Lax pairs on Yang-Baxter deformed backgrounds

We explicitly derive Lax pairs for string theories on Yang-Baxter deformed backgrounds, 1) gravity duals for noncommutative gauge theories, 2) $\gamma$-deformations of S$^5$, 3) Schr\"odinger spacetimes and 4) abelian twists of the global AdS$_5$\,. Then we can find out a concise derivation of Lax pairs based on simple replacement rules. Furthermore, each of the above deformations can be reinterpreted as a twisted periodic boundary conditions with the undeformed background by using the rules. As another derivation, the Lax pair for gravity duals for noncommutative gauge theories is reproduced from the one for a $q$-deformed AdS$_5\times$S$^5$ by taking a scaling limit.


Introduction
The AdS/CFT correspondence is a fascinating subject in the study of string theory. The most famous one among a lot of variations is a duality between 10D type IIB string theory on the AdS 5 ×S 5 background and the 4D N = 4 super Yang-Mills theory at large N limit [1]. A great progress is that an integrable structure has been discovered behind this duality [2]. On the string-theory side, the Green-Schwarz string action on AdS 5 ×S 5 is constructed as a 2D coset sigma model [3] and the Z 4 -grading of the supercoset ensures the classical integrability [4] (For a big review of the AdS 5 ×S 5 superstring, see [5]). Although the essential mechanism of the duality has not been fully understood yet, the integrability has played a crucial role in checking conjectured relations in the AdS/CFT. It would be significant to consider integrable deformations of the AdS/CFT. It may shed light on a deeper structure behind gauge/gravity dualities beyond the conformal invariance.
On the string-theory side, an influential way is to employ the Yang-Baxter sigma model description [6]. This is a systematic way to study integrable deformations of 2D non-linear sigma models. By following this approach, an integrable deformation is specified by picking up a skew-symmetric classical r-matrix which satisfies the modified classical Yang-Baxter equation (mCYBE). The original argument [6] was restricted to principal chiral models. It is generalized to symmetric cosets by Delduc-Magro-Vicedo [7] 1 . Then they succeeded in constructing a q-deformed action of the AdS 5 ×S 5 superstring [12]. This formulation is also based on the mCYBE.
After that, it has been reformulated in [13] based on the (non-modified) classical Yang-Baxter equation (CYBE), where the Lax pair and the kappa transformation should be reconstructed and this generalization is not so trivial. An advantage in comparison to the mCYBE case is that partial deformations of AdS 5 ×S 5 can be considered. This is because the zero map R = 0 is allowed for the CYBE while not for the mCYBE. Furthermore, one can find many solutions of the CYBE. In fact, in a series of papers [14][15][16][17][18][19][20], many examples of (skewsymmetric) classical r-matrices have been identified with the well-known backgrounds such as γ-deformations of S 5 [21,22], gravity duals for noncommutative (NC) gauge theories [23] and Schrödinger spacetimes [24], in addition to new backgrounds [14]. This identification may be called the gravity/CYBE correspondence [15] (For a short summary, see [25]) and indicate that the moduli space of (a certain class of) solutions of type IIB supergravity can be described by the CYBE 2 .
In the recent, this correspondence has been generalized to integrable deformations of 4D Minkowski spacetime [27]. In particular, (T-duals of) 4D (A)dS spaces are reproduced as Yang-Baxter deformations of the Minkowski spacetimes. Furthermore, this development has an intimate connection with kappa-Minkowski spacetime [28] via preceding works e.g., [29].
For a recent argument with a gravity dual, see [20].
It is also remarkable that the gravity/CYBE correspondence seems to be valid beyond the integrability. There are many examples of non-integrable AdS/CFT correspondences.
2 Yang-Baxter deformations of string on AdS 5 ×S 5 We shall give a brief review of Yang-Baxter deformations of the AdS 5 ×S 5 superstring action based on the CYBE case [13] 3 .
The deformed classical action of the AdS 5 ×S 5 superstring is given by where the left-invariant one-form A α is defined as with the world-sheet index α = (τ, σ) . Here the conformal gauge is supposed and the worldsheet metric is taken as γ αβ = diag(−1, +1) . Hence there is no coupling of the dilaton to the world-sheet scalar curvature. The anti-symmetric tensor ǫ αβ is normalized as ǫ τ σ = +1 .
The constant λ c is the 't Hooft coupling. Note that η is a deformation parameter and hence the undeformed action [3] is reproduced when η = 0 .
A key ingredient in our analysis is the operator R g defined as R g (X) ≡ g −1 R(gXg −1 )g , X ∈ su(2, 2|4) , where a linear R-operator R : su(2, 2|4) → su(2, 2|4) is a solution of the classical Yang-Baxter This R-operator is related to a skew-symmetric classical r-matrix in the tensorial notation through the following supertrace operation on the second site:

5)
3 For the mCYBE case [12], see Appendix B. 4 In the original work [13], a wider class of R-operators whose image is given by gl(4|4) has been proposed.
We will concentrate here on a restricted class in which the image is su(2, 2|4) from the beginning, so as to deal with pre-projected quantities like the deformed current J itself, without introducing extra generators.
For general cases argued in [14,17], a more detailed study would be necessary.
where the classical r-matrix is represented by The projection operator d is defined as where P i (i = 0, 1, 2, 3) are projections to the Z 4 -graded components of su(2, 2|4) . In particular, P 0 (su(2, 2|4)) is a local symmetry of the classical action, so(1, 4) ⊕ so (5) . Note that the numerical coefficients are fixed by requiring the kappa-symmetry [13].
It is convenient to introduce the light-cone expression of A α like when we will study Lax pair in the following sections.

The bosonic part of the Lagrangian
Our aim here is to explicitly derive Lax pairs for the bosonic part of deformed actions. Hence it is convenient to rewrite the bosonic part of the deformed Lagrangian (2.1) as where J ± is a deformed current defined as Note here that the factor 2 in front of η comes from the projection operator d given in (2.7) .
By solving the following equations, the deformed current J ± is determined 5 . Then the metric and NS-NS two-form are evaluated from the symmetric and skew-symmetric parts regarding the world-sheet coordinates in (2.9), respectively. 5 In order to derive the metric and NS-NS two-form, it is enough to determine P 2 (J ± ) by solving the projected conditions as done in a series of the previous papers [14][15][16][17][18][19][20]. However, it is necessary here to determine J ± themselves so as to evaluate the form of Lax pair.
Taking a variation of the Lagrangian (2.9), the equation of motion is obtained as By definition, the undeformed current A ± satisfies the flatness condition, Then, in terms of the deformed current J ± , this condition can be rewritten as where we have introduced a new quantity defined as Note that CYBE g (X, Y ) vanishes if the R-operator satisfies the CYBE in (2.4). The relation (2.14) means that J ± also satisfies the flatness condition with the equation of motion E = 0 .
That is, J ± satisfies the flatness condition only on the on-shell, while A ± do even on the off-shell.
It is helpful to decompose J ± with the projection operators P 0 and P 2 like where we have used the completeness condition P 0 + P 2 = 1 . For the concrete expressions of the projection operators, see Appendix A. Then the equation of motion (2.12) can be rewritten into the following form: The flatness condition (2.13) can also be rewritten in a similar way: With the help of the linear independence of the grade 0 and grade 2 parts, one can obtain the following two conditions: − ] + 2η P 0 (R g (E)) = 0 , − ] + 2η P 2 (R g (E)) = 0 .
(2. 19) Note here that the terms proportional to η vanish on the on-shell, i.e., E = 0 .
Then the three conditions in (2.17) and (2.19) can be recast into the following set of the equations C i = 0 (i = 1, 2, 3) : (2.20) Namely, C i = 0 (i = 1, 2, 3) are satisfied on the on-shell and are equivalent to the equation of motion (2.12) and the flatness condition (2.13).

Lax pair
Finally, a Lax pair for the deformed action is given by with a spectral parameter λ ∈ C 6 . Note that the existence of the Lax pair (2.21) is based on the Z 2 -grading of AdS 5 ×S 5 .
As a matter of course, the flatness condition of L ± is equivalent to the equation of motion E = 0 [in (2.12)] and the flatness condition Z = 0 [in (2.13)] . In order to confirm the equivalence, it is helpful to notice that the right-hand side of (2.22) can rewritten in terms of C i as follows: Thus we have shown the equivalence.
In the following sections, we will evaluate explicit forms of the Lax pair (2.21) for some examples of classical r-matrices.

Lax pairs for gravity duals of NC gauge theories
In this section, let us study Lax pairs for gravity duals of noncommutative (NC) gauge theories from the viewpoint of Yang-Baxter deformations. The integrability of this background was recently shown in [16] in the sense of the kinematical integrability. The Lax pair was implicitly derived in [16], but the explicit expression has not been computed yet.
First of all, we evaluate explicit forms of Lax pairs with classical r-matrices in [16].
The resulting Lax pairs depend on two deformation parameters. Next, one may consider another derivation for a special one-parameter case. Then the associated Lax pair can also be reproduced by taking a scaling limit [37] of the one for a q-deformed AdS 5 [12,36].

Lax pairs from Yang-Baxter deformations
In the context of Yang-Baxter deformations, abelian Jordanian classical r-matrices [16] are associated with gravity duals of NC gauge theories [23]. The classical r-matrices in Because the square of the associated R-operator vanishes due to the properties of p µ 's, the classical r-matrices (3.1) are called Jordanian type. The classical r-matrices do not contain any generators in su(4) , and hence only the AdS 5 part is deformed. Therefore we will concentrate on only the AdS 5 part below.

The deformed metric and NS-NS two-form
To derive the metric and NS-NS two-form from the Lagrangian (2.9) , let us introduce a coordinate system through a parametrization of an SU(2, 2) element as follows: By solving the relation in (2.11) , the deformed current J α is explicitly determined as Then the resulting metric and NS-NS two-form are given by This result exactly agrees with the gravity duals of NC gauge theories [23]. When c 1 = c 2 = 0 , the Poincaré AdS 5 is reproduced.

Lax pair
Let us derive the associated Lax pairs. Now L NC ± are explicitly evaluated as (3.5) In the undeformed limit c 1 , c 2 → 0 , the above expressions are reduced to This is nothing but a Lax pair for the Poincaré AdS 5 .

Another derivation of Lax pair
It would be of good significance to describe another derivation of Lax pair (3.5) .
The undeformed current is now given by Then, by comparing the deformed current (3.3) with the undeformed one (3.7) , the deformation under our consideration can be reinterpreted as the following replacement rules: The above concise rules give rise to another simple derivation of the Lax pair (3.5). By applying the rules to the undeformed Lax pair (3.6) , the desired one (3.5) can be reproduced.
This derivation is quite similar to Frolov's construction of Lax pair for string on the γdeformed S 5 [22].

Twisted periodic boundary condition
In fact, due to the rule (3.8) , the deformation can be regarded as a twisted periodic boundary condition with the undeformed AdS 5 ×S 5 , as argued in [22].
For simplicity, suppose c 1 = 0 and c 2 = 0 . The analysis for the case with c 2 = 0 is quite similar, though there is a subtlety for the signature of the metric (For the detail, see [23]).
After performing the Yang-Baxter deformation (equivalently the associated TsT transformation) , the original coordinatesx 2 andx 3 for the undeformed AdS 5 ×S 5 are mapped to x 2 and x 3 . Then the relations are given by These relation indicate the following equivalence of Noether currents where P α i andP α i (i = 2, 3) are conserved currents associated with translation invariance for x i andx i directions, respectively. The τ -component of the relations means that the Then, evaluating the σcomponent of (3.10) leads to the relations: Finally, by integrating these expressions, one can figure out that the deformed background with the usual periodic boundary condition is equivalent to the undeformed AdS 5 ×S 5 with a twisted periodic boundary condition: Here P i are Noether charges for translation invariance in the x i directions.
Thus the Yang-Baxter deformation with the classical r-matrix (3.1) can be reinterpreted as a twisted periodic boundary condition with the usual AdS 5 ×S 5 .

A scaling limit of a Lax pair for a q-deformed AdS 5
In Section 3.1, we have derived the Lax pair (3.5) as Yang-Baxter deformations of AdS 5 .
Here we shall reproduce it as a scaling limit of a Lax pair for a q-deformed AdS 5 .
A scaling limit of a q-deformed AdS 5 ×S 5 We first give a short review of a scaling limit of a q-deformed AdS 5 ×S 5 [37]. In this limit, the metric and NS-NS two-form in (3.4) can be reproduced.
The starting point is the q-deformed metric and NS-NS two-form, Let us next rescale the coordinates as follows: (3.14) Here new coordinates x 0 , x 1 , x 2 , x 3 , z and a real constant ζ 0 have been introduced.
After taking the κ → 0 limit, the resulting metric and NS-NS two-form are given by This result exactly agrees with a one-parameter case of (3.4) through the identification For the S 5 part, this limit is nothing but the undeformed limit.
Lax pair -the third derivation Next, we will derive the Lax pair (3.5) by taking the scaling limit of a Lax pair for the q-deformed AdS 5 . This is the third derivation.
The Lax pair for the q-deformed AdS 5 ×S 5 was originally constructed in [12]. With a coordinate system [36], the Lax pair can be evaluated explicitly, as shown in Appendix B.
The remaining task is to take the scaling limit of the Lax pair (B.22) .
The first is to rewrite the Lax pair (B.22) in terms of the coordinates (3.14) with (3.16) .
For later convenience, the spectral parameter should be flipped as λ → −λ . Then, taking the κ → 0 limit leads to the following expression: In order to see that the result (3.17) is identical to the Lax pair (3.5) , it is necessary to perform a unitary transformation like After that, the transformed Lax pair sgrees with the one (3.5) , namely, Thus the scaling limit works well at the level of Lax pair.
It would be nice to consider this relation at the level of classical r-matrix. One may interpret the scaling limit as a rescaling of Drinfeld-Jimbo type classical r-matrix [38].
4 Lax pairs for γ-deformations of S 5 In this section, we shall study Yang-Baxter deformations with classical r-matrices corresponding to γ-deformations of S 5 . Concretely speaking, the associated Lax pairs are computed explicitly. The resulting expressions nicely agree with the Lax pairs obtained via TsT transformations of S 5 [22]. We will omit the AdS 5 part in the following.
Let us consider abelian classical r-matrices, which have been found in [15] ,

The deformed metric and NS-NS two-form
It is helpful to use the following representative of a group element of SU (4) , By solving the equations in (2.11) , the deformed current Jγ 1 ,γ 2 ,γ 3 α is determined as Here the parametersγ i are defined asγ 4) and the scalar function G(γ i ) is This deformed current (4.3) will play an important role in the following analysis.
Substituting the deformed current (4.3) into the Lagrangian (2.9) leads to the background Here new coordinates ρ i (i = 1, 2, 3) are defined as The metric and NS-NS two-form in (4.6) agree with 3-parameter γ-deformations of S 5 [22].
A particular one-parameter case witĥ corresponds to the Lunin-Maldacena solution [21] described by where the scalar function G is defined as (sin 2 2r + sin 4 r sin 2 2ζ) . This background is a holographic dual of the β-deformation of the N = 4 super Yang-Mills theory [39].

Lax pair
The next task is to evaluate the Lax pair in (2.21) with the classical r-matrix (4.1) . The components Lγ 1 ,γ 2 ,γ 3 ± are given by Note here that, in the undeformed limitγ i → 0, This is just a Lax pair for the undeformed S 5 .

Another derivation of Lax pair
Then, let us consider a simple derivation of the Lax pair (4.11) , as in the previous section.
The undeformed current is given by By comparing the deformed current (4.3) with the undeformed one (4.13) , we can identify the following replacement rules: Due to the replacement rules (4.14) , the deformed Lax pair (4.11) can be reconstructed from the undeformed one (4.12) . In fact, the replacement rules (4.14) are identical to a TsT-transformation [22] 7 , and hence the Lax pair (4.11) is equivalent to the one in [22] 8 .
This result further confirms the correspondence between a Yang-Baxter deformation with (4.1) and a TsT transformation [15]. 7 In our argument, the rules are identified on the off-shell level, but the one in [22] is done on the on-shell. 8 To see this equivalence (up to small differences of convention), we have to perform a gauge transformation h = exp[−iζn 02 ] exp i 2 rγ 2 exp π 2 (n 12 + in 03 ) and a Möbius transformation for a spectral parameter λ → λ+1 λ−1 .
Finally, it is worth mentioning the reinterpretation of the deformation as a twisted periodic boundary condition. This fact was originally shown in [22]. The twisted periodic boundary condition with the undeformed AdS 5 ×S 5 is given bỹ with γ i ≡γ/ √ λ c . Here J i are Noether charges for rotation invariance in the φ i directions.
Integers n i are winding numbers along the φ i directions.

Lax pairs for Schrödinger spacetimes
Let us consider a classical r-matrix which deforms both AdS 5 and S 5 . Such an r-matrix contains generators of both su(2, 2) and su(4) . A simple example is the following [19]: For convention of the generators, see Appendix A. This r-matrix (5.1) is associated with Schrödinger spacetimes realized in type IIB supergravity [24], as shown in [19].

The deformed metric and NS-NS two-form
The bosonic group elements of SU(2, 2) and SU(4) are parameterized as The deformed current J ± can be expanded in terms of the generators of su(2, 2) ⊕ su(4) .
Then, by solving the equations in (2.11) , J ± is determined as Here we have performed a coordinate transformation, With the deformed current (5.3) , the resulting background is given by Here the S 5 metric is written as an S 1 -fibration over CP 2 , Now χ is the fiber coordinate and ω is a one-form potential of the Kähler form on CP 2 . The symbols Σ i (i = 1, 2, 3) and ω are defined as It is remarkable that only the AdS 5 metric is deformed while the S 5 part is not, in spite of the expression of the classical r-matrix (5.1) . On the other hand, the NS-NS two-form carries two indices, one of which is from AdS 5 and the other is S 5 .

Lax pair
In a similar way, one can evaluate the associated Lax pair. The resulting expression is a bit messy but given by It would be helpful to check the undeformed limit. As η → 0 , the above Lax pair L Sch ± is reduced to the following:

Another derivation of Lax pair
Let us consider another derivation of the Lax pair again. One can see the replacement rules by comparing the deformed current with the undeformed one, as before.
The undeformed current is decomposed into the AdS 5 and S 5 components like where A a ± and A s ± are given by Then the Lax pair (5.7) can be reproduced by applying the replacement rules (5.11) to the undeformed Lax pair (5.8) .
Finally, let us mention about the reinterpretation of the deformation as a twisted periodic boundary condition. Similarly, one can see that the following boundary conditioñ with the undeformed AdS 5 ×S 5 is equivalent to the deformed geometry with the usual periodic boundary condition. Here P − and J χ are Noether charges for translation and rotation invariance for the x − and χ directions, respectively. An integer n χ is a winding number for the χ direction. It may be interesting to consider a relation between the above argument and the symmetric two-form studied in [41]. This r-matrix deforms only the AdS 5 part, hence we will omit the S 5 part hereafter.

Another derivation of Lax pair
Even in this case, one can read off the replacement rules as well.
The undeformed current is − cos θ n a 03 . (6.8) Then, by comparing the deformed current (6.3) with the undeformed one (6.8) , the replacement rules are identified as By applying the replacement rules to the undeformed Lax pair (6.7) , one can reproduce the Lax pair (6.6) as well.
Again, one can reinterpret the deformation as a twisted periodic boundary condition.
After performing a similar analysis, the twisted boundary conditioñ with the undeformed AdS 5 ×S 5 is equivalent to the deformed background with a periodic boundary condition. Here J i are Noether charges for rotation invariance in the φ i directions.
Integers n i are winding numbers along the φ i directions.

Conclusion and discussion
We have explicitly derived Lax pairs for string theories on Yang-Baxter deformed backgrounds, 1) gravity duals for NC gauge theories, 2) γ-deformations of S 5 , 3) Schrödinger spacetimes and 4) abelian twists of the global AdS 5 . As another derivation, the Lax pair for gravity duals for NC gauge theories has been reproduced from the one for a q-deformed as in the work of [22]. It would be interesting to study the fermionic sector by following the work [40].
This simple derivation really helps us to check the direct computation of Lax pairs based on Yang-Baxter deformations. In addition, it enables us to derive Lax pairs for Yang-Baxter deformations of Minkowski spacetime [27], for which the universal expression of Lax pair has not been obtained yet. Our procedure can play a significant role in studying along this direction. The result would be reported in another place [42].
A more general question is what is the class of classical r-matrix for which one can deduce the replacement rule. Probably, it would be possible for some restricted r-matrices. This is also concerned with another question, what is the class of classical r-matrices for which the insertion of the operator can be eliminated by changing a boundary condition on the string world-sheet. It would be quite important to answer these questions.
We believe that our concise prescription to construct Lax pairs would be helpful for further understanding of the gravity/CYBE correspondence.

Acknowledgments
We are very grateful to Io Kawaguchi for collaboration at an early stage of this work. We also appreciate Heng-Yu Chen, Takuya

Appendix A Notation and convention
We summarize here our notation and convention of the su(2, 2) and su(4) generators.

The gamma matrices
Let us first introduce the following gamma matrices: To embed su(2, 2) and su(4) into su(2, 2|4) , we follow an 8 × 8 matrix representation as Note that each block of the matrices is a 4 × 4 matrix.
The su(2, 2) and su(4) generators The Lie algebras su(2, 2) ∼ so (2,4) and su(4) ∼ so(6) are spanned as follows: The subalgebras so (1,4) and so (5) in the spinor representation are formed as For a coset construction of Poincaré AdS 5 , it is useful to employ the following basis: Here the generators p µ , k µ and the Cartan generators h 1 , h 2 , h 3 are defined as Since non-Cartan generators of su(4) are not used in our analysis here, we will not write them down explicitly.

The bosonic coset projectors
In deriving the bosonic part of Lax pairs, it is necessary to employ the coset projectors P 0 and P 2 regarding the Z 2 -grading property. The projectors P 0 and P 2 are decomposed into the AdS 5 part and the S 5 part like where P a,s 0 and P a,s 2 are the following coset projectors for so (2,4) and su (4) , These coset projectors can be represented by the su(2, 2) and su(4) generators as follows:  A typical skew-symmetric solution of the mCYBE is Drinfeld-Jimbo type [38]. The classical action of the deformed AdS 5 ×S 5 superstring associated with this r-matrix was constructed by Delduc-Magro-Vicedo [12]. The metric (in the string frame) and NS-NS twoform have been computed in [36]. The deformed background is often called the η-deformed AdS 5 ×S 5 . Some specific limits [43] and a mirror description [44,45] have been studied. For various classical solutions, see [46][47][48][49][50]. Two-parameter generalizations have also been studied in [43,51]. For some arguments towards the complete supergravity solution, see [37,52,53].

B.1 Yang-Baxter deformations from the mCYBE
Let us first give a short review on the Yang-Baxter deformations of the AdS 5 ×S 5 superstring based on the mCYBE case [12].
A q-deformed classical action of the AdS 5 ×S 5 superstring [12] is given by The definition of A α and R g is the same as in Section 2. A main difference is that the linear R-operator should satisfies the mCYBE The projection operators d is slightly different from the CYBE case like Namely, the coefficient in front of P 2 depends on η . This comes from the difference of the kappa transformation.

The bosonic part of the Lagrangian
We consider here the bosonic part of the deformed action (B.1) . The Lagrangian can be rewritten into a simple form, with the deformed current J ± defined as The expression of J ± is determined by solving the following equations: By taking a variation of (B.4) , the equation of motion is obtained as The undeformed current A ± automatically satisfies the flatness condition which can be rewritten in terms of J ± as Note that the quantity Finally, a Lax pair [12] is given by with a spectral parameter λ ∈ C . The flatness condition of L ± B.2 A Lax pair for a q-deformed AdS 5 ×S 5 We shall study the bosonic part with a classical r-matrix of Drinfeld-Jimbo type [38], where E ab (a, b = 1, . . . , 4) and E cd (c, d = 5, . . . , 8) are the fundamental representations of su(2, 2) and su(4) , respectively. This is a solution of the mCYBE (B.2) .