Holography, Probe Branes and Isoperimetric Inequalities

In many instances of holographic correspondences between a d dimensional boundary theory and a d+1 dimensional bulk, a direct argument in the boundary theory implies that there must exist a simple and precise relation between the Euclidean on-shell action of a (d-1)-brane probing the bulk geometry and the Euclidean gravitational bulk action. This relation is crucial for the consistency of holography, yet it is non-trivial from the bulk perspective. In particular, we show that it relies on a nice isoperimetric inequality that must be satisfied in a large class of Poincar\'e-Einstein spaces. Remarkably, this inequality follows from theorems by Lee and Wang.


Introduction
Consider a holographic correspondence between a (d + 1)-dimensional bulk gravitational theory on a conformally compact manifold M and a d-dimensional field theory on its compact boundary X = ∂M . 1 Assume that the correspondence follows by considering the near horizon limit of a large number N of BPS (d − 1)-branes (mentioned simply as branes in the following) [1]. The boundary field theory has N colors and can be interpreted as living on these branes. It is then natural to study the physics associated with probe branes in the bulk geometry. These branes are clearly special since, in some sense, they make up the bulk holographic space itself. Such studies have appeared many times in the literature; particularly instructive results were discussed, for example, in [2].
Recently, a precise construction of the probe brane action S b from the point of view of the boundary field theory was proposed [3]. The main motivation in [3] is to provide purely field theoretic tools to study holography in a wide range of models. It is shown that the probe action naturally describes the motion of the brane in a higher dimensional holographic bulk space, including in the case of the pure Yang-Mills theory where a fifth dimension automatically emerges [3]. In particular, the details of the bulk geometry can be read off from the probe action [4].
The construction in [3] implies an interpretation of the probe brane that seems to depart from the standard lore, which relates the presence of a probe brane in the bulk to some Higgsing of the gauge group on the boundary. Instead, the Euclidean partition function for K probe branes in the bulk is shown in [3] to compute exactly the ratio Z N +K /Z N between the Euclidean partition functions of the boundary theory for N + K and N colors respectively, where we have denoted by Σ the degrees of freedom living on the brane. This point of view has many interesting consequences and seems consistent with the notion of Highly Effective Action described in [5], which corresponds to the special case The aim of the present work is to understand, from the bulk perspective, one of the simplest consequence of Eq. (1.1). Assume that the free energy − ln Z N scales as N γ F at large N , for some exponent γ, with corrections o(N γ−1 ) (for example, in the standard gauge theories considered in [3], γ = 2 and the corrections are of order O(N 0 ) = o(N )). Then ln(Z N +1 /Z N ) = −γN γ−1 (F + o(1)). On the other hand, in the large N limit, the probe brane action S b is very large (for example, it is proportional to N in gauge theory). The right-hand side of (1.1) is then dominated by configurations minimizing S b . If we denote by S * b the minimum value of S b , we obtain in this way ln(Z N +1 /Z N ) = −S * b . If, moreover, we use the standard holographic dictionary of [6,7] that identifies N γ F with the on-shell gravitational bulk action S * g , we get the fundamental identity This is an archetypal holographic identity, relating a bulk quantity on the left-hand side to a surface quantity on the right-hand side.
The reasoning that leads to (1.2) is very robust and we believe that it constitutes an important basic property of holography. The relation (1.2) can be easily generalized to a large variety of situations, including cases with several types of branes (like for example in AdS 3 holography) and cases where α corrections and even finite N effects are included. We let the discussion of some of these extensions to a companion paper [8] and focus presently on the basic conceptual issues in the simplest framework.
At first sight, the equality (1.2) seems rather puzzling, for at least two basic reasons. First, the gravitational action is naively infinite and a holographic renormalization procedure is required to make sense of it [9], whereas the probe brane action is naively finite with no need to renormalize. Yet, equation (1.2) implies that an analogue of the holographic renormalization prescription must exist for the probe brane action and we have to understand what this could be. Second, the gravitational action S g is the sum of the bulk Einstein-Hilbert term and a boundary Gibbons-Hawking term, whereas the brane action S b is a purely surface quantity, sum of Dirac-Born-Infeld and Chern-Simons contributions. To compute the on-shell value S * b , one naively has to solve the field equations on the brane and evaluate S b on the solution. This looks quite complicated and the matching with the very different-looking on-shell gravitational action may seem rather miraculous. Clearly, in view of the claimed extreme generality of (1.2), a simple mechanism must be at work, simplifying drastically the analysis and ensuring consistency.
We shall elucidate these issues in the following in the case of pure gravity, where the bulk space M is a Poincaré-Einstein manifold, We shall prove that the consistency of (1.2) relies on a non-trivial isoperimetric inequality, bounding from below the area A(Σ) of any hypersurface Σ ⊂ M homologous to the boundary by the volume V (M Σ ) of bulk space enclosed by Σ, It is easy to see that this inequality is violated if the Yamabe constant of the boundary is negative. Holography thus cannot be consistent in these cases, a fact that has been known for a long time [10,11] (a negative Yamabe constant simply means that the action for a conformally coupled scalar on the boundary will not be bounded from below, implying that the field theory on the boundary is ill-defined). Quite remarkably, when the Yamabe constant is non-negative, the inequality (1.4) can be derived from the details of the proof of a theorem by Lee [12] and was also proved directly by Wang in [13].
2 A simple example: Schwarzschild-AdS 5 It is very useful to first analyse a simple example. So let us consider the famous Schwarzschild black hole in AdS 5 , which is dual to the N = 4 gauge theory on X = S 3 × S 1 , when the temperature is above the Hawking-Page transition [7,14]. We pick the standard representativeḡ for the conformal class of the metric on X, where t and t + β are identified and dΩ 2 3 is the round metric of radius one on S 3 . The bulk metric can be conveniently written by using the Fefferman-Graham coordinates associated with (2.1) as 2 The full cigar-shaped bulk manifold M = B 2 × S 3 is covered when 0 < r ≤ r h (or 0 < x ≤ 1), with r = 0 corresponding to the boundary and r = r h to the tip of the cigar (horizon). The parameter α belongs to the interval ]0, 1[, ensuring that f > 0.
Smoothness at r = r h yields the relation between α and the inverse temperature β.
Let us now consider a 3-brane, which is a hypersurface Σ in M . In the present section, for simplicity and consistently with the symmetries of the metric (2.2), we limit our discussion to hypersurfaces given by an equation r = constant. The brane action is then a function of r, sum of DBI and CS contributions. The DBI term is simply the area of the hypersurface for the induced metric times the 3-brane tension. A simple calculation yields This term is a monotonically decreasing function of x (or of r). It tends to make the brane shrinks. The Euclidean CS term is where the Ramond-Ramond five-form field strength F 5 = dC 4 is related to the bulk volume form Ω 5 by The · · · represent components on the S 5 part of the ten-dimensional geometry, which must be present because F 5 is self-dual. However, these terms play no role in our discussion, nor does the S 5 . This is why we have not mentioned them up to now, and we shall not mention them any longer. It is straightforward to integrate the volume form of the metric (2.2) to obtain C 4 . The integration generates and arbitrary integration constant c, yielding where ω 3 is the volume form on the unit radius round 3-sphere. Plugging into (2.6), we get for some x-independent constant s (which is proportional to the constant c in (2.8)). The CS term is a monotonically increasing function of x, tending to make the brane inflate towards the boundary x = 0. Adding up (2.5) and (2.9), we finally get This formula has three important basic qualitative features. First, it is a monotonically decreasing function of x: the DBI term wins over the CS term and the brane wants to shrink. The minimum value of the action is obtained for the maximum value x = 1 of the variable x, for which the shrunken brane sits at the tip of the cigar, For most purposes, this ambiguous constant s in the brane action is inoffensive. It can be interpreted as coming from the gauge symmetry C 4 → C 4 + c 4 , for any closed 4-form c 4 . However, for our purposes, it clearly does play a crucial role. Our aim is to find the on-shell value S * b of the brane action and any undetermined constant would allow to shift S * b to any value we like, which is of course nonsense. The only ambiguity in S * b that one can tolerate, and which is actually expected in view of the fundamental relation (1.2) we wish to prove, is the possibility to add finite local counterterms. In the present case, the most general counterterm action, constrained by locality, general covariance and power counting, is of the form for dimensionless renormalization constants c 0 , c 1 and c 2 that may depend on a regulator but not on a or β. These terms correspond to adding a cosmological constant, curvature and curvature squared terms in the boundary theory. These considerations yield the following simple proposal to fix the ambiguity associated with the integration constant s: The brane action, evaluated for a brane worldvolume r = , where r is the Fefferman-Graham radial coordinate and > 0 a regulator, should go to a purely counterterm action near the boundary, up to terms that go to zero when → 0.
This is a very natural prescription, which is consistent with the construction in [3] and the general intuition that going to the boundary of bulk space corresponds to a UV limit in the boundary field theory. In our example, using the well-known formulas for the tension of a D3-brane in type IIB string theory and the relation between the AdS scale L and the string theory parameters [1], we see that the first term in the right-hand side of (2.10) is precisely of the form (2.12) when x → 0. The constant s must thus be of the form (2.12) as well. Putting everything together, we obtain for an arbitrary finite counterterm action S CT . Using (1.2), with an exponent γ = 2 suitable for a free energy scaling as N 2 in gauge theory, we reproduce precisely the correct free energy of the N = 4 Yang-Mills theory [7,14].
Remark : the shrinking of the probe brane to the tip of the cigar geometry might be interpreted as the Euclidean version of a brane falling into the horizon of the Minkowskian black hole geometry. However, we would like to emphasize that this is misleading. As will be clear in the next section, the tip of the cigar is not a special point for the brane. If allowed to deform in arbitrary ways, the brane can shrink at any point on the cigar, thus including at r < r h . Only the minimal value of S b has a physical meaning.

The general case
Let us now consider an arbitrary Poincaré-Einstein bulk space M . We pick a representativeḡ of the conformal structure on the boundary X = ∂M . We denote by r the Fefferman-Graham radial coordinate and by z the coordinates on X. The bulk metric near the boundary reads has the usual near-boundary Fefferman-Graham expansion. We introduce a regulator > 0, denote by Σ the hypersurface r = and by M the interior of Σ , the regulated bulk space. We also denote with the symbol ≡ equalities modulo the addition of local counterterms on the boundary and terms that go to zero when → 0. The gravitational action is the sum of the Einstein-Hilbert and the Gibbons-Hawking terms, which is a surface integral over Σ . It is easy to check, using the expansion (3.2), that the Gibbons-Hawking term is always a pure counterterm. This is a nice consequence of using the Fefferman-Graham coordinate r to regulate the bulk space. Using Einstein's equations (1.3), we thus obtain where G d+1 is the bulk Newton constant and V (M ) the volume of the regulated bulk space.
We now have to define what we mean by probe brane in general. On physical grounds, it is reasonable to consider that a probe brane should be an embedding of the boundary manifold X in M which can be obtained by smoothly deforming Σ . A less stringent requirement would be to consider all hypersurfaces homologous to the boundary. We shall work with this second point of view for simplicity, but we believe that the first point of view should be equivalent for our purposes (at least it is on the specific examples we are aware of). We denote by M Σ the bulk space enclosed by Σ, ∂M Σ = Σ. The DBI term in the brane action is simply τ d−1 A(Σ), where τ d−1 is the brane tension and A its area (worldvolume) for the induced metric on Σ. The Euclidean CS term is −iτ d−1 Σ C d , with dC d = id L Ω d+1 proportional to the volume form of the bulk space, generalizing (2.6) and (2.7). Integrating to get C d produces an arbitrary integration constant s, as in the example of section 2. This constant must be the same for all the probe branes, since they are all homologous to each other. Moreover, up to this constant, Stokes' theorem implies that the CS term is proportional to the volume of M Σ . Overall, we thus obtain The constant s is fixed by using the principle formulated in Sec. 2: we impose that S b (Σ ) ≡ 0, i.e. the brane action on the boundary is a pure local counterterm. Since it is obvious that A(Σ ) ≡ 0, the area being a local cosmological constant term on the boundary, we get, by taking (3.3) into account, To compute the on-shell brane action S * b , we thus have to minimize the functional A − d L V over all probe branes. If we can prove the isoperimetric inequality (1.4), then the minimum value will be zero, which is realized by a shrunken brane. 3 The identity (1.2) would automatically follow, with an exponent Note that γ must be independent of N . For example, in type IIB with the N = 4 theory living on the boundary X, the ten dimensional Newton constant is G 10 = 1 2 π 2 8 s g 2 s = π 4 L 8 2N 2 , and thus, taking into account the volume π 3 L 5 of the S 5 piece in the geometry, the five dimensional G 5 = πL 3 2N 2 . Using (2.13), we see that (3.6) yields γ = 2 as expected. Cases with other values of γ are discussed in [8].
Thus there remains to understand the crucial inequality (1.4). This kind of inequalities have been much studied in mathematics, see e.g. [17]. To build intuition on (1.4), it is very instructive to start by considering a special class of large hypersurfaces. We use the Trudinger-Aubin-Schoen theorem to pick a conformal class representativeḡ on the boundary having constant scalar curvaturē R. It is then straightforward to compute A − d L V for hypersurfaces Σ given by r = constant, at small r, where r is the Fefferman-Graham radial coordinate associated withḡ, by using the expansion (3.2). One finds that it diverges asR/r d−2 , if d > 2, or as −R ln r if d = 2 [10,11]. In particular, ifR < 0, the probe brane action is unbounded from below and S * b = −∞! A crucial requirement is thus thatR ≥ 0. This is equivalent to saying that the Yamabe constant Y ([ḡ]) of the conformal class at infinity is non-negative. Holography will be inconsistent in such cases, precisely due to the emission of large probe branes, as argued in [11,16]. 4 We thus limit ourselves to the cases Y [ḡ] ≥ 0. Remarkably, the inequality (1.4) was then derived in [13], building on the results in [10] and on geometric measure theory. Let us sketch here a more elementary approach, based on some of the results of [12]. The idea is to consider a scalar field φ on M of mass m 2 = (d + 1)/L 2 , thus sourcing an operator of dimension δ = d + 1 on the boundary. As usual, such a scalar field will behave as r d−δ = 1/r near the boundary. For our purposes, we choose the sourceφ = lim r→0 (rφ) to be a strictly positive constant, say equal to one. The field equation (∆ + m 2 )φ = 0 then implies immediately, from the maximum principle, that φ > 0 on M . Moreover, using (1.3), it is not difficult to check that ∆(|dφ| 2 − φ 2 /L 2 ) ≤ 0, where |dφ| 2 = G µν ∂ µ φ∂ ν φ. The maximum principle then implies that on M , as soon as this is valid near the boundary r = 0. But, when r → 0, this inequality can be directly checked by using the expansion (3.2) and the similar wellknown expansion for the scalar field. Using the same conformal class representative as in the previous paragraph, with constant scalar curvatureR, one finds that |dφ| 2 − φ 2 /L 2 −R/(d(d−1)) near the boundary, which is indeed non-positive if Y ([ḡ]) ≥ 0.
This being established, we can proceed as follows. 5 We consider the vector field v µ = ∂ µ ln φ. By using (3.7), we immediately find that We now integrate the second inequality above over M Σ and use Stokes' theorem to find where P(G) denotes the determinant of the induced metric on Σ and n is the unit normal to Σ, pointing outward. The isoperimetric inequality (1.4) then follows from the bounds (3.10) where, in the last step, we have used the first inequality in (3.8).
supported in part by the DFG Transregional Collaborative Research Centre TRR 33 and the DFG cluster of excellence "Origin and Structure of the Universe" as well as the Belgian American Educational Foundation.