Marginal deformations of a class of AdS$_3$ $\mathcal{N}=(0,4)$ holographic backgrounds

We discuss marginal deformations of warped AdS$_3\times$S$^2$ solutions preserving small $\mathcal{N}=(0,4)$ supersymmetry in massive IIA and eleven-dimensional supergravity and obtain a whole family of new solutions. We characterise these new backgrounds by studying some observables like the quantised charges, associated Hannany-Witten brane set-ups and the holographic central charge, the latter is shown to be invariant under the deformation. The study of the preservation of supersymmetry shows that the new backgrounds support an identity structure on the internal five-dimensional space, which is dynamical.


Introduction and summary
Supersymmetric solutions with AdS p+1 factors in type II and eleven-dimensional supergravities play a prominent role in the context of the AdS/CFT correspondence since they provide a holographic description of p-dimensional superconformal field theories (SCFTs) at strong coupling [1]. The discovery of this holographic duality has ever since triggered a number of efforts to construct and classify AdS vacua for any dimension allowed and preserving various amounts of (super)symmetries that have been used to study and characterise SCFTs.
More recently, the case of AdS 3 backgrounds has gained a lot of attention. There are several motivations for this. For instance, the near horizon geometries of five-dimensional extremal black holes have AdS 3 factors. Using SCFT 2 data it is then possible to understand microscopic features of black holes, like their entropy by computing the central charge of the SCFT 2 [2], among other aspects (see for instance [3][4][5][6][7]). On the other hand, twodimensional SCFTs are special on their own since they can, in certain cases, be fully solvable due to the structure of the superconformal algebra. It is therefore interesting to explore deeply each side of this dual pair in order to shed some light on new phenomena via holography. For a sample of works regarding AdS 3 supersymmetric backgrounds in ten and eleven-dimensional supergravity preserving different amounts of supersymmetry and their holographic applications see .
Moreover, on the geometrical side, attempts to constructing and classifying supersymmetric AdS 3 solutions have been mostly focused on the G-structure formalism [42] for which one extracts geometric constraints for the fields of the solutions according to the number of (super)symmetries and geometrical structures, etc, we impose in the internal space. It has also been considered back-reacting D-brane arrangements which are known to produce AdS solutions in the near horizon limit [34,36], among others. However, these efforts have been non-exhaustive due to the many choices we have on the number of supersymmetries, and superconformal algebras, supported by the solutions constraining the internal space submanifolds. Thus the approach has been focussed on searching and classifying all supergravity solutions preserving given amounts of supersymmetry, choices of internal structures, etc. This program has allowed to expand significatively our knowledge of new string backgrounds which may have very interesting applications in the context of holography. In this vein, another possibility to explore the landscape of AdS 3 vacua is to consider AdSpreserving deformations of well-known supergravity solutions. Depending on the details of the deformation these solutions may preserve supersymmetry whilst changing the structure of the internal space, and in some cases escape from presently known classifications of supergravity solutions.
In this work we will use TsT transformations [43] and the analog to eleven dimensions [44] in order to generate a larger class of warped AdS 3 supersymmetric solutions. The seed backgrounds we will consider are a subclass of the solutions constructed in [22] which are solutions of massive IIA supergravity of the warped form AdS 3 ×S 2 ×CY 2 foliated over an interval, the two-sphere realising geometrically the SU(2) R R-charge of the solution. They are given in terms of three linear functions, preserve small N = (0, 4) supersymmetry and an SU(2) structure in the internal five-dimensional space, analogously, for these solutions, an SU(3) structure in the seven-dimensional space transverse to AdS 3 . The above solutions appear in the near horizon limit of D 2 -D 4 -NS5-D 6 -D 8 brane arrangements. D 2 and D 6 branes are colour branes and are suspended between the NS5 branes whilst D 4 and D 8 correspond to localised sources and provide flavour groups attached to the gauge nodes which leave the dual quiver CFT anomaly free [23,24]. For vanishing Romans mass, the uplift to eleven dimensions of the above solution gives rise to a class of AdS 3 × S 3 /Z k ×CY 2 foliated over an interval, which preserve the same amount of supersymmetry and internal structure group [31]. The brane configuration for this solution involves M2 branes and KK monopoles suspended between M5' branes as well as extra flavour M5 branes.
Given the internal symmetries of the seed solutions above, we have two choices which produce inequivalent backgrounds after TsT transformations. Namely, if we consider or not the azimutal direction inside the S 2 for the process. In the latter case we are left with solutions for which supersymmetry is fully preserved. Of course more generic supersymmetric solutions can be generated by a sequence of TsT's not involving the U(1) inside the S 2 , but we will explore this more generic case in the future. For the ten-dimensional solutions, we study the brane configurations that we propose generate our solutions in the near-horizon limit. Holographically, these new backgrounds are dual to marginal deformations of the seed (undeformed) SCFT, both theories having the same central charge in the holographic limit. The latter can be understood since their degrees of freedom, in the aforementioned limit, are associated to the weighted volume of the internal spaces (to be defined below). The deformation changed the internal space enriching the geometric structure but left in-variant its weighted volume. We prove this using the holographic calculation and left the specification of the SCFT for a forthcoming publication.
The content of this paper is organised as follows. In Section 2 we start by briefly reviewing the seed solutions in [22]. We then proceed to apply the TsT transformation in Section 3 in order to obtain the new family of backgrounds in massive IIA. We study the quantised charges and present brane configurations which we argue give rise to our solutions in the near horizon limit. In Section 4 we study the eleven-dimensional analog of the TsT transformation for the solutions in Section 2 with vanishing romans mass uplifted to eleven dimensions. One of the solutions obtained correspond to the uplift of the TsT-deformed IIA solution in the massless case. We then prove the invariance of the central charge under the deformation in Section 5 using the holographic computation. Finally, In Section 6 we study the preservation of supersymmetry for the solutions obtained in Sections 3 and 4 . This analysis suggest the new supersymmetric solutions support a dynamical identity structure in the internal five-dimensional space. Some comments and final remarks are addressed in Section 7. In Appendix A we give our conventions for supersymmetry.

The seed AdS 3 N = (0, 4) holographic backgrounds
In this section we shall briefly review the AdS 3 solutions in massive IIA supergravity preserving small N = (0, 4) supersymmetry obtained in [22]. They will constitute our starting point from which we will obtain the marginally deformed solutions via a transformation involving dualities.
The solutions in [22] are of the warped form AdS 3 × S 2 × M 5 , supporting an SU(2) structure on M 5 , equivalently, for these solutions, an SU(3) structure in seven dimensions. Moreover, the five-dimensional space M 5 locally splits into a four-dimensional piece M 4 and an interval. There are two classes of solutions. In this work we will concentrate on a subclass of class I solutions for which M 4 is (conformally) CY 2 . From now on we will consider CY 2 =T 4 . The NS sector of the solution in the string frame reads where the functions u, h 4 , h 8 are functions of ρ only. This is supported with the following RR field strengths where = ∂ ρ and The above background is a supersymmetric solution of massive type IIA supergravity provided h 4 (ρ) = 0, h 8 (ρ) = 0, u (ρ) = 0, (2.4) the first two away from localised sources. The ρ coordinate parametrising the interval can be taken to be of finite range. This imposes additional constraints on the various functions of the solution. We require for 0 ≤ ρ ≤ 2π(P + 1) that 1 The metric functions obeying the above conditions are then explicitly given in Table 1. Table 1. Piece-wise continuous functions satisfying the conditions in eq. 2.5. The value of u(ρ) is the same in all intervals, as required by supersymmetry.
The set of constants (α j , β j , µ j , ν j , b 0 ) for j = 0, . . . P parametrising the piece-wise continuous functions above are subject to certain constraints imposing continuity of the NS sector along the ρ intervals. The conditions are The supergravity solution is trustable whenever these constants as well as the number P have large values.

The marginally deformed backgrounds
In this section we will construct a family of solutions corresponding to deformations of the supergravity solutions in eqs (2.1)-(2.2). Such deformations are built upon a sequence of T dualities and a change of coordinates [43]. The resulting backgrounds are considered to be holographic duals of the marginally deformed SCFTs dual to the original (undeformed) backgrounds.
In order to proceed, we first pick a two-torus in the geometry. For the solution in eq. (2.1) there are two options which will produce inequivalent solutions. They correspond to where ϕ is the azimuthal angle inside the S 2 and x i the coordinates on T 4 . The deformation is achieved by performing a T-duality in one of the coordinates, a shift with parameter λ in the second and T duality back in the first. The solutions obtained will describe a family of solutions in terms of the functions u, h 4 , h 8 and the parameter λ.
In the first case T 2 : (ϕ, x 1 ), following the T duality rules in [45] , the above procedure generates the following background where the higher fluxes are obtained via the lower ones as indicated in eq (2.2). The background in eq. (3.1) is a solution of massive IIA supergravity if conditions in eq. (2.4) are imposed. We notice the original solution is recovered after turning off the deformation parameter, as expected.
For the second case T 2 : (x 3 , x 4 ), the procedure outlined above produces the following background which is a solution of massive IIA supergravity if conditions in eq. (2.4) are imposed. We notice since the S 2 is a spectator subspace for this deformation, we expect N = (0, 4) supersymmetry will be fully preserved as we will explicitly show in Section 6.

Quantised charges and brane set-ups
In this section we will study the Page charges of the deformed backgrounds. Throughout, we shall use the following We start with the solution in eq. (3.1) and consider the following non-trivial cycles of the geometry The Page charges read If we allow large gauge transformations B 2 → B 2 + πk dvol S 2 , the Page fluxes are those in eq. (3.1) except forf The charges in eq. (3.4) computed in 2πk ≤ ρ ≤ 2π(k + 1) are explicitly, We notice that for finite ρ ∈ [0, 2π(P + 1)] we have P + 1 parallel NS5 branes. From the above expressions we see in particular that no extra NS5' branes were generated by the deformation. In addition, the above charges are well-defined as long as the set of constants α k , β k , µ k , ν k as well as the combination λ(kβ k − α k ), ∈ Z.
As we pointed out before, the first two conditions in eq. (2.4) must be satisfied by the solutions everywhere except at points were we have localised sources. At those points, we have a change in gradient of the piece-wise linear functions proportional to h 4,8 pointing the possible existence of a source for D p branes via the modified Bianchi identities df = j s . From Table 1 we obtain Using this information as well as the Page fluxes of the solution we compute where we have used xδ(x) = 0. From this we conclude that D 4 , D 4 as well as D 8 , having non-zero sources, correspond to flavour branes whilst D 6 and D 2 are colour ones. Thus in addition to the D-branes of the seed solution, the deformation has induced (semi-localised) flavour Q D 4 branes. The brane configuration, before the near horizon limit is taken, we argue is associated to the solution above is shown in Table 2. Table 2. Brane configuration which in the near horizon limit gives the solution in eq. (3.1). We show the world-volume directions the branes are suspended as well as their number in the k-th interval.
For the second solution, we consider the following cycles (3.12) and non-trivial Page formŝ (3.13) An analysis as detailed above shows that in addition to the D-branes of the seed solution, the generated Page charges after the transformation (λ-dependent) are given by This implies the quantisation conditions λβ k ∈ Z, λα k ∈ Z, which requieres rational λ. In order to determine if the above charges correspond to colour or flavour branes, we compute Using then (3.7) we find that the effect of the deformation was to add Q D 4 colour and Q D 6 flavour branes respectively. Therefore the original D 4 -NS5-D 2 and D 8 -NS5-D 6 brane arrangements are modified by the addition of D 4 branes extended along (t, x, x 3 , x 4 , ρ) as well as semi-localised D 6 branes in (AdS 3 , S 2 , x 3 , x 4 ) wrapped on T 2 : (x 3 , x 4 ). The brane set-up corresponding to this configuration is summarised in Table 3. Table 3. Brane configuration which in the near horizon limit gives the solution in eq. (3.2). We also show the world-volume directions the branes are suspended as well as their number in the k-th interval. We see the D 6 and D 4 branes are wrapped on T 2 : (x 3 , x 4 ).

The deformation in eleven dimensions
In this section we will study a generalisation to eleven dimensional supergravity of the TsT transformation studied in the previous section. The seed solutions will be the uplift of the background in eq. (2.1)-(2.2) for vanishing Romans mass. The backgrounds obtained will correspond to a family of supersymmetric solutions which are out of a subclass of the classification for AdS 3 eleven dimensional solutions studied in [31]. In order to proceed, we consider a vanishing Romans mass in the solution of eq. (2.1)-(2.2) which lead us to consider h 8 = k. The uplift of this solution to eleven dimensions was first constructed in [31]. For latter use we will present some details here. We determine the three and one-form potentials to be We notice the 3-form potential above is not globally well-defined. This would be the case if h 4 were a continuous function. Using the usual KK anzats eq. (A.2), the eleven-dimensional solution raeds where ds 2 This solution preserves small N = (0, 4) supersymmetry and supports an SU(2) structure.
We will now generalise this class of solutions by performing an SL(3,R) transformation of coordinates.
For a solution which is SL(3,R) invariant we use the anzats where the a, b indices correspond to the three-torus directions, g µν is the transverse eight dimensional metric and detM=1. We have two possible choices for which we can apply the transformation. Namely T 3 :(x 2 , x 3 , x 4 ) and T 3 :(x 3 , x 4 , x 11 ).
In the first case, the background in eq. (4.2) can be bring into the form of eq. (4.4) provided we identify We then use the transformation rules spelled out in [44] to obtain the new background parametrised by λ. The transformation for the one-form A a , using (4.5), givesÃ a = A a + 1 2 λ abc C (1)bc = 0 and thereforeDφ a = Dφ a = dφ a . On the other hand, the non-trivial transformation associated to τ = −C (0) + i∆ 1/2 readsτ = τ /(1 + λτ ), from which we obtaiñ The deformed background then reads where This background is a solution of 11d supergravity when conditions in eq. (2.4) are imposed, and reduces to the undeformed solution for λ = 0, as expected.
In the second case T 3 : (x 3 , x 4 , x 11 ), the solution obtained following the procedure spelled out above corresponds to the uplift to eleven dimensions of the solution in eq. (3.2). The eleven-dimensional background reads where and Υ was defined in eq. (4.3). This background is a solution of 11d supergravity when conditions in eq. (2.4) are imposed. In Section 6 we will show that the solutions presented in this section preserve N = (0, 4) supersymmetry supporting an identity structure. Before to close this section, it is worth noticing that the solutions in eqs. (4.7) and (4.9) can be used as seed solutions in order to generate other families of supersymmetric solutions. For instance, after appropriate analytical continuations we can generate solutions with AdS 3 /Z k × S 3 factors which further reduction to IIA along the Hopf-fibre direction of AdS 3 will generate new AdS 2 × S 3 solutions in IIA supergravity, which can be further extended to massive IIA, generalising those studied in [31], etc.

Holographic central charge
The main goal of this section will be to compute the central charge characterising the new family of solutions. For the seed solutions this was done in [24,31] and using the analysis of the spin-2 spectrum in [29]. A generic result involving the deformations discussed above is that they leave the internal space volume transverse to AdS 3 -weighted by the dilatoninvariant. We then anticipate the central charges will be the same before and after the deformation.
In order to see this explicitly, we consider the metric of the solutions written in the following way For the ten dimensional solution, since the deformations acted on the internal space of the solutions, we clearly see the quantities a(r, y), b(r) in eq. (5.1) are spectator under the deformations. In addition, we find that , where tilde denotes fields after the deformation. It is then easy to see that e −4Φ det(g ij )ã d = e −4Φ det(g ij )a d , givingc hol = c hol as anticipated. This result goes through for the elevendimensional solutions after considering the relation between the ten and eleven-dimensional quantities in the KK anzats (A.2) and H = d ŷ det(ĝ ij )â d , where quantities with hat are eleven-dimensional ones.
After we have characterised the backgrounds by computing their central charges, the goal is to compare them with the central charges obtained from the putative dual field theories to these solutions, in the holographic limit. Some comments are in order. For instance, in the case of the field theory read off from the brane configuration in Table 3, we can achieve an anomaly free quiver field theory following the rules in [23,24]. Nevertheless, this gives a central charge that is apparently changing due to the extra gauge and flavour group insertions. We would expect cancelations among them that will give the same central charge as before the deformation, or that their contributions are sub-leading in the holographic limit. We will elaborate more on this in a forthcoming publication.

Comments on supersymmetry and G-structure of the solutions
In this section we will study the supersymmetries preserved by the supergravity solutions in eqs. (3.1), (3.2) and (4.7), (4.9), based on the explicit form of the Killing spinors of the original solution (2.1). The conventions we follow for supersymmetry are detailed in Appendix A. The solution in eq. (2.1) preserves small N = (0, 4) supersymmetry by construction. In the conventional approach, this implies the existence of two algebraic conditions on the ten-dimensional Majorana-Weyl (MW) spinor ensuring the vanishing of the supersymmetry variations.
In order to see this explicitly, we decompose the ten-dimensional gamma matrices as follows where ρ α and γ µ are three and seven-dimensional gamma matrices respectively and the σ i are the usual Pauli matrices. In this notation the chirality matrix is Γ 11 = −σ 3 ⊗ I ⊗ I.
After plugging the solution in eq. (2.1) (in the natural frame) into eq. (A.11) we find the MW Killing spinor takes the form where ζ is the AdS 3 Killing spinor and Since the deformation involves a sequence of T dualities, a condition for preserving Killing spinors reduces to their invariance by the action of the Kosmann-Lie derivative along the Killing vector K associated to the isometric direction we picked to perform the duality L K = 0, where is the Killing spinor of the un-dualised solution. By considering K = ∂ y , the above condition reduces to ∂ y = 0 [47]. Moreover, invariance under the change of coordinates in the second direction also requires independence of it on the spinor. Therefore, supersymmetry is compatible with TsT transformations as long as the spinor is uncharged under the directions used for the transformation [48].
For the first solution in eq. (3.1), there is a residual U(1) ϕ which we may think of as a candidate R-charge for N = (0, 2) preserved supersymmetry. However, the spinors (6.2) are charged under this coordinate and T duality along this direction will project out this dependence. The residual U(1) ϕ is therefore a global symmetry and supersymmetry is completely broken. In other words, compatibility with the TsT transformation imposes the projection condition γ 56 χ 1,2 = 0 breaking all supersymmetries. Despite the breaking of supersymmetries, this solution is interesting in its own since it still solves the BPS condition in eq. (2.4).
For the second marginally deformed solution in eq. (3.2), the spinor is independent of the T 4 directions, so we ensure supersymmetry is fully preserved. To be more precise, working with the supersymmetry transformations for the solution in eq. (3.2), we find where tilde denotes fields after the transformation, provided we identifỹ ensuring supersymmetry is preserved as the original solution does. This is along the lines of the generic result in [48], which in addition showed that the entire information of the transformation is encoded in an antisymmetric bi-vector associated to classical r-matrices solving the Yang-Baxter equation.
Let us now turn to the G-structure characterising the above background. To begin with, the solutions in [24] were constructed by imposing that they support an SU(2) structure on the five-dimensional internal space M 5 transverse to AdS 3 × S 2 . For these solutions, this implies that the internal five-dimensional spinors are globally parallel. The deformed Killing spinors break the above condition, each of which defining an SU(2) structure, the intersection of which gives an identity structure. To be more precise, given the rotation of the internal spinor under TsT eq. (6.6), the transformed MW spinor takes the form (6.2) with the internal spinor transformed accordingly χ 1,2 →χ 1,2 . In addition, the sevendimensional spinor can be further decomposed into S 2 × M 5 factors according to eq. (6.3). Namely, χ 1,2 = e A 2 ξ ⊗ η 1,2 , where ξ is a Killing spinor on the S 2 charged under SU(2) R . An SU(2) structure on M 5 implies 3 η 1 = η, η 2 = η. (6.7) Using (6.6) the TsT MW spinors are given by where, using the 2+5 split of the internal spinor χ, we find therefore the spinorsη 1,2 are nowhere parallel defining a point-dependent SU(2)×SU(2) structure, that we will refer to it as dynamical. This is then described in terms of the largest common subgroup, which then defines a dynamical identity structure. Notice we could have also analysed the G-structure of the solution in terms of the seven-dimensional spinor. In this case the seed solution supports an SU(3) structure. It would then be possible to understand the fate of the seven-dimensional G-structure following [49]. The analysis of this section suggest this may give a dynamical SU(3) structure, and will provide a new example of AdS 3 solutions with dynamical SU(3) structure. We plan to report on this in a forthcoming publication. For the eleven dimensional solutions in eqs. (4.7) and (4.9) the preserved Majorana Killing spinors can be ascertained just as we did for the ten dimensional case. Namely, the preserved Killing spinors are those which are independent of the directions along which we performed the transformation. Using the relation between the eleven and ten-dimensional spinors (A.8) together with (A.4) we easily see that small N = (0, 4) supersymmetry is preserved. Once again whenever the deformation parameter is turned offƒ we recover the undeformed Majorana Killing spinor defining an SU(2) structure. In the case at hand we have a dynamical identity structure instead.

Conclusions
In this paper we have presented new solutions in massive IIA and eleven-dimensional supergravity obtained via TsT transformations and the analog in eleven dimensions. The solutions obtained preserve small N = (0, 4) supersymmetry and support a dynamical identity structure on the five-dimensional internal submanifold of the solution, as long as we do not use the azimuthal angle inside the S 2 in the procedure. The new backgrounds in ten and eleven dimensional supergravity constitute a whole family of solutions parametrised by the deformation parameter λ and linear functions satisfying the conditions in eq. (2.4). To the best of our knowledge, a complete classification of these solutions is still missing in the literature. One can in principle follow the same procedure as the one outlined in [22] for the SU(2) structure case. That is to say construct bispinors out of seven-dimensional spinors supporting a (dynamical) identity structure in the internal five-dimensional space and obtain geometrical constraints in the form of the solution from the differential conditions implied by supersymmetry. Moreover, in terms of seven-dimensional G-structure, the seed solutions support an SU(3) structure. The analysis we followed in Section 6 suggests this becomes a dynamical SU(3) structure after the transformation. Progress on classification of AdS 3 geometries supporting a dynamical SU(3) structure was recently reported in [37].
For the ten-dimensional solutions, we studied the Page charges and associated brane configurations. We showed that depending on the two-torus chosen, the deformation adds either colour or flavour branes or both to the seed configuration. Holographically, The backgrounds obtained correspond to marginal deformations of the SCFT dual to the seed solutions. We verified this by computing the central charge of the deformed backgrounds, showing they are the same before and after the transformation. In the field theory side side, we can engineer a dual quiver quantum field theory with the information obtained from the Hannany-Witten brane set-ups associated to the solutions. The specification of the dual quantum field theories and more field theory aspects of the solutions are left for a forthcoming publication.