On non-abelian U-duality of 11D backgrounds

In this letter we generalised the procedure of non-abelian T-duality based on a B-shift and a sequence of formal abelian T-dualities in non-isometric directions to 11-dimensional backgrounds. This consists of a C-shift followed by either a formal U-duality transformation or taking a IIB section. We investigate restrictions and applicability of the procedure and find that it can provide supergravity solutions for the SL(5) exceptional Drinfeld algebra only when a spectator field is present, which is consistent with examples known in the literature.


Introduction
String theory is known to respect a rich set of various symmetries, among which those that transform target space-time keeping physics the same are of special interest. The most known example of such duality symmetries is the perturbative T-duality symmetry of Type II string theory, that acts along toroidal directions of target-space according to the so-called Buscher rules [1,2]. The procedure for recovering background fields transformations from the string partition function is well known. One starts with the string partition function defined by the action S 0 [θ] symmetric under global θ → θ + α with θ corresponding to a circular direction. The symmetry is then gauged by introducing a 1-form field dθ → Dθ = dθ + A and the corresponding Lagrange termθF with F = dA to keep the 1-form pure gauge. The resulting partition function defined by the action S 1 [θ, A,θ] can then be reduced to the initial one, integrating outθ, that sets A = dα. Alternatively, one integrates out the 1-form field A obtaining a string action S 2 [θ] defined on a different background related to the initial one by Buscher rules. The scalar field θ(σ, τ ) gets replaced by the fieldθ(σ, τ ) representing dual string coordinates corresponding to winding modes [3,4]. Transformation of the dilaton ensures that measure in the partition function is invariant at one loop. One can be more general and consider backgrounds of the form M × T d in which case T-duality group will be O(d, d; Z).
A natural question is whether one may consider backgrounds with isometries represented by more involved groups than abelian U(1) d , say a sphere or a non-abelian group manifold. The answer is positive and the corresponding dualisation procedure has been considered in [5]. Essentially nonabelian T-duality of string partition function goes along the same lines as the abelian one. The difference comes from more involved definition of the field strength F = dA + [A, A], that is now an element of the corresponding algebra and hence the Lagrange term reads Tr [θF ]. Hence, one dualises the whole set of group coordinates basically replacing left-invariant 1-forms σ a by dual forms dθ a .
The original procedure for NS-NS fields has been complemented by transformation rules for RR fluxes in [6,7]. Explicit canonical formulation of non-abelian T-duality for principal sigma-model has been provided in [8]. Additionally, the work [7] provided a procedure of non-abelian T-dualisation for coset space geometries G/H based on fixing gauge degrees of freedom corresponding to action of the subgroup H. In contrast to abelian T-duality its non-abelian generalisation does not preserve isometries of the original background (in the usual sense) and hence has many in common with deformations of supergravity backgrounds. In particular, NATD techniques have been widely used to generate new supergravity backgrounds interesting from the point of view of holography, and in [9] some explicit examples of such relation have been provided.
Breaking of the initial background isometries by a non-abelian T-duality transformation is in severe contrast with mechanics of the standard abelian T-duality transformations, where preservation of isometries allows to perform T-duality twice making it an involutive symmetry. For a way out of this problem, one looks at Noether currents of the two-dimensional string sigma-model and their Bianchi identities. Starting with sigma-model on a background with isometry algebra defined by structure constants f ab c one is able to construct conserved Noether currents J a , that satisfy dJ a = 0. (1.1) Non-abelian T-dualising along the isometry directions one ends up with sigma-model on a background with no initial isometries, which however still allows to define Noether currents J a , that satisfy [10] dJ a =f a bc J b ∧ J c .
Here the algebras g andg defined by the structure constants f ab c andf a bc form the so-called Drinfeld double D. This is defined as a Manin triple (D, g,g) with the non-degenerate form given by the O(d, d) invariant metric η. Such algebraic construction allows to reverse the NATD transformation applying a Poisson-Lie T-duality transformation, that basically means solving consistency constraints for the Drinfeld double and constructing a background with such isometries (dressing the generalised vielbein in DFT terms). More details on Poisson-Lie T-duality and NATD can be found in the original works [11,12] and in review papers [13][14][15]. For developments from the generalised geometry side one refers to [16][17][18][19]. Explicit examples of backgrounds resulting from PLTD and/or NATD can be found in [7,[20][21][22][23][24]. Representation of Yang-Baxter bi-vector deformations as a B-shift followed by an NATD transformation has been considered in [25].
To some extent the above constructions generalise to M-theory in the sense of membrane dynamics and 11-dimensional supergravity. From the membrane point of view non-abelian U-duality have been addressed in [26], where in particular an analogue of Bianchi identities for currents of 2-dimensional sigma-model have been derived and implemented into the SL(5) exceptional field theory. The notion of Drinfeld double (Manin triple) have been generalised to the so-called exceptional Drinfeld algebra in the series of works [27,28], which however does not carry the structure of a bi-algebra. Instead, the algebrag dual to the isometry algebra g is defined via tri-algebra structure constantsf a bcd , that is in consistency with the current algebra of [26]. Finally, certain explicit results for non-abelian U-dualised backgrounds and their relation to non-abelian T-duality have been presented recently in [29].
This letter considers a generalisation of the approach of [25] to non-abelian T-duality in the formalism of exceptional field theory. In [25] explicit Buscher rules for non-abelian T-duality transformation have been provided written in terms of undressed fields, that can be represented as O(d, d) transformations of the corresponding generalised metric of double field theory [18,30]. Dependence on parametersx a enters in the final expression that finally get interpreted as dual coordinates. Given the embedding into DFT the procedure can be generalised to M-theory backgrounds in terms of exceptional field theory generalised metrics and dual coordinatesx ab corresponding to winding modes of membranes.
The text is structured as follows. In Section 2 we review the NATD procedure as an O(D,D) rotation for group manifolds. As an explicit example Bianchi II space-time with vanishing dilaton is considered. In Section 3 we generalise the approach to non-abelian U-duality transformations of 11-dimensional backgrounds. In Section 4 we analyse the suggested procedure for ExFT's based on U-duality groups SL(5) and SO (5,5) and derive conditions upon which a solution of 11-dimensional supergravity can be generated.

Sigma-model perspective
Non-abelian T-duality transformations generalise standard T-duality Buscher rules and can be written in a very similar form [25]. The case of our interest here is backgrounds of the form M × G where G is a group manifolds, however the sigma-model procedure can be generalised to coset spaces.
To set up the notations we briefly discuss the procedure of [25] here. One starts with the sigma model action of the form where the vielbein 1-form Eα is defined as usual as Unpacking these notations on may write for the first term in the sigma-model action where one defines metric components (2.5) The 2-form Kalb-Ramond field B is defined as usual as pullback of the corresponding target space- The fields G ab , B ab are usually referred to as undressed fields as these are free of dependence on group coordinates y m , which has all been left in the 1-forms σ a .
The procedure of NATD of the sigma-model action then proceeds with replacing (g −1 dg) a → A a and adding a Lagrange multiplierỹ a F a . Performing integration overỹ a one recovers the initial action, while integrating over A a one turns to a dual action, that now has no dependence on y m since the 1-forms σ a no longer present. Instead, a dependence onỹ a enters the dual background originating where f ab c encode structure constants of g.
This procedure can be summarised nicely by presenting a generalisation of Buscher rules, explicitly providing dual background fields. For that one defines a matrix The transformation rules are then written as follows (2.9) These have been shown in [18] to be upliftable to the double field theory formalism where the transformation of the fields becomes an O(d, d) matrix with d = dim G as expected.

Double field theory perspective
Non-abelian T-duality transformation of a 10-dimensional (group manifold) background as described above is known to be equivalent to a sequence of a B-shift and T-duality transformations, equivalently, O(d,d) reflections [30]. The procedure can be generalised to coset spaces as well, where one chooses d Killing vectors in a d-dimensional space and makes basically the same steps. Crucial is that the symmetry group acts without isotropy. In the present text we focus at the case of group manifolds to illustrate the procedure and to make further analysis of its restrictions simpler. Given the results of [30], generalisation to coset spaces must be straightforward.
One starts with noticing, that to generalise the NATD transformation rules written in the form (2.9) to 11d backgrounds these can be conveniently rewritten in terms of O(d,d) rotation of a DFT background. Following [30] the algorithm is as follows • undress background fields; • perform B-shift B ab → B ab +ỹ c f ab c , withỹ a understood as coordinates dual to y m .
• perform formal abelian T-dualities along all directions of the group manifold to turnỹ a into geometric coordinates.
Schematically the procedure is depicted on Fig.1. For further reference and to setup notations let us consider the procedure in more details. The first step splits coordinate dependence to external coordinates x µ and group manifold coordinates Further B-shift introduces additional dependence dual coordinatesỹ a that is not obvious to check against section constraint. However, one notices that the dependence on y m is of very restricted form hidden in the 1-forms σ a . For this reason working with undressed fields allows to overcome this issue.
Below we show that on explicit examples for both DFT and ExFT, while here we will try to develop some intuition allowing to work with such transformation.
One starts with an abelian T-duality transformation in the DFT formalism that corresponds to replacing x m byx m , or better to say, to switching their roles as geometric and non-geometric coordinates. Most transparently this is seen when considering doubled pseudo-interval 1 Here and in what follows capital Latin indices M, N, . . . label directions of the extended space and in case of DFT run 1, . . . , 2dim G. Assigning to y m andỹ m the roles of geometric and dual coordinates respectively, one thus fixes H mn = g mn . To perform T-duality transformation one keeps the pseudointerval the same, switching instead roles of coordinates. Say y 1 now becomes dual, whileỹ 1 becomes geometric. This implies, that H 11 =g 11 is now component of the transformed metric. This procedure has been employed to generate exotic brane solutions and to unify them into a single DFT/ExFT solution in [31][32][33][34].
For the case in question it is tempting to writes instead where dependence on y m has been recollected into the 1-forms σ a = σ a m dx m . For now, the dual coordinates are still represented by exact forms dỹ a .
The procedure described guarantees, that one ends up with a solution of supergravity equations of motion if started with a solution. Indeed, let us start for simplicity with a background, that depends purely on group manifold coordinates, i.e. G ab = const, B ab = const. Hence, one may encode the where G ab is simply the inverse of G ab . Turning to flux formulation of DFT [35]  For the NATD procedure one starts with the undressed fields packed into the "flat" generalised metric H AB and first preforms a B-shift, that can be encoded as (2.14) Instead of T-dualising all coordinates and checking equations of motion of supergravity, double field theory allows to check H ′ AB explicitly, which is much simpler due to linear dependence on the dual coordinatesỹ a . Indeed, in the above expression the matrix O A B can be understood as a generalised vielbein, and the corresponding generalised flux precisely has the same non-vanishing components F ′ ab c = f ab c as that for E M A . Since the "flat" generalised metric H AB is the same, the background encoded in H ′ AB satisfies equations of motion of double field theory. Note, that the B-shift is arranged is such a way as to generate a background with the same generalised flux F ′ ABC = F ABC , which however depends only on dual coordinates. Finally, performing T-dualities along all ofỹ a 's one obtains a supergravity solution, since replacingỹ ↔ y with the corresponding transformation of fields (Buscher rules) is a symmetry of DFT. It is worth mentioning, that after Tdualities the flux components change and one finds non-vanishing F c ab components, since T-duality along each direction replaces a ↔ a [36].
Before turning to an illustrating example, one observes, that the last step where all directions of the group manifold get T-dualised is crucial for ending up with a supergravity solution. It is clear, that one is always able to perform the necessary set of T-dualities to turn allỹ a into geometric coordinates. Picture however gets more complicated in the case of non-abelian U-dualities and such a set may not exist. We discuss this important point in more details in Section 4.

Bianchi II example
As an explicit illustration of the above procedure, consider the standard examples of Bianchi II cosmological space-time embedded into 10 dimensions. The metric is can be chosen to be where the 1-forms σ a and the functions a a read 16) and the constants are constrained by p 2 p 3 = p 2 1 . In what follows we set p a = 1 to avoid the dilaton. Note, that the 1-forms only depend on the coordinates y 1 , y 2 , y 3 on the group manifold generated by the Heisenberg-Weyl algebra The undressed metric is then Since the time direction x 0 is not dualised and the metric does not have mixed g 0a components, it is enough to focus only at the block 1, 2, 3 and consider O(3, 3) double field theory. The corresponding generalised metric is simply given by with ∆B ab =ỹ c f ab c whose only non-vanishing components are Next one is supposed to perform abelian T-dualities along all directionsỹ a . T-dualising along all three directions renders all x a non-geometric as well as the corresponding forms, and one reproduces the well known result for the dual background [21] ds ′2 = ds 6 − a 1 2 a 2 2 a 3 3 (dx 0 ) 2 , Note thatx a are now proper physical coordinates. The dilaton is recovered from the invariant dilaton where g = det ||g ab || is determinant of the undressed metric.
3 Non-abelian U-duality in SL(5) ExFT Let us now try to generalise the above algorithm of NATD to the case of exceptional field theory.
As the very first example one may take SL(5) exceptional field theory, that is a 7 + 10-dimensional field theory, local coordinate transformations include U-dualities of D = 7 maximal supergravity [37,38] (for a review on exceptional field theories see [39][40][41]  Field content of the theory can be written in irreps of the duality group SL(5) as follows Here d is the invariant dilaton of double field theory,h ab is the 3-dimensional block of the full 10-dimensional metric and the matrix M ij encodes the degrees of freedom of the axion-dilaton The pair of vectors V i a encode internal parts of the NS-NS Kalb-Ramond 2-form B ab and RR field where ǫ abc is the Levi-Civita symbol ǫ 123 = 1 It is important to notice, that the parametrisation used here differs from that of [29] by rescaling of the metric and 2-form fields by certain power of e φ . More precisely, the parametrisation of [29] provides formulation of IIB supergravity explicitly covariant under the SL(2) duality symmetry, that is reflected in the fact, that all dependence on the dilaton is hidden inside the SL(2)/SO(2) matrix. In contrast, the parametrisation given above provides fields T-dual to the IIA fields, that can be obtained from the standard 11D parametrisation.
For the purpose of this paper, the latter is more convenient. Now, following the analogy between DFT and ExFT extended spaces one proposes the following non-abelian U-duality scheme for 11D backgrounds 1. undress the metric and the C-field g mn = σ m a σ n b g ab , C mnk = σ m a σ n b σ k c C abc and compose As before, C-shift turning m AB to m ′ AB can be understood as a generalised vielbein with the following generalised fluxes Although here we restrict ourselves to the case of SL(5) ExFT for simplicity, the first two steps of the procedure have straightforward generalisation to higher U-duality group simply by including more winding coordinates. In contrast, the last step appears to be much more restricted for the 10D: Figure 2: Relationship between backgrounds with spectator fields upon the non-abelian U-duality procedure. Here taking a IIB section represents an uplift of three T-dualities with further reduction to 10 dimensions. In this case the bottom line represents the usual non-abelian T-duality. SL(5) theory than for theories with more winding directions. As we show below, at least for group manifolds full dualisation of all four coordinates is possible only when at least one of the coordinates is an abelian isometry, i.e. corresponds to a spectator field. We will conclude that the described procedure for the SL (5) ExFT is always an uplift of an NATD transformation. Similar observations based on the construction of exceptional Drinfled algebras for the SL(5) theory have been made in [29]. Schematically, this is illustrated on Fig. 2. Geometrically such defined Drinfeld double can be realised by choosing a maximally isotropic subalgebra say g to be a "physical" subalgebra. Group element g = exp x a T a of the corresponding Lie group G defined by generators of the physical subalgebra will define left-invariant 1-forms σ = g −1 dg on the group manifold. In this setup a non-abelian T-duality corresponds to transfer the rope of the physical subalgebra to the dual algebrag and constructing space-time 1-forms from group element g = exp x aT a . Note, that here x a is a physical coordinate and not further T-duality is required.  [30] for more details) One notices, that according to the B-shift+T-dualities procedure, one has to replace all winding coordinates by their geometric partners, which can be done in a unique way for O(d, d) theory (for groups-manifolds that are not a product of Lie groups). This seems to be in tension with the Poisson-Lie T-plurality picture, where a given Drinfeld double can be decomposed into a set of more than two Manin triples [44]. Backgrounds corresponding to such Manin triples generate the same Drinfeld double and hence are indistinguishable from the point of view of the two-dimensional sigma model. Examples of such backgrounds can be found in [22]. In the O(d, d) language Poisson-Lie T-plurality corresponds to performing a rotation by an O(d, d) matrix C A B , preserving the Drinfeld double, which in particular can be a set of d reflections [30]. This latter case is precisely the transformation, that turns all winding coordinates into geometric ones. Hence, in all other cases one would expect backgrounds, which do not solve equations of motion of normal supergravity due to remaining dependence on winding coordinates. Indeed, as has been shown on explicit examples in [22] such procedure in particular gives solutions of generalised supergravity equations. More generally, one always ends up with a DFT background. Simply speakin, equivalent gl(d) embeddings into o(d, d) can be obtained from a given one by O(d, d) rotations and by the external automorphism of the algebra.

More generally a Poisson-Lie T-duality is a constant transformation of the generators T
Only the latter turns the fundamental of a given embedding of gl(d) into the antifundamental of the dual embedding. Crucial here is that no weight belongs to both these representations, which is apparent for the o(d, d) algebra but is not always true for symmetry algebras of exceptional field theories.
To conclude, one starts with an irrep R 1 of the abelian T(U)-duality group in which extended coordinates transform. Upon an embedding of the geometric GL(d) subgroup this decomposes into where d corresponds to geometric coordinates and ellipses denotes irreps under which winding coordinates transform. Now, one considers a different embedding of the geometric GL(d) where d ′ is the fundamental of GL(d) none of whose weights inside R 1 coincide with any of the weights of d. Let us provide more details for U-duality groups SL (5), where this cannot be done, and SO (5,5), that can be shown to allow 11-dimensional NAUD.

U-duality and exceptional Drinfeld algebras
We start with the set of the simple roots of the Lie algebra sl(5) in the canonical ω-basis of fundamental weights are given by the following  In addition, one has the same number of negative roots and four Cartan generators. Weight diagram of the fundamental representation 5 of sl(5) is depicted on Fig. 4, where µ 1 , . . . , µ 5 denote basis vectors. Notations for the simple root of the algebra are chosen in such a way that, say the root α 12 sends the weight vector µ 1 to µ 2 . Or, equivalently, the exponent exp(ωα 12 ) acts by SL(2) rotations on the plane (µ 1 , µ 2 ). It is important to note, that the weights µ 2 , µ 3 , µ 4 belong to a 4 for both of the decompositions, while one of µ 1 , µ 5 becomes a singlet. Following the analogy with the NATD one is interested in embeddings of the physical gl (4) subalgebra related by the external automorphism. In particular for the SL(5) theory we are interested in decomposing the 10 of sl(5) upon two embeddings of gl (4), that are shown in Fig. 5. Consider first the decomposition corresponding to deleting the root α 45 (cutting blue arrows). In this case weight vectors X 5a with a = 1, . . . , 4 belong to the 4 of gl(4) while the rest X ab belong to the 6. In the ExFT language, the former get identified with geometric coordinates, while the latter represent winding modes.
Now, according to the procedure of NAUD described above, one need to find such a different embedding of gl (4), that all weights contributed to the irrep governing geometric coordinates of the  Figure 5: Weight diagram of the 10 of sl(5) with two possible embeddings of the gl(4) subalgebra.
first embedding belong to that governing winding modes. Explicitly, all weights from the old 4 must belong to the new 6, which is impossible, according to Fig. 5.
Indeed, suppose one starts with four left-invariant 1-forms σ a , that depend on four coordinates on the (unimodular) group manifold x 1 , x 2 , x 3 , x 4 . Next one constructs a background with flat metric and C-field given by C abc = −3ỹ d[a f bc] d withỹ ab = 1/2ǫ abcd X cd being coordinates along winding directions.
This has been shown to solve equations of motion of ExFT, however to end up with an ordinary supergravity solution one has introduce such turn allỹ ab into geometric coordinates such that all X 5a become non-geometric. From Fig. 5 one concludes that the automorphism acts by interchanging indices 1 ←→ 5 upon which the directions X 25 , X 35 , X 45 become non-geometric since belong to the new 6, while X 15 belongs to the new 4 and hence must be thought of as a geometric direction.
According to the speculative discussion around (2.12) these directions correspond to 1-forms, rather than coordinates and hence the forms σ 2 , σ 3 , σ 4 must be thought of as "non-geometric" while σ 1 as a "geometric". It is suggestive to understand a (non-)geometric 1-form components as those, which depend on (non-)geometric coordinates. Unless dσ 1 = 0, one ends up with a contradiction, when the same set of coordinates on which σ a depend should be understood as non-geometric and as geometric at the same time.
The above conditions can be fulfilled when T 1 commutes trivially with the rest three generators.
In this case dσ 1 = 0 and it can be chosen to depend say on x 1 one which the other 1-forms σ α with α = 2, 3, 4 do not depend. Indeed, otherwise dσ α would give σ 1 on the RHS generating non-vanishing f 1a α . One concludes, that the described procedure applied to a 4-dimensional group manifold provides a solution of supergravity equations of motion only when at least one spectator field is included. This is in consistency with observations made in [29]. Another option would be to generalise the notion of T-plurality to the case of non-abelian U-duality. From the DFT point of view T-plurality generates backgrounds with dependence on dual coordinates, which in particular cases solve generalised supergravity equations. However, no generalised supergravity extension to 11 dimensions is known, and moreover this is widely accepted to not exist.
More strict and rigorous formulation of these points is required and it is tempting to believe that this can be achieved in the formalism of DFT WZW [16,45,46].More detailed investigation of such formulations is reserved for future work.
Consider now more fruitful case of five dimensions and U-duality algebra so (5,5). Its Dynkin diagrams with two possible deletions of simple roots giving gl (5) is depicted on Fig. 6. This has three simple roots generating vector representation, antisymmetric tensor of second and third rang representations and two spinorial representations. Now, from the diagram it is clear that upon the first embedding the geometric coordinates (equivalently, "physical" generators of the SO(5,5) EDA) correspond to the weights (X 1 , . . . , X 5 ), while the rest correspond to winding modes. Upon the second embedding the "physical" subaglebra of EDA is spanned by generators corresponding to the weights (X 12 , . . . , X 15 ). One notices, that the two sets of physical coordinates do not intersect and one is able to perform such an SO (5,5) transformation as to shift all 1-forms σ a into the non-geometric set. Equivalently, this demonstrates existence of two possible choices of the "physical" subalgebra inside exceptional Drinfeld algebra with SO(5,5) symmetry, which do not conflict.

Discussion
In this letter a generalisation of the non-abelian T-duality Buscher rules for 10D supergravity backgrounds to 11D backgrounds has been proposed. suggests to understand NAUD transformation as a switch between two "physical" algebras gl(d) by external automorphism of the corresponding exceptional symmetry algebra. We show, that for the algebra sl(5) such procedure can generate solutions of the conventional supergravity only when a spectator field presents, which is in consistency with observation made in [29]. Investigating the example of the algebra so(5, 5) one concludes that larger U-duality symmetry groups allow such non-abelian U-dualisation and a solution of equations of motion of 11-dimensional supergravity can be constructed. Investigation of explicit examples based on the SO(5,5) and E 6 exceptional Drinfeld algebra is reserved to future work.
One becomes naturally interested in generalisation of the obtained results to exceptional field theories to general manifolds with isometries along the line of [27,28,30]. In this case symmetries manifest themselves in the algebra of Killing vectors, which can be used to organise an tri-vector shift, in contrast to the 3-form shift in the present paper [47,48]. This provides tri-vector deformations of 11-dimensional backgrounds, which in certain cases follow the same scheme as in Figure 2. E.g. one considers tri-vector deformation of Minkowski space-time, which in the IIB frame is again a Minkowski space-time, while solves equations of motion of generalised supergravity in the IIA frame [47]. More detailed analysis of relations between deformations and non-abelian dualities is required.

Acknowledgements
The author thanks vivid discussions with I. Bakhmatov, K. Gubarev, E. Malek and N. Sadik Deger that motivated this project. The author thanks Yuho Sakatani for useful comments and suggestions. This work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" and by Russian Ministry of education and science (Project 5-100). In part the work was funded by the Russian Government program of competitive growth of Kazan Federal University.