Deformations of $T^{1,1}$ as Yang-Baxter sigma models

We consider a family of deformations of T^{1,1} in the Yang-Baxter sigma model approach. We first discuss a supercoset description of T^{1,1}, which makes manifest the full symmetry of the space and leads to the standard Sasaki-Einstein metric. Next, we consider three-parameter deformations of T^{1,1} by using classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). The resulting metric and NS-NS two-form agree exactly with the ones obtained via TsT transformations, and contain the Lunin-Maldacena background as a special case. It is worth noting that for AdS_5 x T^{1,1}, classical integrability for the full sector has been argued to be lost. Hence our result indicates that the Yang-Baxter sigma model approach is applicable even for non-integrable cosets. This observation suggests that the gravity/CYBE correspondence can be extended beyond integrable cases.


Introduction
A fascinating subject in string theory is dualities between gravitational theories and gauge theories. The original form proposed in [1] is the AdS/CFT correspondence, stating a duality between type IIB string theory on AdS 5 ×S 5 and N = 4 SU(N) super Yang-Mills (SYM) theory in four dimensions. The integrable structure behind AdS/CFT plays a significant role in this duality [2]. It enables one to exactly compute some physical quantities such as anomalous dimensions and scattering amplitudes, even at finite coupling without supersymmetries.
Here we are concerned with the string theory side of the correspondence. In the Green-Schwarz formalism, the classical action for the AdS 5 ×S 5 superstring is given by a 2d σ-model on the coset superspace [3], P SU(2, 2|4) SO(1, 4) × SO (5) . (1.1) Classical integrability for the AdS 5 ×S 5 superstring is closely related to the existence of a Z 4 -grading [4]. For an argument of integrability based on the Roiban-Siegel formalism [5], see [6]. A classification of possible integrable cosets is given in [7].
Recently, there has been progress in the study of integrable deformations of AdS 5 ×S 5 .
The Yang-Baxter sigma model approach [8] (generalized to the coset case in [9]) plays an important role in this direction.
A q-deformed action for the AdS 5 ×S 5 superstring has been constructed in [10]. Since a bosonic subsector of this action exhibits a q-deformed su (2), the full symmetry algebra is expected to be a q-deformed psu(2, 2|4) [9,11] 1 . In the end, the deformation used in [10] is the standard one with the classical r-matrix of Drinfeld-Jimbo type [14,15]. The metric in the string frame and NS-NS two-form were obtained in [16], though the complete supergravity solutions have not been found yet. Some limits of the deformed background are considered in [17,18]. A mirror TBA is discussed in [19]. A non-relativistic limit on the world-sheet is considered in [20]. Notably, the singularity of the metric disappears in this limit. Giant magnons are constructed in [19,21].
In addition to standard q-deformations, one may consider non-standard q-deformations (often called Jordanian deformations) [22,23]. Jordanian-deformed actions for AdS 5 ×S 5 have been constructed in [24]. The deformations are characterized by classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). So far, some r-matrices, corresponding to well-known string backgrounds such as Lunin-Maldacena-Frolov backgrounds [25,26], and 1 It would be nice to show an affine extension of psu(2, 2|4) by following the procedure [12,13].
the gravity duals of noncommutative gauge theories [27,28], have been found in [29] and [30], respectively. A new gravitational solution 2 was also constructed from an r-matrix in [31].
The relation between gravitational solutions and classical r-matrices may be referred to as the gravity/CYBE correspondence, as proposed in [29].
In this paper we consider type IIB superstrings on AdS 5 × T 1,1 . This geometry is realized by taking the near-horizon limit of a stack of N D3-branes sitting at the tip of a conifold [33]. The internal manifold T 1,1 is a Sasaki-Einstein manifold with S 2 ×S 3 topology and a SU(2) × SU(2) × U(1) R symmetry (for details on the conifold see [34], and for a review on aspects of AdS/CFT on this background see [35]). At the present moment, the Green-Schwarz string action on this background has not been constructed. Thus, we will focus only on the bosonic sector. In fact, already at the bosonic level one encounters a difficulty in applying the Yang-Baxter deformation to the usual coset decription of T 1,1 , as we discuss now. The usual description of T 1,1 as a coset is given by Although this describes the space topologically, the coset metric is not the Sasaki-Einstein metric that the space admits 3 (and the one which is required as a proper string background).
Since the class of deformations we are interested in are based on the coset description of the metric, before discussing deformations of T 1,1 we must reconsider the usual coset description (1.2). Our proposal is to describe T 1,1 as the bosonic part of the supercoset: As we shall show, a proper embedding of the denominator U(1)'s leads to the standard Sasaki-Einstein metric on T 1,1 . The appearance of the additional U(1) R in the numerator is rather natural given that it is part of the full flavor symmetry, and the grading is rather natural from the point of view of the N = 1 superconformal symmetry of the dual gauge theory. Furthermore, as we shall see, it allows us to easily construct general three-parameter deformations of this space. It would be interesting to study whether this supercoset is relevant to the construction of the Green-Schwarz action on this background, but this is not addressed here.
Next, we consider a family of three-parameter deformations of T 1,1 as Yang-Baxter sigma models with classical r-matrices satisfying the CYBE. This is analogous to the threeparameter real γ-deformations of S 5 as discussed in [26]. The resulting metric and NS-NS two form exactly agree with the ones obtained via TsT transformations in [38] and it contains the Lunin-Maldacena background [25] as a special case. This agreement indirectly supports that the proposed supercoset description is the appropriate description of bosonic strings on It is worth making a comment regarding the issue of integrability for T 1,1 . Although it is generally believed that an integrability structure is present in some sectors, it was argued in [39] that integrability for the full theory is lost due to the appearance of chaos in a certain subsector. Assuming that this conclusion is correct, our result indicates that the Yang-Baxter sigma model approach is applicable even for non-integrable cosets. This observation suggests that the gravity/CYBE correspondence can be extended beyond integrable cases; integrability is not essential for the correspondence and it is just the tip of an iceberg.
This paper is organized as follows. Section 2 considers a coset construction of T 1,1 . A supercoset description is proposed. In Section 3, we consider a family of deformations of T 1,1 as Yang-Baxter sigma model approach. We first give a short introduction to the Yang-Baxter sigma model approach. Then, the one-parameter deformation of T 1,1 is presented.
Finally, three-parameter deformations are considered. Section 4 is devoted to conclusion and discussion. Appendix A reviews an alternative way to derive the T 1,1 metric. In Appendix B, we give the detailed derivation of three-parameter deformation of T 1,1 .
2 A coset construction of T 1,1 In this section, we consider a coset construction of the T 1,1 metric . Instead of the conventional coset (1.2), we describe the supercoset (1.3) 4 .

A supercoset representation of T 1,1
As we have discussed, the coset representation (1.2) does not lead to the metric (2.1). Consider instead the following coset: The generators of the two su(2)'s and the u(1) R in the numerator of (2.2) are denoted by and M , respectively. Rather than 5 × 5 bosonic matrices, we choose a fundamental representation in terms of (4|1) × (4|1) supermatrices, i.e., Here σ i (i = 1, 2, 3) are the standard Pauli matrices, As we shall discuss below, the appearance of supermatrices-rather than bosonic matrices-is in fact natural from the perspective of the full AdS 5 × T 1,1 coset space.
It is easy to see that the generators satisfy the following relations: Here the structure constant is normalized as ǫ 123 = +1 and the su(2) indices are raised and lowered by the Killing form δ ab . Note that the supertrace of a supermatrix is defined as where A, D are bosonic block matrices and B, C are fermionic blocks. While two u(1)'s in the denominator of (2.2) are generated by respectively. Here T 1 denotes the U(1) in the usual coset description, and we have chosen a particular embedding of the second U(1), T 2 , that leads to the correct Sasaki-Einstein metric.

The T 1,1 metric from a supercoset
Let us first show that the supercoset (2.2) indeed leads to the T 1,1 metric.
It is convenient to introduce the orthogonal basis of the quotient vector space as follows: Here the diagonal element H is defined as With this basis, one may introduce a group element parametrized by Then the left-invariant one-form can be written in terms of the coordinates ψ, θ i and φ i (i = 1, 2) .
With (2.10) , the T 1,1 metric (2.1) is obtained from the simple expression, Here P is a projector to the coset space (2.7) and the associated projected current reads This shows that the supercoset (2.2) indeed leads to the Sasaki-Einstein metric (2.1).

What is the origin of the supercoset?
Finally, it is worth discussing the origin of the supermatrix representations in (2.3) . A possible explanation is the following. It is believed that string theory on AdS 5 × T 1,1 is dual to an N = 1 superconformal field theory in four dimensions [33]. The N = 1 superconformal group is composed of the conformal group SU(2, 2) , two sets of four real supersymmetry generators F A , F A , and the U(1) R -symmetry. These generators can be organized into the supermatrix, not contain the SU(2) × SU(2) flavor symmetry, unlike the case of P SU(2, 2|4) .
Thus, to include flavor symmetry it is necessary to consider an embedding of SU(2) × SU(2) × U(1) R into a bigger supermatrix. A natural candidate is the following (8|1) × (8|1) supermatrix: (2.14) Here P SU(2, 2|1) is located at the four corners of (2.14). Thus, the bosonic sector of the describes the bosonic sector of type IIB strings on AdS 5 × T 1,1 . This is indeed a rather natural description of the full P SU(2, 2|1) × SU(2) × SU(2) symmetry group and it may explain the origin of the supermatrix representation (2.3) 5 . As we shall discuss in Section 3, the Yang-Baxter deformation of this supercoset leads to a family of deformations of the metric and NS-NS two-form that exactly agree with the ones obtained in [38]. The Lunin- Maldacena deformation [25] is contained as a special case. We consider this fact as further support for the supermatrix description. It would be quite interesting to find further support for this interpretation from other points of view.
3 Deformations of T 1,1 as Yang-Baxter sigma models Thus far, we have presented a supercoset construction of the Sasaki-Einstein metric on T 1,1 .
In this section we use this description to study Yang-Baxter deformations.
By specifying classical r-matrices, we first discuss a one-parameter deformation in subsection 3.2 and then a three-parameter deformation in subsection 3.3.

The action of Yang-Baxter sigma models on T 1,1
An interesting class of deformations of nonlinear sigma models are given by Yang-Baxter sigma models [8,9]. The original procedure depends on classical r-matrices satisfying the 5 It would be interesting to study whether turning on the fermions in this supercoset sigma model is relevant for the construction of the Green-Schwarz action in this background, but we do not discuss this here.
modified CYBE (mCYBE). However, this approach is not applicable to partial deformations.
Since here we are interested in deformations of the internal manifold T 1,1 only, we apply the formalism of Yang-Baxter sigma models based on the CYBE [24] instead.
The original motivation is to study type IIB superstrings on AdS 5 × T 1,1 and its deformations. However, since the Green-Schwarz action for these backgrounds have not been constructed, we restrict ourselves to the bosonic sector. For simplicity, we consider deformations of the internal manifold T 1,1 only (the AdS 5 part is untouched) and therefore we focus on this part of the action.
The action is given by where the flat metric γ αβ and the anti-symmetric tensor ǫ αβ on the string world-sheet are normalized as γ αβ = diag(−1, 1) and ǫ τ σ = 1 . The projector P to the coset space is given in (2.12) . Here η is a parameter that measures deformations from T 1,1 . In the η → 0 limit, the action (3.1) reduces to the undeformed T 1,1 .
The left-invariant one-form is defined as usual by The group element g is parameterized as (2.9). Note that the supertrace appears in the action (3.1), even though all the fermions are set to zero in the present case.
The most important ingredient in (3.1) is a linear R-operator. The symbol R g denotes a dressed R-operator, given by the adjoint operation of the group, as: and contains a linear operator R satisfying the CYBE. This R-operator is related to the tensorial notation of a classical r-matrix through In our case, a i and b i are generators in su(2) ⊕ su(2) ⊕ u(1) R .

One-parameter deformation
We now consider examples of r-matrices describing deformations of T 1,1 .
Let us begin with the simplest example. This is provided by the abelian r-matrix, with deformation parameter µ . Here K 3 and L 3 are the Cartan generators of two su(2)'s, respectively. The fundamental representation is given in (2.3) .
Thus, the abelian r-matrix (3.5) is the algebraic origin of the γ-deformation of AdS 5 × T 1,1 .

Three-parameter deformation
It is straightforward to generalize the one-parameter case to the three-parameter case. Since there are three Cartan generators L 3 , K 3 and M , the most generic form for the abelian r-matrix is given by with three deformation parameters µ 1 , µ 2 and µ 3 . Note that the explicit appearance of the U(1) R symmetry-generated by M-in the supercoset (2.2) allows us to consider this threeparameter deformation.
The computation is completely parallel to the one-parameter case. Hence we do not repeat it here but simply give the final result. For details, see Appendix B .

Conclusion and discussion
In this paper we have considered a family of deformations of T 1,1 in the Yang-Baxter sigma model approach.
We first discussed a supercoset description of T 1,1 which leads to the standard Sasaki-Einstein metric. This is a rather natural description from the point of view of the N = 1 7 Here we also normalize the scaling factor in (3.1) as η = 1.
superconformal symmetry of the dual gauge theory. To the best of our knowledge this description has not been given explicitly in the literature.
Next, we considered three-parameter deformations of T 1,1 by using classical r-matrices satisfying the CYBE. The resulting metric and NS-NS two-form perfectly agree with the ones obtained via TsT transformations [25,38].
It was shown in [29] that three-parameter real γ-deformations AdS 5 ×S 5 [25,26] are realized by the Yang-Baxter sigma model approach with abelian classical r-matrices. Thus, the results obtained here may be regarded as a generalization of the work [29], giving further support for the gravity/CYBE correspondence. However, it should be stressed that there is a significant difference between S 5 and T 1,1 . The former is represented by a symmetric coset and it is obvious that a two-dimensional nonlinear sigma model on S 5 is integrable. For the case of T 1,1 , however, the claim that it is not integrable was made in [39], by showing the appearance of chaos in a subsector of the theory. Assuming that this result is correct, the class of deformations considered here are not regarded as integrable deformations. However, this would lead to the stronger statement that the gravity/CYBE correspondence would hold independently of integrability. If so, the implications of this paper would be quite important.
Let us make a few comments on possible further generalizations. An interesting class of metrics on S 2 ×S 3 is given by the well-known Y p,q metrics [40]. However, since these have not been explicitly constructed as coset metrics, it would be difficult to consider deformations in this approach. It would also be interesting to study additional coset spaces which may or may not be integrable. A possible candidate is the Lifshitz spacetime. The coset description was given in [41], and it has been argued to be non-integrable in [42].
Other important supercosets appear in descriptions of type IIA compactifications on AdS 4 , such as ABJM theory [43]. The supercoset description has been given in [44]. A natural question is whether the Yang-Baxter deformation can be applied to these theories, or other general supercoset sigma models.
funded by the Dutch Ministry of Education, Culture and Science (OCW).
A T 1,1 metric from the rescaling of vielbeins As we have discussed, the (SU(2) × SU(2))/U(1) coset description of T 1,1 does not lead to the Sasaki-Einstein metric (2.1) that the space admits. This comes as no surprise, since it is well known that coset spaces are not typically Einstein spaces. However, it was shown in [36] that given a coset space G/H it may be possible to rescale the vielbeins to obtain an Einstein space, without loosing the original symmetry of the coset space. This is in fact the case for T 1,1 , as discussed in [37]. Take the left-invariant current A = g −1 dg with g ∈ SU(2) × SU (2) and rescale the coset space directions by three parameters α, β, γ, as The term proportional to A + is the one projected out by the coset and is not rescaled. For α = β = γ = 1, this current describes a natural metric on the coset space (SU(2) × SU(2))/U (1) but not the Sasaki-Einstein metric. However, for arbitrary values of the parameters one finds 8 ds 2 = α 2 (dθ 2 1 +sin 2 θ 1 dφ 2 1 )+β 2 (dθ 2 2 +sin 2 θ 2 dφ 2 2 )+

B Derivation of three-parameter deformations
It would be useful to present here the detailed derivation of the three-parameter deformed metric (3.20) and NS-NS two-form (3.21) .