D-branes and non-Abelian T-duality

We study the effect of non-Abelian T-duality (NATD) on D-brane solutions of type II supergravity. Knowledge of the full brane solution allows us to track the brane charges and the corresponding brane configurations, thus providing justification for brane setups previously proposed in the literature and for the common lore that Dp brane solutions give rise to D(p+1)-D(p+3)-NS5 backgrounds under SU(2) NATD transverse to the brane. In brane solutions where spacetime is empty and flat at spatial infinity before NATD, the spatial infinity of the NATD is universal, i.e. independent of the initial brane configuration. Furthermore, it gives enough information to determine the ranges of all coordinates after NATD. In the more complicated examples of the D2 branes considered here, where spacetime is not asymptotically flat before NATD, the interpretation of the dual solutions remains unclear. In the case of supersymmetric D2 branes arising from M2 reductions to IIA on Sasaki-Einstein seven-manifolds, we explicitly verify that the solution obeys the appropriate generalized spinor equations for a supersymmetric domain wall in four dimensions. We also investigate the existence of supersymmetric mass-deformed D2 brane solutions. rterrisse@ipnl.in2p3.fr, tsimpis@ipnl.in2p3.fr, cwhiting@bates.edu ar X iv :1 81 1. 05 80 0v 3 [ he pth ] 2 S ep 2 01 9


Introduction
Non-Abelian T-duality (NATD) has recently attracted renewed interest, not least as a tool for generating new supergravity solutions [1,2,3,4,5,6,7,8,9,10,11,12,13]. One problem in this context is that NATD only carries local information: even when the starting point (the "seed") is a globally well-defined solution, NATD will typically generate a highly complicated local solution whose global completion, if it exists, is completely obscured. The main motivation of the present paper is to consider, as seeds of the NATD, full-fledged brane solutions -as opposed to AdS near-horizon solutions, which has been the case before. The point is that, as we will see, being able to follow the interpolation between the near-horizon and spatial infinity limit, gives us a better handle on the brane configuration and the global properties of the dual.
Before NATD, standard intersecting brane solutions (i.e. those following the simple harmonic superposition rule) often interpolate between two asymptotic regions, each of which is an independent supergravity solution in its own right: flat space at spatial infinity, and the near-horizon limit -which in the examples considered here always contains an AdS factor. In fact the brane is not strictly-speaking present in these solutions: the brane backreacts on the flat space in which it is initially inserted and dissolves into flux, so that the resulting solution is without sources. 1 In the interpolating and nearhorizon solutions there is still a remnant of the brane, whose charge can be computed by integrating the flux.
The NATD of the spatial infinity limit of a standard intersecting brane solution is then universal, i.e. it is the same for all standard intersecting brane solutions: it is simply the NATD of flat space. As we will see in detail in section 2.2.2, in this case an SU (2) NATD generates a distribution, whose density can be determined explicitly, of parallel NS5 branes continuously distributed along a half line. However after NATD the notion of spatial infinity limit and near-horizon limit are no longer necessarily meaningful. Still, as previously stated, both limits are genuine solutions so that NATD will generate new solutions out of them. These will be called respectively the spatial infinity and near-horizon limit of the dual. For the case of the D3 brane, discussed in section 2, this definition leads to the commutative diagram of figure 1.
Furthermore, all examples considered here will be seen to be consistent with the common lore that Dp brane solutions give rise to D(p+1)-D(p+3)-NS5 backgrounds under SU(2) NATD transverse to the brane. More generally, for nonstandard brane solutions, such as the D2 branes considered here in sections 4, 5, 6, the geometry may not asymptote to flat space even before NATD. In that case the original solution and its dual will in general contain non-vanishing fluxes even at spatial infinity. Nevertheless, by zooming in near the locus of the NS5, one can still see the presence of a continuous distribution of NS5 branes in accordance with the harmonic superposition prescription for intersecting branes.
Having the full interpolating solution can facilitate reading off the possible global completions of the geometry after NATD. In particular the topology of the slices r = constant can be studied more (d) continuous linear distribution of NS5 branes along a half line. The spatial infinity limit of the dual solution is defined by taking the limit before the NATD. easily by taking the spatial infinity limit r → ∞, where the various expressions simplify. As already mentioned, this becomes most clear in the standard cases where, before the NATD, the space becomes flat at spatial infinity and the NATD is universal.
In order to describe the supersymmetric D2-branes of our paper arising from M2 reductions to IIA on Sasaki-Einstein seven-manifolds, we used the formalism of generalized geometry and the sevendimensional pure spinors of [14]. We explicitly verify that this class of solutions obeys the appropriate generalized spinor equations for a supersymmetric domain wall in four dimensions. We have also investigated the existence of supersymmetric mass-deformed D2 brane solutions and ruled out a large class of Ansätze.
The outline of the paper is as follows. We revisit the case of the D3 brane in section 2. Various D2 brane solutions are examined in sections 4, 5, 6. These are obtained by reduction from eleven dimensional solutions of M2 branes transverse to cones over S 7 or Y p,q reviewed in section 3. In section 7 we examine massive deformations of the supersymmetric massless IIA D2 brane solutions of section 4. We start by casting the supersymmetry equations of the branes in the formalism of generalized geometry in section 7.1. Massive deformations of the resulting pure spinor equations are examined in section 7.2. We conclude in section 8. In the appendix we explain our various conventions and compare them to the literature.
The D3 branes lie along the R 1,3 directions. This can be seen in a probe approach. Consider the same expression as (4) for a flux living now in R 1,9 . The coordinates now refer to the metric (1), but with H = 1. Since in spherical coordinates the S 5 collapses at r = 0, its volume form dΩ 5 is ill-defined. However F 5 is a well-defined current (i.e. a distribution-valued form) and we can compute: This means that a brane is inserted in r = 0. In this coordinate system this is a codimension 6 space, and thus a D3 lying along R 1,3 . In the transverse space R 6 the brane looks like a point.
The D3 now acts as a source for the flux F 5 , which backreacts on the metric through Einstein's equations to give (1). The global geometry has changed, and the S 5 no long collapses. The supergravity equations are solved without sources and the brane cannot be seen anymore. Nevertheless the information about the brane is still present in the charge carried by the flux.

Near-Horizon and spatial infinity
Taking (1) as an ansatz for the metric, the supergravity equations reduce to an equation on H. In the probe interpretation this amounts to saying that H is harmonic in the transverse space. If we further constraint H to depend on r only, the general solution is of the form, where a and b are two integration constants. b can readily be interpreted as the brane charge. Then two limiting cases arise: Spatial infinity: If b = 0, H is a constant and the space is flat without flux: no brane is inserted.
Since H → a when r → ∞, this case is called the spatial infinity limit.
Near Horizon: If a = 0, the solution becomes, which is the well-known AdS 5 × S 5 background. For r → 0, H ∼ b r 4 so that this case is called the near horizon limit.
It is remarkable at first sight that both limits correspond to genuine backgrounds. The reason behind this is that they ultimately correspond to different choices of integration constants. These considerations might seem trivial for now, but they will be relevant in the following, when the brane configurations become more involved.

The NATD
After performing NATD along the SU(2) isometry in the σ i , cf. (3), the background (1), (4) becomes, and nonzero RR fluxes given by, In particular we see that the NATD has resulted in a metric which is singular at α = π 2 . Moreover the duality has generated a nonvanishing Kalb-Ramond field B 2 and a varying dilaton φ.
Note that the background (8) contains a family of solutions, inheriting its degrees of freedom from the D3 solutions before duality: for each choice of harmonic function H, NATD generates a different solution. We will keep the same denomination for the different limits, namely the near-horizon for H = L 4 r 4 and spatial infinity for H = 1 (i.e. L = 0). However their interpretation as different limits of the interpolating dual background is less meaningful. We will study them separately to get a better view on the brane configurations.
For later use let us rewrite the metric in (8) in terms of the coordinates defined by, x = r sin α cos θ ; y = r sin α sin θ ; u = r cos α .
Recalling the ranges of the α, θ coordinates, cf.
(2), we see that u ≥ 0, while x, y ∈ R. Simplifying with r 2 = x 2 + y 2 + u 2 then gives, In these coordinates, the metric is singular at u = 0.

Brane configuration and charges
The non-vanishing fluxes might indicate the presence of branes. Here we could expect NS5, D4 and D6 branes as magnetic sources for H, F 4 and F 2 . The first clue is given by the corresponding charges.
Let us start with the NS flux. An appropriate cycle would be the following: start at constant α = α 0 and integrate along ρ, χ, ξ where ρ goes from 0 to ρ 0 . At ρ = 0 the cycle closes but we need to close it at ρ 0 . To do so, keep ρ constant and vary α from α 0 to π/2. The resulting charge will be independent of α 0 so we can take the limit α 0 → π/2.
Along the cycle Σ 3 = [ρ, χ, ξ], α = π 2 , H 3 simplify to Integrating H 3 yields, In fact the charge will depend only on the value of ρ when the cycle reaches α = π/2. As we will see more explicitly in the simpler case of the spatial infinity limit in section 2.2.2, this suggests a continuous distribution of NS5 branes at α = π/2 along the ρ direction, with constant charge density. For the flux to be quantized we need to close the cycle at quantized values of ρ, namely ρ 0 = 4nπ. The NS5 branes are thus located at the singularity: this can be seen from the form of the metric and NS-NS fields in the limit α → π 2 , which is consistent with the general form expected from the harmonic superposition rule [15]. After defining ν = (π/2 − α) 2 we find, in the α → π 2 limit, The harmonic function in the space transverse to the NS5 is proportional to ν −1 , indicating the presence of NS5 branes at ν = 0. However this is not a point in the transverse space. Since ρ is still unconstrained, this is consistent with a distribution of charge along ρ.
In order to determine the configuration of the remaining branes we follow the same strategy. Recall that in solutions with nonzero B 2 , the quantized charges are the Page charges, defined as integrals of the Page forms,F As can be seen from this definition, the Page charges depend on the cohomology class of B 2 , i.e. they are not invariant under large gauge transformations of B 2 .
Integrating the Page forms in the D3 brane solution gives, which leads to L 4 = 8α 2 N D6 . If we denote by ∆Q D4 the change of D4 brane charge under a large gauge transformation of B 2 , we find, which leads to L 4 = 1 n 8α 2 ∆N D4 . From this we readily see that This is nothing other than the creation of D4 branes via a Hanany-Witten effect [16], as will be reviewed in the following in section 2.2.3. In order to get a probe interpretation of these brane charges we would need to know in which background the branes are inserted, but the situation is not entirely clear here.
The expression for the fluxes suggests that the D6 is transverse to r, α, θ and that the D4 is transverse to r, α, θ, χ, ξ. This would lead to the following brane configuration:

Spatial infinity limit
The spatial infinity limit gives the following background: Here there are no RR fluxes anymore so all the D-brane charges vanish. The configuration is thus much simpler. In fact it will now be possible to understand the exact brane configuration, as is done for the D3. Moreover this background is the NATD of the spatial infinity limit of the D3 brane solution: the NATD (21) is simply the NATD of flat space along an S 3 ⊂ R 4 factor. This decomposition is thus better suited for the spatial infinity limit than the S 5 ⊂ R 6 decomposition of the D3 brane solution. Accordingly the seed metric before NATD reads, which is simply the spatial infinity limit of the metric (1) written in the coordinates of (10). In these coordinates the NATD metric (21) is given by, Let us now make a further change of variable, upon which the metric becomes, where in order to obtain a complete metric on R 4 we must have θ ∈ [0, π]. In the second line above we have introduced the function, which is harmonic in R 4 except for θ = 0. The NS-NS two-form and dilaton are given by, This clearly shows the presence of NS5 branes along the R 1,5 directions, located at θ = 0 (or alternatively at u = 0), in accordance with the harmonic superposition rule [15]. However this is not enough to determine the exact position of the branes since they could be anywhere on this half line. Integrating H 3 on a spherical shell of radius R gives, The branes are thus smeared along the θ = 0 direction, leading to a linear distribution of charge in the transverse space, whose charge density is proportional to 1 √ R or constant in ρ (recall that at θ = 0, √ R = ρ).
More explicitly the NS5 distribution can be read off of the harmonic function f in (27) as follows. First it will be convenient to parameterize the R 4 transverse to the NS5 by introducing the cylindrical coordinates r ∈ R 3 , w := R cos θ, so that R 2 = r 2 + w 2 and, The new coordinates (R, w) are related to (u, ρ) by, Moreover it can easily be verified that the function f can be represented as an integral over the Green's function for the Laplacian on R 4 , with linear charge density σ(w) = w − 1 2 along the half line w ≥ 0. This is depicted schematically in diagram (d) of fig. 1.
An alternative way to find the charge distribution is to compute the source for the H 3 Bianchi identity, However H 3 = dB 2 is closed as a form, and we thus need to consider this equation on currents. Indeed H 3 is not defined for θ = 0, which is precisely the locus where we expect to find the brane. As a current, dH 3 acts as a linear form (distribution) on six-forms. Consider a test six-form Ω, where v 6 is the volume form of R (1,5) . After integration against H 3 , the only components of dΩ we need consider are, so that, Integrating each term by parts (respectively in θ and R), the derivatives cancel out since H 3 is closed as a form. The charge can then be seen in the boundary terms. R 1/2 vanishes at R = 0, ω vanishes at R → ∞ because it is a test function, and cos 3 θ 2 vanishes at θ = π. Note also that at θ = 0, ω cannot depend on χ, ξ. We thus obtain, From this we can read off the current, which gives the exact distribution of NS5 charge. This distribution is remarkable since it is entirely created by NATD from an empty flat space. It will be characteristic of the behavior of NATD near a fixed point of the SU (2) isometry. For instance we found the same kind of distribution when looking close to the α = π/2 singularity in the full dual solution (14).

The near-horizon limit
The NATD of the near horizon solution is, and the nonzero RR fluxes are given by,

Field Theory interpretation of near horizon NATD
In [17] a holographic interpretation of the background (40)-(41) was proposed. It was pointed out that the background belongs to a class of Gaiotto-Maldacena geometries [18] dual to N = 2 superconformal linear quivers with gauge groups of increasing rank. Their argument crucially involved constraining the range of the dual coordinate ρ in quantizing the NS5 brane charge. Let us briefly summarize the main points of the arguments originally presented in [17] and extended to further examples in [19,20]. Related examples with flavor branes include [7,21] and [8,22].
In the NATD a new set of dual coordinates arise, which we have labeled (ρ, χ, ξ). The coordinates (χ, ξ) are naturally interpreted as compact angles on an S 2 , i.e. χ ∈ [0, π], ξ ∈ [0, 2π]. The question remains how to interpret the ρ coordinate, as NATD currently lacks the global information needed to constrain the dual coordinates. Using insight from string theory the authors of [17] were led to impose the boundedness of the following quantity, where in the case of (40) b 0 is maximal along This leads to the coordinate ρ varying in nπ intervals, i.e. ρ ∈ [nπ, (n + 1)π]. To keep the relation (42) satisfied, a large gauge transformation must be performed on B 2 at each nπ interval, i.e.
As reviewed in section 2.2.1, this has the effect of changing the Page charges: quantizing Q D6 and Q D4 by integrating the RR fluxes in (41) above leads to Q D6 = N D6 and Q D4 = 0. However under a large gauge transformation of B 2 , we find ∆Q D6 = 0 and ∆Q D4 = nN D6 , where Q N S5 = N N S5 = n. Putting all this together suggests that there are parallel NS5 branes, each located at a π interval in ρ. Between each π interval n horizontal D4 branes are suspended between them. That is, as we move towards larger ρ, an increasing number of D4 branes appear. In the field theory interpretation this corresponds to an infinite linear quiver with increasing gauge group ranks. Interestingly, the field theory analysis of [17] suggested that there should be a cutoff to the ρ coordinate in order to terminate the quiver with a flavor brane. The intuitive way to see this is to start with parallel NS5 branes and a D6 flavor brane on one of the ends of the array. When one moves this flavor brane across the NS branes, D4 branes are created across the NS branes via the Hanany-Witten effect [16]. This completion of the quiver corresponds to giving ρ a finite range and it was shown that this is necessary to make sense of the dual field theory as a 4d CFT. 2 Thus the "stringy" picture is consistent with the spatial infinity limit of section 2.2.2, provided we replace the supergravity approximation of a continuous linear distribution of NS5 branes along a half line, by a grid of localized NS5's so that there is one unit of NS5 charge per ρ ∈ [nπ, (n + 1)π] interval.

M2 branes
The M2 brane solutions of eleven-dimensional supergravity can be reduced in various ways in order to obtain ten-dimensional IIA D2 brane solutions. Let us start from the M2-brane solution in flat space, whereQ is a constant related to the number of parallel M2-branes, dΩ 2 7 is the metric of the round seven-sphere, and vol 3 is the volume element of R 1,2 . We will adopt the parameterization of the metric on S 7 given by, where µ ∈ [0, π], σ i are the left-invariant SU (2) Maurer Cartan one-forms given in (3), while the Σ i have exactly the same form but with coordinates (θ 2 , φ 2 , ψ 2 ). We will only treat the round S 7 case, i.e. λ = 1. In the near-horizon limit, we have H =Q r 6 and the space becomes AdS 4 × S 7 .
This solution preserves 16 real supercharges (enhanced to 32 in the near-horizon limit), i.e. N = 4 in four dimensions. In (44) we have written the flat metric on the space R 8 transverse to the M2 as an eight-dimensional cone over the seven-sphere. We may replace the base of the cone by any Sasaki-Einstein seven-manifold 3 , and still obtain a solution of eleven-dimensional supergravity. The amount of preserved supersymmetry depends on the number of Killing spinors of the Sasaki-Einstein.
Replacing the round sphere metric dΩ 2 7 by the Y p,q (B 4 ) metric of [24,25], reduces supersymmetry to N = 1 in four dimensions, enhanced to N = 2 in the near-horizon limit. After a change of coordinates to bring us to the conventions of [13], the metric reads, for some functions w, f of θ that will be specified below, where ds 2 (M 6 ) is the metric of the S 2 (B 4 ) bundle, with θ ∈ [0, π], ψ ∈ [0, π] the coordinates of the S 2 fiber; the connection A is a one-form on the base with J the Kähler form of B 4 . Later we will consider the special case B 4 = CP 2 for concreteness and in order to perform an SU(2) NATD.

Brane configuration and charges
We expect the M2 branes to lie along the R 1,2 directions. The transverse space would then be R 8 or the cone over Y p,q depending on the choice of 7-dimensionnal space. In both cases we find: However since the 7-dimensional cycle collapses in the transverse space when r = 0, G is not closed and: We can also compute the M2 brane charge, which is defined by: with the M2 brane tension given by T M 2 = 2π (2πlp) 3 and 2κ 2 11 = (2π) 8 l 9 p , where the Planck length is given by For example, in the Y p,q case, we compute: This relates the constant in the harmonic function to the number of M2 branes, We will now proceed to track the M2, first through dimensional reduction, then through NATD.

Supersymmetric D2 from reduction on Y p,q
Here and in the following section we will need to make the choice B 4 = CP 2 so that a non-abelian SU(2) isometry is manifest in the metric acting on the σ i , where µ ∈ [0, π 2 ], and the σ i are given in (3). Reducing the M2 brane solution (49) to IIA on the circle parameterized by α preserves supersymmetry, as will be explicitly verified in section 7.1. Let us set, so that upon reduction to ten dimensions ds 2 A and the function φ(r, θ) are identified with the IIA string-frame metric and dilaton respectively. Moreover the nonvanishing fluxes of the solution are given by, where A was given below (49). F 2 carries a magnetic charge, but we will not interpret it as coming from a brane. Since this charge is not related to the M2 charge, but rather to the dimensional reduction, we will say that the flux is only geometric, and it will not be of interest here. The flux F 4 on the other hand carries an electric charge and is sourced by a stack of parallel D2 branes filling R 1,2 and placed at r = 0 in the transverse space. Note that the H function is inherited from the M2 solution, so that it does not need to be harmonic in the new transverse space. We also inherit the usual parameters for a brane solution, which allow us to define the near-horizon and spatial infinity limit.
We can then obtain the explicit form of the functions w(θ), f (θ) by taking the near-horizon limit (H =Q r 6 ) of (56), (57) and comparing with [13]: where ds 2 (AdS 4 ) is the metric of an AdS 4 space of unit radius, so that its scalar curvature is normalized to R = −12. Comparing with (2.22), (2.23), (2.24) of [13] we read off, To summarize, the ten-dimensional D2-brane solution is given by (56), (57), where ds 2 (M 6 ) is given in (47), H is given in (49) and f , w are given in (59). In the near-horizon limit the metric becomes a warped AdS 4 × M 6 product, cf. (58).
At spatial infinity (H = 1) the metric becomes a warped product R 1,2 × C(M 6 ), where C(M 6 ) is the metric cone over M 6 , while the remaining fields are given by, It can be verified that this is an exact supergravity solution in its own right. Contrary to the case of the D3 brane, here spacetime is neither flat nor empty at spatial infinity.
The solution (56), (57) describes D2 branes with worldvolume along the R 1,2 , as inherited from the M2 solution. Looking at F 4 , we find: However, contrary to the standard brane configurations (such as the D3 and M2 presented previously), the probe interpretation is not straightforward. In order to understand this configuration we take the transverse space to be the cone over M 6 . There the 6-cycle collapses at r = 0, so that: Here again this equation must be considered on the transverse space. The D2 background is a genuine solution of IIA supergravity, in which the 6-cycle does not collapse anymore. Then F 4 is closed, as required by the equations of motion, and the brane is not visible. We can however compute the brane charge, which requires the D-brane tension T −1 Dp = ((2π) p α (p+1) 2 ) and 2κ 2 10 = (2π) 7 α 4 . We obtain, The flux quantization condition Q D2 = N D2 then leads tô

The NATD
The NATD of the supersymmetric D2 brane is obtained by an SU(2) action on the σ i , cf. (55). The NS-NS sector reads, where we have defined the following one-forms, and included the following definitions, The RR sector is given bŷ where a(θ) = 2(1 + cos 2 θ) and v 4 = − r 2 w(θ)

Brane configuration and charges
The D-brane background before the NATD was a D2 brane solution, therefore we expect to see the presence of D3, D5, and NS5 branes from the general lore Dp→D(p+1)-D(p+3)-NS5. We will follow the same strategy as in section 2.2.1 to better understand the brane configuration.
We first compute the NS5 charge. In the same spirit as in the D3 brane example, cf. (12), we integrate H 3 along the cycle (Σ 3 [ρ, χ, ξ], µ = 0), on which H 3 simplifies as, We get, where we cut off the integration at ρ = ρ 0 . For the charge to be correctly quantized, we need ρ 0 = 8nπ 3 . This is compatible with the condition of boundedness of b 0 given in (42). Modulo a large gauge transformation on B 2 , this condition is satisfied if the range of ρ is taken to be [ 8(n−1)π 3 , 8nπ 3 ]. Once again we can see that there is, at least from the supergravity point of view, a continuous distribution of charge at the singularity created by NATD (here µ = 0). This distribution is smeared along the ρ direction and is constant in ρ. As was the case in section 2.2.1, this can be seen directly in the metric by zooming in at the singularity. Close to µ = 0 and after making the substitution ν = µ 2 , the metric becomes, where ν −1 is the harmonic function in the transverse space for NS5 branes along the (R 1,2 , r, θ, ψ) directions.
For the D-branes we need to consider the Page forms, given by, whereF 1 =F 1 given in (68) is unchanged.
We can readily see that the Page forms have two contributions: one coming from the geometric flux F 2 and the other one from F 4 . 4 Since we want to trace the fate of the M2 branes we will consider the first part as geometric fluxes and focus on the second. The relevant components are thus those proportional toQ (i.e. those that vanish when there is no M2).
Ignoring the geometric fluxes, the only non-vanishing Page charge is Q D5 , which can be found by integratingF 3 . Namely, we keep only the (θ, µ, ψ) term, The quantization condition of Q D5 then leads to a relation between the constant in the harmonic function and the number of D5 branes, However as it was pointed out in the D3 example, the page charges depend on the choice of B 2 , and may change under a large gauge transformation. Here under a large gauge transformation given by, Q D3 receives a new contribution, Evaluating this and comparing to (75), we then find, This is analogous to the relation found in (20) above.
Assuming that the r coordinate still describes the radius of the cycles wrapped by the RR-fluxes in the transverse space of the D-branes, the brane configuration is given by the table below.
As we will find in additional examples throughout this paper, this relationship seems to be universal for D-brane backgrounds generated by SU(2) non-Abelian T-duality. When a Dp-brane background is transformed, the D(p+3) brane charges are easily found from integrating the appropriate term in the Page form. The D(p+1) brane charges then are found from restricting B 2 to a cycle containing (χ, ξ) cycle, performing a large gauge transformation, computing the change in the Page form under this transformation, and finally integrating to obtain ∆Q D(p+1) . This results in the general relation ∆Q D(p+1) = nQ D(p+3) . The new D2 brane examples presented in this paper are not only distinct from the original D3 brane example where this relation was proposed, but also highly nontrivial.

The spatial infinity limit
The spatial infinity limit of the supersymmetric D2 brane NATD solution (65)-(68) is given by, where we have defined the following one-forms, with Q, K defined in (67) and M = 4l 2 p α 2 ρ 2 cos 2 χ + w(θ)Ξ 2 . The RR sector is given bŷ As was already the case for the D2 solution, the spatial infinity is neither flat nor empty. In a probe interpretation, this configuration could be interpreted as the space in which the branes are inserted.
5 Non-supersymmetric D2 from reduction on Y p,q We will now reduce along the "obvious" Sasaki-Einstein S 1 cycle, thereby completely breaking supersymmetry. Let us rewrite the Y p,q (B 4 ) metric (46) as follows, where the baseM 6 is topologically an CP 2 × S 2 with metric given by, and we have defined,Ã := A + 2 w(θ) cos θdα .
Note that, as follows from (59), for the S 2 parameterized by (θ, α) to be smooth α must have period 2π/(g s e 2A 0 ). Alternatively we may redefine α → g s e 2A 0 α, so that α ∈ [0, 2π]. In terms of the redefined coordinate, The corresponding eleven-dimensional solution reads, Reducing along the S 1 cycle parameterized by ψ results in a non-supersymmetric ten-dimensional D2-brane solution given by, In the spatial infinity limit, H = 1, the metric reduces to, while the remaining fields reduce to, Once again we see that, contrary to the D3 case, the spacetime is neither flat nor empty in the spatial infinity limit: rather it is conformal to R 1,2 × C(M 6 ), where the latter factor is the metric cone over M 6 .
Upon dimensional reduction on ψ the M2 branes become D2 along R 1,2 , whose transverse space would be the cone overM 6 . As in section 4 we find, Since the cycleM 6 collapses in the transverse space at r = 0, We compute the quantized D2 charge and obtain a similar result to the supersymmetric case, up to a factor of 2 difference arising from the different ranges of α and ψ, sin θ a(θ) 3/2 S 3 dΩ 3 Since F 2 is not related to the D2 brane charge, we will only consider it as a geometric flux.

The NATD
We can now take the NAT dual of the background (87). The NS-NS sector of the resulting background is given by, where we have defined the following one-form, and included the following definitions, The RR sector is given bŷ Ξd(ρ cos χ) + 2ρ cos µ cos χΛdµ , where a(θ) = 2(1 + cos 2 θ) and v 4 = − r 2

Brane configuration and charges
As in the supersymmetric reduction, spacetime is singular at µ = 0, which corresponds to the fixed locus of the SU (2) isometry before duality. We thus first compute the NS5 charge by integrating H 3 on the cycle (Σ 3 [ρ, χ, ξ], µ = 0), on which H 3 simplifies to, so that, For the charge to be quantized we need ρ 0 = 4nπ 3 . This is compatible with the condition (42) which leads to ρ ∈ [ 4(n−1)π 3 , 4nπ 3 ] and a large gauge transformation on B 2 . We can now examine the metric close to µ = 0, with ν = µ 2 , where ν −1 is the harmonic function for NS5 branes along the (R 1,2 , r, ) directions. As in the previous example, the NS5 branes are located at the singularity µ = 0 and are smeared along the ρ direction.
The Page forms are given bỹ cos µ sin 2 µ sin θdα ∧ dθ ∧ dμ withF 1 =F 1 given in (96). We will focus on the components that are proportional toQ, whereas the remaining term will only be considered as geometric flux. Integrating the (α, θ, µ) term inF 3 we obtain, leading to a relation between the constant in the harmonic function and the number of D5 branes, If we further consider the change in the Page forms under a large gauge transformation in B 2 5 , it is the D3 charge which is created and we find ∆Q D3 = nN D5 .
In the same spirit as before, we would then have the following brane configuration:
We can see the presence of D2 branes from F 4 , where v 6 is the volume form of the 6-dimensional space M 6 (along µ, dΩ 2 and dΩ 3 ). If we take the transverse space to be the cone over M 6 , this cycle collapses at r = 0, where we can see the D2 brane, Upon quantizing the flux, we obtain leading to,Q = 128π 2 l p α 5/2 N D2 .
On the other hand, as is the case for the near-horizon limit, F 2 is sourced by a D6 brane along R (1,2) , r, Ω 3 and located at µ = π 2 , where the 2-sphere Ω 2 collapses. The charge is given by: The brane configuration is thus the following: Note that the 3-sphere Ω 3 , on which we will now dualize, is transverse to the D2 but parallel to the D6. We will now see how both will behave under NATD.
with v 4 = − The Page five-form is given by,

Brane configuration and charges
We first compute the NS5 charge by integrating H 3 on the cycle ([ρ, χ, ξ], µ = 0), and a suitable large gauge transformation on B 2 , the relation (42) is satisfied.
The NS5 branes are also seen by zooming in on the singularity generated by the NATD at µ = 0, This is indeed consistent with the harmonic superposition rule, with harmonic function proportional to ν −1 . This gives the characteristic NS5 brane configuration: along the (R 1,2 , r, Ω 2 ) directions, located at µ = 0 and smeared along ρ.
Next we compute the quantized Page charges. We start with the dual of F 4 to track the D2. This corresponds to the terms proportional toQ. Here only F 3 gives a non zero charge, leading to the quantization condition,Q However, we may perform a large gauge transformation on B 2 and find the resulting change in the Page charge for F 5 , ∆Q D3 , We then find thatQ = 256πlpα 5/2 n ∆N D3 and therefore, ∆Q D3 = nN D5 . We can also track the D6 by looking at the dual of F 2 , i.e. the remaining components of the page forms. We find With a large gauge transformation on B 2 , Q D3 is also modified, in the same fashion as Q D5 , The brane configuration is summarized in the table below.

Domain wall supersymmetry equations
As already mentioned, (60) is a supersymmetric domain wall (DW) solution in four-dimensional space, where the latter is viewed as a foliation, parameterized by r, with R 1,2 leaves. The supersymmetry conditions for N = 1 domain walls were written in [14] in generalized G 2 × G 2 form in eqs. (2.5), (2.6) therein. For our purposes it would be more useful to recast these equations in terms of generalized pure spinors on M 6 . Such a rewriting is indeed given in [14], cf. (2.7) therein. We will now review their results adapting them to our case.
The ansatz for the splitting of the metric and the flux is given by: where the warp factor A and the dilaton φ are not constrained at this point; λ is an involution reversing the order of wedge products. The NS-NS form H is assumed to be internal, i.e. to only have legs along M 7 , and likewise for the internal RR flux F . The total flux F t is then chosen to be self-dual: for F internal we get λF = v 3 ∧ 7 λF and in ten Lorentzian dimensions ( λ) 2 = 1. Unbroken supersymmetry of the solution implies on M 7 the existence of two Majorana spinors χ 1 , χ 2 normalized so that χ † a χ a = 1. This leads us to define a bispinor Ψ, which can also be viewed as a polyform via the Clifford map: where Ψ + and Ψ − are respectively the real-even and imaginary-odd parts of Ψ. We should be careful however about how the identification is imposed: odd dimensional Fierzing does not provide an isomorphism between bispinors and polyforms because the Clifford representation is not faithful, as can be confirmed by a simple count of dimensions. We thus need to choose the range of our identification.
Here we take Ψ to be self-dual as a polyform: −i 7 λΨ = Ψ. This also means that the decomposition (126) is only valid in the polyform space and that Ψ + , Ψ − are not independent: These choices lead to the normalization: We now have all the necessary ingredients to write the supersymmetry for IIA in terms of generalized spinors: In order to match (56), (57) we need to further split M 7 to M 6 plus a transverse direction parameterized by the coordinate r. The metric and fluxes thus decompose as follows: where a depends only on r, and F i , F r , H i , H r only have legs on M 6 . Note also that the expression ds 2 (M 6 ) can depend on r, since it can include a warp factor for instance. The same split must then be performed for the spinors, by expressing 7D spinors in terms of 6D chiral spinors. Since we are splitting along r, γ r (in flat basis) becomes the chirality matrix for spinors of M 6 . Thus we take: where P ± := 1 2 (1 ± γ r ). Introducing the following bispinors on M 6 (which can be viewed equivalently, via 6D Fierzing and the Clifford map, as polyforms or generalized spinors): we get: The factor e 3Z−φ is introduced here for future convenience; it is simply another choice of normalization: We then substitute (130) and (133) into (129), and decompose along dr. We look for an expression solely in terms of polyforms on the internal space M 6 , where r is now considered as an external parameter: where now d H = d + H i ∧, ∂ H r = ∂ r + H r ∧, and d acts only on the coordinates of M 6 . Note also that , now refers to the 6D Mukai pairing.

Supersymmetric D2
We now want to check explicitly that the solution in (56), (57) is compatible with the equations (135). This amounts to defining two polyforms Φ 1 , Φ 2 , whose SU (3) × SU (3)-structure carries the 6D part of the metric (56), and which is solution of (135). First we need to identify the various fields. Comparing (56) and (130) we find, Since the fluxes are not given in the same formalism, we need to retrieve F 6 and F 8 from F 2 and F 4 by Hodge duality, in order to build the total flux polyform F t in the democratic formalism. If we write F nd = F 2 + F 4 , the total flux of the solution (57) we find, This leads to, where v 6 is the volume form of M 6 taking into account the warp factor. We can now use the results from section A.3 to define our polyforms Φ 1 and Φ 2 . Our ansatz will introduce several functions of θ as supplementary degrees of freedom that should enable us to find a solution of the DW equations.
We begin with the local SU (2)-structure, given by the Kähler structure of B 4 . We denote byĵ the Kähler form andω a holomorphic 2-form normalized so that, Note thatĵ is global butω can only be defined locally. Furthermore we define, Finally the polyforms (or, equivalently, the generalized spinors) defining the SU (3) × SU (3)-structure are given by, where the factor √ H has been added to match the normalization (134), We are thus left with five undetermined functions of θ (α, β, ζ, ν and ϕ) that should provide enough freedom for a solution of the DW supersymmetry equations: α and ζ act as rotation of the local SU (2) and, since the SU (2)-structures span a two-sphere, they can be respectively seen as the intrinsic rotation and precession; β is merely a modification of the phase of the vielbein one-form K; the meaning of ν and ϕ is explained in section A.3, recall in particular that ϕ must vanish at θ = 0, π. Note also that a global phase of Φ 2 can be absorbed in ν and α whereas a global phase of Φ 1 can be absorbed in β.

Solution
Note first that in the near horizon limit, the SU (3) × SU (3) is in fact pure SU (3), i.e. ϕ = 0. Thus if our ansatz is correct (ϕ is function of θ only), ϕ should remain constant to match the near horizon limit. Looking at the first equation of (135), the scalar term gives straightforwardly, This is consistent with the ansatz, and also gives information about ν. We get The structure is then pure SU (3) all along the r coordinate. Moreover α and β now play the same role: a global phase shift of the holomorphic 3-form. β can thus be absorbed by a redefinition of α, and be set to 0. At this point, the second and fifth equation of (135) are satisfied. Moreover the three-form part of the fourth equation of (135) leads to 2α = − π 2 . All the remaining terms are then proportional to ζ so that ζ has to be constant. Taking ζ = 0 then solves (135).

Mass Deformation
The only difference between the full interpolating brane solution and its AdS 4 near horizon limit is a modification of the function H(r). The background (56), (57) is a genuine solution under the sole condition that H(r) is harmonic in the transverse space R 7 . The interpolating solution corresponds to the most general choice of H(r), whereas the near horizon limit and the spatial infinity limit correspond respectively to the choices H(r) = Q/r 6 and H(r) = 1. We would then expect that finding a massive deformation of the interpolating solution would amount to adding the correct r-dependence in the different functions of the massive deformation.
Consequences of the massive deformation: For the massive deformation of the near horizon limit, the SU (3) × SU (3) structure is no longer pure SU (3) [26,27], and this will obviously be the case for the interpolating solution. It will now be necessary to switch on the function ν, ϕ and ζ, leading to our first source of complication. It is also important to notice that the base and fiber of M 6 get a different warp factor, and our ansatz should take that into account. We also expect the massive deformation to switch on all fluxes. Switching on the three-form H also impacts equation (138) by twisting the derivative.
This corresponds to the ansatz of (130). The internal metric of M 6 follows from the SU (3) × SU (3), For The Bianchi Identities to be automatically satisfied we will rather define the fluxes by their potentials: the two-form B and the odd polyform C, The fluxes are then, in the formulation of (130), All the warp factors Z, B, C, the dilaton φ, the phases α, β, γ, ζ, ϕ, ν and the fluxes h, f 2 , f 4 , f 6 are allowed to depend on r, θ.
Despite its generality, we have checked that this ansatz does not solve the supersymmetry equations (except for the solutions we already know, which are special cases thereof). Unfortunately, without further input, relaxing the ansatz to allow for a dependence on more variables quickly becomes intractable.

Conclusions
We have seen that having full-fledged brane solutions, interpolating between the near-horizon and spatial infinity limit, gives us a better handle on the brane configurations and the global properties of the NATD. In particular we have seen certain general features emerge. The NATD of the spatial infinity limit of standard intersecting brane solutions is universal: it is given by a continuous linear distribution of NS5 branes along a half line with specific charge density. We have also observed a general relation between the Page charges generated by the NATD and their behavior under a large gauge transformation of the NS flux. Since this behavior results from the non-trivial dependence of the Page charge on the B 2 field, NATD naturally furnishes several examples where the choice of B 2 plays an important role.
More generally we have seen that in cases where the brane configuration before NATD is not flat at spatial infinity, the NATD contains highly nontrivial RR fluxes even at spatial infinity. If the charges before NATD are related to the presence of branes, the latter can be tracked throughout the NATD. Note that the spatial infinity limits of the NATD backgrounds presented here are highly nontrivial exact supergravity solutions in their own right, and they can be considered independently from the full interpolating intersecting brane solution.
We have cast the supersymmetric D2 brane solution, arising from the reduction of M2 branes on sevendimensional Sasaki-Einstein, in the language of generalized geometry pure spinor equations for domain walls. This framework allowed us to look for massive supersymmetric deformations of the D2 brane solutions, and we have been able to rule out a certain class of ansätze. It would be interesting to try to construct these massive deformations explicitly, at least in a perturbative expansion in Romans mass as in [28]. If they exist, these would be full interpolating intersecting brane solutions whose near-horizon limit coincides with the class of massive IIA AdS 4 × M 6 solutions of [27]. 6 It would also be interesting to cast the NATD of the supersymmetric solutions in the generalized geometry formalism for domain walls, thus refining the general results of [33,34]. Besides providing a check of supersymmetry, this might give insight into the global structure of the solutions. In certain cases the duals might fall within the class recently examined in [35]. We hope to return to these questions in the future.
Now we can make the complex structure I explicit, defined in such a way that Ω is (n, 0). For n = 3:

A.2 Pure Spinors
Consider now a six-dimensional Riemannian spin manifold M 6 , with metric g and volume form v 6 . Suppose that M 6 admits a chiral spinor γ 7 η = η, globally defined and nowhere-vanishing. η is also pure and can be normalized to η † η = 1. η thus leads to an SU (3)-structure (J, Ω) by contraction with gamma matrices: The choices here are made to be compatible with (147) and (148). This can be checked by Fierzing, using: where P ± is the projector to positive, negative chirality respectively.

A.3 SU (3) × SU (3)-structures
Suppose now that we have not only one but two spinors η 1 , η 2 of positive chirality on M 6 , normalized so that η † 1 η 1 = η † 2 η 2 = 1. Each defines its own SU (3)-and almost complex structure, according to (151). They both combine to form an SU (3) × SU (3)-structure on M 6 , given by two generalized spinors (i.e. spinors of Spin(3, 3)): Note that, thanks to Fierz isomorphism, generalized spinors can be seen equivalently as bispinors or polyforms (in even dimension). We thus want to express Ψ 1 , Ψ 2 as polyforms. It is also important to notice that η 1 , η 2 are not necessarily independent. Since η 1 is pure, the space of spinors can be constructed from η 1 or η c 1 and (anti-)holomorphic gamma matrices. Note that we call a holomorphic gamma matrix the image of a holomorphic one-form in the Clifford algebra, with respect to the complex structure defined by η 1 . With this choice of conventions, the annihilators of η 1 are the anti-holomorphic gamma matrices, as can be seen by the following calculation. Consider a one-formK such that K · η 1 =K j γ j η 1 = 0, then: It follows that any normalized spinor of positive chirality, such as η 2 in particular, can be written as: η 2 = e iν cos ϕ η 1 + sin ϕ χ .
Here e iν cos ϕ = η † 1 η 2 and χ is a normalized spinor, orthogonal to η 1 , and defined from an antiholomorphic one-formK such thatK · K = 2: In this definition K (and thus χ) does not need to be globally well-defined, provided sin ϕ vanishes whenever the definition of K fails. Moreover, χ (and K) defines locally another SU (3)-structure, "orthogonal" to η 1 's: We can also define a local SU (2)-structure (j, ω): The two orthogonal SU (3)-structures can be reconstructed back from the local SU (2) and K: Now we can compute the generalized spinors using Fierz identities: 2 K∧K e −iν cos ϕ e i j + sin ϕω Ψ 2 =K ∧ e iν cos ϕω − sin ϕ e i j Finally we introduce the Mukai pairing of polyforms: where λ swaps the order of the wedge product. This leads to the following normalization of the generalized spinors: Ψ 1 ,Ψ 1 = Ψ 2 ,Ψ 2 = 8i v 6 (161)

Remarks on the conventions
The conventions adopted here respect the general relation of SU (n)-structures (147) as well as the orientation (148). However this implies that the annihilators of a pure spinor of positive chirality are anti-holomorphic. If we would rather have the annihilators to be holomorphic, we need to change the complex structure. This can be done by swapping Ω andΩ, and I should be modified accordingly (by a sign). This can be done in two ways: 1. Change the sign of J: this will keep the relations (147) intact but will change the orientation (148).
2. Take rather I j i = g jk J ki instead of (149): this will keep the orientation but add a sign in the relation (147).
The first change is the choice made in [37] while the second was chosen in [36]. All cases are made to be consistent with the relation (150), which is also merely a convention. Fortunately, everyone seems to agree on this one.