A Jordanian deformation of AdS space in type IIB supergravity

We consider a Jordanian deformation of the AdS_5xS^5 superstring action by taking a simple R-operator which satisfies the classical Yang-Baxter equation. The metric and NS-NS two-form are explicitly derived with a coordinate system. Only the AdS part is deformed and the resulting geometry contains the 3D Schrodinger spacetime as a subspace. Then we present the full solution in type IIB supergravity by determining the other field components. In particular, the dilaton is constant and a R-R three-form field strength is turned on. The symmetry of the solution is [SL(2,R)xU(1)^2] x [SU(3)xU(1)] and contains an anisotropic scale symmetry.


Introduction
One of the most intriguing subjects in string theory is the AdS/CFT correspondence [1][2][3] and it has well been studied from various aspects with an enormous number of works.
Although it is often supposed to hold as a matter of course in the recent studies, it is still important to elaborate the original form, the duality between type IIB string theory on AdS 5 ×S 5 and the N =4 super Yang-Mills (SYM) theory, to gain deeper insight for the basic origin of AdS/CFT. In this direction, the integrability behind this duality would play an important role (For a comprehensive review, see [4]).
We will concentrate on the string-theory side, type IIB string theory on AdS 5 ×S 5 here.
This coset enjoys the Z 4 -grading property and it leads to the classical integrability [6] 1 .
The classification of possible supercosets, which lead to the classically integrable, consistent string theories, is performed in [9] The next task is to consider integrable deformations. Although there are some kinds of integrable deformations, we will focus upon q-deformations of the AdS 5 ×S 5 superstring.
Then, by applying the YBsM description to the AdS 5 ×S 5 superstring and the q-deformed classical action was presented in an abstract form with the group-theoretical language [30].
In the YBsM description, a linear R-operator is a key ingredient. It is constructed from a skew-symmetric, classical r-matrix satisfying the modified classical Yang-Baxter equation (mCYBE). The coordinate system has been introduced in [31] and the metric in the string frame and NS-NS two-form have been determined. However, the complete gravitational solution has not been fixed yet in type IIB supergravity.
There is another kind of q-deformations, which are called Jordanian deformations [32][33][34] or sometimes non-standard q-deformations (For the case of Lie superalgebras, see  There is another formulation of the AdS 5 ×S 5 superstring action [7]. For the classical integrability in this formalism, see [8]. 2 Some new examples of AdS 2 and AdS 3 have been found in [10]. 38]). In the previous work [39], we have considered Jordanian deformations of the AdS 5 ×S 5 superstring action by using linear R-operators satisfying the classical Yang-Baxter equation (CYBE), rather than mCYBE. The action presented in [39] is also written abstractly in terms of a group element, and explicit examples have not been provided yet.
In this paper, we consider a Jordanian deformation of the AdS 5 ×S 5 with a simple Roperator. The metric and NS-NS two-form are explicitly derived with a coordinate system.
Only the AdS part is deformed and the resulting geometry contains the 3D Schrödinger spacetime as a subspace. In this sense, this study can be regarded as a generalization of the previous works [40,41]. Then we present the full solution in type IIB supergravity by determining the other field components. In particular, the dilaton is constant and a R-R three-form field strength is turned on. The symmetry of the solution is given by and contains an anisotropic scale symmetry.
This paper is organized as follows. In section 2 we give a short review of Jordanian deformations of the AdS 5 ×S 5 superstring action. Then, by taking a simple R-operator satisfying CYBE, the metric and NS-NS two-form are explicitly derived with a coordinate system. The resulting geometry is given by the product of a deformed AdS space and round S 5 . Also for a slightly generalized R-operator, the string action is derived. The resulting metric represents a time-dependent background. In section 3 we present the gravitational solution in type IIB supergravity by finding out the other field components. In particular, the dilaton is constant. Section 4 is devoted to conclusion and discussion. In Appendix A, our notation and convention is summarized. In Appendix B, we list some classical r-matrices and the associated string actions.
2 Jordanian deformations of AdS 5 ×S 5 In this section, we first introduce Jordanian deformations of the AdS 5 ×S 5 superstring. Then by taking a simple example of skew-symmetric, classical r-matrix, the string action is obtained with a coordinate system. Then the metric and NS-NS two-form are derived explicitly.
The resulting metric contains the 3D Schrödinger spacetime as a subspace. A more general example is also presented.

Setup
First of all, we will give a short summary of the work [39]. One may consider Jordanian deformations of the AdS 5 ×S 5 superstring action with linear R operators satisfying CYBE.
The construction follows basically [30] with the help of the YBsM description [18].
The deformed Green-Schwarz string action is given by [39] where the left-invariant one-form A α is given by The projection operators P αβ ± are defined as linear-combinations of the metric γ αβ and the anti-symmetric tensor ǫ αβ on the string world-sheet like and satisfy the following properties, In the action (2.1) the projection P αβ − is utilized. Recall that the Lie superalgebra su(2, 2|4) has a Z 4 automorphism, Ω with Ω 4 = 1 . This automorphism leads to the decomposition of su(2, 2|4) as follows: (2.5) Here the operation of Ω is defined for an element of su(2, 2|4) (n) as Ω(X (n) ) = i n X (n) for X (n) ∈ su(2, 2|4) (n) .
One can introduce projections P n from su(2, 2|4) onto su(2, 2|4) (n) (n = 0, 1, 2, 3) . Then the operator d is defined as The remaining task is to introduce the operation [R Jor ] g . In the first place, we introduce a linear R-operator, R Jor , Note that R Jor does not preserve the real-form condition of su(2, 2|4) in general, even if the domain is restricted to su(2, 2|4) . The real-form condition is not necessary for the classical integrability (i.e., the construction of Lax pair) as shown in [39]. One may expect that the string actions become complex if the real-form condition is not preserved. However, it is not always the case. In fact, some examples of R Jor , which break the real-form condition, give rise to real string actions, as we will see later. That is, the real-form condition should be regarded as a sufficient condition for the reality. Still, we have no general criterion to specify the linear operators that lead to the real string actions. It is an important issue to argue the criterion in the future.
Then the operator [R Jor ] g is defined as a sequence of the adjoint operation Ad g by g , the R-operation and the inverse of the adjoint : This operation is intrinsic to the coset case [18,27].

The tensorial notation of the R-operator
It is helpful to see the tensorial notation of R Jor . In the present case, r Jor is skew-symmetric due to the property 3) . Hence r Jor can be represented by using a skew-symmetrized tensor product of two elements of gl(4|4) , The linear R operator action is associated with the tensorial notation as follows: (2.11) In the tensorial notation, the property 1) is recast into the familiar expression of CYBE, Here the subscripts of r Jor specify vector spaces on which r Jor acts.
Thus a skew-symmetric solution of CYBE is associated with a linear R-operator and is related to an integrable deformation of AdS 5 ×S 5 . So far, the string action (2.1) is written in an abstract form with a group-theoretical language. In the next subsection, we will take an example of skew-symmetric classical r-matrix and express explicitly the action with a coordinate system.

A simple example of the string action
Let us consider an explicit example of Jordanian deformations by taking a skew-symmetric classical r-matrix 3 , where E ij is a 4 × 4 matrix defined as The normalization of r Jor is absorbed by rescaling of η , as one can see from the action (2.1) .
Here it is fixed for later convenience.
The r-matrix (2.13) induces the action of the associated R-operator as This mapping rule is obtained from the relation (2.11) . Note that the r-matrix (2.13) does not preserve the real-form condition of su(2, 2|4) . However, it leads to a real string action, as we will see later.
Let us evaluate the string action (2.1) . For simplicity, we focus on the bosonic part by restricting a group element g to the bosonic subsector. In addition, only the AdS 5 part is deformed in the present example and hence the coset construction for the S 5 part is the usual. Therefore, we concentrate on the coset construction for the AdS 5 part. Then it is convenient to consider the following coset representative, g = e p 0 x 0 +p 1 x 1 +p 2 x 2 +p 3 x 3 e γ 5 ρ/2 ∈ SU(2, 2)/SO (1,4) .
Later, we often use the following quantities instead of x 0 , x 3 and ρ .
With this setup, the classical action (2.1) can be rewritten as Here A α is restricted to su(2, 2) . Then S S represents the usual S 5 part of the string action and we will not touch on this in the present section.
From now on, let us compute the explicit form of A α and After that, the bosonic part of the classical action can be determined explicitly with the coordinate system introduced with the parametrization (2.15) .
First of all, P 2 (A α ) can be evaluated as where each of the coefficients is given by The next task is to evaluate P 2 (J α ) . The relation (2.19) can be inverted as By acting P 2 on both sides, the following expression is obtained, This is just a linear equation for P 2 (J α ) and hence it is straightforward to evaluate P 2 (J α ) .
Note that P 2 (J α ) can be expanded as Here j µ α and j 5 α are unknown functions to be determined. With this expansion, the right-hand side of (2.22) can be rewritten as . (2.24) By comparing (2.24) with (2.20) , the expressions of j µ α and j 5 α are determined as Thus the classical action has been obtained as where the last term in the coupling to NS-NS two-form is a surface term and it can be dropped off. This action is real. From this action and the S 5 part, one can read off the metric in the string frame and NS-NS two-form. In order to determine the string background completely, it is still necessary to fix the other field components by solving the field equations of motion in type IIB supergravity. This will be the issue in the next section.
So far, the r-matrix (2.13) has been considered. Note that the following four r-matirces lead to the same string action (2.25) , up to double Wick rotations and coordinate transformations (For the detail, see Appendix B). Note that r Jor and r (4) Jor are obtained by performing adjoint operations with ∆(E 23 ) and ∆(E 14 ), respectively, to the classical r-matrix of Drinfeld-Jimbo type satisfying mCYBE, as argued in [39]. Thus the present example may be regarded as Jordanian twists [32][33][34], though this fact is not manifest from the expression of the r-matrix (2.13) .

Other examples
Before closing this section, let us present a generalized example 4 of skew-symmetric r-matrix satisfying CYBE, This r-matrix is also regarded as a Jordanian twist [32][33][34]. Note that the r-matrix (2.26) does not preserve the real-form condition of su(2, 2|4) . However, this is also an example which gives rise to a real string action, as we will see below.
We will not show the derivation in detail. With the r-matrix (2.26), only the AdS 5 part is deformed again and the algorithm of the derivation is the same.
The resulting action for the deformed AdS 5 part is given by

A two-parameter deformation
It is also interesting to see a two-parameter deformation of AdS 5 . It can be considered by taking the following r-matrix, where s 1 and s 2 are constant parameters. The resulting string action is given by (2.29) The action is complicated but it is still real.

A solution in type IIB supergravity
In this section we will present a solution in type IIB supergravity containing the metric and NS-NS two-form obtained from (2.25) 5 .

The action of type IIB supergravity
Let us first introduce the equations of motion of type IIB supergravity [42]. Here we will follow the notation of [43]. The action of the bosonic part is given by Here G M N is the 10D metric in the Einstein frame and R is its Ricci scalar. The constant parameter κ is related to the 10D Newton constant G 10 like 2κ 2 ≡ 16πG 10 . The symbol * denotes the 10D Hodge dual operator. Φ is the fluctuation of the dilaton field and C is the axion field. Then B 2 , C 2 and C 4 are the NS-NS two-form, the R-R two-form and the R-R four-form. Their field strengths are defined as The modified field strengths F 3 and F 5 are defined as Note that F 5 has to satisfy the self-dual condition, By taking variations of the action (3.1) , the equations of motion are obtained as The Bianchi identities are given by With this setup, we will consider a gravitational solution in the next subsection.

A Jordanian deformed solution
It is a turn to present a solution corresponding to a Jordanian deformation. From the construction of the string action, the metric is given by Here the metric of round S 5 is expressed as a U(1) fibration over CP 2 , where χ is the local coordinate on the Hopf fibre and ω is the one-form potential for the Kähler form on CP 2 .
The metric of CP 2 and ω are given by 6 where Σ a (a = 1, 2, 3) are defined as Note that the metric contains the 3D Schrödinger spacetime 7 as a subspace with x 1 = x 2 = 0 , while the deformed AdS 5 part itself is not the 5D Schrödinger spacetime 8 .
Note that the metric which appear in the string action is represented in the string frame.
However, the metric for the deformed AdS 5 part is invariant under the following scaling and one may expect that the dilaton should be constant (i.e., Φ = 0). Thus the metric can be regarded as the one in the Einstein frame. For simplicity, we set C = 0 .
By considering the S 5 components of the equation of motion for the metric (3.5) and taking account of the self-duality condition (3.4) , the five-form field-strength is fixed as (3.16) 6 We follow Appendix A.2 of [44]. 7 Therefore, the result of [45] on the fast-moving limit [46] is directly applicable for this background.
Note that the NS-NS two-form also vanishes at x 1 = x 2 = 0 . 8 One may consider whether the deformed AdS 5 can be represented by a coset by following [47].
Thus F 5 is not modified under the deformation.
Then the NS-NS two-form B 2 has also been derived as 17) and the associated field strength is given by From the equation of motion for H 3 , (3.9) , one can notice that F 3 has to be turned on.
The remaining task is to find out F 3 so as to satisfy all of the equations of motion. The resulting F 3 is give by where the associated R-R two-form C 2 is given by In summary, the gravitational solution in the Einstein frame is given by where Φ = C = 0 . The R-R scalar field C may take a non-vanishing constant C = 0 . In this case, the R-R two-form C 2 has to be shifted as C 2 → C 2 + CB 2 .
Note that the Green-Schwarz string action on this solution (at the quadratic order of fermions) is easily obtained by substituting the solution (3.21) into (3.27) of [48]. Recently, the quartic-order action has been derived in [49]. It would also be useful for further studies.
As a matter of course, the total action is real, including the fermionic sector. It would be interesting to argue the world-sheet S-matrix by using the obtained action.
The symmetry of the solution Let us check the symmetry of the solution (3.21).
We first concentrate on the symmetry of the deformed AdS part. It is obvious to see that the solution is invariant under two translations: where a ± are constant parameters. The invariance under the rotation in the 1-2 plane is also manifest. Recall that the solution is invariant under the anisotropic scaling, A less obvious one is the special conformal transformation 9 , In total, the resulting symmetry is given by It seems likely that the solution (3.21) is not supersymmetric because the F 3 flux is the same type of the one considered in [51], where the H 3 flux is considered but the mechanism to break supersymmetries would be identical. It might be interesting to consider a brane-wave deformation, instead of the F 3 flux, as in [52]. Some of the original supersymmetries may be preserved, while the integrability would become unclear.
In comparison to the Jordanian deformed solution (3.21), it seems quite difficult to find out the full gravitational solution corresponding to the standard deformation in type IIB supergravity. The metric in the string frame is obtained in [31], but it involves a curvature singularity and the dilaton would be very complicated. 9 For the derivation of this transformation law, for example, see [50].

The tidal force
It is also important to check whether the solution (3.21) involves a singularity or not. The solution is just regarded as a pp-wave like deformation of the AdS 5 ×S 5 background. Hence no obvious curvature singularity is not found by computing curvature invariants. However, there may be another kind of singularity called pp-singularity [53]. In order to discuss this singularity, it is necessary to check the tidal force.
First of all, one needs to take a time-like world-line and its tangent vector is where the index m runs only for the deformed AdS 5 part and "dot" denotes the derivative with respect to the affine parameter λ . Assume that the affine parameter is chosen so that the tangent vector becomes a unit vector: The dynamics of a particle moving on the solution (3.21) is described by the action The equations of motion for x ± provide two constants of motion, P − and E , Solving P − and E with respect toẋ ± leads to the following expressions, The equations of motion for x 1 and x 2 are given by d dλ Noting that the normalization condition (3.26) is explicitly written as one can solve (3.26) forż and obtain the following expression, Here we have used the expressions ofẋ ± given in (3.28) .
To evaluate the tidal force, it is not necessary to solve equations of motion explicitly.
The tidal force is represented by the components of Riemann tensor in an orthonormal frame which is parallelly transported along the world-line. Thus one just needs to identify a basis e m for the orthonormal frame d dλ e m = Γ m np t n e p . (3.32) The orthonormal system is given by Then the tidal force is defined as 33) and the components of the tidal force are listed below : From the tidal force, one can see that the solution (3.21) is regular at the horizon, z = ∞ , while it hits on a singular at the boundary, z = 0 , except at x 1 = x 2 = 0 .
It is worth noting the similarity to the 5D Schrödinger spacetime with the dynamical critical exponent z c . When z c = 2 , there is no divergence of the tidal force at the horizon and the boundary [54]. But, when z c = 3 , the tidal force diverges at the boundary [54].
The solution (3.21) exhibits an isotropic scaling with z c = 2 , while its asymptotic behavior around the boundary is close to the one with z c = 3 . The divergence of the tidal force at the boundary in the solution (3.21) is similar to the one of the Schrödinger spacetime with z c = 3 .

Conclusion and discussion
We have considered a Jordanian deformation of the AdS 5 ×S 5 superstring action with a simple Though the curvature invariants are not singular, the tidal force diverges at the boundary, except a certain point.
There are many open problems now. The first is to consider a relation to deformed S-matrices on the string world-sheet. The standard q-deformations of the S-matrices are studied in [55][56][57][58], but Jordanian deformed S-matrices have not been argued yet. It would be interesting to study them and compare the results with the string world-sheet S-matrices as in [31]. The most important issue is the deformation of N =4 SYM corresponding to the gravitational solution presented here. Probably, it would be concerned with non-local field theories such as dipole theories [59]. Although we have considered a deformation of the AdS 5 part, it might be possible to consider a similar deformation of the S 5 part. As far as we have tried, the metric contains imaginary parts and it seems difficult to give a physical interpretation. Anyway, because it should be regarded as a marginal deformation, such a complex solution might be related to a complex β-deformation discussed in [60].
The solution presented here is just an example. We expect that many interesting gravitational solutions would be found through Jordanian deformations. The recipe to look for them is given in [39] and this paper. We hope that many integrable solutions are discovered

Acknowledgments
The work of I.K. was supported by the Japan Society for the Promotion of Science (JSPS).

Appendix A Our notation and convention
Our notation and convention is summarized here by basically following [61] .
An element of Lie superalgebra su(2, 2|4) is represented by an 8 × 8 supermatrix: Here m and n are 4 × 4 matrices with Grassmann even elements, while ξ and ζ are 4 × 4 matrices with Grassmann odd elements. These matrices satisfy an appropriate reality condition. As a result, it turns out that m and n belong to su(2, 2) = so (2,4) and su(4) = so(6) , respectively.
For our purpose, it is helpful to prepare an explicit basis of su(4) and su (2,2) . Let us first introduce the following γ matrices: Then n ij (i, j = 1, 2, 3, 4, 5) are given by It is easy to see that γ i 's generate the Clifford algebra of so (5): Thus n ij 's generate the Lie algebra so (5) . Note that are regarded as the generators of so (6) .

B A list of r-matrices and deformed string actions
This appendix gives a list of lsome possible r-matrices and the associated string actions.
The AdS part of the Jordanian deformed action can be rewritten as where the sigma model part L G and the coupling to NS-NS two-form L B are given by The undeformed AdS 5 part is represented by This part is common for all of the deformation, and L B always vanishes in the η → 0 limit.
It would be interesting to classify possible r-matrices and the associated string actions, though the classification here is forcussed upon some simple examples and not complete.
Remarkably, all of the string actions contained in the list are real, up to surface terms appearing in L B , after performing appropriate Wick rotations.
The deformed string actions are classified into the three classes: The three classes are listed below.
The deformed Lagrangian: This is the case with (2.13) considered in the body. The last term in L B is a surface term. It can be ignored without boundaries. The Lagrangian (B.4) is invariant under SL(2, R) × U(1) 2 , which contains the anisotropic scaling invariance under where λ is a constant. For the detail, see subsection 3.2.
(1) r The deformed Lagrangian: This can be obtained from the case (0) by exchanging x ± → x ∓ and flipping η → −η .
Thus this case is equivalent to the case (0).
(2) r The deformed Lagrangian: After performing the double Wick rotation x 2 → ix 2 and x 0 → ix 0 and redefining the light-cone coordinates likex ± = (x 2 ± x 1 )/ √ 2 , this case is identical to the case (0), up to the total derivative.
The deformed Lagrangian: After flipping x 2 → −x 2 and η → −η , this case is equivalent to the case (2). Thus the Lagrangian (B.8) is also equivalent to the case (0), up to the total derivative.
The deformed Lagrangian: After flipping x 2 → −x 2 , this case is equivalent to the case (2), up to the total derivative. Thus the Lagrangian (B.9) is also equivalent to the case (0) .
Note that the actions in the class A are identical, up to the total derivative. If boundaries are taken into account, the class A should be divided into subclasses. But we are interested in closed strings here and will not argue such subclasses.

Class B
(5) r The deformed Lagrangian: The last term in L B is imaginary but just a surface term. Thus the Lagrangian (B.10) is real without boundaries. Note that the Lagrangian (B.10) is invariant under the anisotropic scaling (B.5), the rotation in the 1-2 plane and the shift of x − , i.e., U(1) 3 .
The class B corresponds to the case discussed in subsection 2.3.
Class C (7) r The deformed Lagrangian: By performing a Wick rotation x 2 → −ix 2 , the Lagrangian (B.12) becomes real but contain two time directions. Thus it seems to be unphysical. Note that, in comparison to the other cases, the Lagrangian (B.12) is invariant under the isotropic scaling where λ is a constant. After the Wick rotation, the Lagrangian (B.12) is invariant also under the transformation, wherex ± = (x 2 ± x 1 )/ √ 2 and λ ′ is a constant. This can be understood as the diagonal part of the two Lorentz boosts. In addition, it has the invariance under a "rotation" in the (x + ,x + ) and (x − ,x − ) planes, x ± → cos θ x ± − sin θx ± ,x ± → sin θ x ± + cos θx ± .
Thus the resulting symmetry is U(1) 3 .
The deformed Lagrangian: By exchanging x ± → x ∓ , this case is equivalent to the case (7).
The class C seems to be unphysical because of two time directions. It would be interesting to figure out a general criterion for the physical metric so as to exclude the class C.