Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence

We consider γ-deformations of the AdS5×S5 superstring as Yang-Baxter sigma models with classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). An essential point is that the classical r-matrices are composed of Cartan generators only and then generate abelian twists. We present examples of the r-matrices that lead to real γ-deformations of the AdS5×S5 superstring. Finally we discuss a possible classification of integrable deformations and the corresponding gravity solution in terms of solutions of CYBE. This classification may be called the gravity/CYBE correspondence.


Introduction
The AdS/CFT correspondence [1][2][3] has been well investigated and there would be no doubt for its validity at least in the planar limit. However, it is still important to consider the fundamental structure of the duality in order to make our understanding much deeper and look for a clue of new physics. The discovery of the integrable structure behind it [4] would play an important role along this direction. The integrability provides a guiding principle to extend the AdS/CFT correspondence by elaborating integrable deformations of it.
Our concern here is the integrable structure of type IIB superstring on AdS 5 ×S 5 . The Green-Schwarz string action is constructed from the following supercoset [5]: The Z 4 -grading of this coset ensures the classical integrability [6]. For similar argument based on another coset representation [7], see [8,9]. Possible supercosets, which lead to classically integrable and consistent string theories, are classified in [10,11].
A recent interest is to consider q-deformations of the AdS 5 ×S 5 superstring. There are two kinds of q-deformations, 1) standard q-deformations [12][13][14] and 2) non-standard qdeformations (also called Jordanian deformations) [15,16]. Both of them are based on the Yang-Baxter sigma model approach proposed by Klimcik [17][18][19], 1 where linear R-operators JHEP06(2014)135 constructed from classical r-matrices play the central role in constructing deformed classical actions. As a characteristic property, the former is based on the modified Yang-Baxter equation (mCYBE) and the latter is on the classical Yang-Baxter equation (CYBE).
For standard q-deformations of sigma models, many works have been done so far. Although deformed target spaces are not represented by symmetric cosets 2 and there is no general prescription to argue the integrability, many techniques have been developed and various aspects have been revealed. Especially for squashed S 3 , the Lax pair was presented in [22] and the classical integrable structure has been elaborated in the subsequent works [23][24][25][26][27][28][29][30][31][32][33][34]. As a possible way toward higher-dimensional cases, the Yang-Baxter sigma model approach was proposed by Klimcik [17][18][19]. Though it was originally argued for principal chiral models, Delduc, Magro and Vicedo succeeded to generalize it to symmetric coset cases [35], where the standard q-deformed algebra is presented as a generalization of [27,28]. Then they have constructed a standard q-deformed AdS 5 ×S 5 superstring action with a linear R-operator satisfying mCYBE [36]. The coordinate system was introduced and the metric in the string frame and NS-NS two-form have been determined so far [37]. However the full solution has not been obtained yet in type IIB supergravity. For further discussion with specific values of the deformation parameter, see [38]. A related mirror TBA is also discussed in [39]. It would be an important task to compare the results with the deformed S-matrices [40][41][42][43][44][45][46][47].
For non-standard q-deformations, deformed AdS 5 ×S 5 superstring actions have been constructed with linear R-operators satisfying CYBE [48]. A remarkable point is that partial deformations are possible in comparison to the standard q-deformation. For a simple example of the classical r-matrices deforming only AdS 5 , the metric and NS-NS two-form are obtained by a coset construction with an appropriate coordinate system. Then the complete type IIB gravitational solution has been found [49]. In particular, the solution is real and there is no curvature singularity, while the tidal force diverges at the boundary except for a specific surface. It also contains the three-dimensional Schrödinger spacetime as a subspace and, for the subsector analysis, one can use the results obtained in a series of works [50][51][52][53]. All of the results are consistent to the recent analysis [54].
In this note, we consider γ-deformations of the AdS 5 ×S 5 superstring as Yang-Baxter sigma models with classical r-matrices satisfying CYBE. An essential point is that the classical r-matrices are composed of Cartan generators only and do not satisfy the nilpotency condition in comparison to Jordanian deformations considered in [48]. These generate abelian twists which are particular examples of the Drinfeld-Reshetikhin twists [12,13,55].
We present examples of the r-matrices that lead to real γ-deformations of the AdS 5 ×S 5 superstring. Our analysis is concerned with the metric and NS-NS two-form only. The coincidence gives a strong evidence in favor of the equivalence of the Yang-Baxter sigma models and the gravity solutions. For the definiteness, it is also necessary to show the coincidence of the R-R fields. However, it would be very complicated to extract them from the fermionic sector of the Yang-Baxter sigma models with the supercoset construction. It is an important issue in the future. Finally we discuss a possible classification of integrable JHEP06(2014)135 deformations and the corresponding gravity solution in terms of solutions of CYBE. This classification may be called the gravity/CYBE correspondence. This note is organized as follows. In section 2 we introduce skew-symmetric classical R-operators composed of Cartan generators only. These are solutions of CYBE, but do not satisfy the nilpotency condition. With the R-operators, we present classically integrable and κ-invariant string actions. Section 3 presents simple examples. We show a relation between a classical r-matrix and a TsT transformation which leads to a real γ-deformed AdS 5 ×S 5 . It is straightforward to generalize the classical r-matrix for three-parameter γ-deformations. As a result, the Lunin-Maldacena background is contained as a particular case. Section 4 is devoted to conclusion and discussion. Appendix A explains our notation and convention. In appendix B the metric of the three-parameter γ-deformed AdS 5 ×S 5 is rewritten for our convenience. Appendix C describes in detail the derivation of the metric and the NS-NS two-form from the Yang-Baxter sigma model approach.

Deformed AdS 5 ×S 5 string actions with CYBE
We are concerned here with the deformed Green-Schwarz string action [48], where the left-invariant one-form A α is defined as Here γ αβ and ǫ αβ are the flat metric and the anti-symmetric tensor on the string worldsheet. The operator R g is defined as where a linear operator R satisfies CYBE rather than mCYBE [36]. Note that the scaling factor η may be chosen as η = 1 in our later arguments, due to a peculiarity of CYBE. The R operator is related to the tensorial representation of classical r-matrix through The operator d is given by the following,
For the action (2.1) with a Jordanian R-operator, the Lax pair has been constructed [48] and the classical integrability is ensured in this sense. The κ-invariance has been proven as well [48]. Here it is worth noting that the nilpotency condition is not necessary for the κ-invariance and the classical integrability, though it is a sufficient condition to ensure the existence of 1/(1 − ηR g • d). This will be a key observation for our later discussion.

Classical R-operators for abelian twists
In the previous work [48], we have studied classical r-matrices of Jordanian type, which satisfy the following properties: 1) solutions of the classical Yang-Baxter equation (CYBE), 2) the skew-symmetricity, 3) the nilpotency. The nilpotency condition is a characteristic property of Jordanian type. A simple example to deform only the AdS 5 part [49] is where (E ij ) kl ≡ δ ik δ jl and the skew-symmetrized symbol ∧ is defined as In fact, the associated linear R-operator exhibits the nilpotency R n Jor = 0 for n ≥ 3. One may adopt "the abelian condition" as the third property, instead of the nilpotency. It is easy to construct such r-matrices by using Cartan generators. A typical example is where µ ij = −µ ji are arbitrary parameters and h i are Cartan generators. We refer the r-matrices of this type as to abelian r-matrices because these generate abelian twists which are particular examples of the Drinfeld-Reshetikhin twists [12,13,55]. These commute with each other and hence satisfy CYBE obviously. Note that abelian r-matrices are intrinsic to higher rank cases (rank ≥ 2). For example, for su (2), these become trivial, i.e., r Abe = 0. A remarkable point is that the κ-invariance and the classical integrability are ensured for abelian r-matrices, according to the observation denoted in section 2.1.
3 γ-deformed AdS 5 ×S 5 from classical r-matrix We present here a relation between abelian classical r-matrices and γ-deformed AdS 5 ×S 5 .

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For a particular class of marginal deformations of N =4 SYM [56] called β-deformations, the gravitational duals were presented by Lunin and Maldacena [57]. Their original construction is based on an SL(2, R) symmetry and a single parameter is contained.
Then the solutions were generalized so that three parameters are contained by performing three TsT transformations [58] Here φ i (i = 1, 2, 3) are the Cartan directions in the S 5 metric and the symbol (φ 1 , φ 2 ) TsT , for example, means the following. First, a T-duality is performed along φ 1 . Then φ 2 is shifted as φ 2 +γ 3 φ 1 with a constant parameterγ 3 . Finally a T-duality is taken for φ 1 again.
The resulting metric of three-parameter deformed AdS 5 ×S 5 (in the string frame) and the NS-NS B-field are given by Here there is a constraint 3 i=1 ρ 2 i = 1 and a scalar function G is defined as For the other field components, see [58]. This solution is often called the three-parameter real γ-deformed AdS 5 ×S 5 background. 3 Whenγ 1 =γ 2 =γ 3 ≡γ, the original Lunin-Maldacena background for the real β-deformation is reproduced. In section 3.2, we will present a classical r-matrix corresponding to one of the TsT transformations used above.

One-parameter case
As a warm-up, let us consider a simple example of classical r-matrix, r (µ) Here µ is a deformation parameter and the fundamental representation of Cartan generators of su (4), h 1 and h 2 are defined as The action of the associated linear R Abe operator is given by and hence only the S 5 part of AdS 5 ×S 5 is deformed.
Since we are interested in deformations of S 5 , it is convenient to restrict the current A α ∈ su(2, 2|4) to the su(4) subalgebra as follows:

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With this setup, the S 5 part of the classical action (2.1) is reduced to where L G is the sigma model part and L B represents the coupling to the NS-NS two-form.
Here η has been taken as η = 1.
Then the classical Lagrangian given in (3.8) can be rewritten as For the derivation, see appendix C. By imposing a parameter relation, and performing the following coordinate transformation, ρ 1 = sin r cos ζ , ρ 2 = sin r sin ζ , ρ 3 = cos r , (3.12) we find that the Lagrangian (3.9) and (3.10) are nothing but the ones obtained from the γ-deformed metric (3.1) and the NS-NS two-form (3.2), respectively. Thus we have shown that the classical r-matrix (3.4) corresponds to a TsT transformation (φ 1 , φ 2 ) TsT . It would be interesting to reinterpret this result from the viewpoint of a twisted boundary condition by following [62]. In particular, there should be some relation between the classical r-matrix and the boundary condition. Along this direction, the correspondence of the Lax pairs would play an important role. The Lax pair constructed in [48] with the r-matrix (3.4) should be related to the one constructed in [58], up to a coordinate transformation. Then, it may be possible to find out the relation between the r-matrix (3.4) and the twisted boundary condition through the result [62].

Three-parameter case
Now it would be easy to deduce the classical r-matrix that corresponds to the threeparameter deformed solution, according to the result obtained in the previous subsection.

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The candidate r-matrix is represented by where µ i and h i (i = 1, 2, 3) are deformation parameters and the Cartan generators of su(4). For h 1 and h 2 , see (3.5). The remaining h 3 is defined as Then the action of the associated linear R (µ 1 ,µ 2 ,µ 3 ) Abe is given by With the following parameter identification, 16) and the normalization η = 1, the deformed Lagrangians L G and L B turn out to be L G = − γ αβ 2 sin 2 r ∂ α r∂ β r + (cos r sin ζ∂ α r + sin r cos ζ∂ α ζ)(cos r sin ζ∂ β r + sin r cos ζ∂ β ζ) + (cos r cos ζ∂ α r − sin r sin ζ∂ α ζ)(cos r cos ζ∂ β r − sin r sin ζ∂ β ζ) where the function G is rewritten as G −1 = 1 + cos 2 r sin 2 r γ 2 1 sin 2 ζ +γ 2 2 cos 2 ζ +γ 2 3 sin 4 r cos 2 ζ sin 2 ζ . (3.19) Finally, with the coordinate transformation (3.12), the deformed metric and NS-NS twoform obtained from (3.17) and (3.18) exactly agree with the three parameter γ-deformed metric (3.1) and (3.2), respectively. The derivation of them is described in detail in appendix C. Thus the classical r-matrix (3.13) corresponds to three TsT transformations (φ 1 , φ 2 ) TsT , (φ 2 , φ 3 ) TsT and (φ 3 , φ 1 ) TsT . The gravity dual for the real β-deformation is realized as a particular case witĥ γ 1 =γ 2 =γ 3 =γ. It is wroth noting that complex β-deformations are argued to yield nonintegrable backgrounds [63]. Probably, there would be no classical r-matrix for the complex β-deformations within the class that allows the Lax pair construction. It may be intriguing to look for the corresponding classical r-matrix by admitting that the integrability is broken. Now the relation between classical r-matrices and the γ-deformed geometries has been clarified. For the γ-deformed geometries, various things are understood such as the deformed potential in N =4 SYM [57][58][59], the twisted Bethe ansatz [64,65] and the worldsheet S-matrix [66]. The mirror TBA with twisted boundary conditions is also investigated in [67,68]. It would be interesting to argue the relation between them and the classical r-matrices used in the Yang-Baxter sigma model approach.

Conclusion and discussion
In this note, we have considered γ-deformations of the AdS 5 ×S 5 superstring as Yang-Baxter sigma models with classical r-matrices satisfying CYBE. An essential point is that the classical r-matrices are composed of Cartan generators only and generate abelian twists. They do not satisfy the nilpotency condition in comparison to Jordanian deformations considered in [48]. We have presented examples of the r-matrices that lead to real γ-deformations of the AdS 5 ×S 5 superstring. Based on our result, one may expect that TsT transformed AdS 5 ×S 5 geometries could be classified in terms of classical r-matrices satisfying CYBE. The conjectured relations are summarized in table 1, though it is still necessary to make efforts to get supporting evidence. A support is that the type IIB supergravity solution constructed with a Jordanian twist [49] may be regarded as a null Melvin twist, basically following the argument in appendix C of [69]. We will report on the details in the near future [70]. There are many gravitational solutions obtained as TsT transformed or null Melvin twisted AdS 5 ×S 5 . There should be a classical r-matrix for each of them.
At the beginning, the Yang-Baxter sigma model approach has been regarded as a prescription for standard q-deformations. Now it seems likely that it potentially contains much broader applications to study integrable deformations. It would provide a guiding principle for classifying possible integrable deformations and the corresponding gravity solutions in terms of solutions of CYBE, which should be called the gravity/CYBE correspondence.

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An element of su(2, 2|4) is identified with an 8 × 8 supermatrix: Here m and n are 4 × 4 matrices with Grassmann even elements, while ξ and ζ are 4 × 4 matrices with Grassmann odd elements. These matrices satisfy a reality condition. Then m and n belong to su(2, 2) = so (2,4) and su(4) = so (6), respectively. In this note we are concerned with deformations of the S 5 part. Hence it is helpful to prepare an explicit basis of su (4).

C Derivation of deformed actions
Here we describe in detail the derivation of the deformed action with the classical rmatrix (3.13). The AdS 5 part is not deformed and hence we will concentrate on the S 5 part hereafter. Let us adopt the following coset parametrization [37]: with the matrices Λ, Ξ andǧ r defined as where h i (i = 1, 2, 3) are diagonal matrices given by These correspond to the Cartan generators of su (4).
With this parametrization, the S 5 part of the Lagrangian (2.1) can be rewritten as where A α = g −1 ∂ α g is restricted to su(4) and the R-operator is defined in (3.15) and we have set that η = 1 in (2.1). For later argument, it is convenient to divide the Lagrangian L into the two parts like L = L G + L B , where L G is the metric part and L B is the coupling to the NS-NS two-form, respectively: Here the deformed current J α is defined as This current contains µ i (i = 1, 2, 3) and the normalization factor η.