Spacetimes for \lambda-deformations

We examine a recently proposed class of integrable deformations to two-dimensional conformal field theories. These {\lambda}-deformations interpolate between a WZW model and the non-Abelian T-dual of a Principal Chiral Model on a group G or, between a G/H gauged WZW model and the non-Abelian T-dual of the geometric coset G/H. {\lambda}-deformations have been conjectured to represent quantum group q-deformations for the case where the deformation parameter is a root of unity. In this work we show how such deformations can be given an embedding as full string backgrounds whose target spaces satisfy the equations of type-II supergravity. One illustrative example is a deformation of the Sl(2,R)/U(1) black-hole CFT. A further example interpolates between the $\frac{SU(2)\times SU(2)}{SU(2)}\times\frac{SL(2,R)\times SL(2,R)}{SL(2,R)} \times U(1)^4$ gauged WZW model and the non-Abelian T-dual of $AdS_3\times S^3\times T^4$ supported with Ramond flux.


Introduction
The Wess-Zumino-Witten (WZW) model [1] and the Principal Chiral Model (PCM) [2] for a group manifold G provide two of the most well studied examples of twodimensional integrable systems and are of immense importance in many areas of theoretical and mathematical physics. Key to their simplicity is the underlying group structure; the WZW model is a current algebra Conformal Field Theory (CFT), whereas the equations of motion and Bianchi identities for the currents of the PCM can be combined into a Lax equation for a Lax connection from which an infinite number of conserved quantities can be deduced [3]. It is natural to ask whether it is possible to deform such theories whilst preserving integrability. It was observed in [4] that there exists a one-parameter deformation of the canonical Poisson structure of the PCM which defines two commuting Kac-Moody alegbras and preserves integrablity. Some years later, a first step to finding a Lagrangian description of these deformed theories was made in [5] for the case where the underlying group G = SU (2). Due to technical complexity involved, extending the direct approach of [5] to arbitrary groups seemed rather intractable.
However, very recently [6] the Lagrangian description of the deformed theories for any group G was provided. The approach of [6] was to consider a total action comprised of the sum of PCM parametrized by group elementg ∈ G together with a WZW parametrized by a g ∈ G. The combined action enjoys a G L × G R global symmetry of the PCM and a G L,cur × G R,cur current symmetry of the WZW. The critical step is to then gauge a subgroup of the global symmetry that acts as G L ong and G diag on g. The gauge symmetry can be fixed, for instance by settingg = 1, and the nonpropagating gauge fields may be integrated out. The result is a σ-model that depends on the level k of the WZW and the 'radius' κ 2 of the PCM in the combination (1.1) The opposite limit λ → 1 requires more care; it was shown in [6] that if the group element g is appropriately expanded near the identity the result is to produce a σmodel whose spacetime is the non-Abelian T-dual of the PCM on the group space G with respect to the G L action. So for λ near unity, one can also view this as a regulated version of non-Abelian T-dual resolving global ambiguities. 1 By either a direct calculation of the algebra of non-local charges or via an expansion of the Maillet-type Poisson brackets for the monodromy matrix, these theories can be seen to exhibit the whole Yangian symmetry for all values of the deformation 0 λ 1 [9].
With some modification, this general construction of integrable deformations can be applied to strings on cosets or symmetric spaces giving rise to integrable deformations of coset CFTs [6,10]. Furthermore it has been extended, with obvious applications to AdS/CFT, to Green-Schwarz superstrings on super-cosets [11]. In this later context one may make connections with other known deformations of superstrings in AdS 5 × S 5 . A sequence of works [12][13][14][15] have studied how the symmetries of the world sheet S-matrix may be deformed to a quantum group whilst still satisfying Smatrix axioms. The deformation is labeled by a parameter q and there are two cases to consider. First is q = e η ∈ R which corresponds to the "η-deformation" introduced from the string world sheet perspective in [16,17] building on earlier work in [18].
The η-deformation has been further developed in [19][20][21]. The second case is when q is a root of unity and it was conjectured in [11] that the λ-deformed theories described above give a world sheet realisation for this scenario.
A crucial question is then whether these integrable deformations are marginal and thus give rise to a target space that is a consistent string theory background. Working in a κ-fixed Green-Schwarz style action makes it technical to ascertain the full geometry of such a deformation. Within the context of just bosonic string theory the deformation of the WZW CFT fixed point is certainly not marginal according to the results of [22] (indeed the running of λ was calculated in [23,24] and shown to agree with that of the non-Abelian bosonized Thirring model computed in [25]). Further fermionic field content, coming from the RR sector of the type-II superstring, is needed so that one-loop conformal invariance is preserved for all values of λ. This then provides the motivation to the question we address here: can we embed the target spaces corresponding to integrable λ-deformations as full solutions of type-II supergravity?
We will show, with a number of worked examples, that this is indeed possible. It is, by no means, obvious that this will be the case; indeed a number of simplistic first attempts at this problem yielded no success. Our results come from two observations. First applying the above deformation to a compact group will give a λ dependent positive contribution to the one-loop dilaton beta-function. To counter balance this it seems necessary to perform a similar λ deformation in a non-compact group. Second, at the λ = 1 fixed point which we recall is the G L non-Abelian T-dual of a PCM, we can embed the geometry into a solution of supergravity by the inclusion of Ramond fields determined by group theoretic considerations in [26]. The close relation to non-Abelian T-duality suggests that the techniques of [26] may be generalised to find appropriate supporting Ramond fluxes for all values of λ.
In this paper we will explicitly consider examples of λ-deformations applied to : 1. AdS 3 × S 3 using the SU(2) × SL(2, IR) isometry of the group 2. AdS 2 × S 2 using the SU(2) × SL(2, IR) isometry of the maximal coset 3. AdS 3 × S 3 using the SU(2) × SU(2) × SL(2, IR) × SL(2, IR) isometry of the maximal coset The first example is somewhat simpler since it uses groups rather than cosets and we include it for didactic purposes. However, in this case one is forced to the conclusion that the RR fields must be imaginary and thus constitute a solution of type-II rather than type-II theories [27]. The reader who is dissatisfied with this state of affairs should quickly move to the second and third example which employ the coset generalisation of [6] -though a little more involved, these are real backgrounds of type-II theories.
The rest of the paper is as follows: We begin in section 2 by reviewing the general construction of [6] for group manifolds. In section 3 we provide the first of the examples listed above including a high level of detail and methodology. In section 4 we describe the generalisation to cosets and follow this with the remaining examples in section 5 and section 6.

λ-deformations for groups
In this section we present the background fields for the NS-sector of our models. In order to set up our notation and make our paper self-contained we first briefly review the necessary results and conventions.
Consider a general compact group G and a corresponding group element g parametrized by X µ , µ = 1, 2, . . . , dim(G). The right and left invariant Maurer-Cartan forms, as well as the orthogonal matrix (or adjoint action) relating them, are defined as The matrices T A obey [T A , T B ] = i f ABC T C and are normalized as Tr(T A T B ) = δ AB .
The PCM on the group manifold for an elementg ∈ G is and enjoys a G L × G R global symmetry. The WZW action for a group element g ∈ G is defined by where B is an extension such that ∂B = Σ. The approach of [6] was to consider the sum of the actions in (2.2) and (2.3) and to gauge a subgroup of the global symmetries that acts asg This is achieved by introducing a connection A = A A T A valued in the alebgra of G that transforms as We replace derivatives in the PCM with covariant derivatives defined as and replace the WZW with the G/G gauged WZW given by (2.7) The gauge symmetry can now be gauged fixed by settingg = 1 such that all that remains of the gauged PCM is a quadratic term in the gauge fields. The gauge fields, which are non-propagating are integrated out to result in the σ-model action [6] S k,λ (g) = S WZW,k (g) + k The σ-model of (2.8) is integrable as was proven in [6] by showing that the corresponding metric and antisymmetric tensor fields satisfy the algebraic constraints for integrability of [5] and [28]. A form of the action similar to (2.8) appeared before in [29].
We note that a more general class of actions can be obtained by retaining λ AB as a general constant matrix (or one that depends only on spectator fields and not the X µ ).
Such models are obtained by repeating the same procedure but replacing the inner product κ 2 δ AB occurring in the PCM of (2.2) with a general constant coupling matrix E AB to which λ AB is related with a straightforward extension of (2.9) [6,24]. It remains an open question as to which choices E AB can be made whilst retaining integrability.
With future possibilities in mind many of our derivations are done keeping λ AB as a general matrix.

Limit properties
In the limit of small λ AB the action (2.8) can be approximated by corresponding to the WZW theory perturbed by the current bilinear J A + J B − with arbitrary coupling matrix λ AB . The first two terms define the bosonized anisotropic non-Abelian Thirring model in analogy with the non-Abelian Thirring model [30,31].
Hence, it is reasonable to expect that (2.8) provides an effective all loop action for the bosonized non-Abelian Thirring model. Based on studies of the RG flow and symmetry considerations this has been shown for λ AB = λδ AB in [23] and for general λ AB in [24]. The fact that the model is driven away from the conformal point by the current bilinear in (2.10) is tied to the non-Abelian nature of the group. Had the current bilinear been restricted to the Cartan torus, as for instance in [32], the σ-model would have remained conformal [22]. We remark that other interesting types of marginal deformations have included the so-called asymmetric deformations of the form d 2 zJJ G in whichJ G corresponds to some other U(1) outside the chiral ring of the WZW [33,34].
Heterotic embeddings of these asymmetric deformations were considered in [35].
What motivated the present paper is the behaviour of (2.8) for k 1 and λ → 1 [6].
Then expanding the matrix and group elements near the identity we have that In this limit the action (2.8) becomes which is the the non-Abelian T-dual with respect to the G L action of the σ-model given by the PCM action with general coupling matrix E AB .

Towards a supergravity embedding
The purpose of this paper is to embed models for which the metric and antisymmetric tensor of the NS sector are provided by (2.8) to type-II supergravity. This will be done by supporting these fields with a dilaton as well as with appropriate RR fluxes. The dilaton factor is obtained from integrating out the gauge fields in a path integral and is given by where Φ 0 is the dilaton of the original theory in which the PCM is a part of. Let us now restrict to the simplest cases when λ AB = λδ AB . The target space metric can be read from the σ-model (2.8) and can be conveniently expressed using frame fields two frames e A + and e A − given by [6,23] Both these frame field define the same geometry and are related according to a local frame rotation e − = Λe + given by In the spirit of [26] we will need the orthogonal transformation in the spinor representation Ω found from where Γ A are the ten-dimensional Γ-matrices. An ansatz for the RR fields, completely determined by the group theory, is given by allowing this Ω matrix to act by Clifford multiplication on the RR fields of the original model in much the same as it does for both Abelian and non-Abelian T-duality. We consider the RR sector in the democratic formalism that incorporates fluxes and their Hodge duals equally specified by poly- from which one obtains bi-spinors / F by contracting the constituent p-forms with panti-symmetrised gamma matrices. Then the ansatz we propose for the RR sector is where on the right hand side we have the bispinor / F 0 formed from the RR fields supporting the PCM of (2.2) when embedded into a supergravity background. The RR fields across the deformation are obtained from the left hand side of this relation upon replacing the anti-symmetrised gamma matrices with the wedge of the corresponding frame fields e A given in (2.15). We include in this ansatz a possible multiplicative λ dependent constant coefficient µ(λ) which is, of course, related to the normalisation employed for the dilaton. This constant should necessarily vanish in the limit λ → 0 since then the background, being a CFT, should consist of purely NS fields. We believe this to be the only consistent ansatz for the RR fields which is compatible with the group theoretic structure of the problem and that agrees with the result in the non- Abelian T-dual limit. A first principles derivation of the form of the RR fields should eventually be established using e.g. a Green Schwarz or pure-spinor formalism.

Integrable deformation based on SU(2) and SL(2, IR)
With the aim of constructing an integrable deformation of AdS 3 × S 3 × T 4 we recap the application of the above construction for the case of the SU(2) group manifold given in [6] for the NS sector. We then provide the analytic continuation that gives an analogous result for SL(2, IR) before combining these to give a full supergravity embedding. To prevent feelings of resentment on behalf of the reader, let us state the outcome upfront: in this case the result will be a background of Type-IIB supergravity with imaginary fluxes. To obtain real backgrounds of type-II supergravity we will consider the generalisation of the these λ deformations to coset spaces in later sections.
The Lorentz rotation matrix define by (2.16) is given by and its spinor representation (2.17) is given by In the limit λ → 0 we obtain the metric and the antisymmetric tensor for the SU (2) WZW model with normalization such that R ij = 2g ij . In this case the rotation matrix in the spinor representation (3.6) becomes The fact that it is not the identity is in agreement with the fact that the frames become in that limit e A + = −R A and e A − = L A . This implies that in this limit Λ AB = −D AB , consistent with (2.16).
The non-Abelian limit of the SU(2) PCM is obtained (setting κ = 1) by letting This limiting procedure gives In addition the Lorentz transformation in the spinor representation (3.6) becomes where v = rn. These expressions correspond the non-Abelian T-dual of the SU (2) PCM which in fact has been embedded in supergravity. It was shown in [26] that when supported by appropriate flux fields it is a solution of massive IIA-supergravity and that it represent the non-Abelian T-dual of the background corresponding to the near horizon of the D1-D5 brane system.

Embedding to supergravity
We consider performing the above procedure on SL(2, IR) × SU(2) × T 4 . i.e. we look for an integrable deformation to the model whose target space is AdS 3 × S 3 × T 4 supported in type-IIB supergravity by an RR three-form flux field where the indices 0, 1, 2 and 3, 4, 5 run along the AdS 3 and S 3 directions. 2 Turning to the deformation we note that this will be non-trivial for the SU(2) and SL(2, IR) factors as presented in the previous section. There will be no deformation for T 4 . The metric and the NS antisymmetric tensor are given by and 14) with the obvious notation for the various terms corresponding to (3.3) and (3.11). The geometry will supported by a dilaton field given by The frame we will use are given by so that the metric is then Note that these simple frame fields are not the same as those defined by 2.15.
Combining the SU(2) result of (3.7) with its SL(2, R) counterpart, the Ω matrix relating left and right moving frames, in this basis, has the form Then we use our proposal for the RR fields described by eq. (2.19) in this case with 19) obtained in this way has components 2 cos α cosα(e 012 + e 345 ) − (1 + λ) 2 sin α sinα(e 045 + e 123 ) , Then, the Bianchi identities and equations of motion of the type-II supergravity are solved. However the fluxes are pure imaginary meaning that this should be interpreted in the context of the type-II theory described in [27]. This arises because the Ω matrix involves a Γ 0 and therefore has similar features to performing a time-like T-duality.
In the λ → 0 limit one immediately recovers the geometry AdS 3 × S 3 × T 4 supported by NS flux and in the λ → 1 limit we find the non-Abelian T-dual of AdS 3 × S 3 × T 4 supported by RR flux.

λ-deformations for cosets
In this section we will let {T A } be the generators of G, {T a } be those of some subgroup H ⊂ G and {T α } the remaining generators for the coset G/H. We will also need a second subgroup K ⊂ G and denote its generators by {T m }.
The σ-model on the geometric coset G/H is given by We introduce a connection that transforms as and repeat the analogous steps in the gauging, replacing derivatives in the PCM to covariant ones and replacing the WZW for G with a G/K WZW model, giving the gauged action where Integrating out the A ± 's gives in which theL andD are the left-invariant forms and adjoint matrix for the PCM group elementg and the contracted indices are running over the gauge group K.
Let us focus our attention on the case where K = G, i.e. we gauge the entire global G-symmetry. In that case we may partially fix the gauge by settingg = 1. This will however leave a residual H gauge symmetry that will be used to fix dim(H) degrees of freedom in g. Then the action becomes where For the case where the coset is a symmetric space, 3 integrability of this theory was proved in [10] using the gauged WZW-like origin of the construction of the action. We 3 A symmetric coset G/H is one for which the algebra g of G admits a Z 2 grading g = g (0) ⊕ g (1) where g (0) = h is the algebra of H and with [g (0) , g (0) ] ⊂ g (0) , [g (0) , g (1) ] ⊂ g (1) , [g (1) , g (1) ] ⊂ g (0) . also note that the action (4.7) with the given expression for M AB arises if we simply set in (2.8) the block diagonal part of the matrix λ corresponding to the subgroup H to unity.
The λ → 1 limit, together with appropriate rescalings results in the non-Abelian T-dual of the geometric coset as constructed in [36]. On the other hand the λ → 0 limit of (4.7) gives which is the σ-model corresponding to the gauged WZW model for G/H as expected.
The leading correction that drives the model away from the CFT point is proportional to where A 0 ± are the solution for the gauge fields arising from integrating them out in the action after setting λ = 0. Explicitly This term can be written at a bilinear in the classical parafermions [37,38] and therefore it has a precise CFT interpretation. For the perturbation on SU(2)/U(1) the above considerations were made explicit in [6].
One generalisation of this construction is given by replacing the inner product δ αβ entering in (4.1) with general metric E αβ , however the G invariance condition should still be obeyed. 4 A second generalisation is to consider rather than a single WZW model, multiple WZW factors each at different levels -we will give further details of this in the appendix.
To obtain the supergravity embedding we need to generalise the ansatz for RR fields described in (2.19) from the group to the coset. Fortunately this has been done already in the case of non-Abelian duality (the λ → 1 limit of the present construction) in [36] and is easily extrapolated to the case at hand. If we let X α be dim(G) − dim(H) local coordinates for the σ model (4.7) then frame fields are obtained by defining (4.13) These are related by a Lorentz rotation Λ = N + N −1 − from which the corresponding spinor matrix Ω can be obtained and then the rule in (2.19) can be directly applied. Of course, the exact form of these objects will depend on the way the residual symmetry is gauge fixed.

Deforming the SU(2)/U(1) exact CFT
We follow this procedure and work out the action (4.7) for the case G = SU(2) and H = U(1). We parametrize the group element as g = e i(φ 1 −φ 2 )σ 3 /2 e iωσ 2 e i(φ 1 +φ 2 )σ 3 /2 . (5.1) In both the gauged WZW and its deformation given by eq. (4.7), we can fix the U(1) gauge redundancy by setting φ 2 = 0 (one finds in an explicit calculation that φ 2 enters the action only as a surface term). The metric of the deformed σ-model (4.7) is then given by and zero antisymmetric tensor, where the parameter λ is defined in (2.9) and have renamed φ 1 by φ. The factor that will contribute to the dilaton in the supergravity embedding that will shall do is determined, up to a constant piece, as e −2Φ = sin 2 ω.
These expressions coincide with those found in [6].
For λ 1, i.e. k κ 2 , the dominant term is that corresponding to the exact SU(2)/U(1) coset CFT [37]. It can moreover be shown [6] that the extra term is a parafermion bilinear which corresponds to a relevant perturbation since the parafermions have conformal dimension 1 − 1/k. Hence these parafermions drive the σ-model away from the CFT point in accordance with our general discussion above. This perturbation has been shown to be integrable, massive and argued that in the k → ∞ limit the model flows under the renormalization group to the O(3) σ-model [40]. This is consistent with the fact that SU(2)/U(1) is a symmetric coset space.
Performing a further analytic continuation in (5.4) as we obtain a σ-model with metric zero antisymmetric tensor and the contribution to the dilaton factor e −2Φ = sinh 2 ρ.
This background for λ = 0 corresponds to geometry of the exact SL(2, R)/U(1)) coset CFT. It was globally extended and interpreted as a two-dimensional black hole in [41]. 5 We shall return to the black hole interpretation shortly.

Embedding to supergravity
Consider the ten-dimensional metric arising from combining (5.2) and (5.6) with the six-dimensional flat metric on the T 6 In addition, by combining the corresponding dilaton factor we have for the dilaton The antisymmetric tensor vanishes. In order to satisfy the supergravity equations of motion we need to turn on flux fields. To present them we first define the frames We will also denote by J 2 the Kahler form and by J 3 the real part of the complex differential form of type (3, 0) in R 6 . In a convenient basis we have that (5.10) The NS sector fields can be supported in a full supergravity solution either within type-IIB or within type-IIA supergravity.
Within type-IIB we have the five-form RR flux IIB : Within type-IIA with the two-and four-form RR fluxes are IIA : (5.12) These are real forms, and are solutions of type-II supergravity. The form of the RR fields may be established using the action of Ω = Γ 1 Γ 2 on either the type-IIB or type-IIA embedding of the AdS 2 × S 2 PCM (see [42] for discussion of the GS action for the superstring in this background and its integrability).

The non-Abelian T-dual limit and the near singularity region
Then above geometry is singular for ρ = 0 and for ω = 0 where, for instance, the scalar curvature blows up. We are interested in magnifying the geometry around these points. It turns out that we have to zoom in also at specific sections for the variables t and φ. Indeed after letting (5.13) we obtain that and for the dilaton Compared with (5.8) we have shifted the dilaton so that e Φ gets multiplied by 4k 2 and Φ remains finite in the above limit. That implies that the fluxes are also multiplied by 4k 2 so that the Einstein equation of motion is satisfied. The result for this limiting procedure for the fluxes of type-II supergravity gives IIB : and IIA : In conclusion the non-Abelian T-dual in this case provides the geometry near the singularities when the parameter λ tends to unity.

Global structure
In this section we will study the geometry presented here in the context of the twodimensional black hole solution of [41]. First let us just consider the two-dimensional metric (5.6) and obtain its conformally flat form by making a coordinate transforma- Then (5.6) becomes For λ = 0 this coincides with the global metric found in eq. (28) of [41]. At face value the effect of the λ-deformation is to modify the location of the singularity defined by f (u, v) = 0. This is illustrated in the Penrose diagram of fig. 1. A peculiar feature is that the deformed black-hole singularity curve is no longer a horizontal line extending to null infinities I ± but instead "bends" back on itself in the Penrose diagram to close off into a tear drop shaped ending at future time-like infinity. As a consequence it appears that some portion of the singularity is not protected by a horizon. This is clearly a puzzling feature and warrants a further study. One likely resolution is that this portion of the space time should be excised. The study of this metric is made somewhat more difficult since for λ = 0, 1, it does not admit any isometries as can be seen by direct inspection of the Killing equations. It is interesting to consider the limit as λ → 1 where in the Penrose diagram the singularities curves degenerate into vertical lines. As stated above, if the coordinates are also scaled in this limit one obtains a non-Abelian T-dual geometry that probes the singularity. Instead let us take a different limit under which the coordinates are not scaled. We simply let λ → 1 by sending k → ∞ with κ 2 fixed. Then the metric (5.19) becomes so that an AdS 2 geometry of size κ 2 emerges. Hence, taking this limit in this way one (necessarily) shows that the singularity is removed.
We now turn to a second question: can this full globally extended geometry be supported in type II supergravity? In fact, the answer is no and the reason can be seen even in the context of the undeformed black-hole for λ = 0. In this case the singularity lies on uv = 1 and in crossing the singularity (going from region V to II in the terminology of [41]) the dilaton Φ = − 1 2 ln(1 − uv) picks up a shift of iπ. Of course, when λ = 0 the geometry does not requires any RR flux and so the additive shift to the dilaton can be, to an extent, ignored at the level of solving the supergravity equations.
However when λ = 0, if one wishes to keep the dilaton real, this will necessitate allowing the RR fluxes to become imaginary. Thus there is no global extension that can cover both asymptotic regions whilst keep the solution in type II. 6 An AdS 3 × S 3 coset deformation 6

.1 Deforming the SO(4)/SO(3) coset
We parametrize the group SO(4) element g in the SU(2) × SU(2) decomposition with with the usual determinant constraints. A nice gauge choice that completely fixes the residual H = SU(2) symmetry is and H invariant combinations of the remaining coordinates are given by which are simply | α|, | β| and α · β after gauge fixing (in what follows we prefer to use the invariants α 0 and β 0 rather than α and β since they lead to marginally more concise expressions).
Following the procedure described in section 4 results in an effective target space geometry and The NS two-form potential can be chosen to be zero and the contribution to the dilaton factor is given (up to a constant shift) by e −2Φ = Λ. The contribution to the dilaton beta function equation turns out to be 6 k 1 + λ 2 1 − λ 2 which is just a constant. This is an important consistency check that there is a possibility to embed this into a full supergravity solution by combining with another model that may contribute exactly the opposite.
Note the ranges of the coordinates ensure that on this domain the function Λ 0. These are important to keep in mind since they ensure that the metric has positive signature (that is to say if one uses the metric (6.4) blindly and goes beyond the range of these coordinates, it does not remain positive). There is a manifest Z 2 symmetry swapping α 0 and β 0 just corresponds to a switching of the two SU(2) factors of the SO(4) decomposition.
In the limit λ → 0 the geometry is precisely that corresponding to the G/H gauged WZW model written explicitly in this parametrisation in (4.12) (for r = 1) of [43].
The limit λ → 1 needs to be taken with some care. Defining and taking the limit k → ∞ one recovers (setting κ = √ 2) the metric of the non- Abelian T-dual of G/H = S 3 with respect to G = SO(4) obtained in (4.20) of [36].
The frame fields for the geometry are defined by eq. (4.13) and are given 6 by (we start the frame numbering at 3 for reasons that become obvious momentarily): where the angle ψ is defined through Because the plus and minus frames differ only by reflection in the e 4 and e 5 directions it is evident that the corresponding Lorentz rotation in the spinor representation will have the form Ω ∼ Γ 4 Γ 5 .
There may be several different ways to perform an analytic continuation and these are given essentially by changing the domains of α 0 , β 0 and γ in eq. (6.11). The continuation we seek should be such that the frame e 3 defined in (6.9) becomes time-like and e 4 and e 5 remain space-like. With such a choice, the Ω matrix will contain reflections only in space-like directions and the resulting RR fields will remain real (other choices could lead to solutions of type II ).
Letα 0 ,β 0 andγ be the coordinates for the analytically continued geometry which is given by the same metric as (6.4) but with the replacement of k → −k and domains We define the functionΛ which is positive over its domain and the angleψ through The analytically continued versions of the frame fields in (6.9) are given by where the flat metric has signature (− + +). The Lorentz rotation spinor matrix is this space is given byΩ ∼ Γ 1 Γ 2 .

Supergravity embedding
We introduce frame fields in the T 4 directions e i = dx i , i = 6 . . . 9 such that the full ten-dimensional metric is ds 2 = η ij e i e j with e 0 , e 1 , e 2 given by (6.14) and e 3 , e 4 e 5 by (6.9).
The dilaton is given by e −2Φ = ΛΛ and the NS two-form potential can be taken to be zero.
It is evident that because we have sent k → −k in the analytic continuation, the contributions to the dilaton equation from the 3, 4, 5 directions will cancel exactly with those in the 0, 1, 2 directions. To satisfy the Einstein's equations we need simply an RR three form e 045 + e 123 , (6.15) which is in keeping with the ansatz for RR fluxes given by (2.19) when the spinor rotation matrix is the combination of that in the 3, 4, 5 and 0, 1, 2 directions i.e. Ω ∼ Γ 12 Γ 45 . This flux also solves its Bianchi identity and equation of motion. One can see that all components of F 3 are extended in either or both the non-compact directionsα 0 andβ 0 , for this reason there seems to be no well defined (i.e. finite) charge associated to this flux. A final comment is that since this background only has an F 3 active, the S-dual will have purely NS flux which may be useful for further investigation.

Conclusions and discussion
In this work we have demonstrated very explicitly that the NS of backgrounds corresponding to integrable λ-deformations can be upgraded to full solutions of supergravity supported by appropriate Ramond fluxes. This gives very convincing support that the λ-deformation of a supercoset σ-model (such as that for the AdS 5 × S 5 string) will be an exactly marginal deformation and correspond to the q root of unity quantum deformations as postulated in [11]. Extracting the supergravity background in the specific case of AdS 5 × S 5 will be the subject of future work.
The examples presented within preserve no isometries and are thus very unlikely to be supersymmetric. Also the defomations act equivalently in the AdS and sphere parts of spacetime. It would be extremely interesting if one finds a way to avoid either of these features; preserving even N = 1 supersymmetry would be desirable and acting just in the sphere directions would result in a deformation to the geometry for which the dual field theory would remain conformal. If this is the case one might be able to understand more clearly the consequences of these λ-deformations for holography.
In this work we have been implicitly always thinking of the λ-deformation as being a deformation away from the CFT defined by a (gauged)-WZW model. This is reflected in the fact that our gauge fixing choice always involved fixing the group element defining the PCMg = 1. One consequence of this is that geometrically it is hard in general to see the PCM geometry emerging (it is the non-Abelian T-dual of the PCM that is recovered in a limit in which coordinates are also scaled as λ → 1). Prior to gauge fixing the PCM and the WZW model are treated on equal footings. One may thus like to reconsider the system instead as deformation away from the PCM point.
One might expect that this can be done by adopting a different gauge fixing choice in which the group element defining the PCM is left untouched. In support of this, we saw for the case of Sl(2)/U(1) that by making an appropriate coordinate transformation an AdS 2 space did indeed emerge in the λ → 1 limit. There are however two difficulties with this, firstly it is not possible to completely fix the gauge symmetry in this way and secondly that experience dictates that different gauge fixing choices do no more than generating diffeomorphisms of the target space. This issue warrants further consideration.
We may easily generalize the construction for the case of direct groups. Consider a WZW model on G k 1 × G k 2 × · · · × G k N and we gauge the G action of each of these factors to give a copies of a G/G WZW at all the levels. In addition we consider a PCM on a coset ∏ N i=1 G i /H where H is a diagonal subgroup of G. We find it convenient to work with a block diagonal realization of this set up in which in introduce the composite indexĀ = Ai . The subgroup H is generated by t a = (t a , t a , . . . , t a ) and then the coset is comprised by all generators TĀ, except these. Then the group element is g = (g 1 , g 2 , . . . , g N ), where g i ∈ G i and the gauged WZW is given by the usual formulas but with the modification that the inner product is normalised such that k i appears for each block. As before we consider the case in which group element of the PCM on the coset is fixed tog = 1. Then all the formulae of the previous section can be applied directly.
Then one finds an action after gauge fixing the PCM model given by where k = k 1 + k 2 , s i = k i /k and We make the gauge fixing exactly as in section 6 and find after integrating out the gauge fields a σ-model on a target space with For the case of r = 1 i.e. equal levels, this metric reduces to that of Section 6 and for unequal levels but with the deformation turned off (i.e. λ = 0) they give the geometry [43], i.e. eq. (4.12) (see also [44]) . There is a also a manifest Z 2 invariance under exchange of α ↔ β and k 1 ↔ k 2 .
Frame fields for this geometry are found using the general formula (4.13) and are given by where the angle ψ is given by (6.10). The NS two form potential can be chosen to be zero and the dilaton is given (up to a constant shift) by e −2Φ = Λ. The dilaton beta function equation gives a constant as is required for this to be embedded into a full supergravity solution.

IIB embedding
The SL(2) k 1 × SL(2) k 2 result can be obtained by analytic continuation exactly as described in section 6.1.1; we change the domain of α 0 and β 0 and simultaneously flip the sign of the levels as k i → −k i . Let us denote the corresponding frame fields obtained in this way as e 0 , e 1 and e 3 . These will be given by the expressions in (A. 8) with all quantities replaced by the tilded counterparts and with the arguments in the square roots also flipping signs, i.e. √α 2 − 1. A ten-dimensional metric is completely by appending a T 4 to the six-dimensional space obtained from the SL(2, IR) and SU (2) constructions. The dilaton is then given by whereΛ is as defined in (6.12). The dilaton supergravity equation is solved by construction due to the cancelation between SL(2) and SU(2) factors. The Einstein equation is solved when the geometry is supported by three-form where µ 2 = 1 k 1 r 4 Z 3 N 3 1 + r 4 (1 − Z 2 N 2 ) 2 − 2r 2 (1 + Z 2 N 2 ) , N −1 = (1 + r)(1 − λ) . (A.11) One may check that the constant µ is indeed invariant under the above Z 2 symmetry, albeit not manifestly. This flux solves its Bianchi identity and equation of motion.