Perturbing gauge/gravity duals by a Romans mass
Davide Gaiotto and Alessandro Tomasiello
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Università di Milano–Bicocca and INFN, sezione di Milano–Bicocca, I20126 Milano, Italy
Abstract
We show how to produce algorithmically gravity solutions in massive IIA (as infinitesimal first order perturbations in the Romans mass parameter) dual to assigned conformal field theories. We illustrate the procedure on a family of Chern–Simons–matter conformal field theories that we recently obtained from the theory by waiving the condition that the levels sum up to zero.
1 Introduction
The Romans mass parameter of IIA supergravity [1] is understood from a modern perspective [2] as the Ramond–Ramond (RR) flux . In spite of this, it still retains some aura of mystery. For example, its interpretation in M–theory is still challenging (although see for example [3, 4]). Also, the branes that source it are D8–branes, which have the peculiarity of generating a back–reaction that grows with distance (since there is only one direction transverse to them).
On spaces with boundary conditions with an AdS factor, the AdS/CFT correspondence [5] gives a non–perturbative understanding of string theory. One can then hope to get a non–perturbative understanding of the parameter on such backgrounds. Some non–supersymmetric AdS vacua with were proposed already in [1]; supersymmetric ones were found much more recently, starting from [6] and more recently in [7, 8, 9].
It was also anticipated some time ago [10] that vacua with Romans mass would be dual to field theories with a Chern–Simons term. Recently, many Chern–Simons–matter conformal field theories (CFTs) have found their gravity dual in string theory, starting with the example on AdS in [11]. Those gravity duals do not involve the parameter . However, it was later shown in [12] that the gauge/gravity duality in [11] could be deformed by adding .
In fact, we found in [12] that several ways of introducing were possible, yielding CFTs with varying amounts of supersymmetry, from to . Two theories, with and , had large flavor symmetries (SO(6) and SO(5) respectively). This helped us find their gravity duals, which were presented already in [12]. The and theories had smaller flavor symmetry groups, and their gravity dual could not be immediately identified.
In this paper, we partially fill that gap by finding those duals as infinitesimal first–order deformations of the solution on AdS. To see that the solutions are the right gravity duals, one can at first match the bosonic symmetry group, the amount of supercharges, and the moduli spaces of vacua. One finds, however, that all these matches derive from the match of the abelian superpotential, which actually also guarantees that the solutions are the correct duals, as we will now explain.
In these backgrounds, even a single D2–brane probe feels an effective superpotential . The D2, then, cannot move freely: it will only preserve supersymmetry along some subspace of . By AdS/CFT, should also be the superpotential for the field theory when the gauge group is abelian. indeed does not vanish for the and theories proposed in [12] (unlike for the theory of [11]). In four dimensions, an example of a family of theories whose abelianized superpotential does not vanish is given by the Leigh–Strassler theories [13]; in their gravity dual, D3–brane probes only preserve supersymmetry along some locus. Infinitesimal perturbations of the AdS background with these properties have been obtained in [14] at first order and in [15] at second and third order. (For a particular type of Leigh–Strassler theory, the gravity dual can actually be found exactly by solution–generating symmetries [16]).
After identifying the superpotential felt by a single D2brane probe with the abelian superpotential of the field theory, it turns out that the first order perturbation^{1}^{1}1 is quantized in string theory, but it can still be small compared to the other fluxes in the unperturbed solution. In this sense, it makes sense to work in perturbation theory and to postpone consideration of the flux quantization conditions to when one has the full solution. in of the gravity solution can be found with no extra Ansatz or choice. This is quite general. Suppose one has a supersymmetric solution with , whose CFT dual is known. Suppose one knows that a deformation exists, with a superpotential that does not vanish when abelianized, and with . (The meaning of the latter condition on the field theory side is discussed in [12, 17]). We observe in this paper that, in such a situation, the conditions for the existence of a supersymmetric deformation of the background, at first order in , leave no room to any guesswork. There is a clear procedure that leads to a solution, provided of course one starts with a superpotential which is appropriate for a CFT. This procedure is the AdS analogue of [14], except that there are nontrivial restrictions on the superpotential already at first order. The conditions for AdS solutions are more restrictive than the ones for AdS; for example, the Bianchi identities do not follow from supersymmetry as for AdS solutions in IIB [18].
So, to summarize, we outline a general procedure to deform gauge/gravity duals by an infinitesimal amount of Romans mass , and we illustrate it by finding the perturbations of the solution on AdS [19, 20] dual to the and theories discussed in [12]. In section 2 we review those theories. In section 3 we review the conditions for supersymmetry, and we isolate the function that plays the role of the superpotential for a probe D2–brane. In section 4 we outline the general procedure for finding infinitesimal perturbations; in section 5 we apply it to the and solutions on AdS.
2 Review of the field theories
Although our procedure is general, to fix ideas we will start by introducing the field theories which will provide its concrete applications in section 5.
In [12], we introduced several Chern–Simons–matter theories. The ones of interest for this paper consist of an Chern–Simons theory with gauge group UU, coupled to chiral superfields and vector superfields . The action is
(2.1) 
where
(2.2) 
There is a renormalization group flow in the space of the coefficients , . If , there is a fixed point at , : it is the of [11]. If is small but , it was argued in [12] that a fixed point will still exist for some value of the , although for a different value of the coefficients .
In fact, we argued that there is a fixed line, that passes through two points with enhanced symmetries. In general, the theory (2.1) has supersymmetry, and an SU(2) of flavor symmetry (as well as the R–symmetry SO(2)). For
(2.3) 
supersymmetry is enhanced to , and hence we have SU(2)SO(3) of R–symmetry. It was argued in [12] that the line of fixed point intersects this locus. Also, for
(2.4) 
supersymmetry remains , but the flavor symmetry gets enhanced to SU(2)SU(2) (SO(2)). The line of fixed points should intersect this locus as well.
General arguments predict [12, 17] that the gravity dual of a Chern–Simons–matter theory should have Romans mass
(2.5) 
In this paper, we will confirm this picture by finding the gravity dual to these theories, as an infinitesimal perturbation in of the solution on AdS. In finding these duals, we have been guided by comparing the superpotential in (2.2) with the superpotential of D2 probes, as we now explain.
Even in the abelian case, the superpotential (2.2) is non–vanishing. In the gravity dual, it should be reproduced by the superpotential felt by a single D2 probe extended along the three–dimensional Minkowski and at fixed radius (in Poincaré coordinates). Usually, a single brane probe which is point–like in the internal space and in the radial direction does not feel any superpotential, and the moduli space of its world–volume theory is unrestricted. For example, for AdS SE, where SE is a Sasaki–Einstein five–manifold, the moduli space of a D3 probe is the cone over SE, namely a conical Calabi–Yau, which has real dimension 6. Likewise, for AdS a tri–Sasaki–Einstein, Sasaki–Einstein, or weakly G seven–manifold, the moduli space of an M2 probe is the entire cone over those manifolds – a conical space with special holonomy and of dimension 8.
Going back to AdS, an example of background in which a D3 brane probe is not able to move freely is the Lunin–Maldacena background [16], dual to one of the Leigh–Strassler gauge theories. In that case, the moduli space of the D3 consists of three copies of intersecting at the origin, which reproduces the fact that, in the field theory, there is a superpotential even at the abelian level. In fact, it is not difficult to show that a D3 brane probe can reproduce this abelian superpotential.
In our three–dimensional field theories we expect a similar phenomenon as in the Leigh–Strassler theories: the abelian version of the superpotential (2.2) will be reproduced in the gravity dual by a D2 domain wall. This fact will help us find the gravity duals: in section 3.4 we will derive a general expression for the D2 superpotential in terms of certain differential forms that characterize the gravity solution, which we will now introduce.
3 Supersymmetry equations
We will review here the conditions for unbroken supersymmetry in the most general setting, using the language of generalized complex geometry.
3.1 The equations in general
Let us consider a spacetime of the warped–product form AdS, which means that the metric is of the form . Then this spacetime is supersymmetric in type IIA^{2}^{2}2The conditions for type IIB, that we do not need here, are obtained by . if and only if [21, Sec.7]

There exists an structure on . Here, are polyforms which are pure spinors for Clifford, and which satisfy
(3.1) for any . We have used the Chevalley internal product between internal forms: , .

There exist a closed three–form H, an even–degree polyform (the sum of all the internal fluxes)
(3.2) where is the cosmological constant, and . The last equation is actually the Bianchi identity, which can be generalized to contain –function–like sources (something we will not do in this paper).
If these equations can be solved, determine a metric , a –field, a dilaton and two six–dimensional Weyl spinors . The formulas for the metric and –field in terms of are a bit involved in general [22], but we will see in section 3.2 what they give for the cases that we are interested in. The dilaton is determined by
(3.3) 
where vol is the volume form determined by the metric ; notice that this is not an extra equation to solve; rather, it determines the dilaton once the supersymmetry equations have been solved. The spinors are determined by
(3.4) 
where , and we are confusing a differential form with associated bispinor; one can show [21, Sec. 3] that one can find such that (3.4) is true for any structure . (This fact is crucial in showing that the conditions (3.1),(3.2) above are equivalent to the original fermionic equations for supersymmetry.)
We will call the –field determined by “intrinsic”. A slight imprecision in (3.2) is that only if this intrinsic vanishes, the in (3.2) is the usual Hodge star. This is not a big problem, because one can always obtain a pure spinor pair with vanishing intrinsic by the action .
Moreover, there is also an alternative, equivalent formulation of (3.2) in which does not appear at all. It was found in [23]; here we write a more practical version:
(3.5) 
Here is an operator that depends on alone; it is explained at length in [23]. In some cases, its action is easier to compute than the whole Hodge star. The reason for the appearance of a subscript on in (3.5) is that the physical also receives contribution from the determined via (3.4):
(3.6) 
Notice, however, that it is not necessary to compute in order to solve the equations (3.5). Similarly, the physical RR fields are
(3.7) 
which obey
(3.8) 
In this paper, we will actually be looking for solutions with extended supersymmetry, namely and . This simply means that there should be an SO worth of structures, all obeying (3.2) (or (3.5)) with the same physical fields: the metric , the dilaton and the fluxes , . We will see concretely how this works in section 4.
3.2 Solving the algebraic constraints
We will now analyze the algebraic part of the supersymmetry equations, (3.1).
In full generality, there are three cases to consider. Let us call the type of a pure spinor the smallest degree that appears in the sum; in other words, only contains forms of degree type or higher. It turns out that the type of a pure spinor in dimension 6 can be at most 3. There are then three cases:

has type 0, and has type 3. This is usually referred to as the “ structure” case, for reasons that will become clear soon.

has type 0, and has type 1. This is the most generic case, and for this reason it is sometimes just called “”, or also “intermediate SU(2) structure”.

has type 2, and has type 1. This is called “static SU(2) structure” case.
In this paper, we are considering small deformations of a solution of type . This will fall in the second, generic , case. Hence we will now review briefly the solution of the algebraic constraint in the SU(3) structure case, then move on to the generic case, which is our real interest; and we will not pay any attention at all to the static SU(2) structure case.
In the SU(3) structure case, the condition of purity on each separately determines (up to a –transform)
(3.9) 
with a complex function, a non–degenerate two–form, and a decomposable three–form (one that can be locally written as wedge of three one–forms) such that is never zero^{3}^{3}3We are including in the definition of purity.. The constraint (3.1) then reduces easily to
(3.10) 
These equations define an SU(3) structure, which justifies the name given earlier to case 1. We mentioned after (3.4) that any pair determines an “intrinsic” ; in this case it is zero. It is more laborious, but also possible, to see that any structure determines a metric [22, 21]. In this case, this works as follows. , being decomposable, determines an almost complex structure (it is the one such that is a –form). Then we can just define the metric as . The condition (3.10) implies that the defined in this way is symmetric.
We now come to the case of interest in this paper, namely case 2. To find the solution to this constraint, one can use [24, 25, 26] two different internal spinors in (3.4); as we remarked earlier, any solution of (3.1) can be written as in (3.4), so there is no loss of generality in proceeding this way. One can also [26] solve directly the constraints (3.1). Either way, one gets
(3.11a)  
(3.11b) 
for some (varying) angle , real function , one–form and two–forms satisfying
(3.12) 
which mean that define an SU(2) structure. Actually, from the constraint (3.1), one would get (3.12) wedged with , but one can show [26, Sec. 3.2] that these can be dropped without any loss of generality. The pair (3.11) has a non–zero intrinsic –field (the one defined by (3.4)):
(3.13) 
Notice the difference with the SU(3) structure case, (3.9); there, the –field of the pair is zero when the exponent of is purely imaginary. For (3.11), the exponent of is purely imaginary, but the –field is non–vanishing and is given by (3.13). As we mentioned above, an structure also defines a metric. In this case, we get
(3.14) 
Finally, from the equation (3.3), we see that the dilaton is determined by
(3.15) 
for both cases considered in this subsection, (3.9) and (3.11).
3.3 The differential conditions
In this subsection we will take a first look at the differential equations for supersymmetry (3.2), both for the SU(3) structure case and for the general case.
The SU(3) structure case has been analyzed in [27]. One can also derive the same conditions from (3.2) [21] or (3.5). If we plug (3.9) in (3.5), using (for more details see [22]), we see immediately that . In this paper, we want to perturb SU(3) structure solutions with into structure solutions with . Hence, we only need to give the differential equations for the SU(3) structure case when . For that reason, we take the angle in (3.9) to be
(3.16) 
and we obtain
(3.17) 
with . One could also obtain these equations from Mtheory. Notice that^{4}^{4}4We thank D. Martelli and J. Sparks for discussions on this point. nothing prevents at this point the warping (and hence the dilaton ) from being non–constant, in contrast to the case , in which constancy of (because of its Bianchi identity) implies constancy of . Even though the procedure we outline later for first–order deformations does not require the warping of the undeformed SU(3) structure solution to be constant, it will be so for the explicit examples of section 5.
We will now look at the structure case. We will actually only solve the supersymmetry equations at first order in perturbation theory; a full analysis of the system (3.2) in the structure case is not really necessary. Even so, the study of the structure case is of independent interest; not many attempts have been made so far for negative cosmological constant (for a recent study, using a particular “singlet Ansatz”, see [28]). We collect here some of the relevant formulas.
We will first look at the first equation in (3.2) or (3.5), and substitute the expression (3.11) for the pure spinors.
The one–form part says that
(3.18) 
The three–form part gives, remembering that we choose to be purely imaginary:
(3.19)  
(3.20) 
where we have introduced
(3.21) 
which is none else than times the exponent of (3.11a). Finally, the five–form part can be shown to follow from the one– and three–form parts, (3.18) and (3.20).
Equation (3.19) suggests that we define
(3.22) 
which is such that . We have to remember, however, that the physical –field also contains another contribution, as we saw in (3.6) and (3.13). Hence we get
(3.23) 
up to closed two–forms.
As for the second (and third) equation in (3.2) or (3.5), we will look at them directly in perturbation theory, since the expressions we obtained are lengthy and not particularly illuminating. The only flux that appears to have a reasonably compact expression is . Using the formula for relevant for the pure spinor given in (3.11a),
(3.24) 
after some manipulations we compute
(3.25) 
The expressions for the other fluxes are more conveniently extracted directly from (3.2). Again, we will see them explicitly in perturbation theory later.
3.4 Superpotential for D2 probes
We remarked in section 2 that the abelian version of the superpotential (2.2) should be reproduced by a D2 domain wall, pointlike in the internal manifold and at fixed radius in Poincaré coordinates. In this subsection, we compute this superpotential in terms of pure spinors, in a way similar to [29] for four–dimensional theories, and anticipated in [30] for thre–dimensional theories. The result will be essential later, in section 4, when we will outline the procedure to find infinitesimal perturbations of solutions with no Romans mass.
In massive IIA, let us start with a metric of the form
(3.26) 
where the warping factor is a function of the seven internal coordinates, and the internal metric is so far unrestricted. We will use the internal fluxes as an electric basis; they determine the external fluxes (with legs in the spacetime) via
(3.27) 
One can get the equations for supersymmetry with a computation similar to the one in [31]. These equations were considered in [32] in the case without the warping, in [30, App. B] for the AdS case (which is the one we need here), and they will be presented in general in [33]. For our present purposes, we only need to know that they include
(3.28) 
where , and is part (along with a of no relevance here) of a “generalized G structure” [34].
To specialize the equations (3.28) to a spacetime of the form AdS, we take
(3.29) 
where is the warping from the four–dimensional point of view (the one introduced in section 3). We then decompose
(3.30) 
With this identification, (3.28) reproduces the real part of the first equation in (3.2). The rest of (3.2) can be reproduced too, but we do not need it here.
Now, let us consider a brane that extends along the three external dimensions, and an internal cycle . Such a brane is supersymmetric if and only if
(3.31) 
where denotes pullback to the . This then suggests that the superpotential is
(3.32) 
Notice that this makes sense precisely because is closed, (3.28). If we now consider the case in which is a point, we get that
(3.33) 
Using (3.11a) and the first equation in (3.18), we conclude
(3.34) 
for some proportionality constant . As we stressed earlier, this D2 superpotential should match the superpotential of the abelianized theory. Hence the function , which is one of the data of a gravity solution, is proportional to the abelianized superpotential. This fact should be true for a full solution; but it will be most useful in perturbation theory, as we will now see.
4 The first–order procedure
We will illustrate here how to start from an SU(3)–structure supersymmetric solution with , and perturb it to a first–order solution with . In section 4.1 we will explain how to do so at the algebraic level (namely, as far as the constraints in (3.1) are concerned), and in section 4.2 how to solve the differential equations.
4.1 Perturbing structure in structure
The general form of the pure spinors for the SU(3) structure case and for the generic structure case have been given in (3.9) and (3.11). We will now explain how to take a limit that sends one into the other.
The first thing we want to do is to send the one–form in (3.11b) to zero, since in (3.9) has no one–form part. Calling our first–order deformation parameter, we can write that as
(4.1) 
This creates two potential problems. First, in the exponent of in (3.11a), we see that the second term would seem to go to zero in the limit . But (3.12) implies , which means that is degenerate; since should be non–degenerate, we should not let the term in (3.11a) go to zero. This is accomplished by having start its expansion in at first order:
(4.2) 
We should also remember that, in the SU(3) structure case, we took (see (3.16)); hence, we should take
(4.3) 
From the first equation in (3.18) we also see that . Summing up, for we get
(4.4) 
with
(4.5) 
which is a limit of (3.21). The choice (4.2) also fixes the second problem created by (4.1): that it would have risked sending to zero the entire in (3.11b). One might think that now the five–form part will start with a term of order because of the in the exponent, but that term is , which vanishes thanks to (3.12). The next term in the expansion is order , and vanishes in the limit. The expansion of hence reads
(4.6) 
In particular, at order , we get
(4.7) 
Hence is a (1,0)–form and is a (2,0)–form with respect to the SU(3) almost complex structure defined by . This is consistent with the constraint in (3.12).
4.2 Strategy to solve the differential equations
We now move on to the differential equations for supersymmetry, (3.2). All the equations in this section and in the ones that will follow are to be understood up to orders , since we will only solve the equations at first order in perturbation theory.
We begin by noticing that, in the parameterization (3.1) of the structure that we are using, it is natural to divide the various forms according to their parity under reversal of the angle . The parity transformations of the pure spinors are
(4.8) 
recall that is multiplication by a sign, defined on a –form to be . From (3.2), we also see that then the fluxes , and the warping transform as
(4.9) 
We took equal to the perturbation parameter , at first order (Eq. (4.2)). So at order , we can consider only the forms with parity .
We can now use the expansions in for we obtained in (4.4) and (4.6) in the differential equations (3.2). In fact, the first equation was already analyzed beyond perturbation theory in section 3.3, so we can just use (4.2) and (4.3) in the equations there. Using the remark above about parity under , each of these equations will contribute either to order (in which case it should reproduce one of the equations for the SU(3) structure case, (3.17)), or at order .
The first equation in (3.18) is even in . It now simply gives that , which reproduces of the SU(3) structure case (see (3.15) and (3.17)). The second equation in (3.18) is odd in , and it gives
(4.10) 
Next, rather than reading (3.19), we can jump at the equation giving the total –field, which is odd in and reads at first order
(4.11) 
(3.20) is even in and, at order , it simply gives the second equation in (3.17).
We will now look at the expressions for the RR fluxes (the second equation in (3.2) or (3.5)). We know from (4.9) that the equation for and will simply reproduce, at order , the corresponding equations in (3.17), and that they will not change at order . In contrast, and will vanish at order , but not at order . For , we can just use (3.25):
(4.12) 
We have not given the all–order formula for in section 3.3. We can compute it now by using (4.4) and (4.6) in (3.2):
(4.13) 
Finally, let us look at the Bianchi identities (the third in (3.2)). The one for simply says that it is constant. The one for is
(4.14) 
recall that there is a non–vanishing in the SU(3) structure solution that we want to deform, and that and are given by (4.11) and (4.13).
Notice that and (4.14) are the only differential equations we have seen so far. The others are definitions of the fields provided by the supersymmetry equations. At all orders, there would also be equations on the geometry not involving the flux; but, at first order, we just saw that there is no such equation.
To summarize so far, the equations we have to solve at first order in are (4.14) and that in (4.12) is constant. If one wants to have extended supersymmetry, we remarked at the end of section 3.1 that one is actually looking for a SO worth of pure spinors, but in such a way that the physical fields (the fluxes, the metric and the dilaton) are invariant. In that case, one will then have to impose by hand that , and in (4.11), (4.12) and (4.13) are invariant.
We will now see that there is not much freedom in solving these equations: for an assigned field theory, no guesswork is necessary.
First of all we should remember (3.34). That equation should be true at all orders, but at first order it just says
(4.15) 
Now, follows by combining (4.10) with the fact that it is a (1,0)–form with respect to the almost complex structure of the SU(3)–structure solution:
(4.16) 
where is the Dolbeault operator. We can now find and from the data of the SU(3) structure, and . For , we can use (4.7) combined with the limit of (3.14); for , we can simply invert (4.5):
(4.17) 
At this point the fluxes are going to be determined via (4.11), (4.12) and (4.13); there are no choices to be made. All one has to do is to check that the supersymmetry equations explained earlier hold. In this sense, our procedure is algorithmic. Once one knows from field theory arguments the right , the gravity dual is determined at first order in perturbation theory.
Let us summarize. Suppose one has a CFT dual to a supersymmetric SU(3) structure AdS vacuum of IIA; most AdS/CFT duals known are of this type. Suppose one identifies a new conformal field theory that deforms the old one, in a way which is dual to switching on a Romans mass; examples of such deformations were given in [12]. If the superpotential of this theory is non–vanishing even at the abelian level, the gravity dual will be a solution of structure type, and it will be given, at first order in the Romans mass, by the procedure outlined in the preceding paragraph.
In the next section, we will illustrate this procedure by finding the perturbative solutions dual to the theories reviewed in section 2.
5 Perturbative solutions on
In this section, we will apply the procedure outlined in section 4 to the theories discussed in section 2. We will start by reviewing briefly, in section 5.1 and 5.2, the SU(3) structure solution we want to deform, in two sets of coordinates convenient to our needs. In the remaining subsections, we will find the and gravity duals we promised.
5.1 The solution in homogeneous coordinates
When we discussed the differential supersymmetry conditions for SU(3) structure in section 3.3, we found in equation (3.17) that cannot be closed (recall that ). Hence, it cannot be a Kähler form, and in particular not the usual Fubini–Study Kähler form . Also, one could not even write an which is globally defined and which is (3,0) with respect to the usual complex structure on , since, for that complex structure, . Fortunately, there are other almost complex structures on , with respect to which (so that a globally defined (3,0)–form exists), and so that is not closed. There is an worth of such almost complex structures; each point in this corresponds to a supersymmetry of the solution.
Let us start from , with coordinates , . One can think of as a bundle over (with missing zero section), with projection map . A form on the total space of a bundle with projection is the pull–back of a form on the base space if and only if it is basic, namely if it is vertical (, for any tangent to the fibres of ) and invariant (its Lie derivative with respect to any tangent to the fibres of vanishes, ). In our case, the forms
(5.1) 
are basic: they are annihilated by contraction with both vectors
(5.2) 
and they are closed (which, together with (5.2), implies that they are also invariant). Hence, they are pull–back of forms on . We will use the projector in (5.1) to do computations on using coordinates of .
Another way of thinking about (5.1) is the following: given a form on , one can try to define a basic form by subtracting its non–vertical part. In terms of the one–forms
(5.3) 
in the case of the form , this decomposition reads
(5.4) 
The one–form is dual to above, in that . We can apply the same procedure to the standard Kähler form in :
(5.5) 
The explicit expression of on the right makes it clear that it is basic. One can also see that is vertical from its definition on the left, using that ; using the fact that is quadratic, , one can also see easily that
(5.6) 
which implies that is also invariant under . This is the standard Fubini–Study Kähler form on . As we remarked earlier, however, it is not exactly what we need in the supersymmetry equations.
To construct the supersymmetric , we need to introduce more data. A holomorphic symplectic form in four complex dimensions is a two–form whose square gives the holomorphic volume form :
(5.7) 
In , one has an –worth of holomorphic symplectic forms . From each of these, one can extract the radial and non–radial parts using the vector and , just like in (5.4) and (5.5):
(5.8) 
In components, using the forms (5.1), one can also write
(5.9) 
These forms are vertical by construction, but they are not invariant under . By comparing (5.8) with , one obtains
(5.10) 
and, from this, , . Similarly, if one defines a vertical form by
(5.11) 
one sees that (which is related to the fact that ). In fact, by using our definition (5.7) above, we get
(5.12) 
for any holomorphic symplectic . So does not define a form on .
This, however, suggests a way of defining a different three–form which is both vertical and invariant:
(5.13) 
this time , because the charges of and add up to zero, rather than to as for . (The factor has no particular meaning; it has been selected for consistency of notation with the previous sections.) The new three–form now defines a new almost complex structure , under which it is a (3,0) form. Roughly speaking, we have just conjugated the usual FubiniStudy complex structure in one direction out of three.^{5}^{5}5 can also be thought of as the twistor space of ; the new almost complex structure corresponds then to conjugation on the fibre. This second almost complex structure makes sense on any twistor space [35].
For supersymmetry, we need to complement in (5.13) with a that obeys (3.10). If we decompose as
(5.14) 
the remark we just made about the new almost complex structure defined by (5.13) suggests that we define
(5.15) 
Notice that this form is also well–defined on , because the term is invariant under . Using now (5.10) and some manipulations, it is not difficult to see that (3.10) and (3.17) are satisfied by
(5.16) 
Since this solution works for any holomorphic symplectic form (see (5.7)), and there is an worth of such forms on , we conclude that this solution has .
Before we move on to the perturbative solutions, let us also remark that one can also use homogeneous coordinates to describe the massive solutions in [8]. One simply has to rescale and by a factor of , so that^{6}^{6}6In [8], the solution is recovered for . In this paper, we use slightly different conventions: the solution in (5.16) is obtained by again setting in (5.17), followed by an additional (immaterial) conjugation , , . Also, for consistency with [8] we introduced in (5.17) the curvature radius , which we have set to one in the rest of this paper.
(5.17) 
The formulas for the fluxes can then be found in [8, Eq. (2.2)].
5.2 The foliation
We present here the solution in a different set of coordinates, first used in [36], which are adapted to the foliation of in . These coordinates will allow us to offer, later on, an alternative presentation of one of our solutions, the one with SO(4) isometry group (discussed in section 5.3).
Before we discuss the foliation, let us review some useful forms on , that we will then use on each of the s in (5.23). In terms of the usual holomorphic coordinates on , , we have the one–form
(5.18) 
the round metric is then , and the Kähler form . Also,
(5.19) 
In usual coordinates, ; note that . Of course globally is not exact (it is the Kähler form of ), and the expressions we just wrote are valid in a patch. Finally, notice also that and are related to the SU(2)–invariant forms on via the Hopf fibration: if one adds an angle , one has
(5.20) 
These forms satisfy , as appropriate for left–invariant forms on . Notice also that, in these conventions, the round metric on with radius one is
(5.21) 
In what follows, we will use the forms , , we just introduced on each of the , with a subscript , , denoting which of the two s it refers to.
We will now discuss the foliation of . From the point of view of the field theory, this foliation exists because of a simple relation [37, 38, 39] between the moduli spaces of the Chern–Simons–matter theory and of the four–dimensional theory with the same quiver, which is in this case the conifold theory [40]. On the gravity side, it comes about as follows. The splitting allows one to realize as a fibration of on a segment. We can parameterize the segment as an angle ; the radii of the two s are and :
(5.22) 
We can now rearrange and , and reduce on the angle . Each of the leaves at gets reduced from to . The reduction on is nothing but the Hopf fibration to ; hence we have realized as a foliation whose generic leaves are copies of . Even at the level of the metric we can write:
(5.23) 
where