Supergravity background of the lambda-deformed AdS_3 x S^3 supercoset

We construct the solution of type IIB supergravity describing the integrable lambda-deformation of the AdS_3 x S^3 supercoset. While the geometry corresponding to the deformation of the bosonic coset has been found in the past, our background is more natural for studying superstrings, and several interesting features distinguish our solution from its bosonic counterpart. We also report progress towards constructing the lambda-deformation of the AdS_5 x S^5 supercoset.


Introduction
Integrability is a remarkable property, which has led to a very impressive progress in understanding of string theory over the last two decades (see [1] for review). While initially integrability was discovered for isolated models, such as strings on AdS p ×S q [2], later larger classes of integrable backgrounds have been constructed by introducing deformations parameterized by continuous variables. The first example of such family, known as beta deformation [3], has been found long time ago [4], but recently two new powerful tools for constructing integrable string theories have emerged. One of them originated from studies of the Yang-Baxter sigma models [5,6,7], and it culminated in construction of new integrable string theories, which became known as η-deformations [8,9,10,11,12]. The second approach originated from the desire to relate two classes of solvable sigma models, the Wess-Zumino-Witten [13] and the Principal Chiral [14] models, and it culminated in the discovery of a one-parameter family of integrable conformal field theories, which has WZW and PCM as its endpoints [15,16,17] 1 . This connection becomes especially interesting when the PCM point represents a string theory on AdS p ×S q space, and the corresponding families, which became known as λ-deformations, have been subjects of recent investigations [19,20,21,22].
A close connection between the η and λ deformations has been demonstrated in [20]. In this article we study the λ-deformation for AdS 3 ×S 3 and AdS 5 ×S 5 .
While the metrics for the λ-deformation of AdS p ×S q have been constructed in [17,19], the issue of the fluxes supporting these geometries has not been fully resolved. Although the metric for the deformation can be uniquely constructed starting from the corresponding coset, there are two distinct prescriptions for the dilaton: one is based on a bosonic coset [17], and the other one uses its supersymmetric version [16]. In the first case the deformations for all AdS p ×S q have been constructed in a series of papers [17,19], while in the second case, which is more natural for describing superstrings, only the result for AdS 2 ×S 2 is known [22]. In this article we construct the geometry describing the λ-deformed AdS 3 ×S 3 supercoset and report progress towards finding the deformed AdS 5 ×S 5 solution.
This paper has the following organization. In section 2 we review the procedure for constructing the λ-deformation, which will be used in the rest of the paper. In section 3 we use this procedure to construct the metric and the dilaton for the deformed AdS 3 ×S 3 , but unfortunately construction of Ramond-Ramond fluxes requires a separate analysis. In section 3.3 we determine these fluxes by solving supergravity equations, and in sections 3.4-3.5 we find some interesting connections between the new background and solutions which exist in the literature. Section 4 reports progress towards constructing the λ-deformation for super-coset describing strings on AdS 5 ×S 5 . Specifically, we determine the metric and the dilaton, but unfortunately we were not able to compute the Ramond-Ramond fluxes. The λ-deformation of AdS 2 ×S 2 constructed in [22] is reviewed in Appendix A, and its comparison with higher dimensional cases is performed throughout the article.

Brief review of the λ-deformation
We begin with reviewing the procedure for constructing the NS-NS fields for the λ-deformed cosets. Such deformation belongs to a general class of two-dimensional integrable systems with equations of motion in the form ∂ µ I µ = 0, ∂ µ I ν − ∂ ν I µ + [I µ , I ν ] = 0, (2.1) where currents I µ take values in a semi-simple Lie algebra. Integrability of this system can be demonstrated by writing it as a zero-curvature condition for a linear problem 2 : Two well-known examples of the integrable systems described by equations (2.1) are the Principal Chiral Model (PCM) [14] and the Wess-Zumino-Witten model [13] for a group G: (2.4) and the λ-deformation interpolates between these systems. This deformation utilizes two important symmetries of (2.3) and (2.4): the global G L × G R symmetry of the PCM and the G L,cur × G R,cur symmetry of the current algebra of the WZW.
λ-deformation for groups Let us review the construction introduced in [15], which allows one to interpolate between the systems (2.3) and (2.4) while preserving integrability. To find such λ deformation, one adds the PCM and WZW models (2.3), (2.4) for the same group G and gauges the G L × G diag,cur subgroup of global symmetries. This is accomplished by modifying the derivative in the PCM as 5) and by gauging the resulting WZW model. Integrating out the gauge fields A ± , one arrives at the final action [15] 3 Deformation (2.6) interpolates between the PCM (λ = 1) and the WZW model (λ = 0) while preserving integrability [15].
To extract the gravitational background describing the deformation, one rewrites (2.6) as and compares the result with the action of the sigma model This leads to the metric and to the Kalb-Ramond field: where B 0 is a Kalb-Ramond field of an undeformed WZW model with the field strength Recalling the definition of M and the relation D T D = 1, one can rewrite the metric in terms of convenient frames: Expressions for D and L are given in (2.6).
Dilaton for the λ-deformation Although extraction of the metric and the Kalb-Ramond field for the lambda deformation is rather straightforward, the procedure for calculating the dilaton is controversial. The original proposal of [15] suggested the expression 12) which can be written as , (2.13) where the determinant is taken in the algebra. In [16] it was argued that for supergroups and supercosets an alternative expression is more appropriate: Here the superdeterminant is computed in the full superalgebraf . The difference between (2.13) and (2.14) originates from difference in the gauge fields which have been integrated out.
Recalling that an element of a superalgebra can be written as where (A, D) are even and (B, C) are odd blocks [23,24], the expression (2.14) becomes . (2.16) Heref 0 andf 2 refer to the even subspaces A and D, whilef 1 andf 3 refer to the odd subspaces B and C. In this article we will refer to (2.13) (which is equal to the denominator of (2.16)) as the bosonic prescription, and the numerator of (2.16) would be called the fermionic contribution to the dilaton.

λ-deformation for cosets
The extension of the λ-deformation to cosets G/H is presented in [17]. Separating the generators T A of G into T a corresponding to H ⊂ G and T α corresponding to the coset G/H, one finds the metric The expression for the dilaton is given by the generalizations of (2.13) and (2.16) [17,16,20]: .

(2.19)
Here P λ is a projector which separates the generators of H and the coset G/H, and it has the form [16] Here P 2 is the projector in the bosonic sector, which can be written as The action of fermionic projectors P 1 and P 3 is evaluated on a case-by-case basis, and we will address this question in the sections 3 and 4. Notice that P 2 has already appeared in the matrix M defined in (2.17): We conclude this discussion with reviewing a very interesting observation made in [19]: factorization of the λ-dependence in the determinant of M AB . This technical simplification becomes especially useful in the AdS 5 ×S 5 case, where one has to deal with large matrices. Following [19], we write M AB as a product of two block-triangular matrices: As demonstrated in [19], matrix P has eigenvalues λ −2 ± 1, so the coordinate dependence of the bosonic dilaton (2.18) comes from det A. We find that direct evaluation of the determinant of M is easier than construction of P, but our final results confirm that the coordinate dependence of det M is inherited from det A.

Deformation of AdS ×S 3
Let us apply the procedure reviewed in the last section to AdS 3 ×S 3 . The bosonic part of the sigma model is described by a product of two cosets 1) and the full string theory is described by a super-coset [25] 4 P SU(1, 1|2) 2 SU(1, 1) × SU (2) . (3.2) In section 3.1 we construct the metric and the bosonic contribution to the dilaton for the cosets (3.1), (3.2). While this will give the full answer for (3.1), the dilaton for the supercoset (3.2) also receives a fermionic contribution, which will be evaluated in section 3.2. In section 3.3 we construct the Ramond-Ramond fluxes supporting the λ-deformed supercoset (3.2), and properties of the new geometries are discussed in sections 3.4 and 3.5.

Metric and the bosonic dilaton
The metric is constructed using the bosonic coset (3.1), then S 3 and AdS 3 decouple, and they can be studied separately. We begin with analyzing the sphere, and deformation of AdS 3 can be found by performing an analytic continuation. Deformation of the sphere.
To describe the coset SU (2) l ×SU (2)r SU (2) diag , we use the algebraic parameterization introduced in [17]: where variables α k , β k are subject to the determinant constraints (α k ) 2 = 1, (β k ) 2 = 1. (3.4) Gauging of the diagonal part of SU(2) l × SU(2) r makes the description (3.3) redundant, and to remove the unphysical degrees of freedom we impose a convenient gauge, which was also used in [17]. Acting on g l as g l → h −1 g l h, we can set α 2 = α 3 = 0, then the remaining U(1) transformations h = exp[ixσ 1 ] can be used to set β 3 = 0: Following [17] we introduce a convenient coordinate γ and solve the constraints (3.4) to express all remaining components of g 1 and g 2 in terms of (α 0 , β 0 , γ): (3.6) To simplify notation, we will drop the subscripts of α 0 and β 0 . The elements of SU(2) l × SU(2) r can be represented as block-diagonal 4 × 4 matrices: then the generators corresponding to the subgroup H and to the coset G/H can be written in terms of the Pauli matrices 5 .
Substitution of (3.7)-(3.8), where g l , g r are given by (3.3), (3.5), into the defining relations (2.17) 6 leads to the metric [17] Deformation of AdS 3 . The deformation of the AdS 3 is constructed by performing an analytic continuation of (3.9). The defining relation for g ∈ SU(1, 1) l × SU(1, 1) r is and it can be enforced by starting with an element of SU(2) l × SU(2) r , renaming the coordinates as and changing their range from To view this transition as a proper analytic continuation, one can introduce alternative coordinates (a, b, γ) as (3.14) Then transition from (3.12) to (3.13) amounts to a continuation from real to imaginary (a, b). This changes the signature from (+++) to (−−+), and by changing the sign of k we recover (+ + −). Analytic continuation (3.11) along with the replacement k → −k gives the metric for the λ-deformed AdS 3 Dilaton and RR fields for the bosonic coset.
The deformation of AdS 3 ×S 3 constructed in [17] is described by the metric {(3.9), (3.15)} and the dilaton corresponding to the bosonic prescription (2.18): (3.16) This article also listed the corresponding Ramond-Ramond fields: However, as argued in [16,20,22], the dilaton (2.19) for the supercoset is more natural for describing superstrings, and in the next subsection we will find the appropriate expression and construct the corresponding Ramond-Ramond fluxes.

Fermionic contribution to the dilaton
In this subsection we will construct the dilaton for the supercoset (3.2) using the prescription (2.19). Before focusing on (3.2), we will outline the procedure for applying (2.19) to a supermatrix (2.15) constructed from extending an algebra of the bosonic coset ( A supersymmetric extension of av algebra g 1 × g 2 has the form 18) and to find the supercoset, we should fix the gauge corresponding to subalgebras h 1 , h 2 and evaluate the relevant projectors P λ . This can be done in five steps: 1. Find an automorphism J 1 of algebra g 1 which leaves invariant only the elements of h 1 . In other words, g ∈ g 1 satisfies the condition 3.19) if and only if g ∈ h 1 . Automorphism J 2 in g 2 is defined in a similar way.
2. Construct an automorphism of the super-algebra as 20) and project out the elements M which are left invariant under such automorphism 7 : For bosonic generators this reduces to (3.19) and its counterpart for g 2 , while the projections for the fermionic matrices are 3. Construct the projector P 2 acting on bosonic generators by requiring that [1 − P 2 ] kills the same elements as (3.19) and its counterpart with J 2 . Such P 2 projects on the bosonic part of the supercoset.
4. Construct projector P 3 acting on fermionic generators by requiring that P 3 keeps the same elements as (3.22). The fermionic projector complementary to P 3 is P 1 = 1 − P 3 .

5.
Construct the projector P λ using the definition (2.20). Substitution of this expression into (2.19) or (2.18) and evaluation of the resulting determinant gives the dilaton for the (super)coset.
To apply this procedure to the AdS 3 × S 3 coset (3.2), we observe that g 1 represents the algebra of (3.7), g ∈ g 1 : g = g l 0 0 g r , g l ∈ su (2), g r ∈ su (2), (3.23) while the elements of h 1 = su(2) diag have the form (2). (3.24) This leads to two options for the automorphism J 1 : Expression for J 2 is constructed in a similar way, and putting these results together, we find two options for the automorphism P: The fermionic generators of P SU(1, 1|2) × P SU(1, 1|2) appearing in (3.18) obey the relation with Σ 4 given in (3.10), and projection (3.21) leads to further constraints. It is convenient to decouple f 12 and f 21 by working with holomorphic and anti-holomorphic coordinates.
Relations (3.22) isolate 4 + 4 components of f 12 and f 21 killed by P 1 , while P 3 kills the complementary 4 + 4 components 8 . Extraction of (P 1 , P 2 , P 3 ), construction of P λ via (2.20), and evaluation of superdeterminant (2.19) gives the same dilaton for both choices (3.26): We conclude this section by analyzing the symmetries of the metric {(3.9), (3.15)} and the dilaton (3.28), which will be used for constructing the Ramond-Ramond fluxes. First, it is clear that neither the metric nor the dilaton has continuous symmetries, but all NS-NS fluxes are invariant under several discrete transformations: These symmetries will be used in the next section to select a natural solution for the RR field C 2 .

Ramond-Ramond fluxes
Although the Ramond-Ramond fluxes for the lambda-deformed backgrounds can be extracted from the fermionic part of the sigma model, such problem is notoriously complicated [22]. When similar deformation were analyzed in the past, the RR fluxes were obtained by solving supergravity equations [9,10,22], and in this section we will follow the same route. We will demonstrate that under very weak assumptions, supergravity gives the unique expression for all fluxes.
Since the undeformed AdS 3 ×S 3 geometry is supported by the Ramond-Ramond threeform, we assume that the situation will remain the same after the deformation, so the relevant part of action for the type IIB supergravity reads This leads to the equations of motion 3.33) and the first one is solved by metric (3.9), (3.15) and the dilaton (3.28).
To construct an expression for C 2 , we observe that the left-hand side of the Einstein's equation (3.33) has the structure where Q is given by (3.28), and P is a polynomial in (α, β, γ,α,β,γ). This suggests a natural ansatz for C 2 : where allC µν are polynomials of degree two 9 in (α, β, γ,α,β,γ). This ansatz leaves 6 × 5 2 × 1 + 6 + 6 + 6 × 5 2 = 420 (3.36) undetermined coefficients. We then found the most general solution forC µν following these steps: Once uniqueness of the solution for λ = 0 is demonstrated, we can choose a convenient gauge which respects the discrete symmetries (3.29): This solution is odd under S 1 and S 2 . The uniqueness of the solution in the zeroth order in λ guarantees that, up to a gauge transformation, there is a unique gauge potential C 2 , at least in the perturbative expansion in powers of λ. Making a guess consistent with symmetries (3.29), we arrive at the final solution Notice that, unlike the solution (3.17) with the "bosonic dilaton", the field (3.38) has a complicated lambda dependence, and the situation is similar in the AdS 2 ×S 2 case, which is reviewed in the Appendix A. In particular, while the field (3.17) vanishes at the WZW point (λ = 0), our solution for the supercoset (3.38) goes to a nontrivial limit, and, as we will see in section 4 and in the Appendix A, the same phenomenon persists for AdS 2 ×S 2 and AdS 5 ×S 5 .
To summarize, the λ-deformed version of AdS 3 ×S 3 is described by the metric (3.9), (3.15), the dilaton (3.28), and the Ramond-Ramond two-form (3.38). In the next subsection we will analyze some special cases of this geometry.
The gauged WZW model is obtained by setting λ = 0: This should be contrasted with bosonic gWZW, which has the dilaton e Φ = 1 ΛΛ (3.40) and vanishing C 2 (see (3.17)). A similar contrast is encountered in the AdS 2 ×S 2 and AdS 5 ×S 5 cases, which discussed in section 4 and in the Appendix A.
Another interesting limit is obtained by setting λ = 1. However, this limit should be approached with a great care since denominators contain (λ 2 − 1). We will follow the procedure discussed in [20] adopting it to our coordinates. To arrive at a sensible limit, we rescale the coordinates on the sphere as and send ε to zero. This gives the metric of the η-deformed S 3 [9], and to see this, we introduce the standard coordinates (r, φ, ϕ) by Performing a similar change of variables on AdS 3 along with an analytic continuation φ → ψ, ϕ → t, r → iρ, k → −k, (3.45) and sending ε to zero, we arrive at the metric and the dilaton This geometry describes the η-deformed AdS 3 ×S 3 [9], and similar relations between λ-and η-deformations have been explored in [20].

Alternative parameterizations
In subsections 3.1-3.3 we derived the full supergravity solutions corresponding to the λdeformed supercoset, but the metric for this geometry has already appeared in the literature [17,20]. We used the parameterization of [17], and in this subsection we will discuss the relation with the coordinates used in [20] and discuss one more parameterization which becomes useful for comparing AdS 3 ×S 3 and AdS 5 ×S 5 solutions. To find the relation between our parameterization and the coordinates used in [20], we observe that the action by H = SU(2) diag changes components of g l and g r in (3.3), but three expressions remain invariant: Although the gauge used in [20] was different from ours, we can find the map between two sets of coordinates by matching the expressions (3.47) in two descriptions. The authors of [20] used parameterization in terms of the Euler's angles: Evaluating the invariants (3.47) for parameterizations (3.5)-(3.6) and (3.48), and comparing the results, we arrive at the map 10 α = cos(ϕ + φ) cos ζ, β = cos(ϕ − φ) cos ζ, γ = cos 2ϕ − cos 2ϕ + cos 2φ 2 cos 2 ζ. (3.49) Another interesting coordinate system comes from parameterizing the coset SO(4)/SO(3) in terms of a three-dimensional vector X and an anti-symmetric 3 × 3 matrix A [29,19]. Such parameterization of SO(n + 1)/SO(n) will be used in the next section for studying the deformed AdS 5 ×S 5 , so it is important to introduce similar coordinates in the present case to make comparisons. The detailed discussion of parameterization and the gauge fixing is presented in section 4.1, here we just write the result 11 :  (4), so to relate them we should compare quantities which don't depend on the representation. We have already encountered such an object before: ( 3.51) To establish the map between generators, we recall that the subgroup H = SU (2) and its inverse . (3.55) The AdS coordinates are obtained by the replacement In coordinates (Y 1 , Y 2 , a,Ỹ 1 ,Ỹ 2 ,ã) the dilaton becomes In particular, for the gauged WZW model (λ = 0) we find (3.58) Notice that this expression does not depend on coordinates a andã, and the same phenomenon is encountered in the AdS 5 ×S 5 case, see the last factor in (4.27).

Towards the deformation of AdS 5 ×S 5
In this section we apply the procedure described in section 2 to construct the λ-deformed AdS 5 ×S 5 supercoset. Our final result includes the metric and the dilaton, but since the latter looks rather complicated, we were not able to solve the equations for the Ramond-Ramond fluxes. Superstrings on AdS 5 ×S 5 are described by a sigma model on the supercoset [28] P SU(2, 2|4) SO(4, 1) × SO (5) .
The corresponding superalgebra is represented by 4 × 4 matrices, and an explicit parameterization is presented in the appendix B. The bosonic part of the supercoset (4.1) is given by (6) SO (5) , (4.2) and, as in the AdS 3 ×S 3 case, the two subgroups decouple in the metric (2.9) and in the bosonic contribution to the dilaton (2.18). While these objects have been computed in [19], to evaluate the fermionic contribution to the dilaton we will have to use a different parameterization, so we begin with specifying our coordinates, finding the metric and the bosonic dilaton for them, and comparing the results with [19]. The fermionic contribution to the dilaton will be evaluated in section 4.2.

Metric and the bosonic dilaton
To apply the procedure outlined in section 2, we need an explicit form of the coset (4.2). The most natural way to parameterize the sphere S 5 = SO(6)/SO (5) is to use the Euler angles, and such description has been used in [19], but unfortunately these coordinates make the evaluation of the ferminonic contribution to the dilaton nearly impossible. Thus we use the alternative coordinates introduced in [29,19], in which all expressions remain algebraic 12 . Specifically, we write the element of SO (6) as where X i is a five-dimensional vector and h m n is an element of SO (5). The defining condition for SO (5), h T h = I, can be solved by writing h in terms of an anti-symmetric matrix A as The SO (5) rotations act on A and X as To fix this gauge freedom, we follow the procedure discussed in [29]: first we rotate A to a block form 13 : (4.6) and then we use the remaining [SO(2)] 2 rotations to set X 2 = X 4 = 0. The so(6) algebra has 15 generators, first ten of them form so (5), while the last five correspond to the coset. Specifically, in our parameterization, the coset generators are 14 Application of the procedure (2.17) leads to the bosonic contribution to the dilaton (2.18) 4.8) 12 It appears that the authors of [19] used the same coordinates while computing the metric and rewrote the final answers in terms of the Euler's angles. We find the algebraic coordinates more convenient. 13 Notice that there is a slight difference in gauge fixing between SO(n)/SO(n − 1) for odd and even n: matrix A has [(n − 1)/2] independent components, and there are [n/2] independent X.
14 Recall that throughout this article we use hermitian generator, so the element of a group is constructed where we defined Note that the lambda dependence factorizes in (4.8), and this is a general feature of the bosonic dilaton, as discussed in the end of section 2. Specifically, in the present case, matrix P defined in (2.23) has the form This matrix has eigenvalues λ −2 ± 1 and (4.10) The metric for λ-deformation is constructed using (2.17), and the result reads where e β (0) refer to the frames describing the gauged WZW model (λ = 0): , (4.12) The AdS 5 counterparts of the metric and the dilaton are obtained by an analytic continuation 13) and the corresponding frames are denoted by e 1 (0) ,. . . ,e 5 (0) .

Fermionic dilaton: general discussion
Although the SO(6)/SO(5) representation (4.3) of the five-dimensional sphere is very intuitive, the construction of the supercoset (4.1) requires embedding of SO (6) into SU(4) and identifying the fermionic degrees of freedom corresponding to the supercoset. We begin with finding the SU(4) matrices in parameterization (4.3).
The SU(4) matrices g describe a representation of SO (6), which acts on anti-symmetric 4 × 4 matrices A as (4.14) Specifically, starting with the fundamental representation of SO (6) acting on six-dimensional vectors (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ), one can construct matrix A as The generators of SU(4) are hermitian 4×4 matrices, and to proceed with the coset construction, we need to identify the elements t α corresponding to the generators (4.7). Comparing the action T α on (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) and the action of g ∈ su(4) on (4.15), we find (4.16) All generators of SU(4), including (4.16), are hermitian, while generators of SU(2, 2) satisfy the modified hermiticity relation For example, the counterparts of the coset generators (4.16) are obtained by an analytic continuation To proceed we need to construct an automorphism J 1 which satisfies (3.19) for all generators g ∈ su(4) with the exception of (4.16). While it is easy to find this J 1 for 6 × 6 matrices and coset generators (4.7) (specifically, J 1 = ±diag (−1, 1, 1, 1, 1)), such matrix does not exist in the four-dimensional representation of so (6), and the closest analog of (3.19) is (4.19) This means that condition (3.21) will be modified as (4.20) and such grading is a familiar feature of P SU(2, 2|4) (see, for example, [23] for a detailed discussion). In our parameterization, (4.21) and the detailed discussion of fermions projected out by (4.21) and relation to other conventions used in the literature is presented in the Appendix B. Here we only mention that if 8 × 8 supercoset matrix is written as 22) then projector P 3 entering P λ (2.20) selects the components satisfying an additional relation (B.14): The last ingredient for constructing the fermionic contribution to the dilaton is the explicit expression for the element of SU(4)/SO (5) in the gauge (4.3), (4.6): The element of SU(2, 2)/SO (4,1) is obtained by making the analytic continuation (4.13) in the last expression. Notice that the symmetry 25) which was obvious in the SO(6) parameterization (4.3), (4.6), is less explicit in (4.24). Evaluation of the fermionic contribution to the dilaton involves a straightforward but tedious calculation of the determinant 26) and the results are rather complicated. We collect them and discuss some of their features in the next two subsections.

Dilaton for the gauged WZW model
Geometry with λ = 0 describes the gauged WZW model, and the solution in this case is given by the frames (4.12), along with their AdS 5 counterpart and the dilaton The bosonic contribution to the dilaton is obtained by dropping the expression in the brackets, and the bosonic coset does not require any Ramond-Ramond fluxes. The situation for the supercoset is different, as we have already seen in the AdS 3 ×S 3 case: the Ramond-Ramond fluxes are turned on even at λ = 0. In the present case we were not able to construct the fluxes explicitly, but we verified that the solution (4.12)- (4.27) can be supported by F 5 .
Recall that the stress-energy tensor for the self-dual five-form, (4.28) satisfies the Rainich conditions [30] 15 : T m m ≡ Tr T = 0, Tr T 3 = 0, Tr T 5 = 0, Tr T 7 = 0, Tr T 9 = 0, (4.29) and for geometry supported only by the dilaton and the metric the T mn can be expressed as 16 The right-hand side vanishes for the "bosonic" dilaton, while for the full solution (4.27) it gives a nontrivial result which satisfies the constraints (4.29). It would be very interesting to find the corresponding flux F 5 .

Special cases for λ = 0
Although the dilaton for arbitrary values of λ can be computed by evaluating the appropriate determinants, unfortunately the results are not very illuminating. In this subsection we will collect the answers for some special cases which give manageable expressions. Since the general expression for the bosonic dilaton is already given by (4.8), we will focus only on the fermionic contribution to (2.16): First we observe that at λ = 0 the expression for e 2Φ F depends only on X k andX k . While this property does not hold for general values of λ, setting a =ã = b =b we still find an interesting result: In the opposite case, where all X are switched off, the expression is much more complicated, for example at λ = 1 it has the form In particular, we observe that the last expression is fully symmetric under interchanging the elements of the list (a, b,ã,b). This property persists for all values of λ, as long as X m =X m = 0, but the general expression is not very illuminating, so we will not write it here. The last two interesting cases corresponds to looking only at the sphere or only at the AdS space: The complexity of our results beyond λ = 0 suggests that the full solution for the λdeformation of AdS 5 ×S 5 cannot be constructed unless one finds better coordinates, and we leave this problem for future investigation.

Discussion
In this article we have constructed the supergravity background describing the λ-deformation of AdS 3 ×S 3 supercoset and reported some progress towards the analogous result for AdS 5 ×S 5 . Our main result is summarized by equations (3.15), (3.9), (3.28), (3.38). In the AdS 5 ×S 5 case we have constructed the metric and the dilaton describing the supercoset, and while the results presented in section 4 are rather complicated, there are striking similarities with lower-dimensional cases. For example, at the WZW point, where the expression (4.27) for the ten-dimensional dilaton is rather simple, one finds a very close analogy with the sixdimensional case (3.58), and we hope that a further exploration of such analogies will lead to construction of full gravity solution for the deformed AdS 5 ×S 5 .
B Parametrization of psu(1, 1|2) and psu (2, 2|4) In this appendix we briefly summarize the parameterization of psu(1, 1|2), psu(2, 2|4), and their cosets used in sections 3 and 4. We will mostly follow the notation of [23,24], although our parameterization of fermions differs from the one in [23], and we will comment on the difference. The Lie superalgebras psu(n, n|2n) can be defined in terms of (4n) × (4n) supermatrices with even (2n) × (2n) blocks A, D and odd (2n) × (2n) blocks B, C. The graded Lie bracket is defined as Matrix M is subject to the hermiticity condition where Σ is a hermitian matrix of signature (n, n). Convention for su(n, n) represented by A fixes the matrix Σ and the parameterization of fermions B, C.

(B.5)
To construct the algebra for the coset P SU(1, 1|2) l × P SU(1, 1|2) r SU(1, 1) diag × SU (2)  The top left block of this matrix describes AdS space, the bottom right block describes the sphere, and the matrix P corresponding to this supercoset is given by (3.26): (B.10) In particular, this matrix does not mix the B i and C i components, so in section 3.2 we computed the fermionic contribution to the dilaton by treating the holomorphic and antiholomorphic components (b ij and b † ij ) as independent variables. Let us now discuss the psu(2, 2|4) superalgebra, which emerges in the description of strings on AdS 5 ×S 5 [28]. In this case equation ( This choice leads to a relation between 2 × 2 blocks of B and C in (B.1): A choice of holomporhic and anti-holomorphic fermions is no longer convenient since the coset projection mixes them. As discussed in section 4.2, for the psu(2, 2|4) supercoset, the condition (3.21) is replaced by (4.20) (B.13) with P given by (4.21). An explicit calculation shows that projection (B.13) chooses the elements which satisfy B.14) in addition to (B.12). The coset corresponds to the generators included in (B.12), but not in (B.14). In other words, generators satisfying both (B.14) and (B.12) survive under projection P 3 , and P 1 is defined as P 1 = 1 − P 3 . We conclude this appendix by relating our conventions with notation used in [23]. We chose a different embedding of the coset into SU(4) × SU (2,2), and this led to a following relation between our generators and the ones used by Arutyunov and Frolov (AF) [23]: While our generators are convenient for evaluating the λ-deformation, the generators of Arutyunov and Frolov are better suited for imposing kappa symmetry. Specifically, elimination of this freedom in the notation of [23] gives while in our notation The expressions for kappa symmetry are not used in this paper.