AdS solutions and $\lambda$-deformations

We elevate $\lambda$-deformed $\sigma$-models into full type-II supergravity backgrounds. We construct several solutions which contain undeformed $AdS_n$ spaces, with $n=2,3,4$ and $6$, as an integrable part. In that respect, our examples are the first in the literature in this context and bring $\lambda$-deformations in contact with the AdS/CFT correspondence. The geometries are supported by appropriate dilaton and RR-fields. Most of the solutions admit non-Abelian T-dual limits.


Introduction
In recent years there has been a lot of activity in constructing classes of two-dimensional quantum field theories having a group theoretical basis, remarkable properties and controllable behavior. One such class comprises theories that generically come under the name of λ-deformations. In the prototype example [1] the model is characterized by a conformal field theory (CFT) having an current algebra symmetry at level k and a Lagrangian realization in terms of a Wess-Zumino-Witten (WZW) model [2], as well as by a matrix λ. For small entries of this matrix it is just the action of the WZW theory perturbed by current bilinears λ ab J a + J b − . The full action is non-linear in λ, effectively takes into account all loop quantum effects arising from the self-interaction of the chiral and anti-chiral currents of the original unperturbed WZW theory and is integrable for λ proportional to the identity [1]. Since then, this construction has been extended to cover cases with more than one current algebra theories, mutually and/or self-interacting [3][4][5][6]. These models have a very rich structure and their RG flow possesses several fixed points. The use of non-trivial outer automorphisms in this context was put forward for the case of a single group in [7]. A major reason that the λ-deformed models have attracted attention is due to the fact that they are integrable for specific choices of the deformation matrix λ. Such cases exist first for isotropic deformation matrices [1,[4][5][6] (for the SU(2) group case, integrability was shown in [8]) as well as for some anisotropic model cases [9,10] and for subclasses of the models of [5,6]. Integrable deformations based on cosets having a Lagrangian description in terms of gauged WZW models, symmetric and semi-symmetric spaces have also been constructed in [1], [12,13] and [11].
In parallel to the above, a class of integrable σ-models was introduced in [14 -16] and [17][18][19] for group and coset spaces, respectively. These are ultimately connected to the Principal Chiral Model (PCM) for groups and cosets which are integrable. There is relation between λ-deformations and η-deformations for group and coset spaces as was discussed in [20,21], [10,[22][23][24]. In particular, in these works it was shown that for the isotropic case the λ-deformed model is related via Poisson-Lie T-duality [25] and appropriate analytic continuations to the η-deformed model. For a comprehensible review concerning certain aspects of λ-and η-deformations see [26].
Having a σ-model, let alone an integrable one, it is very important to elevate it into a full supergravity solution. That is, support the metric and the antisymmetric tensor fields of the two-dimensional σ-model with the appropriate dilaton and Ramond-Ramond (RR) fields so that the equations of motion are satisfied. Nevertheless, this is in general a challenging task to perform. Several examples in this direction have been already presented in the literature [27,28,21,[29][30][31][32][33][34][35]. A common, characteristic of these works, no matter whether it concerned λ-or η-deformations, is that the entire nontrivial part of the space-time is deformed. Hence, undeformed spaces like AdS were not found so far as part of full supergravity solutions. This is the task we successfully overtake in the present paper by constructing type-II supergravity solution containing both λ-deformed and AdS spaces. Therefore, they can be used in the context of the AdS/CFT correspondence.
The outline of this paper is as follows: In Sec. 2 we review the λ-deformed models based on the coset CFTs SU(2)/U(1), SL(2, R)/U(1) and SL(2, R)/SO(1, 1). In Sec. 3 we continue by constructing solutions of the type-IIA and type-IIB supergravity theories with geometry M 6 × CS 2 λ × CH 2,λ . 1 We found that the corresponding geometries admit AdS 2 and AdS 3 factors , respectively. In the same spirit, in Sec. 4 we present various type-IIB backgrounds with geometry M 8 × CS 2 λ which include AdS 2 , AdS 4 and AdS 6 factors. In Sec. 5 we find solutions of the type-IIB supergravity, whose metric is of the form M 8 × CH 2,λ and includes AdS 2 factors. Next, in Sec. 6, after reviewing the deformed CFTs on SO(4)/SO(3) and SO(3, 1)/SO(2, 1) we construct a type-IIB background with geometry M 4 × CS 3 λ × CH 3,λ . This solution allows for an AdS 2 factor. Finally, Sec. 7 contains our concluding remarks. We complete the presentation with two appendices. In Appendix A we give an account of the equations of motion of the type-IIA and type-IIB supergravities, while in Appendix B we discuss the non-Abelian T-duality limit of the deformed models and explicitly perform it to a solution of Sec. 3.

λ-deformed model based on the coset SU(2)/U(1)
The geometry of the λ-deformed SU(2)/U(1) is given by [1] When λ = 0 one recovers the undeformed geometry corresponding to the SU(2)/U(1) exact CFT [36] ds 2 = k dβ 2 + cot 2 β dα 2 . (2. 2) The geometry (2.1) is supported by a non-trivial dilaton arising from integrating out gauge fields in the construction [1] which we note is λ-independent. We will denote the background corresponding to (2.1) and (2.3) as CS 2 λ . It is convenient to define the following combinations of constants It is also useful to express the deformed geometry using the following frame 2 The background CS 2 λ is invariant under the following symmetries [37] which leaves the above frames invariant. There is an additional symmetry that has been unnoticed so far in the literature, acting as On the frame it acts as e 1 → e 2 and e 2 → −e 1 , leaving of course the metric invariant. 2 Notice that the frame can be written as e 1 = e Φ dx 1 and e 2 = e Φ dx 2 where x 1 = λ − sin α cos β and x 2 = −λ + cos α cos β. As a result the metric is conformally flat and the dilaton plays the rôle of the conformal factor, i.e. ds 2 = e 2Φ dx 2 1 + dx 2 2 . In these coordinates the dilaton becomes e −2Φ = The dilaton beta function is Another useful combination that plays significant rôle in the following sections is where η ab is the Minkowski metric with signature (−, +). This will be instrumental in constructing type-II supergravity solutions in which the background CS 2 λ will be an integrable part. Note also that (2.9) is frame-dependent since the right hand side is proportional to η ab even though the metric is of Euclidean signature. That implies that an SO(2) rotation of the frames (2.5) will not leave (2.9) invariant. Similar to (2.9) relations hold for the other two-and three-dimensional λ-deformed models we use in this paper. It is likely that a similar relation, differing only on the numerical coefficient of the right hand side, is true for all λ-deformed coset CFTs SO(n + 1)/SO(n) and their analytic continuations.

λ-deformed model based on the coset SL(2, R)/U(1)
The various expressions for the fields for the λ-deformed SL(2, R)/U(1) model can be derived from the one above by applying the analytic continuation This operation transforms the metric to (2.11) and the dilaton to The undeformed metric is obtained for λ = 0 corresponding to the SL(2, R)/U(1) exact CFT [38]. We will denote the background corresponding to (2.11) and (2.12) as CAdS 2,λ .
The dilaton beta function simply acquires an overall minus sign compared to the one for the CS 2 λ case, i.e.
while the Ricci tensor and the dilaton satisfy the following relation Note again that the Kronecker δ ab appears in the right hand side of this relation even though the signature of the space is Minkowski. Hence, as (2.9) this is a frame dependent relation.

λ-deformed
Under this transformation the line element becomes whereas the dilaton is given by (2.12). The undeformed metric is found by setting λ = 0 which simply gives corresponding to the SL(2, R)/SO(1, 1) exact CFT [38]. The symmetries (2.6) and (2.7) still hold with the appropriate renaming. We will denote the above background as The corresponding frame for the deformed line element now is and it is associated to a metric with signature (+, +). As in footnote 2 the combina- The dilaton beta function is the same as the one given for . Finally, the Ricci tensor and the dilaton now satisfy the relation 3 Solutions with geometry of the form M 6 × CS 2 λ × CH 2,λ We start our constructions by looking for solutions whose geometry is given by the direct product of a six dimensional manifold and the λ-deformed geometry CS 2 λ × CH 2,λ . Thus the ansatz for the ten-dimensional line element is ds 2 = ds 2 (M 6 ) + e 6 2 + e 7 2 + e 8 2 + e 9 2 , (3.1) where we take e 6 and e 7 to be identified with e 1 and e 2 of eq. (2.5) respectively and e 8 and e 9 with e 1 and e 2 of eq. (2.20). We will also complete the NS sector by considering a dilaton given by the sum of the corresponding λ-deformed spaces The above information allows one to get already an idea of the curvature of the space M 6 . This can be done by re-writing the dilaton equation (A.4) as which due to (2.9) and (2.21) implies for its Ricci scalar that

Type-IIA solutions containing AdS 2 factors
By proposing an appropriate ansatz for the RR-sector we can obtain a series of type-IIA solutions with AdS 2 factors. In the following lines we describe each case separately. Let us consider backgrounds supported by the following set of RR fields with c 1 , . . . , c 8 being constants. With this choice the expression for F 2 mixes the two λ-deformed backgrounds. The expression for F 4 mixes the entire λ-deformed background with a two-dimensional part of M 6 . Using the observation in footnote 2, we immediately see that the Bianchi identity (A.2) for F 2 is satisfied while that for F 4 implies that (provided that F 4 is not identically zero) Moreover, the first of the flux equations in (A.6) gives while from the last two we get The next step is to investigate the structure of the Ricci tensor on M 6 by analyzing the Einstein equations. It turns out that the non-vanishing components of the symmet-ric tensor T I I A ab in eq. (A.5) are (3.9) The first two lines imply that the Ricci tensor on M 6 must have the following structure (3.10) Clearly r 1 > 0, while the condition (3.4) implies Therefore, the curvature r 2 = r 1 /2 > 0. Hence, we see that M 6 can be written as the direct product of two Einstein spaces, a two-dimensional one of negative curvature and a four-dimensional one of positive curvature. We will denote this direct product structure as 3 The two conditions (3.6) and (3.8) are obviously satisfied being the volume forms of the corresponding spaces. Hence, we see the existence of an AdS 2 factor in the geometry. The positive part of the six-dimensional geometry M + 4 will not mix with the rest. It can be any regular four-dimensional manifold with curvature r 1 /2. Moreover, from the last four lines of the equation (3.9) we obtain the following condi-tions for the constants c 1 , . . . , c 8 In order to determine the solution we have to solve the constraints given in eqs (3.7), (3.10) and (3.13). This is a system of seven quadratic equations with eight unknown parameters. Let us now restrict ourselves to the following cases:

Solution 1:
In this case the constants are (3.14) Reality requires that c 2 3 0 which imposes the following bound on r 1 Hence, in the undeformed limit λ = 0 (µ → 0) is enough that the curvature is positive.
However, in the non-Abelian T-duality limit (described in detail in App. B) where λ approaches unity and simultaneously k → ∞ we have that µ → 2. Then, the bound becomes r 1 8. Similar comments hold for the solutions 2 & 3 below.

Solution 2:
In this case we have that We need to impose that c 2 8 0 in order for the RR fields to be real. As a result Note that in the above three solutions the signs of the non-vanishing c i 's are all independent. Also note that, even for λ = 0 the RR-fields still remain non-vanishing. In that case, the four-dimensional transverse space corresponds to the exact CFT SU(2)/U(1) × SL(2, R)/SO(1, 1). Then, this space supports the embedding of AdS 2 × M + 4 to type-IIA supergravity. Similar comments hold for other solutions in this paper. Unlike the previous solutions, here the curvature is bounded from above as well. In the conformal limit the curvature r 1 must vanish and thus the corresponding geometric six-dimensional space becomes flat. On the other hand, in the non-Abelian T-duality limit 2 r 1 8. Similar comments hold for the remaining solution below.

Solution 4: The constants are
Solution 5: Now we have that

Type-IIB solutions containing AdS 3 factors
We turn our attention to type-IIB solutions. We try for the self-dual RR-form the ansatz (3.26) The first two lines imply that the Ricci tensor on M 6 must have the structure Obviously this geometry contains AdS 3 factors.
The last four lines of eq. (3.26) imply the following constraints for the parameters Reality implies that c 2 3 0 giving a lower bound for r, i.e. r µ . (3.31) The possible signs of c 3 and c 4 are all independent.

Solution 2:
In this case λ = 0 and the constraints are solved for where now c 1 is arbitrary. Reality requires that r 2c 2 1 . Here the signs of the constants c i are correlated through the last two equations in (3.29).
Note that in the above solutions the expression for the RR five-form completely mixes the three-dimensional subspaces of M 6 with the λ-deformed parts of the geometry.

Solutions with geometry of the form
Another interesting category of backgrounds is that with geometries written as direct products of an eight-dimensional manifold and a two-dimensional deformed space.
This allows more room to obtain higher than three-dimensional AdS spaces as part of the full supergravity solution. In this section we will take the deformed part to be CS 2 λ , thus for the ten-dimensional geometry we make the ansatz given by the line element ds 2 = ds 2 (M 8 ) + e 8 2 + e 9 2 , (4.1) where we take e 8 and e 9 to be identified with e 1 and e 2 of eq. (2.5), respectively. We also assume that the NS two-form vanishes while the dilaton is given by (2.3). From the dilaton equation (A.4) we get the following for the curvature of M 8 Using (2.8) this implies that M 8 is a constant negative curvature space We will now study the geometric characteristics of M 8 by proposing specific ansatze for the RR fields.

Type-IIB solutions containing AdS 2 & AdS 4 factors
Let us focus on the case of type-IIB backgrounds whose RR sector consists of the following fields Next we compute the non-vanishing components of T I IB ab which are (4.6) From the first two lines we can read the structure of the Ricci tensor on M 8 R ab = − c 2 1 + c 2 2 + c 2 3 + c 2 4 η ab = −r 1 η ab , a, b = 0, 1, 2, 3 , R ab = − c 2 1 − c 2 2 + c 2 3 + c 2 4 δ ab = r 2 δ ab , a, b = 4, 5, 6, 7 . (4.7) Notice that r 1 > 0. This tells us that M 8 can split into a direct product of two fourdimensional Einstein spaces one of which has negative curvature, i.e.
Restricting ourselves to AdS solutions we can think of two possibilities. One is to take M − 4 to be the direct product of AdS 2 with R ab = −r 1 η ab and a two-dimensional space of negative curvature M − 2 with R ab = −r 1 δ ab and Euclidean signature, e.g. H 2 . The second, perhaps more interesting, possibility is to take M − 4 to be AdS 4 normalized such that R ab = −r 1 η ab . Moreover, the condition (4.3) relates the curvatures r 1 and r 2 with the deformation parameter λ as The last two lines of (4.6) can be thought of as constraints on the parameters c 1 , . . . , c 4 The solution is where s i = ±1 satisfy the condition s 1 s 2 + s 3 s 4 = 0. This is a one-parameter family of solutions since the constants c 1 , . . . , c 4 depend on r 1 .
In order to ensure the reality of the solution we need to impose that all the arguments in the square roots of the previous expressions are positive. Thus we have the following possibilities: Range of λ ∈ 0, 4 − √ 15 : In that case the curvature r 1 is restricted inside the interval Range of λ ∈ 4 − √ 15, 1 : Now the curvature r 1 lies in the interval Here it is also possible to consider the non-Abelian T-duality limit. In the above analysis we chose the arbitrary parameter to be r 1 . . This is done through a non-trivial RR five-form.
In the special case where λ = 0 we obtain another solution

Type-IIB solutions containing AdS 6 factors
Another solution of the type-IIB supergravity whose geometry supports an AdS 6 factor can be derived by the following ansatz The last suggests that the Ricci tensor on the manifold M 8 has the form (4.20) As a result, the eight-dimensional space can be written as a direct product of a sixdimensional Einstein space of negative curvature and another two-dimensional Einstein space, that is Clearly one possibility that fits in this category is M 8 = AdS 6 × S 2 . Also, we can re-write the condition (4.3) as Two more constraints on the parameters c 1 , . . . , c 4 arise from the last two lines of eq.  also that r 2 0, which means that M 2 has negative curvature.

Solution 2:
There is also a second solution where now the various parameters are Notice that, the second solution has always a non-vanishing RR three-form. For the first one this is also true except when λ = 4 − √ 15. The existence of a non-trivial RR three-form mixes the λ-deformed part with a two-dimensional subspace of M 8 spanned by (e 6 , e 7 ). The AdS 6 part, more generally M − 6 , stands on its own.

The M 8 × CH 2,λ case
Another possible class of backgrounds could be those whose geometry is the direct product of an eight-dimensional manifold and CH 2,λ . In principle these can be con- The geometric properties of M 8 are determined depending on the specific ansatze for the RR fields to which we now turn. We are going to consider a type-IIB background with non-trivial RR form fields given by where as before c 1 , . . . , c 4 are constants. Obviously the Bianchi eq. (A.9) for F 1 is satisfied automatically, while that for F 3 gives the condition a, b = 2, 3, 4, 5, 6, 7 . (5.6) Obviously, the eight-dimensional space can be written as a direct product of a twodimensional Einstein space of negative curvature and another six-dimensional Einstein space, that is It is clear that this structure allows the existence of an AdS 2 factor decomposing the eight-dimensional space as M 8 = AdS 2 × M 6 . Also, we can re-write the condition (5.1) as The parameters c 1 , . . . , c 4 obey two more algebraic equations which come from the last two lines of eq. (5.5). These are The background can be completely determined by solving the eqs (4.18), (5.6), (5.8) and (5.9). It turns out that we can only have one real solution which is

Embedding the λ-deformed coset CFT SO(4)/SO(3)
Here we are going to apply our strategy in order to embed the λ-deformed coset CFT SO(4)/SO(3) and its non-compact counterpart SO(3, 1)/SO(2, 1) [28] into the type-IIB supergravity. We first summarize the field content of the λ-models that correspond to these spaces. The line element of the deformed SO(4)/SO(3) CFT which we denote as CS 3 λ , is given in terms of the following frame (see eq. (A.5) of [28]) where we have defined for convenience the functions Notice that the above frame corresponds to a metric with signature (+, +, +). The geometry is also supported by the non-trivial dilaton (see eq. (A.6) of [28]) whereas the NS two-form is zero. Some key formulas allowing for a good ansatze for the RR sector of the type-II supergravity are Moreover, the dilaton beta-function for this model reads Another useful relation is where now η ab is the three-dimensional Minkowski metric with signature (−, +, +).
Note that, as in the case of the two-dimensional λ-deformed coset spaces this is a frame depended relation as well.
The non-compact version of CS 3 λ , which we will denote as CH 3,λ , can be obtained via an analytic continuation, namely As a result, the right hand sides of (6.5) and (6.6) flip signs.
We will also make the hypothesis that the NS sector consists of a dilaton, which is given by the sum of the dilatons of the λ-models CS 3 λ and CH 3,λ . Due to the fact that the dilaton beta functions for CS 3 λ and CH 3,λ come with opposite sign the dilaton equation (A.4) implies that Focusing on type-IIB backgrounds we take as the only non-vanishing RR-field that (6.12) The first two lines in eq. (6.12) suggest that the Ricci tensor of M 4 is As a result the four-dimensional space M 4 splits into a direct product of two twodimensional Einstein spaces, one of which is of negative curvature and spanned by the directions (e 0 , e 1 ) while the second has positive curvature and is spanned by (e 3 , e 4 ).
From now on we will denote this splitting as M 4 = M − 2 × M + 2 . Clearly one can choose M − 2 to be AdS 2 , i.e. M 4 = AdS 2 × M + 2 . From eq. (6.13) one can also check that eq. (6.9) is indeed valid. Moreover, from the last two lines of eq. (6.12) we obtain one more constraint (6.14) To fully determine the background we need to solve eqs (6.13) and (6.14). Since the total number of independent equations that we need to solve is two and the number of parameters we have is three (r, c 1 , c 2 ) one of the parameters will be free. Choosing the free parameter to be r the final solution is then for all possible choices of the signs s 1,2 = ±1. The solution is real provided r µ/2.

Concluding remarks
In this work we constructed solutions of the type-II supergravity based on the λ- where c 1 , . . . c 6 are constants. An exhaustive analysis of all possible solutions is beyond the scope of this paper.
Our solutions seem to be non-supersymmetric. We have checked the dilatino equation for the first solution in Sec. 3.1 as well as for the solution in Sec. 3.2. Indeed, the variation of the dilatino vanishes only when the Killing spinor is trivial, suggesting that the supersymmetry is completely broken. We expect that this is the case for the other solutions we found in this paper. The two-dimensional σ-models corresponding to our supergravity solutions can easily be made integrable. The reason is that all factors, i.e. the λ-deformed, as well as the AdS spaces, are. In addition, we may certainly choose the rest of the factors, e.g. M + 4 in (3.12) to correspond to an integrable σ-model as well. However, the RR-fields may spoil integrability for the full string solution.
The reader might have noticed that we have not provided solutions with an AdS 5 factor, the reason being that we did not find an appropriate ansatz for the RR-fields.
Perhaps this can be done by allowing warping to take place between the various factors in the metric. Our type-IIA solutions can be trivially lifted to eleven-dimensional supergravity. Nevertheless, it will be very interesting to find solutions of elevendimensional supergravity with no type-IIA origin.
One might also wonder if the same method can be applied for deformed CFTs based on group spaces. The easiest option would be to try to embed the λ-model on SU(2). This seems more challenging than the coset cases due to the non-vanishing NS two-form. In a past attempt to embed these to supergravity led to solutions within the type-II*-theory [27] in which the RR-fields are purely imaginary. Perhaps progress can be made by allowing warping between the different factors participating in the ansatz.
Another direction that would be interesting to follow, is to take advantage of these new solutions in order to bring the λ-deformed σ-models in contact with the ideas of the AdS/CFT correspondence. In the non-Abelian T-duality limit this opened new ways to understanding dual field theories, e.g. [39][40][41][42][43][44][45]. Unlike the non-Abelian Tduality case here we have one-parameter families of supergravity solutions, thus making the use of the AdS/CFT correspondence more appealing. In that respect, let's recall that one of the original motivations for constructing the λ-deformed σ-models was to understand [1] the global issues of supergravity backgrounds obtained by applying the non-Abelian T-duality transformation. The way that the non-Abelian T-duality limit is taken, the originally compact variables become non-compact, e.g. see Appendix B. Our results are important steps forward in the quest to understand global issues in non-Abelian T-duality, by using instead of the non-Abelian T-dual backgrounds the corresponding λ-deformed ones.
Finally, it will be very interesting to consider the plane-wave limit of our solutions, taken in such a way that there is still a left over λ-dependence in the resulting solution.
We intent to address some of these ideas in the near future.

Acknowledgments
We would like to thank H. Nastase and D.C. Thompson for useful remarks. The work of G.I. is supported by FAPESP grant 2016/08972-0 and 2014/18634-9.

A The type-IIA and type-IIB supergravities
Here we give a brief account of the equations of motion for the type-IIA and type-IIB supergravities.

A.1 The type-IIA supergravity
The type-IIA supergravity is described by the following ten-dimensional action in the string frame where the RR fields can be written in terms of the NS potentials B, C 1 , C 3 as From the last we can easily verify the Bianchi identities From the action (A.1) one can derive the equation of motion for the dilaton, which is The equation of motion for the metric leads to the Einstein equations 24 . (A.5) Finally, the equations of motion for the potentials give us the flux equations below (A.6)

A.2 The type-IIB supergravity
The action for the type-IIB supergravity in string frame is where the RR fields can be written in terms of the NS potentials B, C 0 , C 2 , C 4 as From the last we can easily verify the Bianchi identities From the action (A.7) one can derive the equation of motion for the dilaton which turns to be the same as in the type-IIA supergravity, i.e. it is given by the formula 24 . (A.10) Finally, the equations of motion for the potentials give us the flux equations below (A.11)

B The non-Abelian T-duality limit
In the cases where the deformation parameter can approach the identity one can consider the non-Abelian T-duality limit. For the geometries CS 2 λ , CAdS 2,λ and CH 2,λ this limit is obtained by considering large values of k after the following rescaling To see this in practice we will consider the non-Abelian T-duality limit of the first solution in section 3. where c 3 = r 1 8 − 1 and c 4 = r 1 8 + 1. Notice that we rescaled the dilaton appropriately in order to absorb powers of k.
The above background can be obtained by performing an non-Abelian T-duality transformation along S 2 and H 2 of the following type-IIA solution ds 2 = ds 2 M 2 + ds 2 M 4 + ds 2 S 2 + ds 2 H 2 , where c 3 and c 4 are given below eq. (B.2).
The non-Abelian T-duality limit can be applied in all solutions we found in the main text, provided the deformation parameter can approach identity for large values of k. However, we will not exhaust all the cases studied.