Non-abelian tri-vector deformations in d=11 supergravity

A truncation of the SL(5) Exceptional Field Theory that allows to describe spacetimes of the form $M_4 \times M_7$ with the 4-form flux on $M_4$ is constructed. The resulting theory is used to test the recently proposed tri-vector generalisation of Yang-Baxter deformations applied to the AdS${}_4 \times \mathbb{S}^7$ solution in $d=11$ supergravity. We present two new supergravity solutions corresponding to non-abelian non-unimodular tri-vector deformations of AdS${}_4 \times \mathbb{S}^7$.


Introduction
Supergravity interpretation of integrable deformations of string theory σ-models has seen rapid progress in the recent years. Yang-Baxter deformations [1][2][3], η-deformed [4,5] and λ-deformed [6,7] σ-models may all be represented by combinations of T-dualities [8,9], as well as their non-abelian [10][11][12] and Poisson-Lie [13] extensions. An element of the T-duality group O(d, d), acting on a supergravity background, can be conveniently represented by the so-called β-shift, parametrised by a bivector β [14,15]. Basic building blocks of integrable deformations in the supergravity language, the Lunin-Maldacena (TsT) [16,17] transformations, correspond to constant β [18]. General Yang-Baxter deformations result from using r-matrix solution to the classical Yang-Baxter equation as a deformation bi-vector. The transformation of the NSNS supergravity background fields (g, b) → (G, B) is given by the Seiberg-Witten open/closed string map [19], with β playing the role of an anticommutativity parameter [20,21]. Generalised to include the b-field in the initial background, this map takes the form: (1.1) Here β = 1 2 r αβ k α ∧ k β is the deformation bi-vector written in terms of a constant antisymmetric rmatrix and the Killing vectors of the initial background, that obey the isometry algebra [k α , k β ] = f αβ γ k γ . Assuming that r αβ satisfies the classical Yang-Baxter equation is sufficient for the deformed fields G, B to be a solution of supergravity [22,23]. This allows to view the transformation (1.1) as a supergravity solution generating method, valid for generic spacetimes with isometries [24,25]. The reason that the classical Yang-Baxter equation (1.2) is instrumental in the d = 10 deformation prescription is ultimately that the two-dimensional string worldsheet theory exists behind the scenes of the supergravity approximation. Similarly, it is natural to expect that some fundamental properties of M-theory would become manifest if a generalisation of Yang-Baxter deformations to d = 11 supergravity were constructed. In the absence of an M-theory version of σ-model deformation narrative, we propose that supergravity symmetries can be employed to build such a generalisation.
Supergravity formulations that are natural to look at in this context are Double [26] and Exceptional [27] Field Theories (DFT and ExFT, respectively). Specifically designed to render supergravities in various dimensions covariant under T-and U-duality groups at the expense of extending the spacetime dimension, they are useful in describing Yang-Baxter deformations [15,[28][29][30][31] and Poisson-Lie T-duality [32][33][34][35][36][37]. The proof of [23] that (1.1), (1.2) is a supergravity symmetry relied upon DFT techniques, in particular the β-supergravity formalism [38][39][40]. The map (1.1) is then viewed as expression of the intrinsic freedom of frame choice in DFT, which admits straightforward extension into the ExFT realm, and hence to d = 11 supergravity. The deformation bi-vector β becomes a dynamical field, and it can be shown that the CYBE (1.2) is sufficient to put β on-shell.
In [41] a tri-vector deformation prescription for d = 11 supergravity was proposed, based on the freedom of frame choice in the SL(5) ExFT [42]. The NSNS 2-form b and the deformation bi-vector β are replaced by rank 3 tensors C and Ω, with a Killing tri-vector ansatz for the latter, and a slightly more involved deformation prescription (3.16) instead of the open/closed map. When the Killing vectors form a U(1) 3 subgroup, this prescription reproduces an uplift of TsT to d = 11 [43,44]. For a single commuting ∂ * the tri-vector can be written as Ω = ∂ * ∧ β, and non-abelian YB deformations w.r.t. β can be recovered after dimensional reduction. Whether any intrinsically 11-dimensional deformations exist has been left an open question because of a technical restriction imposed by the formalism. Namely, the simplified SL(5) ExFT setup of [41] required that there be no flux of the 3-form C inside the 4-dimensional submanifold, where the deformation acts. Thus, the consideration was essentially restricted to flat space or a sphere and it was hard to come up with an isometry algebra nontrivial enough to provide a completely non-abelian Ω. It is one of the aims of the present paper to overcome these restrictions. We adopt the approach similar to that of [23], only now instead of β-supergravity field equations one has to deal with the dynamical equations of a certain truncation of the SL (5) ExFT. This theory is designed to describe the mechanics of U-duality within the 4-dimensional submanifold in a 4 + 7 split. Thus, it can become a natural tool in studying tri-vector deformations of AdS 4 within the Freund-Rubin solution. Conformal algebra of AdS 4 is nontrivial enough to harbour non-abelian tri-vectors, so that the resulting deformations of AdS 4 × S 7 cannot be interpreted as uplifts of d = 10 Yang-Baxter deformations in any obvious manner. Using the generators of momentum P a , angular momentum M ab , and dilatation D, we study nonabelian deformations corresponding to Ω ∼ P ∧ P ∧ M and D ∧ P ∧ P . Such Ω cannot be represented in the form Ω = ∂ * ∧ β such that ∂ * commutes with the generators of β. More importantly, one shows that the 11-dimensional analogue of the I vector of generalised supergravity is non-zero for these backgrounds. Hence, although these two deformed backgrounds are solutions of the conventional d = 11 supergravity, one might expect them to connect to solutions of d = 10 generalised supergravity upon reduction. This might be a hint of non-existence of an analogue to generalised supergravity in d = 11.
The paper is structured as follows. After briefly introducing the SL(5) exceptional field theory in Section 2, we derive explicit relations between the fields of d = 11 supergravity in the 4 + 7 split and the ExFT. In Section 3 we truncate the theory to backgrounds of the form M 11 = M 4 × M 7 with the metric on M 7 that does not depend on coordinates of M 4 . We define the deformation map for background with the 3-form flux on M 4 and provide equations of motion that the deformed background must satisfy. In Section 4 the developed formalism is applied to the AdS 4 ×S 7 background and the deformed solutions are presented. We discuss the results in Section 5, and comment on the tentative d = 11 generalisation of the CYBE that has yet to be determined.

Exceptional field theory: SL(5) group
The SL(5) exceptional field theory describes the supergravity dynamics, while being explicitly covariant under transformations of the SL(5) U-duality group. The theory is formulated in terms of fields that depend on 7 coordinates y µ parametrising the so-called 'external' space and 10 coordinates X M N paramterising the so-called 'internal' space (M, N = 1, . . . , 5 are fundamental SL(5) indices and X M N is antisymmetric. We refer to the appendix A for our index conventions). h µν is the external space metric, while m M N , collection of scalars from the d = 7 theory viewpoint, forms what is called the ExFT generalised metric. The internal space is endowed with the structure of extended geometry [45,46] with SL(5) as the group of local coordinate transformations. Under these a generalised vector V M of weight λ transforms as defined by the following generalised Lie derivative where T M N represent generators of SL (5). Algebra of such local generalised diffeomorphisms closes upon imposing the section constraint where bullets represent any combination of any fields. In what follows we will always assume the solution of the section constraint ∂ mn = 0, that corresponds to d = 11 supergravity [47] by removing the dependence on six out of ten extended coordinates (m, n = 1, . . . , 4). Lagrangian of SL(5) exceptional field theory reads [47] where e = (det h µν ) 1 2 and the scalar part part is given by (2.5) Here and further in the text the upper sign corresponds to the case where the external d = 7 space has Lorentzian signature, while the lower corresponds to the Euclidean signature. Splitting the fundamental SL(5) index as M = 1, . . . , 4, 5 = (m, 5), the components of generalised metric can be parametrised as with V m = 1 3! ε mnkl C nkl and h = det h mn . The variables h mn and C mnk will be later related to the components of the ordinary metric and the 3-form on the d = 4 subspace. Under generalised Lie derivatives the metric transforms as that is as a generalised tensor of weight λ[m M N ] = 0. To provide explicit relation between 11-dimensional fields and those of the full SL(5) exceptional field theory one first performs Kaluza-Klein decomposition under the 7+4-split and rearranges the resulting fields into combinations covariant under generalised Lie derivative of the SL(5) theory. Along these lines we closely follow the E 6 discussion of [48] with minor changes relevant for the SL(5) group. One starts with the fields h µν and h mn which are related to the full d = 11 metric by the usual Kaluza-Klein ansatz for the vielbein E (μ,â = 1, . . . 11): Here h µμ is the 7-dimensional vielbein defined as h µν = h µμ h νν ημν, h mm is the 4-dimensional vielbein defined as h mn = h mm h nn ηmn and h = det h mn . Note the factor h − 1 10 that is needed to end up with the correct Einstein-Hilbert action in 7-dimensions.
An important point is that the external metric is not a scalar under generalised Lie derivative, as it transforms under its four-dimensional part Λ m as Hence, h µν is a weighted scalar of weight λ[h µν ] = 2 5 . This is crucial for defining the procedure of deformation in analogy with the d = 10 case as in [23,49], namely as a rotation of the generalised metric by a specific matrix taking values in the symmetry group (SL(5) in our case). To perform this consistently we rescale the generalised metric so as to absorb the degrees of freedom contained in det h µν . As explained below, this makes a connection between the full exceptional field theory defined above and its truncated version described in [50].
Similarly one applies the general prescription of the Kaluza-Klein reduction to the d = 11 3-form to obtain the following tower of p-forms (2.10) Note that in contrast to [48] we are using the conventions where the 3-form kinetic Lagrangian has the prefactor −1 48 This together with the action for the Kaluza-Klein vector A µ m following from the eleven-dimensional Einstein-Hilbert action produce kinetic terms for all p-forms in the theory. Generalised metric encodes scalar degrees of freedom encoded in the metric h mn and the gauge field C mnk . The corresponding terms in the d = 11 action read Here we denote L EH the Lagrangian and define the combinations covariant under the internal diffeomorphisms Substituting the explicit form of the generalised metric (2.6) it is straightforward to check that the above can be written in an SL(5) covariant form as (2.5). Note how the above expression differs from that of [42] in the part that includes only m M N and its derivatives. The reason is that the above reproduces the action with h − 1 5 as a prefactor, following from the proper Kaluza-Klein ansatz, rather than the action √ h(R[h (4) ] − 1 48F 2 ) as one would expect in a truncated theory. More details on that in the following section.
Although using the dualisation procedure it is possible to provide full identification between the 11-dimensional action and the SL(5) covariant action, for the purposes of the present paper we are not interested in topological terms of the SL(5) theory. Moreover, in what follows we will truncate the theory to describe only a special class of backgrounds, relevant for the discussion.

Truncation to extended space
The general procedure for deforming supergravity background in the ExFT/DFT formalism is based on switching between geometric and non-geometric frames encoding the same generalised metric, and further interpretation of the non-geometric tri/bi-vector as a deformation tensor rather than a fundamental field. In this approach the deformation tensor can include only Killing vectors of the 'internal' part of the background, using the terms of exceptional field theory.
To simplify further discussion we consider only such backgrounds, whose metric can be represented in a block-diagonal form, i.e. M 11 = M 4 × M 7 , where the internal metric h mn does not depend on the external coordinates y µ . This allows to significantly simplify the equations of motion by truncating the theory to purely scalar SL(5) extended geometry, similar to that of [42,45], however keeping track of geometry of the external space. Taking this into account equations of motion following from the full SL(5) exceptional field theory are truncated to the case when Moreover, given the structure of the theory the second line above can be applied already at the level of the Lagrangian. This simplifies the setup of exceptional field theory leaving us with only the d = 7 Einstein-Hilbert term and the scalar potential for the generalised metric in the action where R[h (7) ] is the Ricci curvature scalar of the metric h µν . It is important to note, that such truncation is background dependent, based on the specific ansatz (3.1) and does not provide full consistent truncation of the theory. However, taking a specific initial solution of the form (3.1) one is guaranteed to end up with a valid d = 11 solution when making a tri-vector deformation, as long as the chosen Killing vectors do not introduce any dependence on the external coordinates y µ . Note that although we truncate the Lagrangian, the structure of the couplings is such that the truncation at the level of equations of motion would be equivalent.
In what follows we are interested in the case where a deformation results in rescaling of the 7dimensional part of the metric by a single x m -dependent factor. The d = 7 metric before deformation will be restricted to the form h µν (y µ , x m ) = e −2φ(x m ) h 1 5h µν (y µ ), which allows to hide the x m dependence φ(x m ) inside a properly rescaled generalised metric. To achieve this, define the rescaling as follows This implies that the Lagrangian L = eR[h (7) ] + L sc can be rewritten as follows where M = det M M N = e 5φ h −1 2 andē = (deth µν ) 1 2 . For the rescaling (3.3) the d = 11 vielbein can be written in the following nice form while the generalised metric becomes with V m = 1 3! ε mnkl C nkl and h = det h mn . Substituting this into (3.4) one gets for the Lagrangian Note, that when R[h (7) ] = 0 the covariant Lagrangian (3.4) reproduces the SL(5)×R + Lagrangian of [42] up to full derivative terms.

Deformation map
The rescaled metric M M N proves to be the most convenient to define deformations by a generalisation of β-shift of DFT that we call the Ω-shift. The membrane generalisation of the open-closed string map that is behind the deformation procedure in 11-dimensional supergravity used in [41] can be understood as the following transformation, acting on the generalised vielbein where ǫ mnkl is epsilon symbol and Ω mnk are tensor components of the deformation tri-vector Ω = 1 3! ρ αβγ k α ∧ k β ∧ k γ . Deformation in this form is completely frame independent and allows to define deformations for background with fluxes. Consider the initial background comprised by the internal metric g mn , gauge 3-form field encoded by v m = 1 3! ε mnkl c nkl and the 7 × 7 block of the 11-dimensional metric g µν = e −2φ(x)ḡ µν (y). The generalised metric is given by (3.6) where g m a is vielbein for the metric g mn and g = det g mn . The deformation map (3.8) transforms the initial generalised vielbein into E = e . The latter is now understood as generalised vielbein written in the C-frame for the new deformed background parametrised by deformed fields G mn , V m and G µν = e −2Φ(x)ḡ µν (y) (recall that we consider only the deformations, where the external metric changes only by an x m -dependent factor). In terms of generalised metrics one writes where W m = 1 3! ε mnkl Ω nkl . The first matrix above is just the result of multiplication (3.8), while the second matrix already contains the deformed fields. Equality between these two generalised metrics is what defines the deformation in terms of d = 11 fields (g mn , g µν , c mnk ) → (G mn , G µν , C mnk ).
To recover explicit relations for d = 11 fields we follow the same procedure as in [41] and start with equating determinants in both frames to obtain where G = det G mn . Next, equating the generalised metrics block-by-block one writes Taking determinant of the first line and using the algebraic identity one obtains This defines transformation rule e Φ = K 1 6 e φ for the field φ and hence for the external metric. Understanding K as a function of the deformation parameter W m , the equations in (3.13) express the deformed fields in terms of the original metric g mn , gauge field v m and deformation tensor W m . Altogether, deformation rules can be summarised as follows: (3.16) Note that indices of C mnk are raised by the deformed metric G mn , while indices of c mnk are raised by the corresponding initial metric g mn . It is worth reminding that the external G µν and internal G mn blocks of the full d = 11 metric are defined by the following interval 17) and the external metric has the form G µν (y, x) = e −2Φ(x)ḡ µν (y) for the initialḡ µν that does not depend on the internal coordinates.

Equations of motion
Consider now the dynamical equations that control the deformation tensor W m , given that the initial and the deformed backgrounds satisfy equations of motion of the full d = 11 supergravity, or equivalently of the truncated theory.
For technical reasons we consider equations governing deformation of the AdS 4 × S 7 background in the C-frame. Hence, equations on the deformation tensor W m are implicit in this case. One starts with the Lagrangian of the truncated SL(5) ExFT in the C-framē Equations of motion for the dynamical fields φ, h mn and V m then become These prove to be much simpler for further calculations than the original equations of motion of eleven-dimensional supergravity. The external space is always fixed to be the 7-sphere with the metricḡ µν up to a prefactor e −2φ . Any supergravity solution of the form (3.1), before or after the deformation, must also be a solution to these equations. To derive explicit equations on the deformation tensor for the AdS 4 × S 7 background, one has to work in the mixed (C − Ω)-frame using the generalised metric (3.11) in the Lagrangian (3.4). This provides formulation of eleven-dimensional supergravity in terms of both C mnk and Ω mnk , however, with the restriction that Ω mnk is non-dynamical and rather encodes deformations. Given the complicated form of the generalised metric (3.11) this appears to be a technically involved procedure, and hence stays beyond the scope of the present paper. Explicit construction of such formulation for both DFT and ExFT is an open problem.

AdS × S 7 background
As an application of the developed formalism, let us look at the deformations of AdS 4 × S 7 spacetime. We will study deformations, corresponding to Ω ∼ P ∧ P ∧ M, and D ∧ P ∧ P . Geometry of the initial eleven-dimensional solution may be expressed as with a unit metric on the seven-sphere. We consider the AdS part as the 'internal' space for the SL(5) ExFT. Denoting the AdS coordinates as x m = (x 0 , x 1 , x 2 , z), the metric is as usual The only component of the flux and the corresponding 3-form gauge potential then become In this work we are interested in tri-vector deformations of generalised Yang-Baxter type where k α are Killing vectors of the initial background, in our case AdS 4 ×S 7 . As it has been mentioned in the previous section, the deformation matrix O[Ω] does not depend on the frame chosen, which implies that one may use Killing vectors of AdS 4 in the C-frame 1 . Hence, we list Killing vectors of the AdS 4 space in the C-frame P a = ∂ a , K a = x 2 ∂ a + 2x a D, where a, b = 0, 1, 2 and m, n = 0, 1, 2, z, and we define x 2 = η mn x m x n and x a = η ab x b .
To proceed with explicit examples of deformed AdS 4 ×S 7 backgrounds systematically, we consider such combinations of the Killing vectors, that the resulting Ω is polynomial of order 0, 1, etc. in powers of AdS coordinates. Applying the transformation rule (3.16) we derive the deformed metrics G µν , G mn and the 3-form C mnk from their undeformed initial values g mn , g µν , c mnk and the deformation tensor W m defined by the given choice of Ω mnk . To check whether a deformation gives a solution of equations of motion of 11-dimensional supergravity we substitute the deformed background written in terms of the fields Φ, G mn , V m into the equations of motion (3.19) of the truncated ExFT. Since the S 7 part only receives a correction encoded in the prefactor e −2φ , using truncated equation proves technically much simpler than the full d = 11 theory.

P ∧ P ∧ P
Start with the tri-vector as a polynomial of order 0 in coordinates, that corresponds to the trivial abelian P ∧ P ∧ P deformation defined as The deformation tensor and the prefactor K then become (4.7) Following the prescribed procedure one finds for the deformed background which is a solution of the equations (3.19) and hence of the d = 11 equations of motion. For this deformation the Q-flux Q m nkl = ∂ m Ω nkl can be checked to have no trace Q m mnk = 0, hence the solution can be consistently reduced to a solution of the 10-dimensional type IIA theory. In fact, this P ∧ P ∧ P deformation is abelian in the sense that there exists a generator ∂ * (any of the P a in the present case) that commutes with the other two. This implies that the deformation (4.8) can be understood as a result of dimensional reduction of the initial AdS 4 × S 7 to IIA along x * , then a TsT deformation w.r.t. a bi-vector β, such that Ω = ∂ * ∧ β: and finally an uplift back to d = 11. As expected, this reflects the fact that the corresponding deformation is simply a d = 11 extension of a TsT [43,44].

P ∧ P ∧ M
The very next example with Ω being a polynomial of order 1 in x a provides a nonabelian deformation. Using the coefficients with the symmetry ρ ab,cd = ρ [ab],[cd] , consider where ρ 0 = R 3 4 (ρ 02,01 − ρ 01,02 ), and we have introduced a numerical coefficient for convenience. It is easy to see that there is no such generator that commutes with all the others, which means that this deformation is non-abelian. The deformation tensor is 12) and the resulting deformed background then is given by Using (3.19) one can check that this provides a solution to 11-dimensional equations of motion for arbitrary values of the constants ρ a . In contrast to the previous example, trace of the Q-flux is non-zero and reads (4.14) Upon dimensional reduction from ExFT in the Ω-frame to β-supergravity one expects that nonvanishing trace Q m mkl generates non-vanishing trace of the Q-flux of β-supergravity. The latter is known [51] to correspond to the vector I of generalised supergravity.

D ∧ P ∧ P
Another way to build a tri-vector first order in powers of x m is to use the dilatation generator D together with momenta. For the conformal algebra of AdS 4 there are three possible pairs of P a , P b . It is convenient to parametrise a generic tri-vector of the form D ∧ P ∧ P as with ρ a corresponding to the three independent components of the ρ-matrix. Using this Ω, the deformation tensor and the prefactor are where we define ρ 2 = ρ a ρ b η ab . The deformed background is then given by By checking either (3.19) or the field equations of d = 11 supergravity one can show that this background is a solution, if parameters ρ a form a null vector: This is reminiscent of the d = 10 Yang-Baxter deformation with Θ = τ a M ab ∧ P b , also parametrised by a null vector τ . The exact manner in which the condition (4.18) arises is completely analogous to the way in which the Yang-Baxter equation is encoded in d = 10 supergravity. One simply finds a factor of ρ 2 out front every field equation after some simplifying algebra. We take this as a hint, that the condition (4.18) may be an elementary example of a generalised Yang-Baxter equation, as applied to the tri-vector (4.15). Similar to the P ∧ P ∧ M case, this background is an example of a deformation with vanishing R-flux, but non-vanishing trace of the Q-flux. For the latter one calculates Following the same arguments as in the previous subsection we conclude that the obtained deformed background cannot be reduced to a solution of conventional d = 10 supergravity. Moreover, since the tri-vector Ω is non-ablelian the P ∧ P ∧ M and D ∧ P ∧ P deformations cannot be put to the form Ω = ∂ * ∧ β. The conclusion is that both these solutions are proper 11-dimensional deformations that cannot be accessed via 10-dimensional techniques.

D ∧ K ∧ K
The outer automorphism of the conformal algebra can be realised geometrically by an inversion, which is an isometry of AdS spacetime: (4.21) Applying this map to the D ∧ P ∧ P -deformed background (4.17), one should be able to recover the deformation with Ω ∼ D ∧ K ∧ K. Given the geometric symmetry, one expects this D ∧ K ∧ K deformation to also be a solution. Note that the tri-vectors are in fact very closely related, Explicit calculation shows, however, that already the second equation in (3.19), which states ∇ m V m e −7φ = const, does not hold for the obtained background. This negative result makes it very intriguing to derive explicit equations for the deformation tensor, that is the equations of motion (3.19) in the mixed (C − Ω)-frame, and investigate the reason of such unexpected behaviour more closely.

Conclusions and discussions
In this work we studied tri-vector deformations of the AdS 4 × S 7 solution of 11-dimensional supergravity, generalising the results of [41] to the case of non-abelian deformations. Working in the formalism of SL(5) exceptional field theory properly truncated to describe backgrounds of the form M 4 × M 7 , we generalise the deformation map of [41] to the case of backgrounds with nonvanishing 3-form flux and provide two examples of non-abelian deformations. The corresponding tri-vector deformation parameter is schematically given by Ω ∼ P ∧ P ∧ M and Ω ∼ D ∧ P ∧ P , where D, P a , M ab stand for generators of the AdS 4 algebra. Both these deformations are non-abelian, that is one cannot represent the tri-vector in the form Ω = ∂ * ∧ β where ∂ * commutes with the generators of β. This implies, that the deformed backgrounds cannot be obtained by reducing to 10 dimensions, performing a bi-vector deformation and uplifting back to d = 11 (see e.g. [44]). Our proposed procedure may be used to further investigate AdS 4 × S 7 background in search for more non-abelian deformations, as well as to address the deformations of the sphere part of AdS 7 × S 4 . Part of motivation for constructing the non-abelian tri-vector deformations was to test the proposals for generalised Yang-Baxter equation that have appeared recently. In [25] it has been shown using techniques of Double Field Theory and β-supergravity that for a bi-vector deformation β = 1 2 r αβ k α ∧k β to generate a solution to the field equations of d = 10 supergravity, it is sufficient that the matrix r αβ satisfy the classical Yang-Baxter equation. The same condition is imposed by assuming that the R-flux vanishes. Turning to M-theory backgrounds one naturally considers tri-vector instead of bi-vector. In [41] the vanishing of the ExFT R-flux R m,nklp = Ω mq[n ∂ q Ω klp] was proposed as the condition for a tri-vector deformation to be a solution. Assuming the tri-Killing ansatz for Ω (1.3), Explicit check shows that for the P ∧ P ∧ M and D ∧ P ∧ P deformations the R-flux indeed vanishes. However, at least for the D ∧ P ∧ P this is not sufficient to end up with a solution to d = 11 equations of motion, and a stronger algebraic constraint on ρ αβγ (4.18) is required.
Recently using a generalisation of Poisson-Lie T-duality to the U-duality setup and to M-theory brane dynamics [36,52], a proposal for the sufficient algebraic constraint for ρ αβγ has been made in [36]. Both non-abelian deformed solutions described in the present work are in the non-unimodular class, meaning ∂ m Ω mnk ≠ 0, therefore the corresponding ρ αβγ cannot satisfy the equations of [36] as the latter suppose unimodularity. It is then natural to expect that the algebraic constraints for the tri-vector components ρ αβγ , such as (4.18), are manifestations of the M-theory generalisation of the CYBE with non-unimodularity properly taken into account. Note that while in the d = 10 case both unimodular and non-unimodular deformations are required to satisfy the same classical Yang-Baxter equation, this seems not to be the case for M-theory. Moreover, the condition of vanishing R-flux, which is equivalent to the CYBE in d = 10, appears to be only a part of the equations of [36].
Given these results, searching for the general algebraic equations for ρ αβγ that generalise the classical Yang-Baxter equation appears to be an interesting direction of further research. From the algebraic point of view a natural generalisation is to replace the CYBE, relevant for the scattering of particles in 1 + 1 dimensions with the tetrahedron equation describing scattering of strings in d = 1 + 2 [53,54]. Depending on the labeling scheme, the tetrahedron equation may be referred to as Zamolodchikov or Frenkel-Moore equation. Deriving a representation independent form of semi-classical limit of the tetrahedron equation and comparing the results to that of [36] is an open problem.
More transparent is the algebraic interpretation of the vanishing R-flux condition. Following [55,56] one notices that the M2-brane world-volume dynamics brings about a non-commutativity parameter given by a tri-vector Ω mnk , as well as the following Nambu-Poisson 3-bracket {x m , x n , x k } = Ω mnk . (5. 2) The fundamental identity for such bracket, is precisely the vanishing R-flux condition of the SL(5) theory. Indeed, when written in terms of W m = 1 3! ε mnkl Ω nkl the fundamental identity is proportional to ε mnkl W [n ∂ k W l] = 0, that is R m,ijkl ε ijkl = 0. Given this observation and the fact that all particular examples of tri-vector deformations are R-fluxless, it is reasonable to conjecture that any sensible M-theory background must have vanishing R-flux.
As the final remark we notice that in contrast to the approach of [23], in the present work we did not derive explicit equations for the deformation tensor Ω mnk from exceptional field theory, rather working in the C-frame. The dynamical differential equations for Ω seem to be the optimal starting point for deriving the algebraic constraints on the deformation parameters ρ αβγ . However, to address backgrounds with fluxes one should go to the mixed (C − Ω)-frame, which we leave for future work.

A Notations and conventions
In this paper we use the following conventions for indiceŝ µ,ν, = 1 . . .