Yang-Baxter deformations of the AdS5 × T1,1 superstring and their backgrounds

We consider three-parameter Yang-Baxter deformations of the AdS5× T1,1 superstring for abelian r-matrices which are solutions of the classical Yang-Baxter equation. We find the NSNS fields of two new backgrounds which are dual to the dipole deformed Klebanov-Witten gauge theory and to the nonrelativistic Klebanov-Witten gauge theory with Schrödinger symmetry.


Introduction
The AdS/CFT correspondence conjectures that certain gauge theories have a dual description in terms of string theories. The first case of the AdS/CFT correspondence states that N = 4 supersymmetric Yang-Mills theory on a four-dimensional flat spacetime is dual to type IIB superstring theory propagating in AdS 5 × S 5 [1]. One of the most important features of the AdS/CFT correspondence is its integrability which in the string theory side is associated to the existence of a Lax connection ensuring the existence of an infinite number of conserved charges. In the case of AdS 5 × S 5 superstring, the theory is described by a σ-model on the supercoset PSU(2,2|4) SO(1,4)×SO(5) [2] and the Z 4 -grading of the psu(2, 2|4) superalgebra is an essential ingredient to get a Lax connection [3]. The same happens for the AdS 4 × CP 3 superstrings [4], partially described by the supercoset UOSp(2,2|6) SO(1,3)×U(3) [5,6], which also has Z 4 -grading and is integrable [5].
Another way to get integrable theories is to start with an integrable model and then deformed it in such a way that integrability is preserved. This is accomplished by introducing r-matrices that satisfy the Yang-Baxter equation [7]. When applied to the AdS 5 × S 5 case [8,9] the superstring will propagate on what is called a η-deformed background which is not a solution of the standard type IIB supergravity equations [10,11], leading to the proposal of generalized supergravities [12,13].
Deformations based on r-matrices that satisfy the classical Yang-Baxter equation (CYBE) can also be considered [14]. When applied to superstrings in AdS 5 × S 5 [15][16][17][18][19]  The Klebanov-Witten gauge theory is obtained by putting N D3-branes on the singularity of M 1,4 × Y 6 , where M 1,4 is the four-dimensional Minkowski space and Y 6 a Ricci flat Calabi-Yau cone C(X 5 ) with base X 5 [50]. Near the horizon the geometry becomes AdS 5 × X 5 , where X 5 is a compact Sasaki-Einstein manifold, i.e., an odd-dimensional Riemannian manifold such that its cone C(X 5 ) is a Calabi-Yau flat manifold [56]. Taking X 5 as T 1,1 , 5 only 1/4 of the supersymmetries are preserved so that we have N = 1 supersymmetry in four dimensions. The superpotential has a SU(2) × SU(2) × U(1) symmetry, with U(1) being part of the R-symmetry that gives the N = 1 supersymmetry, and SU(2) × SU(2) being a flavor symmetry which is not included in the N = 1 superconformal group in four dimensions PSU(2, 2|1) [58][59][60]. Thus, the full isometry group is PSU(2, 2|1)×SU(2)×SU (2). The bosonic part of the superalgebra g = psu(2, 2|1) on which we construct the σ-model is su(2, 4) ⊗ u(1). The generators of psu(2, 2|1) can be written as supermatrices which are formed by blocks that correspond to bosonic (diagonal) and fermionic (anti-diagonal) generators, The isometry group of AdS 5 × T 1,1 is given by the coset which is not the bosonic part of any supercoset [61,62]. Besides that, the coset for T 1,1 does not lead to the standard Sasaki-Einstein metric for T 1,1 . This happens because neither the bosonic subalgebra su(2) ⊗ u(1) nor the isometry group (2.2) captures the full isometries of the theory. All this can be overcome by extending the coset (2.2) to [54] AdS where the U(1) R now appears as part of the global symmetries and a second U(1) was added in order to preserve the number of parameters that describe the space. Thus, in terms of this extended Z 2 -graded algebra, the symmetric coset for AdS 5 × T 1,1 is taken as (2.4) The supermatrix has the block structure

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where the dashed lines split the algebras corresponding to the subspaces AdS 5 and T 1,1 , while the solid lines split the M 8×8 and M 1×1 bosonic blocks.
The algebra for the global symmetry of the AdS 5 space is and Str (K m K n ) = η mn , m, n = 0, 1, 2, 3, 4.
The T 1,1 space can be written as the coset in We also have Str (K m K n ) = − 1 3 δ mn , m, n = 5, . . . , 9. (2.20) The where T 1 generates the original U(1) in (2.2). An appropriate coset representative is then The coset representative that will allow use for AdS 4 × T 1,1 is then The projector P 2 on g (2) can be defined as Applied to A = g −1 dg, the Maurer-Cartan one-form, we get Then, we can compute the AdS 5 × T 1,1 metric from and where (θ 1 , φ 1 ) and (θ 2 , φ 2 ) parametrize the two spheres of T 1,1 and 0 ≤ φ 3 ≤ 2π.
The metric (2.30) was first obtained in [63] and describes the basis of a six-dimensional cone. It can be understood as the intersection of a cone and a sphere in C 4 such that its topology is S 2 × S 3 , and that the metric is a U(1) bundle over S 2 × S 2 . Besides that, SO(4) ∼ = SU(2) × SU(2) acts transitively on S 2 × S 3 and U(1) leaves each point of it fixed so that T 1,1 is described by the coset (SU(2) × SU(2)) /U(1).

Yang-Baxter deformed backgrounds
In this section we present some r-matrices satisfying the CYBE and build the corresponding deformed background identifying its gravity dual. As mentioned before, the background can be deformed partially by choosing generators on each subspace. The bosonic Yang-Baxter deformed action is [8] where A = g −1 dg ∈ g, γ αβ is the worldsheet metric and ε αβ is the Levi-Civita symbol. P 2 was defined in (2.24) and the deformed current one-form is where η is the deformation parameter. The dressed R operator R g is defined as

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Moreover, we can compute P 2 (J) in (3.1) by defining the action of P 2 as The coefficients j m can be calculated from where the matrix components C n m are those of The matrix Λ has components defined as Then, from (3.1), we can read off the metric and the B-field as [49] The three-parameter β-deformed of T 1,1 was obtained in [54] by a Yang-Baxter deformation and in [55] by a TsT transformation in perfect agreement. In this case the r-matrix was In the following subsections we will introduce two more r-matrices and the corresponding deformations they produce.

Dipole deformed Klebanov-Witten theory
Let us first consider an Abelian r-matrix like where X 3 , Y 3 and M are the Cartan generators of su(2)⊕su(2)⊕u(1) and µ i , i = 1, 2, 3, are the deformation parameters. 6 In this case (3.11) combines generators of both subspaces, which will lead to a deformation of the entire AdS 5 × T 1,1 background. The nonzero components of Λ n m in (3.7) are (3.12) 6 The deformation parameter η can always be absorbed in the r-matrix such that it is present in the µi's.

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It is worth mentioning that the choice of generators in (3.11) is dictated by the place where we want put the two-tori from the TsT perspective. In the present case we have one coordinate in AdS 5 and a combination of the U(1)'s in T 1,1 . The resulting metric (3.16) has deformations along the x 3 -direction in AdS 5 and along the angles φ 1 , φ 2 and φ 3 in T 1,1 .

Nonrelativistic Klebanov-Witten theory
In order to construct this deformation we must write the AdS 5 space in light-cone coordinates. Thus, the coset representative is now while for the T 1,1 we keep the same form as in (2.22). The AdS 5 metric is then while the T 1,1 metric is given by (2.30).
Let us now consider the r-matrix (3.11) with p 2 replaced by p − , 7 where X 3 , Y 3 and M are Cartan generators of the algebra. Taking the same steps as in the previous case we find that the nonzero components of Λ n m are

22)
7 In this case we identify x− ∼ x− + 2πr − , such that p − = i∂x − can be interpreted as the number operator p − = N/r − . Moreover, if we consider x+ to be the time then p + is the energy [64].

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while the nonzero elements of C n m are now where The deformed metric is then where now (3.27) The first two terms in (3.25) is the metric of a Schrödinger spacetime. 8 The choice of generators in (3.21) is very similar to the one in (3.11). Now, however, the two-tori defined by the TsT transformation takes the x − coordinate and a combination of the internal U(1)'s in T 1,1 and does not introduce any noncommutativity in the dual field theory. The metric (3.25) coincide with the Sch 5 × T 1,1 obtained in [47] for µ 1 = n 1 /2, µ 2 = n 2 /2 and µ 3 = −n 3 , where n i (i = 1, 2, 3) are the deformation parameters.

Conclusions
In this paper we have derived the metric and the B-field for the gravity duals of the dipole-deformed and the nonrelativistic Klebanov-Witten theory as Yang-Baxter deformations. We made use of an extended coset description of AdS 5 × T 1,1 which simplified the computation of the undeformed background and its deformation. We considered two abelian r-matrices with three-parameter satisfying the classical Yang-Baxter equation. The first r-matrix was composed by a momentum generator in AdS and a combination of the three U(1)'s generators of the internal space which lead to the gravity dual of the dipoledeformed Klebanov-Witten theory which should be obtained by TsT transformation of the AdS 5 × T 1,1 background. In second case we have also a momentum operator in AdS and a combination of the three U(1) generators in T 1,1 . It produced the Sch 5 × T 1,1 background which, having Schrödinger symmetry, corresponds to the nonrelativistic Klebanov-Witten theory [45]. The next step is to compute the RR fields of the deformed backgrounds. To get them we have to consider the fermionic sector as in [19]. The fact that we have not included the fermionic sector of the supercoset does not mean that we are unable to check the supergravity equations for the new backgrounds. Since the r-matrices that we used in the bosonic background are abelian they satisfy trivially the unimodularity condition, which is a sufficient for the background to satisfy the supergravity equations [12,13,33].
Another interesting case which deserves further study is the dual of the dipole deformation of N = 1 SU(N ) × SU(N ) Yang-Mills theory as well as its nonrelativistic limits. 9 The dynamical z factor is the exponent in the power of the radial direction in the z −2z dx 2 + term. To have Schrödinger symmetry we must have z = 2. The relativistic symmetry corresponds to z = 1. 10 The harmonic function is denoted in general as Φ , where 1, 2 are labels for the SU(2)'s, and r and q are U(1) charges [58,66]. 11 In [47], the harmonic function Φ is defined as the non-negative length square of the Killing vector K on T 1,1 , Φ = K 2 = gijK i K j with i, j = 1, 2, 3, where K = (µ1∂ φ 1 , µ2∂ φ 2 , µ3∂ φ 3 ).

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A A basis for so(2, 4) algebra Let us choose the following representation for γ µ