On the stability of $AdS$ backgrounds with $\lambda$-deformed factors

We investigate the stability of the non-supersymmetric solutions of type-IIB supergravity having an unwarped $AdS$ factor and $\lambda$-deformed subspaces found in arXiv:1911.12371. Among the plethora of solutions we study the perturbative stability of backgrounds with an $AdS_n$, with $n = 3,4,6$, factor. Our analysis is performed from a lower dimensional effective theory which we construct. We uncover the regions and isolated points in the parameter space of potential perturbative stability.

While these works have been further developed in various directions, in the present work we are focused in aspects concerning their embedding to type-II supergravity.
In general this is a challenging task to perform and necessarily involves turning on a dilaton as well as Ramond-Ramond (RR) fields that supplement the two-dimensional σ-model metric and antisymmetric tensor fields. Most of the examples constructed in the literature [16][17][18][19][20][21][22][23][24][25], have the entire space-time deformed and therefore they lack an AdS space-time as part of the full supergravity solution. The only exception is the work of [26], where (unwarped) AdS solutions with λ-deformed internal subspaces have been explicitly constructed.
Applying the machinery of AdS/CFT correspondence is the natural next step in order to enlighten the field theory side of the solutions of [26]. However, the two important characteristics of these backgrounds, namely the absence of supersymmetry and the presence of an unwarped AdS factor, combined with the Ooguri-Vafa conjecture [27], suggest that an investigation of their stability, at least perturbatively, is imminent. Specifically, this conjecture states that any non-supersymmetric AdS vacuum is unstable and it is a stronger version of the weak gravity conjecture [28]. Therefore the tedious effort of checking the stability of the AdS backgrounds with λ-deformed factors can be put in the wider context of satisfying or disproving this conjecture.
What makes the models of [26] attractive in this context is their relative simplicity and the fact that the various factors correspond to integrable subspaces from a twodimensional σ-model point of view. The latter property empirically and intuitively, if not guaranteeing stability, it makes instability less likely to be the case. In that respect, we note that the parametric space in some solutions is two-dimensional, leaving enough room for stability in part(s) of it. However notice that, even if we could prove perturbative stability of the λ-deformed backgrounds, to disprove the conjecture we should also exclude possible non-perturbative instabilities.
The state of the art in the field of investigating perturbative stability is coming from a method that is based on exceptional field theory [29] and was developed in [30,31] for computing Kaluza-Klein spectra of maximal gauged supergravity vacua. In [32] this method was applied to prove perturbative stability for the non-supersymmetric G 2 -invariant AdS 4 × S 6 background of massive-IIA supergravity 1 , through the analytic computation of the full Kaluza-Klein spectrum. The aforementioned calculation combined with the absence of a brane-jet instability [35], i.e. a recently introduced non-perturbative instability, for these backgrounds [36], sets an interesting challenge for the Ooguri-Vafa conjecture.
The plan of the paper is as follows: In section 2 we present in a minimalistic way the fluctuation analysis for a wide class of effective theories containing only gravity and scalars. Since this structure is a common characteristic of all the dimensionally reduced solutions that we study in the current paper, we present the results in a unified framework. We perform a perturbative analysis around an AdS solution having constant scalar fields. Analysing stability boils down to determining the eigenvalues of a mass matrix, which is constructed using the scalar potential and the constant metric in the scalar field space.
In section 3 we perform a stability analysis of the type-IIB AdS 3 × S 3 × CS 2 λ × CH 2,λ solution, where CS 2 λ and CH 2,λ are the λ-deformed (two dimensional) coset CFTs SU(2)/U(1) and SL(2, R)/SO(1, 1), respectively [1,26]. We adopt an initial reduction ansatz containing only gravity and scalars which is also consistent with the Bianchi identities and flux equations for the RR-fields. The resulting equations of motions are eventually derived from a lower dimensional effective action of the type studied in section 2. From this action we construct the mass matrix. The analysis is depicted in figures 1 and 2 and reveals regions of stability/instability separated by a critical line between the two.
In section 4 we perform a stability analysis of the type-IIB AdS 4 × N 4 × CS 2 λ where the four-dimensional space N 4 can be any of the following spaces (S 4 , H 4 , T 4 ) [26]. We follow the same line of reasoning as before and arrive to a mass matrix. In figure 3 we plot its eigenvalues as a function of the deformation parameter when N 4 is either S 4 or H 4 . Both cases are proven unstable. The case of N 4 = T 4 provides a range of potential stability.
In section 5 we perform the stability analysis of the type-IIB AdS 6 × N 2 × CS 2 λ where the two-dimensional space N 2 can be any of the following (S 2 , H 2 , T 2 ) [26]. In figure 4, we plot the eigenvalues as a function of the deformation parameter when N 2 is either S 2 or H 2 which reveals instability. The case of T 2 , existing for zero deformation parameter, is also unstable.
In section 6 we gather our results and discuss potential future directions. The main text is supplemented with two useful appendices. In appendix A, to set up the basis for the notation, we list the equations of motion of type-IIB supergravity in the string frame. In appendix B we summarise all the supergravity solutions of [26] that we analyse in the main text.

Gravity and scalars
Our stability analysis of the type-II solutions will be based on particular dimensional reductions of the ten-dimensional fields. These reductions will give rise to lower dimensional theories of gravity coupled to scalars. In all cases the background values for the scalars are constants while the background geometry is an AdS space. In this section we study the linearized equations of motion up to first order in the fluctuations for this class of effective theories.
Consider a theory of gravity coupled to n scalars in D dimensions described by the action where the vector X = (X 1 , . . . , X n ) encodes the n scalars and γ ij is an n × n constant symmetric matrix, playing the rôle of the metric in the scalar field space.
The Einstein equations arising from varying this action with respect to the metric g µν The trace of the Einstein equations gives Eliminating the Ricci scalar from the Einstein equations we end up with Moreover the equations of motion for the scalars are where γ ij is the inverse of γ ij .

Perturbations around AdS D
We will linearise the equations of motion (2.4) and (2.5) assuming an AdS D background solution where early Greek indices run through α, β = 0, 1, . . . , D − 2 and the Minkowski metric is mostly plus. Note the bar above the background metric and scalars, a notation that we will follow in our paper.
Then, using (2.6) in (2.4) and (2.5) implies that (2.7) We will now consider the fluctuations Using the above we next study the perturbations of (2.4) and (2.5) to linear order.
From the definition of the Riemann tensor we have that Therefore, using the transversality condition we have that ∇ ρ ∇ σ δg ρν =R λ σ δg λν +R λνρσ δg ρλ . (2.13) In addition, the Riemann tensor of the AdS D geometry is (2.14) Using the above properties From this and the transverse-traceless gauge, (2.10) becomes which is the equation for a massless graviton in AdS D . We conclude that the metric fluctuations are stable under small perturbations. Any potential instability, will arise solely from the particular form of the scalar potential V(X), that enters the analysis of the scalar fluctuations.

Scalar fluctuations
We now move to the perturbations of (2.5). To linear order these read where we have defined the mass squared matrix Suppose now that we find a matrix P diagonalising M 2 , i.e.
with PδX = ( f 1 , . . . , f n ). If we assume for f i a plane wave dependence on the coordinates x α , i.e. e ik (i) α x α and expand the Laplacian we find that where m 2 i = −k (i) · k (i) . We will look the behaviour of f i with i = 1, . . . , n for large r. Thus we assume f i ∼ r −∆ i . If we plug this into the above differential equation and keep the dominant terms we end up with an algebraic equation for ∆ i which is This is the well known mass-dimension formula obtained in the context of the AdS/CFT correspondence in [37]. Reality of the scaling dimensions ∆ i requires that which is known as the Breitenlohner-Freedman (BF) bound [38].

The AdS solution
In this section we examine the stability of the type-IIB solution on AdS 3 × S 3 × CS 2 λ × CH 2,λ , where we use the notation CS 2 λ and CH 2,λ for the λ-deformed cosets SU(2)/U(1) and SL(2, R)/SO(1, 1), respectively. The solution depends on the deformation parameter λ, the scale of the AdS 3 and the level k of the undeformed CFTs. It turns out that, the parametric space of the mass matrix arising from the stability analysis is two-dimensional since the k and will appear via the combinationˆ = k . Details of the solution can be found in appendix B.1. We will show that demanding perturbative stability restricts the allowed parametric space.

The reduction ansatz
We start by introducing a reduction ansatz for the solution in appendix B.1. The reduction will be along the three-sphere and the two λ-deformed spaces of the tendimensional space (B.4). Thus, for the metric we adopt the following ansatz (our approach is similar in spirit to that in [39][40][41]) M 3 + e 2ψ dθ 2 1 + sin 2 θ 1 dθ 2 2 + sin 2 θ 1 sin 2 θ 2 dθ 2 2 +e 2φ y λ 2 + e 2χ 1 dy 2 1 + λ 2 − e 2χ 2 dy 2 2 + e 2φ z λ 2 + e 2χ 3 dz 2 where M 3 is a three-dimensional space with metric The scalars A, ψ, χ 1 , . . . , χ 4 can only depend on the coordinates x µ of M 3 . For the NS three-form and the dilaton we take where φ y (y) and φ z (z) are given by (B.2) and (B.3), respectively. Finally, for the RRsector the ansatz is where Vol(M 3 ) is the volume form on M 3 and Vol(S 3 ) is given explicitly by (B.7).
The constants c 1 and c 2 are The solution of appendix B.1 is obtained by taking the space M 3 to be an AdS 3 with line element normalised asR µν = − ḡ µν and by setting the scalars tō

The equations of motion
To find the equations of motion for the scalars A, ψ, χ 1 , . . . , χ 4 and the metric g µν on Tensors constructed below and all contractions are performed with respect to the metric g µν on M 3 .

The dilaton equation:
The dilaton equation (A.5) reduces to The directions along M 3 : Restricting ourselves to the components of the Einstein equations (A.5) along the M 3 directions we get Taking the trace of the equation above and using it in order to eliminate the Ricci scalar from (3.8) we obtain that It is convenient to use this equation instead of the equivalent one in (3.8).
The directions along S 3 : It turns out that the only non-vanishing components of the Einstein equations along the sphere directions are the diagonal ones. As expected, due to symmetry, we get a single equation given by (3.11) The y-directions along the λ-deformed spaces: Focusing on the y-components of (A.5), turns out that the ones surviving are along the diagonal directions (y 1 y 1 ) and (y 2 y 2 ) resulting at and respectively.
The z-directions along the λ-deformed spaces: As before, the non-vanishing components of the Einstein equations (A.5) are along (z 1 z 1 ) and (z 2 z 2 ) leading to (3.14) and respectively.

The mixed directions:
There is also a number of mixed components that are nontrivial. These are along the (µy)and (µz)-directions and they give rise to the following first order equations, respectively where the integration constants were fixed by requiring consistency with the background values (3.7).
The constraints (3.17) tell us that from the six scalars only four of them are independent. In addition, using them one can easily see that (3.12) and (3.13) are equivalent and similarly for (3.14) and (3.15). Hence, we remain with the metric g µν and, by making a specific choice among the scalars, with A, ψ, χ 1 , χ 3 . The independent set of equations now will be (3.9) together with (3.10), (3.11), (3.12) and (3.14).

A change of frame and the stability analysis
Our equations can be further simplified in a different frame metric given by In this frame, the equations of motion for the rescaled metric g µν and the scalars A, ψ, χ 1 , χ 3 can be derived from an action of the form (2.1) where now D = 3 and the scalars are encoded in a four-vector X = (A, ψ, χ 1 , and the potential V(X) reads (3.20) The vacuum of appendix B.1 corresponds to the background values (3.6) and (3.7).
Hence, the background value for g µν is related to that of g µν in (3.6) viā and using (2.6) amounts to an AdS 3 space with radius In order to proceed with the stability analysis we need the eigenvalues of the mass matrix square M 2 which is a 4 × 4 matrix. The expressions are quite complicated and not illuminating for general values of λ. Nevertheless, for λ = 0 they become tractable so that we examine this case first. The result we obtain give us an insight of what to expect in general.
The undeformed case with λ = 0 : The matrix M 2 is with eigenvalues From (3.24) and (2.22) we find the associated scaling dimensions All of them, apart from ∆ ± 4 , are manifestly real. To ensure reality of ∆ ± 4 we demand thatˆ 8 3 . (3.27) Therefore the radius of the AdS 3 requires a minimum value so that stability is not excluded, even though classicallyˆ > 0.
The general case: Classically, reality of the supergravity solution requires a minimum value for the AdS 3 scale that isˆ (3.28) Our findings above for λ = 0 suggest that stability may require a stricter bound than (3.28), which is explicitly evaluated below.
To proceed we define the matrix with eigenvalues written in terms of d i (the eigenvalues of M 2 ) as The positivity is required for stability according to (2.23). The characteristic polynomial of the matrix B turns out to be From the factorisation of p 4 (s) clearly one eigenvalue of B is unity, say b 1 = 1. The constant term of the polynomial p 3 (s) coming with a minus sign equals the product In the desired scenario of stability all of the eigenvalues (b 2 , b 3 , b 4 ) must be non-negative, so must be their product. This tells us that a necessary but not sufficient condition for stability is thatˆ satisfies the inequalitŷ which is clearly a stricter bound than that in (3.28). This guarantees that b 2 b 3 b 4 0, but not the positivity of each eigenvalue separately. There is always a possibility that one eigenvalue is positive and two negative. However this can not be true in our case. In order to show this we assume, without loss of generality, that b 2 0 and , which together with our assumption implies that b 2 > 19. In addition, it can be shown that the coefficient of the linear term in p 3 (s) is positive for all values The latter contradicts to our initial assumption where b 4 < 0.
Hence we conclude that wheneverˆ satisfies (3.32) all eigenvalues b i , i = 1, 2, 3, 4 are non-negative which according to (2.23) ensures the reality of the scaling dimensions.
The analysis above is also illustrated in figures 1 and 2. In figure 1 we plot the eigenvalues b 2 and b 3 of the matrix B as a function of λ andˆ parameters. The latter are confined between the horizontal axis and the curve in red, which is defined by the equality in (3.28). In this domain the eigenvalues b 2 and b 3 are positive, as it can be seen from the two contour plots, and thus they are not associated to unstable modes.

The AdS solutions
In this section we perform a stability analysis of a class of type-IIB solutions with geometry AdS 4 × N 4 × CS 2 λ where the four-dimensional space N 4 can be any of the following spaces (S 4 , H 4 , T 4 ). All of the aforementioned cases are summarised in appendix B.2 and have only one free parameter, that is λ, in addition to the level k of the undeformed CFTs. When the four-dimensional space is either the sphere S 4 or the hyperboloid H 4 we strictly have λ = 0. When N 4 = T 4 , we can construct one background with λ 0 and one with λ = 0 but with an additional free parameter. The latter case shows signs of stability.  We will adopt a reduction ansatz that fits all the solutions that are mentioned in appendix B.2. The reduction takes place on the λ-deformed CS 2 λ and the four-dimensional spaces N 4 . Hence, for the metric we have the ansatz where M 4 is a four-dimensional space with metric and ds 2 N 4 can be any of the line elements in (B.9). The scalars A, ψ, χ 1 , χ 2 are taken to depend only on the coordinates x µ of M 4 . For the NS three-form and the dilaton we consider the ansatz with φ y (y) given in (B.2). For the RR sector we assume that normalised asR µν = − 1ḡµν and by setting the scalars tō A =ψ =χ 1 =χ 2 = 0 .
Taking the trace of the previous equation and combining it with (4.7) in order to eliminate the Ricci scalar we find that (4.9) We will use this equation instead of the equivalent one in (4.7). (4.10) The directions along the λ-deformed space: Focusing on the y-components of (A. 5) we find that the diagonal ones, i.e. the (y 1 y 1 ) and (y 2 y 2 ), contribute, respectively, as along the mixed (µy) directions give rise to the first order equation This is integrated to where the integration constant is fixed by the background values (4.6). Due to this constraint, (4.11) and (4.12) are equivalent.

A change of frame and the stability analysis
The solutions I-IV: The equations can be further simplified if we change the metric frame as Then, it turns out that the equations of motion for the metric g µν and the scalars ψ, χ 1 , χ 2 can be obtained by an action of the form (2.1) where now D = 4 with the vector for the scalars being X = (ψ, χ 1 , χ 2 ). In addition, the matrix γ ij is whereas the potential V(X) is (4.18) The vacua associated with the solutions I-IV of appendix B.2 correspond to (4.5) and (4.6). Notice that, although we changed the frame according to the eq. (4.16) the background values for g µν and g µν are the same, i.e.
g µν =ḡ µν , (4.19) and that amounts to an AdS 4 of radius (4.20) To study the stability of the fluctuations around the vacua I-IV we define, similarly to Obviously the mode associated to the eigenvalue b 3 is unstable as it violates the BF bound. We can use this and (4.15) to obtain ψ = 1 2 (χ 1 + χ 2 ). Thus we stay only with the metric on M 4 and the scalars χ 1 , χ 2 . Again we can go to the convenient Einstein frame letting g µν = e −3(χ 1 +χ 2 ) g µν . (4.26) In the new frame, the equations of motion for the metric g µν and the scalars χ 1 , χ 2 can be derived from the four-dimensional analogue of the action (2.1) with X = (χ 1 , χ 2 ), the matrix γ ij being γ ij = 1 2 17 15 15 17 (4.27) and the potential V(X) (4.28) Using (2.18) we can construct the 2 × 2 mass matrix and from its eigenvalues we evaluate b 1 and b 2 via (4.22) giving The mode associated with b 2 always violates the BF bound as it is negative for all the values of λ in the range [0, 2 − √ 3).

The solution VI:
In this example a more interesting structure appears. Initially we observe that the constraint (4.13) is trivially satisfied when the constant c 1 , i.e. the free parameter entering in the solution VI through the coefficients of the RR fields F 1 and F 5 in (B.24)) takes the value c 1 = ± 2 k . When this happens one can safely use the four-dimensional version of the action (2.1) with γ ij given in (4.17) and V(X) in (4.18).
As usual, the matrix M 2 is computed from eq. (2.18) and has the form The eigenvalues of this matrix and those for the matrix B defined in (4.21) are related via (4.22). We find that Since b 3 < 0 we conclude that the corresponding mode is unstable.
Another option for the parameter c 1 is to allow |c 1 | < 2 k . In this case the constraint (4.13) implies that A and ψ are related as in eq. (4.25). Substituting the relation between A and ψ in (4.10) we arrive to the following new constraint This is satisfied in two ways: Either when c 1 = ± 1 k , which, as explained below (B.25), is identical with the λ = 0 limit of the solution V and thus it has one unstable mode or, if c 1 = ± 1 k , by taking χ 1 = χ 2 := χ. Therefore, we end up with a fourdimensional system of gravity with one scalar described by the action The scalar χ has mass squared M 2 = 4 k and thus the BF bound is not violated. This is the only case in the whole analysis of the λ-deformed gravity solutions with an AdS 4 factor that an instability is not present.
Summarising, we arrive at the following conclusion: The stability analysis of the λ-deformed type-IIB backgrounds with an AdS 4 factor uncovered an unstable mode for all solutions I-V, and only for the solution VI there are islands of potential stability.
Precisely, in the case of the AdS 4 × T 4 × CS 2 background (λ=0), with the values of the free parameter c 1 satisfying |c 1 | < √ 2/k and excluding the values c 1 = ± √ 1/k, our stability analysis did not detect any sign of instability. Technically, the feature that distinguishes the case VI from the other five (I-V) is that one is forced, cf. (4.33), to take the two scalars equal, i.e. χ 1 = χ 2 . In turn, this gives rise to a positive mass term around the minimum of the single scalar potential.

The AdS 6 solutions
Following the same line of the previous sections we examine the stability of a class of type-IIB solutions with geometry AdS 6 × N 2 × CS 2 λ , where now the two-dimensional space N 2 is one of the following spaces (S 2 , H 2 , T 2 ). More details on these solutions can be found in appendix B.3. Likewise to the preceding section, λ is the only independent parameter, apart from the level k of the undeformed CFT.

The reduction ansatz
A reduction ansatz for the metric accommodating the three solutions of appendix B.3 has the form where M 6 is a six-dimensional space with metric and ds 2 N 2 can be any of the line elements in (B.27). Also, the scalars A, ψ, χ 1 , χ 2 are taken to depend exclusively on the coordinates x µ of M 6 . For the NS three-form and the dilaton we consider where φ y (y) is given by (B.2) and for the RR sector we assume that Tensors constructed below and all contractions are performed with respect to the metric g µν on M 6 .

The dilaton equation: The dilaton equation (A.5) reduces to
The symbol takes the values 0, ±1 according to (B.28). give The trace of the last equation combined with (5.7) serves to eliminate the Ricci scalar from it giving It is convenient to use this equation instead of the equivalent one in (5.7).
The directions along N 2 : Along these directions the only non-trivial components of Einstein's equations are the diagonal ones leading to the same expression The directions along the λ-deformed space: The diagonal y-components of (A.5), i.e. the (y 1 y 1 ) and (y 2 y 2 ), give respectively. The off-diagonal component is trivially satisfied.

The mixed components:
The integration of the equation arising from the mixed (µy) directions provides that in accordance with the background values (5.6). This can be used to solve for A and eliminate it from the equations. Hence, the metric g µν on M 6 and the independent scalars ψ, χ 1 , χ 2 must satisfy equations (5.8), (5.9), (5.10) and (5.11), where we note that (5.12) is equivalent to (5.11).

A change of frame and the stability analysis
If we further change metric frame as g µν = e −ψ− χ 1 +χ 2 2 g µν , (5.14) then, the equations of motion for the metric and the scalars can be obtained by an action of the form (2.1) where now D = 6 and the scalars are encoded into the vector Also the potential V(X) is The corresponding matrix B constructed from (5.19) has the eigenvalues lution is not the smooth λ → 0 limit of I (one should also replace the sphere by the torus) the result of the stability analysis is qualitatively the same: one of the eigenvalues leads to an unstable mode. At this point, there is a qualitative difference compared to the AdS 4 result, and the reason (as we discussed at the end of the previous section) is that the λ = 0 limit in the AdS 4 case is accompanied with a simplification in the reduction ansatz (i.e. χ 1 = χ 2 = χ), something that is absent in the AdS 6 case.

Conclusions
In the current paper we study a wide class of relatively simple supergravity backgrounds constructed in [26]. These are non-supersymmetric solutions with an unwarped AdS factor and originate from embedding integrable λ-deformed σ-models to type-II supergravity. All of these solutions contain a continuous parameter λ, taking values in a certain range between 0 and 1. The absence of supersymmetry, the presence of the AdS factor and the Ooguri-Vafa conjecture [27], as well as the potential usefulness within the AdS/CFT correspondence, were the mains motivations to investigate the stability of these solutions, starting with the perturbative one.
Among the different solutions constructed in [26] we chose to analyse those with an AdS n factor of n = 3, 4, 6. All of them fall into the class of solutions that the Ooguri-Vafa conjecture applies. In the present work we examined their perturbative stability.
Our study was based on a consistent dimensional reduction of these solutions to a lower dimensional theory with scalars coupled to gravity and a subsequent perturbative stability analysis of the corresponding lower dimensional theory. The study of the different backgrounds revealed that most of them are perturbatively unstable except for the λ-deformed solution with geometry AdS 3 × S 3 × CS 2 λ × CH 2,λ and the undeformed one with geometry AdS 4 × T 4 × CS 2 (solution VI in appendix B.2). Below we mention the solutions resisting the characterisation unstable. In the case of AdS 4 × T 4 × CS 2 there is another free parameter that it is called c 1 appearing in the RR sector. It turns out that when |c 1 | < √ 2/k (excluding the values with |c 1 | = 1/ √ k), our stability analysis does not detect unstable modes violating the BF bound. As such, this background serves as another candidate for a more elaborate study.
Equally important with the perturbative instabilities are the non-perturbative ones.
Current progress in that front is coming from a novel decay channel, introduced in [35], and comes under the name brane-jet instability. In this set-up, a probe brane is placed in the background and the force acting on the brane has to be examined. In case this force is repulsive, the vacuum is characterised unstable. In [35], one of the few gravity backgrounds that challenged the Ooguri-Vafa conjecture, i.e. the nonsupersymmetric SO(3) × SO(3) invariant AdS 4 vacuum of the 4-dimensional N = 8 SO(8) gauged supergravity [42], was proved brane-jet unstable. In a parallel line of research, tachyonic Kaluza-Klein modes were found for the same background [43].
The fact that the perturbative analysis of the AdS 3 and AdS 4 solutions that survived leave a substantial part of the parametric space free of instabilities suggests that a brane-jet calculation could be useful to further constrain the window of potential stability. However, such a computation for the AdS 3 case is not conclusive since it is not easy to extract the potential generating the forces on the brane. This is due to the involved dependence of the RR potential on the internal coordinates. As such, we can not infer about the existence of possible non-perturbative instabilities based on a brane-jet argument. The same reasoning holds for the AdS 4 example as well. The

A Type-IIB supergravity
In this appendix we review the field content and equations of motion of type-IIB supergravity. It is described by the following string frame action where S t.t. is a topological term given by The field content of type-IIB supergravity consists of the metric G MN on the tendimensional space M 10 , a dilaton Φ, a NS two-form B 2 whose field strength is H 3 and the RR potentials C 0 , C 2 , C 4 which give rise to the higher-rank forms F 1 , F 3 , F 5 through Thus, the form fields satisfy the following Bianchi identities The five-form F 5 is self-dual, i.e. F 5 = F 5 which is imposed by hand.
The equations of motion arising from variations of the dilaton and the metric are while those arising from the variations of the RR potentials are

B Supergravity solutions with AdS and λ-deformed spaces
Here we summarise the supergravity backgrounds found in [26], whose the perturbative stability is analysed in the main text. Before doing that let us introduce the following set of parameters which appear often in this study In the above, k is the level of the associated CFTs we mention below, which is a positive number and in addition an integer in the compact case. The deformation parameter λ in principle takes values in the interval [0, 1). However, each solution below may impose further restrictions on the allowed values of it.
Since all solutions are based on the λ-deformation of the gauged WZW models corresponding to the exact coset CFTs, SU(2)/U(1) and SL(2, R)/SO(1, 1), it is also useful to introduce the metrics and the dilatons for these spaces. For the λ-deformed model on SU(2)/U(1) we have that where the coordinates (y 1 , y 2 ) are restricted inside the unit disc y 2 1 + y 2 2 < 1. This space will be denoted as CS 2 λ . Similarly, the λ-deformed model on SL(2, R)/SO(1, 1) is where now the coordinates (z 1 , z 2 ) lie outside the unit disc, i.e. z 2 1 + z 2 2 > 1. This space will be denoted as CH 2,λ .

B.1 The
The NS sector of this solution contains a metric that takes the form ds 2 = 2 − r 2 dt 2 + r 2 dx 2 + dr 2 r 2 + dθ 2 1 + sin 2 θ 1 dθ 2 2 + sin 2 θ 1 sin 2 θ 2 dθ 2 3 + ds 2 where is a constant and the line elements for CS 2 λ and CH 2,λ are given in (B.2) and (B.3), respectively. There is also a dilaton whose expression is where the functions φ y (y) and φ z (z) are given in (B.2) and (B.3). The NS two-form B 2 is trivial and so is its field strength H 3 .

B.2 The
The space N 4 is normalised such that The RR sector can be written in a universal fashion as F 1 = c 1 λ + dy 1 + c 2 λ − dy 2 , F 3 = 0 , (B.12) Notice that when c 1 = ± 1 k one obtains the λ = 0 limit of (B.22). On the other hand, when c 1 = 0 one finds the λ = 0 limit of (B.18) with H 4 replaced by T 4 , while whenever c 1 = ± 2 k one recovers the λ = 0 limit of (B.16) with S 4 replaced by T 4 .

B.3 The AdS
The last class of solutions of interest in the present work, is that with geometry containing an AdS 6 part and the λ-deformed space CS 2 λ . The NS-sector of these backgrounds includes a metric ds 2 = 5 1 − r 2 dt 2 + r 2 dx 2 1 + r 2 dx 2 2 + r 2 dx 2 3 + r 2 dx 2 4 + dr 2 r 2 + ds 2 N 2 + ds 2 with φ y (y) being the function in (B.2).
The field content of the RR-sector is Note that, there is an equivalent solution arising from interchanging the coordinates y 1 and y 2 .