AdS$_2$ Type-IIA Solutions and Scale Separation

In this note we examine certain classes of solutions of IIA theory without sources, of the form AdS$_2\times {\cal M}^{(1)}\times \dots \times {\cal M}^{(n)}$, where ${\cal M}^{(i)}$ are Riemannian spaces. We show that large hierarchies of curvatures can be obtained between the different factors, however the absolute value of the scalar curvature of AdS$_2$ must be of the same order or larger than the absolute values of the scalar curvatures of all the other factors.


Introduction
The question of scale separation of AdS vacua in string theory [1][2][3] is the subject to a lot of recent discussion. In many of the previous works, backgrounds of the form AdS d × M (p) were considered, and special emphasis was given to the case with d = 4 and p = 6. Scale separation is possible whenever the radius L d of AdS d is parametrically larger than the inverse Kaluza-Klein (KK) mass scale L p ≃ 1 m KK of M (p) , i.e. L d ≫ L p . In the case where scale separation is possible, the theory possesses a limit in which the solution can be regarded as d-dimensional AdS d space. On the other hand, if scale separation is not possible, i.e. if L d ≃ L p , the solutions are not really d-dimensional, and the gravitational background has to be considered as (d + p)-dimensional. Whether scale separation in supergravity backgrounds of the above form is possible or not has also profound consequences for the holographically dual CFT in (d − 1) dimensions.
We do not want to review all arguments which were given in favor or against scale separation for AdS d × M (p) background spaces. Some general arguments against scale separation were given in [2]. On the other hand, one of the early papers addressing this issue is the work of DGKT [4], where it was argued that in the presence of orientifold planes scale separation is possible. This discussion was recently refined and extended in [5][6][7][8][9][10]. The question of scale separation was also recently addressed in the general context of the quantum gravity swampland discussion [11], namely as the AdS Distance Conjecture (ADC) [12]. This conjecture states that the limit of small AdS cosmological constant, Λ ≃ 1 L 2 d → 0, is at infinite distance in the space of AdS metrics, and that it is related to an infinite tower of states with typical masses that scale as, ADC : m ∼ Λ α , (1.1) with α = O(1). The strong version of the ADC proposes that for supersymmetric backgrounds α = 1 2 , and that in this case scale separation is not possible, since L d ∼ 1 m .
In this paper we will consider several AdS 2 solutions in string theory, where the total space is of the form AdS 2 × M (1) × · · · × M (n) . Assuming that the scale, or radius, is related to the scalar curvature, or cosmological constant, via Λ ≃ 1 L 2 , we will see that scale separation for these backgrounds will never be possible in the sense that for all the considered cases the radius L 2 of AdS 2 can never be much larger than at least one of the radii of the other factors. E.g. if M (1) is a two-sphere S 2 of radius L ′ 2 , then L 2 ≤ L ′ 2 . However there are cases where the rest of the radii, e.g. the radius of a Ricci-flat space M (6) , can be much smaller than L 2 , L ′ 2 . In particular, L 2 = L ′ 2 ≫ L 6 is possible. This means that there can be scale separation between AdS 2 × S 2 and M (6) , even within a regime of weak coupling and curvature where the supergravity approximation is valid. The reason this is possible is that in the Ricci-flat case the radius is no longer related to the inverse of the scalar curvature (which vanishes). Instead the radius becomes a free parameter of the solution, only constrained by flux quantization.
The case of AdS 2 × S 2 is of special interest, since it corresponds to the near horizon geometry of four-dimensional extremal, supersymmetric black holes. The radii L 2 = L ′ 2 are directly related to the entropy S of the corresponding black hole solutions: As it was recently discussed in [13], the limit of large entropy, S → ∞, is at infinite distance in the space of 4D black hole metrics. Therefore, similarly to the ADC, a black hole entropy conjecture (BHEC) was put forward in [13], stating that the large entropy limit of black holes is also accompanied by a tower of light modes. However these modes cannot be given in terms of the internal KK modes of M (6) . This was already seen in [13] from the so-called attractor equations, since, as a function of the electric and magnetic black hole charges, S can be made large, while keeping the internal scale L 6 small. Here we will confirm this result by investigating the supergravity solutions of the corresponding intersecting D-branes, and reading off from the supergravity solutions the corresponding length scales. However, as we will discuss, there are other classes of AdS 2 supergravity solutions, where scale separation is only possible in the other direction: namely there are solutions where the AdS 2 space and some of the internal factors are more highly curved than the rest of the internal factors. We call this the "wrong" kind of scale separation.
The paper is organized as follows. In the next section we will briefly review the background spaces of supergravity p-branes in ten spacetime dimensions. In §3 we will then discuss the construction of supergravity solutions of intersecting D-branes, which lead to supersymmetric 4D black holes with AdS 2 × S 2 near-horizon geometry. We will see that scale separation L 2 = L ′ 2 ≫ L 6 is possible within the validity regime of the supergravity approximation. We also compare these results with those of [13]. In §4, we discuss various generalizations of spaces M 6 and show that scale separation works in a different way than before. The case of M 6 Ricci-flat, discussed in §4.4, includes the brane set up of §3.1 as a special case. In §4.5 the case of backgrounds of the form is analyzed. In §4.5.1 we discuss solutions of the form AdS 2 ×S 2 ×S 2 ×S 2 ×T 2 , AdS 2 ×S 2 ×S 2 ×T 4 , or AdS 2 ×S 2 ×S 2 ×K3, obeying flux quantization within the validity regime of the supergravity approximation. We conclude with a discussion in §5.

p-branes in D = 10
For a review of brane solutions see e.g. [14][15][16]. In ten dimensions, in the string frame, p-branes are solutions of the action S = S bulk + S sources where, g 10 is the determinant of the metric g M N , M, N = 0 . . . 9, F p+2 = dC p+1 is the abelian (p + 2)-form field strength, and φ is the dilaton. The square of a q-form A q is defined by where Σ p+1 is the world-volume of the p-brane with coordinates ξ i , i = 0 . . . p, and ı * [·] the pull-back to Σ p+1 . The gravitational constant and the tension are given by, where α ′ = l 2 s , with l s the string length. For BPS sources as here, one has µ p = T p . The p-brane solutions in the string frame are then given by, with x i=0...p the coordinates along the brane, vol p+1 = dx 0 ∧ . . . ∧ dx p , and y m=p+1...9 the coordinates of the space transverse to the brane. H( y) is a harmonic function with localized source in the (unwarped) R 9−p space transverse to the p-brane, where the brane is located at y 0 in the transverse space. For D p -branes we have, The constant e φ 0 = g s can be used to define the string coupling as the value of the dilaton at asymptotic infinity, where the harmonic functions tend to unity. However, once the nearhorizon limit is taken (see section 3.1), the asymptotic region is no longer accessible. More generally one should think of g s as a free parameter of the solution, related to the string coupling via (2.4).

The brane configuration
One can form superpositions of brane solutions according to the harmonic superposition rule [17,18]. Consider the following system of intersecting D4/D0-branes: where y m=1,...,6 are assumed to parameterize a T 6 , and x i=1,2,3 are coordinates of R 3 . We use the notation D4 α , α = 1, 2, 3 to distinguish the three different types of D4-branes in the configuration of the table above. We shall assume there are n 0 D0-branes located at x 0 ∈ R 3 , and n α D4 α -branes located at x α , α = 1, 2, 3.
The explicit form of the metric reads, where H α , α = 1, 2, 3, are the harmonic functions of the D4 α -branes, and H 0 is the harmonic function of the D0-branes. We have, for α = 0, . . . , 3, and we took into account that (2.5) implies c = − Q 4π , for the case where the transverse space is R 3 . The N α are proportional to the number of D-branes n α . The precise relation will be derived below using flux quantization. 1 More explicitly,

The fluxes read,
where dΩ 2 is the volume form of the unit 2-sphere, and we have taken into account that the time coordinate has been rescaled as indicated below (3.4). All fluxes, as well as their Hodge-duals, can readily be seen to be everywhere well-defined and closed, dF = d ⋆ F = 0, indicating the absence of sources. In other words, the near-horizon limit is a pure gravity background, all branes having dissolved into fluxes in the limit.
We shall assume that the areas, in units of string length, of the three 2-tori Σ i spanned by the internal coordinates are given by three moduli The flux quantization conditions, then relate N α to the number of D-branes n α ∈ N, which have dissolved into flux quanta in the near-horizon geometry, where we have substituted (3.5) into (3.7), taking (3.3), (3.6) into account.
The geometry of (3.4) is AdS 2 ×S 2 ×T 6 . The radii L 2 , L ′ 2 of AdS 2 , S 2 respectively, can be seen to be equal to each other. The 4D part of the geometry, AdS 2 ×S 2 , is thus characterized by a radius L 4 := L 2 = L ′ 2 . The latter and the radius L 6 of T 6 can be read off of (3.4), Unlike the individual values of the radii L 4 , L 6 , their ratio is frame-independent, The dilaton is constant,

Validity
The metric, fluxes and dilaton (3.4), (3.5), (3.11) give an exact bulk supergravity solution without sources. The solution is parameterized by the parameters g s , n a , v i , which can be tuned independently.
Let us denote by T i the effective areas (in string frame and in string units) of the three 2-tori in the near-horizon limit, Taking (3.3), (3.8) into account, this is equivalent to, For the validity of the supergravity regime we must require, In addition we must require that the radius of curvature of the background is much larger than the string scale, and that the string coupling obeys 16) in order for the higher-order flux corrections to be under control. In the second inequality above, the norm of the p-form flux is given by Conditions (3.14), (3.15), (3.16) are necessary and sufficient for the supergravity solution given in (3.4), (3.5), (3.11) to be within its regime of validity.

Scale separation
Let us first note that any rescaling of the v i 's can be cancelled by a corresponding rescaling in g s . So in the following we can keep v i fixed without loss of generality.
Suppose there is a solution parameterized by {g s , n a , v i }. Let us moreover rescale, for some non-vanishing p, q ∈ N * , t ∈ R + . Under this rescaling we have, where i = 1, 2, 3. Scale separation (L 4 ≫ L 6 ) is thus equivalent to taking t ≫ 1.
If in addition we want to respect conditions (3.14), (3.15), we also must take p ≫ q. Then the second of the two inequalities in (3.16) is automatically satisfied. To satisfy the first inequality in (3.16), it suffices to take p = q 2 ≫ 1, and t = q r , with 0 < r < 1 4 . We are then guaranteed to be within the validity regime of the supergravity approximation.

Comparison with [13]
In order to compare with [13] let us first redefine the constant g s → g s (v 1 v 2 v 3 ) − 1 2 , and also set l s = 1, so that the formulae of section 3.1 become, 19) and the 4d dilaton reads, where V := L 6 6 .
We see that the dependence on the 2-tori areas v i disappears, having been absorbed in the independent constant g s . This is consistent with the attractor mechanism according to which the near-horizon geometry is fixed by the charges, and in particular is independent of the values of the Kähler moduli at asymptotic infinity. The latter correspond to the areas of the 2-tori, v i . On the other hand, the values of the Kähler moduli at the horizon are given in (3.13), and correspond to the effective values of the areas of the 2-tori at the horizon. Indeed the T i 's here are essentially the same as defined in eq. (14) of [13].
The ratio L 4 /L 6 here, cf. (3.19), corresponds to m KK (S N =2 ) 1/2 of [13], up to a numerical factor of order one. We find agreement with [13], cf. eq. (20) therein, provided we identify the constant c −2/3 of that reference with the constant g s here.

Generalizations
We will look for solutions of (massive) IIA supergravity the form M 2 × M ′ 2 × M 6 , where M 2 is a two-dimensional maximally-symmetric Lorentzian manifold (i.e. R 1,1 , dS 2 , AdS 2 ), M ′ 2 is a two-dimensional maximally-symmetric Riemannian manifold (i.e. R 2 , S 2 , H 2 ) or discrete quotients thereof, and M 6 is a six-dimensional nearly-Kähler (NK), Calabi-Yau (CY), or Einstein-Kähler manifold. In §4.5 we will also consider the case where M 6 is a product of two-dimensional Riemannian manifolds.

M 6 nearly-Kähler with m = 0
The ansatz of the present section can be obtained from the consistent truncation of [19] §5 therein, by setting the scalars to constants and taking the one-forms therein to obey γ = χα. We are following the conventions of that reference. The ten-dimensional metric reads, ds 2 (10) = g µν dx µ dx ν + g ij dx i dx j + g mn dy m dy n , where the signs of Λ 1,2 are unconstrained at this point. The NK manifold M 6 possesses a real two form J and a complex three-form ω obeying, With these conventions, the constant ω ∈ R above is related to Λ in (4.2) via, We will assume that the dilaton is constant. Moreover, our ansatz for the forms reads, where χ, ϕ, ξ are real constants, and vol 2 , vol ′ 2 are the volume forms of M 2 , M ′ 2 respectively. 2 The two-form β is given by, where f , f ′ are real constants. It can then be seen that (4.5) automatically obeys the Bianchi identities (A.6). Plugging the above ansatz into the ten-dimensional equations of motion we obtain the following: the internal (m, n)-components of the Einstein equations read, The (µ, ν)-components read, (4.8) The (i, j)-components read, where we have taken (4.8) into account. All the mixed (µ, m), (i, m), (µ, i) components are automatically satisfied.
The dilaton equation reads, The F -form equation of motion is automatically satisfied. The H-form equation reduces to the following three equations, and, The G-form equation of motion reduces to, For ω, m = 0 this system of equations implies ξ = χϕ and f = f ′ = 0, from which we see in particular that Λ 1 = Λ 2 , so that no scale separation is possible. There are three classes of solutions, as given in [20] §11.4 therein. 3 Explicitly we have: First class: (4.14) Second class: Third class: We only expect the third class, given in (4.16) above, to be supersymmetric: it can be obtained from the solutions of [21], which are special cases of [22], by replacing AdS 4 by an AdS 2 ×H 2 (or AdS 2 × Σ g ) space subject to Λ 1 = Λ 2 , cf. footnote 3. A similar substitution of AdS 4 by an AdS 2 ×H 2 (or AdS 2 × Σ g ) space can also be performed for all known AdS 4 solutions. The converse is not true, however, as the AdS 2 ×H 2 space allows for more general fluxes, which would otherwise break the symmetries of AdS 4 . As already mentioned, scale separation is not possible in any of the three classes of solutions above, since all curvatures are of the same order.

M 6 nearly-Kähler with m = 0
The ansatz for the forms reads, In this case the equations of motion can be solved to give a one-parameter solution of the form AdS 2 ×S 2 , without scale separation, where the sign of χ is correlated with the sign of f /f ′ . Note that |f ′ /f | < 1 as it should, since Λ > 0 for a nearly-Kähler manifold, cf. (4.4). Scale separation is again not possible.

M 6 Kähler Einstein
The manifold M 6 is now assumed to be Kähler-Einstein with Kähler form J, dJ = 0. The form ansatz reads, with χ, ϕ, ξ, β as before, while the two-form γ is given by, where g, g ′ are real constants. It can then be seen that (4.19) automatically obeys the Bianchi identities (A.6).
Plugging the above ansatz into the ten-dimensional equations of motion we obtain the following: the internal (m, n)-components of the Einstein equations read, The (µ, ν)-components read, The (i, j)-components read, where we have taken (4.22) into account. All the mixed (µ, m), (i, m), (µ, i) components are automatically satisfied.
The dilaton equation reads, and, One way to solve this system of equations would be to view (4.25), (4.26) as a linear system of three equations for three unknowns f , f ′ , χ. 4 The solution can then be substituted into (4.24) to obtain one constraint on the remaining unknowns: g, g ′ , m, ϕ, ξ. Equations (4.21)-(4.23) then simply determine the curvatures Λ 1 , Λ 2 , Λ.
Let us now examine whether we can obtain a hierarchy between the curvature scales. From (4.21)-(4.24) we obtain, Moreover from (4.23), (4.27) it follows that Λ 1 ± Λ 2 ≤ 0, so that, If Λ 1 , Λ 2 = 0, this then implies that the external space M 2 is at least as highly curved as the internal space M ′ 2 .
• If Λ 2 = 0 then from (4.27) it follows that |Λ 1 | = 3Λ. Hence the curvatures of the external space M 2 and the internal space M 6 are of the same order.
• The equations of motion can easily be solved (e.g. numerically) for |Λ 1 | ≈ 3Λ ≫ |Λ 2 |, so that the spaces M 2 , M 6 are much more highly curved than M ′ 2 . Of course this is the "wrong" kind of scale separation.

M 6 Ricci-flat
As a special case of §4.3 we can impose that M 6 is Ricci-flat, Λ = 0, so that (4.27) implies, The remaining equations of motion can then be solved to give a two-parameter solution without scale separation, where the constants f , f ′ , g, g ′ are constrained to obey, The solution corresponding to the brane configuration of section 3 is a special (supersymmetric) case of the above solution: as we can see by comparing (3.5) with (4.19), taking (4.6), (4.20), (4.29) into account, it corresponds to setting f ′ = g = 0. Indeed this is a solution of (4.31) above, provided f = ±g ′ .
In this case, to make contact with §3, we can associate to the curvatures Λ 1 , Λ 2 , radii L 2 , L ′ 2 via: , and L 4 := L 2 = L ′ 2 . Moreover, in the case where M 6 is Ricci-flat, the radius L 6 is a free parameter (only subject to flux quantization) and does not enter the equations of motion (4.29)-(4.31).

M
Let us now consider ten-dimensional spacetimes of the form M is a two-dimensional maximally-symmetric space of Lorentzian signature whereas M (i) 2 , for i = 2, . . . , 5, are two-dimensional maximally-symmetric spaces of Euclidean signature, or discrete quotients thereof. The solutions we present here generalize those in [23,24], in which the choice of flux is not the most general.
Let us set, where we have denoted by R mn(i) , g mn(i) the Ricci tensor, resp. the metric components along M (i) 2 , i = 2, . . . , 5. The form ansatz will be taken to be, 2 , and f (i) , g (ij) are constants obeying: g (ij) = g (ji) , g (ii) = 0. This gives, It is also useful to list the Hodge duals, The equations of motion are as follows: the Einstein equations reduce to, (4.36) The dilaton equation reads, Equivalently, the modified dilaton equation reads, The equations of motion for the RR forms F , G are automatically satisfied. The H-form equation of motion reads, (23) g (45) + g (24) g (35) + g (25) g (34) where in the second equation above it is assumed that j < k < l and j, k, l = 1, i.
This system of equations can be solved in a similar fashion as that of §4.3: 5 in general we can solve the linear system of five equations (4.39) for the five unknowns f (i) . The solution can then be substituted into (4.37) to obtain one constraint on the remaining unknowns g (ij) , m. Equations (4.36) then simply determine the curvatures Λ (i) , while (4.38) is automatically satisfied.
It can easily be seen that the system of equations (4.36)-(4.39) admits solutions such that the curvatures Λ i are not necessarily equal. However, similarly to the case of §4.3, it is impossible to achieve |Λ 1 | < |Λ i |, for i = 2, . . . , 5. This can be seen as follows: equations (4.36) can be solved for Λ 1 , Λ i , in terms of the fluxes. Then using the dilaton equation (4.37) we find that −Λ 1 and −Λ 1 ± Λ i , i = 2, . . . , 5, can all be expressed as sums of squares, so that, for all i = 2, . . . , 5. Therefore, assuming the Λ i are not all vanishing, we conclude that the AdS 2 radius is bounded above by at least one of the radii of the internal factors. If we set Λ i = 0 for i = 2, . . . , 5, so that the internal space is flat and the radii of the internal factors become free parameters, then (4.38) would also imply Λ 1 = 0, so that all the fluxes vanish and we obtain a solution with flat ten-dimensional spacetime.

Flux quantization
For the supergravity solutions to be promoted to full-fledged solutions of the quantum theory, flux quantization must be imposed. For simplicity, let us set the 2πl s = 1 in the following. For i, j = 2, . . . , 5, flux quantization constrains the constants f (i) , g (ij) in (4.33) to obey, where n i , n ij ∈ Z and V i : . The constants f (1) , g (1i) are constrained to obey, where n 1 , n 1i ∈ Z. Moreover, the Romans mass is constrained to be an integer, m ∈ Z.
Let us also note that for the two-sphere S 2 , or a discrete quotient of the hyperbolic space H 2 , the volume is related to the scalar curvature by the Gauss-Bonnet theorem. Indeed, a (compact) Riemann surface Σ g of genus g > 1 can be obtained as a discrete quotient of the two-dimensional (non-compact) hyperbolic space H 2 , Σ g =H 2 /Γ, where Γ is a discrete subgroup of SO (1,2). Let the Ricci tensor of Σ g be given by R mn = Λg mn . The Gauss-Bonnet theorem then implies, where V = Σg vol is the volume of Σ g . For the two-sphere S 2 the corresponding relation reads, We have not been able to find a solution with general flux to the system of equations of motion subject to (4.41), (4.42). However, as we now show, special solutions are possible for m = 0, g (ij) = 0, i.e. for vanishing Romans mass and four-form flux. In this case the dilaton equation, (4.37) reads, Taking this into account, the Einstein equations (4.36) read, for i = 2, . . . , 5, while the remaining equations of motion are automatically satisfied.
We have already discussed solutions, obeying flux quantization, of the form AdS 2 ×S 2 ×T 6 . Let us instead suppose that the curvatures of the internal manifolds are all strictly positive, Λ i > 0, for i = 2, . . . , 5. Flux quantization, (4.41), (4.42), taking (4.44) into account, implies, 2 =T 2 is not related to its curvature, and thus does not enter the equations of motion. Flux quantization now implies, This equation simply determines V 5 in terms of the flux quanta, and always admits a solution. Therefore solutions of the form AdS 2 ×S 2 ×S 2 ×S 2 ×T 2 , are possible in the quantum theory. Note also that by taking large enough quanta we can make sure we are in the regime of small curvature and large volume (in string units). Moreover the dilaton is a free parameter of the solution, and can be tuned to weak coupling so as to ensure we remain within the validity regime of the supergravity approximation.
Similarly one can show that solutions of the form AdS 2 ×S 2 ×S 2 ×T 4 , obey flux quantization.
In the latter case we may also replace T 4 by a K3 surface.

Discussion
We have investigated superstring and supergravity backgrounds of the form M (1) ×· · ·×M (n) with special emphasis on the question of whether or not scale separation between the different factors is possible. We have seen that in all our solutions the scalar curvature of AdS 2 (in absolute value) must be of the same order or larger than the curvatures of all the other factors. Moreover, the other factors cannot all be (Ricci-)flat: in the solutions presented here this would also force the curvature of AdS 2 and all the flux to vanish. One might therefore invoke the relation between the radius, L, and the curvature: Λ ≃ 1 L 2 , to conclude that the radius of AdS 2 will be of the same order or smaller than the radius of at least one of the other factors.
Aside from the fact that the relation between scalar curvature and radius is more involved than what the previous paragraph suggests (many different definitions of the "radius" of a space are possible), there is a caveat to the argument of the previous paragraph: taking (possibly singular) discrete quotients of the internal spaces considered here, would leave invariant their local properties such as their curvature, while changing their global properties such as the radius. I.e. curvature hierarchies only concern the local properties of the spaces and do not immediately translate to corresponding hierarchies of radii. One way to directly address the question of the radius of the internal space is to study the spectrum of the (scalar) Laplacian on that space, whose first non-vanishing eigenvalue in particular can serve to read off the radius.
Indeed this was the approach used in [1,25] to establish the absence of scale separation in the vacua of [26].
For the supergravity solutions of §4 to be promoted to full-fledged superstring solutions, flux quantization must be imposed. As we saw in section §4.5.1, this is indeed possible to carry out in special cases, notably when the internal space includes a T 2 , K3 or CY factor. However the general problem seems rather involved and we have been unable to find a solution obeying flux quantization in the case of the most general flux ansatz. It would be interesting to examine whether this can be addressed algorithmically with the help of a computer.
A possible issue with the supergravity solutions of §4 is their potential instabilities, given the fact that we expect them to be non-supersymmetric in general. For example the solutions of §4.1, for which a supersymmetry analysis has been performed, come in three distinct classes only one of which is supersymmetric. On general grounds we would expect the nonsupersymmetric solutions to be unstable [27,28]. It would also be interesting to establish the supersymmetry (or absence thereof) of the remaining solutions of §4.