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PHYSICAL REVIEW D 103, 066021 (2021)

Polyvector deformations in eleven-dimensional

† Kirill Gubarev * and Edvard T. Musaev Moscow Institute of and Technology, Institutskii pereulok 9, Dolgoprudny 141700, Russia

(Received 8 December 2020; accepted 3 March 2021; published 26 March 2021)

We consider 3- and 6-vector deformations of 11-dimensional supergravity backgrounds of the form

M5 × M6 admitting at least three Killing vectors. Using flux formulation of the E6ð6Þ exceptional field theory, we derive (sufficient) conditions for the deformations to generate a solution. In the group manifold case, these generalizations of the classical Yang-Baxter equation for the case of r-matrices with three and

six indices are shown to reproduce those obtained from exceptional Drinfeld algebra for E6ð6Þ. In general, we see an additional constraint, which might be related to higher exceptional Drinfeld algebras.

DOI: 10.1103/PhysRevD.103.066021

I. INTRODUCTION transformations that keep the sigma-model in a consistent vacuum, however changing it in a controllable Vacua of understood as a perturbative way. Particularly interesting examples are based on mani- formulation of the nonlinear two-dimensional sigma-model folds with an AdS factor known to be holographically dual to are known to be represented by a vast landscape of 10-dimensional manifolds equipped by various gauge superconformal field theories. While in general, CFTs are an fields: Kalb-Ramond 2-form and Ramond-Ramond p-form isolated point in the space of couplings corresponding to a gauge fields. In the full nonperturbative formulation, one fixed point of renormalization group flow, SCFTs belong to finds that the space of string vacua is mostly populated a family of theories connected by varying couplings. Adding by 11-dimensional manifolds, and 10-dimensional back- exactly marginal operators to a theory will preserve a grounds represent points with a small string coupling at the quantum level and move the constant (see [1] for a more detailed review). A set of corresponding point in the space of couplings along the so- string vacua possesses a huge amount of various sym- called a conformal manifold [15]. Certain progress in metries that prove useful in better understanding of its understanding of the structure of conformal manifold can structure. In particular, one finds T-duality symmetries: be made by investigating the gravitational side of the Abelian [2,3], non-Abelian [4], and more generally, AdS=CFT correspondence. Indeed, given a set of exactly Poisson-Lie dualities [5–7], which are related to back- marginal operators that deform a SCFT keeping it on a grounds that are indistinguishable from the perturbative conformal manifold, there exists a family of dual AdS string point of view. Nonperturbatively symmetries get solutions related by the deformations of metric, , enhanced to (Abelian) U-dualities, which can be under- and p-form gauge fields. A well-known example is provided stood as transformations related to toroidal backgrounds by β-deformations of D ¼ 4N ¼ 4 SYM, whose gravity equivalent from the point of view of the membrane [8,9]. dual is a bi-vector Abelian (TsT) deformation along two of 5 Certain progress toward defining non-Abelian generaliza- three U(1) isometry directions of S [16]. In a similar fashion 7 tion of U-dualities have been made recently in [10–14]. considering AdS4 × S , one is able to pick three U(1) At the level of low-energy theory of background fields directions of the seven-sphere to construct a trivector such duality symmetries appear as solution-generating deformation of the ABJM theory. For a general formula transformations. More generally, one is interested in for TsT transformations of gauge theories, see [17]. Generalizing the results known for TsT deformations, one naturally gets interested in bi- and tri-vector deforma- * [email protected]; Also at the Institute for Exper- tions along with a set of noncommuting Killing vectors. As imental and Theoretical Physics. † [email protected]; Also at Kazan Federal University, the most symmetric example here, one finds deformations Institute of Physics. of two-dimensional sigma models preserving integrability, e.g., η-deformation of the Green-Schwarz superstring 5 Published by the American Physical Society under the terms of on AdS5 × S constructed in [18,19]. The choice of the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to for the corresponding superalgebra gives the author(s) and the published article’s title, journal citation, rise to the so-called ABF background [20], which solves and DOI. Funded by SCOAP3. equations of motion of generalized supergravity [21–23] or

2470-0010=2021=103(6)=066021(17) 066021-1 Published by the American Physical Society KIRILL GUBAREV and EDVARD T. MUSAEV PHYS. REV. D 103, 066021 (2021) to normal supergravity backgrounds. To depart from back- Ti → Ti; grounds given by group manifolds and coset spaces, one Ti → Ti rijT ; 3 generalizes the procedure and for a general deformation þ j ð Þ parametrized by a bi-vector Θ ¼ rijk ∧ k obtains i j and requiring the deformed generators to also form a Drinfeld double algebra. Similarly, deforming generators ðg þ bÞ−1 ¼ðG þ BÞ−1 þ Θ; ð1Þ of an exceptional Drinfeld algebra (EDA) by tri- and six- vector tensors ρi1…i3 , ρi1…i6 and restricting the deformed generators to form an EDA, one arrives at a set of where G, B and g, b are the metrices and the 2-form fields conditions on ρ-tensors. The exceptional Drinfeld algebra for the initial and deformed backgrounds, respectively based on the SL(5) group has been constructed in [10,11] – [24 26]. Although at this level, the deformation is consi- and for the group E6 6 , this has been done in [35]. dered as a transformation of (generalized) supergravity ð Þ Conditions on the 3-vector deformation tensor ρi1i2i3 solutions without a direct reference to two-dimensional derived for the SL(5) EDA in [10] are equivalent to the sigma-models, both initial and deformed field configura- unimodularity condition tions could be understood as consistent sigma-model backgrounds. Both the sigma-model (for coset spaces) ρkl½i1 f i2 ¼ 0: ð4Þ and the field theory (for general manifolds with Killing kl vectors) approaches show that a deformed background This condition is the only one to appear in the SL(5) case is a solution of (generalized) supergravity equations when 4 ij due to the dimension d ¼ of the internal manifold, which the r-matrix r satisfies classical Yang-Baxter equation appears to be too small to embed 3-vector deformations. (CYBE) [27,28] The situation is the same as for bi-vector deformations in dimension d ¼ 3, where one gets only the unimodularity k½i1 i2jlj i3 r r fkl ¼ 0: ð2Þ condition. In contrast, the deformations considered inside the E6ð6Þ Here, f k are structure constants of the algebra of Killing EDA are subject to a nontrivial constraint that is supposed ij to generalize classical Yang-Baxter deformation. In this vectors. work, we investigate whether this condition is sufficient for Notably, the transformation (1) appears essentially non- a deformation to be a solution generating a transformation linear only due to a bad choice of variables. It becomes a as for the bi-vector case. To this end, we start with linear O(10,10) transformation when written in terms of providing a short review of the E6 6 extended geometry generalized metric i.e., a representative of the coset ð Þ Oð10; 10Þ=Oð1; 9Þ ×Oð9; 1Þ. For this reason, deriving in Sec. II. In Sec. III, we define a deformation map for E6ð6Þ classical Yang-Baxter equation from the standard super- generalized vielbein and investigate the transformation of gravity formalism faces huge technical difficulties and has generalized fluxes under the map. We find the generalized been done in [27,29] only in the second order in Θ. The Yang-Baxter equation of [35] as a sufficient condition for formalism of double field theory [30,31] appears more the fluxes to stay undeformed. Also we observe an addi- natural, with the generalized metric being the canonical tional constraint whose algebraic interpretation is yet variable, which allowed a full proof in [28] that CYBE is unknown. sufficient to end up with a solution. Moving to tri-vector deformations, one naturally employs the exceptional field II. TRUNCATION OF 11D SUPERGRAVITY theory formalism (ExFT) for precisely the same reasons: The bi-vector deformation of backgrounds of a nonlinear the generalized metric transforms linearly under deforma- sigma-model is given by the nonlinear map (1), whose form tions. The generalization of the deformation map (1) cannot be called self-evident. Due to a nonlinear nature of obtained in the formalism of SL(5) ExFT in [32] finds the map explicit check that this is a solution generating a the same interpretation as an open-closed membrane map transformation is a highly nontrivial task, as it has been [33]. Examples of non-Abelian tri-Killing deformations demonstrated in [27]. Choosing the correct representation based on the open-closed membrane map or equivalently a of degrees of freedom allows turning the deformation map specially defined SL(5) transformation have been provided into a linear transformation. Hence, for bi-vector deforma- 7 for the AdS4 × S background in [34]. tions, one organizes NS-NS fields into the generalized One may naturally ask whether the condition that a tri- metric of the double field theory parametrizing the coset vector deformation generates a solution of supergravity is Oðd; dÞ=OðdÞ × OðdÞ and the invariant dilaton, and R-R equivalent to some algebraic condition generalizing CYBE. fields into an Oðd; dÞ spinor. In this case, the transforma- One first notices that CYBE appears when deforming tion becomes simply an Oðd; dÞ rotation, and showing its i generators of a Manin triple ðTi;T ; ηÞ representing a solution-generating nature becomes a straightforward task Drinfeld double algebra by r-matrix (see [26,36,37] for more detail).

066021-2 POLYVECTOR DEFORMATIONS IN ELEVEN-DIMENSIONAL … PHYS. REV. D 103, 066021 (2021)

Similarly, as it has been shown in [32] that tri-vector Amnk ¼ Cmnk; deformations of the 11-dimensional background can be k Aμ ¼ Cμ − Aμ C ; defined as an SL(5) transformation of generalized metric mn mn kmn − 2 n n k of the corresponding exceptional field theory. Below, we Aμνm ¼ Cμνm A½μ Cνmn þ Aμ Aν Cmnk; show that for 3-vector deformations within the SL(5) m m n Aμνρ ¼ Cμνρ − 3A μ Cνρ þ 3A μ Aν Cρ theory to generate a solution unimodularity is sufficient. ½ m ½ mn m n k Hence, no generalized Yang-Baxter equation is generated − Aμ Aν Aρ Cmnk: ð6Þ both in the field theory and in the SL(5) EDA. Here, we consider the E6ð6Þ exceptional field theory that allows The resulting fields 3- and 6-vector deformations and, according to the μ¯ m a analysis of the exceptional Drinfeld algebra of [35],a fgμ ;Aμ ;em ;Amnk;Aμnk;Aμνk;Aμνρgð7Þ nontrivial generalization of the classical Yang-Baxter equation exists. transform appropriately under splitted 11-dimensional ξμˆ ξμ Λm A detailed presentation of the E6ð6Þ exceptional field ¼ð ; Þ. To organize these into theory can be found in [38–40]. The presentation includes multiplets of E6ð6Þ, one has to dualize all forms to the bosonic and fermionic field content, trans- lowest possible rank. Hence, the two forms Aμνm get m formations, full Lagrangian and truncations to the dualized intro 1-forms and can be collected with Aμ M 11-dimensional, and Type IIB supergravities. For a review and Aμmn into the vector Aμ of ExFT transforming in the of the recent progress, see [41,42]. For our purposes here, 27 of E6ð6Þ; the 3-form Aμνρ after dualization contributes to we stress the following defining features of the theory: the scalar coset. The 6-dimensional space gets extended to (i) ExFT is an EdðdÞ-covariant background independent include coordinates corresponding to the winding modes of theory combining full 11-dimensional and Type IIB the M2- and M5- and is now parametrized by XM supergravities (no reduction); transforming as the 27 under global U-duality transforma- (ii) field content is represented by tensors of GL(11-d) tions. Local coordinate transformations on such extended taking values in certain representations of the space are given by the so-called generalized Lie derivative U-duality group; (iii) section condition, restricting dependence of fields on M K M K M LΛV ¼ Λ ∂ V − V ∂ Λ the total 5 þ 27 coordinates is required. In what K K follows, we assume the standard solution of the 1 þ 10dMKRd VN∂ ΛL − λ − VM∂ ΛK; section constraint leaving only dependence on 5 þ 6 NLR K 3 K coordinates. ð8Þ Hence, for the purposes of this work, the exceptional field theory simply provides a convenient rewriting of degrees of where VM denote the components of some generalized freedom of the 11-dimensional supergravity, turning poly- vector of weight λ and dMNK, and d are the invariant vector deformations into a linear map. MNK tensors of E6ð6Þ. For such defined transformations to form a The construction of fields and the Lagrangian of the E6 6 ð Þ closed algebra, one imposes the section constraint exceptional field theory starting from the 5 þ 6 split of the 11-dimensional supergravity is given in detail in [38].To dMNK∂ • ∂ • 0; 9 introduce notations and for further reference, we provide a N M ¼ ð Þ brief overview of the setup. The bosonic field content of the ˆ αˆ where the bullets denote any fields and their combinations. 11-dimensional supergravity consists of the elfbein Eμˆ and The above condition has two (maximal) inequivalent the 3-form potential Cμˆ 1μˆ 2μˆ 3 . Keeping full dependence on all μˆ solutions corresponding to embeddings of the 11- of the 11 coordinates x , one splits the fields into tensors in dimensional and Type IIB 10-dimensional supergravity. 5-dimensions and organizes them into multiplets of E6ð6Þ. We will be working with the former, i.e., always take into For the latter, one has to follow the dualization prescription account the decomposition of e6ð6Þ irreps under its sub- of [43]. Decomposing the 11-dimensional indices as algebra glð6Þ. For the coordinates XM, this reads μˆ ¼ðμ;mÞ, αˆ ¼ðμ¯;aÞ, one parametrizes the elfbein in the following upper-triangular form M m m¯ X ¼ðx ;xmn;x Þ; ð10Þ −1 μ¯ 3 m a where m; n ¼ 1; 6, m;¯ n¯ ¼ 1; 6. We refer to Appendix A for ˆ αˆ e gμ Aμ em Eμˆ ¼ ; ð5Þ the used index notations and conventions and notice here 0 a em 6 that the barred indices label the same of E6ð6Þ, and these are distinguished from the unbarred small Latin indices for and redefines the fields arising from the 3-form potential as technical convenience. In what follows, we always assume

066021-3 KIRILL GUBAREV and EDVARD T. MUSAEV PHYS. REV. D 103, 066021 (2021)

mn M K M K M ∂m¯ f ¼ 0; ∂ f ¼ 0; ð11Þ LΛE A ¼ Λ ∂KE A − E A∂KΛ 1 10 MKR ∂ ΛL EM ∂ ΛK for any field f of the theory. Upon such decomposition, þ d dNLR A K þ 3 A K : ð16Þ nonvanishing components of the symmetric invariant ten- sor can be written as In this work, we focus only on the scalar sector of the 1 theory, i.e., only on the fields entering the generalized dnn¯ ¼ pffiffiffi δ n¯ δ n; vielbein. As in the SL(5) theory [34], consistent truncation m1m2 5 ½m1 m2 requires to also keep track of determinant of the external 1 ffiffiffi ϵ metric gμν. Below, we discuss the rescaling of the gener- dn1n2n3n4n5n6 ¼ p n1n2n3n4n5n6 ; 4 5 alized metric that combines all these degrees of freedom 1 and decouples them from the rest of the fields. m1m2 ½m1 m2 d¯ ¼ pffiffiffi δ ¯ δ ; nn 5 n n 1 A. Generalized flux formulation dn1n2n3n4n5n6 ¼ pffiffiffi ϵn1n2n3n4n5n6 : ð12Þ 4 5 For simplification of further discussion, we consider only such backgrounds that can be presented as Finally, the (bosonic) E6ð6Þ ExFT field content reads M11 ¼ M6 × M5, where the internal metric gmn, the

3-form Cm1m2m3 , and the 6-form Cm1m2m3m4m5m6 do not μ M depend on the external coordinates y . Also, we take fgμν; MMN;Aμ ;BμνMg; ð13Þ M Aμ ¼ 0 and BμνM ¼ 0. Hence, we consider the following ansatz where gμν is the metric of the external space, MMN is the so-called generalized metric parametrizing the scalar coset, M Aμ is the generalized connection, and BμνM is set of μ m m gμν ¼ gμνðy ;x Þ; MMN ¼ MMNðx Þ; two-forms. M Aμ ¼ 0;Bμν ¼ 0: ð17Þ It is convenient to turn from the generalized metric MMN M EA to generalized vielbeins M defined as Since the external metric is a scalar of nonzero weight M E AE BM MN ¼ M N AB; ð14Þ under the generalized Lie derivative and, as we discuss later, the deformations are given by E6ð6Þ transformations, where MAB is the constant matrix (unity, for concreteness). the external metric transforms by rescaling. It is convenient In terms of fields of the 11-dimensional supergravity, the to explicitly factor out the part of nonzero weight e2ϕ of the EA generalized vielbein M and its inverse can be parametrized external metric and further restrict the coordinate depend- μ −2ϕ m 2 μ as follows m ðx Þ 9−d ¯ 6 ence as gμνðy ;x Þ¼e e gμνðy Þ (d ¼ for E6ð6Þ). 2 3 −2ϕðxmÞ −1 1 ¯ 1 ¯ The factor e is possible to combine with the ea pffiffi eb C ea U þ eaC Vnb1b2 m 2 m ba1a2 2 m 4 n mb1b2 generalized metric of ExFT MMN to define 16 7 6 m1 m2 1 m1 m2 ¯ 7 EA ¼ e3 0 e e − pffiffi e e Vb1b2a ; M 4 ½a1 a2 2 b1 b2 5 −1 ¯ μ m −2ϕðxmÞ 2 μ 00 e ea gμνðy ;x Þ¼e e9−dg¯μνðy Þ; 2 m¯ 3 m 00 −2ϕ 2 ea M ¼ e e9−dM : ð18Þ 6 7 MN MN 1 1 a1 a2 M − 6 ffiffi n ½ 0 7 E 3 p e C em1 em2 A ¼ e 4 2 a nm1m2 5: ¯ ¯ ¯ ¯ 4 − e m e l mkn peffiffi a1a2m m Applied to the SL(5) theory, one has d ¼ , and the metric 2 ea U þ 4 eaClknV 2 V eea¯ MMN will be precisely that of the truncated theory of [44]. ∈ Rþ ð15Þ Such rescaled generalized metrics MMN E6ð6Þ × and can be represented in terms of the generalized vielbein a m1m2m3 A ∈ Rþ where e ¼ detðemÞ and the tri-vector V ¼ EM E6ð6Þ × as usual −1 e ϵm1m2m3n1n2n3 3! Cn1n2n3 . The scalar degree of freedom U ¼ −1 e ϵn1…n6 6! Cn1…n6 comes from dualization of the three-form A B MMN ¼ EM EN MAB: ð19Þ Aμνρ and the procedure directly relating to these two can be found in [38]. Such defined generalized vielbein is a generalized vector of weight λ ¼ 0, which is necessary In components, the generalized vielbein and its inverse for the full Lagrangian of ExFT to be invariant read

066021-4 POLYVECTOR DEFORMATIONS IN ELEVEN-DIMENSIONAL … PHYS. REV. D 103, 066021 (2021) 2 3 −1 1 ¯ 1 ¯ ea pffiffi eb C ea U þ eaC Vnb1b2 metric are those coming from the scalar potential 6 m 2 m ba1a2 2 m 4 n mb1b2 7 ˜ ϕ 1 L ðM Þ plus the “cosmological term” coming from A 6 0 m1 m2 − ffiffi m1 m2 b1b2a¯ 7 sc MN EM ¼ e 4 e e p e e V 5; ½a1 a2 2 b1 b2 the curvature scalar of the external space R½g¯μν. 00 e−1ea¯ 2 m¯ 3 em 00 a δL˜ 1 1 6 7 scðMMNÞ 1 ½a1 a2 − L˜ MN ¯ MN M −ϕ6 pffiffi n 0 7 scðMMNÞM ¼ R½gμνM : ð25Þ eaCnm1m2 em1 em2 EA ¼ e 4 2 5: δMMN 18 18 ¯ ¯ ¯ ¯ − e m e l mkn peffiffi a1a2m m 2 ea U þ 4 eaClknV 2 V eea¯ ð20Þ In what follows, we assume that these equations are satisfied for the undeformed background and search for One lists some useful relations: g ¼ detðgμνÞ¼ conditions upon which the deformation does not spoil this. −2ϕ 2 5 μ − 5 μ − 1 1 In what follows, it proves convenient to turn to the so- ðe e3Þ g¯ðy Þ¼M 27g¯ðy Þ, gμν ¼ M 27g¯μν M2 ¼ E ¼ ϕ −1 27 1 called flux formulation of the scalar sector of the excep- A 3 M2 E EA 1 M detðEMÞ¼ðe e Þ , ¼ ¼ detð MÞ¼ , MN ¼ 1 tional field theory and to rewrite the above Lagrangian in −27 M MMN. terms of generalized fluxes as in [46]. Indeed, written EM EM C Due to rescaling, the generalized vielbeins A and A completely in terms of the components of fluxes F AB ,to M 1 M transform with different weights λ½E A¼3, λ½E A¼0 be defined below, the equations of motion will be guaran- and hence, one has teed to hold after the deformation if the latter preserves the flux components. Given the deformation is an E6ð6Þ trans- M K M K M LΛE A ¼ Λ ∂KE A − E A∂KΛ formation, this would simply be a requirement for the flux þ 10dMKRd EN ∂ ΛL: ð21Þ components to transform covariantly. NLR A K In [47], for the generalized fluxes of double field theory, Dropping all terms in the full Lagrangian of the excep- this requirement has been equivalent to the classical Yang- tional field theory that do not give contributions to the Baxter equation. The same idea is applicable here. Hence, one defines equations of motion of the generalized metric and det gμν M upon Aμ ¼ 0, BμνM ¼ 0, one has L EM ¼ EK ∂ EM − EK ∂ EM −1 EA B A K B B K A g 2L R gμν L M ;gμν ¼ ½ þ scð MN Þð22Þ MKR N L C M þ 10d dNLRE B∂KE A ¼ F A;B E C; R F C 2 C K ∂ M where ½gμν is the usual Ricci scalar, and the scalar A;B ¼ EM E ½Aj KE jB potential is given by MKR C N L þ 10d dNLREM E B∂KE A: ð26Þ L M scð MN;gμνÞ 1 Note that here one does not require the fluxes to be constant ¼ − MMN∂ MKL∂ M 24 M N KL as it is done in the generalized Scherk-Shwarz reduction of 1 1 [46]. The latter generates the scalar potential of the − MMN∂ MKL∂ M − −1∂ ∂ MMN 2 M L NK 2 g Mg N maximal D ¼ 5 gauged supergravity, where the preserva- 1 1 tion of supersymmetry requires the so-called linear con- MN −2 MN μν − M g ∂ g∂ g − M ∂ g ∂ gμν: ð23Þ F C ∈ 27 ⊕ 351 4 M N 4 M N straint A;B . In components, one has the C trombone θA ∈ 27 and Z-flux ZAB ∈ 351 [48,49]. C Upon rescaling as above, the Lagrangian can be written in The components F A;B defined in (26) automatically the following simple form satisfy this constraint, and the corresponding components read 1 − 1 L ¯ 2 18 R ¯ L˜ ¼ g M ð ½gμνþ scðMMNÞÞ; ð24Þ −27θ 9∂ M B∂ M L˜ A ¼ ME A þ EM AE B; where scðMMNÞ is the same as in the case of the nonlinear C 10 MKR C N C∂ L realization of E6ð6Þ ExFT [45]. This is due to the fact that ZAB ¼ dNLRd EM E ðBjEM KE jAÞ: ð27Þ the generalized metric obtained in the nonlinear realization has the same rescaling symmetry as our truncated met- rics MMN. The second line here can be further simplified. Define first ABC Now, one applies the same logic as in [34], and notices the symmetric invariant tensors dABC and d by explicitly that the equations of motion of the rescaled generalized listing the components as in (12)

066021-5 KIRILL GUBAREV and EDVARD T. MUSAEV PHYS. REV. D 103, 066021 (2021) 2 3 1 δ n 00 aa¯ ffiffiffi δ a¯ δ a m d b1b2 ¼ p ½b1 b2 ; 6 7 1 nm1m2 m1 m2 5 N 6 − pffiffi Ω δ δ 0 7 O ¼ 2 ½n1 n2 ; 1 M 4 5 ffiffiffi ϵ e−1 n e−1 nkl e−ffiffi1 n¯ da1a2a3a4a5a6 ¼ p a1a2a3a4a5a6 ; δ ¯ J þ Ω W ¯ − p W ¯ δ ¯ 4 5 2 m 4 mkl 2 n1n2m m 1 ð33Þ b1b2 ½b1 b2 d¯ ¼ pffiffiffi δ ¯ δ ; aa 5 a a e ϵ Ωm1…m6 1 where we define J ¼ 6! m1…m6 and Wm1m2m3 ¼ a1a2a3a4a5a6 a1a2a3a4a5a6 e m4m5m6 d ¼ pffiffiffi ϵ : ð28Þ ϵ … Ω . The barred small Latin indices are the 4 5 3! m1 m6 same as the unbarred indices, which are introduced for Due to the nonvanishing weight of the rescaled generalized technical convenience to distinguish between the blocks of M ∈ Rþ the generalized vielbein. vielbein E A E6ð6Þ × , the invariant tensors in the MNK Hence, one defines the deformation of the generalized curved indices d and dMNK are related to dABC with flat vielbein and of its inverse as follows indices as A → N A δ N Ω N A M N P −3ϕ EM OM EN ¼ð M þ M ÞEN ; dMNPE AE BE C ¼ e edABC: ð29Þ M → −1 M N δM Ω˜ M N E A ðO Þ NE A ¼ð N þ NÞE A; ð34Þ A Note that for EM , one has the expected relation where one singles out the unity matrix and introduces the M N P Ω N Ω˜ M dMNPE AE BE C ¼ dABC: ð30Þ covariant deformation tensors M and N. These tensors contain only 3- and 6-vector deformation param- Using this relation, we finally obtain the following expres- eters, and one naturally decomposes the deformation of the Ω’ Ω N sion for Z-flux generalized fluxes in powers of s. While for M , the ˜ M expression is self-evident, for Ω N, one obtains Z C ¼ 5d dMKRE Cðe−3ϕe∂ E D þ E D∂ ðe−3ϕeÞÞ; AB ABD M K R R K 2 −1 −1 3 1ffiffi nm1m2 e n e nkl 0 p Ω − δ ¯ J þ Ω W ¯ ð31Þ 6 2 2 m 4 mkl 7 ˜ N 6 −1 7 Ω M ¼ 4 00 epffiffi 5: 2 Wn1n2m¯ which will be used in what follows. 00 0 ð35Þ III. POLYVECTOR DEFORMATIONS A. Transformation of fluxes One now considers the transformation of generalized In the nonlinear realization approach to the construction fluxes under such defined deformations. The simplest option of the generalized metric of the exceptional field theory, is when generalized fluxes (in flat indices) are invariant upon ρ one finds that the generators of positive level tm1m2m3 , a condition on the 3- and 6-index -tensors. In the flux tm1…m6 generate lower triangular generalized vielbein, and formulation of the truncated exceptional field theory this the corresponding 3- and 6-forms give the standard field invariance guarantees that the deformation does not spoil the content of supergravity. Working with negative level equations of motion, due to all the equations of motion of the scalar sector are written purely in terms of fluxes. For the generators tm1m2m3 , tm1…m6 , one replaces p-forms with p-vectors generalizing the β-frame of the double field double field theory, such invariance of generalized fluxes has theory [50] that proves convenient in describing a non- been observed to be precisely the case when the classical geometric background (see, e.g., [51]). Since both such Yang-Baxter equation appears [26]. Moreover, the condition realizations give a properly defined element of the coset on generalized fluxes in flat indices to transform as scalars G=K, with G and K being the global and local U-duality appears as a natural condition for fluxes in curved indices to groups, respectively, it is convenient to define a poly- transform covariantly. Indeed, consider vector deformation as a linear transformation given by the F 0 K ¼ E0 AE0 BE0 KF 0 C following E6ð6Þ element MN M N C AB P Q −1 K F L ΔF K     ¼ OM ON ðO Þ L PQ þ MN ; ð36Þ 1 1 Ωmnk Ωm1…m6 O ¼ exp tmnk exp tm1…m6 : ð32Þ K 3! 6! where the additional noncovariant term ΔF MN comes from C a nontrivial transformation of F AB . Hence, one concludes ∈ C In components, the deformation map O E6ð6Þ can be that generalized fluxes F AB must naturally be scalars under written as follows such defined E6ð6Þ transformations.

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In general, one might be interested in deriving conditions useful to consider the transformation order-by-order in the mnk m1…m6 N on general Ω and Ω imposed by the above deformation tensor ΩM or equivalently, in ρ-tensor, where constraints on transformations of fluxes. However, here ρi1…i6 is understood as order two. Hence, one obtains at we restrict the narrative to only 3- and 6-Killing deforma- order one tion, i.e., when ðδZ CÞ ¼ð…Þρi1i2i3 f i4 ; ð41Þ 1 AB 1st i2i3 Ωm1…m3 ρi1…i3 m1 m3 ¼ ki1 ki3 ; 3! where, as previously stated, the terms in brackets contain Ωm1…m6 ρi1…i6 m1 m6 ¼ ki1 ki6 : ð37Þ only Killing vectors. Note the absence of antisymmetriza- tion in ½i1i4. From the explicit form of the transformation m Here, fki g are the Killing vectors of the initial unde- provided in Appendix B, one concludes that no combina- formed background, and ρi1…i3 , ρi1…i6 are the constant tion of ρ-tensors found in terms of higher order can be used antisymmetric tensors generalizing the classical r-matrix. to cancel such first order terms. This forces to strengthen The Killing vectors satisfy the algebra defined by the usual the unimodularity constraint found in the trombone flux expression and demand

m∂ n − m∂ n i3 n ρi1i2i3 i4 0 ki1 mki2 ki2 mki1 ¼ fi1i2 ki3 : ð38Þ fi2i3 ¼ : ð42Þ

Consider now the transformation of the trombone flux Note that the condition of precisely this form was found θ A, for which one finds in [35]. At order two, one finds the following δθ … ρ½i1ji2i3 ji4 … ρ½i1i2i3i4ji5i6 ji7 A ¼ð Þ fi2i3 þð Þ fi5i6 : ð39Þ δ C … ρ½i1i2i3i4ji5i6 ji7 ð ZAB Þ2nd ¼ð Þ fi5i6 Here, the terms in brackets contain the Killing vectors and various expressions in vielbein and do not contain deriv- … ρi1i2i6 ρi3i4i5 i7 þð Þ½i2i3i4i7 fi5i6 atives of the Killing vectors. Hence, one may naturally impose the following sufficient conditions 3 − ρi2i3i4i5i6i7 f i1 ð43Þ 2 i5i6 ρ½i1ji2i3 ji4 0 fi2i3 ¼ ; ρ½i1i2i3i4ji5i6 ji7 0 where the terms in the first line reproduce the previously fi5i6 ¼ : ð40Þ found unimodularity condition for ρi1…i6, and we left the Recall, for the deformation of the 10d supergravity, one indices ½i2i3i4i7 on the terms in parentheses in the second has very similar conditions coming from the trombone line to show explicitly the antisymmetry. One finds the i1i2 i3 0 terms in brackets projected on the antisymmetric part in flux r fi1i2 ¼ , where both lower indices of the structure constants get contracted with the r-matrix. ½i2i3i4i7 is the condition sufficient for the second order This condition guarantees unimodularity of the dual transformation to vanish. For further comparison with the algebra, and being contracted with the Killing vectors, results of [35], let us rewrite this condition in a more gives the vector Im of generalized supergravity equations convenient form. Denoting the projection by subscript, of motion. Hence, this unimodularity constraint regulates we have whether one has a generalized or an ordinary supergravity  3  background upon the deformation, given CYBE is ρi1i2i6 ρi3i4i5 f i7 − ρi2i3i4i5i6i7 f i1 i5i6 2 i5i6 satisfied.  ½i2i3i4i7 One finds it natural to call the conditions (40) unim- ρi1i2i6 ρi3i4i5 i7 − 4 ½i1 ρi2i3i4i5i6i7 ¼ fi5i6 fi5i6 odularity constraints, which, indeed, ensures tracelessness ½i2i3i4i7  of the dual structure constants. One may speculate here on 4 ¼ ρi1i2i6 ρi3i4i5 f i7 − ρi2½i1ji6jρi3i4ji5jf i7 generalized supergravity in 11-dimensions as well as on 3 i5i6 i5i6  relaxing this condition, while keeping the flux invariant. − 3 ½i1 ρi2i3i4i5i6i7 fi5i6 : ð44Þ Postponing this discussion to Section IV, we notice that the ½i2i3i4i7 above condition was also found in the analysis of the exceptional Drinfeld algebra and hence, seems the most Here, in the first line, we used the identity natural (see below). While for the invariance of the trombone flux the 5 ½i1 ρi2i3i4i7i5i6 4 ½i2 ρi1ji3i4i7ji5i6 − i1 ρi2i3i4i7i5i6 0 fi5i6 þ fi5i6 fi5i6 ¼ unimodularity condition is sufficient, the transformation C ð45Þ of the flux ZAB becomes more subtle. Here, one finds it

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ρi1i2i3 i4 0 and the unimodularity condition on the 6-tensor. In the fi2i3 ¼ ; second line, we used the identity ρ½i1i2i3i4ji5i6 ji7 0 fi5i6 ¼ ;

0≡ ½i2 ρi1i3ji5jρi4i7i6 ρi1½i2ji6jρi3i4ji5j i7 − ρi2½i1ji6jρi3i4ji5j i7 fi5i6 fi5i6 fi5i6 i2 ρi1i3i5 ρi4i7i6 −4 ½i1jρi2ji3ji5 ρji4i7i6 − 3f ½i1 ρi2i3i4i5i6i7 0; ¼ðfi5i6 fi5i6 Þ½i2i3i4i7; ð46Þ i5i6 ¼ ρi1i2½i8 ρi3i4i5i6i7i9 i10 0 fi8i9 ¼ : ð50Þ where the most left-hand side (lhs) is due to antisymemtry in i5i6 , and the right-hand side (rhs) provides a convenient ½ First three conditions above are precisely the same as in the decomposition of the antisymmetrisation i1i2i3i4i7 . ½ EDA approach of [35], whereas the last line is an additional Hence, the sufficient condition that takes into account condition, which cannot be seen in the E6 6 exceptional the symmetries of the remaining terms in the transforma- ð Þ Drinfeld algebra description of group manifold back- tion of the generalized flux reads grounds. The equation in the third line above is quadratic in ρijk and can be referred to as the generalized Yang- ρi1½i2ji6jρi3i4ji5jf i7 − ρi2½i1ji6jρi3i4ji5jf i7 i5i6 i5i6 Baxter equation. Its relation to the construction of excep- − 3 ½i1 ρi2i3i4i5i6i7 0 fi5i6 ¼ : ð47Þ tional Drinfeld algebras is discussed further in the text. Meaning of the fourth equation as well as its algebraic This has precisely the same form as the condition on the origin is not as clear. ρ-tensors obtained in [35] from the analysis of the excep- tional Drinfeld algebra; however, up to some additional B. Relation to the algebraic approach of EDA identifications to be explicitly provided in Sec. III B. Let us now turn to the analysis of the terms in the The supergravitational analysis above describes the C deformation map O N encoding a 3- and 6-vector defor- transformation of the flux ZAB of the last third order where M one finds an additional constraint mation of an 11-dimensional background. Using explicit relations between 11-dimensional fields and those of the δ C … ρi1i2½i8 E6ð6Þ exceptional field theory, the corresponding general- ð ZAB Þ3rd ¼ð Þ i3i4i5i6i7i10  ½  ized metric is defined, whose deformation is a linear ρi3i4i9 ρi5i6i7 i10 − 18ρi3i4i5i6i7i9 i10 × fi8i9 fi8i9 ; transformation. Let us now provide an explicit relation between the r-tensors and the constraints on them appear- ð48Þ ing in the deformation of supergravity backgrounds and those appearing from a constraint on the consistency of the where we drop all the terms proportional to the unim- deformations of the E6ð6Þ exceptional Drinfeld algebra [35]. odularity and the quadratic constraint (47), and leave One first notices that the parametrization of the 27 explicit indices ½i3i4i5i6i7i10 of the terms in brackets. employed for the generalized metric here is different from Hence, one might take the expression cubic in ρ-tensor the one used to define the E6ð6Þ EDA in [35]. Indeed, here, as an additional constraint sufficient for the E6ð6Þ fluxes to ½m1;m2 we use T ¼fT ;T ;T¯ g, whereas in [35], one has be invariant M m m T ¼fT ;T½m1;m2;T½m1;m2;m3;m4;m5g: ð51Þ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{≡0 M m ρi1i2½i8 ρi3i4i9 ρi5i6i7f i10 − 18ρi1i2½i8 ρi3i4i5i6i7i9f i10 0; i8i9 i8i9 ¼ In this form, the origin of the last 6 from the windings of the ð49Þ M5- is more transparent. Hence, the relation is simply given by the invariant tensor of SL(6) probably with additional antisymmetrization in 1 ½i3i4i5i6i7i10. Notice that the above condition has 7 ½m1m2m3m4m5 T ¯ ¼ T pffiffiffiffi ϵ ¯ : ð52Þ antisymmetrized indices labeling generators of the sym- m 5! m½m1m2m3m4m5 metry algebra, defined by Killing vectors. This is the reason N why such a condition cannot appear in the approach of [35] Next, one compares the deformation map OM here and 1 … 6 N B restricted to group manifolds, when i; j ¼ ; ; .One the matrix CM (denoted CA in the Eq. (4.3) of [35]). N expects to find this additional constraint in the E7ð7Þ While the deformation map OM acts on the generalized exceptional field theory. vielbein, and one requires such deformed generalized N Hence, we find the following conditions that are suffi- vielbein to encode a solution, the matrix CM acts on cient for the generalized fluxes of the E6ð6Þ exceptional field generators fTMg of the exceptional Drinfeld algebra theory to be invariant and hence, for a deformation to be (EDA), and one requires the transformed generators to solution-generating form an EDA again. One has

066021-8 POLYVECTOR DEFORMATIONS IN ELEVEN-DIMENSIONAL … PHYS. REV. D 103, 066021 (2021) 2 3 n δm 00 6 1 7 − pffiffi Ωnm1m2 δm1 δm2 0 N 6 n1 n2 7 OM ¼ 4 2 ½ 5; −1 −1 −1 ¯ e δ n e Ωnkl − epffiffi δ n 2 m¯ J þ 4 Wmkl¯ Wn1n2m¯ m¯ ; 2 2 3 m δn 00 6 1 7 N 6 pffiffi ρ¯nm1m2 δa1 δm2 0 7 CM ¼ 4 2 ½n1 n2 5; ð53Þ 1 … 5 20 pffiffiffi ρ¯nm1 m5 pffiffiffi ρ¯n½m1m2 ρ¯m3m4m5 pffiffiffiffiffiffi δ½m1 δm2 ρ¯m3m4m5 δm1 …δm5 5! þ 5! 2!5! n1 n2 ½n1 n5

e ϵ Ωm1…m6 ρ where again J ¼ 6! m1…m6 and Wm1m2m3 ¼ i.e., the -tensors now acquire space (curved) indices. Now e ϵ Ωm4m5m6 μ …μ 0 3! m1…m6 . For group manifolds, Killing vectors using ½ 1 7¼ , one rewrites m m can be identified with the vielbein ei ¼ ki , and one has e e δ n δ n ϵ ρn1…n6 ϵ ρnn1…n5 1 1 m¯ J ¼ m¯ n1…n6 ¼ mn¯ 1…n5 : ð55Þ Ωm1…m3 ¼ ρi1i2i3 k m1 k m2 k m3 ¼ ρm1m2m3 ; 6! 5! 3! i1 i2 i3 6 Ωm1…m6 ρi1…i6 m1 … m6 ρm1…m6 ¼ ki1 ki6 ¼ ; ð54Þ Hence, the deformation map for group manifolds becomes

2 3 n δm 00 6 1 7 N 6 − pffiffi ρnm1m2 δm1 δm2 0 7 OM ¼ 4 6 2 ½n1 n2 5: ð56Þ 1 … 1 1 ¯ ϵ ρnn1 n5 ρnklϵ ρn1n2n3 − pffiffi ϵ ρn1n2n3 δ n 5!2 mn¯ 1…n5 þ 4·63 mkln¯ 1n2n3 36 2 m1m2mn¯ 1n2n3 m¯

N Comparing this map to the matrix CM above and using (52), one finds

ρm1m2m3 ¼ −6ρ¯m1m2m3 ; ρm1m2m3m4m5m6 ¼ 2ρ¯m1m2m3m4m5m6 : ð57Þ

Now, having at hands the equivalence between transformations encoded by the matrices O and C, one may investigate the relations between the constraints on the ρ-tensors. The consistency constraints for the generators TM transformed by the N matrix CM to form an EDA derived in [35] read

ρ¯m1m2m3 m4 0 fm1m2 ¼ ; unimodularity3; ρ¯m1m2½m3…m6 m7 0 fm1m2 ¼ ; unimodularity6; 3 ½n1 ρ¯n2n3k1 ρ¯m1m2k2 − 5 ½m1 ρ¯m2½n1n2 ρ¯n3k1k2 − ½m1 ρ¯m2n1n2n3k1k2 0 “ ” fk1k2 fk1k2 fk1k2 ¼ ; gen: CYBE : ð58Þ

While the unimodularity conditions in the first two lines are C. The short SL(5) story precisely the same as the ones following from the flux Let us now briefly look at 3-vector deformations in the invariance, the third line requires more work. Let us expand formalism of the SL(5) exceptional field theory and show the antisymmetrization in ½n1n2n3k1k2 of the third line that the only condition appearing both in the algebraic and above and reorganize it in the following more symmetric ExFT approaches is the unimodularity constraint. form Here, we work in the same conventions as those in [32] and for the sake of brevity, we will avoid a lengthy 6 ½n1 ρ¯n2n3jk1jρ¯m1m2k2 − 6 ½n1 ρ¯n2n3jk1jρ¯m2m1k2 description of the SL(5)-covariant exceptional field theory. fk1k2 fk1k2 For the details of the construction of the SL(5) exceptional ¼ f ½m1 ρ¯m2n1n2n3k1k2 : ð59Þ k1k2 field theory, see [52,53]. However, it is important to mention the generalized Lie derivative of the generalized Upon (57) this becomes precisely (47). vielbein

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1 M KL M MK L Hence, the unimodularity constraint is, indeed, sufficient LΛE ¼ Λ ∂ E þ ∂ Λ E A 2 KL A KL A for the flux to be invariant and for the deformation to 1 generate a solution. ∂ ΛKL M þ 4 KL E A; ð60Þ At the algebraic side, one considers the SL(5) excep- tional Drinfeld algebra developed in [10,11] and deforms whose explicit form reflects the fact that we work in the generators, as follows: truncated theory. Here, M; N; K; … ¼ 1; …; 5 label the … 1…5 coordinate indices, whereas A; B; C; ¼ label Tm → Tm; the flat indices. As previously accepted, the small Latin mn → mn ρmnk indices m; n; k; … and a; b; c; … label the directions of the T T þ Tk: ð66Þ “internal” space, which is four-dimensional in this case. Explicitly, the generalized metric and the deformation Requiring the deformed generators to also form an SL(5) map are given by exceptional Drinfeld algebra, one derives the following 2 3 constraints (see [10] for more detail) −1=2 a 1=2 a e em e v ϕ6 7 ρm1m2m3 m4 0 fm2m3 ¼ ; A 2 6 7 EM ¼ e 4 5; ð61Þ ½m1 m2k1½n1 n2n3k2 ½n1 n2n3k1 m1m2k2 1 2 4f ρ ρ þ 3f ρ ρ ¼ 0: 0 e = k1k2 k1k2 ð67Þ 2 3 δ n 0 6 m 7 The first line is simply the unimodularity constraint whose N 6 7 OM ¼ 4 5; ð62Þ appearance was expected. The second line is a quadratic 1 pqr 3! ϵmpqrΩ 1 constraint, which is, however, equivalent to the first line. Indeed, since the indices m; n; k ¼ 1; …4, it is natural to −1 define m e mnkl m1m2m3 where v ¼ 3! ϵ Cnkl and, as before, Ω ¼ 1 ρi1i2i3 k m1 k m2 k m3 . The generalized flux is defined as 3! i1 i2 i3 ρ ¼ ϵ ρnkl: ð68Þ structure constants of the corresponding Leibniz algebra, m mnkl as follows: Substituting this into the second line of (67) and contracting the indices m1m2 and n1n2 with epsilon-tensor, one L EM ¼ F DEM ; ð63Þ EAB C AB;C D rewrites the constraint as MN 2 M N where EAB ¼ E½A EB .In[54], it was shown explic- ρ ρ m1 0 m1 ½n1 fn2n3 ¼ ; ð69Þ itly that such defined flux contains only the 10, 15, and 40 of SL(5), which in tensor notations are encoded by the which vanishes identically upon the unimodularity con- θ AB;C ρ trombone ½AB and by fluxes YðABÞ and Z . Explicitly, straint. Note that while one is able to define such m only on one defines the irreducible flux components as YðBCÞ ¼ group manifold, we see that the unimodularity constraint is F A θ F A F D D θ δD still sufficient for a tri-vector SL(5) deformation to generate AðB;CÞ , ½BC ¼ A½B;C , ½ABC ¼ ZABC þ ½AB , C solutions. which gives One notices, however, that the non-Abelian deforma- tionprovidedin[34] are all nonunimodular, but still θ −1 MN ∂ − M ∂ N AB ¼ E E AB MNE E ½A MNE B; provide solutions. This perfectly illustrates the fact that M ∂ N YAB ¼ E ðA MNE BÞ; such obtained conditions are only sufficient rather than 1 necessary. At the same time, this observation again raises Z D ¼ EMEN ED∂ EK þ ð2EM ∂ EN the question of searching for nontrivial tri-vector defor- ABC ½A Bj K MN jC 3 ½Aj MN jBj mations, now satisfying the generalized classical Yang- M N −1∂ δD þ E½AEBjE MNEÞ jC: ð64Þ Baxter equation.

A Here, we denote E ¼ det E M. Following the same pro- IV. CONCLUSIONS AND DISCUSSIONS cedure as for the E6 6 case, one finds that the trans- ð Þ In this work, we investigate 3- and 6-Killing deforma- formation of all components of the generalized flux at all tions of general backgrounds of the 11-dimensional super- ρi1i2i3 orders in can be written as gravity admitting at least three Killing vectors, not necessarily commuting. The formalism of the exceptional δF D … ρi1i2i3 i4 AB;C ¼ð Þ fi2i3 : ð65Þ field theory provides more natural degrees of freedom for

066021-10 POLYVECTOR DEFORMATIONS IN ELEVEN-DIMENSIONAL … PHYS. REV. D 103, 066021 (2021) the deformations than the conventional formulation of The same analysis of generalized fluxes of the truncated supergravity. In these terms, tri-vector deformations appear SL(5) theory shows that the sufficient condition is simply i1i2i3 to be encoded by an EdðdÞ element generated by a 3- and the unimodularity constraint on ρ , i.e., 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, ρi1i2i3 f i4 ¼ 0: ð72Þ these transform linearly. Schematically that means i2i3

… … i1i2i3 Φ M1 MN → M1 MN Φ N1 NN We notice that the quadratic constraint on ρ derived in μ1…μp ON1 ONN μ1…μp : ð70Þ [10] is satisfied identically upon the unimodularity con- Here, the indices μ ¼ 1…11 − d, the indices M, N label straint, which provides consistency between supergravity certain irrep of EdðdÞ and the matrix O is defined as in (32), and the EDA picture. and we restrict the polyvectors to be proportional to Killing It is worth here to return back to the result of [32], where 7 vectors of the initial background explicit examples of tri-vector deformations of AdS4 × S based on the SL(5) approach were presented. One checks 1 that all non-Abelian examples there are nonunimodular, Ωm1…m3 ¼ ρi1…i3 k m1 k m3 ; 3! i1 i3 which naively seems to contradict the above result. Ωm1…m6 ¼ ρi1…i6 k m1 k m6 : ð71Þ However, since the lhs of the unimodularity condition gets i1 i6 contracted with subtle expressions in vielbeins, 3-forms and Killing vectors, the condition is far from being necessary. Working for concreteness in the E6 6 exceptional field ð Þ One concludes that the results of [32] provide examples of theory, we consider its truncation to the scalar sector as tri-Killing deformations that are not (generalized) Yang- in [32]. This leaves us with only the generalized metric þ Baxter in the above sense. To our knowledge, the literature parametrizing the coset E6 6 × R =Uspð8Þ. Explicitly, the ð Þ does not contain examples of such non-Yang-Baxter bi- corresponding generalized vielbein is given in (20). While vector deformations relevant for two-dimensional sigma- the generalized vielbein transforms under the deformations models. It would be interesting to search for examples of linearly, the transformation of the supergravity fields enter- such deformations in more space-time dimensions, where ing its definition are nonobvious. The covariant formalism one has nontrivial CYBE in addition to the unimodularity allows us to refrain from attempting to provide explicit constraint. formulas for the deformations of supergravity fields, as it Given the explicit check of the invariance of (internal) was done in [32]. Such analysis, however, would be generalized fluxes of the E6 6 exceptional field theory, necessary for one to generate explicit examples of defor- ð Þ where at the end of the day subtle and long terms boil mations within the E6 6 theory. ð Þ down to relatively simple expressions containing the alge- Truncated exceptional field theory can be completely braic equation, one might expect the same to happen for all written in terms of generalized fluxes, as it was explicitly fields of any exceptional field theory. Since, as we discussed shown in [46] for the E6ð6Þ . The corre- above, the examples of tri-vector deformations of [32] sponding equations of motion for the generalized vielbein stand slightly apart from the standard narrative of Yang- can as well be written in terms of fluxes with a single factor Baxter deformations, it is still an interesting problem to M of the inverse vielbein EA . Hence, the sufficient condition find examples of deformation that do satisfy the generalized for the deformation to map a solution of such equations Yang-Baxter equation (50). Another interesting direction for of motion to a solution is the condition for generalized further research is to investigate tri-vector deformations of fluxes to transform covariantly. Hence, fluxes with local the 10-dimensional supergravity and in particular, two- F C USp(8) indices AB must be invariant. We investigate the dimensional sigma-models, which raises intriguing ques- transformation of generalized fluxes under such defined tions of integrability of such deformations. This is work in transformations and find out that one may ensure the progress, and results will be reported in further publications. invariance imposing conditions on the constant tensors ρi1i2i3 and ρi1…i6 . These tensors are the generalization of the classical r-matrix, and the condition is believed to be the ACKNOWLEDGMENTS generalization of the classical Yang-Baxter equation. We This work was supported by Russian Science show that the condition sufficient for the deformation to be Foundation Grant No. RSCF-20-72-10144. The authors a solution-generating transformation is precisely derived acknowledge discussions with I. Bakhmatov, E. Malek, ρ via the -deformation of the E6ð6Þ exceptional Drinfeld N. S. Deger, who motivated this project, and thank E. M. algebra in [11]. Bazanova for useful comments on the text.

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APPENDIX A: NOTATIONS AND CONVENTIONS In this paper, we use the following conventions for indices

μˆ; νˆ; ¼ 1…11 eleven directions; curved; αˆ; βˆ; ¼ 1…11 eleven directions; flat; μ; ν; ρ; … ¼ 1…5 external directions of ExFT; curved; μ¯; ν¯; ρ¯; … ¼ 1…5 external directions of ExFT; flat; k; l; m; n; … ¼ 1; …; 6 internal six directions; curved; k;¯ l;¯ m;¯ n;¯ … ¼ 1; …; 6 dual internal six directions; curved; a; b; c; d; … ¼ 1; …; 6 internal six directions; flat; a;¯ b;¯ c;¯ d;¯ … ¼ 1; …; 6 dual internal six directions; flat; M; N; K; L; … ¼ 1; …; 27 fundamental ExFT indices; curved; A; B; C; D; … ¼ 1; …; 27 fundamental ExFT indices; flat; i; j ¼ 1; …;N indices labeling Killing vectors; ðA1Þ

APPENDIX B: DERIVATION OF FLUX TRANSFORMATIONS Tedious part of calculations involved with explicit substitution of the deformed vielbein into flux expressions, decomposition, and preliminary simplification of the result were done using the computer algebra program Cadabra v.1.4 [55,56] (note that v.2 is also available [57]). The corresponding Cadabra files can be found here [58]. One starts with the trombone flux whose transformation becomes

1 1 ¯ δθ ¼ −3 pffiffiffi E k mρ½i1ji2i3 f ji4k n þ Ek ϵ¯ k lk mk nk pρi1i2i3 ρ½i4ji5i6 f ji7k q A 2 mnA i1 i2i3 i4 96 A klmnpq i1 i2 i3 i4 i5i6 i7

3 ¯ − Ek ϵ¯ k lk mk nk pρ½i1i2i3i4ji5i6 f ji7k q 16 A klmnpq i1 i2 i3 i4 i5i6 i7 1 ¯ þ Ek ϵ¯ k lk mk nk pρi1i2i3 ρi5i6½i7jf ji4k q 96 A klmnpq i1 i2 i3 i4 i5i6 i7 ∼ ð…Þρ½i1ji2i3 f ji4 þð…Þρ½i1i2i3i4ji5i6 f ji7: ðB1Þ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}i2i3 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}i5i6

unimodularity3 unimodularity6

One observes that terms of both first and second order in the ρ-tensors are proportional to the unimodularity constraint. Note that although here the constraint comes with antisymmetrization of the upper indices, further analysis of the Z-flux will require a more strict version of the constraint without antisymmetrization

ρi2i3i4 i1 0 fi2i3 ¼ : ðB2Þ

The transformation of the Z-flux is convenient to analyze order-by-order in the ρ-tensor assuming that ρi1i2i3 is of order one, while ρi1…i6 is of order two. Hence, at first order, one has 5e C C D m¯ i2i3½i4j ji1 n ½D C m¯ i2i3i4 i1 n ðδZ Þ1 ¼ pffiffiffiffiffi d E ¯ E k ρ f k þ E ¯ E k ρ f k AB st 6 10 ABD m n i1 i2i3 i4 m n i1 i2i3 i4 1 þ EklCEmnDϵ k pρi2i3½i1 f i4k q ; ðB3Þ 4 klmnpq i1 i2i3 i4 which again vanishes upon the unimodularity constraint.

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Next, at order two, one writes e 1 C ðDj lmjCÞ k n p q i1½i2i3j i5i6ji7 i4 ðδZ Þ ¼ pffiffiffi d ϵ E E i1 ρ ρ f AB 2nd 24 5 ABD lmnpqr 12 k i2 i3 i4 i5i6 1 1 þ E DElmCK k n p qρi2i3i4 ρi5i6½i1 f i7 þ E ½DjElmjCK k p q nρi5i6i7 ρi1i2i3 f i4 18 k i1 i2 i3 i4 i5i6 36 k i1 i5 i6 i4 i2i3 1 1 þ E ½DjElmjCK k n p qρi1½i2ji6 ρji3i4ji5 f ji7 − E DElmCK k n p qρi2i3i4i5i6i7 f i1 6 k i1 i2 i3 i4 i5i6 4 k i1 i2 i3 i4 i5i6 − ðDj lmjCÞ k n p qρi1½i2i3i4ji5i6 ji7 r Ek E Ki1 i2 i3 i4 fi5i6 i7 ; ðB4Þ where, for clarity of expressions, we define

… m1 mp m1 m K … ¼ k …k p : ðB5Þ i1 ip i1 ip

Given the unimodularity constraint, the first three lines vanish, and the above expression becomes e 1 C ½Dj lmjC k n p q i1½i2ji6 ji3i4ji5 ji7 ðδZ Þ2 ¼ pffiffiffi d ϵ E E K ρ ρ f AB nd 24 5 ABD lmnpqr 6 k i1 i2 i3 i4 i5i6 1 r − E DElmCK k n p qρi2i3i4i5i6i7 f i1 − E ðDjElmjCÞK k n p qρi1½i2i3i4ji5i6 f ji7 : ðB6Þ 4 k i1 i2 i3 i4 i5i6 k i1 i2 i3 i4 i5i6 i7

Using the identity

5 1 ρi1½i2i3i4ji5i6 f ji7 ¼ ρ½i1i2i3i4ji5i6 f ji7 − ρi2i3i4i7i5i6 f i1 ; ðB7Þ i5i6 4 i5i6 4 i5i6 and performing a series of algebraic manipulations, the final expression can be “massaged” to

pffiffiffi 5 5e C ðDj lmjCÞ k n p q r ½i1i2i3i4ji5i6 ji7 ðδZ Þ2 ¼ − d E E ϵ K ρ f AB nd 96 ABD k lmnpqr i1 i2 i3 i4 i7 i5i6 pffiffiffi 5e 3 þ d E ½DjElmjCϵ K k n p q r ρi1½i2ji6 ρji3i4ji5 f ji7 − ρði2i3i4i5i6i7 f i1Þ : ðB8Þ 3 · 48 ABD k lmnpqr i1 i2 i3 i4 i7 i5i6 2 i5i6

The yellow expression in the first line is the unimodularity condition for the tensor ρi1…i6, whereas the black terms in the second line compose the generalized classical Yang-Baxter equation. The latter is equivalent to the constraint obtained in [35]. Finally, the terms of third order in the ρ-tensors read e 5 C C D k l m n p q r i1i2i3 i4i5i6 i7i8i9 i10 t ðδZ Þ3 ¼ pffiffiffiffiffi d E E ϵ K ρ ρ ρ f AB d 96 10 ABD 54 k l mnpqrt i1 i10 i2 i3 i4 i5 i6 i7i8 i9 5 þ E ðCE DÞϵ K k l m n p q r tρi1i2i3 ρi5i6i9 ρi10i7i8 f i4 54 k l mnpqrt i1 i10 i2 i3 i4 i5 i6 i9 i7i8 1 þ E CE Dϵ K k l m n p q r tρi2i3i4i5i6i9 ρi7i8½i10 f i1 18 k l mnpqrt i1 i10 i2 i3 i4 i5 i6 i9 i7i8 pffiffiffi − 5 2 D lmCϵ k n p q rρi2i3i4i5i6i7 i1 Ek E lmnpqrKi1 i2 i3 i4 i7 fi5i6

066021-13 KIRILL GUBAREV and EDVARD T. MUSAEV PHYS. REV. D 103, 066021 (2021) pffiffiffi − 4 2 ðDj lmjCÞϵ k n p qρi1½i2i3i4ji5i6 ji7j r Ek E lmnpqrKi1 i2 i3 i4 fi5i6 i7 5 þ E CE Dϵ K k l m n p q r tρi1½i3ji8 ρi2ji4ji9 ρji5i6i7 f i10 27 k l mnpqrt i1 i2 i3 i4 i5 i6 i7 i10 i8i9e 5 − E CE Dϵ K k l m n p q r tρi1½i3ji8 ρji10i4ji9 ρji5i6i7f i2 54 k l mnpqrt i1 i2 i3 i4 i5 i6 i7 i10 i8i9e 5 þ E CE Dϵ K k l m n p q r tρi1½i3ji8 ρji7i4ji9 ρji5i6ji2 f ji10 18 k l mnpqrt i1 i2 i3 i4 i5 i6 i7 i10 i8i9e 1 þ E CE Dϵ K k l m n p q r tρi1i2i9 ρ½i3i4i5i6i7ji8 f ji10 : ðB9Þ 3 k l mnpqrt i1 i2 i3 i4 i5 i6 i7 i10 i8i9e

The first five lines vanish because of the unimodularity constraint, whereas the rest can be neatly packaged, as follows:

pffiffiffi 5e C C D k l m n p q r t i1½i3ji8 i2ji4ji9 ji5i6i7 i10 ðδZ Þ3 ¼ pffiffiffi d E E ϵ K ð2ρ ρ ρ f AB rd 5184 2 ABD k l mnpqrt i1 i2 i3 i4 i5 i6 i7 i10 i8i9

− ρi1½i3ji8 ρji10i4ji9 ρji5i6i7 i2 3ρi1½i3ji8 ρji7i4ji9 ρji5i6ji2 ji10 18ρi1i2i9 ρ½i3i4i5i6i7ji8 ji10 fi8i9 þ fi8i9 þ fi8i9 Þ: ðB10Þ

Now, using the identity

2 1 ρi1½i3ji8 ρji2i4ji9 ρji5i6i7 f i10 ¼ ρi1½i3ji8 ρi2ji4ji9 ρji5i6i7 f i10 − ρi1i2i8 ρ½i3i4ji9 ρji5i6i7 f i10 i8i9 7 i8i9 7 i8i9 3 1 − ρi1½i3ji8 ρji7i4ji9 ρji5i6ji2 f ji10 − ρi1½i3ji8 ρji10i4ji9 ρji5i6i7f i2 ; ðB11Þ 7 i8i9 7 i8i9 for the first two terms in (B10), we obtain

pffiffiffi e 5 C C D k l m n p q r t i1½i3ji8 ji2i4ji9 ji5i6i7 i10 ðδZ Þ3 ¼ pffiffiffi d E E ϵ K ð7ρ ρ ρ f AB rd 5184 2 ABD k l mnpqrt i1 i2 i3 i4 i5 i6 i7 i10 i8i9

6ρi1½i3ji8 ρji7i4ji9 ρji5i6ji2 ji10 ρi1i2i8 ρ½i3i4ji9 ρji5i6i7 i10 − 18ρi1i2i8 ρ½i3i4i5i6i7ji9 ji10 þ fi8i9 þ fi8i9 fi8i9 Þ: ðB12Þ

Due to the high amount of index symmetries provided by contraction with the epsilon tensor, one is able to show that the first two terms are zero, whereas the third and the fourth can be rewritten in a fine form. Indeed, start with the first term

7 k l ½m n p q r tρi1½i3ji8 ρji2i4ji9 ρji5i6i7 i10 Ki1 i2 i3 i4 i5 i6 i7 i10 fi8i9 7 k ½l m n p q r tρi1i3i8 ρi2i4i9 ρi5i6i7 i10 0 ¼ Ki1 i2 i3 i4 i5 i6 i7 i10 fi8i9 ¼ ; ðB13Þ due to the antisymmetrization in seven indices ½lmnpqrt¼0 each running from 1 to 6. For the second term, we will use the generalized Yang-Baxter equation obtained from (B8), and the following identity

1 4 0 ¼ ρ½i5i6ji2 ρji3i7i4ji9i8ji10 f i1 ¼ ρ½i5i6ji2 ρji3i7i4ji9i8ji10f i1 − ρ½i5i6ji2 ρi1ji7i4ji9i8ji10 f i3: ðB14Þ i8i9 7 i8i9 7 i8i9

ρ½i3i6ji8 ρji7i4i9 i10 0 Taken together with the identity fi8i9 ¼ this implies

Kklmnpqrt ρi5i6i2 ρi3½i1ji8 ρji7i4ji9 f ji10 0: B15 i1i2½i3i4i5i6i7i10 i8i9 ¼ ð Þ

Hence, we write for the second term in (B21)

066021-14 POLYVECTOR DEFORMATIONS IN ELEVEN-DIMENSIONAL … PHYS. REV. D 103, 066021 (2021)

6 k l ½m n p q r tρ½i5i6ji2 ρi1ji3ji8 ρji7i4ji9 ji10 Ki1 i2 i3 i4 i5 i6 i7 i10 fi8i9 gen:CYBE 9 k l ½m n p q r tρ½i5i6ji2 ρji3i7i4ji9i8ji10 i1 ¼ Ki1 i2 i3 i4 i5 i6 i7 i10 fi8i9 36 k l ½m n p q r tρ½i5i6ji2 ρi1ji7i4ji9i8ji10 i3 ¼ Ki1 i2 i3 i4 i5 i6 i7 i10 fi8i9 36 k l ½m n p q r tρi5i6i2 ρi1i7i4i9i8i10 i3 ¼ Ki1 i2 i3 i4 i5 i6 i7 i10 fi8i9 genCYBE 36 k l ½m n p q r tρi5i6i2 ρi3½i1ji8 ρji7i4ji9 ji10 0 ¼ Ki1 i2 ½i3 i4 i5 i6 i7 i10 fi8i9 ¼ : ðB16Þ

Where the last identity is due to (B15). Next, we consider the third term in (B21). Again using the unimodularity condition and the identity ρ½i3i6ji8 ρji7i4i9 i10 0 fi8i9 ¼ , one derives the following useful relation 2 K k l ½m n p q r tρi1i2i8 ρi3i4i9 ρi5i6i7 f i10 ¼ K k l ½m n p q r tρi1i2½i8 ρi3i4i9 ρi5i6i7f i10 : ðB17Þ 7 i1 i2 i3 i4 i5 i6 i7 i10 i8i9 i1 i2 i3 i4 i5 i6 i7 i10 i8i9

This allows rewriting the third term as 7 K k l ½m n p q r tρi1i2i8 ρ½i3i4ji9 ρji5i6i7 f i10 ¼ K k l ½m n p q r tρi1i2½i8 ρi3i4i9 ρi5i6i7f i10 : ðB18Þ i1 i2 i3 i4 i5 i6 i7 i10 i8i9 2 i1 i2 i3 i4 i5 i6 i7 i10 i8i9

Finally, for the last term in (B21), we will need the identity 2 K k l ½m n p q r tρi1i2i8 ρi3i4i5i6i7i9 f i10 ¼ K k l ½m n p q r tρi1i2½i8 ρi3i4i5i6i7i9f i10 ; ðB19Þ 7 i1 i2 i3 i4 i5 i6 i7 i10 i8i9 i1 i2 i3 i4 i5 i6 i7 i10 i8i9 which is true upon the unimodularity constraint on ρi1…i6 . This gives for the fourth term

−18 k l ½m n p q r tρi1i2i8 ρ½i3i4i5i6i7ji9 ji10 − k l m n p q r t63ρi1i2½i8 ρi3i4i5i6i7i9 i10 Ki1 i2 i3 i4 i5 i6 i7 i10 fi8i9 ¼ Ki1 i2 ½i3 i4 i5 i6 i7 i10 fi8i9 : ðB20Þ

Altogether, the third order terms in the transformation of the Z-flux can be written as pffiffiffi 7 5 C e C D k l m n p q r t δðZ Þ3 ¼ pffiffiffi e6d E E ϵ K AB rd 10368 2 ABD k l mnpqrt i1 ½i2 i3 i4 i5 i6 i7 i10

ρi1i2½i8 ρi3i4i9 ρi5i6i7 i10 − 18ρi1i2½i8 ρi3i4i5i6i7i9 i10 × ð fi8i9 fi8i9 Þ: ðB21Þ

We see that the terms are organized into the expression in brackets, which is antisymmetric in seven indices i labeling Killing vectors. For group manifolds that have i ¼ 1; …; 6, the third order terms would identically vanish. However, for general 6-dimensional manifolds with at least 7 Killing vectors, we find an additional constraint. Notably that the terms in brackets cannot, in general, be reduced to the generalized Yang-Baxter equation (47). Indeed, in the second term due to the i1…i6 i1…i3 i1…i6 antisymmetrization the indices i8, i9 stay either both on ρ or separate between ρ and ρ . The latter case is evidently beyond something that can be massaged into (47).

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