AdS3 solutions in massive IIA, defect CFTs and T-duality

We establish a map between AdS3×S2 and AdS7 solutions to massive IIA supergravity that allows one to interpret the former as holographic duals to D2-D4 defects inside 6d (1,0) CFTs. This relation singles out in a particular manner the AdS3×S2 solution constructed from AdS3×S3×CY2 through non-Abelian T-duality, with respect to a freely acting SU(2). We find explicit global completions to this solution and provide well-defined (0,4) 2d dual CFTs associated to them. These completions consist of linear quivers with colour groups coming from D2 and D6 branes and flavour groups coming from D8 and D4 branes. Finally, we discuss the relation with flows interpolating between AdS3×S2×T4 geometries and AdS7 solutions found in the literature.


Introduction
Defect QFTs play an important role in our current understanding of Quantum Field Theories. Of particular interest is the situation when the ambient QFT is a CFT with a holographic dual. In this case, introducing appropriate branes in the dual geometry it is possible to construct the gravity dual of the defect QFT, that can then be studied holographically [1][2][3]. When the defect QFT is a CFT, the explicit AdS dual geometry can be constructed in terms of the fully backreacted geometry [4,5], if the number of defect branes is sufficiently large.
In this work we will be interested in an extension of the first realisation. We will show that a sub-class of the local solutions constructed recently in [11], preserving small N = (0, 4) supersymmetry on a foliation of AdS 3 ×S 2 ×CY 2 over an interval, can be used to construct globally compact solutions dual to 2d (0,4) SCFTs that have an interpretation in terms of D2-D4 defects in 6d (1,0) CFTs. More precisely, we will be using the word defect to indicate the presence of extra branes in Hanany-Witten brane set-ups that would otherwise arise from compactifying higher dimensional branes. This provides a new scenario in which 2d (0,4) CFTs appear in string theory.
In [11] AdS 3 ×S 2 ×M 4 solutions in massive IIA supergravity preserving N = (0, 4) supersymmetry with SU(2)-structure were classified. These solutions are warped products of AdS 3 ×S 2 ×M 4 over an interval, with M 4 either a CY 2 or a Kahler manifold. The CFT duals of the first class were studied in [34,35]. They are described by (0,4) quiver gauge theories with gauge groups n i=1 SU(k i ) × SU(k i ). SU(k i ) is the gauge group associated to k i D2 branes stretched between NS5 branes and SU(k i ) is the gauge group associated tok i D6-branes, wrapped on the CY 2 , also stretched between the NS5 branes. On top of these there are D4 and D8 branes that provide flavour groups to both types of nodes of the quiver. These quivers are a generalisation of the linear quivers studied in [26], where the D6 branes are unwrapped and are thus non-dynamical. In this paper we give an interpretation to our brane systems as D2-D4 brane defects in the D6-NS5-D8 branes associated to 6d (1,0) CFTs.
The organisation of the paper is as follows. In section 2 we review the main properties of the AdS 3 ×S 2 ×CY 2 solutions constructed in [11], and summarise the key features of their 2d dual CFTs, following [35]. In section 3 we construct a mapping that relates a sub-class of these solutions with the AdS 7 solutions in massive IIA supergravity constructed in [36]. Using this map we can interpret the 2d dual CFTs as associated to D2-D4 defects in the D6-NS5-D8 brane set-ups dual to the AdS 7 solutions, wrapped on the CY 2 . This suggests that it should be possible to construct RG flows that interpolate between these two classes JHEP12(2019)013 of solutions. In section 4 we discuss the AdS 7 solution that describes the 6d linear quiver with gauge groups of increasing ranks terminated by D6 branes, in relation to the map constructed in section 3. By means of this study we rediscover the non-Abelian T-dual (NATD) of the AdS 3 ×S 3 ×CY 2 geometry, constructed in [37] (see also [30]), as the leading order in an expansion on the number of gauge groups, of this solution. Then in section 5 we start a detailed study of the non-Abelian T-dual solution. We show that it provides a simple explicit example in the general classification in [11], that describes a 2d (0,4) CFT with two families of gauge groups [35] with increasing ranks. As in other AdS solutions generated through non-Abelian T-duality, the solution is non-compact, and this renders and infinitely long dual quiver CFT. Remarkably, we are able to provide explicit global completions of the solution that have associated well-defined 2d (0,4) dual CFTs, that we describe. This solution thus provides a useful example where it is possible to use holography in a very explicit way to determine global properties of non-compact solutions generated through non-Abelian T-duality, following the ideas in [38][39][40][41][42]. In section 6 we attempt to make connection with RG flows in the literature that connect AdS 3 geometries in the IR, with an interpretation as 2d defect CFTs, with AdS 7 solutions in the UV [6,7]. Our results are negative, and thus exclude the RG flows constructed in these references as interpolating between the AdS 3 solutions in [11] and the AdS 7 solutions in [36]. Section 7 contains our conclusions and future directions. Appendix A contains some explicit derivations useful in section 5. Appendix B contains details of the BPS flow constructed in [6], upon which section 6 is built.
2 AdS 3 ×S 2 ×CY 2 solutions in massive IIA and their CFT duals In [11] AdS 3 ×S 2 solutions in massive IIA with small (0,4) supersymmetry and SU(2) structure were classified. Two classes of solutions that are warped products of the form AdS 3 ×S 2 ×M 4 ×I were found, for M 4 either a CY 2 manifold, class I, or a family of Kahler 4 manifolds depending on the interval, class II. The solutions in the first class provide a generalisation of D4-D8 systems involving additional branes, while those in the second class are a generalisation of the (T-duals of the) solutions in [28], based on D3-branes wrapping curves in F-theory. In this paper we will be interested in the first class of solutions, that we now summarise.
The explicit form of the NS sector of the solutions referred as class I in [11] is given by: Here Φ is the dilaton, H the NS 3-form and ds 2 is the metric in string frame. The warpings are determined from three independent functions h 4 , u, h 8 . h 4 has support on (ρ, CY 2 ) while u and h 8 have support on ρ, with u = ∂ ρ u. The reason for the notation h 4 , h 8 is that JHEP12(2019)013 these functions may be identified with the warp factors of intersecting D4 and D8 branes when u = 1. 3 The 10 dimensional RR fluxes are with the higher fluxes related to these as Supersymmetry holds whenever which makes u a linear function. Hereˆ 4 is the Hodge dual on CY 2 . In turn, the Bianchi identities of the fluxes impose away from localised sources. In this paper we will be interested in the subclass of solutions for which the symmetries of the CY 2 are respected by the full solution. This enforces H 2 = 0 and a compact CY 2 . Thus, we will be dealing with T 4 or K3. The supersymmetry and Bianchi identities are then all solved for h 8 , u, h 4 arbitrary linear functions in ρ.
The magnetic components of the Page fluxesF = F ∧ e −B 2 , are given bŷ where we have included large gauge transformations of B 2 of parameter n, such that (2.9) The 2d CFTs dual to this class of solutions were constructed in [35]. They are described by (0,4) supersymmetric quivers with gauge groups associated to D2 and D6 branes, the JHEP12(2019)013 Figure 1. Generic quiver field theory whose IR is holographic dual to the solutions discussed in this section. The solid black line represents a (4,4) hypermultiplet, the grey line a (0,4) hypermultiplet and the dashed line a (0,2) Fermi multiplet. (4,4) vector multiplets are the degrees of freedom at each node. 0 1 2 3 4 5 6 7 8 9 D2 x x x D4 x  figure 1. (x 0 , x 1 ) are the directions where the 2d CFT lives, (x 2 , . . . , x 5 ) span the CY 2 , on which the D6 and the D8-branes are wrapped, x 6 is the direction along the linear quiver, and (x 7 , x 8 , x 9 ) are the transverse directions on which the SO(3) R symmetry is realised. latter wrapped on the CY 2 manifold, stretched between NS5 branes. Having finite extension in this direction, the field theory living in both the D2 and D6 branes is two dimensional at low energies compared to the inverse separation between the NS5-branes. It was shown in [35] that these quivers are rendered non-anomalous with adequate flavour groups at each node, coming from D4 and D8 branes. Remarkably, the flavour groups associated to gauge groups originating from D2 branes arise from D8 branes (wrapped on the CY 2 ) while those associated to the gauge groups originating from wrapped D6-branes arise from D4-branes. The corresponding quiver is depicted in figure 1. The underlying brane set-up is summarised in table 1.
The 2d CFTs dual to the solutions in class I thus generalise the (0,4) quivers studied in [26] from D2, NS5 and D6 branes, in two ways. First, the D6 branes are compact, and therefore give rise to gauge, as opposed to global, symmetries. Second, there are D8 branes between the NS5 branes that can give rise to different flavour groups to each gauge group coming from D2 branes [44,45]. Non-compact D4 branes provide the necessary flavour groups that render the nodes associated to the new, colour, D6 branes non-anomalous. Our quivers also generalise the (0,4) quivers constructed in [32] from D3-brane box configurations to gauge nodes with different gauge groups. JHEP12(2019)013 1 8 -BPS brane set-ups such as the one depicted in table 1 were discussed in [7] in the context of 2d defect CFTs originating from D2-D4 branes living in 6d (1,0) CFTs. In the next section we find that it is indeed possible to give an interpretation to some of the CFTs studied in [35] in these terms. We will discuss the connection with the solutions constructed in [7] in section 6.
3 A map between AdS 3 ×S 2 and AdS 7 solutions in massive IIA In [36] an infinite class of AdS 7 solutions in massive IIA was constructed, 4 preserving 16 supersymmetries (eight Poincare and eight conformal) on a foliation of AdS 7 ×S 2 over an interval. In this section we show that they can be related to our solutions in [11], preserving (0,4) supersymmetries on a foliation of AdS 3 ×S 2 ×CY 2 over an interval, through a map that reduces supersymmetry by half. As opposed to the mappings in [47] between AdS 7 solutions and the AdS 5 and AdS 4 solutions in [48,49], this mapping is not one-to-one, due to the presence of D2-D4 defects, whose backreaction introduces new 4-form and 6form fluxes.
We start by briefly summarising the solutions constructed in [36]. Using the parametrisation in [50], these solutions can be completely determined by a function α(z) that satisfies the differential equation Where F 0 is the Ramond zero-form. Explicitly, the metric and fluxes are given by These backgrounds were shown to arise as near horizon geometries of D6-NS5-D8 brane intersections [51,52] (see also [50,53] for previous hints), from which 6d linear quivers with 8 supercharges can be constructed [44,45]. In these quivers anomaly cancelation implies that for every gauge group the number of flavours must double the number of gauge multiplets, N f = 2N c [53]. In reference [50] a prescription was given to calculate the function α(z) that encodes the explicit AdS 7 solution dual to a given 6d quiver diagram. In this quiver diagram the NS5 branes are located at different values of z, the D6-branes are stretched between them along this direction and the D8 branes are perpendicular. The corresponding brane set-up is depicted in table 2.
After this brief summary we can introduce the mapping that relates these solutions to the solutions in class I in [11], summarised in the previous section. The mapping reads Using these relations one can match the B 2 field, dilaton, F 0 and F 2 fluxes of the two solutions, as well as the S 2 × I components of the metric. For the rest of the metric one must consider the mapping Besides, the F 4 and F 6 fluxes, which would violate the symmetries of the AdS 7 solution, must be disregarded when using the mapping from AdS 3 to AdS 7 . These fluxes clearly sign the presence of a D2-D4 defect in the AdS 3 solution. As we discuss below, its backreaction has also the effect of modifying the dependence of the different functions on both sides of (3.7)-(3.9) on the respective field theory directions (related through (3.6)). Indeed, (3.7) and (3.9) relate linear functions in ρ with a cubic function of z. This mapping is therefore essentially different from the mappings found in [47], where other than the replacements of AdS 5 × Σ 2 or AdS 4 × Σ 3 with AdS 7 , the internal space is just distorted by some numerical factors. This difference is due to the presence of the D2-D4 defect in the AdS 3 solution, which is also responsible for the reduction of the supersymmetry from 1/2 BPS to 1/4 BPS.
Using (3.8) and (3.6) it is possible to obtain the AdS 7 solution related to a particular AdS 3 ×CY 2 solution. One finds from which α(z), and thus, the explicit AdS 7 solution in [36], can be determined. This mapping does not however give the expressions for the u and h 4 functions that define the AdS 3 solution. Still, one can exploit (3.11) to show that the D8-brane charges of the AdS 7 and AdS 3 solutions, determined, respectively, from h 8 and − ... α/(162π 3 ), agree, and JHEP12(2019)013 that the same holds for the D6-brane charges, given that the correspondingf 2 Page fluxes satisfŷ (3.12) This implies that the D6-NS5-D8 sector of the AdS 3 solution is simply obtained by compactifying on the CY 2 the D6-NS5-D8 branes that underlie the AdS 7 solution.
However, as we have mentioned, the u and h 4 linear functions needed to fully specify the AdS 3 solution, cannot be determined from the AdS 7 solution using this mapping, other than the fact that they have to be proportional to each other. 5 This was to be expected, since, as we showed in [11], these functions encode the information of the additional D2-D4 branes present in the AdS 3 solution. This is, once more, essentially different from the mappings between AdS 5 and AdS 4 and AdS 7 solutions found in [47], where it is not possible to identify 6 and 4-cycles on which additional D2 or D4 brane charges can be defined. In this case this is possible due to the non-trivial CY 2 4-cycle in the internal space of the AdS 3 solutions.
The symmetry between the D6-NS5-D8 and D2-NS5-D4 sectors, manifest in the expressions of the RR Page fluxes of the AdS 3 solutions, stress the role of both D2 and D6 branes as colour branes in the 2d CFT dual to the AdS 3 solution, and of D4 and D8 branes as flavour branes [35]. The resulting 2d (0,4) CFT thus contains two types of nodes, associated to the gauge groups of D2 and compact, wrapped on the CY 2 , D6 branes. This is the generalisation of the (0,4) quivers discussed in [26] that we found in [35]. Note that compactification on the CY 2 of the 6d CFT living in D6-NS5-D8 branes preserves (4,4) supersymmetries. 6 The D2-D4 branes further reduce the supersymmetries by one half [7] (see also [59]). Alternatively, one could start with the D2-NS5-D4 Hanany-Witten brane set-ups discussed in [60,61], realising 2d (4,4) field theories, and intersect them with wrapped D6 and D8 branes, which would also reduce the supersymmetries by a half. The resulting 1 8 BPS configuration (increasing to 1 4 at the near horizon) is the one that we depicted in table 1. 5 We will see below that this guarantees that the two solutions share the same singularity structure, or, in other words, that the S 2 shrinks in the same way to produce topologically an S 3 . 6 Gauge theories with (4, 4) supersymmetry in two dimensions may be viewed as the dimensional reduction of 6d (1, 0) gauge theories. The six dimensional gauge theories have an SU(2)R R-symmetry. Upon dimensional reduction to two dimensions there is an additional SO(4) = SU(2)r × SU(2) l symmetry acting on the four reduced dimensions. This is also an R-symmetry since the supercharges are a spinor of this SO(4) group; the left-moving (positive chirality) supercharges are in the (2,1,2) representation of SU(2) l × SU(2)r × SU(2)R while the right-moving (negative chirality) supercharges are in the (1, 2, 2) representation [54,55].

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Let us now discuss the physical reason for the condition h 4 ∼ u, implied by (3.9) and (3.7), in the AdS 3 solutions. As we have mentioned, the functions u and h 4 , needed to completely determine the AdS 3 ×CY 2 solution, cannot be computed from (3.7) and (3.9), due to the different dependence on ρ and z of these functions and α(z), respectively. Rather, the relation h 4 = 81u/8 has to be seen as a restriction on the class of AdS 3 ×CY 2 solutions that can be interpreted as defects in the CFTs dual to AdS 7 solutions. This restriction comes from the condition that both solutions share the same singularity structure. In order to see this we note that in both solutions the range of the interval is determined by the points at which the S 2 shrinks, such that the S 2 ×I space is topologically an S 3 . In AdS 7 there is a D6 brane when α = 0,α = 0, and a O6 whenα = 0, α = 0. In turn, when α = 0,α = 0 the S 2 shrinks smoothly [48]. Similarly, for AdS 3 solutions satisfying h 4 ∼ u there is a D6 brane when u ∼ h 4 = 0, h 8 = 0 and a O6 when h 8 = 0, u ∼ h 4 = 0. In turn, the S 2 shrinks smoothly for u ∼ h 4 = 0, h 8 = 0 [11]. The role played by the D6 branes terminating the space as flavour branes is discussed in section 4.
Let us summarise our findings so far in this section. We have shown that a subclass of the solutions in [11] 7 can be interpreted as arising from D2-D4 defect branes inside the D6-NS5-D8 brane intersections underlying the AdS 7 × S 2 × I solutions in [36], wrapped on the CY 2 of the internal manifold. 6d (1,0) CFTs compactified in CY 2 manifolds give rise to 2d (4,4) field theories that are not conformal [54,55]. Therefore, AdS 3 solutions cannot be obtained from the AdS 7 solutions in [36] simply by extending the construction of AdS 5 and AdS 4 solutions in [47] to 4d manifolds. As we showed in [11] extra D2 and D4 branes are needed, that further reduce the supersymmetries down to 1/8 BPS and the AdS 3 solutions to 1/4-BPS. These branes backreact in the compactified geometry, and modify the simple mappings found in [47] such that the dependence of the functions defining the AdS 3 and AdS 7 solutions change, due to the backreaction. One can thus think of the 2d CFT associated to the AdS 3 solutions as comprised of two sectors, one coming from D6-NS5-D8 branes wrapped on the CY 2 , which by itself does not give rise to a 2d CFT, and one coming from extra, D2-D4 branes, which would not give rise either to 2d CFTs together with the NS5-branes [60]. One can in this sense interpret the D2-D4 branes as defects inside D6-NS5-D8 brane systems. We would like to stress that this defect interpretation is essentially different from the defect interpretation in terms of punctures that can be given to the Gaiotto theories in 4d [56], dual to the Gaiotto-Maldacena geometries [57]. In this last case both the field theory in the absence of punctures (dual to the Maldacena-Nunez solution [58]) and the ones with punctures are well-defined 4d CFTs, in contrast with the 2d CFTs dual to our AdS 3 solutions.
Further light on the relation between the 2d (0,4) CFTs dual to the AdS 3 solutions and compactifications on CY 2 of the 6d (1,0) CFTs dual to the AdS 7 solutions comes from comparing their respective central charges, following [62]. The holographic central charge of the 6d CFTs dual to the AdS 7 solutions was computed in [63]:

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In turn, the holographic central charge of the 2d CFTs dual to the AdS 3 ×CY 2 solutions is [35] c Using the mapping given by (3.6)-(3.9) this becomes Thus, there exists a universal relation between the central charges associated to both types of solutions. Similarly, in [64] (see also [65]) AdS 3 × Σ 4 solutions of massive IIA were constructed whose 2d (0,1) and (0,2) CFT duals arise as compactifications of the 6d (1,0) theories dual to the AdS 7 solutions. Their respective free energies were shown to satisfy the relation where Σ 4 is the compactification manifold and X IR is a constant that characterises the AdS 3 solution. 8 Our result is thus in agreement with an interpretation of the 2d CFTs dual to our solutions as compactified 6d (1,0) theories in CY 2 manifolds, with extra degrees of freedom coming from the 2d defects. It would be very interesting to obtain explicit flows connecting the AdS 3 ×CY 2 solutions in the IR with the AdS 7 solutions in the UV. In particular, it would be interesting to clarify whether these involve R 1,1 ×CY 2 warped product geometries, which would be the natural extension of the flows constructed in [62,64,65], or wrapped AdS 3 subspaces, more directly related to defects, as in [6][7][8]. In [7] different limits of the D2-D4-D6-NS5-D8 intersections depicted in table 1 were studied, giving rise to either AdS 7 or AdS 3 ×S 3 ×I' geometries, associated to the UV or IR limits of the intersection, respectively. In particular, AdS 3 ×T 4 geometries should arise when the branes are smeared on the T 4 . In section 6 we explore the connection between the BPS flows constructed in [6,7] and the subclass of AdS 3 ×T 4 solutions defined by the mapping discussed in this section.

The linear quiver with infinite number of nodes
As we have mentioned, the mapping found in the previous section is formal, in the sense that it relates α, a cubic function in z, to h 4 ∼ u, which are linear in ρ (with z and ρ related as in (3.6)). In this section we discuss a particular instance in which α and h 4 ∼ u can be explicitly related.
Consider an AdS 7 solution in which the S 2 ×I geometry is smooth at z = 0 and terminates at z = P + 1, such that For this we need N (P + 1) D8-branes at z = P , given that As shown in [63], for a particular choice of the integration constants such that α(0) = α(P + 1) = 0, and α and α are continuous functions, we have 3) and the dual CFT is a linear quiver with gauge group finished with a SU((P + 1)N ) flavour group, represented by the D8 branes. The brane set-up associated to this quiver is depicted in figure 2. Now, consider the situation in which P is very large, so that the region of interest reduces to 0 ≤ z ≤ P and we can take α(z) = 27π 2 N 2 (P (P + 2)z − z 3 ) for all P . 9 Redefining z = P (P + 2)x, we can write the solution in this region as This solution can be expanded close to x = 1 (the end of the space) by defining x = 1 − v.
We then have a metric and dilaton that for small values of v read, It is thus clear that close to v ∼ 0 or x ∼ 1, in the end of the space, we have D6 branes that extend along AdS 7 . As discussed in [48], these D6 branes can play the role of flavour JHEP12(2019)013 branes even when their dimensionality is the same as that of the colour branes. They differ in that the colour branes are extended along the six Minkowski directions of AdS 7 plus a bounded interval, while the flavour D6-branes are extended on the whole AdS 7 . Being noncompact they can act as flavour branes, as happens in many other (qualitatively different) examples, like [66,67]. Now, we would like to use the mapping between AdS 3 and AdS 7 solutions described by (3.6)- (3.9). This tells us that we should identify, This is not a solution of the equations of motion of the AdS 3 system. Nevertheless, if we take P → ∞ or, equivalently, ρ → 0, we have which defines a non-compact AdS 3 solution. This is the solution constructed in [37] acting with non-Abelian T-duality on the AdS 3 ×S 3 ×CY 2 solution dual to the D1-D5 system [54,55,[68][69][70].
As we discuss in the next section, the non-compact nature of the non-Abelian T-dual solution is reflected in the dual CFT in the existence of an infinite number of gauge groups of increasing ranks. In this section we have rediscovered it as the leading order of the solution defined by (4.5), dual to a well-defined six dimensional CFT. 10 Since we are working at very small values of z (equivalently, very small values of ρ), we do not see the flavour D6 branes, and the space is rendered non-compact. Conversely, taking P → ∞ we see no sign of these branes closing the space.
We discuss the non-Abelian T-dual solution in detail in the next section, and describe other possible ways to define it globally using AdS 3 /CFT 2 holography.

The non-Abelian T-dual of AdS 3 ×S 3 ×CY 2
In this section we discuss in detail one of the simplest solutions in the classification of AdS 3 ×S 2 geometries in [11], with a focus on the description of its 2d dual CFT, following [35]. This solution arises acting with non-Abelian T-duality on the near horizon of the D1-D5 system, and was originally constructed in [37]. In reference [30] it was shown that the (4,4) supersymmetry of the D1-D5 system is reduced to (0,4) upon dualisation, and that the solution can be further T-dualised and uplifted to M-theory such that it fits in the class of AdS 3 ×S 2 ×S 2 ×CY 2 solutions in [71]. 11 This solution is particularly interesting in the study of the interplay between non-Abelian T-duality and holography, since it allows for simple explicit global completions of the geometry using field theory arguments. 10 To be more precise, (4.8) selects a particular non-Abelian T-dual solution, with a given relation between the D2 and D6 brane charges. We give more details in the next section. 11 Actually, it provides the only known example in this class with SU(2) structure.

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In this section we also discuss another solution in the class in [11] that arises from the D1-D5 system, and that can be obtained as a limit of the non-Abelian T-dual solution [38,39,72]. This is the Abelian T-dual (ATD) of AdS 3 ×S 3 ×CY 2 along the Hopf-fibre of the S 3 , and orbifolds thereof, that also preserve (0,4) of the supersymmetries of the original D1-D5 system. The orbifold solutions describe the D1-D5-KK system, and are dual to (0,4) CFTs that have been discussed in the literature [18-21, 25, 32].

The NATD solution
The non-Abelian T-dual (NATD) of AdS 3 ×S 3 ×T 4 with respect to a freely acting SU(2) subgroup of its SO(4) R-symmetry group was constructed in [37]. As in other NATD examples, the space dual to S 3 becomes, locally, R × S 2 . The SO(4) R-symmetry is reduced to an SU(2) R-symmetry, and the solution is rendered (0,4) supersymmetric [30]. Due to our lack of knowledge of how non-Abelian T-duality extends beyond spherical worldsheets [73], the space is globally unknown. In this section we will resort to holography in order to construct a compact internal space for which a well-defined 2d dual CFT exists, following the strategy in [38][39][40][41][42].
We start generalising the solution constructed in [37] to arbitrary D1 and D5 brane charges and a compact CY 2 four dimensional internal space. The most general solution reads The corresponding D1 and D5 brane charges are given by The NATD with respect to a freely acting SU(2) group on the S 3 reads

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It is easy to see that this solution fits locally in the class of AdS 3 ×S 2 ×CY 2 solutions constructed in [11], with the simple choices u = 4L 4 M 2 ρ (5.14) These functions define a regular, albeit non-compact, solution. We will shortly be discussing various possibilities that define it globally. For now let us analyse the associated quantised charges.
We start discussing the relevance of large gauge transformations. Close to ρ = 0 the 3d transverse space is R 3 , while for large ρ it is R × S 2 . This implies that for finite ρ there is a non-trivial S 2 on which we can compute S 2 B 2 , which needs to satisfy For B 2 as in (5.9) this implies that a large gauge transformation needs to be performed as we move in ρ, such that B 2 → B 2 + nπvol S 2 for ρ ∈ [ρ n , ρ n+1 ], with The non-compactness of ρ is then reflected in the existence of large gauge transformations of infinite gauge parameter n. Moreover, taking into account large gauge transformations, we see that even if the 2-form and 6-form Page fluxes vanish identically, implying the absence of D6 and D2 brane quantised charges, there is a non-zero contribution when n = 0, such that N 8 = 2πF 0 = 2πL 2 (5.20) These conserved charges suggest that the D1-D5 system that underlies the Type IIB AdS 3 ×S 3 ×CY 2 solution has been mapped under the NATD transformation onto a brane system consisting on n D2-D6 branes at each [ρ n , ρ n+1 ) interval, dissolved in a D4-D8 bound state, due to the non-vanishing B 2 -charge. The corresponding brane distribution is depicted in table 3. x x x D4 x Table 3. Distribution of branes compatible with the quantised charges of the NATD solution. (y 0 , y 1 ) are the directions where the 2d CFT lives, (y 2 , . . . , y 5 ) parameterise the CY 2 , y 6 = ρ, y 7 is the radius of AdS 3 and (y 8 , y 9 ) span the S 2 .
This configuration is the same as the one underlying the solutions constructed in [7], and, as in that case, it can be related to the 1 8 -BPS brane set-up depicted in table 1, where the SU(2) R symmetry is manifest, through a rotation in the (x 6 , x 7 ) subspace. Due to the non-compactness of ρ the brane system is however infinite. This suggests a relation with the linear quiver with infinite gauge groups discussed in section 4, that we can now make more explicit.
Indeed, given that h 4 and u, as given by (5.15) and (5.14), satisfy the condition h 4 ∼ u, the NATD solution fits in the class of solutions that can be related to AdS 7 solutions, discussed in section 3. Both solutions are related explicitly through the mapping with P as introduced in (4.3). This selects the NATD solution with M 2 = 3 4 2 L 2 , 12 as the one related to the 6d (1,0) linear quiver discussed in section 4. These relations show that in the supergravity limit L 1 the D6-branes are sent off to infinity. In this way we can think of the NATD solution as the leading order in an expansion in P , of the AdS 7 solution dual to the 6d linear quiver with gauge groups of increasing ranks, terminated with flavour D6-branes.
In the next subsections we define other ways of completing the NATD solution with compact AdS 3 solutions. This will be valid for arbitrary values of the charges.

2d (0,4) dual CFT
As we have seen, the quantised charges of the NATD solution are compatible with an infinite brane system consisting on D2 and D6 branes stretched between NS5 branes. The D6 branes are wrapped on the CY 2 , and thus share the same number of non-compact directions of the D2 branes.
General 2d (0,4) quiver theories associated to the 1/8-BPS D2-D4-D6-D8-NS5 brane configurations depicted in table 1 were constructed in [35]. For the particular configuration corresponding to the NATD solution the quiver contains two infinite families of nodes, associated to D2 and wrapped D6 branes, with gauge groups of increasing ranks, and no flavours. This quiver is depicted in figure 3. We next summarise its main ingredients (the reader can find more details in reference [35]): • To each gauge node corresponds a (0,4) vector multiplet plus a (0,4) twisted hypermultiplet in the adjoint representation of the gauge group. In terms of (0,2) multiplets, the first consists on a vector multiplet and a Fermi multiplet in the adjoint, and the second to two chiral multiplets forming a (0,4) twisted hypermultiplet, also in the adjoint. The (0,4) vector and the (0,4) twisted hypermultiplet combine to form a (4,4) vector multiplet. They are represented by circles.
• Between each pair of vertical nodes there are two (0,2) chiral multiplets forming a (0,4) hypermultiplet, in the bifundamental representation of the gauge groups. They are represented by grey solid lines.
• Between each gauge node and any successive or preceding node there is one (0,2) Fermi multiplet in the bifundamental representation. They are represented by dashed lines.
• Between each gauge node and a global symmetry node there is one (0,2) Fermi multiplet in the fundamental representation of the gauge group. They are again represented by dashed lines.
The previous fields contribute to the gauge anomaly of a generic SU(N i ) gauge group as: • A (0,2) vector multiplet contributes with a factor of −N i . Figure 4. Completed quiver with a finite number of gauge groups.
• A (0,2) Fermi multiplet in the adjoint representation contributes with a factor of −N i .
• A (0,2) Fermi multiplet in the fundamental or bifundamental representation contributes with a factor of − 1 2 .
Following these rules it is easy to see that the coefficient of the anomalous correlator of the symmetry currents < J A µ (x)J B ν (x) >∼ kδ µν δ AB vanishes for each gauge group (see [35] for more details) -hence the gauge anomalies vanish. By assigning R-charges to the different multiplets (see [35] for the precise assignation), we can calculate the U(1) R anomaly (for U(1) R inside SU(2) R ). The correlation function < j µ (x)j ν (y) > for two U(1) R currents is proportional to the number of N = (0, 4) hypermultiplets minus the number of N = (0, 4) vector multiplets. This result is conserved when flowing to lower energies. In the far IR, when the theory is proposed to become conformal the R-symmetry anomaly is related to the central charge as indicated below.

Central charge
Let us now discuss the central charge associated to this quiver. We compute it using the formula (see [35,43]) c = 6(n hyp − n vec ), (5.26) where n hyp counts the number of fundamental and bifundamental hypermultiplets and n vec of vector multiplets. Clearly, these numbers are infinite for our quiver in figure 3. However, since they are subtracted in the computation of the central charge, they could still render a finite value. Terminating the space at a given n = P and analysing the behaviour when P goes to infinity we show however that this is not the case. Anomaly cancellation enforces that flavour groups must be added to both gauge groups at the end of the quiver. The resulting quiver is the one shown in figure 4. This quiver was discussed in [35], as one of the anomaly free examples analysed therein. For completeness we reproduce here the computation of its central charge.
The hypermultiplets that contribute to the counting of n hyp are the two chiral multiplets in each solid horizontal line, plus the two chiral multiplets in each vertical line.

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They give (5.27) Vector multiplets come from each node in the quiver, such that: This gives for the central charge To leading order in P we have, The central charge thus diverges with P 3 for the infinite quiver dual to the NATD solution. Still, it is useful to show that (5.30) coincides with the holographic central charge for ρ ∈ [0, ρ P ], with ρ P satisfying (5.18). Note that for large P we can simply take ρ P = 2πP . Using (3.16) we find for ρ ∈ [0, 2πP ], in agreement with the field theory result. Our calculation shows the precise way in which the central charge diverges due to the non-compact field theory direction. It also gives us a possible way to regularise the infinite CFT dual to the NATD solution. Indeed, the quiver depicted in figure 4 describes a welldefined 2d (0,4) CFT, that can be used to find a global completion of the non-Abelian T-dual solution. This completion is obtained glueing the non-Abelian T-dual solution at ρ P = 2πP to another solution in [11] that terminates the space at ρ = 2π(P + 1). We present the details of this completion in the next subsection. In section 5.3.2 we present a different completion, which makes manifest that this procedure is not unique and that one can device different global completions of the NATD solution, as stressed in [38].

Completions
In this section we present two possible completions of the NATD solution. The AdS 3 example is particularly useful in this respect, because the completed solution is not only explicit but also extremely simple, as opposed to other examples in higher dimensions [38,39,41].

Completion with O-planes
The simplest way to complete the NATD solution is by terminating the infinite linear quiver at a certain value of ρ, as we have done in the previous subsection. We take this to JHEP12(2019)013 be ρ = 2π(P + 1), with P ∈ Z, and choose the u, h 8 and h 4 functions such that: 2πP ≤ ρ ≤ 2π(P + 1).
The explicit form of the metric, dilaton and fluxes in the 2πP ≤ ρ ≤ 2π(P + 1) region can be found in appendix A. One can check that the NS sector is continuous at ρ = 2πP . The 2-form and 6-form Page fluxes are also continuous once large gauge transformations are taken into account. They are given bŷ nπ 0 ≤ n ≤ P P π(P + 1 − n) P ≤ n ≤ P + 1 (5.35) . nπ 0 ≤ n ≤ P P π(P + 1 − n) P ≤ n ≤ P + 1, (5.36) so they vanish at n = P + 1, where the geometry terminates. We show below that at this point the background has a singularity associated to O6-O2 planes. In turn there is a discontinuity in F 0 and F 4 at n = P that is translated into (P + 1)N 8 and (P + 1)N 4 additional flavours connected to the nodes corresponding to P N 4 D2 and P N 8 D6 branes, respectively. This is exactly as in the quiver depicted in figure 4. The expressions of the metric and dilaton in the 2πP ≤ ρ ≤ 2π(P + 1) region, given by equations (A.1), (A.2) in appendix A, show that close to ρ = 2π(P + 1) they behave as where x = ρ − 2(P + 1). This singular behaviour corresponds to the intersection of an O6 fixed plane lying on AdS 3 ×CY 2 with O2-planes lying on AdS 3 and smeared on CY 2 ×S 2 . Even if it is not clear what this object is in string theory, the fact that the solution has a well-defined dual CFT suggests that it should be possible to give it a meaning.

Glueing the NATD to itself
Another interesting way of defining globally the NATD solution is by glueing it to itself. In this case we take:

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The explicit form of the metric, dilaton and fluxes in the 2πP ≤ ρ ≤ 4πP region can be found in appendix A. One can check that the NS sector is continuous at ρ = 2πP . The 2-form and 6-form Page fluxes are also continuous once large gauge transformations are taken into account. They are given bŷ Therefore, they are both continuous at n = P and vanish at n = 2P . The corresponding quantised charges are: (5.43) and where N 6 denotes anti-D6 brane charge, N 2 D2-brane charge and N 8 = ±2πF 0 in the two regions. For N 4 we have in the two regions. Thus, the D2 and D6 brane charges increase linearly in the 0 ≤ n ≤ P region, corresponding to the NATD solution, and decrease linearly in the P ≤ n ≤ 2P region, till they vanish at n = 2P , where the geometry terminates. At this point the S 2 shrinks smoothly. The discontinuity of N 8 and N 4 at n = P is translated into 2N 8 and 2N 4 additional flavours at the nodes with flavour groups P N 4 and P N 8 , respectively. The associated quiver is the one depicted in figure 5. The 2N 8 and 2N 4 flavour groups contribute each with one (0,2) Fermi multiplet in the fundamental representation of the corresponding gauge group. As for the quivers constructed in [35], the flavour group introduced at the node associated to D2-branes arises from D8-branes while that introduced at the node associated to D6-branes arises from D4-branes. The central charge of this quiver is given by To leading order in P this gives which one can check is in agreement with the holographic central charge. Figure 5. Symmetric quiver associated to the NATD solution glued to itself.

The Abelian T-dual limit
The non-Abelian T-dual solution defined in ρ ∈ [ρ n , ρ n+1 ] gives rise to the Abelian T-dual, along the Hopf-fibre of the S 3 , of the original AdS 3 ×S 3 ×CY 2 background, in the limit in which n goes to infinity [38,39,72]. In this subsection we will be interested in the ATD solution, and orbifolds thereof, in its own right, as another explicit example in the class in [11]. The ATD solution is given by where ψ is the ATD of the Hopf-fibre direction, normalised such that ψ ∈ [0, 2π]. Upon dualisation, the (4,4) supersymmetries of the original solution are reduced to (0,4) [30], and the solution fits in the classification in [11]. The corresponding u, h 4 and h 8 functions are given by The quantised charges are, One can check that this is reproduced from the NATD solution for ρ ∈ [ρ n , ρ n+1 ] and n large, using that N 2 = nN 4 and N 6 = nN 8 in this interval. The brane set-up describing the ATD solution consists on N 2 D2-branes and N 6 D6-branes, wrapped on the CY 2 , stretched along the ψ circular direction between two NS5-branes that are identified.
Orbifolds of this solution can be constructed taking ψ ∈ [0, 2πN ]. They are Tdual to the AdS 3 ×S 3 /Z N ×CY 2 solution in Type IIB that describes the D1-D5-KK system [18][19][20][21]25]. The Type IIA brane realisation of this system is depicted in figure 6. From this quiver we have that One then obtains a central charge c = 6(n hyp − n vec ) = 6N 2 N 6 N + 12N. (5.59) This gives in the large N 2 , N 6 limit, in agreement with the central charge of the D1-D5-KK system [18]. 13 For N = 1 the quiver in figure 6 reduces to the quiver depicted in figure 7. The (4,4) hypermultiplets connecting N 2 nodes and N 6 nodes among themselves become (4,4) hypermultiplets in the adjoint representation. In turn, the (0,2) Fermi multiplets connecting each JHEP12(2019)013 N 2 (N 6 ) node with adjacent N 6 (N 2 ) nodes combine into (0,4) Fermi multiplets connecting each N 2 node with its respective N 6 node, which together with the (0,4) hypermultiplets between them give (4,4) hypermultiplets in the bifundamental. In this way supersymmetry is enhanced to (4,4), and the quiver describes the D1-D5 system in terms of the D2 and D6-brane charges of the Abelian T-dual solution. 14 6 Relation with the AdS 3 ×S 2 flows of Dibitetto-Petri In [6,7] Dibitetto and Petri (DP) constructed various BPS flows within minimal N = 1 7d supergravity that are asymptotically locally AdS 7 . These flows are described by warped AdS 3 solutions triggered by a non-trivial dyonic 3-form potential. A particularly interesting solution was constructed in [6], which was shown to interpolate between asymptotically locally AdS 7 and AdS 3 ×T 4 geometries. The UV AdS 7 limit is (asymptotically locally) the reduction to 7d of the AdS 7 solutions of massive IIA constructed in [36]. In this subsection we would like to explore the 10d uplift of the IR AdS 3 ×T 4 limit, in connection with the subclass of solutions discussed in section 2, in the case in which CY 2 =T 4 .
The AdS 3 solution constructed in [6] reads (see appendix B for the details), ds 2 7 = e 2U(r) ds(AdS 3 ) 2 + e 2V (r) dr 2 + e 2W (r) ds(S 3 ) 2 , X = X(r), where X, U , V , W , k and l are functions of r discussed in the appendix B. This solution is asymptotically locally AdS 7 when r → ∞ , while when r → 0 it flows to an AdS 3 ×T 4 non-singular limit, given by, 15 and 14 See [75], section 4, for this analysis in Type IIB. 15 As compared to [6], we write the 7d metric in terms of an AdS3 space of radius one.

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As the AdS 7 asymptotic limit, this geometry is not a solution of 7d N = 1 minimal supergravity by itself, but rather the IR leading asymptotics of the flow. In the discussion that follows it will be useful to recall from appendix B that the values of the 7d scalar X in the r → ∞ and r → 0 limits are X = 1 and X 5 = 2 2 /3, respectively. 7d N = 1 minimal supergravity can be consistently uplifted to massive IIA on a squashed S 3 [64]. Using the uplift formulae provided in appendix B, a family of AdS 3 solutions to massive IIA can thus be constructed from the DP flow. This gives rise in the r → ∞ limit to 10d geometries that asymptote to the AdS 7 ×S 2 ×I family of solutions in [36]. In turn, the geometry that is obtained in the AdS 3 ×T 4 limit reads 16 As in 7d, the uplift of the r → 0 limit of the DP flow is not a solution to massive IIA by itself, but rather its IR leading asymptotics. We would like to see whether it can be completed by an AdS 3 ×T 4 solution in the class of [11], with the same asymptotics. For that it is easy to realise that one can absorb the constant X that causes the distortion of the internal space (we are referring to (B.25)-(B.29) in appendix B) by simply modifying the mapping for the h 4 function in (3.9) as h 4 = 81 8 X 5 u ↔ 81 8 X 5 α. We then have for the IR geometry given by (6.4)-(6.9), ρ ↔ 2πz (6.10) u ↔ α (6.11) This gives for the AdS 3 ×T 4 subspace (6.14) 16 Here we have taken g 3 = 8 √ 2, which is the value for which the internal space and fluxes of the AdS7 solutions in [36] are recovered.

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The result is a bonna fide AdS 3 ×T 4 solution to massive IIA, supplemented with F 4 and F 6 fluxes satisfying (2.7) and (2.8). The resulting 7d metric does not share however the asymptotics of the 7d metric arising from (6.4). Thus, the IR limit of the DP flow cannot be completed into an AdS 3 ×T 4 solution in the class of [11], that shares its same asymptotics. This result excludes the RG flows constructed in [6] as solutions interpolating between AdS 3 ×T 4 geometries (in the subclass defined in section 3) and the AdS 7 solutions constructed in [36]. Still, it should be possible to construct these flows, perhaps as R 1,1 ×CY 2 warped product geometries, as the ones discussed in [64].

Conclusions
In this paper we have discussed some aspects of the class of AdS 3 ×S 2 solutions with small N = (0, 4) supersymmetry and SU(2)-structure constructed in [11]. We have focused our analysis on a sub-set of solutions contained in "class I" of [11], which are warped products of AdS 3 ×S 2 ×CY 2 over an interval with warpings respecting the symmetries of CY 2 . 2d (0,4) CFTs dual to these solutions have been proposed recently in [34,35].
We have established a map between the previous solutions and the AdS 7 solutions in [36], that allows one to interpret the former as duals to defects in 6d (1,0) CFTs. More precisely, the 2d dual CFT arises from wrapping on the CY 2 the D6-NS5-D8 branes that underlie the AdS 7 solutions, and intersecting them with D2 and D4 branes. In this sense it combines wrapped branes and defect branes. The D2-branes are stretched between the NS5branes, as the D6-branes, and the D4-branes are perpendicular, as the D8-branes. They give rise to (0,4) quivers with two families of gauge groups connected by matter fields [35]. Each family is described by a (4,4) linear quiver and is connected with the other family by (0,4) and (0,2) multiplets, rendering the final quiver (0,4) supersymmetric.
The previous mapping suggests that it should be possible to construct flows connecting the AdS 3 ×CY 2 solutions in the IR with the AdS 7 solutions in the UV. The presence of D2-D4 defects suggests that one should look at warped AdS 3 flows, as the ones discussed in [6], which interpolate between asymptotically locally AdS 3 ×T 4 geometries, with an interpretation as 2d defect CFTs, and AdS 7 solutions. We have found however that our solutions have different asymptotics than the IR AdS 3 geometries considered in [6]. This discrepancy could originate on the wrapped branes present in our solutions, more suggestive of an R 1,1 ×CY 2 flow [62], as the one constructed in [64]. It would be very interesting to find the explicit flow that interpolates between these two classes of solutions.
We have provided a thorough analysis of the AdS 3 ×S 2 ×CY 2 solution that arises from the Type IIB solution dual to the D1-D5 system through non-Abelian T-duality. Using the map between AdS 3 and AdS 7 solutions derived in the first part of the paper, we have rediscovered this solution as the leading order of the AdS 7 solution in the class in [36] dual to a 6d linear quiver with gauge groups of increasing ranks, terminated by D6-branes. Secondly, we have provided two explicit global completions with AdS 3 solutions in the class in [11]. One of these completions is obtained glueing the non-Abelian T-dual solution to itself, in a sort of orbifold projection around the point where the space terminates. This solution has a well-defined 2d dual CFT that we have studied. Orbifolds have previously JHEP12(2019)013 played a role in the completion of NATD solutions, remarkably in the example discussed in [41], but this is the first time the explicit completed geometry has been constructed. The AdS 3 example provides indeed a very useful set-up where to test the role played by holography in extracting global information of NATD in string theory, following the ideas in [38][39][40][41][42].

A Completions of the NATD solution
Completion with O-planes. The metric, dilaton and fluxes of the NATD solution completed as indicated in section 5.3.1 read, in the 2πP ≤ ρ ≤ 2π(P + 1) region, The solution discussed in section 6 was obtained in [6] taking the following ansatz: where r = log ρ and κ and L satisfy, In these expressions g and h are the gauge coupling of the vector fields 17 and the topological mass of the 3-form potential, respectively, of minimal N = 1 7d supergravity.

B.1 The r → ∞, AdS 7 limit
When r → ∞ the previous solution is asymptotically locally AdS 7 , for any values of κ and L respecting the constraint given by their equation (4.27). The explicit way in which AdS 7 arises is as follows. in terms of unit radius S 3 and AdS 3 spaces. In turn, the 3-form potential is given by, and thus asymptotes to that of an AdS 7 space of radius 1/h. The geometry in the UV can thus be completed by an AdS 7 space with vanishing 3-form potential, that solves the equations of motion and gives rise to an AdS 7 solution to massive IIA supergravity upon uplift to ten dimensions [64]. As discussed in [6], this geometry is not a solution of 7d N = 1 minimal supergravity by itself, but rather the IR leading profile of the flow for L and κ given by (B.19), (B.21).

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B.3 Uplift to massive IIA 7d N = 1 minimal supergravity can be consistently uplifted to massive IIA on a squashed S 3 [64]. The uplift formulae were provided in that reference. They read, in the parameterisation used in [50] and for vanishing vector fields: where in the last expression we have used the odd-dimensional self-duality condition [76] X 4 * 7 F 4 = −2hB 3 .

(B.30)
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