Supercoset construction of Yang-Baxter deformed AdS$_5\times$S$^5$ backgrounds

We proceed to study Yang-Baxter deformations of the AdS$_5\times$S$^5$ superstring with the classical Yang-Baxter equation. We make a general argument on the supercoset construction and present the master formula to describe the dilaton in terms of classical $r$-matrices. The supercoset construction is explicitly performed for some classical $r$-matrices and the full backgrounds including the Ramond-Ramond (R-R) sector and dilaton are derived. Within the class of abelian $r$-matrices, the perfect agreement is shown for well-known examples including gravity duals of non-commutative gauge theories, $\gamma$-deformations of S$^5$ and Schr\"odinger spacetimes. It would be remarkable that the supercoset construction works well, even if the resulting backgrounds are not maximally supersymmetric. In particular, three-parameter $\gamma$-deformations of S$^5$ and Schr\"odinger spacetimes do not preserve any supersymmetries. As for non-abelian $r$-matrices, we will focus upon a specific example. The resulting background does not satisfy the equation of motion of the Neveu-Schwarz-Neveu-Schwarz (NS-NS) two-form because the R-R three-form is not closed.


Introduction
The Yang-Baxter deformation [1] is a systematic way to study integrable deformations of non-linear sigma models in two dimensions. Given a classical r-matrix satisfying the classical Yang-Baxter equation (CYBE), an integrable deformation is determined and the associated Lax pair follows automatically. This correspondence between a deformed geometry and a classical r-matrixindicates a profound connection between a differential geometry and a finite-size matrix. Hence it is significant to make the understanding of the Yang-Baxter deformation much deeper from the viewpoints of theoretical physics and pure mathematics.
The Yang-Baxter deformation was originally invented for principal chiral models with the modified classical Yang-Baxter equation (mCYBE). Now that it is generalized to symmetric cosets [2] and the homogeneous CYBE [3], one can study Yang-Baxter deformations of symmetric coset sigma models with a lot of examples of classical r-matrices. For the related affine algebras, see the series of works [4][5][6].
The most interesting coset sigma model is type IIB string theory on AdS 5 ×S 5 in the context of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [7]. The classical string action has been constructed in the Green-Schwarz formulation based on a supercoset [8] P SU(2, 2|4) SO(1, 4) × SO (5) .
This coset enjoys the Z 4 -grading property and ensures classical integrability [9] (for a nice review, see [10]). The integrability plays an important role in checking the conjectured relation in AdS/CFT (for a comprehensive review, see [11]).
By employing the Yang-Baxter deformation, Delduc, Magro and Vicedo constructed the classical action of a q-deformed AdS 5 ×S 5 superstring [12]. This deformation comes from the classical r-matrix of Drinfel'd-Jimbo type satisfying the mCYBE [13]. The string-frame metric and Neveu-Schwarz-Neveu-Schwarz (NS-NS) two-form were derived by Arutyunov, Borsato and Frolov [14]. Then they performed the supercoset construction and derived the remaining sector [15] (for earlier attempts, see [16,17]). As a result, the full background does not satisfy the equations of motion of type IIB supergravity, although it is related to a complete solution [18] via T-dualities apart from the dilaton part. In particular, the dilaton cannot be separated so that the Ramond-Ramond (R-R) flux should satisfy the Bianchi identity. Recently, Arutyunov et al. proposed an exciting conjecture that type IIB supergravity itself would get deformed, for example, the definition of R-R field strength may be modified [19]. This "modified gravity conjecture" may be connected to our result presented here.
One may also consider Yang-Baxter deformations of the AdS 5 ×S 5 superstring with classical r-matrices satisfying the homogeneous CYBE [20]. A strong advantage in this case is that partial deformations of AdS 5 ×S 5 are possible. In fact, for well-known backgrounds including gravity duals of noncommutative gauge theories [21,22], γ-deformations of S 5 [23,24], Schrödinger spacetimes [25], the associated classical r-matrices have been identified in a series of works [26][27][28][29][30][31][32] (for short summaries, see [33]). However, the analysis has been limited to the bosonic sector so far, and it is still necessary to confirm the R-R sector and dilaton by performing the supercoset construction explicitly.
The goal of this present work is to perform the supercoset construction and present the resulting backgrounds for some classical r-matrices. We will first give a general treatment basically by following the seminal paper by Arutyunov, Borsato and Frolov [15]. Then we derive the backgrounds for some classical r-matrices. As a byproduct, we present the master formula to describe the dilaton in terms of classical r-matrices.
Within the class of abelian classical r-matrices, the perfect agreement is shown for well-known examples including gravity duals of non-commutative gauge theories [21,22], γ-deformations of S 5 [23,24], and Schrödinger spacetimes [25]. It is worth noting that the supercoset construction works well, even though the resulting backgrounds are not maximally supersymmetric. More strikingly, three-parameter γ-deformations of S 5 and Schrödinger spacetimes do not preserve any supersymmetries [25]. Hence it seems likely that the supercoset construction works well with the class of the abelian classical r-matrices. It is consistent with the interpretation as TsT transformations [6,24,32,[34][35][36].
As for non-abelian classical r-matrices, we will focus on a specific example discussed in [29,30]. The resulting background does not satisfy the equation of motion of the NS-NS two-form because the Bianchi identity of the R-R three-form is broken, namely the field strength is not closed. It is also remarkable that this background is different from the one proposed in [29,30] and hence the identification made in [29,30] was not correct. Anyway, this result indicates that there would be some potential problems in the non-abelian cases.
This is really intriguing but just an example. It is of importance to study extensively other non-abelian r-matrices.
This paper is organized as follows. Section 2 introduces the classical action of Yang-Baxter deformed AdS 5 ×S 5 superstring based on the CYBE. In Sect. 3, we discuss the supercoset construction, by following the procedure of [15]. Most of the argument does not rely on specific expressions of classical r-matrices and is quite general. We present the conjectured master formula to describe the dilaton in terms of classical r-matrices. In Sect. 4, we present the resulting backgrounds for concrete examples of classical r-matrices. Section 5 is devoted to the conclusion and discussion. Appendix A provides a matrix representation of the superalgebra su(2, 2|4).
2 Yang-Baxter deformed AdS 5 ×S 5 superstring In this section, we give a short introduction to the classical action of Yang-Baxter deformed AdS 5 ×S 5 superstring based on the homogeneous CYBE [20]. This construction basically follows from the work with the mCYBE [12].
The deformed classical action of the AdS 5 ×S 5 superstring is given by where the left-invariant one-form A a is defined as with the world-sheet index a = (τ, σ). Here, the conformal gauge is supposed and the world-sheet metric is taken to be the diagonal form γ ab = diag(−1, +1). Hence there is no coupling of the dilaton to the world-sheet scalar curvature. The anti-symmetric tensor ǫ ab is normalized as ǫ τ σ = +1. The constant λ c in front of the action (2.1) is the 't Hooft coupling. The deformation is measured by a constant parameter η, and the undeformed AdS 5 ×S 5 action [8] is reproduced when η = 0.
A key ingredient in Yang-Baxter deformations is the operator R g defined as where a linear operator R : su(2, 2|4) → su(2, 2|4) is a solution of the CYBE 1 , This R-operator is connected to a skew-symmetric classical r-matrix in the tensorial notation through the following formula: In the original work [20], a wider class of R-operators is argued and their image is given by gl(4|4). The gl(4|4) image is restricted on su(2, 2|4) under the coset projection d, as pointed out in [31].

Supercoset construction
In this section, we shall consider the supercoset construction, starting from the deformed action (2.1). The following argument will be undertaken without fixing a specific expression of classical r-matrices and hence will be quite general. Our purpose here is to extract the R-R fluxes and dilaton, and hence we will investigate the deformed action at the quadratic level of fermions.

The su(2, 2|4) superalgebra
For the subsequent argument, it is necessary to determine our convention and notation for the su(2, 2|4) superalgebra. Hereafter, we will work with the following algebra [8]:

A group parametrization and the left-invariant current
Then let us introduce a parametrization of the group element g ∈ SU(2, 2|4) as follows: Herred, g b is a bosonic element and parametrized with an appropriate coordinate system, depending on the backgrounds we are concerned with, as in the previous works [26][27][28]. We assume that the bosonic element is parametrized as The coordinates x µ and z describe the Poincaré AdS 5 , and r, ζ, φ i=1,2,3 parametrize the round S 5 . Then g f is a group element generated by the supercharges as follows: where Q I θ I ≡ (Qαα) I (θαα) I (α = 1, . . . , 4;α = 1, . . . , 4) .
Then the left-invariant one-form A can be expanded as [8] A = (e m + i 2θ where the covariant derivative for θ is defined as Here, the last term represents the contribution of the R-R five-form field strength.
For later convenience, it is helpful to rearrange the above expansion of A with respect to the order of θ as follows: Here, A (p) is the p-th order of θ and the explicit expressions of A (p) are given by Thus we have prepared to write down the undeformed action of the AdS 5 ×S 5 superstring at the quadratic order of θ.

Decomposing the deformation operator
In the case of the deformed action (2.1), it is further necessary to expand the deformation operator in terms of θ, because R g contains the adjoint operation with g as denoted in (2.3).
We will basically follow the strategy of [15] hereafter.
Let us introduce the following operator O and expand it in terms of θ as Then the inverse operator O inv can also be expanded as Here, due to the relation O • O inv = 1, each of the components O inv (p) (p = 0, 1, 2) can be expressed as follows: In the following, for simplicity, we will concentrate only on the bosonic deformations 2 generated by bosonic generators a i , b i like Then the action of R g b • d can be evaluated as Here, from the relation in (2.5), λ n m and λ np m are expressed as (2) can be examined as follows.
The action of O inv (0) is given by where k m n is determined by the following relation: Then the action of O inv (1) is written as Here, the terms proportional to J mn are not explicitly written down because they do not contribute to the final expression.
Finally, the action of O inv (2) is evaluated as Here, the terms proportional to J mn have not been written down on the same reasoning.
Furthermore, the explicit expression of O inv (2) (Q I ) is not necessary for our argument because it always leads to higher-order contributions with O(θ 4 ) in the resulting Lagrangian.
Next is to evaluate the Lagrangian using the formulae obtained above.

The deformed Lagrangian at order θ 2
Let us now examine the deformed action at the second order of θ. Now the Lagrangian in (2.1) can be rewritten as whered is defined asd This Lagrangian can be expanded in terms of θ as Here, L (0) does not include any θ. The second-order term L (l,m,n) contains two θs. The set of subscripts (l, m, n) indicates the numbers of θ included ind(A a ), O inv and A b , respectively.
For example, in the case of L (2,0,0) , the two θs are included ind(A a ), and there is no θ in In the following, let us see each term of the expansion (3.24). The first one is L (0) and does not contain any fermions. This can be rewritten into the standard form as follows: Here we have used the relation e m a = e m µ ∂ a X µ , and the X µ s are the target-spacetime coordinates. The last expression (3.26) should be compared with the standard bosonic string with the spacetime metric G and NS-NS two-form B. Then one can obtain the following relations: Here we have introduced the vielbeinsẽ m M for the deformed metric for our later convenience. Note that the index M is raised and lowered by G M N and G M N , respectively.
Then let us evaluate the combination L (2,0,0) + L (0,0,2) . From the point of view of symmetry, this combination is convenient and can be evaluated as By the same reasoning, it is helpful to evaluate the combination L (1,1,0) + L (0,1,1) . The resulting expression is given by Finally, L (0,2,0) and L (1,0,1) are evaluated as, respectively, So far, we have derived the deformed Lagrangian at the quadratic level of θ. However, the resulting sum of the components evaluated above is quite intricate and we still need to recast it into the canonical form via coordinate transformations.

The canonical form of the Lagrangian
Here, let us perform coordinate transformations in order to realize the canonical form of the Lagrangian. This process is mainly composed of two steps, 1) the shift of X and 2) the rotation of θ.

The canonical form
First of all, let us present the canonical form of the Lagrangian at order θ 2 [37]: Here, the Γ m s are 32 × 32 gamma matrices composed of γm and γm as follows: Here, C is a charge conjugation matrix defined as The remaining task is to rewrite the Lagrangian L expanded above by performing a shift of X and a rotation of θ . We will explain each of the steps below. One can realize that the terms should appear with δ IJ from the expression of the canonical form (3.33). There is no obstacle for (a), however (b) involves terms like √ λ c 2θ I γ ab σ IJ 1 e m a k (mn) η λ n ,pq γ pq ∂ b θ J . (3.37) Such terms do not appear in the canonical form (3.33) and hence must be removed somehow.
A possible resolution is to shift X as [15] X µ −→ X µ +θ I δX µ IJ θ J , δX µ IJ ≡ 1 4 σ IJ 1 e n µ η λ n pq γ pq . (3.38) While this shift removes the problematic terms, it generates additional ones: Note here that these terms do not involve derivatives of θ.
At this stage, the quadratic Lagrangian including γ ab is written down as The next step is to see the terms with ǫ ab ∂ b θ in L. This part has the terms involving σ IJ 1 as well. Fortunately, the sift of X in (3.38) can eliminate the problematic terms simultaneously, while some additional terms including ǫ ab are again generated. Then the quadratic Lagrangian including ǫ ab is written down as ǫ IK e m a k nm η λ np γ p e q b k q r γ r + i 4 σ IK 1 e m a k nm η λ n,pq γ pq e r b k r s η λ s tu γ tu For the next step, it is convenient to switch from the 16 × 16 gamma matrices γ to the 32 × 32 ones Γ, and hence we will work in the 32 × 32 notation in the following. The lift-up rule is summarized in Appendix A, and it is straightforward to rewrite the Lagrangian.

Rotation of θ
After shifting X, the resulting derivative terms of θ take the following form: and depend on the index I. Hence we need to perform a Lorentz transformation for the spinor θ to remove the I dependence.
The first step is to determine the I-independent form of the vielbeins as a reference frame.
Hereafter, it is fixed by taking I = 1 in (3.43) as e m a = e p a k (pn) η nm + 2ηλ nm .  It seems difficult to present simple formulae of U (I) andŪ (I) that work for an "arbitrary" classical r-matrix. As a matter of course, given an explicit expression of a classical r-matrix, these quantities can be computed concretely.
However, at least for a simple class of classical r-matrices, we can propose the following concise forms: After all this, we have obtained the canonical form of the Lagrangian. In the actual derivation of R-R fluxes and the dilaton, we still need to use a concrete expression of a classical r-matrix and computation software like Mathematica or Maple, at least at the current level of understanding. We will present the resulting backgrounds for some example classical r-matrices in Sect. 4.

The master formula for the dilaton
We propose the master formula for dilaton, in which the dilaton is described in terms of the classical r-matrix directly. That is, just by putting the elements of the classical r-matrix, the associated dilaton is obtained directly, without passing through the supercoset construction.

Examples
Let us consider some examples of classical r-matrices satisfying the homogeneous CYBE.
Then it is possible to complete the supercoset construction and derive the resulting backgrounds.
For the following argument, let us introduce the terms "abelian" and "non-abelian" classical r-matrices. Suppose that a classical r-matrix is given by r = a ∧ b. It is called "abelian" when a and b commute with each other. If not, it is "non-abelian".

Gravity duals of noncommutative gauge theories
Let us discuss gravity duals of noncommutative gauge theories as Yang-Baxter deformations with the following classical r-matrix [27]: Here it is assumed that P µ are naturally embedded into 8 × 8 matrices like   P µ 0 4 This is an abelian classical r-matrix and satisfies the homogeneous CYBE.
The bosonic part has already been studied in [27], where the string-frame metric and NS-NS two-form are reproduced with the r-matrix (4.1). The R-R sector and dilaton can be determined by performing the supercoset construction.
The supercoset construction can be carried out by following the general argument in Sect. 3. In the present case, the key ingredient λ mn is given by Then U (I) andŪ (I) are obtained by using the formulae in (3.48).
With the general argument in Sect. 3, one can read off the following background: This is nothing but the solution found in [21,22] as a gravity dual of noncommutative gauge theories. Note that the dilaton can be reproduced by using the master formula (3.49) with (4.3).

γ-deformations of S 5
We shall discuss three-parameter γ-deformations of S 5 with the following classical r-matrix [26]: Here ν i (i = 1, 2, 3) are real constant parameters, and h a (a = 1, 2, 3) are the Cartan generators of su(4) embedded in 8 × 8 matrices as the lower diagonal block (for their matrix representation, see Appendix A). This is an abelian classical r-matrix and satisfies the CYBE.
The bosonic part has already been studied in [26]. The remaining task is to perform supercoset construction in order to determine the R-R sector and dilaton.
The quantities U (I) andŪ (I) are determined by Then, by following the general discussion, the full solution presented in [23,24] can be reproduced as Here we have introduced a scalar function G andγ i (i = 1, 2, 3) defined as Three coordinates ρ i satisfying the constraint 3 i=1 ρ 2 i = 1 are parametrized by two angle variables α and θ through the relation: ρ 1 ≡ sin α cos θ , ρ 2 ≡ sin α sin θ , ρ 3 ≡ cos α . (4.10) It should be remarked that the resulting background is non-supersymmetric other than for exceptional cases like ν 1 = ν 2 = ν 3 . But the supercoset construction still works well.
Note that the dilaton can be reproduced by using the master formula (3.49) with (4.6).
Note here that the classical r-matrix (4.11) contains a tensor product of an su (2,2) generator and an su(4) one. Hence the rotation of θ should be a ten-dimensional Lorentz transformation, and becomes intricate. The quantities U (I) andŪ (I) are given by, respectively, Note that λmň can be obtained from λmn, because λ mn is anti-symmetric.
After all, the full solution [25] has been reproduced as (4.13) and the other fields are zero. Here, the light-cone coordinates are defined as The S 5 metric is given by (4.14) Namely, the round S 5 is expressed as an S 1 -fibration over CP 2 , where 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 Σ 1 = 1 2 (cos ψ dθ + sin ψ sin θ dφ) , Σ 2 = 1 2 (sin ψ dθ − cos ψ sin θ dφ) , Σ 3 = 1 2 (dψ + cos θ dφ) , ω = sin 2 µ Σ 3 . It should be remarked that the R-R sector has not been deformed and the dilaton remains constant, though the expression of the fermionic sector is very complicated in the middle of the computation. The cancellation of the deformation effect is really non-trivial. Note also that the background (4.13) does not preserve any supersymmetries [25]. It may sound surprising that the supercoset construction still works well without the help of supersymmetries.
The constant dilaton of this background can be reproduced by using the master formula (3.49) with (4.12) as well.

A non-abelian classical r-matrix
So far, we have considered abelian classical r-matrices, for which it seems likely that supercoset construction works well even though the resulting background is non-supersymmetric.
The next significant issue is to study non-abelian classical r-matrices.
As for non-abelian classical r-matrices, there is no well-known example of the associated background. A nice candidate for non-abelian classical r-matrices is given by Note here that 1 4 is included in the expression and hence the image is extended from su(2, 2|4) to gl(4|4). However, it can be ignored due to the presence of the projection operator in the classical action as pointed out in [31].
To ensure that the resulting metric and NS-NS two-form are real, it is necessary to impose the reality condition [30] It is now convenient to introduce a 1 , a 2 as follows: Note here that the classical r-matrix (4.16) is non-abelian in general. The case that c 1 is pure imaginary (i.e., a 1 = 0) is exceptional and it becomes abelian.
The bosonic part has already been studied well [29][30][31], and the remaining task is to determine the R-R sector and dilaton by performing supercoset construction. The quantities U (I) andŪ (I) are obtained by using After all this, one can read off the resulting background: 4 Thus the classical r-matrix (4.16) does not lead to a solution of type IIB supergravity.
It is worth noting that the pathology vanishes when a 1 = 0. This is an exceptional case in which the classical r-matrix becomes abelian and the background (4.20) is reduced to the Hubeny-Rangamani-Ross solution [38]. This correspondence was originally argued in [30] and elaborated in [31]. Note here that the constant dilaton can be reproduced by using the master formula (3.49) with (4.19) again.
It would be valuable to see that the background (4.20) is different from the one constructed in [29]. The former (4.20) does not contain the R-R fluxes with the S 5 indices, while the latter does. It was conjectured in [29,30] that the classical r-matrix (4.16) should be associated with the latter, but it was not correct. Our supercoset construction has revealed that the classical r-matrix (4.16) should be associated with the background (4.20).
The result that the Bianchi identity is broken is similar to the q-deformed AdS 5 ×S 5 [15].
In fact, the background (4.20) satisfies the generalized type IIB supergravity equations of motion proposed in [19]. Detailed analysis will be presented in a future paper [39].

Conclusion and discussion
In this paper, we have discussed the supercoset construction in the Yang-Baxter deformed In particular, three-parameter γ-deformations of S 5 and Schrödinger spacetimes do not preserve any supersymmetries. For non-abelian r-matrices, we have concentrated on a specific example. The resulting background does not satisfy the equation of motion of the NS-NS two-form because the Ramond-Ramond three-form is not closed. Thus, at least so far, it seems likely that there would be no problem for abelian classical r-matrices, while there are some potential problems in the non-abelian cases. We will report on results on other examples of abelian and non-abelian classical r-matrices in the near future [39].
There are many open problems. The Yang-Baxter deformation has diverse applicability.
For example, it can be applied to the AdS 5 × T 1,1 background [40]. In this case, the Green-Schwarz string action has not yet been constructed. However, at least for the bosonic sector 5 , it has been shown that three-parameter γ-deformations of T 1,1 [23,42] can be reproduced as Yang-Baxter deformations with abelian classical r-matrices [40]. It is remarkable that the AdS 5 × T 1,1 background is not integrable because chaotic string solutions exist [43]. It would be of significance to construct the AdS 5 × T 1,1 superstring action and then investigate its Yang-Baxter deformations by following the procedure presented here.
It is also interesting to consider a supersymmetric extension of Yang-Baxter deformations of Minkowski spacetime [41]. As a toy model along this direction, it is easier to study the Nappi-Witten model [45]. Yang-Baxter invariance of this model has been discussed in [46].
It would be nice to argue its supersymmetric extension by employing the work [47] and further generalization with general symmetric two-forms [48].
We believe that our supercoset construction could capture the tip of an iceberg, namely the gravity/CYBE correspondence that denotes a non-trivial relation between type IIB supergravity and the classical Yang-Baxter equation.
A representation of su (2,2) It is convenient to introduce the following basis of su(2, 2) ≃ so (2,4): Here, γ µ are gamma matrices satisfying the Dirac algebra: where η µν is the four-dimensional Minkowski metric with mostly plus. It is convenient to adopt the following matrix realizations of the γ µ :

A conformal basis
It is also helpful to use the conformal basis: Here, the translation generators P µ , the Lorentz rotation generators L µν , the dilatation D, and the special conformal generators K µ are represented by, respectively, The non-vanishing commutation relations are given by A representation of su (4) It is easy to see that n ij = 1 4 [γ i , γ j ] (i, j = 1, . . . , 5) (A.7) generate so(5) by using the Clifford algebra Note that so (6)  Here we have used the following notation, A lift up to the 32 × 32 matrix representation In Sect. 3, it is necessary to rewrite the deformed Lagrangian in terms of ten-dimensional 32 × 32 gamma matrices Γ in order to read off the component fields of type IIB supergravity.
Hence we introduce a concise rule to switch from the 16 × 16 notation to the 32 × 32 one.
(A. 24) Note here that the right-hand side of (A.24) contains the factor Γ 01234 , which is evaluated as The insertion of this factor is necessary for an appropriate lift-up. For the detail of the lift-up, see, for example, [49].