Non-isometric U-dualities

I study generalisations of U-duality transformations which do not rely on the existence of isometries. I start by providing more details of a recently proposed generalised U-duality map between solutions of type IIA supergravity of the form $\text{M}_7 \times \text{S}^3$, with NSNS flux, and solutions of 11-dimensional supergravity, in which the three-sphere is replaced by a four-dimensional geometry which encodes three-algebra structure constants. I then show that when $\text{M}_7$ admits two abelian isometries, TsT deformations on the IIA side become six-vector deformations in the 11-dimensional setting. These six-vector deformations involve an action of $\mathrm{E}_{6(6)}$ on both isometric and non-isometric directions. I discuss the algebraic interpretation of these deformations, and compare and contrast them with (generalised) Yang-Baxter deformations in supergravity.


Introduction
Supergravity inherits T-and U-duality symmetry from string and M-theory. These dualities can be used to map solutions with abelian isometries to other solutions with abelian isometries.
Broader classes of transformations between solutions with more complicated or even no isometries can be considered to be generalised dualities. The most studied examples are generalised T-dualities such as non-abelian [1] and Poisson-Lie [2,3] T-duality. These have well-established applications as solution generating mechanisms in supergravity and close links to integrable deformations such as the Yang-Baxter deformation [4]. More recently, thanks in large part to progress in understanding the generalised geometric/double field theory interpretation of these generalised T-dualities [5][6][7][8], progress has been made extrapolating features of generalised dualities to the case of U-duality [9][10][11][12][13][14][15][16][17][18]. The motivation for this paper is to further study the realisation of new non-isometric U-dualities in 11 dimensions.
The following guideline or definition for a generalised duality in supergravity can be extracted such that reducing theory 1 on the d-dimensional geometry M d gives the same lower-dimensional theory as reducing theory 2 on thed-dimensional geometry Md. 1 The appropriate sense of dimensional reduction required is that of a consistent truncation, so that solutions of the common lower dimensional theory can be lifted to solutions of either theory 1 and theory 2 in higher dimensions. These uplifted solutions are then regarded as 'dual', at the very least providing a solution generating mechanism in supergravity.
In this paper, I study an example of this type of map, developed in [12,18], of the form: with NSNS flux with '3-algebra flux' (1.2) This can be used as a solution generating mechanism in the following way. Start with the geometries on the left-hand side. These are solutions of ten-dimensional type IIA supergravity of the form M 7 × S 3 , with NSNS flux. These consistently truncate to 7-dimensional CSO(4, 0, 1) gauged supergravity [19,20]. Then (in principle) new solutions of 11-dimensional supergravity are constructed (locally) on the right-hand side by uplifting using the alternative consistent truncation described in [12,18].
These new solutions are linked to an underlying non-Lie algebraic structure, which is that of a three-algebra. By this I mean an algebra with a totally antisymmetric bracket involving three, rather than two, elements, such that generators t a obey [t a , t b , t c ] =f abc d t d . The map (1.2) involves the four-dimensional case withf abc d = ǫ abce δ ed , and it is these structure constants which control the form of the new four-dimensional geometry Mf 4 . (Despite this, the role, if any, of an actual three-algebra symmetry is unclear, and will make no further appearance in this paper.) This can be viewed as an M-theoretic generalisation of the sort of solutions which appear after 1 I denote the dimensions as d andd to allow for duality between 10-and 11-dimensional theories.
applying non-abelian T-duality [1,21], in which the dual geometry is locally determined by the existence of an underlying Lie algebra symmetry.
A proof of concept of the map (1.2) has been demonstrated in [18], where the F1-(near horizon) NS5 brane solution of type IIA was taken as the initial solution on the left-hand side of (1.2) and used to generate a novel 11-dimensional solution. However, explicit general formulae applicable to arbitrary backgrounds of the form M 7 × S 3 were not presented. The first result of this paper is to improve this situation, by providing the complete expressions starting with purely NSNS solutions of type IIA. These are given in section 3.1.
I then investigate and explain an interesting feature noticed in the new solution obtained in [18]. In that case, the F1 near horizon limit of the initial IIA solution was of the form M 7 = AdS 3 × T 4 . The corresponding new 11-dimensional solution in the same limit then had the (local) form AdS 3 × T 4 × Mf 4 . 2 The interpolation away from the F1 near horizon region, which can be realised as a TsT transformation, or bivector deformation, was shown in [18] to become an I then discuss, in section 4, the algebraic interpretation of this non-isometric duality. The full underlying algebraic structure behind the right-hand side of (1.2) is the Poisson-Lie (or Nambu-Lie) U-duality construction of [9,10]. This introduces an underlying four-dimensional Nambu-Lie group equipped with a trivector, π abc . In the case (1.2), this group is abelian and the trivector obeys L v d π abc =f abc d , where v a are the left-invariant vectors of the group. The geometry of Mf 4 is then determined via the Nambu-Lie group data (left-invariant vectors and forms, and the trivector) using a generalised frame in SL(5) generalised geometry/exceptional field theory. This frame generates a generalised parallelisation realising an algebra dubbed the exceptional Drinfeld algebra.
The six-vector parameter deforming the 11-dimensional solutions on the right-hand side of (1.2) is: where a = 1, . . . , 6 labels a set of six vectors. This set comprises the two commuting Killing vectors k α inherited from the isometries of the original type IIA background, as well as the four vectors v a associated with the underlying Nambu-Lie group structure. These are not Killing (their action on the spacetime fields is determined in terms of the three-algebra structure constants linked to Mf 4 [11]). The fact that this deformation (1.3) nonetheless generates a solution can be linked to the fact that it can be viewed as a transformation leaving invariant the underlying exceptional Drinfeld algebra. 2 The solution in this limit could be identified as belonging to a class of M-theory AdS3 solutions obtained in [22]. These solutions admit holographic duals similar to the holographic duals of solutions obtained by non-abelian T-duality, again suggesting that these three-algebra geometries are 11-dimensional analogues of such solutions, and also giving one way to globally complete the solution. Conventional U-duality deformations of the solutions of [22] were studied in [23].
This can be compared and contrasted with existing approaches to polyvector deformations.
Inspired by the recasting of the Yang-Baxter deformation [4,[24][25][26] in terms of a non-constant bivector parameter [27][28][29][30][31][32][33], polyvector deformations involving p Killing vectors, of the form r a 1 ...ap k a 1 . . . k ap , have been studied in both 10-and 11-dimensions [9,10,13,[34][35][36]. The transformation (1.3) further generalises these generalised Yang-Baxter deformations by dropping the assumption that all vectors involved are Killing. This is in fact implicitly what has been used in identifying Yang-Baxter deformed backgrounds and non-abelian T-duals as T-folds, as in [37,38]. There the background is patched by a global T-duality transformation which does not act on isometric directions. This generalises to the U-fold interpretation of the 11-dimensional solution appearing on the right-hand side of (1.2). These examples suggest the existence of a broader landscape of polyvector deformations applicable in backgrounds admitting underlying (but nonisometric) Poisson-Lie or Nambu-Lie symmetry.
The complete outline of this paper is as follows: I first discuss in section 2 the necessary technical language surrounding generalised metrics, polyvectors and generalised duality. six-vector deformations in section 3.3. I further briefly discuss here the fate of more general T-duality transformations, which can also be interpreted as non-isometric E 6(6) transformations generically with a less clear geometric interpretation than the six-vector deformation.
In section 4, I discuss the algebraic interpretation of the six-vector deformation, and its similarities to, and differences with, other sorts of polyvector deformations.
In section 5, I conclude with a discussion.
Appendix A describes my conventions relating to forms and to generalised geometry.
Appendix B describes the derivation of the map (1.2) in detail.
Appendix C highlights a non-trivial self T-duality property of AdS 3 × T 4 × S 3 with NSNS flux.
2 Generalised geometry and duality 2.1 Generalised duality from generalised geometry A convenient way to formulate generalised duality as a map of the form (1.1) is to use double or exceptional geometric approaches [5][6][7][8][9][10], where supergravity is reformulated -in arbitrary backgrounds -in terms of variables on which O(d, d) or E d(d) transformations act linearly [39,40], see [41,42]  To describe generalised duality of more general backgrounds, a little more detail is needed.
First of all, it is required that there exists a factorisation in which the generalised vielbein E A and the scalar ∆ depend only on the coordinates on M d , while M AB is independent of these coordinates. Under certain differential conditions on E A , this permits a dimensional reduction to a lower dimensional gauged maximal supergravity. These differential conditions have a very simple interpretation using the generalised Lie derivative (see equation (A.5)) of generalised geometry, and in terms of the inverse generalised frame E A have the form for constants F AB C and θ A . This defines a 'generalised parallelisation' and permits a consistent truncation where F AB C and θ A become the gaugings of the lower-dimensional gauged supergravity. (The case of interest to this paper has θ A = 0.) The map (1.1) is possible when there exists a different background Md allowing for a generalised vielbein E A obeying the same differential constraints and permitting an alternative consistent truncation to lower dimensions. Then both theory 1 and theory 2 reduce, on M d and Md respectively, to a common lower dimensional theory, and solutions of this theory (given here by a specific M AB ) can be uplifted to solutions of either theory 1 or theory 2. While a generalised geometric frame is known realising this consistent truncation [44,45], the original formulae of [19,20] in fact suffices to formulate (1.2) explicitly, as used in [18] (and here in appendix B). TsT transformations [46,47]. Secondly, coordinate dependent polyvectors will be used when formulating generalised vielbeins meeting the conditions leading to the existence of the generalised duality map (1.1). Lastly, a constant E 6(6) polyvector will appear when analysing the fate of TsT deformations after generalised duality. This will involve E 6(6) acting on a combination of isometric and non-isometric directions. I now review the background needed to understand each of these cases.

O(d, d) T-duality and bivectors
In the T-duality case, the generalised tangent bundle is ) of a vector v and one-form ω (1) . I introduce a metric G and two-form B on M d . These are encoded in a generalised metric defined on the generalised tangent bundle: 3) The matrix involving the two-form B is itself an O(d, d) element. I can introduce a bivector β in the form of the 'transposed' transformation: (2.5) I will realise TsT transformations, or bivector shifts, by acting with O = U β , with a constant bivector. In my conventions, a bivector with non-zero components β 12 = −β 21 = λ corresponds to the combination of: T-duality in x 2 , shift x 1 → x 1 + λx 2 , T-duality back onx 2 . Alternatively: T-dualise in x 1 and x 2 , shift the B-field as B 12 → B 12 + λ and T-dualise back onx 1 andx 2 .
where H = dB is the original field strength for the 10-dimensional 2-form. Here I assume explicitly that x α are adapted coordinates for isometries, and use the subscript (p) to denote a transforms as a vector.
I now specialise to the case d = 2. I can let B αβ = Bǫ αβ . 5 Under a bivector transformation with β αβ = λǫ αβ , with the other fields invariant.
Later on, I will need to use the following factorisation of the O(2, 2) generalised metric. Definē (via the SL(2) inversion element) and bivector shifts, with the latter acting as Note also that T-duality on one direction corresponds to a Z 2 transformation swapping τ and ρ.

SL(5) (generalised) U-duality and trivectors
In the U-duality case, with M n ×M d a background of 11-dimensional supergravity, the generalised generalised metric is: The conformal factor ensures that det M = 1, otherwise the generalised metric will not be a true SL(5) element. As a generalised vector consists of a vector and a two-form, the indices on the term g −1 g −1 are to be understood as being antisymmetrised, thus The matrix involving the three-form is an E d(d) transformation, and I introduce a trivector as the transposed transformation: A non-constant trivector is used to define the right-hand side of the generalised U-duality map (1.2). This trivector appears as part of the auxiliary geometric data defining the right-hand side of (1.2) via the Poisson-Lie or Nambu-Lie U-duality construction of [9,10]. This introduces an underlying Nambu-Lie group G, which in this case is four-dimensional, with associated leftinvariant vector fields, v a , and forms, l a , obeying f ab c are the structure constants of the Lie algebra g of G. Here a, b = 1, . . . , 4. In addition, the group is equipped with a trivector, denoted π abc . By definition, this obeys: wheref abc d are structure constants for a three-algebra. The additional constant L a is identified with the derivative of an additional scalar, L va ln α = 1 3 L a . Using the above differential conditions (see [9,10] for full details) the generalised frame defined by then obeys the algebra (2.2) with structure constants of the so-called exceptional Drinfeld algebra (EDA). This algebra is defined by following (not necessarily antisymmetric) brackets: (2.16) The subalgebra generated by the generators T a is a Lie algebra. This is the distinguished isotropic subalgebra whose selection corresponds to a particular choice of decomposition of the EDA, leading to the construction of the frame (2.15).
The right-hand side of the generalised U-duality map (1.2) corresponds [12] to the case where the group G is abelian, so f ab c = 0, but the three-algebra is take to be the Euclidean three-algebra withf abc d = ǫ abce δ ed . 6 Introducing coordinates x i , i = 1, . . . , 4, the corresponding geometric data are then The algebra (2.16) then corresponds to CSO(4, 0, 1), which is to say the Euclidean Poincaré algebra. The generators T a correspond to the abelian translational subalgebra, while T ab generate the rotational so(4) Lie algebra. 7 The generalised frame (2.15) built using (2.17) then allows for a consistent truncation to 7-dimensional CSO(4, 0, 1) gauged supergravity.
To describe this consistent truncation fully, I should introduce the additional SL(5) covariant fields needed to capture all the degrees of freedom of 11-dimensional supergravity. This is presented in appendix B.
. It is often convenient to dualise the five-form index, letting V¯i = 1 5! ǫ ij 1 ...j 5 V j 1 ...j 5 . The fields of 11-dimensional supergravity restricted to M d are the metric, which I denote φ ij , the three-form, C ijk , and the dual six-form, C i 1 ...i 6 . The latter has only 1 independent component, so I write it as C i 1 ...i 6 = C 6 ǫ i 1 ...i 6 . These fields can be encoded in a generalised metric [48,49], with the following parametrisation: In addition, La = 0. This implies that the 'trombone' gauging θA in (2.2) also vanishes. 7 This is clearest to see by defining dualised generators T ab = 1 2 ǫ abcd T cd and noting that the Euclidean metric δ ab is encoded in the three-algebra structure constants, again by dualisation with ǫ abcd .
The dual polyvector variables are now a trivector, Ω ijk , as well as a six-vector, These can be introduced using the following E 6(6) valued matrix: (2.20) I will use matrices of this form to describe the action of six-vector deformations. I can also embed the trivector appearing in the SL(5) generalised frame in a E 6(6) matrix of the form (2.20). Note that trivector and six-vector transformations commute.
The complete E 6(6) -covariant description of 11-dimensional supergravity also requires fields carrying five-dimensional indices [50]. For instance, the 11-dimensional metric is decomposed as: 1) . The five-dimensional metric g µν is invariant under E 6(6) . In addition there are gauge fields. The most relevant is the one-form A M (1) which contains the 'Kaluza-Klein vector' A i (1) appearing in the metric decomposition, as well as the components of the three-and six-form carrying one five-dimensional index. (The subscript (p) now denotes a p-form on M 5 , the five-dimensional part of the spacetime.) I will only need the two-form field strength of this gauge field 9 , which on general grounds can be written as: 1) , and F (2)ij and F (2)i 1 ...i 5 are related to components of the 11-dimensional four-form and dual seven-form (F (7) = ⋆F (4) ) by field redefinitions involving A i (1) , as in equation (A.11). This general form of F (2) M is determined by the requirement that it transform covariantly under generalised diffeomorphisms, which fixes it to be 'twisted' by the potentials in the same manner as the generalised metric.
3 11-dimensional solutions and six-vector deformations from TsT 3.1 11-dimensional solutions from M 7 × S 3 I now write down the generalised U-duality solution generating mechanism (1.2). The existence of this map was pointed out in [12], and demonstrated for a particular example in [18], but a general expression for the resulting 11-dimensional backgrounds was not given. In appendix B I derive the map for general pure NSNS solutions of type IIA SUGRA on S 3 that admit a consistent truncation to CSO(4, 0, 1) gauged supergravity. In this section, I restrict to the case of a direct product solution M 7 × S 3 , with string frame metric and three-form field strength given by 10 with I, J = 0, . . . , 6. In (3.1), I suppose that the seven-dimensional metric, G IJ , the sevendimensional restriction of the three-form, h (3) , and the dilaton, ϕ, are independent of the coordinates on the sphere. This background can then be consistently truncated to a solution of seven-dimensional CSO(4, 0, 1) gauged maximal supergravity, following [20], and then uplifted to a new solution of 11-dimensional supergravity, following [12,18]. This uplift uses the EDA generalised frame (2.15) with data (2.17). The process of uplift in SL(5) covariant variables followed by extraction of 11-dimensional fields is recounted in appendix B.
The resulting 11-dimensional metric has the form: and the field strength and its dual are: where ρ 2 ≡ δ ij x i x j , and The Bianchi identity for F (4) implies it is possible to introduce a two-form B (2) and a three-form In the initial type IIA picture, the two-form B (2) is the original NSNS two-form gauge field (restricted to M 7 ), while B (3) arises from the dual six-form (wrapping the S 3 ). Using these, the 11-dimensional three-form can be expressed as: In applications below, I will need to make explicit use of the dual six-form. This is defined by (4) . A direct calculation produces: (3.10) The third line vanishes by the original type IIA equation of motion for the dilaton ϕ. The first line and second line together define the components of C (6) restricted to the 7-dimensional part of spacetime inherited from the type IIA geometry. The final line defines the mixed components of C (6) which include both 7-dimensional and 4-dimensional contributions. The crucial part which will appear often below is the 'internal' contribution proportional to the volume form on the 4-dimensional space: It can be convenient in calculations to define spherical coordinates, letting x i = ρµ i with δ ij µ i µ j = 1. The following elementary results: along with x i dx i = ρdρ, allow the solution to be rewritten in these coordinates. The metric and field strength are then:

(3.13)
A final comment is that the calculation of dC (6) also provides the Page charge density, Q Page = (4) . Integrating this over (spatial) seven-cycles calculates the M2 charge. From (3.10) it follows that the Page charge is a total derivative and so vanishes up to large gauge transformations. On the other hand, the form of F (4) suggests the presence of M5 charge.

TsT to six-vector deformations: AdS 3 × T 4 × S 3 examples
My main goal in this paper is to study how T-duality transformations of the original type IIA solution then generate additional 11-dimensional solutions. I will restrict to T-duality transformations which preserve the form of the ansatz (3.1): this means that I will only consider transformations which do not touch the three-sphere. These transformations will be elements of acting non-trivially on the metric G IJ , 2-form B IJ and dilaton ϕ.
The observation underlying this goal was the discovery in [18] that an O(2, 2) TsT transformation of AdS 3 × T 4 × S 3 was transmuted into a more complicated six-vector transformation of the dual 11-dimensional solution. This six-vector deformation could be realised as an E 6(6) element. The unusual feature about this E 6(6) transformation was that it acted both on isometric and non-isometric directions. A natural conjecture is that this would be true for other TsT transformations.
In the next subsection, I will present a general demonstration of this fact applicable to any background of the form (3.1) in which M 7 admits two commuting isometries. However, here I first focus on this AdS 3 case, in order to motivate this goal and review the observation of [18].
The original type IIA background is: with db = Vol S 3 and where I take the AdS metric to be with η αβ the 2-dimensional Minkowski metric. This evidently is of the form (3.1) with and so gives an 11-dimensional solution (fitting into the class of solutions found in [22]) with I now will deform the background (3.14) by carrying out three types of TsT deformations, using the formulae of (2.9) and carry the result over to 11-dimensional solutions.

AdS-AdS deformation
This is the deformation already analysed in [18]. The O(d, d) bivector has only non-zero component β αβ = − 1 2 λǫ αβ . In 10 dimensions this produces: which (after setting λ = 1 by a coordinate rescaling) corresponds to the extremal F1-NS5 brane solution in the near horizon limit of the NS5. In eleven dimensions the result is: (3.20) It was shown in [18] how to view this as a six-vector deformation of the λ = 0 solution, with Verifying this involves making a 5+6 dimensional split of the coordinates, with x i = (x α , x i ) corresponding to the directions on which E 6(6) acts. Under this decomposition, the five-dimensional metric and the generalised metric suffice to consider all physical degrees of freedom, as the twoand three-form field strengths are zero and the four-form field strength is dual (in five dimensions) to the derivative of the generalised metric. The λ-dependence on the generalised metric can then be shown to correspond to a six-vector deformation (in [18] this was done by first switching to spherical coordinates x i → (ρ, θ α ), and noting there was a natural 3+3 decomposition of the background in terms of (x α , ρ) and (θ α ). For such backgrounds, general formulae for the action of the six-vector deformation on the generalised metric can be straightforwardly obtained and applied -see the appendix of [18]).

Torus-torus deformation
I finally take the O(d, d) bivector to have only non-zero components in the torus directions. I label these as (y 1 , y 2 , y 3 , y 4 ), and consider the case β y 1 y 2 = λ. In 10 dimensions, this produces (3.25) The new 11-dimensional metric is ds 2 11 = 1 + λ 2 + ρ 2 1/3 (1 + λ 2 ) −2/3 ds 2 T 2 (12) + 1 + λ 2 + ρ 2 1/3 (1 + λ 2 ) 1/3 ds 2 AdS 3 + ds 2 T 2 (34) 26) and the field strength is This can again be viewed as a six-vector deformation, but is a more complicated case than the AdS-AdS deformation. Making again a 5+6 dimensional split, the directions on which E 6(6) acts are x i = (y 1 , y 2 , x i ). As the field strength F (4) and its dual have components proportional to Vol T 4 , this decomposition leads to a non-vanishing two-form field strength of the form (2.22). It can then be shown, using similar techniques to those applied for the AdS-AdS deformation in [18] that the five-dimensional metric, generalised metric and this field strength can all be viewed as a six-vector deformation of the λ = 0 solution (these fields suffice to capture all physical degrees of freedom), where the six-vector has components: Rather than describe the particulars of this example in detail, I instead move on to the general explanation of how these six-vector deformations are induced.

TsT to six-vector deformations: general case
where Dx α = dx α + A µ α dx µ . Using the general expression (2.21), the E 6(6) invariant metric is then is the O(2, 2) invariant dilation. As G µν is also invariant under O(2, 2), it follows that g µν is likewise invariant.
I now focus on the E 6(6) generalised metric encoding the six-dimensional components of the metric, three-form and six-form. Explicitly, these ingredients are: where I let B αβ ≡ Bǫ αβ as before. The generalised metric components can then be straightforwardly, if tediously, computed. In order to give the results in a usable fashion, it helps to remember that the dependence on the x i coordinates must enter through a twist by a trivector, according to the SL(5) EDA generalised frame (2.15), (2.17). This is easily embedded in E 6(6) .
(This is not a solution on its own.) This can in turn be factorised as where G and U C are defined as in (2.19) with This generalised metric has the following block structure: The structure of the generalised metric becomes clear on identifying the following tensor product structure: where H ρ as defined in (2.10) is one of the SL(2) generalised metrics appearing in the factorisation which also naturally appears in the above expressions. The SL(2) τ T-dualities are geometric in nature and do not act in an interesting manner. I therefore consider the fate of generic SL(2) ρ T-dualities.
In the above decomposition, these act on H ρ , and leave invariant |G| −1/2 G αβ , the generalised dilaton, and hence the rest of the E 6(6) generalised metric M AB .
A general SL(2) ρ transformation acting on H ρ embeds into an E 6(6) transformation of M AB of the form: N , which has the form: whereπ ijk = + 1 3! ǫ ijkl 1 ...l 3 π l 1 l 2 l 3 . The O(2, 2) bivector shift (2.11) is the case a = 1, d = 1, b = 0, c = −λ. This is the special case where O M N in (3.45) is constant, and corresponds to a E 6(6) -valued six-vector transformation of the form be seen to correspond to the following shifts of the three-form and six-form: The case a = 0 = d, b = ±1, c = ∓1 corresponds to Buscher duality on two directions, and gives the non-trivial E 6(6) transformation: I could also consider the Buscher duality transformation on one direction, which swaps τ and ρ. In general, I would not expect this to be realisable as a E 6(6) transformation, as this is a transformation which exchanges type IIA and type IIB, and the solution generating map (1.2) specifically starts with a type IIA solution.
I next look at the transformation properties of the field strengths. I make the standard redefinition (A.11) of the field strengths. The components of the redefined field strengths carrying two five-dimensional indices are: where h µνρ and h µνα are obtained from the Kaluza-Klein decomposition of h IJK mimicking (2.7).
Using the expression for the E 6(6) -covariant two-form field strength given by (2.22), I can verify that these components appear in the following (twisted) field strength: The effect of the trivector twist is simply to induce a non-zero component Referring to ( (2) ρ T-dualities, these commute with the trivector twist defining the 11-dimensional solution, and so can be viewed as constant E 6(6) transformations. Nonetheless these transformations have an unusual character, in that they involve the would-be U-duality group E 6(6) acting in both isometric and non-isometric directions.

Algebraic interpretation of the deformation
I now discuss in more detail the geometric and algebraic characterisation of these six-vector transformations.
The six-vector deformation and E 6(6) EDA The non-isometric directions involved in the deformation are those of the four-dimensional geometry Mf 4 . This geometry has special characteristics resulting from the underlying algebraic structure. The failure of the geometry to admit translational isometries can be precisely characterised as follows. Denoting by g IJ the 7-dimensional block of the metric, and g ij the four-dimensional block, the vector fields v a = δ i a ∂ i , a = 1, . . . , 4, associated to translations in x i are not Killing but rather obey [11] The vector fields v a are associated with the Lie algebra which is selected as the physical subalgebra of the CSO(4, 0, 1) exceptional Drinfeld algebra. This corresponds to the abelian translational subalgebra of the Lie algebra of CSO(4, 0, 1). Given an 11-dimensional geometry admitting two commuting Killing vectors, k α , and the set of four vectors v a associated to the EDA structure, I group these together and let Then, the six-vector parameter appearing in the 11-dimensional solutions can be naturally identified with with r a 1 ...a 6 = λǫ a 1 ...a 6 . This gives a covariant formula for the six-vector deformation parameter.
In adapted coordinates for the isometries, and in the original coordinates used to define the 11-dimensional solution, this six-vector will be a constant. In other choices of coordinates, such as spherical coordinates on Mf 4 as used in the analysis of the AdS-AdS deformation in [18] it will not be constant.
The six-vector transformation can also be viewed as acting on the underlying CSO(4, 0, 1) exceptional Drinfeld algebra, embedded into an E 6(6) EDA. The E 6(6) EDA is constructed in [13].
I list the resulting brackets for the special case where the only non-zero structure constants appearing are those of a 3-algebra, f a 1 ...a 3 b , with the assumption that f a 1 a 2 b b = 0:  To connect transformations of the algebra with transformations of the geometry, I adopt the following general viewpoint. Given the E 6(6) factorisation of the generalised metric, as in equation (3.34), and the associated structure constants F AB C generated by the generalised frame (E EDA ) A , I can consider a solution generating transformation of the following form: for O A C a constant E 6(6) transformation. This in turn implies a transformation of the E 6(6) EDA algebra, with the generators transforming as  Letting alsof a 1 a 2 a 3 ,b ≡ 1 3! ǫ a 1 a 2 a 3 c 1 c 2 c 3f c 1 c 2 c 3 b , the algebra is: This implies that I can then take E ′ A = (E EDA ) A , i.e. I can construct the deformed background using the original frame, such that: This can alternatively be viewed simply as a transformation of the generalised frame which preserves the algebra: Either way, this then induces the transformations studied in the previous section.

Comparison with Yang-Baxter deformations
The six-vector deformation with parameter (4.3) brings to mind existing approaches in the literature involving polyvector deformations.
A basic example of this sort is the (homogeneous) Yang-Baxter deformation [4,[24][25][26], which has a particularly nice interpretation as an O(d, d) transformation corresponding to a bivector deformation [29,31,33]. 12 I review this following [9,33]. The starting point is a background admitting a (generically non-abelian) group isometry, with Killing vectors k a (corresponding to the right-invariant vectors of the group). Let r ab be a constant antisymmetric matrix and define the following bivector: The bivector deformation corresponds to the following solution generating mechanism. First carry out a constant bivector shift acting on the data that would appear after consistent truncation, in the same manner as (4.5): This corresponds to the following transformation of the generators of the Drinfeld double: (4.14) Assuming the homogeneous Yang-Baxter equation f de [a r b|d r |c]e = 0, this leads to a new algebra withf ab c = 2r d[a f cd b] . A new generalised frame E ′ A , of standard Poisson-Lie type, can then be constructed to realise this algebra, and is used to construct the new 10-dimensional deformed solution: The transformation (4.14) produces the following algebra: with This deformation therefore turns on 'R-flux' structure constants, so that the algebra is no longer of the Drinfeld double type. This sort of algebraic deformation has appeared previously in the literature. Normally it is required that the R-flux vanishes in order to stay within the confines of the Drinfeld double, for which the construction of associated generalised frames is known.
The latter is again a (not necessarily constant) bivector shift with β mn = r ab v a m v b n . This is therefore directly analogous to the six-vector deformation (4.3).
An explicit example of this type, i.e. a choice of r ab which gives an invariance of the algebra, is provided by: for some n c . Then R abc = 0 by the Jacobi identity forf ab c . One reason why I am spending time on this example is that the deformation (4.20) is what is needed to view non-abelian T-duals, or Yang-Baxter deformed backgrounds, as T-folds. The Tfold interpretation of such backgrounds was proposed in [37,38]. This generalises to the U-fold interpretation of the 11-dimensional backgrounds studied in this paper (discussed already in [18]). This uses a generalised frame involving a trivector linear in the coordinates. Indeed, returning to the SL(5) EDA (2.16) with f ab c = 0, I can apply the following transformation, corresponding to a trivector shift This leaves invariant the brackets involving T ′ a , but produces Hence this is generically a non-trivial deformation of the EDA. The new structure constant R a,bcde can be interpreted as an 11-dimensional R-flux [57]. However, for certain parameters r abc the deformation vanishes (this presumably can be phrased in terms of three-algebra cohomology).
For example, a constant shift of the coordinates x i appearing in the generalised frame corresponds to taking r abc =f abc d n d .

Discussion
In this paper, I developed further the generalised U-duality transformation (1.2), previously investigated in [12,18], mapping solutions of type IIA on S 3 to new 11-dimensional solutions. I The particularly interesting feature here is that the six-vector deformation involves an action of E 6(6) on isometric and non-isometric directions. Hence it could also be thought of as a generalised U-duality transformation, with the special property being that it commutes with the trivector twist used in constructing the solution (this trivector twist introduces the coordinate dependence which breaks the isometries on which the E 6(6) U-duality group would otherwise act).
This six-deformation also preserves the form of the underlying exceptional Drinfeld algebra. While the six-vector deformation was my main focus, I also briefly discussed how more general T-dualities result in trivector-dependent (and hence coordinate-dependent) E 6(6) transformations of the form (3.45). These could also be argued to be non-isometric or generalised U-dualities, A potentially important point here is that as these transformations do not act on isometric directions, it is not immediately clear that they should be regarded as being on the same footing as conventional T-dualities acting on toroidal backgrounds with abelian isometries. The precise nature of the T-/U-fold interpretations mentioned above is therefore an interesting question.
This paper considered supergravity (and algebraic) aspects of generalised dualities only. Their implications for brane or field theories is an intriguing direction for future work.
On the brane side, the expectation from the structure of the fluxes of 11-dimensional solution that IIA strings in M 7 should be identified with M5 branes wrapping the four-dimensional geometry Mf 4 in the new 11-dimensional setting. 13 So one can consider the question of whether or how properties of (deformations of) strings on type IIA are directly mapped to M5 branes in M-theory. It is well-appreciated for instance that bivector shifts can be suggestively recast as open string non-commutativity parameters (as for example noted or used in the Yang-Baxter context in different ways in [27,28,32,53,54]). A corresponding interpretation of the six-vector deformation (and also the trivector twist) in terms of M-theory five-branes and membranes would be interesting.
On the field theory side, the obvious questions concern the original example of AdS 3 ×T 4 ×S 3 .
In section 3.2 I noted that AdS-AdS and AdS-torus bivector deformations of AdS 3 × T 4 × S 3 gave the same geometry after a simple coordinate shift. This is commented on in [51], though there it is viewed as an accidental feature of the extremal solution. It would be interesting to understand whether this has any deeper significance. To this end I have included a short appendix C describing the chain of dualities under which the AdS 3 × T 4 × S 3 geometry is invariant. Now, the context for [51] is the study of the thermodynamic properties of the CFTs dual to the deformed spacetime geometry. This is of interest as TsT transformations of AdS 3 × T 4 × S 3 correspond to ('single-trace') TT and JT deformations of the dual CFT 2 [59][60][61] (the AdS-AdS deformation is TT , the AdS-torus one is JT orJT , and torus-torus example is JJ). It is only natural to then speculate about the existence of an analogous deformation of the dual CFTs of the 11-dimensional AdS solutions constructed by the generalised U-duality (1.2). These dual CFTs could be those analysed in [22], which are dual to a possible global completion of the local solutions produced by (1.2). A starting point could be to analyse the thermodynamics of the non-extremal version of the new 11-dimensional solution, which was already written down in [18].
generalised Lie derivative is defined by: (2) , L u ω (5) − ι v dλ (5) + ω (2) ∧ dλ (2) ) . (A.5) Comparing with (A.4) motivates decomposing the generalised vector V as whereV = (v,ω (2) ,ω (5) ) is gauge invariant. (This geometric 'twisting' is related to the definition of a choice of generalised frame [49] which naturally accomodates the gauge potentials on M d . The signs in [49] are slightly different as they take gauge parameters with opposite signs.) From this I read off the matrix U M N appearing in the factorisation of the generalised metric in terms of the p-form fields. It is given by: In order to describe the full 11-dimensional geometry, I need to make a split of the 11-dimensional spacetime as M 11−d × M d . Let x i denote d-dimensional coordinates on M d , and x µ denote the remaining 11 − d coordinates. The 11-dimensional metric is decomposed as: where φ ≡ det φ ij . The three-form and its four-form field strength are decomposed as follows: with implicit wedge products, and where the (p) subscript denotes an (11−d)-dimensional p-form.
U-duality mixes the components of the three-form and its dual. Hence C (6) and F (7) are expanded similarly to (A.10) and (A.11). In this appendix I describe in more detail the derivation of the map (1.2), following the logic of [12,18]. This involves a reduction of type IIA on a three-sphere to CSO(4, 0, 1) gauged supergravity in seven dimensions, following by uplift to 11-dimensional supergravity on the space Mf 4 . I restrict to the NSNS subsector of type IIA on a three-sphere. This leads to the expressions for the 11-dimensional solutions presented in section 3.1.
[42]. This uplift uses an SL(5) generalised vielbein represented here as a five-by-five matrix: where F ij ≡ δ i a δ j b F ab . I now need the parametrisation of these fields in terms of the components of the usual 11-dimensional fields. The SL(5) generalised metric appears here in the five-dimensional representation with where C i ≡ 1 6 ǫ ijkl C jkl and φ ≡ det φ ij . The SL(5) covariant field strengths can be identified with components of the 11-dimensional four-form and its seven-form dual as follows: (B.9) The signs are fixed in order that the SL(5) covariant Bianchi identities match those of 11dimensional supergravity.
I identify g IJ with the 7-dimensional SL(5) invariant metric appearing in the decomposition (A.9) (on replacing the indices µ, ν and i, j there with I, J and i, j here). To reconstruct the full 11-dimensional metric, I also require the uplift of the one-form gauge potential: where on the right-hand side I introduce the Kaluza-Klein vector and components of the threeform. Using the above expression for the SL(5) generalised metric, the internal four-dimensional metric can be extracted. Referring to the metric decomposition (A.9), this leads to the full 11-dimensional metric: ds 2 11 = Φ −5/4 + g 2 M kl x k x l 1/3 Φ 1/12 g IJ dx I dx J + M ij + Φ 5/4 g 2 M ip M jq x p x q Φ −5/4 + g 2 M kl x k x l 2/3 Φ −5/12 (dx i + gx r A ir (1) )(dx j + gx s A js (1) ) .

(B.11)
The generalised metric also delivers the internal components of the three-form: (B.12) From this and (B.10), two of the components of the four-form field strength in the decomposition (A.11) are determined, namely Here d (1) means the exterior derivative with respect to the seven-dimensional coordinates only, and L A (1) means the four-dimensional Lie derivative with respect to A i (1) . These can be computed more explicitly, but the expressions are not immediately illuminating. The dictionary for the field strengths, see (B.9), then determines other components of both the four-form and its dual: (B.14) The latter can be dualised to obtain F (4) . Alternatively, using J (4) , one finds where ⋆ 7 is defined using g IJ . The full 11-dimensional four-form field strength then follows according to (A.11).

(C.3)
• Bivector shift with β xy = γ followed by a coordinate transformation with t ′ = t + γy. This produces the background: The backgrounds (C.2) and (C.4) agree on identifying β = γ 2 . As U −1 β = U −β , this implies the following transformation is an invariance of the original background (C.1): This could be interpreted as a generalised Killing isometry acting on doubled coordinates as (C.7) The simplifed discussion above then embeds into the NSNS AdS 3 × S 3 × T 4 solution, for which with f (r) = r 2 . The above shows that this background is invariant under: TsT on (x, y i ) with parameter γ, coordinate shift on (t, y) with parameter γ, TsT on (t, x) with parameter γ 2 2 .