Polyvector deformations in eleven-dimensional supergravity

We consider 3- and 6-vector deformations of 11-dimensional supergravity backgrounds of the form $M_5\times M_6$ admitting at least 3 Killing vectors. Using flux formulation of the E${}_{6(6)}$ exceptional field theory we derive (sufficient) conditions for the deformations to generate a solution. In the group manifold case these generalisations of the classical Yang-Baxter equation for the case of r-matrices with 3 and 6 indices are shown to reproduce those obtained from exceptional Drinfeld algebra for E${}_{6(6)}$. In general we see an additional constraint, which might be related to higher exceptional Drinfeld algebras.


Introduction
Vacua of string theory understood as a perturbative formulation of the non-linear two-dimensional sigma-model are known to be represented by a vast landscape of 10-dimensional manifolds equipped by various gauge fields: Kalb-Ramond 2-form and Ramond-Ramond p-form gauge fields. In the full non-perturbative formulation one finds that the space of string vacua is mostly populated by 11-dimensional manifolds and 10-dimensional backgrounds represent points with small string coupling constant (see [1] for more detailed review). The set of string vacua possesses huge amount of various symmetries that prove useful in better understanding of its structure. In particular one finds T-duality symmetries, both abelian [2,3] and non-abelian [4], which relate backgrounds that are indistinguishable from the perturbative string point of view. Non-perturbatively symmetries get enhanced to (abelian) U-dualities, that can be understood as transformations relating toroidal backgrounds equivalent from the point of view of the membrane [5,6]. Certain progress towards defining non-abelian generalisation of U-dualities have been made recently in [7][8][9][10][11].
At the level of low-energy theory of background fields such duality symmetries appear as solution generating transformations. More generally one is interested in transformations that keep the string sigma-model in a consistent vacuum, however changing it in a controllable way. Particularly interesting examples are based on manifolds with an AdS factor, which are known to be holographically dual to superconformal field theories. While in general CFTs are isolated point in the space of couplings corresponding to fixed point of renormalisation group flow, SCFTs belong to a family of theories connected by varying couplings. Adding exactly marginal operators to a theory will preserve conformal symmetry at the quantum level and move the corresponding point in the space of couplings along the so-called conformal manifold [12]. Certain progress in understanding of the structure of conformal manifold can be made by investigating the gravitational side of the AdS/CFT correspondence. Indeed, given a set of exactly marginal operators that deform a SCFT keeping it on a conformal manifold, there exists a family of dual AdS solutions related by deformations of metric, dilaton and p-form gauge fields. The well-known example is provided by β-deformations of D = 4 N = 4 SYM whose gravity dual is a bi-vector abelian (TsT) deformation along two of three U (1) isometry directions of S 5 [13]. In the similar fashion considering AdS 4 × S 7 one is able to pick three U(1) directions of the seven-sphere to construct a trivector deformation of ABJM theory. For a general formula for TsT transformations of gauge theories see [14].
Generalising the results known for TsT deformations one naturally gets interested in bi-and tri-vector deformations along a set of non-commuting Killing vectors. As the most symmetric example here one finds deformations of two-dimensional sigma models preserving integrability, e.g.
η-deformation of the Green-Schwarz superstring on AdS 5 × S 5 [15]. This gives rise to the so-called ABF background [16], which solves equations of motion of generalised supergravity rather than the ordinary Type II supergravity [17]. To depart from backgrounds given by group manifolds and coset spaces one generalises the procedure and for a general deformation parametrised by a bi-vector Θ = r ij k i ∧ k j obtains (g + b) −1 = (G + B) −1 + Θ, (1.1) where G, B and g, b are the metrices and the 2-form fields for the initial and deformed backgrounds respectively [18]. Although at this level the deformation is considered as a transformation of (generalised) supergravity solutions without direct reference to two-dimensional sigma-models, both initial and deformed field configurations could be understood as consistent sigma-model backgrounds. Both the sigma-model (for coset spaces) and field theory (for general manifolds with Killing vectors) approaches show that a deformed background is a solution of (generalised) supergravity equations when the r-matrix r ij satisfies classical Yang-Baxter equation (CYBE) [19,20] r k[i 1 r i 2 |l| f kl i 3 ] = 0. (1. 2) Here f ij k are structure constants of the algebra of Killing vectors.
It is important to notice, that the transformation (1.  [19] only in the first order in Θ. More natural appears the formalism of Double Field Theory [21,22], where the generalised metric is the canonical variable and which allowed full proof in [20] that CYBE is sufficient to end up with a solution. Moving to tri-vector deformations one naturally employs the formalism of exceptional field theory (ExFT) for precisely the same reasons: the generalised metric transforms linearly under deformations. The generalisation of the deformation map (1.1) obtained in the formalism of SL(5) ExFT in [23] finds the same interpretation as an open-closed membrane map [24]. Examples of non-abelian tri-Killing deformations based on the open-closed membrane map, or equivalently a specially defined SL (5) transformation, have been provided for the AdS 4 × S 7 background in [25].
One may naturally ask, whether the condition that a tri-vector deformation generates a solution of supergravity is equivalent to some algebraic condition generalising CYBE. One first notices that CYBE appears when deforming generators of a Manin triple (T i , T i , η) representing a Drinfeld double algebra by r-matrix and requiring the deformed generators to also form a Drinfeld double algebra. Similarly deforming generators of an exceptional Drinfeld algebra (EDA) by tri-and six-vector tensors ρ i 1 ...i 3 , ρ i 1 ...i 6 and restricting the deformed generators to form an EDA, one arrives at a set of conditions on ρ-tensors.
Exceptional Drinfeld algebra based on the SL(5) group has been constructed in [7,8] and for the group E 6(6) this has been done in [26]. Conditions on the 3-vector deformation tensor ρ i 1 i 2 i 3 derived for the SL(5) EDA in [7] are equivalent to the unimodularity condition that is due to the dimension d = 4 of the internal manifold, which appear to be too small to embed 3-vector deformations. The situation is the same as for bi-vector deformations in dimension d = 3 where one gets only the unimodularity condition.
In contrast, deformations considered inside the E 6(6) EDA are subject to a non-trivial constraint, which is supposed to generalise classical Yang-Baxter deformation. In this work we investigate whether this condition is sufficient for a deformation to be a solution generating transformation as for the bi-vector case. For this we start with providing short review of the E 6(6) extended geometry in Section 2. In Section 3 we define deformation map for E 6(6) generalised vielbein and investigate transformation of generalised fluxes under the map. We find, the generalised Yang-Baxter equation of [26] as a sufficient condition for the fluxes to stay undeformed, as well as an additional constraint.

Truncation of 11D supergravity
Bi-vector deformation of backgrounds of non-linear sigma-model is given by the non-linear map (1.1), whose form cannot be called self-evident. Due to this non-linear nature of the map explicit check that this is a solution generating transformation is a highly non-trivial task, as it has been demonstrated in [19]. Choosing correct representation of degrees of freedom allows to turn deforma-  [27][28][29] for more details).
Similarly, as it has been shown in [23] tri-vector deformations of 11-dimensional background can be defined as an SL (5)  Detailed presentation of the E 6(6) exceptional field theory can be found in [30][31][32] that includes bosonic and fermionic field content, supersymmetry transformations, full Lagrangian and truncations to the 11-dimensional and Type IIB supergravities. For our purposes here we stress the following defining features of the theory: • ExFT is an E d(d) -covariant background independent theory combining full 11-dimensional and Type IIB supergravities (no reduction); • field content is represented by tensors of GL (11-d) taking values in certain representations of the U-duality group; • section condition, restricting dependence of fields on the total 5 + 27 coordinates is required. In what follows we assume the standard solution of the section constraint leaving only dependence on 5 + 6 coordinates.
Hence, for the purposes of this work exceptional field theory simply provides a convenient rewriting of degrees of freedom of 11-dimensional supergravity, turning polyvector deformations into a linear map.
Construction of fields and the Lagrangian of the E 6(6) exceptional field theory starting from the 5+6 split of the 11-dimensional supergravity is given in details in [30]. To introduce notations and for further reference we provide a brief overview of the setup. Bosonic field content of 11-dimensional supergravity consists of the elfbeinÊμα and the 3-form potential Cμ 1μ2μ3 . Keeping full dependence on all of the 11 coordinates xμ one splits the fields into tensors in 5-dimensions and organises them into multiplets of E 6 (6) . For the latter one has to follow the dualisation prescription of [33]. Decomposing 11-dimensional indices asμ = (µ, m),α = (µ, a) one parametrises the elfbein in the following upper- and redefines fields arising from the 3-form potential as (2. 2) The resulting fields transform appropriately under slpitted 11-dimensional diffeomorphisms ξμ = (ξ µ , Λ m ). To organise these into multiplets of E 6(6) one has to dualise all forms to the lowest possible rank. Hence, the two forms A µνm get dualised intro 1-forms and can be collected with A µ m and A µmn into the vector where V M denote components of some generalised vector of weight λ and d M N K and d M N K are the invariant tensors of E 6 (6) . For such defined transformations to form a closed algebra one imposes the section constraint where bullets denote any fields and their combinations. The above condition has two (maximal) inequivalent solutions corresponding to embeddings of the 11-dimensional and Type IIB 10-dimensional supergravity. We will be working with the former, i.e. always take into account decomposition of e 6(6) irreps under its subalgebra gl (6). For the coordinates X M this reads where m, n = 1, 6,m,n = 1, 6. We refer to appendix A for the used index notations and conventions and notice here, that barred indices label the same 6 of E 6(6) and these are distinguished from unbarred small Latin indices for technical convenience. In what follows we always assume for any field f of the theory. Upon such decomposition non-vanishing components of the symmetric invariant tensor can be written as n , d n 1 n 2 n 3 n 4 n 5 n 6 = 1 4 √ 5 ǫ n 1 n 2 n 3 n 4 n 5 n 6 , n , d n 1 n 2 n 3 n 4 n 5 n 6 = 1 4 √ 5 ǫ n 1 n 2 n 3 n 4 n 5 n 6 . (2.8) Finally, the (bosonic) E 6(6) ExFT field content reads where g µν is the metric of the external space, M M N is the so-called generalized metric parametrising the scalar coset, A µ M is a generalized connection and B µνM is a set of two-forms.
It is convenient to turn from the generalized metric M M N to generalized vielbeins E A M defined as where M AB is a constant matrix (unity, for concreteness). In terms of fields of 11-dimensional supergravity the generalized vielbein E A M can be parametrized as follows where e = det(e a m ) and V m 1 m 2 m 3 = e −1 3! ǫ m 1 m 2 m 3 n 1 n 2 n 3 C n 1 n 2 n 3 . The scalar degree of freedom U = e −1 6! ǫ n 1 ...n 6 C n 1 ...n 6 comes from dualization of the three-form A µνρ and the procedure directly relating these two can be found in [30]. For the inverse vielbein one has Such defined generalised vielbein is a generalised vector of weight λ = 0, that is necessary for the full Lagrangian of ExFT to be invariant In this work we will be focusing only at the scalar sector of the theory, i.e. only at the fields entering the generalised vielbein. As in the SL(5) theory [25] consistent truncation requires to also keep track of determinant of the external metric g µν . Below we discuss a rescaling of the generalised metric, that combines all these degrees of freedom and decouples them from the rest of the fields.

Generalised flux formulation
For simplification of further discussion we consider only such backgrounds, that can be presented don't depend on the external coordinates y µ . Also we take A µ M = 0 and B µνM = 0. Hence, we consider the following ansatz Since the external metric is a scalar of non-zero weight under generalised Lie derivative and, as we discuss later, the deformations are given by E 6(6) transformations, the external metric transforms by a rescaling. It is convenient to explicitly factor out the part of non-zero weight e 2φ of the external metric and further restrict coordinate dependence as g µν (y µ , x m ) = e −2φ(x m ) e 2 9−dḡ µν (y µ ) (d = 6 for E 6(6) ). The factor e −2φ(x m ) is possible to combine with the generalised metric of ExFT M M N to Applied to the SL(5) theory one has d = 4 and the metric M M N will me precisely that of the truncated theory of [34]. Such rescaled generalized metrics M M N ∈ E 6(6) × R + and can be represented in terms of the generalised vielbein E M A ∈ E 6(6) × R + as usual In components the generalised vielbein and its inverse read (2.17) One lists some useful relations: Dropping all terms in the full Lagrangian of exceptional field theory which do not give contributions to the equations of motion of the generalised metric and det g µν upon A µ whereR[g µν ] is the usual Ricci scalar and the scalar potential is given by (2.20) Upon the rescaling as above the Lagrangian can be written in the following simple form whereL sc (M M N ) is the same as in the case of the non-linear realisation of E 6(6) ExFT [35]. This due to the fact that the generalized metric obtained in the non-linear realisation has the same rescaling symmetry as our truncated metrics M M N . Now one applies the same logic as in [25], that is to notice that equations of motion of the rescaled generalised metric are those coming from the scalar potentialL sc (M M N ) plus 'cosmological term' coming from the curvature scalar of the external space In what follows we assume that these are satisfied for the undeformed background and search for condition upon which the deformation does not spoil this.
In what follows it proves convenient to turn to the so-called flux formulation of the scalar sector of exceptional field theory and to rewrite the above Lagrangian in terms of generalised fluxes as in [36].
Indeed, written completely in terms of components of fluxes F AB C , to be defined below, equations of motion will be guaranteed to hold after a deformation if the latter the flux components. Given the deformation is an E 6(6) transformation this would simply be a requirement for the flux components to transform covariantly.
In [28] for generalised fluxes of Double Field Theory this requirement has been shown to be equivalent to the classical Yang-Baxter equation. The same idea is applicable here. Hence, one defines with generalised flux components Note that here one does not require the fluxes to be constant as it is done in generalised Scherk-Shwarz reduction of [36]. The latter generate the scalar potential of the maximal D = 5 gauged supergravity, where preservation of supersymmetry requires the so-called linear constraint F A,B C ∈ 27 ⊕ 351. In components on has the trombone θ A ∈ 27 and Z-flux Z AB C ∈ 351 [37,38].
The components F A,B C defined in (2.23) automatically satisfy this constraint and the correspond- (2.25) The second line here can be further simplified. Define first the symmetric invariant tensors d ABC and d ABC by explicitly listing the components as in (2.8) (2.26) Due to the non-vanishing weight of the rescaled generalised vielbein E M A ∈ E 6(6) × R + the invariant tensors in curved indices d M N K and d M N K are related to that with flat indices as (2.28) Using this we finally obtain the following expression for Z-flux  Working with negative level generators t m 1 m 2 m 3 , t m 1 ...m 6 one replaces p-forms with p-vectors generalising the β-frame of double field theory [39], that proves convenient in describing non-geometric background (see e.g. [40]). Since both such realisations give properly defined element of the coset G/K with G and K being the global and local U-duality groups respectively, it is convenient to define a polyvector deformation as a linear transformation given by the following E 6(6) element In components the deformation map O ∈ E 6(6) can be written as follows Hence, one defines deformation of the generalised vielbein and of its inverse as follows field theory this has been observed to be precisely the case when classical Yang-Baxter equation appears [28]. Moreover, the condition on generalised fluxes in flat indices to transform as scalars appears as a natural condition for fluxes in curved indices to transform covariantly. Indeed, consider where the additional non-covariant term ∆F M N K comes from a non-trivial transformation of F AB C .
Hence, one concludes that generalised fluxes F AB C must naturally be scalars under such defined E 6(6) transformations.
In general one might be interested in deriving conditions on general Ω mnk and Ω m 1 ...m 6 imposed by above constraints on transformations of fluxes. However here we restrict the narrative to only 3and 6-Killing deformation, i.e. when Consider now transformation of the trombone flux θ A , for which one finds Here terms in brackets contain Killing vectors and various expressions in vielbein, and do not contain derivatives of Killing vectors. Hence, one may naturally impose the following sufficient conditions Recall, for deformation of 10d supergravity one has very similar conditions coming from the trombone One finds it natural to call the conditions (3.9) unimodularity constraints, which indeed ensure tracelessness of the dual structure constants. One may speculate here on generalised supergravity in 11-dimensions as well as on relaxing this condition, while keeping the flux invariant. Postponing this discussion to the Section 4 we notice, that the above condition has been also found in the analysis of exceptional Drinfeld algebra and hence seems the most natural (see below).
While for invariance of the trombone flux the unimodularity condition is sufficient, transformation of the flux Z AB C becomes more subtle. Here one finds it useful to consider the transformation orderby-order in the deformation tensor Ω M N or equivalently in ρ-tensor, where ρ i 1 ...i 6 is understood as order 2. Hence, one obtains at order 1 concludes, that no combination of ρ-tensors found in terms of higher order can be used to cancel such first order terms. This forces to strengthen the unimodularity constraint found in the trombone flux and demand Note, that the condition of precisely this form has been found in [26].
At order two one finds the following where terms in the first line just reproduce the previously found unimodularity condition for ρ i 1 ... .

(3.13)
Here in the first line we used the identity and the unimodularity condition on the 6-tensor. In the second line we used the identity where the most LHS is due to antisymemtry in [i 5 i 6 ] and the RHS provides a convenient decomposition of the antisymmetrisation [i 1 i 2 i 3 i 4 i 7 ]. Hence, the sufficient condition that takes into account symmetries of the remaining terms in the transformation of the generalised flux reads This has precisely the same for as the condition on the ρ-tensors obtained in [26] from analysis of exceptional Drinfeld algebra, however up to some additional identifications to be explicitly provided in Section 3.2.
Let us now turn to the analysis of terms in transformation of the flux Z AB C of the last third order where one finds an additional constraint where we drop all terms proportional to the unimodularity and the quadratic constraint (3.16), and leave explicit indices [i 3 i 4 i 5 i 6 i 7 i 10 ] of terms in brackets. Hence, one might take the expression cubic in ρ-tensor as an additional constraint sufficient for the E 6(6) fluxes to be invariant This is the reason why such condition cannot appear in the approach of [26] restricted to group manifolds, when i, j = 1, . . . , 6. One expects to find this additional constraint in the E 7(7) exceptional field theory.
Hence, we find the following conditions which are sufficent for generalised fluxes of the E 6 (6) exceptional field theory to be invariant and hence for a deformation to be solution-generating First three conditions above are precisely the same as in the EDA approach of [26], while the last line is an additional condition, which can not be seen in the E 6(6) exceptional Drinfeld algebra description of group manifold backgrounds.

Relation to the algebraic approach of EDA
The supergravitational analysis above describes the deformation map O M N encoding a 3-and 6vector deformation of an 11-dimensional background. Using explicit relations between 11-dimensional fields and those of the E 6(6) exceptional field theory the corresponding generalised metric is defined, whose deformation is a linear transformation. Let us now provide explicit relation between the rtensors and constraints one them appearing in deformation of supergravity backgrounds and those appearing from constraint on consistency of deformations of the E 6(6) exceptional Drinfeld algebra [26].
One first notices, that parametrisation of the 27 employed for the generalised metric here is different from that used to define the E 6(6) EDA in [26].   While the unimodularity conditions in the first two lines are precisely the same as the ones following from flux invariance, the third line requires more work. Let us expand the antisymmetrisation in [n 1 n 2 n 3 k 1 k 2 ] of the third line above and reorganise it the following more symmetric form Upon (3.26) this becomes precisely (3.16).

The short SL(5) story
Let us now briefly look at 3-vector deformations in the formalism of the SL(5) exceptional field theory and show that the only condition appearing both in the algebraic and ExFT approaches is the unimodularity constraint.
Here we are working in the same conventions as that of [23] and for the sake of brevity we will avoid lengthy description of the SL(5)-covariant exceptional field theory. Important however is to mention the generalised Lie derivative of the generalised vielbein whose explicit form tells that we are working in the truncated theory. Here M, N, K, · · · = 1, . . . , 5 label coordinate indices while A, B, C, · · · = 1 . . . 5 label flat indices. As before small Latin indices m, n, k, . . . and a, b, c, . . . label directions of the "internal" space, that is four-dimensional in this case.
Explicitly the generalised metric and the deformation map are given by where v m = 1 3! ǫ mnkl C nkl and as before Ω m 1 m 2 m 3 = 1 Generalised flux is defined as structure constants of the corresponding Leibniz algebra as follows (3.32) Here we denote E = det E A M . Following the same procedure as for the E 6(6) case one finds that transformation of all components of the generalised flux at all orders in ρ i 1 i 2 i 3 can be written as Hence the unimodularity constraint is indeed sufficient for the flux to be invariant and for the deformation to generate a solution.
At the algebraic side one considers the SL(5) exceptional Drinfeld algebra developed in [7,8] and deforms generators as follows T m → T m , Requiring the deformed generators to also form an SL(5) exceptional Drinfeld algebra one derives the following constraints (see [7] for more details) The first line is simply the unimodularity constraint whose appearance was expected. The second line is a quadratic constraint, which is however equivalent to the first line. Indeed, since the indices m, n, k = 1, . . . 4 it is natural to define Substituting this into the second line of (3.35) and contracting the indices m 1 m 2 and n 1 n 2 with epsilon-tensor one rewrites the constraint simply as which vanishes identically upon the unimodularity constraint. Note that while one is able to define such ρ m only on group manifold, we see that the unimodularity constraint is still sufficient for a tri-vector SL(5) deformation to generate solutions.
One notices however, that the non-abelian deformation provided in [25] are all non-unimodular, while still provide solutions. This nicely illustrates the fact that such obtained conditions are only sufficient, not necessary. At the same time this observation again raises the question of searching for non-trivial tri-vector deformations, now satisfying generalised classical Yang-Baxter equation.

Conclusions and discussions
In this work we investigate 3-and 6-Killing deformations of general backgrounds of 11-dimensional supergravity admitting at least three Killing vectors, not necessarily commuting. The formalism of exceptional field theory provides more natural degrees of freedom for that than the conventional formulation of supergravity. In these terms tri-vector deformations appear to be encoded by an E d(d) element generated by a 3-and 6-vector. Since the field content of ExFT is given by tensor of GL (11-d) taking values in irreps of the U-duality group, these transform linearly. Schematically that  Working for concreteness in the E 6(6) exceptional field theory we consider its truncation to the scalar sector as in [23]. This leaves us with only the generalised metric parametrising the coset E 6(6) × R + /Usp (8). Explicitly the corresponding generalises vielbein is given in (2.17). While the generalised vielbein transforms under the deformations linearly, transformation of the supergravity fields entering its definition are very non-obvious. The covariant formalism allows us to refrain from attempting to provide explicit formulae for deformations of supergravity fields as it was done [23].
Such analysis however would be necessary for one to generate explicit examples of deformations within the E 6(6) theory.
Truncated exceptional field theory can be completely written in terms of generalised fluxes, as it has been explicitly shown in [36] for the E 6(6) symmetry group. The corresponding equations of motion for the generalised vielbein can as well be written in terms of fluxes with a single factor of the inverse vielbein E A M . Hence, the sufficient condition for a deformation to map a solution of such equations of motion to a solution is the condition for generalised fluxes to transform covariantly.
Hence fluxes with local USp (8) indices F AB C must be invariant. We investigate transformation of generalised fluxes under such defined transformations and learn that one may ensure invariance if a condition on the constant tensors ρ i 1 i 2 i 3 and ρ i 1 ...i 6 . The latter are a generalisation of the classical r-matrix and the condition is believed to be a generalisation of the classical Yang-Baxter equation.
We show that the condition sufficient for the deformation to be a solution-generating transformation is precisely that derived via ρ-deformation of the E 6(6) exceptional Drinfeld algebra in [8].
The same analysis of generalised fluxes of the truncated SL(5) theory shows that the sufficient condition is simply the unimodularity constraint on ρ i 1 i 2 i 3 , i.e.
We notice that the quadratic constraint on ρ i 1 i 2 i 3 derived in [7] is satisfied identically upon the unimodularity constraint, that provides consistency between supergravity and EDA picture.
It is worth here to return back to the result of [23] where explicit examples of tri-vector deforma-  [42,43] (note that v.2 is also available [44]). The corresponding Cadabra files can be found here [45] One starts with the trombone flux whose transformation becomes One observes that terms of both first and second order in the ρ-tensors are proportional to the unimodularity constraint. Note, that although here te constraint comes with antisymmetrisation of the upper indices, further analysis of Z-flux will require more strict version of the constraint without the antisymmetrisation Transformation of the Z-flux is convenient to analyse order-by-order in the ρ-tensor assyming that ρ i 1 i 2 i 3 is of order one while ρ i 1 ...i 6 is of order two. Hence, at first order one has which again vanishes upon the unimodularity constraint.
Next, at order two one writes where for clarity of expressions we define Given the unimodularity constraint the first three lines vanish and the above expression becomes and performing a series of algebraic manipulations the final expression can be massaged to The yellow expression in the first line is the unimodularity condition for the tensor ρ i 1 ...i 6 , while the blue terms in the second line compose the generalised classical Yang-Baxter equation. The latter is equivalent to the constraint obtained in [26].
Finally, terms of third order in the ρ-tensors read Due to the high amount of index symmetries provided by contraction with the epsilon tensor, one is