Bosonic $\eta$-deformed $AdS_4\times\mathbb{CP}^3$ Background

We build the bosonic $\eta$-deformed $AdS_4\times\mathbb{CP}^3$ background generated by an $r$-matrix that satisfies the modified classical Yang-Baxter equation. In a special limit we find that it is the gravity dual of the noncommutative ABJM theory.


Introduction
A Hamiltonian system is integrable if there exists an infinite number of conserved charges in involution or, equivalently, if a Lax connection can be constructed. The required involution of these conserved charges leads to a particular form of the Poisson bracket of the Lax connection in terms of an r-matrix [1][2][3]. In the context of string theory, the integrability of the AdS 5 ×S 5 superstring described by a σ-model on the supercoset psu(2, 2|4)/so(1, 4) ⊕ so(5) [4,5] is a remarkable property that also shows up in less symmetric cases like the AdS 4 ×CP 3 background which is partially described by the supercoset uosp(2, 2|4)/so(1, 3) ⊕ u(3) [6,7].
Since there is no generic and systematic way to construct integrable theories it is quite natural to look for deformations of known integrable theories which still preserve integrability. For the AdS 5 × S 5 superstring this has been extensively analyzed by adapting techniques used to deform integrable sigma models [8]. The strategy for building these deformations, called q-deformations, is to construct a Poisson bracket that preserves the relation between the Lax matrix and the undeformed Hamiltonian producing a deformed Hamiltonian while keeping the dependence of the Lax matrix on the currents. From it we can derive a Lagrangian which is integrable and depends on the R-operator that satisfies the modified Classical Yang-Baxter equation (mCYBE). This procedure was applied to deform the AdS 5 × S 5 superstring [9] and it was found that its η-deformed background is not a solution of the standard type IIB supergravity equations despite the presence of κ-symmetry [10,11].
For superstrings propagating in other backgrounds only a few results are known. Recently some examples of integrable deformations of the AdS 4 × CP 3 background were given based on abelian solutions of the Classical Yang-Baxter equation (CYBE), which also have an interpretation in terms of TsT transformation [12,13]. These deformed backgrounds are duals of noncommutative, dipole and β-deformed ABJM theory as well as a nonrelativistic limit having Schrödinger symmetry.
In this paper we will consider deformations of the AdS 4 ×CP 3 space based on a solution of the mCYBE whose R-operator is where E ij , with i, j = 1, . . . , 10, are the gl(4|6) generators [14]. Its main characteristic is that the r-matrix, the map associated to the R-operator, is composed by the roots of the superalgebra. This is in contrast to the r-matrices that solve the CYBE which are given by the commuting generators of the superalgebra. Here we give the first steps by deriving the bosonic part of the deformed AdS 4 × CP 3 background via the standard coset construction based on the superalgebra g = uosp(2, 2|6). We then take a special undeformed limit of this background that leads to the gravity dual of the noncommutative ABJM theory [13].
Our results can be used towards to the computation of the full η-deformed AdS 4 × CP 3 background, which, once obtained, would allow us to explore a new family of integrable backgrounds, including those of a new type of generalized type IIA supergravity. This paper is organized as follows. In section 2 we present briefly the main steps needed to construct an η-deformed superstring σ-model. Next, in section 3, we review the coset construction of the undeformed bosonic AdS 4 × CP 3 background and in section 4 we derive the η-deformed bosonic sector of uosp(2, 2|4)/so(1, 3) ⊕ u(3). In section 5 we conclude and discuss our results.

η-deformed Superstring Sigma Models
The action for the η-deformed superstring σ-model on g is [9,15] where A = g −1 dg ∈ g, g ∈ G, γ αβ is the string worldsheet metric with det γ = 1 and κ 2 = 1. The Z 4 -grading of g allows the split of A as The operatorď is defined as the following combination of projectors P i (i = 1, 2, 3) on the gradings of gď 3) The absence of P 0 is required for (2.1) to be g (0) -invariant. The deformed current is then where the operator R g is and R : g → g, which is the operator associated to the r-matrix required for integrability, must satisfy the Yang-Baxter equation (YBE) where M, N ∈ g. In (2.1) and (2.6) the parameter c refers to either the classical Yang-Baxter equation (CYBE), c = 0, or to the modified classical Yang-Baxter equation (mCYBE), c = 1. The case c = 1, which is also known as non-split R-matrix [16], can be solved as [10], and was considered in [9]. This type of deformation has been explored for superstrings in AdS 5 ×S 5 [10,11] but not for superstrings in AdS 4 ×CP 3 . In this work we will present some results concerning the bosonic η-deformed background based only on the bosonic roots of the algebra as done in [10].

Coset Construction of the Bosonic AdS 4 × CP Background
The isometry group of AdS 4 × CP 3 is the coset [6,7]. The supergroup G = U OSp(2, 2|6) has the superalgebra g = uosp(2, 2|6) on which the σ-model can be constructed. The bosonic sector of g can be expressed as [12,13] (3.2) As supermatrices, the elements of g can be written as where the dashed lines divide the supermatrix into the blocks corresponding to the subspaces AdS 4 , CP 3 and Q,Q ∈ g f = g (1) ⊕ g (3) .
We can now find a special undeformed limit which is a solution of the standard supergravity in a similar way as has been done for AdS 5 × S 5 [11]. We first rescale the AdS 4 coordinates where ζ 0 is a parameter, and then set χ = 0 to get and B = 1 2 This is the gravity dual of the noncommutative ABJM theory with deformation parameter sin ζ 0 [13] which is a type IIA supergravity solution. The connection between the η-deformed background and the gravity dual of the noncommutative ABJM theory can be understood as an infinite boost of the r-matrix that generates the η-deformed background which gives a r-matrix with parameter sin ζ 0 leading to the gravity dual of the noncommutative theory [19].

Conclusions and Outlook
In this paper we derived the bosonic η-deformed AdS 4 × CP 3 background using the same technique developed for the bosonic Yang-Baxter deformed AdS 4 × CP 3 background [13].
We have also shown that in the special limit χ → 0 it is the the gravity dual of the noncommutative ABJM theory. In the AdS 5 × S 5 case there is also the so-called "maximal deformation" limit χ → ∞ [20] that in our case should leads to the mirror AdS 4 × CP 3 background. This would allow us to study the finite size thermodynamic Bethe ansatz. However, in our case, the construction of the double Wick rotated background that generates the mirror undeformed AdS 4 × CP 3 background is not simply the interchange of the metric elements corresponding to the coordinates involved in the light-cone gauge fixing [21]. To do that we have to take coordinate t in AdS 4 and the coordinate ψ in CP 3 . This happens because CP 3 is not block diagonal so that the double Wick rotation constructed in [20] (see also [22,23]) must be adapted to our case. After that, it will be possible to get the maximal deformation of our η-deformed metric and B-field and see if it also has the mirror background as a limit. Another interesting limit is χ → i, which establishes the connection to the Pohlmeyer reduced model for the undeformed AdS 4 × CP 3 [24]. These are important topics that deserve further study.
The deformed fermionic sector can in principle be obtained in the same way as in the AdS 5 × S 5 case [11] but there are important subtleties that must be taken into account. The superalgebra uosp(2, 2|6) does not describe all fermionic degrees of freedom of the AdS 4 × CP 3 string [6] which are needed to build the RR sector. A fermionic factor g f will appear in the coset representatives needed to obtain the fermionic currents A (1) , A (3) and the fermionic contributions to the operator O −1 in (2.4). Then they can be expanded up to quadratic order to get the deformed fluxes of the theory. The first problem one must face to get the full deformed background is to reincorporate those 32 fermions. Since not all of them are part of the supercoset, a possibility is to start from a supercoset that contains them.
The R-operator (2.7) was also considered in AdS 5 × S 5 [10] where it was found that the deformed background does not solve the standard type IIB supergravity equations but a kind of generalized supergravity equations [25,26]. We then expect that our NSNS fields (4.8), (4.9) and (4.12) are also part of some generalized supergravity. Besides that, the R-operator (2.7), called reference R-operator in [27], can be associated to a r-matrix which is not unimodular [28]. However, as shown in [27], it is possible to find a permutation of the R-operator which gives an unimodular inequivalent R-operator such that the η-deformed background is a solution of type IIB supergravity with the same NSNS fields. It will be very interesting to find out whether the AdS 4 × CP 3 case has the same property.
A A Basis for the so(2, 3) Algebra The 10 generators of SO(2, 3) can be written as and satisfy where i, j, k, = 0, 1, 2, 3, 4. We choose the following representation for the SO(2, 3) Γ i matrices with η ij = diag(− + + + −), and γ a being the gamma matrices in a Dirac representation of SO(1, 3) [29] (see [6] for a different choice) and In order to make explicit the conformal group let us split the indices as A list of non-vanishing structure constants can be found in [30]. In this representation the Cartan generators are given by λ 3 , λ 8 and λ 15 .