Supersymmetric backgrounds from λ -deformations

We provide the ﬁrst supersymmetric embedding of an integrable λ -deformation to type-II supergravity. Speciﬁcally, that of the near horizon of the NS1-NS5 brane inter-section, geometrically corresponding to AdS 3 × S 3 × T 4 . We show that the deformed background preserves 1/4 of the maximal supersymmetry. In the Penrose limit we show that it preserves no-more than one half of the maximal supersymmetry.


Introduction
The near horizon limit of the NS1-NS5 brane-system [1,2] corresponds geometrically to the space AdS 3 × S 3 × T 4 .Together with appropriate three-form flux fields proportional to the volume forms of the AdS 3 and S 3 factors, it is a solution of the common NS sector of type-IIA and type-IIB supergravities.This near horizon solution preserves half of the maximal supersymmetry [3,2,4].Moreover, each of the factors AdS 3 and S 3 with the flux fields corresponds to a current algebra exact conformal field theory (CFT) having a Lagrangian description in terms of a WZW model [5] for the groups SL(2, IR) and SU(2), respectively.A systematic procedure to deform WZW models for a general semi-simple group by breaking conformal invariance but preserving integrability has been devised in [6] and is known as λ-deformation.This also includes the extensions to deformations based on more than one current algebra or even cosets CFTs [7][8][9][10][11][12][13][14].
A natural question is whether or not such deformations, when applied to the simplest case of the SL(2, IR) × SU(2) WZW model, can be promoted, with the inclusion of the T 4 factor, to a solution of type-II supergravity by turning on appropriate RRfields.This will be then naturally called the type-II supergravity embedding of the λdeformation of the near horizon NS1-NS5 brane intersection background.Moreover, can such an embedding be supersymmetric by preserving a fraction of the original half-supersymmetric near-horizon NS1-NS5 configuration?A supergravity embedding was indeed achieved in [15] albeit within the type-IIB* theory of [16] in which the RR-fields are purely imaginary which is not satisfactory. 1In the present work, we manage to overcome this problem and provide the first such supersymmetric embedding in type-IIB supergravity, of the aforementioned λ-deformed near horizon NS1-NS5 brane intersection background.
The plan of the paper is as follows: In Section 2, we construct our 1 /4 supersymmetric type-IIB solution by turning on RR-fields.We check its supersymmetry and make contact with recent literature on one-quarter supersymmetric solutions with warped AdS 2 and S 2 factors.In Section 3, we consider the Penrose limit around a null geodesic and the supersymmetry of the associated plane-wave solution.Details on the Killing spinor equations are contained in the Appendix A.

The supergravity solution
In this section we present the embedding of the λ-deformed model based on SL(2, R) × SU(2) in type-IIB supergravity.A version of this was first found in [15] but the embedding was in the type-IIB* theory of [16].However, in that work it was overlooked that the background admits an analytic continuation of the coordinates mapping it to a solution of type-IIB supergravity with real RR-fields.In the notation of [15] this 1 Besides this work type-III supergravity embeddings of λ-deformations based on (super) cosets have been constructed in [17][18][19][20][21].Such embeddings are not supersymmetric even in the limit of no deformation.Moreover, supergravity embeddings have been constructed for the closely related ηdeformed models [22][23][24][25] in [26][27][28][29][30].The closest in spirit to our present work is the supersymmetric deformation of the AdS 3 × S 3 × T 4 solution studied in [31].This deformation also preserves one-quarter of the maximal supersymmetry and can be found by applying TsT transformations to the one-parameter Yang-Baxter integrable deformation [22][23][24].
This analytic continuation does not change the signature of the spacetime and no change on the sign of the WZ level parameter k is needed.
Alternative to the analytic continuation, one may use the general λ-deformed σ-model action [6] with the group elements for SU(2) and SL(2, IR) given by (2.2) In the above, the SU(2) group element is connected to the identity element reached for α = 0. Instead, the SL(2, IR) group element is not connected to the identity element for any real values of the angles α, β and γ.Hence, we conclude that the identity is not part of the group manifold in the type-IIB supergravity solution we present below.
This will be important when we will consider the non-Abelian T-duality (NATD) limit below.
The field content is summarized below and various interrelations are depicted in Fig- The metric: The geometry in ten dimensions is the direct sum of the target space metric of the λ-deformed model on SL(2, R) × SU(2) and a four-dimensional torus.
The line element of this configuration is 2 An alternative analytic continuation which results to a type-IIB solution is In this paper we prefer to work with (2.1) as it is more appropriate for studying Penrose limits.

NATD
N ATD(AdS where The first line in (2.3) corresponds to the target space metric of the λ-deformed model on SL(2, R), while the part spanned by (α, β, γ) to that of the SU(2) λ-model.The four-torus is parametrized by the coordinates x i (i = 1, . . ., 4).
Notice that when λ = 0 the six-dimensional space transverse to the torus is just Although the presence of the deformation breaks the isometry of AdS 3 × S 3 , a subspace with topology AdS 2 × S 2 is still present.When α approaches ±∞ the subspace parametrized by (α, β, γ) looks like R × AdS 2 .On the other hand the topology of the subspace parametrized by (α, β, γ) looks different in the neighborhood of the points α = 0, π and α = π /2.Specifically, near α = 0, π it looks like R 3 , while near For later convenience we also introduce the orthogonal frame As usual this is associated to the ten-dimensional tangent frame metric on R 1,9   ds 2 = η ab e a e b , g µν = η ab e a µ e b ν , e a = e a µ dx µ , η = diag(−1, 1, • • • , 1) , (2.6) where Greek indices denote the curved ones, while Latin ones those of the tangent space.
The dilaton: The NS sector contains also a non-trivial dilaton arising from integrating out the gauge fields in the construction [15].This takes the simple form (2.7) The inclusion of the λ-depended factor is arbitrary and affects the form of the RR-flux fields below by a related overall factor.We have chosen this factor in such a way that all the background fields are invariant under the non-perturbative symmetry found in the general contest of the λ-deformed models in [32].This symmetry is satisfied by construction by the metric (2.3) and the NS two-form (2.9) below.

The NS two-form:
The NS two-form has contribution both in the SL(2, R) λ and in the SU(2) λ directions, namely (2.9) The RR-forms: This solution has all the RR forms turned on when λ = 0.These are better expressed in terms of the frame (2.5) as where F 5 is self-dual 3 while µ depends on λ and k through It is interesting to point out that the background described above interpolates between the type-IIB solution on AdS 3 × S 3 × T 4 with only NS fields, and the type-IIB solution arising from a non-Abelian T-duality (NATD) transformation on AdS 3 × S 3 × T 4 with a RR three-form and vanishing NS form.The former corresponds to λ = 0 while the latter can be seen as a correlated limit λ → 1, k → ∞ combined with a zoom-in along α and α.Moreover, as we will see later, when λ = 0 the supergravity solution preserves 16 supercharges while for λ = 0 supersymmetry is broken to 8 supercharges.This is in agreement with the fact that the deformation breaks the isometries of the background.

The non-Abelian T-dual limit
It is known that when λ approaches the identity, the λ-deformed model becomes singular.It order to make sense of the geometry a correlated limit in which k is also taken to infinity has to be taken.Then, the result on a group reduces to the non-Abelian Tdual of the corresponding Principal Chiral Model (PCM) as long as, at the same time, a zoom-in near the identity element of the group, by rescaling the group parameters appropriately with inverse powers of k, is also taken [6].
In turns that the background (2.3), (2.7) and (2.10) does not admit a non-Abelian Tdual limit.The reason is that, as noted before, there are no real value for the angles α, β and γ for which the identity element of SL(2, IR) group element in (2.2) can be reached.Nevertheless, at the supergravity level described in this section, the NATD limit can be taken by setting and then sending k to infinity. 4This analytic continuation can be easily seen to effec- 3 Our conventions for the Hodge dual on a p-form in a D-dimensional spacetime are that , where ε 0...9 = 1.From the above we find ⋆ ⋆ F p = s(−1) p(D−p) F p , where s is the signature of the spacetime which in our case is taken to be mostly plus, see Eq. (2.6). 4 Curiously, a coordinate transformation of the form α → i π 2 + α leads to a solution of type-IIB * for tively connect the SL(2, IR) group element in (2.2) to the identity which is a necessary condition for the non-Abelian T-duality limit to exist.Moreover, in the NATD limit the dilaton diverges with an imaginary constant.Shifting the dilaton with a k-dependent constant absorbs this divergence.In doing so, one has to also rescale the RR-forms by an appropriate imaginary k-dependent constant.This guarantees that the NATD limit solves the supergravity equations of motion and eventually provides a real solution of type-IIB supergravity with the various fields presented below.
The NS sector: After the NATD limit (2.12) the metric, the dilaton and the NS twoform read Notice that in order for the metric to preserve the correct signature, the coordinate ρ must take values |ρ| > 1.For later convenience in comparing with the literature, we have multiplied the resulting from the limit expression for e −2Φ by a factor of 16.
That affects the expressions for the RR-flux fields below which, in comparison with the results one gets from the limit, have been multiplied by a factor of 4.
The RR sector: Taking into account the rescaling of the RR forms due to the shift of the dilaton, the limit (2.12) results to It is worth to mention that the NATD solution falls in the class of AdS 2 × S 2 × CY 2 backgrounds with 8 supercharges constructed in [33].One can see this explicitly by setting ĥ4 = h 8 = u 2 and u = −r , in equations (4.6) and (4.7) of that paper (note the all values of λ except for the non-Abelian T-dual limit (2.12) as it is explained below.

Supersymmetry
We turn next to the analysis of the supersymmetry for the solution given in (2.3), (2.7), (2.9) and (2.10).Due to the fact that the RR fields vanish when λ = 0 it is instructive to distinguish this case from the λ = 0 one.
The λ = 0 case: As it is explained in appendix A.2 when λ = 0 the dilatino equation is solved by imposing a single projection, namely the one given in (A.5).Also, the gravitino equations can be easily integrated resulting to the Killing spinor 5 where the indices in the Gamma-matrices are in accordance with the frames defined in (2.5) and η is a constant spinor satisfying the projection The latter means that the type-IIB solution on AdS 3 × S 3 × T 4 with trivial RR sector preserves 16 supercharges, already known in the literature [3].
The λ = 0 case: The presence of the RR fields when λ = 0 requires that we impose an extra projection in order for the dilatino equation to vanish.Indeed, together with (A.5) one has also to consider (A.7).Moreover, after imposing the aforementioned projections and the chirality condition of the type-IIB Majorana-Weyl spinors one can integrate the gravitini.Doing so, leads to an expression for the Killing spinor which 5 We have simplified the expression omitting the tensor product.More precisely, we assume now depends non-trivially on (2.17) Again, η is a constant spinor which now satisfies together with the independent projector (2.16).The fact that one is forced to impose the second projection (2.18) when λ = 0 suggests that the deformed solution preserves 8 supercharges.Therefore the deformation breaks the original supersymmetry by half.

Supersymmetry of the NATD:
In the case of the NATD solution given in (2.13) and (2.14) one can analyse the dilatino and gravitino equations from scratch or simply take the limit (2.12) in (2.17) and (A.7).Either way one finds the Killing spinor with η being a constant spinor satisfying (2.16) and (2.18).We see that the NATD solution also preserves 8 supercharges.
It is worth making the connection with a new type-IIB class of solutions on a warped product of AdS 2 × S 2 × CY 2 × Σ 2 preserving 8 supercharges was constructed recently in [34].This class generalizes the results of [33] and is governed by a system of partial differential equations that is reminiscent of D3-D7-brane configurations and a harmonic function on Σ 2 .The NS sector of these solutions is given in eq. ( 5.35) of [34], while the RR fields can be obtained by combining eq.(5.36) and (2.1).It turns out that the solution discussed here, namely the one given in (2.3), (2.7), (2.9) and (2.10), fits in the class mentioned above.This is in agreement with the outcomes from the supersymmetry analysis that we present later.For the moment we provide the relation between our solution and the class of [34] (2.20) To avoid confusion, we denote the coordinates on Σ 2 by y 1 , y 2 instead of x 1 , x 2 that is used in [34].Also, for simplicity we take k = 1.With the above identifications we see that the metric (2.3) and the dilaton (2.7) match those in eq. ( 5.35) of [34].
However, for the NS three-form we find a difference by an overall minus sign.The same happens with the RR three-forms.Therefore, no problem occurs in view of the symmetry (H 3 , F 3 ) ↔ (−H 3 , −F 3 ) of the type-IIB equations of motion.For the RR one-and three-forms we also find precise agreement.Some discrepancies for the fiveform are explained by the fact that in our case F 5 is self-dual while in [34] anti-self-dual (in our conventions for Hodge duality, as described in footnote 3).

Penrose limit
We continue with the study of null geodesics in the geometry of the λ-deformed background and the derivation of the corresponding plane-wave solutions through a Penrose limit [35].

Null geodesics
In order to find null geodesics in the geometry of Section 2 we consider a particle that is moving along the U(1) isometries realized by shifts in the coordinates γ and γ.
In other words the velocity of the particle has non-vanishing components only along those directions.Requiring that there is no acceleration, i.e. γ = γ = 0, the equations of motion of the particle reduce to where c γ = γ and c γ = γ are the velocities along γ and γ, respectively, and the index µ labels the directions of the ten-dimensional spacetime.To arrive to the above result, we took into account that the metric is diagonal and that it does not depend on γ and γ.This condition implies that Having in mind that the U(1) directions γ and γ should not shrink to zero size prevents us from considering α = β = 0. Thus the only solution of the equations of motion above that makes sense is The null condition simply translates to considering equal velocities along the directions γ and γ, i.e. c γ = c γ .

Plane wave solution
Here we derive the plane wave solution that arises by applying the Penrose limit around the geodesic (3.3) of the λ-deformed background presented in section 2. For this reason we set The fact that the coefficients in front of u for γ and γ are equal ensures that the geodesic is null.After applying this to the supergravity fields of the solution and considering large values for k the background drastically simplifies to where we note that F 1 and F 5 in (2.10) have not survived the Penrose limit.The above content provides a solution of the type-IIB supergravity on a pp-wave geometry.Notice that the mass terms (coefficient of du 2 in the line element) break the O(8) global symmetry in the transverse directions to O(4) × O(2) × O(2) when λ = 0.For λ = 0 the symmetry of the metric enhances to O(4) × O(4) and for λ = 1 to O(6) × O(2).Also note that (3.5) is invariant under λ → 1 /λ which is the left over of the nonperturbative symmetry (2.8) in the Penrose limit.

Supersymmetry
At this point we would like to inquire whether the pp-wave solution found above admits supernumerary supercharges.We already know that when λ = 0, supersymmetry on the pp-wave enhances from 16 (which is the minimum) supercharges to 24 [36].The question is what happens when λ = 0.This can be inferred by examining the dilatino and gravitino variations for the background (3.5).Below we provide a summary of our findings, and we leave the details in Appendix A.3.
Here Ω is a matrix linear in the coordinates z i , x i (i = 1, . . ., 4) which can be represented in terms of the Γ's as in (A.33).
Recall that we work in the frame (A.28).Additionally, χ(u) is expressed in relation to a constant spinor η as shown below Furthermore, χ(u) and η are constrained by (A.37) and (A.38).
Looking at (A.37) and (A.38) we can distinguish the cases λ = 0 and λ = 0. Obviously, when λ = 0 (A.37) are trivially satisfied, while (A.38) becomes This amounts to 8 supernumerary supercharges or 24 in total.On the other hand, when λ = 0, the conditions (A.37) are satisfied only when This automatically solves (A.38).Consequently, in this case there are no supernumerary supercharges and the pp-wave background preserves the minimal supersymmetry, i.e. 16 supercharges.

Conclusions
We promoted the integrable λ-deformed model based on SL(2, R) × SU(2) to a solution of the type-IIB supergravity by including the necessary RR fluxes.This solution utilizes a group element disconnected from the identity and can be obtained as an analytic continuation of the type-IIB * solution found in [15].In the limit where the deformation parameter is zero, we recover the geometry of AdS 3 × S 3 × T 4 supported only by a NS three-form.This geometry originates from the near-horizon limit of the NS1-NS5 brane setup and preserves 16 supercharges.Although in the presence of the deformation, the full group of isometries of AdS 3 × S 3 is broken, the geometry has a manifest AdS 2 × S 2 topology.We showed that supersymmetry is reduced by half, and in fact the ten-dimensional background fits into the class of solutions recently found in [34].This is the first known example of a supersymmetric λ-deformed supergravity solution.Finally, we consider the Penrose limit along a null geodesic in the deformed spacetime.This results in a type-IIB solution on a plane-wave geometry that captures a dependence on the deformation parameter.In the absence of deformation, there exist 8 supernumerary supercharges [36], while for non-zero deformation, the pp-wave background preserves the minimum amount of supersymmetry of 16 supercharges.It turns out that this is not the only supersymmetric embedding of integrable λ-deformations to supergravity, since along similar lines, the embedding of the λ-deformed AdS 3 × S 3 × S 3 × S 1 can also be constructed [37].
The aforementioned features leave room for further investigations of the deformed background.It is interesting to consider whether the deformed solution admits supersymmetric embeddings of probe branes [38][39][40] by analyzing the κ-symmetry condition.Additionally, we have already observed that the deformed geometry maintains an AdS 2 subspace.Therefore, it is natural to explore the holographic dual of our construction and study the thermal effects by constructing the corresponding black hole solution, which amounts to turning on temperature in the holographic dual system.Finally, it would be worth checking whether our supersymmetric λ-deformed AdS 3 × S 3 × T 4 background can be derived from the integrable λ-deformed supercoset σ-model constructed in [41], also argued to preserve 8 supercharges.

A.1 Conventions
Let us introduce our conventions for the dilatino 6 and gravitino supersymmetry variations [42] δλ where (iσ 2 ) is kept in parenthesis to illustrate that it is real and the slash means contraction of the spacetime indices with antisymmetric products of Γ-matrices.In particular, we take / ∂ = Γ µ ∂ µ = e µ a Γ a ∂ µ and / A n = A µ 1 ...µ n Γ µ 1 ...µ n .Also, we have abbreviated tensor products of Gamma-matrices according to footnote 5.Moreover, σ i (i = with ω µab = e c µ ω cab being the spin-connection, which is antisymmetric in the last two indices for the tangent frame metric.

A.2 Supersymmetry variations for the λ-deformed solution
We move to the analysis of the supersymmetry for the background of the Section 2.
Starting with the dilatino equation (A.1), this greatly simplifies by imposing the pro-jection in the tangent frame Γ 0...5 ǫ = −ǫ .(A.5) It can be easily checked that this solves the dilatino equation (A.1) for λ = 0.However, for generic λ = 0, after using (A.5) we arrive at where µ is defined in (2.11) and vanishes for λ = 0.For λ = 0 setting (A.6) to zero is solved by imposing the extra projection where we have defined (A.8) The matrices P α and P α satisfy the properties (A.9) Using the above, one can also show P α iσ 2 P α 2 = P α iσ 2 P α 2 = 1 .(A.10) Shifting to the gravitino variation (A.1), we provide simplified expressions for its components after imposing the type-IIB chirality condition (A.3) and the projections

Figure 1 :
Figure 1: Relation of the various solutions.The pure NS solution corresponds to λ = 0.When λ approaches one, we find the NATD background.