Jordanian deformations of the AdS5 × S5 superstring

We consider Jordanian deformations of the AdS5 × S5 superstring action. These deformations correspond to non-standard q-deformations. In particular, it is possible to perform a partial deformation, for example, of the AdS5 part only, or of the S5 part only. Then the classical action and the Lax pair are constructed with a linear, twisted and extended R operator. It is shown that the action preserves the κ-symmetry.


Introduction
One of the fascinating topics in string theory is the AdS/CFT correspondence [1][2][3]. The most well-studied example is the duality between type IIB superstring on the AdS 5 ×S 5 background [4] (often called the AdS 5 ×S 5 superstring) and the N = 4 SU (N ) super Yang-Mills (SYM) theory in four dimensions (in the large N limit). It has been revealed that an integrable structure exists behind the duality and it plays a fundamental role in testing the correspondence of physical quantities (for a comprehensive review, see [5]).
Recently, a q-deformed AdS 5 ×S 5 superstring action was constructed [42] by generalizing the result in [36]. Then the bosonic part of the action was determined and, by using this action, the world-sheet S-matrix of bosonic excitations was computed in [43]. The resulting S-matrix exactly agrees with the q-deformed S-matrix in the large tension limit. Thus the two approaches are now related to each other and there are many directions to study q-deformations of the AdS 5 ×S 5 superstring.
In this paper, we consider how to twist the q-deformed AdS 5 ×S 5 superstring action. This twisting is regarded as a non-standard q-deformation. Indeed, it would also be seen as a higher-dimensional generalization of 3D Schrödinger sigma models in which the qdeformed Poincaré algebra [44,45] and its infinite-dimensional extension are realized as shown in a series of works [30][31][32]. In particular, it is possible to perform a partial deformation, for example, of the AdS 5 part only, or of the S 5 part only. This would make the resulting geometry much simpler. Some extensions of the twisted R operators are also discussed. Then the classical action and the Lax pair are constructed with a linear, twisted and extended R operator. It is shown that the action preserves the κ-symmetry.
The paper is organized as follows. Section 2 is a short review of the q-deformed AdS 5 ×S 5 action. Section 3 describes how to twist the q-deformed action. Then we construct the Jordanian deformed action of the AdS 5 ×S 5 superstring preserving the κsymmetry. The Lax pair is also presented. Section 4 is devoted to conclusion and discussion. Appendix A describes the notation of the superconformal generators. In appendix B, the notation of the classical R-matrix is explained. A general prescription to twist the classical r-matrix for the standard q-deformation of Drinfeld-Jimbo type is also provided.
2 A review of the q-deformed AdS 5 ×S 5 superstring In this section, we will give a short review of the q-deformed AdS 5 ×S 5 superstring action constructed in [42], using the notation therein.

The linear R operator
A key ingredient in the construction is the classical R-matrix, which is a linear map R : g → g over a Lie algebra g satisfying the modified classical Yang-Baxter equation (mCYBE);

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where M, N ∈ g and c is a complex parameter. When c = 0, the parameter is regarded as a scaling of the R-matrix. When c = 0, the mCYBE is nothing but the classical Yang-Baxter equation (CYBE). The standard q-deformation of the superstring action presented in [42] is described by the following R-matrix, where E ij (i, j = 1, · · · , 8) are the gl(4|4) generators. For the standard notation of the superconformal generators, see appendix A. The parity of the indices is given byī = 0 for i = 1, · · · , 4 andī = 1 for i = 5, · · · , 8. The associated tensorial r-matrix is where the super skew-symmetric symbol is introduced as The relations between the linear R operator and the tensorial notation r are summarized in appendix B. The classical r-matrix given in (2.3) describes the standard q-deformation of Drinfeld-Jimbo (DJ) type [39][40][41].
Note that the linear R operator is defined here as a map from gl(4|4) to gl(4|4) , while the original action of the AdS 5 ×S 5 superstring is concerned with su(2, 2|4) , rather than gl(4|4). The compatibility of R operator with the real-form condition fixes the normalization factor in (2.3) as c = i up to real scalar multiplication.

The classical action and the Lax pair
With the help of the linear R operator defined in (2.2), the q-deformed classical action S is given by 2 Here τ and σ are time and spatial coordinates of the string world-sheet and the periodic boundary conditions are imposed for the σ direction. The real constant η ∈ [0, 1) measures the deformation. 3 The super Maurer-Cartan one-form A α is defined as Here we have normalized the parameter as c = 1 in (2.2). 3 Since the deformation is measured by η , it is often called "η-deformation". On the other hand, η is related to the q parameter of the standard q-deformation by Drinfeld-Jimbo [39][40][41] as shown in [36]. Hence we will refer this deformation as to q-deformation, following [42].

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and A α takes the value in the Lie superalgebra su(2, 2|4). The action of the R-matrix (2.2) on A α is induced from gl(4|4) by imposing a suitable reality condition. Note that A α automatically satisfies the flatness condition, The projection operators P αβ ± are defined as Then operators d andd are linear combinations of the projection operators P i (i = 1, 2, 3) , (2.7) The symbol R g indicates a chain of the adjoint operation and the linear R operation, Note that the usual AdS 5 ×S 5 superstring action is reproduced from (2.5) when η = 0. For a pedagogical review of the undeformed AdS 5 ×S 5 superstring, see [46]. It is convenient to introduce the following notations, Then the equations of motion are written in a simpler form, The Lax pair is given by where λ is the spectral parameter that takes a complex value. The flatness condition (2.6) can be rewritten in terms of J α − and J α + like With the definition L α ≡ L +α + M −α , the zero-curvature condition is equivalent to the equations of motion given in (2.10) and the flatness condition (2.12).

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3 Jordanian deformations of the AdS 5 ×S 5 superstring In this section we shall consider Jordanian deformations of the AdS 5 ×S 5 superstring action. The deformations correspond to a non-standard q-deformation and contain twists of the linear R operator. The twist procedure is realized as an adjoint operation for the linear R operator with an arbitrary bosonic root. We first explain how to construct Jordanian R operators by twisting the linear R operator used in the q-deformed AdS 5 ×S 5 superstring action (2.5). There are two remarkable features of Jordanian R operators. The first is that they satisfy the CYBE rather than the mCYBE (2.1). The second is the nilpotency of them. That is for M, N ∈ g. Then, by using the Jordanian R operators, the Jordanian deformed action with the κ-symmetry and the Lax pair are presented.

Jordanian R operators from twists and their extension
We shall give a description to twist the linear R operator for basic examples of Jordanian R operators here. Then some extensions of twisted R operators are discussed. First of all, note that the classical r-matrix of Drinfeld-Jimbo type (2.3) has vanishing Cartan charges, where the coproduct is given by On the other hand, one may introduce a classical r-matrix which has non-zero Cartan charges for the deformation of AdS 5 × S 5 superstring. In this sense, we refer to these as to Jordanian r-matrices. In general, such an r-matrix can be constructed by a twist of r DJ with an arbitrary bosonic root E ij with i < j , One may also consider twists by negative bosonic roots E ij (i > j), but the corresponding r-matrix has the same property because gl(4|4) algebra enjoys the automorphism Thus positive roots E ij (i < j) are enough for our later argument. The twisted, linear R operator is defined as

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It is straight forward to read off the R operator from the tensorial r-matrix via (3.6) and inner product (A.2). So far, we have constructed the Jordanian R operators via twists of r DJ . One may also consider the extension of the twisted R operators by adding bilinear terms of fermionic root generators. It should be noted that the latter cannot be obtained with the twists. Thus there are the two classes: 1) Jordanian R operators stemming from twists and 2) extended Jordanian R operators. We will introduce some examples below.

1) Jordanian R operators from twists
The first example is twists by simple roots. Then the corresponding subsectors of the superstring action are deformed. For instance, let us consider twists by positive simple root generators E k,k+1 (k = 1, . . . ,4, . . . , 7). 4 Then the associated classical r-matrix is given by The twists give rise to deformations of the AdS 3 or S 3 subspace. For each of the values k = 1, 2, 3 , the resulting geometry is given by a deformed AdS 3 spacetime. It would contain a three-dimensional Schrödinger spacetime and may be regarded as a generalization of the previous works [30][31][32]. The explicit relation is presented in [47].
More interesting examples are deformations of either AdS 5 or S 5 . These partial deformations are realized by twists with the maximal bosonic generators E 14 = P 14 in su(2, 2) and E 58 = R 58 in su(4) , respectively; 5 The deformation of S 5 should be interesting because it would provide a simpler geometry without deforming AdS 5 . The associated linear operator acts on the generators as follows: Remarks. More generally, the Reshetikhin twist [48] or the Jordanian twist [49,50] is closely related to the present prescription. The Jordanian twists for Lie superalgebras are considered in [51][52][53][54]. This relation will be elaborated somewhere else.

2) Extended Jordanian R operators
Let us now consider some extensions of the twisted classical r-matrices given in (3.7), (3.8) and (3.9). Recall that these are obtained by twisting r DJ . Here we are concerned with some extensions of the twisted r-matrices, which are not described as twists.
It is easy to see that a linear combination of (3.8) and (3.9) Furthermore, these r-matrices may be extended to contain supercharges in their tails, including two parameters, like Here α, β, α ′ , β ′ are arbitrary parameters. The extended r-matrices satisfy the CYBE (3.1). As a remark, it would not be obvious that the multi-parameter deformations lead to consistent string theories. The vanishing β-function has not been confirmed even for the single parameter case. The multi-parameter case would be much more difficult.

Comments on fermionic twists
One may think of twists by fermionic generators called fermionic twists. An example is given by the maximal root E 18 , (Also see appendix B.2) (3.14) Note that c is a Grassmann odd element [51], so that the r-matrix is Grassmann even. This is a solution of the CYBE. For this fermionic twist, we have no clear understanding for the physical interpretation because the deformation is measured by a Grassmann odd parameter. It would be interesting to interpret the fermionic twist in type IIB supergravity. Generically the r-matrices of the fermionic twists do not satisfy the CYBE (3.1). As an example, let us consider E 45 =S 45 . This is a simple root generator but it gives rise to JHEP04(2014)153 the maximal twist. That is, the corresponding geometry is also maximally deformed. The associated classical r-matrix is given by However it is not a solution of the CYBE.

Jordanian deformed action
Next we consider Jordanian deformations of the classical action of the AdS 5 ×S 5 superstring. Although the construction is almost parallel to the one in [42], it is necessary to take account of small modifications coming from the fact that the Jordanian linear operator R Jor satisfies the CYBE rather than the mCYBE. In the following, R Jor is used as a representative of arbitrary (extended) Jordanian R operators. 6 The detail expression of R Jor is not relevant to the subsequent analysis.
The Jordanian deformed classical action is given by Here, by using Jordanian R-matrix R Jor , a chain of the operations [R Jor ] g is defined as In the present case, d andd are not deformed and do not contain η like and the overall factor of the action (2.5) is not needed to be multiplied. As in the case of [42], the equations of motion can be written simply with the following quantities: There are two ways to rewrite the action given in (3.16). The first is based on J α and the action is written as

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The second is based on J α and the action becomes The two expressions are useful to discuss the Virasoro conditions and the κ-invariance.
Then equations of motion are given by 22) and the flatness condition is represented by Note that the flatness condition does not contain η 2 terms, in comparison to the one given in (2.12). This modification comes from the fact that the Jordanian operator R Jor satisfies the CYBE, rather than the mCYBE. For later computations, it is convenient to decompose the equations of motion (3.22) and the flatness condition (3.23) as follows: Then the Lax pair is given by L +α +M −α , it is an easy task to show that the zero curvature condition (2.13) is equivalent to the equation of motion (3.22) and the flatness condition (3.23). The next is to consider the Virasoro conditions. The expression given in (3.20) leads to the Virasoro conditions, On the other hand, the expression in (3.21) gives rise to The above two representations of the Virasoro conditions given in (3.27) and (3.28) should be equivalent.

κ-symmetry
Let us consider the κ-symmetry of the action (3.16).
We consider a fermionic local transformation (called the κ-transformation) of g given by where ρ (1) and ρ (3) are arbitrary functions on the string world-sheet to be determined later, and hence ǫ also depends on the world-sheet coordinates. Then the variation of the action given in (3.16) is described as Here the following relations have been used in the second equality, Now let the forms of ρ (1) and ρ (3) be +α + J

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The derivation is the following, The second equality comes from the fact that J −τ is proportional to J −σ . Similarly, one can show the relation, (3.34) Furthermore, for any grade 2 traceless matrix A (2) ±α , the following relation is satisfied [46] , by using the matrix representation, where Υ is the following 8 × 8 matrix: Then we will show that this variation is canceled out with the variation of the action with respect to the world-sheet metric γ αβ . Let the variation of γ αβ be

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Then, by using the expressions of the classical action given in (3.20) and (3.21) , the variation of the action is evaluated as In order to show the second equality, the following relations have been used, Thus, the total variation of the classical action (3.16) becomes zero, That is, the action (3.16) is invariant under the κ-transformation.

Comment on the real-form condition
Here we would like to discuss the real-form condition of su(2, 2|4). So far, we are working with a linear R operator from gl(4|4) to gl(4|4) , hence the image is not necessarily su(2, 2|4) , even if the domain is restricted to su(2, 2|4). In the case of the standard qdeformation [36], the real-form condition is preserved. On the other hand, in the case of Jordanian deformations, it is not preserved. However, this fact does not always lead to serious problems like complex actions. Preserving the real-form condition is just a sufficient condition for real actions and it is not necessary to impose it inevitably. In fact, without preserving the real-form condition, one can get the real actions for some Jordanian deformations as shown in [47]. In particular, different r-matrices may give rise to identical string action, up to coordinate transformations and (double) Wick rotations. So far, we have not found the general criterion for which of Jordanian deformations lead to real and physical actions. It would be interesting to specify it in order to classify the physical Jordanian deformations.
As a matter of course, Jordanian deformations contain some unphysical ones where there are two time directions or the action contains imaginary parts. For example, a Jordanian deformed S 5 contains imaginary parts but it might have some gauge-theoretical interpretations as a complex integrable deformation.

Conclusion and discussion
We have discussed Jordanian deformations of the AdS 5 ×S 5 superstring action. The description to construct Jordanian R operators via twists has been explained in detail. Notably

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the Jordanian R operators satisfy the CYBE rather than the mCYBE and they have nonvanishing Cartan charge. Then we have constructed the Jordanian deformed action that preserves the κ-symmetry. The Lax pair has also been presented.
It should be remarked that partial deformations are possible in our procedure. This fact implies that one may perform a deformation of the AdS 5 part only, or of the S 5 part only, for example. Then the background geometry for the deformed S 5 would be much simpler because the AdS 5 part is not modified and the gauge-theory dual would be identified with a deformation of the scalar sector such as Leigh-Strassler deformations [55]. A promising way is to consider a twist of the q-deformation of the SO(6) sector argued in [56,57]. As a matter of course, even for the maximal twist, the metric of the twisted geometry can be determined, for example, by following [43]. The background geometries associated with simple Jordanian twists are presented in [47] as well as the solution of type IIB supergravity.
In principle, it should be possible to classify all of the skew-symmetric classical rmatrices of gl(4|4) and its real form su(2, 2|4). This classification would enable us to reveal all of the possible deformations of the AdS 5 ×S 5 superstring from the algebraic point of view.
We believe that the study of integrable deformations of the AdS 5 ×S 5 superstring will shed light on new aspects of the integrable structure behind the AdS/CFT correspondence.

B Constant classical R-matrix
We summarize here the notation of the classical R-matrix, which is independent of the spectral parameter (For example, see [58]).

B.1 Classical Yang-Baxter equation
Let g be a bosonic Lie algebra over C. For a i , b i ∈ g, an element denoted by where the action of r ij is extended to three sites g ⊗ g ⊗ g such as Suppose that there exists the invariant non-degenerate symmetric bilinear form , on g. With the bilinear form, the linear operator R : g → g can be introduced though the following relation;

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The twists by the bosonic roots E αβ and E ab with α < β and a < b (α, β = 1, · · · , M and a, b = M + 1, · · · , N + M ) are given by where the coproduct is defined in (3.1). These are solutions of the CYBE rather than the mCYBE. We will call them the bosonic twists.
Also, one may consider a twist by a fermionic root, which is referred as to a fermionic twist. An example is given by E 1,M +N , where c is a Grassmann odd parameter rather than a complex number, so that the r-matrix should be Grassmann even [51]. This is a solution of the CYBE. When M = N = 4, it reproduces (3.14).
In general, the fermionic twist by the fermionic root E α,b gives rise to However it does not seem to be a solution of the (m)CYBE except for (B.11).
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