String backgrounds of the Yang-Baxter deformed AdS4 × ℂℙ3 superstring

We build string backgrounds for Yang-Baxter deformations of the AdS4 × ℂℙ3 superstring generated by r-matrices satisfying the classical Yang-Baxter equation. We obtain the metric and the NSNS two-form of the gravity dual corresponding to noncommutative and dipole deformations of ABJM theory, as well as a deformed background with Schrödinger symmetry. The first two backgrounds may also be found by TsT transformations while for the last background we get a new family of non-relativistic ABJM theories with Schrödinger symmetry.


Introduction
Integrability is a very important feature found in several instances of the AdS/CFT correspondence. On the string theory side, formulated as a two-dimensional field theory, the notion of integrability is associated to the existence of a Lax connection which ensures the existence of an infinite number of conserved charges. In the case of AdS 5 × S 5 superstring, the theory is described as a σ-model on the supercoset P SU(2,2|4) SO(1,4)×SO (5) [1] and the Z 4 -grading of the psu(2, 2|4) superalgebra is a fundamental ingredient to obtain the Lax connection [2].
Recently, techniques to deform integrable theories while keeping their integrability have been developed. One of them is based on r-matrices that satisfy the Yang-Baxter equation. This kind of deformation was proposed by Klimcik as a way to obtain an integrable deformation of the Principal Chiral Model [3,4]. He used a Drinfeld-Jimbo r-matrix [5,6] which satisfies the modified classical Yang-Baxter equation (mCYBE). This deformation was also applied to symmetric cosets σ-model [7] and furthermore to the AdS 5 × S 5 σ-model [8,9]. In this last case the supercoset construction was performed and the background was named η-deformed AdS 5 × S 5 [10,11]. The important feature of this deformed background is that it does not satisfy type IIB supergravity field equations which led to a proposal for generalized supergravity equations [12,13]. However, it was shown recently that the standard supergravity equations are satisfied by an η-deformed background if the Drinfeld-Jimbo r-matrix associated to this deformation is constructed in a specific form [14].

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It is also possible to consider r-matrices that are solutions of the classical Yang-Baxter equation (CYBE). The deformation of the symmetric coset σ-model was performed in [15] and for superstrings in AdS 5 × S 5 in [16]. The interesting property of these deformations is that they lead to several known backgrounds of type IIB supergravity [17][18][19][20][21], namely Lunin-Maldacena-Frolov [22,23], Hashimoto-Itzhaki-Maldacena-Russo [24,25] 1 and Schrödinger spacetimes [27][28][29], which can also be obtained via TsT transformations [30]. In these cases the r-matrices are all Abelian. These results were extended to the nonabelian case [31] and it was conjectured [32] that deformations using solutions of the CYBE are equivalent to nonabelian T-duality transformations [33,34]. An interesting feature of these deformations is that they can generate partial deformations, i.e. either in AdS 5 or in S 5 for the AdS 5 × S 5 case. In terms of TsT transformations this can be explained by considering two-tori with directions either along the brane or transverse to it, or with one direction along the brane and the other transverse to it [35]. In the first case the directions of the two-torus will be generated by momenta operators and this corresponds to a noncommutative two-torus in the field theory, i.e. a noncommutative field theory. By choosing one direction along the brane and one U(1) direction, generated by a Cartan generator, transverse to the brane we get the so-called dipole deformation which gives nonlocal properties to the dual field theory without noncommutativity. If there are three U(1) possible transverse directions, that is three Cartan generators, we can perform the deformation in three ways or more, generally a combination of them. This type of deformation breaks supersymmetry and Lorentz symmetry in the dual field theory. If the two-torus is in the transverse directions we get U(1) × U(1)-type deformations called βdeformations (or γ-deformation if the parameter is real). In general, having three U(1) directions in the transverse space allow us to construct the transverse two-torus in three ways to get different deformations or a combinations of them. This type of deformations is marginal, i.e. it does not affect the conformal symmetry of the dual field theory but reduces the number of supersymmetries from N = 4 to N = 1. There is another way of getting a deformation from the TsT perspective: the null Melvin twist or TsssT transformation [36]. This is similar to the procedure for dipole deformations and can be extended to multiple parameters if the internal space has more than one U(1) direction, as it is the case of S 5 and CP 3 . In general, by deforming the AdS subspace we expect, by the AdS/CFT correspondence, to obtain either nonconformal or noncommutative or nonlocal theories as duals. On the other hand, by deforming the internal subspace we expect, in general, to reduce the supersymmetries on the field theory side. Thus, a combination of these deformations in the gravity side leads to interesting dual field theories.

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deformations of the σ-model on this supercoset has been considered [40]. A solution of the CYBE for an abelian r-matrix, in which only the CP 3 subspace was deformed, was found. This deformation leads to a three-parameter β-deformation of the AdS 4 × CP 3 background that can be obtained also by using TsT transformation [35]. Motivated by this work our aim is to study other possible integrable deformations of this background.
The r-matrices satisfying the CYBE are constructed in terms of combinations of generators of the superalgebra uosp(2, 2|6). We discuss r-matrices that lead to gravity duals of noncommutative ABJM theory as well as its three-parameter dipole deformation. These backgrounds were found initially by performing TsT transformations on the AdS 4 × CP 3 background [35]. In addition to this, we also present r-matrices that lead to gravity duals of a nonrelativistic ABJM theory in a Schrödinger spacetime.
This paper is organized as follows. In section 2 we present a short review on the construction of Yang-Baxter deformed σ-models and in section 3 we derive the bosonic deformed background for a generic Yang-Baxter deformation. In section 4 we build the coset for AdS 4 × CP 3 paying attention to the relevant subalgebras that will be necessary in the following section. In section 5 we discuss the NSNS sector of the new backgrounds obtained by deforming AdS 4 × CP 3 and in section 6 we discuss our results and future perspectives.

Yang-Baxter deformed σ-models
The action for the Yang-Baxter σ-model on a Lie superalgebra g of a supergroup G with Z 4 grading is given by [8] where the Maurer-Cartan one-form A = g −1 dg ∈ g, with g ∈ G, having Z 4 -grading, splits as where γ αβ is the worldsheet metric with det γ = 1, and κ 2 = 1 as required by kappa symmetry. The operator d and its transpose 2 d are where P i (i = 1, 2, 3) are the projectors on each subalgebra of g except P 0 on g (0) . This is required in order to have a g (0) -invariant action. The deformed current and its transpose are defined as

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where the operator R g is R is the operator associated to the Yang-Baxter equation (YBE) which can be written, in its modified version, as where M, N ∈ g. In (2.1) and (2.8) the parameter c refers to either the classical Yang-Baxter (CYBE) equation or to the modified classical Yang-Baxter equation (mCYBE).

Bosonic backgrounds
In order to discuss the bosonic backgrounds we switch off the fermionic generators so that we now have g b = g (0) ⊕ g (2) . For the case of the CYBE we get from (2.3) and (2.4) that d = d = 2P 2 where P 2 is the projector on g (2) . The deformed Lagrangian in (2.1) reduces to where we have chosen κ = 1 for convenience. Let us consider the case where g (2) is a coset subalgebra g (2) = g b /g (0) . Thus, g (0) has the generators of the local bosonic symmetries while g (2) contains the generators which will allow us to construct the background. With this in mind, the projector P 2 can be written as where K m are the generators of g (2) , H i are those of g (0) and X ∈ g. The currents in (2.5) and (2.6) can be recast as and we can find P 2 (J) by projecting (3.3) into g (2) , We then find that P 2 acts as In addition, we are going to need the following projection To do that we define from (2.5) From (3.6), (3.7), (3.8) and (3.9) we find Then, from (3.1), we can read off the metric and the B-field as

Coset construction for AdS × CP 3
Arutyunov and Frolov [38] and Stefanski [39] proposed a supercoset σ-model formulation for type IIA superstrings in AdS 4 × CP 3 along the same lines as what it was done for type IIB superstrings in AdS 5 × S 5 . In this formulation type IIA superstring theory in AdS 4 × CP 3 is described by a σ-model on the supercoset UOSp(2, 2|6)/ (SO(1, 3) × U(3)). However, in this supercoset the spinors have 24 components instead of the usual 32 components so that it does not describe the full superstring. This means that the missing components have been gauged away so that this coset has a partially fixed κ-symmetry [38]. The isometry group of AdS 4 × CP 3 is the coset This bosonic group is part of the supercoset UOSp(2, 2|6)/ (SO(1, 3) × U (3)). The supergroup G = UOSp(2, 2|6) has superalgebra g = uosp(2, 2|6) on which the σ-model can be constructed. The generators of g can be written as supermatrices formed by blocks that correspond to bosonic and fermionic generators. The bosonic part of g can be written as Then a supermatrix on g then has the following form where Q,Q are the fermionic blocks corresponding to the fermionic sector of g, g f = g (1) ⊕ g (3) . As discussed in [40], in order to get the Fubini-Study metric for CP 3 , we have to extend the coset of CP 3 with an su(2) algebra. Then the coset for AdS 4 × CP 3 is now written as (4.4) The supermatrix for g will then have the following structure where the dashed line splits the algebras corresponding to the subspaces AdS 4 and CP 3 , and the solid line splits the M 6×6 and M 4×4 bosonic blocks. Notice that this extension does not mix with the original fermionic blocks. The basis of so(2, 3) ⊕ su(2) ⊕ su(4) that we will consider is composed of so(2, 3) generators denoted by M ij , i, j = 0, 1, 2, 3, 4, su(2) generators denoted by M a , a = 1, 2, 3, and su(4) generators denoted by L m , m = 1, . . . , 15, where m ij are the ten 4×4 antisymmetric matrices representing the generators of isometries of AdS 4 and σ a and λ m are, respectively, the conventional 2 × 2 Pauli and 4 × 4 Gell-mann matrices of su(2) and su(4) (see appendix B and appendix A). The commutation relations and supertraces are with η ij = diag (−, +, +, +, −, +, +, +, +, +), and The global symmetry algebra of AdS 4 space can be written as (4.12) and Str (K m K n ) = 1 4 η mn , m, n = 0, 1, 2, 3. where The CP 3 space can be written as the coset in A basis for this coset is where and H = L 6 + L 9 + M 1 , with L m given in (4.6). We also have Str (K m K n ) = 1 2 δ mn , m, n = 4, . . . , 9. An appropriate coset representative is then Therefore, the full bosonic representative for AdS 4 × CP 3 is In order to find out the undeformed background we will use the procedure taken in section 3. Now the bosonic algebra is so(2, 3)⊕su(2)⊕su(4) so that the projector (3.2) becomes

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and (3.6) turns into P 2 (A) = E m K m , m = 0, 1, . . . , 9. Using the parametrization (4.14) and (4.20) we find (4.24) Then the undeformed AdS 4 × CP 3 metric 3 can be found from (3.15) with η = 0 so that from (2.5) we have J = A and then where (θ 1 , ϕ 1 ) and (θ 2 , ϕ 2 ) parametrize the two spheres of CP 3 , the angle ξ, 0 ≤ ξ ≤ π/2, determines their radii and 0 ≤ ψ ≤ 2π. This background comes from the eleven-dimensional M-theory on AdS 4 × S 7 /Z k corresponding to N = 6 U(N ) × U(N ) superconformal Chern-Simons theory at levels k and −k when we take large N and k [37]. Due to the Z k quotient the S 7 can be expressed as a S 1 fibration over CP 3 . And since the radius of S 1 becomes small as k increases the internal space reduces to CP 3 which gives the AdS 4 × CP 3 tendimensional spacetime. The CP 3 metric (4.26) is a particular form of the Fubini-Study metric [41].

Yang-Baxter deformed backgrounds
In this section we present some r-matrices satisfying the CYBE and build the corresponding AdS 4 × CP 3 deformed backgrounds identifying their gravity duals.

Noncommutative ABJM theory
Let us first consider an Abelian r-matrix like involving momenta operators on AdS 4 along x 1 and x 2 with µ the deformation parameter. 4 The nonzero components of Λ n m in (3.7) are while the nonvanishing elements of C n m in (3.11) are where The deformed metric can then be obtained from (3.15) The B-field is obtained from (3.16), which introduces the noncommutativity in the (x 1 , x 2 )-plane as x 1 , x 2 ∼ µ [25,[42][43][44][45]. The commutative theory is recovered when µ = 0 since M = 1 and B = 0. This result agrees with the gravity dual of noncommutative ABJM obtained by TsT transformations, with γ = 2µ, where γ is the TsT deformation parameter [35]. Other choices for the r-matrix like p 0 ∧ p 1 and p 0 ∧ p 2 leads to noncommutativity in (x 0 , x 1 ) and (x 0 , x 2 ) directions, respectively, but they correspond to non-unitary and non-causal quantum field theories [46,47]. This is in agreement with the fact that Yang-Baxter deformations of AdS, i.e. involving only generators of the conformal algebra, are dual to conformal twists that give rise to general noncommutative structures in the field theory side [48][49][50].

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while the B-field is It is worth mentioning that the choice of generators in (5.8) 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 4 and a combination of the U(1)'s in CP 3 . The resulting metric (5.12) has deformations along the x 2 -direction in AdS 4 and along the (ϕ 1 , ϕ 2 , ψ) angles in CP 3 . For µ 1 = µ 2 = 0 and µ 3 = µ the last two lines in (5.12) vanish and we obtain (5.14) and This agrees with the dipole deformed ABJM theory obtained by a TsT transformation [35]. The B-field (5.15) does not give rise to noncommutativity in the dual field theory because it has legs in AdS 4 as well as in CP 3 . Instead, the components B 2i for i being the labels for the CP 3 coordinates ϕ 1 , ϕ 2 , ψ, give rise to a dipole vector that introduces nonlocality in the field theory [51,52].

Nonrelativistic ABJM theory
In order to construct this deformation we must use light-cone coordinates in AdS 4 . Then the coset representative is now with while for CP 3 we keep the same form as (4.20). The AdS 4 metric is then while the CP 3 metric is still given by (4.26).

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with (5.24) The first term in (5.23) is a Schrödinger spacetime. 5 The B-field is now The choice of generators in (5.20) is very similar to the one in (5.8). Now, however, the two-tori defined by the TsT transformation takes the x − coordinate and a combination of the internal U(1)'s in CP 3 and does not introduce any noncommutativity in the dual field theory. The metric (5.23) has the form of a Schrödinger spacetime with dynamical exponent two [55,56]. 6 This type of background corresponds to gravity duals of nonrelativistic field theories in M-theory [58]. The nonrelativistic ABJM theory has its origin in Mtheory where it is described as the dual field theory of M2-branes in an orbifold space. It has been suggested that there are several nonrelativistic gravity duals with Schrödinger symmetry [59]. The number of Sch 4 × CP 3 spaces is equal to the degeneracy of a scalar harmonic function Φ on CP 3 , for which −∇ 2 Φ = λ k Φ with λ k = 4k (k + 3). For each λ k we have deg(λ k ) = 1 12 (1 + k) 2 (2 + k) 2 (2k + 3) [60]. The value of k is fixed by requiring that the R ++ component of the Ricci tensor, which involves Φ, vanishes [61]. Our background has Φ = M/r 2 and −∇ 2 Φ = 4λ 1 Φ, 7 with λ 1 = 16, which has multiplicity deg(λ 1 ) = 15. This indicates that the Schrödinger background we obtained is one of a family of fifteen solutions. The others solutions can be obtained from the fourteen r-matrices of the form r = p − ∧ i ν i T i , where T i are generators of su(2) ⊕ su(4) that are not Cartan generators and ν i are the deformation parameters. The r-matrices generating these others Sch 4 × CP 3 solutions have the form r = µ i p − ∧ L i , with i = 1, 2,4,5,6,7,9,10,11,12,13,14 and r = µ a p − ∧ M a , with a = 1, 2. In the first case, with i = 1 we have The Schrödinger symmetry is the maximal symmetry group of the free Schrödinger equation. It is the nonrelativistic version of the conformal algebra [53,54]. This symmetry is realized geometrically as Schrödinger spacetimes. 6 The dynamical z factor is the exponent in the power of the radial direction in the r 2z dx 2 + term. To have Schrödinger symmetry we must have z = 2. The relativistic symmetry corresponds to z = 1. A Schrödinger spacetime Sch4 × CP 3 with dynamical exponent three was reported in [57]. 7 The factor 4 in the Laplace-Beltrami equation comes from the fact that the radius of CP 3 is twice the radius of AdS4.

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and we get with For the second family of r-matrices, taking a = 2 we have and we find These deformed backgrounds are all Sch 4 × CP 3 and belong to the family of nonrelativistic ABJM theories mentioned above.

Conclusions
We computed explicitly backgrounds generated by some r-matrices which satisfy the CYBE in deformed AdS 4 × CP 3 . By considering an abelian Jordanian r-matrix we obtained the metric and B-field of the gravity dual of the non-commutative ABJM theory. By choosing a r-matrix with three Cartan generators, one in AdS 4 and a combination of two generators in CP 3 , we obtained backgrounds for a dipole deformed ABJM theory. These backgrounds coincide with those obtained via TsT transformations [35]. We also considered a r-matrix built in a similar way but using a light-cone component of the momenta. Such deformed backgrounds are the gravity duals of nonrelativistic ABJM theories in Schrödinger spacetime. These backgrounds are also expected to be obtained by an appropriate null Melvin twist [36]. An interesting point is to compute the corresponding TsT null Melvin twist of the undeformed AdS 4 × CP 3 background since it should give the same Yang-Baxter JHEP01(2021)056 deformed backgrounds we derived. All these deformations of AdS 4 × CP 3 lend support to the relation between TsT transformations and solutions of the CYBE known as the gravity/CYBE correspondence [17].
We have focused only on the NSNS sector of the background. To compute the RR fields we must take into account the fermionic degrees of freedom. Since the RR fields can be obtained by TsT transformations [35] it should be straightforward to obtain them by just including fermions in our parametrization as it was done for the AdS 5 × S 5 case [21].
A A basis for the so(2, 3) algebra The 10 generators of SO(2, 3) can be written as where i, j, k, = 0, 1, 2, 3, 4. We choose the following representation for the SO(2, 3) Γ i matrices A list of non-vanishing structure constants can be found in [63]. In this representation the Cartan generators are given by λ 3 , λ 8 and λ 15 .