Schr\"odinger geometries arising from Yang-Baxter deformations

We present further examples of the correspondence between solutions of type IIB supergravity and classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). In the previous works, classical $r$-matrices have been composed of generators of only one of either $\mathfrak{so}(2,4)$ or $\mathfrak{so}(6)$. In this paper, we consider some examples of $r$-matrices with both of them. The $r$-matrices of this kind contain (generalized) Schr\"odinger spacetimes and gravity duals of dipole theories. It is known that the generalized Schr\"odinger spacetimes can also be obtained via a certain class of TsT transformations called null Melvin twists. The metric and NS-NS two-form are reproduced by following the Yang-Baxter sigma-model description.


Introduction
The gauge/gravity correspondence provides a fascinating arena for studying new aspects of string theory. A well-studied example is the duality between type IIB superstring theory on AdS 5 ×S 5 and the N =4 super Yang-Mills (SYM) theory in four dimensions [1]. This duality enjoys a powerful property, integrability (For a big review, see [2]). It enables one to compute some physical quantities exactly even at finite coupling without supersymmetries.
Thus the integrable structure behind the duality will be more and more important in the future study.
The integrable structure may provide another insight as well. It is integrable deformations of the AdS/CFT correspondence. It is well recognized that type IIB string theory on AdS 5 ×S 5 [3], which is often called the AdS 5 ×S 5 superstring, is classically integrable [4].
A systematic way to study the deformations is to employ the Yang-Baxter sigma-model formulation, which was originally proposed by Klimcik [5] for principal chiral models 1 and generalized by Delduc-Magro-Vicedo [9] to coset sigma models. In this formalism, a deformed target space is determined by specifying the associated classical r-matrix satisfying the modified classical Yang-Baxter equation (mCYBE). This formalism has been extended to the case with the Wess-Zumino-Witten (WZW)-like term [10] 2 . One may consider a generalization of it to the standard classical Yang-Baxter equation (CYBE) [13]. A threedimensional Schrödinger spacetime is contained here as a simple case with the rank 1 and for this case the classical integrable structure has been recognized as Jordanian twists [14] 3 .
The formalism of the Yang-Baxter deformations is applicable to the AdS 5 ×S 5 superstring [17]. Then it has been generalized to the CYBE case [18]. In comparison to the mCYBE, there are two advantages. The first is that partial deformations of AdS 5 ×S 5 are possible.
For example, one may consider a deformation of either AdS 5 or S 5 , while it seems likely, so far as we see the current achievement, that both of them are inevitably deformed in the case of mCYBE [17]. The second is that there are a variety of solutions of the CYBE. In fact, we have already found various examples of classical r-matrices which correspond to solutions of type IIB supergravity. For example, the Lunin-Maldacena-Frolov background [19,20] and the gravity duals of non-commutative gauge theories [21,22] have been reproduced in [23] and [24], respectively, by following the Yang-Baxter sigma-model description. Thus, the correspondence of this kind may be called the gravity/CYBE correspondence [23] (For a short summary see, [25]).
A remarkable point of the gravity/CYBE correspondence is the connection to TsT transformations [26,27]. That is, a certain class of classical r-matrices satisfying the CYBE is related to a solution generation techniques in type IIB supergravity. More strikingly, it is shown in [28] that this connection holds even for Yang-Baxter deformations of a nonintegrable background, AdS 5 × T 1,1 [29]. Motivated by these developments, our purpose here is to pursuit the relation to TsT transformations and discover further examples of the gravity/CYBE correspondence.
Indeed, we will present further examples of the gravity/CYBE correspondence. In the previous works, classical r-matrices have been composed of generators of only one of either so (2,4) or so(6) . In the preset paper, we consider some examples of r-matrices with both of them, which are schematically of the form; The classical r-matrices of this kind contain (generalized) Schrödinger spacetimes and gravity duals of dipole theories. It is known that the generalized Schrödinger spacetimes can also be obtained via a certain class of TsT transformations called null Melvin twists as discussed in [30] and [31,32]. The metric and NS-NS two-form are reproduced by following the Yang-Baxter sigma-model description.
This paper is organized as follows. Section 2 gives a short review of Yang-Baxter deformations of the AdS 5 ×S 5 superstring based on the CYBE. Section 3 presents classical r-matrices associated with (generalized) Schrödinger spacetimes. Section 4 considers dipole deformations of the AdS 5 ×S 5 background with the corresponding r-matrices. Section 5 is devoted to conclusion and discussion. In Appendix A, we summarize our convention and notation of the so(2, 4) and so (6) generators. In Appendix B, the T-duality rules are listed.
Appendix C gives a derivation of three-parameter dipole deformations of AdS 5 ×S 5 .
2 Yang-Baxter deformations of AdS 5 ×S 5 Let us first recall the formulation of Yang-Baxter sigma models.
The Yang-Baxter sigma-model description was originally developed for purely bosonic non-linear sigma models [5,9]. It is now generalized to supersymmetric cases, and integrable deformations of the AdS 5 ×S 5 superstring can be described based on the mCYBE [17] and the CYBE [18].
Here we consider the latter case [18], where the deformed classical action is given by where the left-invariant one-form is defined as When η = 0 , the classical action (2.1) is reduced to the undeformed one [3]. Here the string world-sheet metric is taken to be flat i.e., γ αβ = diag(−1, 1) . The anti-symmetric tensor ǫ αβ is normalized as ǫ τ σ = 1 .
The operator R g is defined as where a linear operator R is a solution of CYBE rather than mCYBE. The R-operator is related to a classical r-matrix in the tensorial notation through The generators a i , b i are some elements of su(2, 2|4) . The supertrace STr of (4|4) × (4|4) supermatrix is defined as where each of the blocks A, B, C, D is a 4 × 4 matrix of complex numbers, which plays a crucial role in our argument.
To evaluate the action, it is convenient to rewrite the metric part and NS-NS two-form coupled part of the Lagrangian (2.1) into the following form, where J α is a projected current defined as One can read off the deformed metric and NS-NS two-form from L G and L B , respectively.
It is quite messy but straightforward. It would be an easy exercise to derive the metric and NS-NS two-form by following the previous examples [23][24][25][26][27].

Integrability of Schrödinger spacetimes
In this section, we will deduce Shcrödinger spacetimes from classical r-matrices depending on both so(2, 4) and so (6) . Consequently, the classical integrability of the spacetimes automatically follows from the Yang-Baxter sigma model formulation. After presenting the well-known Shcrödinger spacetime in subsection 3.1 , we consider three-parameter generalizations in subsection 3.2 .
More complicated examples are presented in subsection 3.3 .

Schrödinger geometries as Yang-Baxter deformations
Inspired by the spirit of the gravity/CYBE correspondence, we have found that the following classical r-matrix corresponds to the Schrödinger background. Here p − is the light-cone generator in so (2,4) and h 4 , h 5 and h 6 are the Cartan generators in so (6) . Hence the r-matrix in (3.1) is indeed of the form (1.1). The parameter β measures the deformation 4 . For the details of our convention and notation of the generators, see Appendix A.
To find the metric and NS-NS two-form from (2.7) , we need to evaluate the projected deformed current P 2 (J α ) . By solving the equation which is obtained from the definition (2.8) , the deformed current is evaluated as with the coefficients 5 4 The parameter η is not an essential deformation parameter because it is canceled in (2.8) . 5 For the conventions of the gamma-matrices, see Appendix A.
Plugging the above expression of P 2 (J α ) with (2.7) , the resulting metric and NS-NS two-form turn out to be Here the line element ds 2 S 5 is measured by the metric of S 5 with the coordinates (χ, µ, ψ, θ, φ) , 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 The metric of the deformed AdS 5 part is nothing but the metric that was originally proposed in [33,34]. This metric preserves a non-relativistic conformal symmetry called the Schrödinger symmetry [35]. This metric can be reproduced via a coset construction [15].
As a result, we have proven the classical integrability of the string sigma model whose target space is given by the Schrödinger background (3.5) in the sense that the Lax pair has been constructed 6 .

Three-parameter generalizations
In the previous subsection, we have considered a one-parameter deformation. Then, one may consider multi-parameter deformations. An example of three-parameter deformation is described by the following classical r-matrix, where β ≡ (β 1 , β 2 , β 3 ) are the real deformation parameters.
Since the calculations to find the metric and NS-NS two-form are completely parallel to the previous section, we do not repeat them here. With the S 5 metric (A.17), the final expressions are written as where f ( β) and K( β) are a scalar function and a one-form on S 5 , respectively, depending on β . They are explicitly defined as (3.10) The above expressions are quite messy and it is convenient to rewrite them in terms of another coordinate system of S 5 . In fact, by following the argument in Appendix A, one can rewrite them into the following simpler form: Here µ 1 , µ 2 and µ 3 are the S 5 coordinates defined as µ 1 = cos ζ sin r , µ 2 = sin ζ sin r , µ 3 = cos r . the dependence of this type has been studied in [37]. It would be interesting to study whether the classical integrability is preserved for the deformation argued in [37]. The stability of the solution might be related to the classification of possible classical r-matrices.
Note that when the deformation parameters take the special values, the background (3.11) reduces to the Schrödinger geometry (3.5) by taking account of the relation, Finally it is worth noting that the Schrödinger spacetimes discussed so far are nonsupersymmetric for generic values. This is basically because T-dualities are taken for Rsymmetry directions and then all of the supersymmetries are broken. For supersymmetric Schrödinger backgrounds, see the following subsection. To present the examples, the following polar coordinates of S 5 are more convenient: n 1 = µ 1 cos ψ 1 , n 2 = µ 1 sin ψ 1 , n 3 = µ 2 cos ψ 2 , n 4 = µ 2 sin ψ 2 ,
The above coordinates satisfy the following condition: (n 2 1 + n 2 2 ) + (n 2 3 + n 2 4 ) + (n 2 5 + n 2 6 ) = µ 2 1 + µ 2 2 + µ 2 3 = 1 . The first examples is a classical r-matrix composed of p − and γ s 1 like This r-matrix gives rise to the following metric and NS-NS two-form: 2) classical r-matrix with n 13 The second example is a classical r-matrix, The resulting metric and NS-NS two-form are given by 3) classical r-matrix with n 15 The third example is a classical r-matrix, The resulting metric and NS-NS two-form are given by classical r-matrix with n 12 , n 23 and n 34 The fourth example is a classical r-matrix, r = 1 4 p − ∧ (α 1 n 12 + α 2 n 23 + α 3 n 34 ) . (3.23) The resulting metric and NS-NS two-form are given by
Thus the existence of the associated classical r-matrix (3.25) indicates that the string theory defined on the background [32] is classically integrable in the sense that the Lax pair is constructed.
According to the Killing spinor analysis in [32], for values satisfying the condition two real supersymmetries are preserved. When, in addition to the condition (3.27), at least one of the α i (i = 1, 2, 3) vanishes, four real supersymmetries are preserved. Thus, depending on classical r-matrices, the remaining supersymmetries should be different. Hence it would be interesting to study the relation between classical r-matrices and the classification of super Schrödinger algebras [38]. It would also be nice to study the relation to warped AdS 3 geometries, for example, along the line of [39].

Integrability of gravity duals for dipole theories
In this section, we shall derive gravity duals of dipole theories [40][41][42][43] from the viewpoint of the Yang-Baxter deformations. We find out classical r-matrices associated with the dipole backgrounds and derive the deformed metric and NS-NS two-form both for a one-parameter case and a three-parameter case, in subsection 4.1 and 4.2 , respectively.

A one-parameter case
Let us first consider a one-parameter dipole background. It can be obtained by a TsTtransformation (x 3 , ψ 1 ) α 1 , where α 1 is a shift parameter α 1 [44]. From this information on the TsT-transformation, one can easily guess the corresponding classical r-matrix based on the knowledge obtained [26].
A natural candidate of the associated r-matrix is given by where p 3 is a Poincaré generator in so(2, 4) and h 4 is a Cartan generator in so (6) . In fact, the classical r-matrix leads to the deformed metric and NS-NS two form given by, respectively, Here the scalar function G 1 is defined as We have also used the coordinates of S 5 given in (3.12) . Indeed, the background with (4.2) perfectly agrees with (6.30) in [44]. Thus it has been shown that the classical r-matrix

A three-parameter case
The next is to consider a three-parameter generation of the classical r-matrix (4.2) .
Since so(6) has the three Cartan generators h 4 , h 5 and h 6 , one may consider the following three-parameter generalization; with three real constants α 1 , α 2 and α 3 . Note that, when α 2 = α 3 = 0 , it reduces to the oneparameter case (4.2) . After some computations, the resulting metric and NS-NS two-form are given by Here the scalar function G 3 is defined as When α 1 = α 2 = α 3 = 0 , G 3 becomes 1 and B 2 vanishes.
The deformed background with (4.5) can also be obtained by a sequence of three TsTtransformations: 1) (x 3 , ψ 1 ) α 1 , 2) (x 3 , ψ 2 ) α 2 and 3) (x 3 , ψ 3 ) α 3 . The derivation is straightforward but the result has not been listed in [44]. Hence we have derived the explicit expressions in Appendix C. The resulting metric and NS-NS two-form completely agree with the expressions in (4.5).

Conclusion and Discussion
We have presented further examples of the gravity/CYBE correspondence. In the previous works, classical r-matrices have been composed of generators of only one of either so (2,4) or so (6) . In this paper, we consider some examples of r-matrices with both of them. The r-matrices of this kind contain (generalized) Schrödinger spacetimes and gravity duals of dipole field theories. It is known that the generalized Schrödinger spacetimes can also be obtained via a certain class of TsT transformations called null Melvin twists. The metric and NS-NS two-form are reproduced by following the Yang-Baxter sigma-model description.
Thus, this agreement shows that these backgrounds are classically integrable at the bosonic sigma-model level in the sense that the Lax pairs exist. We have to make efforts to prove this statement including the fermionic sector.
So far, the gravity/CYBE correspondence has been confirmed only for the metric (in the string frame) and NS-NS two-from. Hence the remaining task is to check the dilaton and R-R sector. In general, the dilaton is not constant and the R-R sector is also very complicated.
Thus it does not seem so easy to check the remaining sector by explicitly evaluating the operator insertion into the classical action. However, the Schrödinger spacetime with a one-parameter is a bit special. Although the metric is deformed and the NS-NS two-form is newly turned on, the dilaton is still constant and the R-R sector is not modified. This result indicates that the Schrödinger spacetime would be a nice laboratory to check the gravity/CYBE correspondence at the full-sector level. We hope that we could report the result in the near future [45].
There are various applications of the results presented here. An exciting issue is to consider applications to integrable deformations of type IIA string theory on AdS 4 × CP 3 [46]. This system is dual to the N =6 SU(N) × SU(N) Chern-Simons matter system in three dimensions [47]. This system was proposed by Aharony-Bergman-Jafferis-Maldacena (ABJM) [47] and it is often called the ABJM model. Hence the deformations of the stringtheory side should correspond to deformations of the ABJM model, and there should be the associated classical r-matrices in the spirit of the gravity/CYBE correspondence.
In particular, the most significant one is a non-relativistic limit of the ABJM model [48].
This system preserves a super Schrödinger symmetry and the internal symmetry is also revealed. However, the gravity dual for this non-relativistic ABJM model has not been constructed yet 7 . Thus, it may be interesting to try to find out the gravity dual by employing the Yang-Baxter deformations of type IIA string theory. The information on the isometry obtained in [48] would be a key ingredient to find out the corresponding classical r-matrix.
We hope that our results shed light on the gravity dual for the non-relativistic ABJM model.

A Notation and convention
We summarize here the notation and convention of the so(2, 4) and so(6) generators, and a coset representation of AdS 5 ×S 5 .

The gamma matrices
In the following, we use the gamma-matrices represented by Here each block of the matrices is a 4 × 4 matrix.

The bosonic generators
Then, the Lie algebras so(2, 4) and so(6) are spanned by the bases: Note that the subalgebras so(1, 4) and so (5)  A parameterization of the bosonic group elements We are now ready to parametrize bosonic group elements of P SU(2, 2|4) . The group elements of SO(2, 4) and SO(6) are parametrized as Here the light-cone coordinates and the associated generators are given by Thus, a bosonic element g of P SU(2, 2|4) is represented by

Coset projector
To derive the metric of AdS 5 ×S 5 from the left-invariant one-form, it is necessary to introduce the coset projector; For any x ∈ so(2, 4) ⊕ so (6) , it is explicitly defined as where the range of the indices are µ, ν = 0, 1, 2, 3 and i, j = 1, 2, 3, 4, 5 . Then the fourdimensional Minkowski metric is given by where ds 2 AdS 5 and ds 2 S 5 are the metrics of AdS 5 and S 5 respectively, This is the standard representation of the AdS 5 ×S 5 metric. It is noted that the ranges of S 5 coordinates are restricted as follows: The coordinate systems of S 5 When we argue the Schrödinger geometry, it is more convenient to rewrite the S 5 metric (A.15) as an S 1 -fibration over CP 2 : where χ is the S 1 -fiber coordinate and ω is the one-form potential of the Kähler form on CP 2 . The symbols Σ i (i = 1, 2, 3) and ω are defined by Σ 1 = 1 2 (cos ψdθ + sin ψ sin θdφ) , Σ 2 = 1 2 (sin ψdθ − cos ψ sin θdφ) , Σ 3 = 1 2 (dψ + cos θdφ) , ω = sin 2 µΣ 3 . (A.18) The coordinate transformation from (A.15) to (A.17) is given by The T-duality rules The rules of T-duality [51,52] are summarized here. We basically follow the rules of [52].
The transformation rules between type IIB and type IIA supergravities are listed below.
Note that the T-duality is performed for the y-direction and the other coordinates are denoted by a, b, a i (i = 1, . . .) . The fields of type IIB supergravity are the metric g µν , NS-NS twoform B 2 , dilaton Φ , R-R gauge fields C (2n) . The ones of type IIA supergravity are denoted with the tilde, the metricg µν , NS-NS two-formB 2 , dilatonΦ , and R-R gauge fieldsC (2n+1) .
From type IIB to type IIÃ g yy = 1 g yy ,g ay = B ay g yy ,g ab = g ab − g ya g yb − B ya B yb g yy , B ay = g ay g yy ,B ab = B ab − g ya B yb − B ya g yb g yy ,Φ = Φ − 1 2 ln g yy , where the anti-symmetrization for indices is defined as, for example, The symbol |y| inside the anti-symmetrization means that the indices other than the index y are anti-symmetrized.
From type IIA to type IIB C Derivation of dipole deformations of AdS 5 ×S 5 Let us here derive gravity duals for dipole theories by performing TsT transformations for AdS 5 ×S 5 . In the following, we will concentrate on the NS-NS sector. The T-duality rules we utilize is summarized in Appendix B.

C.1 One-parameter deformation
First of all, we consider a one-parameter deformation of AdS 5 ×S 5 . The starting point is the following metric, NS-NS two-form B 2 and dilaton Φ : For this background, we will perform a TsT transformation (x 3 ,ψ 1 ) α 1 . For this purpose, the relevant part is The first step is a T-duality for the x 3 -direction. The resulting background is given by Then, by shifting ψ 1 as ψ 1 =ψ 1 − α 1x 3 , the background (C.3) is rewritten as Finally, a T-duality is performed for thex 3 -direction. Together with the undeformed part, the resulting background is given by Here we have removed the tilde from ψ 1 and a scalar function G 1 (z) is defined as G 1 (z) ≡ 1 + α 2 1 µ 2 1 z −2 .