large masses leaving the gauge fields as the only low energy
degrees of freedom.
On the supergravity side, this proposal corresponds to considering
spacetimes which are asymptotically AdS locally, but not globally. That is,
one spatial direction is compactified on a circle asymptotically.
If the spacetime topology is globally a simple product with an factor,
the standard approaches [5] should still yield a positive energy
theorem (see, e.g., [7,8]).
However, if one considers more general
topologies, e.g., for which the asymptotic
circle is contractible in the interior, those techniques will not
apply and hence it is uncertain if a positive energy
theorem will hold. It is known that in the case of asymptotically flat
spacetimes,
it does not:
These boundary
conditions allow nontrivial zero [9] and negative [10] energy solutions.
In particular, (for a fixed size circle at
infinity) there are nonsingular
solutions to Einstein’s vacuum field equations with arbitrarily negative
energy! Therefore, this sector of the theory is completely
unstable.^{†}^{†} This general result applies for any theory involving
Einstein gravity in higher dimensions, including superstring theory
[10]. The closely related positive action conjecture is also
false for spacetimes which are only locally asymptotically Euclidean [11].

It is important to determine whether a similar instability arises for spacetimes which are asymptotically locally AdS. From a mathematical viewpoint, the latter seems rather likely [12]. Negatively curved spaces tend to be less constrained than those with positive (or zero) curvature [13]. One expects that anything that is true for asymptotically flat spacetimes should also be true for asymptotically AdS spacetimes. Of course, if the result was true for the AdS case, it would have serious consequences in the context of the AdS/CFT correspondence. The straightforward interpretation would be that the supergravity analysis is making the rather dramatic prediction that the nonsupersymmetric, strongly coupled gauge theory is unstable. However, another possibility is that this result is an indication that the correspondence fails with the nonsupersymmetric boundary conditions. In the latter case, it would spoil the hope of using supergravity to learn about ordinary gauge theory.

We will show that there is a static nonsingular solution (to Einstein’s equation with negative cosmological constant) with these boundary conditions which has negative total energy. Rather than invalidate the AdS/CFT correspondence, this particular solution has a natural interpretation in the gauge theory. Since supersymmetry is broken by the antiperiodic boundary conditions on the fermions, the gauge theory on is expected to have a negative Casimir energy. Comparing the negative energy computed from supergravity and the Casimir energy in the weakly coupled gauge theory, we find close agreement. They have the same dependence on all parameters and disagree only by an overall factor of . This is similar to the factor of that was noticed previously in comparisons of the entropy of the near-extremal three-branes [14]. We will show that in fact these two factors have the same origin.

The key question is whether the solution described above is the lowest energy solution with these boundary conditions. If so, there must be a new positive energy theorem which ensures that the energy of all solutions is greater than or equal to this negative value, with equality only for our particular solution. At first sight, this seems very unlikely, since the solution we discuss does not have constant curvature, supersymmetry, or any other distinguishing property which have previously characterized minimum energy solutions in general relativity. Nevertheless, we will present evidence in favor of this new positive energy theorem. We will show that the solution is a local minimum of the energy, i.e., it is stable to small fluctuations. The existence of this new theorem can be viewed as a highly nontrivial prediction of the AdS/CFT correspondence. A complete proof would provide strong evidence for the correspondence.

The outline of this paper is as follows: In the next section, we review the definition of energy for spacetimes that are asymptotically AdS. In section 3, we present our solutions with negative total energy and discuss their relation to the CFT. Section 4 contains the statement of the new positive energy conjecture and some evidence in favor of it. In section 5, we consider some generalizations of the conjecture, and further discussion is given in section 6.

2. Energy in Anti-de Sitter Spacetime

The definition of total energy for spacetimes which asymptotically approach
AdS was first discussed in ref. [4]. In the following, we will adopt an
equivalent definition derived in [15] (see also [16]). The total
energy in general relativity is always defined relative to a background
solution which has a time translation symmetry.
Let the norm of the timelike Killing
field be^{†}^{†} When there is more than one timelike Killing field,
there are additional conserved quantities. The energy is then
one component of a conserved vector (or tensor).
We will focus on one timelike component and call it the energy.
. The energy depends only on and on the metric of
a spacelike surface which asymptotically approaches
the background geometry.
Starting from the action and deriving the Hamiltonian keeping track
of surface terms, one finds [15]:

where the integral is over a surface near infinity,
is the trace of the extrinsic curvature of this surface,
and is the trace of the extrinsic curvature of a surface with the same
intrinsic geometry in the background or reference spacetime.^{†}^{†} To leading
order, will be the same for both the given metric and the background
space. Higher order differences between and
will not affect the result for the energy [15].
This definition is very general and works for both asymptotically flat
and asymptotically AdS spacetimes.

Let us illustrate this definition with a few examples. Consider the Schwarzschild-AdS solution in four dimensions:

where is related to the negative cosmological constant by . Consider a spatial slice of constant in this space. At fixed , one has a round two-sphere with area . The integral of the trace of the extrinsic curvature of this sphere is easily computed as

where is the unit radial vector normal to the sphere. The background or reference spacetime is just anti de Sitter space, i.e., (2.2) with . At fixed , the boundary surface in the background with the same intrinsic geometry as above is again a two-sphere at the same value of the radial coordinate . Thus is simply given by (2.3) with . In either case, is constant on the sphere, and asymptotically approaches . Substituting these expressions into (2.1) yields as expected (where is Newton’s constant in four-dimensions).

This calculation is easily extended to arbitrary dimensions with the black hole metric

where is the metric on a unit -sphere, and . Also note that for the above metric. The final result for the energy is

where -sphere, and is the (+2)-dimensional Newton’s constant. is the area of a unit

Next consider the following asymptotically AdS metrics:

where . For certain values of , these metrics arise in the near horizon geometry of -branes (see, e.g., [1]). With , these metrics correspond to AdS space in horospheric coordinates [17]. Once again we consider a surface of constant . If we introduce as the coordinate volume of the surfaces parameterized by , then the area of a surface at fixed large is simply . Computing the energy as before yields

corresponds to the energy density of the field theory in the CFT/AdS correspondence.

There is a slight subtlety in computing the mass of the above metrics
(2.6). If the
directions along the brane are not identified (i.e., are noncompact),
then the constant
can be changed by rescaling the coordinates
in an appropriate way. Hence the energy
(2.7) is
not well defined. This is not surprising, since the energy is conjugate
to asymptotic time translations, and so if one rescales the time, the energy
should change.^{†}^{†} In the full asymptotically flat -brane
solution, this is not a problem, since the scale for is picked out by the
requirement that be a unit time translation at infinity.
It is this time which corresponds to time translation in the gauge theory.
In the previous metrics (2.4), the scale of is fixed
by requiring the spheres of constant radius to have an area given by
.
In the following, we will be interested in the case where at least one of the
spatial directions
is compactified.
If we fix the periodicity of the circle (corresponding to
fixing the size of the circle in the gauge theory) then cannot be
rescaled. However, when some of the ’s are
compactified, the background spacetime with
has a conical singularity at . We will not worry about
this singularity, since it is likely that string theory resolves it
without changing the asymptotic form of the metric, which is all that is needed
to compute the energy. More importantly, the lower energy solution we
describe in the next section is completely nonsingular.

3. Negative Energy Solutions

We begin by reviewing the negative energy solutions in the asymptotically flat context [10]. It is easy to describe the initial data for these negative energy solutions. For five dimensional solutions, the initial data consists of a four-dimensional Riemannian manifold which asymptotically approaches the flat metric on . Of course, within general relativity, this initial data must satisfy a number of constraint equations. However, if we set the conjugate momentum to zero, these constraints reduce to the condition that the scalar curvature vanish. As initial data, we consider the euclidean Reissner-Nordström metric

To avoid a conical singularity at , we must periodically identify and so (3.1) has the desired asymptotic geometry. It also satisfies the constraint, because the Einstein tensor is proportional to the Maxwell stress tensor, which is trace-free in four dimensions. We now analytically continue the parameter . (Since we are interested only in the metric (3.1) and do not include a Maxwell field, we do not have to worry about the latter becoming complex.) It is now clear that we can take the mass parameter without the metric becoming singular. Since the size of the circle at infinity is just the period of which depends on both and , one can keep this fixed as becomes arbitrarily negative. In fact, one finds, for a fixed period, that the curvature remains bounded as the mass becomes increasingly negative. Therefore one may conclude that nonsupersymmetric compactifications in asymptotically flat spacetimes are unstable. This analysis is quite general and may be applied to any theory involving Einstein gravity in higher dimensions, including superstring theory [10].

We now want to know if an analogous result holds for spacetimes which are asymptotically AdS. The first thing to try is the obvious generalization of the above procedure using the euclidean AdS Reissner-Nordström metric. When the charge in the latter is analytically continued, the metric becomes

As before, this satisfies the vacuum constraints (now with negative cosmological constant), if the momenta are set equal to zero. However, there is an important difference with the asymptotically flat case. In (3.2), the proper length of the circles parameterized by grows with . This means that the area of the surface at infinity grows like just like the uncompactified five-dimensional AdS. As a result, the mass is determined by the terms in the metric, rather than the term. The appropriate physical boundary conditions – see the discussion in section 5 – require that . Thus, this construction only yields the following one parameter family of finite energy initial data:

Note that one cannot change the sign of without introducing a naked singularity at . With the above sign, the radial coordinate is restricted to where is the largest root of . To avoid a conical singularity at , is identified with period

The metric (3.3) can also be obtained by analytically continuing the five-dimensional Schwarzschild AdS solution and restricting to the equatorial plane of the three-spheres. One might thus expect that its mass would be positive. However, we now show that it is negative!

The area of a surface at large is

The integral of the extrinsic curvature becomes

The background metric is simply (3.3) with , which is
four-dimensional hyperbolic space with periodic identifications^{†}^{†} Even
though the spacetime resulting from this initial data is locally AdS, it is
not globally static. So extra restrictions are needed to ensure energy
conservation. This will not be the case for our main example discussed below..
In this reference space, we need to choose a boundary surface with
the same intrinsic geometry as the at fixed above.
For the geometry to agree, the radial coordinate of the surface
in the background must be the same
as in the original spacetime. For the proper distances along the factors
to agree, the periodicity of in the background is related to
by

The integral of the extrinsic curvature in the background is simply

Using (3.7), , and the definition of the energy (2.1), one finds that

From this, one clearly sees that the energy is negative, but it is difficult
to see the dependence on the size of the circle since
is implicitly related to through (3.4) and the definition of .
For , ,
and so one obtains^{†}^{†} For a given , there is another solution to
(3.4) with a smaller value of , but it corresponds
to a configuration for which the energy which is less negative.

Through the AdS/CFT correspondence, this analysis should be related to a gauge theory on where has radius , has period , and supersymmetry breaking boundary conditions are imposed along the . For small , this is expected to have a negative Casimir energy density proportional to (at least at weak coupling). Hence the total energy would be negative and proportional to , in agreement with the above supergravity calculation.

The preceding calculations can be extended to arbitrary dimensions with the initial data metric

which satisfies the constraint equation . Again, the geometry smoothly closes off at , which is now the largest root of , provided that is identified with period

The final result for the energy is

where the final formula again holds for large . Where applicable, these results should be related to a quantum field theory on . Again, a negative energy proportional to can be expected to arise through the Casimir effect in the field theory.

To make a more precise comparison of the energy in AdS and the gauge theory, it would be useful to have an example of a solution with negative energy that asymptotically had topology . This is easily obtained by a double analytic continuation of the near-extremal -brane solution (2.6). That is, we analytically continue this metric with both and . In the following, we will refer to this spacetime as the AdS soliton. The metric becomes

where there are now ’s. Again, the coordinate is restricted to and must be identified with period to avoid a conical singularity at . Note that this spacetime metric is globally static and completely nonsingular. For fixed and , the area of a surface is , where denotes the volume of the transverse ’s. The appropriate background is (3.14) with , which corresponds to AdS space with periodic identifications. It is easy to see that for the background, . Following the above procedure, the energy is again found to be negative

Using the above relation between and , this result can be rewritten as

For certain values of , we can express this in terms of the string theory coupling and Ramond-Ramond charge . For example, for the three-brane, , we have , and is the ten-dimensional Newton’s constant, , divided by the volume of a five sphere of radius , . We thus obtain an energy density

We wish to compare this with the ground state energy of the gauge theory on , where the length of the is . This can only be calculated directly at weak gauge coupling, where to leading order, it reduces to the problem of determining the Casimir energy of the free field theory. The field theory is super-Yang Mills, which contains an SU(N) gauge field, six scalars in the adjoint representation, and their superpartner fermions. In the present case, the latter fermions are antiperiodic on the . The stress-energy tensor for this theory may be found in [18]. The leading order Casimir energy may be calculated by point-splitting the fields in the energy density (i.e., ) with the appropriate free-field Green’s function and then removing the vacuum divergence before taking the limit of coincident fields [19]. The final result is

Thus we find that the negative energy density of the supergravity solution is precisely of the Casimir energy of the weakly coupled gauge theory! This is very reminiscent of the earlier results showing that the entropy of the near-extremal three-brane is precisely the entropy of the weakly coupled gauge theory at the same temperature [14].

In retrospect, it is not surprising that the ground state energies differ by exactly the same factor as the thermal entropies, as both results can be derived as different interpretations of a common euclidean calculation. On the field theory side, consider the euclidean functional integral for the (weakly coupled) Yang-Mills theory on where the circle has period and antiperiodic boundary conditions for the fermions. One natural interpretation is as a thermal field theory calculation at temperature , and the partition function yields the free energy as . Alternatively, one may interpret one of the noncompact directions as euclidean time . In this case the same partition function, evaluated between two surfaces separated by a large difference of euclidean time , yields the ground state energy as . On the supergravity side, consider the euclidean instanton obtained by analytically continuing the AdS soliton metric (3.14) with . This instanton is, of course, identical to that obtained from the black hole metric (2.6) with and identifying with the appropriate period . (We also trivially rename .) In the latter context, the instanton describes a thermal equilibrium of the black hole [20] at temperature , and the euclidean action is interpreted as giving the black hole contribution to the free energy as . On the other hand, in the context of the AdS soliton, the same euclidean action, is simply related to the total energy (3.16) via . Now from the analysis of three-branes [14] (), it follows that when the temperatures of the black hole and Yang-Mills theory are equated, their free energies are related by:

and hence

Hence from the preceding discussion, it also follows that we must find the same factor in relating the total energies, , and the energy densities above, as well.

The factor of discrepancy between the two calculations does not contradict the AdS/CFT correspondence. Rather the supergravity result (3.17) corresponds to the energy density of the gauge theory in a regime of strong coupling. To extrapolate the AdS results to weak coupling, one must include all of the higher order (in the string scale) corrections to the geometry induced by the Type IIB string theory. The leading order correction to (3.14) has recently been computed in [21]. The net effect is that the euclidean action became more negative. Thus from the preceding discussion, as expected, the energy and energy density of the corresponding AdS soliton becomes slightly more negative, improving the agreement with the weak coupling results (3.18).

In order to construct these negative energy solutions (3.14), we need so there will be one spatial direction in (2.6) to analytically continue. The case , corresponding to three spacetime dimensions, is special. All solutions have constant curvature and hence are locally AdS. The metric (2.6) with is the (nonrotating) BTZ black hole [22]. We showed above that if you take the positive mass black hole and double analytically continue, then you get a solution with less mass than the zero mass black hole. However, it is known that three-dimensional AdS spacetime itself has less mass than the black hole [22]. In fact, as we will now show, the double analytically continued black hole is precisely AdS globally, with no extra identifications. We start with the black hole metric (2.6) with

By rescaling , we can set . Now analytically continue in and as before to get

Finally set and to put this metric into the standard AdS form

Note that after the analytic continuation, should be periodically identified to avoid a conical singularity. In (3.23), this is simply the statement that is an angle with the standard periodicity of .

4. A New Positive Energy Theorem?

4.1. The conjectures

The above qualitative and quantitative agreements between AdS energy and Casimir energy in the CFT seem to support the AdS/CFT correspondence in the nonsupersymmetric case. A crucial question though is whether the AdS soliton (3.14) is the lowest energy solution with the given boundary conditions. The aforementioned agreement would be put in peril by the existence of metrics with even lower energies.

For definiteness, let us focus on the case in the following. This corresponds to the near horizon geometry of Dirichlet three-branes, for which the AdS/CFT correspondence is understood in most detail. The AdS soliton metric (3.14) then becomes

where and has period . For the remainder of this section, we will use (4.1) as our reference metric and measure energy relative to it. We will consider metrics which asymptotically approach (4.1) in the sense that for large

Derivatives of are required to fall off one power faster. In (4.2), denotes the AdS soliton (4.1) (or metrics obtained from it as indicated below). Note that even though these boundary conditions allow metrics with different constants asymptotically, the periodicity of is fixed.

The AdS/CFT correspondence together with the expected stability of the nonsupersymmetric gauge theory, suggests that the energy of any solution with these boundary conditions should be positive relative to (4.1). Hence we are lead to formulate a new positive energy conjecture. Below we present three different forms of this conjecture, starting with the most general and becoming more specialized. The simpler conjectures may be easier to prove, but would still be of great interest. Conjecture 1: Consider all solutions to ten-dimensional IIB supergravity satisfying (4.2) (with denoting the product of (4.1) with a five sphere of radius ). Then , with equality if and only if . The self-dual five form must be nonzero to satisfy the asymptotic boundary conditions. If we make the reasonable assumptions that the other supergravity fields will only increase the energy, and spacetimes which are not direct products with will also have higher energy, then the above conjecture can be reduced from ten dimensions to five as follows.

Conjecture 2: Consider all solutions to Einstein’s equation in five dimensions with cosmological constant satisfying (4.2) (with denoting the metric (4.1)). Then , with equality if and only if .

In the above conjectures, the solutions are required to have at least one nonsingular spacelike surface, since otherwise one could easily construct counterexamples with naked singularities. If we assume that there is a surface with zero extrinsic curvature (i.e., a moment of time symmetry) then the constraint equations reduce to the statement that the scalar curvature is constant. We thus obtain:

Conjecture 3: Given a nonsingular Riemannian four manifold with satisfying (4.2) (with denoting the metric on a constant surface in (4.1)), then with equality if and only if .

As we mentioned in the introduction, at first sight these conjectures seem unlikely to be true. The solution (4.1) does not have constant curvature, supersymmetry, or any other special property usually associated with minimum energy solutions in general relativity. It is possible that the above conjectures fail, but there is another solution of minimum energy. However, this is unlikely, since one expects the minimum energy solution to be static and translationally invariant around the circle. One could then double analytically continue this metric to produce a new black hole solution. The “black hole uniqueness theorems” (which have not been proven for this case, but still are believed to be true) would then imply that this solution must be identical to (2.6) with , which corresponds to the analytic continuation of (4.1). Previous experience would suggest that there are time dependent solutions of arbitrarily negative energy — see, e.g., [10].

Nevertheless, in this section we present some evidence that the above conjectures are indeed true. First we note that under perturbations of the metric (4.1), the energy is unchanged to first order. This result in fact applies for any metric that is globally static [23], and can be seen as follows. The gravitational Hamiltonian is a function of the spatial metric and conjugate momentum, and takes the form . where is the lapse-shift vector and are the constraints. Suppose we start with a static solution and choose to generate evolution along the time translation symmetry. Consider the variation of with respect to . On the one hand, this is which vanishes since the background is static. On the other hand, the variation of the constraint will vanish whenever the perturbation solves the linearized constraints. Hence the variation of the energy must also vanish. Since is independent of the conjugate momenta, this is sufficient to establish that the energy is an extremum.

4.2. Perturbative stability

While we have established that the mass of the AdS soliton is an extremum, we would like to show that it is actually a global minimum. Unfortunately, given the (nonsupersymmetric) spin structure on the asymptotic geometry, we can not apply the spinor techniques of [24] and [5] to argue that this is the case. Instead we must be satisfied with showing that the AdS soliton gives a local minimum of the AdS energy functional. Our perturbative approach here follows that of [4], where the stability of the AdS spacetime itself was considered. We refer the interested reader there for a detailed discussion of the technique. Their general analysis is based on the construction of conserved charges for background solutions with Killing symmetries, in particular a time translation symmetry, and hence may be applied to the AdS soliton.

One begins by dividing the metric (globally) in a manner similar to (4.2)

where is the AdS soliton and represents a deviation satisfying the above boundary conditions. For the moment we will work with the general -dimensional case, and later specialize to . In order that is still a solution, must satisfy an equation which may be represented as a linearized Einstein equation with a nonlinear source term. Written in this form, the terms nonlinear in may be taken to define the energy-momentum density of the gravitational field, . By virtue of the field equations, this density is covariantly conserved in the background metric, i.e., . Now given a Killing vector of the background solution, one finds then that is a covariantly conserved current and hence

is a conserved charge. If is a timelike vector, this quantity defines the Killing energy, i.e., the mass of the new metric (4.3) with respect to the background solution. Further, Abbott and Deser show that the integrand of eq. (4.4) is a total divergence, and so may be written as a flux integral over a ()-dimensional surface at infinity. The details of this calculation are not important here, however, we note that with this flux integral form one may show that this Killing energy (4.4) agrees with our previous definition of the energy (2.1)[15].

Instead we wish to construct (or rather the energy density ) directly to second order in the fluctuations . Abbott and Deser [4] turn to the framework of canonical gravity for this purpose. One may view their construction as evaluating the second order term in the field equations by making a third order expansion of the action with a judicious choice of variables. The canonical variables provide a judicious choice for several reasons. First, since one wishes to focus on , the third order expansion need only be carried out for terms linear in the lapse or shift, and quadratic in the spatial metric and conjugate momenta. Second, for the present background solutions of interest, we will be evaluating on a time symmetric slice, and so the background momentum variables vanish. Finally, the background shift vector also vanishes in the solutions considered here. The net result is that one must calculate

with the quantity in square brackets evaluated to second order in the deviations of the spatial metric and the conjugate momentum deviations . Here is the intrinsic curvature scalar of the initial data surface. For convenience, the following gauge conditions are imposed on the fluctuation fields [4]

where is the ()-dimensional covariant derivative on the initial data surface with the background metric. The deviations are also required to satisfied the constraint equations to linear order, which imposes

One can see that together (4.6) and (4.7) ensure that the
fluctuations are transverse and traceless with respect to the background
metric. Evaluating (4.5) subject to these constraints, one arrives
at the following expression^{†}^{†} One must integrate by parts to arrive
at this expression, however, the above boundary conditions (4.2) will
ensure the vanishing of any boundary contributions.

Here it is immediately apparent that the momenta make a manifestly positive contribution to the energy density. Hence if we are interested in lowering the energy of the background solution we should set and focus on the spatial metric fluctuations. For the latter, there is a gradient energy density, which is also positive, and a potential energy density, which a priori has no definite sign. If we consider the background to be anti-de Sitter space, for which

the potential becomes

where . Hence in AdS space, the potential energy and hence the total energy contribution of the spatial metric fluctuations is also manifestly positive, and we may conclude that AdS space is perturbatively stable. Of course, spinor techniques [24] allow one to show that AdS is in fact the absolute minimum energy state within that sector of the theory, i.e., for solutions which admit asymptotically constant spinors. The AdS soliton is not included in this sector, as the spin structure on the asymptotic boundary differs. Further the latter metric does not have a Riemann tensor with the maximally symmetric form of eq. (4.9).

As for our conjecture, the remainder of the discussion will be restricted to the AdS soliton with , i.e., spacetime dimension . In this case, the metric on a constant time slice in (4.1) is

In the following, we will actually refer all indices to the obvious orthonormal frame. Now the curvature of this spatial slice is given by

where . Now the potential term in eq. (4.8) becomes

Given that , we see that the potential ensures the stability of all fluctuations in the off-diagonal components of the metric. Now in evaluating the potential for the diagonal fluctuations, we first impose the traceless condition of eq. (4.7) with ––. Then defining , the remaining potential terms may be written as

where

It is straightforward to determine the eigenvalues of this matrix to be

One easily shows that and are positive in the range of interest, i.e., , and hence the corresponding eigenvectors correspond to manifestly stable metric fluctuations. The most interesting case is that of for which one finds for but for . Hence the potential energy density for this eigenvector

becomes negative in a small region near the center of the space, i.e., .

Thus metric fluctuations with a form where the are dominated by this eigenmode and only have support in this small region near seem to have the potential to be unstable, i.e., lower the energy of the background solution. However for such fluctuations, there is a competition between the manifestly positive gradient contributions and the potential terms in the energy density (4.8). We argue below that the former terms dominate and hence these fluctuations are also stable.

Imagine that we are considering a metric fluctuation which we might characterize as where is a profile, which we assume takes its maximum at , monotonically decreases, in order to minimize the gradient energy, and vanishes outside . The potential energy density (4.14) becomes . With the assumption that the profile takes its maximum at , the minimum of the potential energy density is

While the complete expression for for this fluctuation has a complicated analytic form, for the purpose of the reader’s intuition we note that to within an accuracy of a few percent one may approximate this expression in the region of interest, i.e., , by the simple expression . So if one imagined that the profile was constant, the potential energy density would decrease linearly as a function of reaching zero at .

We estimate the gradient energy as follows

where we further estimate the gradient of the profile by its maximum divided by the proper distance between and , i.e., between and . The latter distance turns out to be . In the region of interest, varies from –1 at to at so we will simply fix it to in our estimate of . Hence we arrive at the following estimate

Comparing eqs. (4.18) and (4.20), we see that this average gradient energy already exceeds the minimum value of the potential energy. Hence it must be that these potentially unstable metric fluctuations in fact have a positive total energy. While our estimate of the gradient contribution may seem crude, a more detailed examination shows that in fact it greatly underestimates the energy. Properly evaluating the covariant derivatives in eq. (4.19) accounting for the tensor properties of the fluctuations adds more positive terms to this expression, which are roughly the same order of magnitude as those considered, i.e., . Further we have not accounted for the gauge fixing constraint (4.6) in our calculations. This constraint fixes the form of the profile for the fluctuation considered above through , which is the only nontrivial component. One finds that the profile must decay more slowly than estimated above, but that it cannot vanish at . Rather it has infinite support, vanishing as in the asymptotic region. While this decreases the local gradient energy density, it also adds a positive potential energy density in the region . Hence the final conclusion that these fluctuations are stable remains correct in a detailed analysis.

5. Generalizations

Although we have focussed on the case above, one can extend the conjecture to other dimensions, including the case which is directly related to four dimensional nonsupersymmetric gauge theories [6]. We believe the solutions (3.14) for all are perturbatively stable, although a detailed analysis of the fluctuations has not yet been carried out.

Consider the following modification [25] of the metric (2.4):

where

is the metric on a -dimensional unit hyperboloid.
This metric is also a solution of Einstein’s equation with negative
cosmological constant, and is asymptotically AdS.
It is unusual since the black hole metric (5.1) (without any analytic continuation) can have negative energy [25].
Indeed, the energy
is still given by (2.5) (with denoting the area of the unit
hyperbolic
space^{†}^{†} One can either compactify this space so that its area is finite, or
work with the energy density.) but now one can let the parameter
be negative and still have a
horizon. There is, however, a minimum energy possible for the black hole.
For example, in the case ,
this occurs when , corresponding
to an energy

The Hawking temperature of these minimum energy black holes is zero. Are they the minimum energy configurations with these boundary conditions?

One can double analytically continue this metric, and . As usual, must be periodically identified with period

so a constant time surface asymptotically approaches the product of hyperbolic space and a circle. The energy is again given by (3.13) and hence is negative for positive values of . For large , the energy is the same as the metric (3.11), since the size of the circle is much smaller than the scale of the curvature on the orthogonal space.

One might conjecture that the metrics (5.1) represent the minimum energy
configurations for these boundary conditions.
Since these solutions are static, they are extrema of the energy. It is likely
that they are also local minima of the energy. In fact the calculations
in section 4 show that this is the case for . This is due to the
remarkable fact that the components of the curvature of (5.1) (in an
orthonormal frame) are identical to (2.4)!^{†}^{†} This
provides a counterexample
to the popular idea that if two metrics have the same curvature, they are
locally isometric. The covariant derivatives of the curvature are different,
showing that the metrics are indeed inequivalent.
We thank S. Ross for discussions on this point. If one analytically
continues in and and restricts to a constant time surface, the
curvatures are still the same. Since the potential term in the
quadratic fluctuations depends only on the curvature, it will again be
positive.

From the CFT viewpoint, the above metrics should describe the CFT on
a product of and a hyperbolic space. Since the scalars couple
to the curvature, a negative curvature space would seem to lead to an
instability.^{†}^{†} We thank J. Maldacena for pointing this out.
Thus, in this case, the apparent stability of the supergravity
solution seems in contrast to the expected CFT result. We do not yet
understand the resolution of this puzzle.

6. Discussion

We have shown that the AdS soliton (3.14) has lower energy than AdS itself. Rather than producing a contradiction with the recently conjectured AdS/CFT correspondence, these results find close agreement with the negative Casimir energy of nonsupersymmetric field theory on .

Since the AdS soliton has extended translational symmetry, one can define not just the total energy, but a full boundary stress-energy tensor. One finds that the agreement in the energy densities, discussed here, extends to agreement of all of the components of the stress tensor [26]. In the example of the gauge theory on , one finds that the factor of relating the energy densities calculated with supergravity and in the weakly coupled gauge theory is, in fact, an overall factor relating the full stress tensors calculated in these two regimes.

It is examining the full stress-energy tensor which motivated in part our choice of physical boundary conditions (4.2) in the positive energy conjecture and in our discussions in the earlier sections. For example, if one retains the term in our first negative energy example (3.2), one might expect that the energy would be divergent. One finds though that there is a precise cancellation in the calculation so that the final result remains exactly the same as in (3.9), which was derived for . This is also true for the translationally invariant solutions. However, if one considers the full boundary stress tensor, a nonvanishing produces divergences in the spatial components of the stress-energy. One should expect that a less symmetric choice of the initial data surface would yield an energy density which mixes the various components of the latter tensor, and hence diverges. Our physical boundary conditions (4.2), which rule out including , ensure that the energy density will remain finite for any choice of time slicing.

A precise comparison of the supergravity and gauge theory energies was only attempted for because this is the case in which the AdS/CFT correspondence is best understood. For this dimension, we only considered the AdS soliton solution, which corresponds to the gauge theory on . However, one might also consider the initial data (3.3), which would correspond to the gauge theory on . We calculated the supergravity energy (3.9), and need only translate it to a gauge theory expression, as in going between (3.15) and (3.17). The final result is that (3.9) yields

where the function satisfies =1. Hence to leading order for small , the energy density is precisely the same as for . This is not surprising as this result is valid in the limit where the radius of the sphere is much bigger that the period of the circle , and so the factor looks essentially like a flat on the latter scale. One should note that this energy is measured relative to the standard AdS background with periodic identifications, which naturally includes the curved factor in its asymptotic geometry. Thus this negative energy does not include the positive contribution which might be expected to appear as the Casimir energy of the two-sphere. It is not difficult to calculate the complete function

Taylor expanding for small would yield higher order corrections to the energy density. It would be interesting to see to what extent the coefficients of the higher order terms are reproduced in the weakly coupled regime of the gauge theory. This function is only defined for since these are the allowed values of from (3.4). (The solutions (3.3) only exist if the circle is small enough.) This suggests that there might be a jump in the ground state energy of the strongly coupled gauge theory on , analogous to the Gross-Witten-like phase transition [27] discussed in [3].

The agreement of the negative energy of the AdS soliton with the expected negative Casimir energy appears to support the AdS/CFT correspondence even in the nonsupersymmetric case. But this is true only if the AdS soliton is the lowest energy solution with the given boundary conditions. This gives rise to a new type of positive energy conjecture. Although this conjecture seems unlikely to be true from a purely mathematical standpoint, we have presented evidence to support it, by showing that the AdS soliton is indeed a local minima of the energy (for the case ). It is natural to extend this conjecture to other dimensions and other asymptotic boundary conditions as we discussed in the previous section.

Our analysis of the perturbative stability of the AdS soliton with respect to metric fluctuations is closely related to the recent calculations of glueball masses in large N QCD [28]. General arguments have been given that massless supergravity fields propagating on the AdS soliton background will have a discrete spectrum [6]. Further, by the AdS/CFT correspondence, these fluctuations should correspond to various glueball states in the large N gauge theory [6,29]. In the context of our first form of the conjecture, the given calculations of glueball spectra [28] verify that the AdS soliton is stable against fluctuations of many of the bosonic supergravity fields, e.g., the dilaton, and Neveu-Schwarz and Ramond-Ramond antisymmetric tensors. While our perturbative calculations have not produced a precise spectrum, they do verify that a positive mass gap exists for the metric fluctuations, i.e., the spin-two glueballs. It is interesting that amongst the metric excitations, our calculations indicate that the lowest energy state (i.e., the mode described as potentially unstable) must in fact contain a scalar contribution (with respect to ) which should actually decouple as the ultraviolet regulator is removed.

Lest the readers imagine that the AdS/CFT correspondence guarantees the local stability of all (static) supergravity solutions, we remind them that this is not the case. Typically there are many unstable stationary points in the scalar potential of the gauged supergravity theory – see, e.g., [30]. For these stationary points, there will be an AdS background, but the cosmological constant will have a value such that supersymmetry is completely broken. In the supergravity analysis, the instability arises because some of the scalars have curvature couplings which exceed the Breitenlohner-Freedman bound [31], i.e., scalar fluctuations around the stationary point have masses which are (too) negative. It would be interesting to understand what the corresponding physics in the gauge theory is.

Acknowledgments

We thank G. Gibbons and the participants of the String Duality program at the Institute for Theoretical Physics, Santa Barbara for helpful discussions. GTH was supported in part by NSF Grant PHY95-07065. RCM was supported in part by NSERC of Canada and Fonds FCAR du Québec. Both GTH and RCM were supported by NSF Grant PHY94-07194 while at the ITP.

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