Integrability of classical strings dual for noncommutative gauge theories

We derive the gravity duals of noncommutative gauge theories from the Yang-Baxter sigma model description of the AdS_5xS^5 superstring with classical r-matrices. The corresponding classical r-matrices are 1) solutions of the classical Yang-Baxter equation (CYBE), 2) skew-symmetric, 3) nilpotent and 4) abelian. Hence these should be called abelian Jordanian deformations. As a result, the gravity duals are shown to be integrable deformations of AdS_5xS^5. Then, abelian twists of AdS_5 are also investigated. These results provide a support for the gravity/CYBE correspondence proposed in arXiv:1404.1838.

1 This note is organized as follows. Section 2 gives a short summary of the Yang-Baxter sigma model description of the AdS 5 ×S 5 superstring with classical r-matrices satisfying CYBE. Then we introduce three classes of skew-symmetric solutions of CYBE. A new class of r-matrices induces abelian Jordanian deformations. Section 3 presents examples of abelian Jordanian type, which lead to the gravity duals of NC gauge theories. In section 4, we consider a deformation of AdS 5 with an abelian r-matrix concerned with a TsT transformation of AdS 5 . Section 5 is devoted to conclusion and discussion. We argue some implications of this result and future directions in studies of the gravity/CYBE correspondence. In Appendix A our notation and convention are summarized. Appendix B presents the gravity duals of NC gauge theories with six deformation parameters. Appendix C describes the detailed computation of three-parameter abelian twists of AdS 5 .

Integrable deformations of the AdS 5 ×S 5 superstring
We introduce here integrable deformations of the AdS 5 ×S 5 superstring based on the Yang-Baxter sigma model description with CYBE [21] . After giving a short review on the general form of deformed actions, we present three classes of classical r-matrices.

Deforming the AdS 5 ×S 5 superstring action with CYBE
A class of integrable deformations of the AdS 5 ×S 5 superstring can be described with classical r-matrices satisfying CYBE [21]. The deformed action is given by 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 world-sheet.
The operator R g is defined as where a linear operator R satisfies CYBE rather than mCYBE [14] . The R-operator is related to the tensorial representation of classical r-matrix through The operator d is given by the following, where P i (i = 0, 1, 2, 3) are the projections to the Z 4 -graded components of su(2, 2|4) . P 0 , P 2 and P 1 , P 3 are the projectors to the bosonic and fermionic generators, respectively. In particular, P 0 (su(2, 2|4)) is nothing but so(1, 4) ⊕ so (5) .
For the action (2.1) with an R-operator satisfying CYBE, the Lax pair has been constructed [21] and the classical integrability is ensured in this sense. The κ-invariance has been proven as well [21] .

A classification of classical r-matrices
According to the construction of the deformed string action, one may expect the correspondence between integrable deformations of AdS 5 ×S 5 and classical r-matrices, called the gravity/CYBE correspondence [29] . To study along this direction, it would be valuable to classify some typical class of skew-symmetric solutions of CYBE.
There are three types of classical r-matrices: i) Jordanian, ii) abelian, and iii) abelian Jordanian. In particular, the third class will play a crucial role in the next section.
In order to study deformations of AdS 5 later, let us consider the case of su(2, 2) .
The first class is classical r-matrices of Jordanian type, where (E ij ) kl ≡ δ ik δ jl are the fundamental representation of su(2, 2) . The characteristic property of Jordanian type r-matrices is the nilpotency. Indeed, we could verify that the associated linear R-operator exhibits (R Jor ) n = 0 for n ≥ 3 .
Jordanian deformations of the AdS 5 ×S 5 superstring are considered in [21] . A simple example of the corresponding type IIB supergravity solution is presented in [25] . Only the AdS 5 part is deformed and it contains a three-dimensional Schrödinger spacetime as a subspace. Hence it may be regarded as a generalization of [30][31][32]. It seems likely that the resulting metric is closely related to a null Melvin twist [26].
The second class is abelian r-matrices composed of the Cartan generators as follows: where µ ij = −µ ji are arbitrary parameters. Since these commute with each other and hence satisfy CYBE obviously. The abelian r-matrix is a particular example of the Drinfeld-Reshetikhin twists [15,16,22]. Note that abelian r-matrices are intrinsic to higher rank cases (rank ≥ 2).
It has been shown in [29] that abelian r-matrices lead to γ-deformed backgrounds [28], which include the Lunin-Maldacena background [27] as a particular case. In section 4, we will consider an abelian twist of AdS 5 with a single parameter. For multi-parameter cases, see Appendix C .
The third class is composed of r-matrices which are nilpotent and abelian. These should be called abelian Jordanian r-matrices. A typical example takes the following form, In the next section, we will show that classical r-matrices of abelian Jordanian type correspond to the gravity duals of NC gauge theories [33,34].

Examples -gravity duals of NC gauge theories
Let us consider examples of classical r-matrices of abelian Jordanian type. These lead to the gravity duals of NC gauge theories [33,34] . Hereafter we will concentrate on the AdS 5 part and S 5 is not deformed.
A possible example is given by where µ, ν are deformation parameters. Here p µ (µ = 0, 1, 2, 3) are the upper triangular matrices defined as For our convention of γ µ and the su(2, 2) generators , see Appendix A. It should be emphasized that p µ 's are upper triangular and satisfy the following property: Thus the classical r-matrix (3.1) is of abelian Jordanian type and trivially satisfies CYBE.
To evaluate the Lagrangian (2.1) , let us take the following coset parametrization [25] : Then the AdS 5 part of (2.1) can be rewritten as Here A α = g −1 ∂ α g is restricted to su(2, 2) and the associated R-operator R AJ with (3.1) is determined by the relation (2.4) .
It is convenient to divide the Lagrangian L into two parts like L = L G + L B , where L G is the metric part and L B is the coupling to an NS-NS two-form, respectively: To derive the explicit form of L , it is sufficient to compute the projected current P 2 (J α ) rather than J α itself. Hence the computation is reduced to solving the following equation, Note that P 2 (A α ) is expanded with γ matrices as follows: Then, by combining (3.9) with (3.8) , P 2 (J α ) can be obtained as The resulting forms of L G and L B are given by, respectively, (3.12) Here two deformation parameters µ, ν and one normalization factor η are contained.
It is easy to see the metric and the NS-NS two-form from (3.11) and (3.12) . By introducing new parameter a and a ′ through the identification, one can find that the resulting metric and two-form exactly agree with the ones of the gravity duals of NC gauge theories presented in [33,34], up to the coordinate change z = 1/u and the Wick rotation x 0 → ix 0 . This result shows that the gravity duals of NC gauge theories [33,34] are integrable deformation of AdS 5 .

Abelian twists of AdS 5
As another kind of integrable deformation of AdS 5 , we consider an abelian twist of AdS 5 with a single parameter 3 . For a three-parameter generalization, see Appendix C .
Let us consider an abelian r-matrix, r (µ) with a deformation parameter µ . Here h i (i = 1, 2) are two of the Cartan generators of su(2, 2) and belong to the fundamental representation, Then, the AdS 5 part of the Lagrangian (2.1) is given by where the current A α is su(2, 2)-valued and the R-operator associated with (4.1) is defined by the rule (2.4).
The projected current P 2 (J α ) is to be determined by solving the equation, By using the coset parameterization (C.5) , P 2 (A α ) is expanded with respect to γ matrices, Then, by plugging (4.6) with (4.5) , P 2 (J α ) can be obtained as with the coefficients Finally, the resulting expressions of L G and L B are given by, respectively, Here a new deformation parameterγ is defined aŝ γ ≡ 8η µ . Now one can read off the metric and NS-NS two-form from (4.9) and (4.10) . By performing the coordinate transformation, ρ 1 = cos ζ sinh ρ , ρ 2 = sin ζ sinh ρ , the resulting metric and NS-NS two-form are given by (4.14) Here there is a constraint 3 i=1 ρ 2 i = −1 . These expressions are quite similar to a one-parameter γ-deformed S 5 [27,28] and thus the solution with the metric (4.13) and the NS-NS two-form (4.14) may be regarded as a single parameter γ-deformation of AdS 5 .

Conclusion and discussion
We have shown that the gravity duals of NC gauge theories [33,34] can be derived from the Yang-Baxter sigma model description of the AdS 5 ×S 5 superstring with classical r-matrices.
The corresponding classical r-matrices are 1) solutions of CYBE, 2) skew-symmetric, 3) nilpotent and 4) abelian. These should be called abelian Jordanian deformations. As a result, the gravity duals are found to be integrable deformations of AdS 5 ×S 5 . Then, abelian twists of AdS 5 have also been investigated. These results provide a support for the gravity/CYBE correspondence proposed in [29].
Our main result here is the integrability of N =4 super Yang-Mills (SYM) theory on noncommutative (NC) spaces. Now there are an enormous amount of arguments on the integrability for scattering amplitudes of N =4 SYM. Integrable deformations of it would be found on NC spaces. Our analysis has revealed a relation between classical r-matrices and deformations parameters of NC spaces. There may be a close connection to deformation quantization of Kontsevich [35]. Thus one may expect a deep mathematical structure behind the correspondence. We hope that our result could shed light on new fundamental aspects of integrable deformations.

Acknowledgments
We would like to thank Io Kawaguchi for useful discussions and collaborations at the earlier

A Notation and convention
We shall here summarize our notation and convention, which basically follow [36] . We are concerned with deformations of AdS 5 . An explicit basis of su(2, 2) is the following.
The γ matrices are given by and satisfy the Clifford algebra The Lie algebra so(1, 4) is formed by the generators and then so(2, 4) = su(2, 2) is spanned by the following set: B Multi-parameter deformations of AdS 5 We present here multi-parameter deformations of AdS 5 by using the Yang-Baxter sigma model description with classical r-matrices. These may be regarded as a multi-parameter generalization of the gravity duals of NC gauge theories discussed in [33,34]. In the original construction [33,34] based on twisted T-dualities, it would be intricate to perform T-dualities many times. A technical advantage of the Yang-Baxter sigma model description is that a single r-matrix gives the corresponding metric and NS-NS two-form in a more direct way.
By following the analysis in section 3 , it is straightforward to get the deformed string action.
For simplicity, we shall write down only the resulting metric and NS-NS two-form, Here a scalar function G and a constant parameter K are defined as By taking the following identification of the parameters and performing a Wick rotation x 0 → ix 0 , one can reproduce the metric and NS-NS two-form of the two-parameter case [33,34]. C Three-parameter abelian twists of AdS 5 Let us consider here a three-parameter generalization of the abelian deformation of AdS 5 discussed in section 4 .
We will consider the following classical r-matrix, with deformation parameters µ i . Here h i are the three Cartan generators of su(2, 2) and belong to the fundamental representation, By using the r-matrix (C.1), the AdS 5 part of (2.1) can be rewritten as where A α = g −1 ∂ α g is restricted to su(2, 2) and the R-operator associated with (C.1) is determined by the rule (2.4) .
The one-parameter result in section 4 is reproduced by setting the parameters aŝ