A $\mathcal{N}=2$ Supersymmetric $AdS_4$ Solution in M-theory with Purely Magnetic Flux

We find a new $\mathcal{N}=2$ $AdS_4$ solution in M-theory supported by purely magnetic flux via a sequence of abelian and non-abelian T-dualities. This provides the second known example in this class besides the uplift of the Pernici and Sezgin solution to 7d gauged supergravity constructed in the eighties. We compute the free energy of the solution, and show that it scales as $N^{3/2}$. It is intriguing that even though the natural holographic interpretation is in terms of M5-branes wrapped on a special Lagrangian 3-cycle, this solution does not exhibit the expected $N^3$ behavior.


Introduction
In recent years non-Abelian T duality (NAT duality) has been very successfully applied as a generator of new supergravity backgrounds that may have interesting applications in the context of the AdS/CFT correspondence [1]- [17]. While some of these backgrounds represent explicit new solutions to existing classifications [1,4,5,12], some of them have been shown to fall outside known classifications [18] or to provide the only explicit solution to some set of PDEs [3].
Type IIB were studied in [11]. This solution contains two U(1) ′ s, one of which can be further used to (Abelian) T-dualize back to Type IIA without breaking any of the supersymmetries. Finally, the solution is uplifted to eleven dimensions, where it can be shown to fulfill the conditions for 11d N = 2 AdS 4 solutions with purely magnetic flux, derived in [37] 4 .
The paper is organized as follows. In section 2 we recall briefly the IIB solution constructed in [11] through non-Abelian T-duality acting on the AdS 4 × CP 3 IIA background. In section 3 we construct its IIA Abelian T-dual, and discuss some properties of the associated dual CFT of relevance for the CFT interpretation of the 11d solution. Section 4 contains the uplift to M-theory. Here we discuss some properties of the CFT associated to the 11d solution, that are implied by the analysis of the supergravity solution as well as its IIA description. We compute the holographic central charge and show that, as expected, it coincides with the central charge of the IIB solution written in terms of the 11d charges. Thus, it scales with N 3/2 , contrary to the expectation for M5-branes. We argue that the field theory analysis that we perform suggests that there should be Kaluza-Klein monopoles sourcing the background, and that M5-branes should only play a role in the presence of large gauge transformations (in a precise way that we define). This is intimately related to the existence of a non-compact direction inherited by the NAT duality transformation, which, as discussed at length in the NAT duality literature (see for instance [8,11,17]), represents the most puzzling obstacle towards a precise CFT interpretation of this transformation. Finally, in section 6 we present our conclusions. Here we discuss further our result for the free energy, as well as the view that we have taken to try to give a CFT meaning to the non-compact direction. We have relegated most of the technical details to three appendices. In Appendix A we include some details of the derivation of both the NAT and T dual solutions presented in sections 2 and 3. These details are especially relevant for the supersymmetry analysis. In Appendix B we review the G-structure conditions for preservation of supersymmetry of AdS 4 × M 6 solutions to Type II supergravities. In Appendix C we perform the detailed supersymmetry analysis of the solutions in IIA, IIB and M-theory.

The IIB NAT dual AdS 4 solution
This solution was constructed in [11], where some properties of the associated dual CFT were also analyzed. We refer the reader to this paper for more details. In this section we present the background for completeness. More technical properties of the derivation that will be useful for the study of the backgrounds constructed from this one in the following sections are presented in Appendix A.
Together with this we find the RR sector: are the fluxes associated to the Page charges. Note that after the dualisation a singularity has appeared at the fixed point ζ = 0, where the squashed S 3 used to dualise shrinks to zero size. This singularity is associated to the component of the metric on the r-direction, and is always compensated with the singularity in the dilaton in computations of physical quantities such as gauge couplings, internal volumes, etc. We will see that it will be inherited by the IIA and M-theory solutions where physical quantities will however be perfectly well defined as well.

The IIA NAT-T Dual AdS 4 solution
Following the steps in Appendix A we get the following solution in Type IIA after dualizing the previous background along the φ 1 direction, that we will simply rename as φ 5 where (3.5) The NS 2 form is given by Notice that this blows up at ζ = 0 indicating that the geometry is singular here, which is confirmed when one studies the curvature invariants. The RR sector is given bŷ where the gauge invariant fluxes are expressed in terms of these viaF = F ∧ e −B 2 .

Supersymmetry
It was shown in [11,13] that the NAT dual of ABJM preserves N = 2 supersymmetry in 3d, which means the R-symmetry is U(1) in the dual CFT. The argument relies on a proof from [13]. In order to see this we must package all the dependence of the original geometry on the SU(2) isometry in a canonical frame 6 where each left-invariant one form ω a appears only once, and x µ are coordinates on some 7d base which fibers the squashed 3-sphere containing the SU(2) isometry. Then there is a bijective map between spinors independent of the SU(2) directions in this frame and those preserved by the NAT dual solution. The map acts on the 10 dimensional MW Killing spinors asǫ with the matrix 7 where v a are dual coordinates in the NAT dual geometry, which we are expressing elsewhere in terms of spherical or cylindrical polar coordinates v 1 = y 1 cos ξ, v 2 = y 2 sin ξ, v 3 = y 2 . In Appendix C.1 we derive a spinor for ABJM independent of the SU(2) directions. This may be written in terms of 6 dimensional MW spinors on CP 3 as in Appendix B where (η 1,2 + ) * = η 1,2 − with the sign labeling chirality. It is possible to decompose the 6d spinors in terms of two in linearly independent parts η + = π + +π + obeying the projections of eq (C.32). We can then make the coordinate dependence explicit as π + = e ζγ 34 π 0 + ,π + = e −ζγ 34π 0 + , (3.14) where we have introduced linearly independent constant spinors obeying the projections γ 1456 π 0 + = −π 0 + , γ 2345 π 0 + = π 0 + , γ 1456π0 + = −π 0 + , γ 2345π0 + =π 0 + , (3.15) in the frame of eq (A.3). The 10d spinor is constructed as in eq (B.3), but all dependence on the CP 3 directions is in eq (3.14), which is clearly independent of the SU(2) directions.
So N = 2 is preserved under the NAT duality transformation. We show in Appendices C.2,C.3 that the solution does this by mapping a U(1)'s worth of the SU(3)-structures supported by CP 3 to a U(1)'s worth of dynamical SU(2)-structures defined on the dual internal spaceM 6 . Of course only two of these dual objects are truly distinct: those that are defined in terms of the two linearly independent Killing spinors onM 6 . These may be summed to give the internal part of the N = 2 Killing spinor in IIB, namelŷ where κ ⊥ and κ satisfy κ 2 ⊥ + κ 2 = 1 and are given in eq (C.27). The Matrix J is defined as in the frame of eq (A. 3), where e Ca are given in eq (A.2). However for what follows it is only important that J is independent of φ.
It turns out that the amount of preserved supersymmetry is left invariant when we perform an additional T-duality on ∂ φ . As proved in [49] (see also [13]), to see this it is sufficient to show that the Killing spinors are independent of φ in the canonical frame of T-duality, where φ only appears in the vielbein e φ = e C(x) dφ + A(x)). (3.18) We get to such a frame by performing a SO(4) transformation R in eq (A.14), on the canonical NAT dual vielbeins in eq (A.8). The action of this rotation on the 10d Majorana Killing spinor will be ǫ → Sǫ, where The matrix S will be complicated but will not depend on φ because R does not, which is all that matters. This together with the fact that eq (3.16) is φ independent ensures all supersymmetry must be preserved. Indeed in the next section (see also Appendix C.4) we see that upon lifting to M-theory the NAT-T dual solution preserves N = 2 supersymmetry in the form of a local SU(2) structure in 7d of the form given in [39].

Properties of the CFT
In this section we briefly discuss some properties of the CFT associated to the NAT-T dual solution. The discussion follows very closely the analysis in reference [11] for the IIB NAT solution. For this reason we will omit most of the explicit computations. The reader is referred to this reference for more details. It was shown in [8,11] that the presence of large gauge transformations in NAT dual backgrounds allows to constrain quite non-trivially their global properties. In our particular background (see [11]) it is easy to see that at the singularity ζ = 0 the NS 2-form given by (3.6) reduces to 20) while the space spanned by (ζ, S 2 ) becomes conformal to a singular cone with boundary S 2 . Therefore large gauge transformations can be defined on this non-trivial 2-cycle, which must render in the fundamental region. For this, B 2 must transform into B 2 → B 2 + nπVol(S 2 ) when r ∈ [nπ, (n + 1)π]. In turn, theF 3 andF 5 field strengths lying on the ζ, θ, φ and ζ, θ, φ, S 2 directions of the NAT dual solution in [11] give rise after the T-duality toF 2 andF 4 field strengths lying on the ζ, θ and ζ, θ, S 2 directions, with the second one non-vanishing only in the presence of large gauge transformations. Accordingly, a Page charge associated to the ζ, θ components ofF 2 is generated in IIA from the quantization condition (3.22) As in [11] this charge is to be interpreted as the rank of the gauge group of the CFT dual to the solution in the r ∈ [0, π] interval. We indeed get N 6 = N 5 , with N 5 given by (3.18) in [11]. Specifically this fixes L to satisfy Note that this means that color branes are now D6-branes spanned on the R 1,2 ×M 1 ×S 1 φ ×S 2 directions. One can indeed check that these branes are BPS when placed at the ζ = 0 singularity. As in [11] the combination e −φ √ g rr in the DBI action is non-singular, rendering well-defined color branes.
In the presence of large gauge transformations with parameter n there is also a nonvanishing D4-brane charge N 4 = nN 6 , equal to the N 3 charge in [11]. Indeed one can check that D4-branes are also BPS when placed at ζ = 0. These branes should also play a role as color branes for n = 0, that is, in the r ∈ [nπ, (n + 1)π] intervals. The physical interpretation is that N 4 charge is created in the worldvolume of a D6 when it crosses n NS5. It is then plausible that the field theory dual to the solution in the [nπ, (n + 1)π] intervals arises in a (D4, D6) bound state -NS5 intersection. A similar realization was suggested in [17] for AdS 3 duals.
The charge interpreted as level in ABJM is also doubled under the NAT duality transformation. As a result, after the new Abelian T-duality we get two charges, k 6 , k 4 , associated to the (r, θ) and (r, θ, S 2 ) components ofF 2 andF 4 , respectively. They thus correspond to D6 and D4 branes or, equivalently, to D4-branes carrying both monopole and dipole charges. We may express the levels in terms of k and the number of large gauge transformations performed as Finally, it is easy to check that particle-like brane configurations can be associated to each of the charges with an interpretation as either rank or level in the IIA background. These branes are in all cases D2 or D4 branes wrapped on different sub-manifolds of the internal space. In particular: As for the IIB AdS 4 solution (see [11] for the details) the di-monopoles and 't Hooft monopoles have to sit at ζ = 0 while the di-baryons sit at r = 0. The previous analysis suggests a dual CFT in the r ∈ [0, π] region defined in terms of a U(N 6 ) k 4 × U(N 6 ) −k 4 quiver gauge theory with N = 2 supersymmetry, sourced by D6-branes spanned on the R 1,2 × M 1 × S 1 φ × S 2 directions 8 . In turn, for r ∈ [nπ, (n + 1)π] the gauge theory would arise from (D4, D6) -NS5 intersections. It was argued in [11] that invariance under large gauge transformations would imply that the seemingly different CFTs dual to the solution as the non-compact internal direction increases, could be related by some kind of duality, as in [33], with the essential difference that in this case the flow parameter would not be the energy scale but the non-compact internal direction. Reference [17] proposed an alternative mechanism which, applied to our solution, would imply that new U(N 6 ) × U(N 6 ) gauge groups would be created by some kind of un-higgsing mechanism, also not related to an energy scale, every time a NS5-brane is crossed. It would be interesting to understand better these proposals for the dual CFT as r increases.
In any case, keeping in mind that there is no a priori reason to expect that the new geometry makes sense as a string theory background 9 , we can just take these proposals as stringy inspired arguments in favor of the existence of a fundamental region in which the dual CFT would contain the same number of gauge groups as the original one.
Restricting ourselves to the r ∈ [0, π] region, a candidate brane realization of the dual CFT would then be the T-dual of the brane picture proposed in [11] for the NAT dual of ABJM: where 5 2 2 denotes the IIA exotic brane that arises after a T-duality transformation along a worldvolume direction of the 5 2 2 exotic brane of the IIB configuration [11], and z 1 and z 2 denote the two special Killing directions of this brane [43,44]. In our notation the (5 2 2 , k 4 D4) bound state would be extended along the 0124 and x 5 cos θ + x 9 sin θ directions, and its relative orientation w.r.t. the 5 2 2 -brane in the 59 plane would depend on k 4 . Note that the previous picture implies that in M-theory the corresponding AdS 4 geometry would be sourced in the fundamental region r ∈ [0, π] by Kaluza-Klein monopoles, as we discuss in the next section.

The purely magnetic AdS solution in M-theory
In this section we lift the solution of the previous section to M-theory and show that it falls into the general class of solutions with purely magnetic flux considered in [37]. The analysis of quantized charges suggests a dual CFT arising from Kaluza-Klein monopoles and M5-branes wrapped on the Taub-NUT direction of the monopoles. We compute the central charge and show that it scales with (N 5 + N 6 /2) 3/2 , where N 5 is the number of wrapped M5-branes and N 6 the number of Kaluza-Klein monopoles. This becomes simply N 3/2 6 in the fundamental region r ∈ [0, π].

Fluxes
The RR potentials of the IIA solution are given by C 1 gives rise to the g µz /g zz component of the 11d metric, where z denotes the eleventh direction. Given that there is a magnetic charge associated to C 1 in IIA, a quantized Taub-NUT charge arises in 11d. The brane that carries Taub-NUT charge is the Kaluza-Klein monopole, which is connected to the IIA D6-brane upon reduction along the eleventh, Taub-NUT direction. Since the IIA solution was sourced by D6-branes in the fundamental region r ∈ [0, π], Kaluza-Klein monopoles should play the role of color branes in M-theory in this region. As we will discuss, BPS KK-monopoles spanned on the R 1,2 ×M 1 ×S 1 φ ×S 2 directions can indeed be constructed in 11d that give rise to the D6 color branes in IIA upon reduction.
(C 3 − B 2 ∧ C 1 ) gives rise in turn to the 3-form 10 in 11d. Note thatĈ 3 has no components along the eleventh direction. This will be of relevance in our later discussion. The M-theory 4-form flux is then given by which, as we can see, is purely magnetic. Therefore there will be no M2-branes sourcing the 11d solution.
As we have noted,Ĉ 3 is by construction transverse to the eleventh direction. This potential couples in the worldvolume of M2-branes constrained to move in the space transverse to the Killing direction and in the worldvolume of M5-branes wrapped on this direction [40]. Moreover, its magnetic components are associated to wrapped M5-branes. Indeed one can show that these branes are BPS in the 11d background and are to be interpreted as color branes. They give rise upon reduction to the color D4-branes of the IIA background. Other field theory observables that we will be able to describe holographically will be constructed in terms of M2-branes transverse to the Killing direction or M5-branes wrapped on this direction.

Geometry and local SU (2) structure
In [39] it was shown that the most general N = 2 preserving solution in M-theory with an AdS 4 factor supports an SU(2) structure in 7d. As the M-theory 4-form G 4 is purely magnetic it actually falls into the more constrained class of solutions originally considered in [37]. In this section we show that we can uplift the IIA solution to M-theory and fit it into this class of solutions.
The metric ansatz of [39] is of the form where we have so that Ricci(AdS 4 ) = −12 g(AdS 4 ) to match the conventions of [39]. The metric on M 6 is mearly the internal part of the IIA metric in eq (3.1).
It is possible to express the internal 7d metric in the form where ds 2 (SU (2)) is the metric on a 4 manifold supporting a canonical SU(2)-structure with associated real 2-form J = J 3 and holomorphic 2-form Ω = J 1 + iJ 2 , satisfying Since G 4 is purely magnetic it is possible to define local coordinates such that [37] where ξ parametrizes the U(1) R-symmetry and ρ is defined through the associated Reeb vectorξ as |ξ| 2 = e −6∆ ρ 2 . Supersymmetry then requires that the SU(2) forms satisfy and the flux be given by We find that the uplift of the IIA solution fits into the above parametrisation. All the forms are defined in terms of the internal M-theory Killing spinors derived in Appendix C.4, one needs only to plug them into the bi-linears in Appendix B of [39]. Performing these steps with some liberal application of Mathematica, we find the local coordinate ρ = k 8 L 4 y 1 sin 2 2ζ sin θ, (4.12) and the solutions specific vielbein 1 − e −6∆ ρ 2 kL 2 256 sin 2 2ζ L 4 sin 2 2ζ sin θdθ + 64y 2 dφ − 64(y 1 dy 1 + y 2 dy 2 ) cos θ + where C 1 is the potential giving rise to the IIA RR 2-form, which may be found in eq (4.1).

Properties of the CFT
As in the previous section, some properties of the CFT dual can be inferred by analyzing the 11d supergravity solution. The picture that arises is simply the 11d realization of the IIA picture described in subsection 3.2, apart from some subtleties that have to do with the existence of the special Taub-NUT direction. Indeed, all brane configurations that play a role in 11d will be either transverse to this direction or wrapped on it.
The non-trivial S 2 of the IIA geometry is also present in the 11d uplift. Therefore one can define large gauge transformations for the uplift of the B 2 field, which is the 11d 3form potential with a component along the Taub-NUT direction, i k C 3 . Thus, as in the IIA background, we need to divide r in intervals of length π in order to have i k C 3 lying in the fundamental region. From here the discussion parallels exactly the IIA discussion.
In 11d we find quantized charges N 6 and N 5 = nN 6 , equal to the N 6 and N 4 , respectively, in IIA. N 6 is associated to KK-monopoles and N 5 to M5-branes wrapped on the Taub-NUT direction of the monopole. The interpretation is that M5-brane charge (with the M5-brane wrapped in the Taub-NUT direction of the monopole) is created in the worldvolume of the KK-monopole when it crosses M5-branes transverse to the Taub-NUT direction 11 . Using the worldvolume effective action that describes a KK-monopole in 11d [42,41] one can easily check that it is BPS when placed at ζ = 0. The calculation parallels the D6-brane calculation in IIA with the only difference that the action is now written in terms of eleven dimensional fields. Similarly an M5-brane wrapped on the Taub-NUT direction is also BPS at this location.
As in IIA, the charge interpreted as level in 11d is also doubled, and we get two values k 6 and k 5 equal to the k 6 and k 4 charges, respectively, in IIA. These are now associated to wrapped M5-branes carrying monopole and dipole charges 12 .
Similarly, we find particle-like brane configurations, which are either M2-branes transverse to the Taub-NUT direction or M5-branes wrapped on this direction. These branes are wrapped on the same sub-manifolds of the internal space as in IIA. Namely, As for the IIB AdS 4 solution (see [11] for the details) the di-monopoles and 't Hooft monopoles have to sit at ζ = 0 while the di-baryons sit at r = 0. In these derivations we have used the action that describes M2-branes transverse to the Taub-NUT direction of the monopole. In this action i k C 3 couples in both the DBI and CS parts, so the M2-branes are sensitive to large gauge transformations. The details of this action can be found in [40].
Putting together this information, and in analogy with the IIA discussion, we expect a field theory in the r ∈ [0, π] region described by a U(N 6 ) k 5 × U(N 6 ) −k 5 quiver with N = 2 supersymmetry, sourced by KK-monopoles spanned on the R 1,2 × M 1 × S 2 × S 1 φ directions, and with Taub-NUT direction z. A possible brane realization in the fundamental region r ∈ [0, π] could be where z denotes the eleventh direction, the M6 is the Kaluza-Klein monopole with Taub-NUT direction z and 5 3 is the exotic brane that gives rise to the IIA 5 2 2 brane upon reduction [43,44].

Free energy
We can now calculate the free energy on a 3-sphere in the CFT dual to the solution in M-theory. This is expressed in terms of the effective 4 dimensional Newton constant G 4 as One can determine G 4 via a dimensional reduction of supergravity on the internal space M 7 , the result is 1 where we work in units such that l p = 1. For the case at hand the relevant quantity is e 9∆ Vol(M 7 ) = k 2 L 6 32 r 2 sin 3 ζ cos 3 ζ sin θ sin χdζ ∧ dθ ∧ dφ ∧ dr ∧ dχ ∧ dξ ∧ dz. Integrating this in the region r ∈ [nπ, (n + 1)π], z ∈ [0, 2π] and using eqs (3.23,3.24) we arrive at This reproduces, as expected, the result in IIB, with N 5 , N 3 → N 6 , N 5 [11]. Essentially we have F S 3 ∼ N 5 + N 6 2 3/2 which reproduces the N 3/2 behavior of ABJM. In particular, in the fundamental region r ∈ [0, π], where N 5 = 0, we find (4.23) This is not a surprising result, given that the dependence of the free energy in type II theories, like the central charge and entanglement entropy of the strip, depends on the internal directions only through the quantity and is thus invariant under Abelian T-duality 13 and uplift to 11d. So, quite surprisingly, we have found an AdS 4 M-theory solution with purely magnetic flux that falls in the general classification of [37], that originates in M5-branes wrapped on calibrated 3-cycles, but whose free energy does not exhibit the expected N 3 behavior. We leave a further discussion on this issue for the conclusions.

Conclusions
In this work we have presented a new warped AdS 4 solution of M-theory preserving N = 2 supersymmetry, giving the only known solution in this class other than the uplift of Pernici-Sezgin. A legitimate question to ask is whether this solution is truly distinct from Pernici-Sezgin, indeed this solution was generated by performing first a NAT then T duality on AdS 4 × CP 3 , and some geometries derived via NAT duality have been shown to fall within the ansatz of previous solutions. This does not seem to be the case with this example: The quickest thing to note is that the free energy of Pernici-Sezgin scales as N 3 while this solution scales as N 3/2 . Additionally the uplift of Pernici-Sezgin is everywhere non singular while the curvature invariants of this solution blow up in certain regions of parameter space. One might still wonder if this solution approximates Pernici-Sezgin at least locally away from the singularity, as was argued in [1] to be the case for the NAT dual of AdS 5 × S 5 and the Gaiotto-Maldacena geometries [47]. This also does not seem to be the case. Sfetsos and Thompson were able to find an additional solution to the Gaiotto-Maldacena Toda equation which gave their solution. The differential equations giving rise to Pernici-Sezgin are more simple and are solved uniquely. So this solution is truly distinct.
In this work, following on [11], we have taken the view that the range of r is restricted to lie in a specific cell of length π after n large gauge transformations of B 2 . The reason is to ensure that 0 < | S 2 B 2 |< 4π 2 , a restriction motivated by string theory. However this does present an issue for the geometry, we are choosing to end it at a regular point which would usually demand the inclusion of extra localized sources. From a purely geometric view point we might choose to take 0 < r < ∞, however this would be very undesirable from an AdS/CFT perspective. A continuous r would lead to, among other things, a CFT dual with operators of continuous conformal dimension [8]. An attractive resolution to these issues is that the NAT duality generates a solution which approximates a better defined solution free of these pathologies. At any rate, regardless of these potential criticisms, it seems likely that one could use this work as a stepping stone to further populate the solution space of purely magnetic M-theory solutions.
Supersymmetric probe branes in the 11d uplift of the Pernici-Sezgin solution were considered in [46], with an aim at introducing punctures on the Riemann surface along the lines of [47]. The BPS configurations were shown to preserve two U(1) ′ s, one more than required by the R-symmetry of the 3d N = 2 SCFT. This second U(1) corresponds to a global U(1) in the 3d field theory, and seems to play a key role in the 3d-3d correspondence [48]. It was argued in [46] that a large number of supersymmetric M5's would ultimately backreact on the Pernici-Sezgin geometry to produce a new AdS 4 solution with a U(1) 2 isometry. It would be interesting to show whether the AdS 4 solution obtained in this paper, containing a U(1) 2 isometry, could be related to this physical situation.
That the free energy of our purely magnetic AdS 4 solution scales like F S 3 ∼ N 3/2 rather than N 3 is a little puzzling. It was proved in [39] that the presence of M2 branes, whether accompanied by M5 ′ s or not, always gives rise to N 3/2 behavior. However we know that our solution cannot contain M2 branes, indeed it is not possible to accommodate M2 branes in a purely magnetic flux ansatz, so what are we to make of this apparent contradiction. Firstly it should be noted that, at least as far as the authors are aware, there is no proof F S 3 ∼ N 3 holds universally for all wrapped M5 brane solutions. However this seems like an inadequate evasion of a confusing result. More likely is that the solution we present does not correspond to wrapped M5 branes. Indeed, the ansatz taken in [37] to derive the purely magnetic solutions is defined by requiring the Killing spinors to satisfy the same projection conditions as the wrapped branes. Yet the solutions need not describe in general M5-branes wrapped in 3d manifolds in the near horizon limit. The metric we have obtained is rather complicated and it seems difficult to identify a 3-cycle in the internal geometry that such branes might wrap. This together with the fact that the free energy does not scale with N 3 is suggesting that this is indeed the case for our solution.
On the other hand, even if the CFT interpretation of the solution is yet very preliminary, we seem to have found that there are quantized charges associated to both KK-monopoles and M5-branes, with the first being the only sources of the geometry in the r-region that we have defined as the fundamental region. This is also suggestive of a geometry not originating from wrapped M5-branes.
Finally, let us comment on something slightly tangential. In the process of discussing the supersymmetry preserved by purely magnetic M-theory solutions we analised the G-structure preserved by the NAT dual of ABJM. We showed in appendix C.3 that this IIB solution preserves a U(1)'s worth of dynamical SU(2)-structures in 6d. We note that, it is possible to take the intersection of two of these and define an identity structure. However, given that a complete systematic study of AdS 4 solutions to type II supergravity preserving N = 2 supersymmetry is currently absent form the literature, we have not pursued this here. Even so we know that, as with the better studied AdS 5 , N = 1 cases [55,56], supersymmetry should be preserved in terms of either a local "SU(2)-structure" or "identity structure" on the internal co-dimensions of the isometry dual to the U(1) R-symmetry. The NAT dual of ABJM will certainly fall into the latter class.

A Some details on the NAT and T duality transformations
In this Appendix we give some details on the derivation of the solution in section 3. The starting point is the AdS 4 × CP 3 metric written as a Hopf fibration where ω a are SU(2) left-invariant 1-forms satisfying dω a = 1 2 ǫ abc ω b ∧ ω c and ds 2 (M 7 ) = L 2 4 ds 2 (AdS 4 ) + 4dζ 2 + cos 2 ζ dθ 2 + cos 2 θdφ 2 , where the AdS radius is 1. Specifically we introduce the vielbeins The dilaton of this solution is constant and set to e Φ = k L , while the non trivial fluxes are where the only non zero components are

A.2 The IIA NAT-T Duality
We would now like to perform a T-duality on the global U(1) corresponding to ∂ φ . To do this we can once more make use of the results of [13] (see [49] for the original derivation).
In order to do this we need to express the metric and B 2 as dŝ 2 = dŝ 2 (M 9 ) + e 2C (dφ + A 1 ) 2 , Clearly B 2 is already in this form, while the same can be achieved for the metric with a rotation of the vielbein basisê → Rê, giving where Ξ = ∆ sin 2 θ + y 2 1 sin 4 ζ cos 2 θ. (A.13) A rotation that achieves this is which acts on 2456. We have introduced the following expressions With the rotated vielbein basis, we may give the RR forms in [11] in terms of them and then use [49,13] to read off the T-dual solution, getting then the results in section 3.

B Type-II G-structure conditions for AdS 4 solutions
In this Appendix we review the G-structure conditions for supersymmetric AdS 4 × M 6 solutions, which is a slight modification 15 of what may be found in [50,51], but with notation more akin to [57,58]. The metric can be cast in the form where the AdS radius is 1 and the dilaton has support only in M 6 . The fluxes have the same direct product structure, which in terms of the RR polyform we may express as We use a real representation of the 10d gamma matrices 16 in which the Dirac and ordinary conjugates coincide. A 4+6 split is performed on the 10d MW Killing spinors where ǫ = (ǫ 1 , ǫ 2 ) T and Γ (10) ǫ = σ 3 ǫ so that we can write where ± labels chirality in 4 and 6 dimensions, so that the upper/lower signs should be taken in IIA/IIB and (η + ) * = η − and we take the internal, η 1,2 spinor to have unit norm.
where once more the upper/lower signs should be taken in IIA/IIB and The G-structure on M 6 can either be an SU (3), when η 1 + and η 2 + are globally parallel, or SU(2) when they are not. Using a Fierz identity and the Clifford map it is possible to express Φ ± as polyforms. In the SU(3)-structure case we can write this in terms of a complex 2-form J and a holomorphic 3-form Ω hol as where the forms may be expressed in terms of the internal spinors as and the forms obey J ∧ J ∧ J = 3i 4 Ω hol ∧Ω hol , J ∧ Ω hol = 0.
For the SU(2)-structure case the internal spinor may be expressed as where χ † + η + = 0 and κ 2 + κ 2 ⊥ = 1. The pure spinors may then be expressed in terms of a holomorphic 1-form z, a real 2-form j and a holomorphic 2-form ω hol as The various forms may be extracted from the spinor via and obey the conditions, Finally it should be noted that the above conditions are actually the conditions for N = 1 in 3d. We will be concerned with N = 2 supersymmetry which implies a CFT dual with U(1) R-symmetry. This will manifest itself in the fact that there should be a U(1)'s worth of pure spinors satisfying eq (B.5), two of which are independent 17 .

C Detailed supersymmetry analysis
In this appendix we shall look at how the Killing spinors are transformed under the sequence of dualities we perform to reach the M-theory solution of section 4.2. We shall begin by identifying a set of spinors on CP 3 that are uncharged under the SU(2) on which the NATduality is performed.

C.1 A SU (2) T-duality invariant Killing spinor on AdS 4 × CP 3
We express the metric of ABJM in terms of the vielbein basis of eq (A.3). Supersymmetry is preserved in type IIA when the variations of the dilatino and gravitino vanish. For ABJM which has a constant dilaton and zero Romans mass these constraints are and Inserting the fluxes into the variation of the dilatino and manipulating leads to This constraint preserves a maximum of 24 real supercharges, however one finds that such a Killing spinor, which also solves the gravitino variation, must depend on the SU(2) directions [11]. Here we take a different approach and impose the projection which preserves only half the supercharges. Turning attention to the gravitino variation, one finds that the components along the AdS 4 directions give which is a standard Killing spinor equation for AdS 4 which one can solve without any constraint.
Using the projection or eq (C.5) it is possible to show that the gravitino variation along the CP 3 directions reduce to a differential equation and an additional projection ∂ ζ ǫ + Γ 6 (iσ 2 )ǫ = 0, (C.7) Γ 1456 ǫ = − cos 2ζ + sin 2ζΓ 6 (iσ 2 ) ǫ. (C.8) These are solved by ǫ = e −Γ 6 (iσ 2 ) ǫ 0 (C.9) where ǫ 0 is a spinor which depends only on the AdS 4 coordinates and obeys Thus we have found a Killing spinor preserving 8 real supercharges which gives N = 2 supersymmetry in 3d. This is the most general spinor which is independent of the SU(2) directions (in the prefered frame) and so [13] informs us that 8 supercharges are preserved under a SU(2) NAT duality transformation.
(C. 35) Clearly eq (C.34) supports a U(1) of dynamical SU(2)-structures, as was the case in type-IIB, which we will not explicitly derive.
(C. 36) In the conventions of [39] the M-theory spinors are where e∆ 2 = eÃ 2 − Φ 6 and e 2Ã is a modified warp factor of AdS 4 in IIA such that Ricci(AdS 4 ) = −12g(AdS 4 ). Thus if we identify the AdS 4 spinors of IIA with those of eq (C.37) we see that which clearly satisfyχ 1 χ 1 =χ 2 χ 2 = 1, and from these one can construct spinors of charge ±2 under the U(1) R-symmetry It is then simply a matter of plugging the χ ± of this section into the spinor bi-linears in appendix B of [39], and rotating the frame to reproduce the results of section (4.2). Note that the frame used in this section needs to be rotated as in eq (A.14) to reach the vielbein basis where flat directions 2456 may be identified with G 1,2,3,4 of eq (3.2) and the rest with eq (A.3).