A supersymmetric and smooth compactiﬁcation of M-theory to AdS 5

We obtain smooth M-theory solutions whose geometry is a warped product of AdS 5 and a compact internal space that can be viewed as an S 4 bundle over S 2 . The bundle can be trivial or twisted, depending on the even or odd values of the two diagonal monopole charges. The solution preserves N = 2 supersymmetry and is dual to an N = 1, D = 4 superconformal ﬁeld theory, providing a concrete framework to study the AdS 5 / CFT 4 correspondence in M-theory. We construct analogous embeddings of AdS 4 , AdS 3 and AdS 2 in massive type IIA, type IIB and M-theory, respectively. The internal spaces have generalized holonomy and can be viewed as S n bundles over S 2 for n = 4, 5 and 7. Surprisingly, the dimensions of spaces with generalized holonomy includes D = 9. We also obtain a large class of solutions of AdS × H 2 .


Introduction
AdS 5 spacetime arises naturally in type IIB supergravity, which provides a non-trivial and relatively simple framework for examining the holographic principle via the AdS 5 /CFT 4 correspondence [1][2][3]. The embedding of AdS 5 spacetime in eleven-dimensional supergravity has also been studied in the past. A smooth but nonsupersymmetric compactification of eleven-dimensional supergravity to AdS 5 was obtained in [4] where the internal space is a Kähler manifold. More recently, an internal background of CP 2 × T 2 was found in [5]. Although the compactification is not supersymmetric at the level of supergravity, it was argued in [5] that it is fully supersymmetric at the level of M-theory, since it is T-dual to the AdS 5 × S 5 of type IIB theory. In the above E-mail address: jvazquez@ulb.ac.be (J.F. Vázquez-Poritz). 1 Research supported in part by the Federal Office for Scientific, Technical and Cultural Affairs through the Interuniversity Attraction Pole two examples, the AdS spacetime and internal manifold are direct products without warp factors. Smooth but non-supersymmetric M-theory solutions have been constructed in [6], which are warped and twisted products of AdS 5 × S 2 or AdS 5 × H 2 with a squashed four-sphere.
In [7,8], supersymmetric embeddings of AdS 5 in M-theory were found as warped geometries with a compact internal metric. This construction can be understood from that the fact that S 5 can be expressed as a foliation of S 3 and S 1 . One can then T-dualize the AdS 5 × S 5 of type IIB theory on the U(1) bundle of the S 3 and obtain a solution in M-theory [9]. However, there is a naked singularity in such a construction, since the U(1) circle of the S 3 can shrink to zero. Supersymmetric and smooth embeddings of AdS 5 in M-theory were obtained in [10]. The eleven-dimensional metric is a warped product of AdS 5 with an internal metric that can be viewed as an S 4 bundle over H 2 , a hyperbolic 2-plane. The construction can give rise to both N = 2 and N = 4 supersymmetry.
In this Letter, we report a supersymmetric and smooth compactification of M-theory to AdS 5 , with the internal space being an S 4 bundle over S 2 . The construction is only possible for N = 2 supersymmetry, and hence it gives rise to the minimum AdS 5 gauged supergravity coupled to matter. This solution provides a supergravity dual to N = 1, D = 4 superconformal field theory. We also obtain supersymmetric and smooth compactifications of Mtheory to AdS 2 and type IIB to AdS 3 . The internal space is an S p bundle over S 2 , where p = 7 and 5, respectively. We also construct a supersymmetric compactification of massive IIA to AdS 4 , which is singular.

AdS 5 × S 2 in M-theory
We begin by considering the sector of D = 7 gauged supergravity with two diagonal U(1) vector fields. The relevant Lagrangian is given by The scalar potential V is given by [11] (2.3) The potential can be expressed in terms of the superpotential We now consider a 3-brane ansatz where the constant takes the values 1, −1 and 0, if dΩ 2 2 is the metric for a unit S 2 , hyperbolic H 2 or 2-torus T 2 . Ω (2) is the corresponding volume form. The system admits the following first-order equations provided that the constraint (2.7) λ 2 = (m 1 + m 2 )g is satisfied. This set of first-order equations were derived for the case of H 2 in [10] by studying the Killing spinor equations of D = 7 gauged supergravity. In [12], a different method was used to obtain them for H 2 , S 2 and T 2 . The equations of (2.6) were analyzed in detail in [12]. Here, we report on only a subclass of solutions where φ and v are constants. In this case, for = 0, the solution is nothing but AdS 7 written in Poincaré coordinates.
For = ±1, we find that the solution is given by This solution is invariant under the simultaneous interchanges of m 1 ↔ m 2 and φ 1 ↔ −φ 1 . The reality conditions of the solution constrain the constants m i and g, as well as the choice of ± in the solution. Let us first consider the case = −1, corresponding to dΩ 2 2 as the metric of a unit (non-compact) hyperbolic 2-plane. In this case, the reality of the solution implies that m 1 m 2 0. This includes the choice of m 1 = 0 (or m 2 = 0) and m 1 = m 2 , which were discussed in [10]. The first case gives rise to N = 4 supersymmetry in D = 5, whilst the second case gives rise to N = 2 supersymmetry.
We are particularly interested in a compact internal space. Thus, we now turn to the choice of = +1, corresponding to dΩ 2 2 as the metric of S 2 . In this case, the reality conditions for (2.8) imply that m 1 m 2 < 0. The condition (2.7) implies further that m 1 = −m 2 . Therefore, the AdS 5 × S 2 solution can only have N = 2 supersymmetry, but cannot arise from the pure D = 7 minimal gauged supergravity.
If we define a charge parameter q = 2m 1 /(m 1 + m 2 ), then the condition for having S 2 versus H 2 can be summarized as It is straightforward to lift this solution to D = 11 by using the ansatz obtained in [11]. Since the solutions for general m i are rather complicated to present, we only consider a representative example with m 1 = 5g and m 2 = −3g. The M-theory metric is given by where c = 10 −2/5 and µ i are spherical coordinates which satisfy µ 2 0 + µ 2 1 + µ 2 2 = 1. The warp factor ∆ is given by The AdS 5 metric is given by where the AdS radius is given by R = 1 2cg . The 4-form field strength in D = 11 can also be obtained using the reduction ansatz in [11]. It is given by * F (4) where (5) is the volume form for the AdS 5 metric. Thus, the internal space of the D = 11 metric can be viewed as an S 4 bundle over S 2 , with two diagonal U(1) bundles. In general, the internal metric can be labeled by (q 1 , q 2 ) = ( 2m 1 m 1 +m 2 , 2m 2 m 1 +m2 ). In the specific example above, (q 1 , q 2 ) = (5, −3) and the solution is smooth everywhere. For general (m 1 , m 2 ), the metric does not have a power-law singularity. However, it could have a conical orbifold singularity, which is absent only if (q 1 , q 2 ) are integers. Since the q i satisfy the constraint q 1 + q 2 = 2, it follows that they are either both even or both odd integers. In the even case, the bundle is topologically trivial, whilst it is twisted for the odd case.

AdS 4 × S 2 in massive type IIA
The scalar potential in gauged supergravity with two U(1) isometries was obtained in [13]. From this, we deduce that the relevant Lagrangian involving the two U(1) vector fields is given by The scalar potential is given by where X 0 = (X 1 X 2 ) −3/2 . As in the previous case, the scalar potential can be expressed in terms of the superpotential We consider a membrane solution of the type given by (2.5). The system admits the following first-order equations provided that the constraint λ 2 = (m 1 + m 2 )g is satisfied. The solutions were analyzed in detail in [12]. Here, we shall only consider the subset of solutions with constant scalars. For = 0, one just reproduces the AdS 6 metric in Poincaré coordinates. For = ±1, we have As in the D = 7 result, we can define a charge parameter q = 2m 1 m 1 +m 2 . We have H 2 or S 2 depending on the following condition: When q = 0 or q = 2, the system has N = 4 supersymmetry. Otherwise, we have N = 2 supersymmetry. Using the reduction ansatz in [13,14], it is straightforward to lift the solution back to D = 10, giving rise to a solution of massive type IIA supergravity. The metric is given by where ∆ = 2 α=0 X α µ 2 α > 0 and µ 2 0 + µ 2 1 + µ 2 2 = 1. Thus, the D = 10 metric is a warped product of AdS 4 with an internal six-metric, which is an S 4 bundle over S 2 or H 2 , depending on the charge parameter p, according to the rule (3.7).

AdS 3 × S 2 in type IIB
Let us now consider the D = 5 minimal gauged supergravity coupled to two vector multiplets. The Lagrangian is given by The scalar potential is given by The scalar potential V can also be expressed in terms of the superpotential We find that the string solution of the type given by (2.5) admits the following first-order equations provided that the constraint λ 2 = (m 1 + m 2 + m 3 )g is satisfied. The general solution for this system was analyzed in [12]. Here, we consider only the solutions with constant scalar fields. For = 0, the solution is AdS 5 in Poincaré coordinates. For = ±1, fixed-point solutions exist only for non-vanishing m i . The solution is given by The reality condition of the solution implies that when three vectors with the magnitudes |m i | can form a triangle, dΩ 2 2 should be the H 2 metric. On the other hand, when they cannot form a triangle, the metric should be that of S 2 . 4 If any of the m i vanish, there is no fixed-point solution, except when one vanishes with the remaining two equal [12].
Using the reduction ansatz obtained in [11], we can easily lift the solution back to D = 10. Since the solution with general m i is complicated to present, we consider a simpler case with m 2 = m 1 . The D = 10 type IIB metric is , ∆= c 1/3 cos 2 θ + c −2/3 sin 2 θ > 0, dA (1) = Ω (2) , We have introduced the charge parameters q i = 2m i /(m 1 + m 2 + m 3 ), and hence they satisfy the constraint q 1 + q 2 + q 3 = 2. In the above solution, if |m 3 | < 2|m 1 |, we should have = −1, corresponding to H 2 ; if |m 3 | > 2|m 1 |, we should have = 1, corresponding to S 2 . In general, the internal metric is an S 5 bundle over S 2 or H 2 , depending the values of the q i according to the above rules.

AdS 2 × S 2 in M-theory
Let us now consider the U(1) 4 gauged N = 2 supergravity in four dimensions. The Lagrangian is given by The scalar potential is given by which can also be expressed in terms of the superpotential The magnetic black hole solution of the type given by (2.5) admits the following first-order equations provided that the constraint λ 2 = g 4 i=1 m i is satisfied. The general solution for this system was analyzed in [12]. Here, we consider only solutions with constant scalar fields. For = 0, the solution is AdS 4 in Poincaré coordinates. For = ±1, we have not obtained the general solution for arbitrary m i , although we have found many examples of specific solutions. We do find a class of special solutions by setting m 2 = m 3 = m 4 . This enables us to consistently set φ 1 = φ 2 = φ 3 ≡ φ. For this truncation, fixed-point solutions for = ±1 are given by The reality condition of the solution implies that for = −1, corresponding to H 2 , we must have either m 2 > 0 and 0 < m 1 m 2 or m 2 < 0 and m 2 m 1 < −3m 2 . For = 1, corresponding to S 2 , we must have m 2 0 and m 1 > −3m 2 . The AdS 2 × H 2 has also been found in [16].
Using the reduction ansatz obtained in [11], we can easily lift the solution back to D = 11 with the metric (1) = Ω (2) , In general, the internal metric is an S 7 bundle over S 2 or H 2 , depending the values of the m i .

Conclusions
We have obtained a large class of supersymmetric embeddings of AdS spacetime in M-theory, as well as type IIB and massive type IIA theories. The internal spaces can be viewed as S n bundles over S 2 or H 2 . Similar solutions have been discussed in [17][18][19]. In particular, we have found a smooth embedding of AdS 5 in M-theory, with a compact internal space of an S 4 bundle over S 2 . The bundle can be trivial or twisted, depending on the two diagonal monopole charges. The solution preserves N = 2 supersymmetry; it is a supergravity dual to an N = 1, D = 4 superconformal field theory on the boundary of AdS 5 . This provides a concrete framework to study AdS 5 /CFT 4 from the point of view of M-theory.
The internal spaces of these embeddings may be regarded as concrete realizations of spaces with generalized holonomy groups advocated in [20], since they are not Ricci flat and involve a form field. An especially interesting example is the S 7 bundle over S 2 or H 2 , which is nine-dimensional. While nine-dimensional Ricci-flat manifolds do not have an irreducible special holonomy group, our aforementioned solutions are explicit examples of ninedimensional spaces which have generalized special holonomy.
The embeddings of AdS spacetimes in M-theory and string theories discussed in this Letter all involve warp factors. We expect that there are many further examples of such solutions. It is of interest to explore them, both from the AdS/CFT perspective as well as for a more concrete understanding and classification of spaces with generalized special holonomy.