1 Introduction

Over the last ten years there has been tremendous progress in our understanding of the AdS/CFT correspondence [1] in the presence of unbroken supersymmetry. We have witnessed the discovery of many highly non-trivial supersymmetric solutions of supergravity, together with a rather detailed understanding of their gauge theory duals. Supersymmetric solutions with an anti-de Sitter (AdS) factor are particularly important, as they are dual to superconformal field theories, in an suitable limit. Comprehensive studies of general supersymmetric AdS geometries, in different dimensions, have been carried out in [2, 3, 4, 5, 6]111The generality of the ansatz used in [4] was proven in [7]. and led to a number of interesting developments. These results have all been obtained using the technique of analysing a canonical -structure in order to obtain necessary and sufficient conditions for supersymmetry [8]. In this paper we will systematically study the most general class of AdS solutions of supergravity. Supersymmetric AdS solutions of supergravity have been discussed before in the literature [9, 10]. However, these references contain errors, and reach incorrect conclusions that miss important classes of solutions.

Our main motivation for studying AdS solutions of supergravity in particular is that, starting with the seminal work of [11, 12, 13], over the past few years there has been considerable progress in understanding the AdS/CFT correspondence in M-theory. In particular, with supersymmetry there is good control on both sides of the duality, and this has led to many new examples of AdS/CFT dualities, including infinite families, along with precise quantitative checks. On the gravity side, the simplest setup is that of Freund-Rubin AdS backgrounds of M-theory, where is a Sasaki-Einstein manifold222Particular cases with include three-Sasakian manifolds and orbifolds of the round seven-sphere.. These are conjectured to be dual to the theory on a large number of M2-branes placed at a Calabi-Yau four-fold singularity. Rather generally, these field theories are believed to be strongly coupled Chern-Simons-matter theories at a conformal fixed point. With supersymmetry the partition function of such a theory on the three-sphere localizes [14, 15, 16], reducing the infinite-dimensional functional integral exactly to a finite-dimensional matrix integral. This can then often be computed exactly in the large limit, where is typically related to the rank of the gauge group, and compared to a gravitational dual computation which is purely geometric. Such computations have now been performed in a variety of examples [17, 18, 19, 20], with remarkable agreement on each side.

Thus far, almost all attention has been focused on AdS solutions. This is for the simple reason that very few AdS solutions outside this class are known. An exception is the Corrado-Pilch-Warner solution [21], which describes the infrared fixed point of a massive deformation of the maximally supersymmetric ABJM theory on M2-branes in flat spacetime. This solution is topologically AdS, but the metric on is not round, and there is a non-trivial warp factor and internal four-form flux on the . This has more recently been studied in [22, 23, 24, 19], and in particular in the last reference the free energy of the superconformal fixed point was shown to match the free energy computed using the gravity dual solution. The Corrado-Pilch-Warner solution also has a simple generalization to massive deformations of M2-branes at a CY four-fold singularity, where CY denotes an arbitrary Calabi-Yau three-fold cone singularity.

In this paper we systematically study the most general class of AdS solutions of M-theory. These have an eleven-dimensional metric which is a warped product of AdS with a compact Riemannian seven-manifold . In order that the isometry group of AdS is a symmetry group of the full solution, the four-form field strength necessarily has an “electric” component proportional to the volume form of AdS, and a “magnetic” component which is a pull-back from . We show, with the exception of the Sasaki-Einstein case, that the geometry on admits a canonical local -structure, and determine the necessary and sufficient conditions for a supersymmetric solution in terms of this structure. In particular, is equipped with a canonical Killing vector field , which is the geometric counterpart to the R-symmetry of the dual superconformal field theory.

Purely magnetic solutions correspond physically to wrapped M5-brane solutions, and we correspondingly recover the supersymmetry equations in [25] from our analysis. There is a single known solution in the literature, where is an bundle over a three-manifold equipped with an Einstein metric of negative Ricci curvature. On the other hand, solutions with non-vanishing electric flux have a non-zero quantized M2-brane charge , and include the Sasaki-Einstein manifold solutions as a special case where the magnetic flux vanishes. For the general class of solutions with non-vanishing M2-brane charge, we show that supersymmetry endows with a canonical contact structure, for which the R-symmetry vector field is the unique Reeb vector field. A number of physical quantities can then be expressed purely in terms of contact volumes, including the gravitational free energy referred to above, and the scaling dimension of BPS operators dual to probe M5-branes wrapped on supersymmetric five-submanifolds . These formulae may be evaluated using topological and localization methods, allowing one to compute the free energy and scaling dimensions of certain BPS operators without knowing the detailed form of the supergravity solution.

In our analysis we recover the Corrado-Pilch-Warner solution as a solution to our system of -structure equations. We also show that this solution is in a subclass of solutions which possess an additional Killing vector field. For this subclass the supersymmetry conditions are equivalent to specifying a (local) Kähler-Einstein four-metric, together with a solution to a particular second order non-linear ODE. We show that this ODE admits a solution with the correct boundary conditions to give a gravity dual to the infrared fixed point of cubic deformations of M2-branes at a CY four-fold singularity. In particular, when CY equipped with its flat metric, this leads to a new, smooth supersymmetric AdS solution of M-theory.

The plan of the rest of this paper is as follows. In section 2 we analyse the general conditions for supersymmetry for a warped AdS background of eleven-dimensional supergravity, reducing the equations to a local -structure when is not Sasaki-Einstein. In section 3 we further elaborate on the geometry and physics of solutions with non-vanishing electric flux, in particular showing that solutions admit a canonical contact structure, in terms of which various physical quantities such as the free energy may be expressed. This section is an expansion of material first presented in [26]. Finally, in section 4 we analyse the supersymmetry conditions under the additional geometric assumption that a certain vector bilinear is Killing. In addition to recovering the Corrado-Pilch-Warner solution, we also numerically find a new class of cubic deformations of general CY backgrounds. Section 5 briefly concludes. A number of technical details, as well as the analysis of various special cases, are relegated to four appendices.

Note: shortly after submitting this paper to the arXiv, the paper [27] appeared, which contains a supersymmetric solution that appears to coincide with the solution we present in section 4.

2 The conditions for supersymmetry

In this section we analyse the general conditions for supersymmetry for a warped AdS background of eleven-dimensional supergravity.

2.1 Ansatz and spinor equations

The bosonic fields of eleven-dimensional supergravity consist of a metric and a three-form potential with four-form field strength . The signature of the metric is and the action is


with the eleven-dimensional Planck length. The resulting equations of motion are


where denote spacetime indices.

We consider AdS solutions of M-theory of the warped product form


Here denotes the Riemannian volume form on AdS, and without loss of generality we take .333The factor here is chosen to coincide with standard conventions in the case that is a Sasaki-Einstein seven-manifold. For example, the AdS metric in global coordinates then reads , where denotes the unit round metric on . In order to preserve the invariance of AdS we take the warp factor to be a function on the compact seven-manifold , and to be the pull-back of a four-form on . The Bianchi identity then requires that is constant. The case in which will turn out to be quite distinct from that with .

In an orthonormal frame, the Clifford algebra is generated by gamma matrices satisfying , where the frame indices , and , and we choose a representation with . The Killing spinor equation is


where is a Majorana spinor. We may decompose via


where and are orthonormal frame indices for AdS and respectively, , , and we have defined . Notice that our eleven-dimensional conventions imply that .

The spinor ansatz preserving supersymmetry in AdS is


where is a positive chirality Killing spinor on AdS, so , satisfying


The superscript in (2.6) denotes charge conjugation in the relevant dimension, and the factor of is included for later convenience. Substituting (2.6) into the Killing spinor equation (2.4) leads to the following algebraic and differential equations for the spinor field on


For a supergravity solution one must also solve the equations of motion (2.1) resulting from (2.1), as well as the Bianchi identity .

Motivated by the discussion in the introduction, in this paper we will focus on supersymmetric AdS solutions for which there are two independent solutions , to (2.1). The general Killing spinor ansatz may be written as


In general the two Killing spinors on AdS satisfy an equation of the form


Multiplying by on the left it is not difficult to show that is necessarily a constant matrix. Using the integrability conditions of (2.10),


one can verify that, without loss of generality, by a change of basis we may take to be the identity matrix. Thus and may both be taken to satisfy (2.7).

In this case with supersymmetry there is a R-symmetry which rotates the spinors as a doublet. It is then convenient to introduce


which will turn out to have charges under the Abelian R-symmetry. In terms of the new basis (2.12), the spinor equations (2.1) read


2.2 Preliminary analysis

The condition of supersymmetry means that the spinors , in (2.9) are linearly independent. Notice that we are free to make transformations of the pair , since this leaves the spinor equations (2.1) invariant. We shall make use of this freedom below.

The scalar bilinears are and , which may equivalently be rewritten in the basis (2.12). The differential equation in (2.1) immediately gives , so that using we may without loss of generality set . Setting in (A.3), the algebraic equation in (2.1) thus leads to


where . We immediately conclude that for we have


When this statement is not necessarily true. The case with and not identically zero is discussed separately in appendix D, where we show that there are no regular solutions in this class. We may therefore take (2.15) to hold in all cases.

It is straightforward to analyse the remaining scalar bilinear equations. In particular, is constant, and using the remaining freedom one can without loss of generality set .444In the special case that one can show that , which in turn leads to only supersymmetry. In the basis (2.12) we may then summarize the results of this analysis as


Here is a complex function on , while it is convenient to define to be the real function


Notice that in the limit we have , while for instead is nowhere zero. We also define the one-form bilinears


Here we have denoted . A priori notice that and are complex, while is real.

2.3 The R-symmetry Killing vector

The spinor equations (2.1) imply that


where we have used (2.15). Thus in fact is real, and it is then straightforward to show that is a Killing one-form for the metric on , and hence that the dual vector field is a Killing vector field. More precisely, one computes


Using the Fierz identity (A.6) one computes the square norm


In particular when we see from (2.17) that is nowhere zero, and thus defines a one-dimensional foliation of . In the case that this latter conclusion is no longer true in general, as we will show in section 2.7 via a counterexample.

The algebraic equation in (2.1) leads immediately to , and using both equations in (2.1) one can show that


It follows that


provided the Bianchi identity holds.555In fact this is implied by supersymmetry when , as we will show shortly in section 2.4. Thus preserves all of the bosonic fields.

One can also show that


so that have charges under . Perhaps the easiest way to prove this is to use the remaining non-trivial scalar bilinear equation


to show that


Since preserves all of the bosonic fields, we may take the Lie derivative of the spinor equations (2.1) to conclude that satisfy the same equations, and hence are linear combinations of . The Lie derivatives of the scalar bilinears, in particular (2.26), then fix (2.24).666More precisely, this argument is valid provided is not identically zero. However, when we necessarily reduce to the Sasaki-Einstein case, as shown in appendix C. In that case (2.24) also holds. We thus identify as the canonical vector field dual to the R-symmetry of the SCFT.

2.4 Equations of motion

Given our ansatz, the equation of motion and Bianchi identity for reduce to


where denotes the Hodge star operator on . We begin by showing that supersymmetry implies the equation of motion, and that for it also implies the Bianchi identity.

The imaginary part of the bilinear equation for the three-form leads immediately to


Thus for we deduce that is closed. On the other hand, the bilinear equation for the two-form


gives, via taking the exterior derivative,


where in the second equality we have combined with equation (2.28). We thus see that supersymmetry implies the equation of motion in (2.27).

Finally, using the integrability results of [28] one can now show that the Einstein equation is automatically implied as an integrability condition for the supersymmetry conditions, once the -field equation and Bianchi identity are imposed. In particular, note that the eleven-dimensional one-form bilinear is dual to a timelike Killing vector field, as discussed in [26] and later in section 3.4. We thus conclude

For the class of supersymmetric solutions of the form (2.1), supersymmetry and the Bianchi identity imply the equations of motion for and the Einstein equations. Moreover, when the Bianchi identity is also implied by supersymmetry.

Note that similar results were obtained in both [6] and [3]. In fact we will see in section 2.7 that the supersymmetry conditions also imply the Bianchi identity, although the arguments we have presented so far do not allow us to conclude this yet.

2.5 Introducing a canonical frame

Provided the three real one-forms defined in (2.2) are linearly independent, we may use them to in turn define a canonical orthonormal three-frame 777We use here, as opposed to , since is invariant under the R-symmetry generated by . In particular, from the definitions in (2.2), and using (2.24), (2.26), we have that .. More precisely, if these three one-forms are linearly independent at a point in , the stabilizer group of the pair of spinors at that point is , giving a natural identification of the tangent space with . Here the structure group acts on in the vector representation. If this is true in an open set, it will turn out that we may go further and also introduce three canonical coordinates associated to the three-frame .888Just from group theory it must be the case that the one-form in (2.2) is a linear combination of and , and indeed one finds that .

We study the case that are linearly dependent in appendix C. In particular, for we conclude that at least one of or holds at such a point. If this is the case over the whole of (or, using analyticity and connectedness, if this is the case on any open subset of ) then we show that is necessary Sasaki-Einstein with . Of course, in general the three one-forms can become linearly dependent over certain submanifolds of , and here our orthonormal frame and coordinates will break down.999This is sometimes referred to as a dynamical structure. By analogy with the corresponding situation for AdS solutions of type IIB string theory studied in [29], one expects this locus to be the same as the subspace where a pointlike M2-brane is BPS, and thus correspond to the Abelian moduli space of the dual CFT, although we will not pursue this comment further here.

Returning to the generic case in which are linearly independent in some region, we may begin by introducing a coordinate along the orbits of the Reeb vector field , so that


The equation (2.26) then implies that we may write


This defines the real functions and , which will serve as two additional coordinates on . The factor of has been included partly for convenience, and partly to agree with conventions defined in [25] that we will recover from the limit in section 2.7. Using (2.25) together with the Fierz identity (A.6), one can then check that


are orthonormal. Here is a local one-form that is basic for the foliation defined by the Reeb vector field , i.e. , . Note here that


is the square length of the Reeb vector field. The metric on may then be written as


We may now in turn introduce an orthonormal frame for , and define the -invariant two-forms


Of course, such a choice is not unique – we are free to make rotations, under which , , transform as a triplet, where the structure group is , and is the spin group associated to .

2.6 Necessary and sufficient conditions

Any spinor bilinear may be written in terms of , , having chosen a convenient basis101010Notice that using the definition of the two-form bilinears in (B.3), and the fact that , we see that also the are invariant under , namely . for the . Having solved for the one-forms in (2.33), the remaining differential conditions arising from -form bilinears, for all , then be shown to reduce (after some lengthy computations) to the following system of three equations


where in addition the flux is determined by the equation


Notice this is the same equation (2.29) we already used in proving that the equation of motion for follows from supersymmetry. The bilinear on the right hand side is given in terms of our frame by


One can invert the expression for the flux using these equations to obtain


Notice that although we have written these equations in terms of the three real functions , and , in fact they obey (2.34), where is given by (2.17). Regarding as a coordinate, there is then really only one independent function in these equations, which may be taken to be the warp factor . We also note that the connection one-form , defined via the orthonormal frame (2.33), has curvature determined by the first equation in (2.37), giving


Proof of sufficiency

It is important to stress that the set of equations (2.37), where the three-frame is given by (2.33), are both necessary and sufficient for a supersymmetric solution. In order to see this, we recall that our structure can be thought of in terms of the two structures defined by the spinors , (or equivalently , ). Each of these determines a real vector , real two-form , and complex three-form , where recall that also . In fact , so that the vectors determined by each structure are equal and opposite, and determine two structures on the transverse six-space .

Let us now turn to the Killing spinor equations in (2.1). We have two copies of these equations, one for each structure determined by the spinors . We shall refer to the first equation in (2.1) as the algebraic Killing spinor equation (it contains no derivative acting on the spinor itself). Using this notice that we may eliminate the term in the second equation, in order to get an equation linear in ; we shall refer the resulting equation as the differential Killing spinor equation. For each choice of , the latter may be phrased in terms of a generalized connection , where is the Levi-Civita connection. The intrinsic torsion is then defined as for each structure, and may be decomposed into irreducible -modules as a section of . Since , the intrinsic torsion may be identified as a section of . It is then a fact that the exterior derivatives of , , determine completely the intrinsic torsion – the identifications of the irreducible modules are given explicitly in section 2.3 of [10]. Our equations (2.37) certainly imply the exterior derivatives of both structures, since they imply the exterior derivatives of all -form bilinears, for . It follows that from our supersymmetry equations we could (in principle) construct both , and hence write down connections which preserve each spinor, so . In other words, our conditions then imply the differential Killing spinor equations for each of the supersymmetries.

For the algebraic Killing spinor equation, note first that forms a basis for the spinor space for each . Thus in order for the algebraic equation to hold, it is sufficient that the bilinear equations resulting from the contraction of the algebraic Killing spinor equation with and hold, where is either of . However, this is precisely how the identities in appendix A were derived. We thus find that the algebraic Killing spinor equation in (2.1) is implied by the two zero-form equations


and the one-form equations


with similar equations for . Notice that the first equation in (2.6) is simply the scalar bilinear in (2.2) which determines . The reader can find explicit expressions for the real two-form and three-form , in terms of the -structure, in appendix B. Using these expressions, one can show that (2.37) imply the remaining scalar equation in (2.6) and both of the equations in (2.43), thus proving that our differential system (2.37) also implies the algebraic Killing spinor equations. The computation is somewhat tedious, and is best done by splitting the equations (2.37) into components under the decomposition implied by the three-frame (2.33). This decomposition is performed explicitly in section 2.8. In the second equation in (2.6) we note that each term is in fact separately zero. We also note that the first equation in (2.43) may be rewritten as


The left hand side is essentially the Lee form associated to the -structure defined by .111111Therefore (2.44) has the geometrical interpretation that the transverse six-dimensional space is conformally balanced.

To conclude, we have shown that (2.37) are necessary and sufficient to satisfy the original Killing spinor equations (2.1).

2.7 M5-brane solutions:

It is straightforward to take the limit of the frame (2.33), differential conditions (2.37), and flux given by (2.40). Denoting , , and we obtain the metric


with corresponding differential conditions


The flux in (2.40) then becomes


These expressions precisely coincide with those in section 7.2 of [25]. Of course, this is an important cross-check of our general formulae.

Notice that the Bianchi identity for is satisfied automatically from the expression in (2.47). In fact for the general class of geometries the Bianchi identity and equation of motion for read


Defining the conformally related metric , the equation of motion for becomes . It follows that is a harmonic four-form on . In particular, imposing also flux quantization we see that defines a non-trivial cohomology class in , which we may associate with the M5-brane charge of the solution.

When there is no “electric” component of the four-form flux , and these AdS backgrounds have the physical interpretation of being created by wrapped M5-branes. Indeed, as we shall see in section 3, when there is always a non-zero quantized M2-brane charge , with the supergravity description being valid in a large limit. The supergravity free energy then scales universally as . One would expect the free energy of the M5-brane solutions, sourced by the internal “magnetic” flux , to scale as , where the cohomology class in defined by scales as . However, the lack of a contact structure in this case (see below) means that a proof would look rather different from the analysis in section 3.

In section 9.5 of [25] the authors found a solution within the class, solving the system (2.7), describing the near-horizon limit of M5-branes wrapping a Special Lagrangian three-cycle . In fact this is the eleven-dimensional uplift of a seven-dimensional solution found originally in reference [30]. The internal seven-manifold takes the form of an fibration over , where the latter is endowed with an Einstein metric of constant negative curvature. As one sees explicitly from the solution, the R-symmetry vector field acts on by rotating the factor in the latter decomposition. In particular, there is a fixed copy of , implying that does not define a one-dimensional foliation in this case. Notice this also implies there cannot be any compatible global contact structure, again in contrast with the geometries. The flux generates the cohomology group .

As far as we are aware, the solution in section 9.5 of [25] is the only known solution in this class. It would certainly be very interesting to know if there are more AdS geometries sourced only by M5-branes.

2.8 Reduction of the equations in components

In this section we further analyse the system of supersymmetry equations (2.37), extracting information from each component under the natural decomposition implied by the three-frame (2.33). Since we have dealt with the equations in the previous section, we henceforth take in the remainder of the paper.

We begin by defining the one-form


which appears in the frame element in (2.33), so that


and further decompose


where . Since also and are orthogonal to , it follows that is a linear combination of , , the orthonormal frame for the four-metric in (2.35). It is also convenient to rescale the latter four-metric, together with its structure, via


so that correspondingly .121212This scaling is different from the scaling used in section 2.7, where . Notice this makes sense only when , so that is nowhere zero.

Given the coordinates defined via (2.33), it is then natural to decompose the exterior derivative as


where from now on hatted expressions will (essentially) denote four-dimensional quantities. We may then decompose the exterior derivatives and forms in the supersymmetry equations (2.37) under this natural splitting.

Beginning with the first equation in (2.37), the utility of the definition (2.49) is that this first supersymmetry equation becomes simply


where to simplify resulting equations it is useful to define the function