Supergravity solutions with AdS4 from non-Abelian T-dualities

We present a large class of new backgrounds that are solutions of type II supergravity with a warped AdS4 factor, non-trivial axion-dilaton, B-field, and three- and five-form Ramond-Ramond fluxes. We obtain these solutions by applying non-Abelian T-dualities with respect to SU (2) or SU (2) /U (1) isometries to reductions to 10d IIA of 11d sugra solutions of the form AdS4 × Y7, with Y7 = S7/ℤk, S7, M1,1,1, Q1,1,1 and N (1, 1). The main class of reductions to IIA is along the Hopf fiber and leads to solutions of the form AdS4 × K6, where K6 is Kähler Einstein with K6 = ℂℙ3, S2 × ℂℙ2, S2 × S2 × S2; the first member of this class is dual to the ABJM field theory in the ’t Hooft limit. We also consider other less symmetric but susy preserving reductions along circles that are not the Hopf fiber. In the case of N (1, 1) we find an additional breaking of isometries in the NAT-dual background. To initiate the study of some properties of the field theory dual, we explicitly compute the central charge holographically.


JHEP02(2016)061 1 Introduction
The AdS/CFT correspondence [1] conjectures a complete equivalence between a field theory and a string theory. This correspondence provides an explicit recipe to connect observable quantities on both sides of the correspondence and it is particularly potent when the field theory is conformal [2][3][4]. A central conceptual and intuitive role is played by the geometerization of conformal invariance which on the gravity side is realized as the presence of an AdS factor whose isometries coincide with the conformal group of the theory living on the boundary.
More precisely, the conformal group of a field theory in d dimensions is SO(d, 2) for d > 2; this group is precisely the isometry group for AdS d+1 . This relation has prompted the search for solutions in type II supergravities and M-theory that contain AdS d+1 as a space-time factor. Most of the effort has, naturally, been concentrated on searching for solutions with an AdS 5 factor (see, for example, [5,6] and [7]). There are, however, strong efforts in other dimensions. Recent efforts include AdS 7 [8]. In the context of theories connected to massive IIA, this search can be extended to AdS 6 [9,10] and more recently to AdS 5 [11] and AdS 4 [12]. Another recent approach to the construction of gravity duals to conformal backgrounds has been presented in [13][14][15]. There has also been recent progress in the direct search for solutions with AdS d+1 factors; interesting examples are the geometries interpolating between AdS 7 and AdS 5 presented in [16,17] and further studied and generalized in [18,19].
Recently, there has been a revival of NATD [20][21][22] including its systematic extension to the Ramond-Ramond sector [23,24]. This resurrected symmetry has already been used to generate solutions from various seed backgrounds in the context of the AdS/CFT correspondence . NATD is a powerful tool in the search for solutions because it can be considered as a solution generating technique. Even more importantly, as shown in for example [39], it leads to supersymmetric solutions with very few apparent isometries other than those of AdS. 1 This makes NATD a unique technique which is able to probe the space of solutions in directions that cannot be accessed by various general classification approaches which are based on explicit symmetries.
The main goal of this paper is to further use T-duality and NATD to construct supergravity backgrounds that contain an AdS 4 factor in the metric. In a very direct sense, this manuscript is an extension of [39] which was devoted to solutions with an AdS 5 factor. There are various interesting applications for conformal field theories in d = 3. For example, they have historically played an important role in our understanding of critical phenomena through the expansion of the Wilson-Fisher fixed point. More formally and recently, 3d conformal theories have helped clarify aspects of the M2 brane theory. We hope that the backgrounds presented in this manuscript will add some interesting new examples and motivate studies of the corresponding field theory duals which are arguably conformal or superconformal field theories in 3d. The quintessential example in this class is the ABJM field theory [48]; the NATD to the gravity background dual to the ABJM theory has been discussed explicitly in [37]. Figure 1. A schematic description of some properties of the supergravity solutions discussed in this manuscript and their field theory duals.

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We summarize our strategy and main results in figure 1. Namely, we generate various families of 10d Type IIB solutions containing an AdS 4 factor by first starting with a Freund-Rubin type solution in 11d supergravity of the form AdS 4 × Y 7 and subsequently reduce the solution to IIA, which is followed by applying a non-Abelian T-duality to either an SU (2) or an SU(2)/U(1) isometry. Generically, the final Type IIB backgrounds have all of the RR fluxes turned on, with the exception of the very symmetric case of S 7 . Finally, we compute the Page charges of these backgrounds and the central charge of the corresponding dual field theories. We also investigate the change in the Page charges under large gauge transformations in B 2 .
The paper is organized as follows. In section 2 we sketch the procedure that we apply to generate the various explicit solutions in the rest of the paper. Sections 3, 4, 5, 6 and 7 correspond respectively to the application of our solution generating techniques to: S 7 /Z k , S 7 , M 1,1,1 , Q 1,1,1 and N (1, 1). In section 8, we discuss our recipe for computing the central charge of the dual field theory holographically and present the results for one representative background from each section. We also propose a method for determining the ranges of the dual coordinates in backgrounds generated from NATD with coset space isometries, where B 2 is absent. We discuss some arguments for the preservation of supersymmetry in some of the new backgrounds and conclude in section 9. We relegate to appendix A the rules for non-Abelian T-duality and appendix B the form of the 11d and 10d equations of motion as we used them to verify that the new backgrounds obtained in the manuscript are solutions.

Freund-Rubin seeds for solution generating techniques using non-Abelian T-duality
In this section we review a construction of solutions to 11-dimensional supergravity based on the Freund-Rubin Ansatz [49]. Essentially every Sasaki-Einstein 7d manifold provides a supersymmetric solution to 11d supergravity. A fairly complete description of solutions of seven dimensional manifolds, providing Freund-Rubin solutions to 11d supegravity, was cataloged in Duff-Nilsson-Pope [50]. The list includes further specification about those JHEP02(2016)061 which are supersymmetric and states what fraction of the supersymmetry is preserved. An exhaustive list of Sasaki-Einstein seven-dimensional manifolds is presented in [51]. Here we focus on a particular but wide set of solutions, hoping to elucidate some of the most generic aspects for other AdS 4 solutions. We start by considering Freund-Rubin type solutions to 11d supergravity, where ds 2 7 is an Einstein metric on Y 7 . In the case when Y 7 is also Sasaki, the corresponding solution is supersymmetric. In this manuscript we are particularly interested in solutions that are supersymmetric. The metric ds 2 4 could also contain an asymptotically AdS 4 black hole but we do not consider explicitly that case as it breaks supersymmetry. If the metric on Y 7 admits a U(1) isometry, which is always the case in the backgrounds we consider in this paper, it can be written as where A is a connection on K 6 and the y n represent coordinates on K 6 . Then we can reduce the corresponding 11d solution to 10d Type IIA via the following rules: If one is concerned with the correct dimensional scalings while performing the reduction, it is important to note that the 11d backgrounds have l p as the natural length scale. Therefore the dilaton must pick up this scale to compensate for the factors of the 11d radius, R, inherited from the a(y n ). This can be seen in the examples provided below.
Once we arrive at a Type IIA supergravity background using the prescription in eq. (2.3), we can further apply a Non-Abelian T-duality (NATD) to the corresponding background, provided that the appropriate symmetries are present. We will find that there are generically two cases where we can perform an NATD. In the first case there is a structure resembling an S 3 and in this case we apply the rules of NATD directly as in [23,25]. In the second case we merely have one or more S 2 subspaces on which we may implement NATD by exploiting the coset structure of S 2 and following the prescriptions and results of [24].
Before we begin the presentation of the backgrounds, let us first define some notational conventions used, and clarify a few points about the NATD gauge fixing. In all of the backgrounds presented in this paper, we choose a gauge fixing for NATD such that all three of the Lagrange multipliers are kept as coordinates in the dual background, i.e. g = 1. We will introduce them in terms of spherical polar coordinates, (θ i , φ i , ψ) → (ρ, χ, ξ). We have also left arbitrary the gauge fixing constant that we label m 2 , which arises from v i → m 2 v i . We direct the reader to appendix A or [25] for more on gauge fixing in NATD. In section 8 we will use the fact that m 2 may be absorbed into the definition of ρ.

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Note that we will adopt the following conventions for AdS 4 , dΩ AdS 4 and 2-spheres throughout this document: We begin our presentation with the background AdS 4 × S 7 /Z k , which is given by, The metric on S 7 /Z k is: 2 ) + sin 2 α 2 ds 2 (Ω 2 ) The angle τ parameterizes the U(1) fiber over CP 3 , defined in the brackets of eq. (3.2) above. The ranges of the angles are 0 ≤ α, θ 1 , θ 2 ≤ π, 0 ≤ φ 1 , φ 2 ≤ 2π, 0 ≤ ψ ≤ 4π and 0 ≤ τ ≤ 2π k . This geometry is supported by an F 4 = − 3 8R dΩ AdS 4 . After a reduction to Type IIA via τ , we recover the familiar AdS 4 × CP 3 background, which we label ⇓ τ (AdS 4 × S 7 /Z k ) to match our labeling conventions. In addition to the metric, the supergravity background has a dilaton, and 4-form and 2-form field strengths from the Ramond-Ramond (RR) sector, Here dΩ AdS 4 is the volume form on AdS 4 and F 2 is proportional to the Kähler form on CP 3 . This IIA background is dual to a supersymmetric field theory with N = 6 susy in 3D; therefore the IIA background preserves the same number, 24, supercharges [48].
In this section, we perform an NATD along the (θ 1 , φ 1 , ψ) directions defining an SU (2) isometry. This NATD was performed in [37], however, the NATD there acted along the (θ 2 , φ 2 , χ) SU(2) isometry. It is true that the two NATD backgrounds are related by a simple map (in the notation of [37] it is ζ → ζ + π 2 ). Our results differ slightly, particularly in the factors of k, due to the presence of the warp factor e − 2 3 Φ left over from the reduction to JHEP02(2016)061 IIA, absent from [37]. Our choice here leads to a slightly different structure of singularities in the resulting background.

Background from AdS × S 7
In this section we consider a reduction to IIA along the ψ 2 direction of the AdS 4 × S 7 background, which we define as: where σ i and Σ i are left-invariant SU(2) Maurer Cartan 1-forms given by,

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and similarly for the Σ i , but with coordinates (θ 2 , φ 2 , ψ 2 ). The range of µ is 0 ≤ µ ≤ π, as given in [50]. This background is supersymmetric for λ = 1, corresponding to the round S 7 , and λ = 1 √ 5 corresponding to the squashed S 7 . We will focus on the round S 7 . Here we perform the reduction along the U(1) angle ψ 2 defined in the σ's above. 2 The 10D background NS and RR sectors take the form: 2 ) + dµ 2 , This Type IIA background has Einstein frame Ricci scalar, The singularity at µ = π, however, can be understood by the presence of D6 branes. Indeed, the presence of the F 2 flux suggests a dual C 7 potential along the directions of AdS 4 and Σ i . More importantly, one can introduce near µ = π the following coordinate: (π − µ)dµ 2 /2 = dr 2 / √ r. In this new coordinate, near the singular point, the metric takes the form which coincides, precisely, with the metric near a D6 brane source. Under the subsequent NATD transformations this singularity will persist, but we know its origin.
We will now perform a NATD along the SU(2) isometry defined by the Σ i . The NATD metric, B 2 , and dilaton have the following form, The non-trivial dual RR Fluxes are, 2 Note that it is only U(1) for the round S 7 . For the squashed S 7 only φ1 and φ2 are U(1) angles.

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After this NATD, the Einstein frame Ricci scalar is, , where here we can again see that the singularities that appear correspond to the zeros of ∆ (µ = 0, π), plus the singularity at µ = π, inherited from the Type IIA background.

Backgrounds from AdS
The space we are concerned with in this section is a U(1) bundle over CP 2 × S 2 with characteristic numbers n 1 and n 2 and metric given by, The supersymmetric case corresponds to n 1 = 3, n 2 = 2; in the notation used by Duff-Nilson-Pope [50] this space is naturally denoted by M (−3, 2). Here, however, we follow a slightly more modern notation widely used in the literature, M 1,1,1 . The corresponding 11D geometry has Ricci scalar R = − Λ R 2 and admits a 4-form Flux, and 0 ≤ µ ≤ π 2 . In the IIA reduction along τ , we obtain, This implies that, C IIA 3 = C 3 , and C (1) = l p (−n 1 sin 2 µσ 3 + n 2 cos θ 1 dφ 1 ). This reduced metric has Einstein frame Ricci Scalar, R E = 0. Note that the metric on the six dimensional subspace, which we denote ⇓ τ (M 1,1,1 ), is simply the product of S 2 × CP 2 . The solution is supported by the Kähler forms which appear as dC (1) . The field theory of this background was examined in [52] where it was shown that this τ reduction of AdS 4 × M 1,1,1 is N = 2 supersymmetric.

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We are now poised to perform a Non-Abelian T-duality along the SU(2) isometry made explicit by the SU(2) invariant one-forms σ.
The RR sector contains, This background has singularities at µ = 0 and µ = π 2 , simultaneous with ρ = 0 or χ = 0, all generated by the NATD. JHEP02(2016)061 After the reduction to IIA of AdS 4 ×M 1,1,1 we have a six-dimensional Einstein-Kähler manifold ⇓ τ (M 1,1,1 ) = CP 2 × S 2 . In the previous section we performed an NATD along the SU(2) isometry present in CP 2 (see its explicit form in eq. (5.3)). Since S 2 is a coset manifold and there is a prescription for applying NATD on coset manifolds [24], we now proceed to apply this coset NATD along the S 2 factor. We review this method in appendix A.2. The dual NS fields are given bŷ with the RR sector given by, This background has a singularity at ρ = 0, as can be seen in the behavior of the dilaton. This is generated by the collapsing cycle in eq. (5.3) given by θ 1 → 0.

Alternate reduction:
In this section we present an alternate, supersymmetric reduction to Type IIA of AdS 4 × M 1,1,1 presented originally in [53]. We then perform an NATD on this background, thus providing a new supersymmetric Type IIB solution with an AdS 4 factor. In [53] it was shown that reducing along the U(1) φ 1 angle yields a supersymmetric Type IIA background of the following form, 3 9) and A = c 2 2 n 2 2 cos 2 θ 1 + c 2 3 sin 2 θ 1 .

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The RR sector contains the F 4 = − 3c 3 1 R dVol(AdS 4 ) flux proportional to the differential volume on AdS 4 , while the reduction produces an additional C 1 and F 2 : Now we move on to present the Non-Abelian T-dual with respect to an SU(2) isometry given in the σ's of eq. (5.9) above.
This background has singularities at ρ = 0 and χ = 0, both generated by the NATD.

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6.1 Alternate reduction: ⇓ φ i (AdS 4 × Q 1,1,1 ) In this section, we perform a different reduction to Type IIA on the AdS 4 × Q 1,1,1 background, in such a way that two SU(2) isometries are manifest. In order to see these isometries, we must set the parameters appearing in the Kähler form, α, β, γ, to zero. We must also make use of the relations between the t i 's such that t 1 = 2t 3 , and t 2 = √ 2t 3 . (NB: the equations of motion are satisfied for any value of t 3 .) Here we will present explicitly the reduction along φ 3 , however from the symmetry of Q 1,1,1 it is evident that reducing along any of the φ i will yield backgrounds of the same general form.
In this section we consider the metric on AdS 4 ×N (1, 1) with λ 2 = 1 2 , which corresponds to the case of N = 3 supersymmetry. There are several U(1) angles with which one can perform the reduction to IIA. We have examined all of the possible reductions. The one that leads to a tractable solution is the U(1) isometry given by the linear combination of ψ 1 + ψ 2 . We label α to be the coordinate corresponding to ψ 1 − ψ 2 .
The reduced background is given by, 2 )+ in which we have recovered an SU(2) isometry in (θ 1 , φ 1 , α), characterised by the Maurer-Cartan forms s i . We also define Z = 3 + cos 2µ. The RR fluxes are given by, N (1, 1))) In this section, we present the results of a Non-Abelian T-duality applied to the SU(2) isometry in eq. (7.2) above. This background is somewhat unique compared to the other backgrounds presented in this paper and other NATD backgrounds presented thus far in the literature. It is unique in the sense that there is mixing between the spectator coordinates (θ 2 , φ 2 ) with the Maurer Cartan forms, s 1 , s 2 . Mixing terms with s 3 have been fairly common (ex. the ABJM, Klebanov Witten backgrounds), however, the non-symmetric mixing with s 1 , s 2 leads to a rich Type IIB NATD background. An interesting consequence of this mixing is the breaking of the U(1) isometry normally found in the ξ coordinate after NATD.

Dual CFT central charge
As a first step into the interpretation of the new backgrounds we produce in this manuscript we perform an analysis of the central charge of the dual field theories. In order to compute the dual field theory central charge, we first consider the quantized Page charges [56,57], defined by where κ 2 11 = (2π) 8 l 9 p , κ 2 10 = (2π) 7 α 4 and the brane tensions are T M 2 = 1 (2π) 2 l 3 p , and . We will present the general results of the Page charges and central charge for a few sample backgrounds considered in this manuscript. In particular, we discuss the importance of the factors of the 11d radius R, and their role in the N scaling of the central charge. Our method for computing the central charge is based on [58], as adjusted and generalized in [39]. The main modifications take into consideration a potential dependence on the coordinates of the internal manifold perpendicular to the field theory directions. A generic string frame metric in type II string theory, dual to a QFT in (d + 1)-dimensions is defined to have the following form,

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As in [39], we define the modified internal volume to be, (8.4) so that the functionĤ is in general given by, Then, the central charge for a QFT in (d + 1) spacetime dimensions is defined to be [58]: The G N factor is needed to cancel the length dimensions inĤ. We will use G 11 = l 9 p , and G 10 = l 8 s = α 4 . In our case of AdS 4 , the functions are, In the case of 11 spacetime dimensions, we obtainV int ∼ R 9 ,Ĥ ∼ R 18 r 3 , therefore, c ∼ L 9 l 9 p .
Due to our choice of Ansatz, eq. (2.1), the relevant objects are always M2 branes, whose normalization yields a scaling relation, R 6 ∼ l 6 p N M 2 , which means c ∼ N 3/2 M 2 , as known previously in [4], for example. In all of the reductions to Type IIA considered in this paper the F 4 flux comes through the reduction unaffected and an F 2 = dC (1) flux is generated. The normalization of the D2 branes leads to a relation, R 6 ∼ l p α 5/2 N D2 . The 'raw' central charge after the reduction now scales like, c IIA ∼ R 9 lpα 4 , therefore c IIA ∼ √ lp α 1/4 N 3/2 D2 . We also note the presence of D6 branes in the Type IIA backgrounds, however they are independent of R and instead are scaled with factors of l p only, suggesting they are topological charges. In the case of ABJM, the factor of k present in F 2 has the field theory interpretation as the level number of the dual gauge theory.
The relevant objects to the central charge after the NATD are the D5 branes, as they always inherit a term with the R 6 scaling factor. As observed previously in [37] for the ABJM background, we see that the effect of NATD is to preserve the N 3/2 scaling, but to change the numerical coefficient of the central charge.
After the NATD we wish to constrain the range of the dual coordinate ρ, so that we may compute the internal Volume,V int . Following the prescription first hinted at in [32], and further discussed in [37,39], we compute the periodic quantity b 0 defined by, The relevant two-cycle is,

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In each case we restrict to a submanifold, which we specify below: N (1, 1)) µ = 0, ρ fixed We should point out that in all of these cases the cycle is placed on a singularity, which could be problematic. However, the strategy still leads to reasonable results and was also observed in the case of NATD of AdS 5 × S 5 in [39]. Presumably in a full homology theory, it would be possible to show independence of this procedure on position in the manifold, but we leave that question to future investigations. For the remainder of this section we assume the prescription can be trusted, and move on to compute, where 0 ≤ χ ≤ π, 0 ≤ ξ ≤ 2π, and we have absorbed the m 2 gauge fixing constant into the definition of ρ. However, as discussed in [46], there is one NS-five brane every time ρ crosses integer multiples of π. Therefore we take the range of ρ from 0 ≤ ρ ≤ (n + 1)π. Next we compute H 3 = dB, Following [46], we normalize the NS flux to N N S5 using T N S5 = 1 (2π) 5 , We are now in a position to compute the Page charges and dual field theory central charge after the NATD. The results are summarized in tables 1, 2, 3, 4, and 5 below for the backgrounds with S 7 /Z k , 4 S 7 , M 1,1,1 , Q 1,1,1 , and N (1, 1), respectively. In the first column we present the flux normalization result, obtained from eq. (8.1) and the corresponding cycle used to integrate. The second column contains the 'Raw' central charge, which is directly computed from eq. (8.6). The third and final column is the substitution of the flux normalization into the 'Raw' central charge. In the cases where we have left some of the numerical constants generalized, we also present the result with the values of these constants replaced. As noted in [37], NATD maps integer charges onto non-integer charges, due to a violation of the condition, T p−n = (2π) n T p on the D brane tensions. In fact, we see a generic difference of π 2 between the D2 and D5 brane normalizations in these cases. In the

Background Page Charge
Cycle 'Raw' CC CC Table 1. Results for AdS 4 × S 7 /Z k and its supergravity duals.
case of AdS 4 ×S 7 /Z k , we note, as in [37], that after the NATD, k is no longer a well-defined level. We define a new level for the dual theory according to, where the cycle of integration is Σ 3 = [ρ, θ 1 , φ 1 ], and ρ is integrated over [0, (n + 1)π]. This leads to a N 2 N S5 scaling 5 in the central charge, which differs from the other cases. In the cases without k, a factor of N 3 N S5 arises from integration over ρ 2 inV int . Finally, we consider changes in the Page charges under large gauge transformations in B 2 , particularly ∆B 2 = nπα sin χdχ ∧ dξ. The Page charge associated to the D5 branes is always either zero or independent of R on the cycle of interest. Therefore, we compute ∆Q D3 using and In all of the cases examined, we find the relation,

Dual CFT central charge for coset NATD
In this section, we present a proposal to investigate the effect of large gauge transformations on cases where the NATD is performed on an SU(2)/U(1) coset isometry. The same method used above in eq. (8.8) cannot be directly applied here, as there is no 2-cycle with which to compute b 0 and thus, restrict the range of the dual coordinates. We propose JHEP02(2016)061    Table 5. Results for AdS 4 × N (1, 1) and its supergravity duals.

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to introduce a nonzero B 2 which is closed, and therefore does not affect the equations of motion:B 2 = α dρ ∧ dχ. This is equivalent to a large gauge transformation on B 2 . We where ρ and χ are integrated over [0, ρ o π] and [−χ o π, χ o π], respectively. Demanding b 0 = [0, 1] restricts the range of ρ and χ. Normally for polar coordinates, we would expect the χ angle to take the range 0 ≤ χ ≤ 2π (or − π 2 ≤ χ ≤ π 2 ), however the modified internal volumeV int computed from eq. (8.4), restricts the range of χ to be − π 2 ≤ χ ≤ π 2 , due to a factor of cos χ. Then we see that ρ would have to take the range 0 ≤ ρ ≤ 4π. Note that this condition was different in the cases of SU(2) NATD.
Considering the case of the τ reduction of AdS 4 × Q 1,1,1 as an example, we compute the Page charge for D3 branes, Integrating over the cycle containing ρ and the two 2-spheres, we obtain the normalization condition, 1 2κ 2 10 T D3 After the NATD, we find that the central charge is, If we had instead integrated over the χ plus the two 2-spheres cycle, the final central charge would be different by a factor of √ 2.

Discussion and conclusions
In this manuscript we have used Non-Abelian T-duality to produce new supergravity backgrounds. We have focused on solutions that preserve an AdS 4 factor. These solutions are particularly relevant in the context of the AdS/CFT correspondence since they describe strongly coupled conformal field theories in three dimensions. One of the prominent members of the class we consider as a seed case is related to the ABJM field theory. Another interesting example was provided by the N (1, 1) spaces, where the NATD apparently destroyed all of the isometries, other than the isometries of AdS and a residual U(1) isometry. We expect that these new backgrounds will ultimately enrich the number of entries in the AdS/CFT dictionary. We have only briefly mentioned the supersymmetry in the case of NATD(⇓ τ (AdS 4 × S 7 /Z k )), which we have argued preserves N = 2 supersymmetry. In the other cases, we

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propose that the reductions along the Hopf fibre coordinate, τ , preserve supersymmetry. In light of the recent work [38] on supersymmetry and SU(2) NATD we argue that the NATD backgrounds also preserved some fraction of supersymmetry. The main result in [38] shows that, at least for backgrounds with Bianchi IX symmetry, supersymmetry is preserved if the Killing spinors of the original background do not depend on the SU(2) isometry directions. A more covariant way to re-state the independence of the Killing spinor on certain coordinates is the Kosmann derivative; this is akin to the Lie derivative being the covariant way of stating that the metric has some invariance. For the reduction along U(1)'s other than the Hopf fiber we do not have a general geometric argument and relied on some explicitly known cases. It would be interesting to systematically and explicitly study the supersymmetry of the resulting backgrounds.
One interesting open problem would be to track, on the field theory side, the effects of various NATD's. In particular, the analysis we have presented of the holographic central charges seems to point to marked differences when one considers reducing along the Hopf fiber or along some other, susy preserving, U(1) direction. The implications for the dual quiver field theories and its potential cascading phases are only hinted at in the gravity side and it would be quite interesting to explore those phases.

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considered in this paper have B 2 = 0, so for simplicity we omit its contribution from the general procedure. The Lagrangian density for the NS sector fields (omitting the dilaton contribution) is given by, where A, B = 1, . . . , 10 and We then gauge the SU(2) isometry by changing derivatives to covariant derivatives and introduce gauge fields, A ± according to ∂ ± g → D ± g = ∂ ± g − A ± g. The next step is to add a Lagrange multiplier term to eq. (A.2) to ensure that the gauge fields have vanishing field strength. The Lagrange multiplier term is given by, Since the dimension of SU(2) is three, we have introduced three new dynamical variables in the form of the Lagrange multipliers, v i , so we must eliminate three of the variables by making a gauge fixing choice. A natural choice is g = I, so that all of the Euler angles in the SU(2) are zero and all three of the Lagrange multipliers become dual coordinates. The last step is to integrate out the gauge fields to obtain the dual Lagrangian density, where we can read off the dual components ofQ AB from, We have additionally defined: We can identify the dual metric and the generatedB 2 field as the symmetric and antisymmetric components ofQ AB , respectively. The transformation of the dilaton is given bŷ where and We also need to define ζ a = κ a i z i with κ a i κ a j = g ij and z i = 1 det κ (b i + v i ).

A.2 Coset spaces: SU(2)/U(1) isometry
Here we present a quick review of performing the dualization procedure on a background with an SU(2)/U(1) isometry. A more complete description of generating the Non-Abelian T-dual of a background with coset space isometries was first presented in [24]. We start with a metric of the form of eq. (A.1), where µ, ν = 1, . . . 8 and i, j = 1, 2, and the L i 's are replaced with L 1 = dθ, L 2 = sin θdφ. The procedure is essentially the same as in appendix A.1 above, but we modify, E ij = diag(g ij , λ), (A.14) where g ij is 2-dimensional. Due to this, we must additionally gauge fix one of the three Lagrange multipliers. In all of the cases considered, we choose (v 1 = 0, v 2 = m 2 α ρ cos χ, v 3 = m 2 α ρ sin χ). After inverting M ij , we take λ → 0 and construct the dual frames from e = κM −T dv.

B Review of supergravity EOM
In this appendix we will briefly review the 11 and 10D supergravity equations of motion, presented in the convenient form of [25,59]. The 11D supergravity action is given by [60], The 11D Einstein's equations and Bianchi identities can be expressed as,

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The action of massive Type IIA supergravity in string frame without sources is given by, In this case, the Bianchi identities are given by, which can be deduced from the definitions of the fluxes, The equations of motion that follow from varying (B.4) with respect to the metric are, , (B.7) and the dilaton equation is, The action of Type IIB supergravity in string frame without sources is given by, √ g e −2Φ R + 4(∂Φ) 2 − H 2 12 − 1 2 F 2 1 + Here, the definition of the fluxes take the form, The equations of motion that follow from varying (B.9) with respect to the metric are,

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.