Poisson-Lie U-duality in Exceptional Field Theory

Poisson-Lie duality provides an algebraic extension of conventional Abelian and nonAbelian target space dualities of string theory and has seen recent applications in constructing quantum group deformations of holography. Here we demonstrate a natural upgrading of Poisson-Lie to the context of M-theory using the tools of exceptional field theory. In particular, we propose how the underlying idea of a Drinfeld double can be generalised to an algebra we call an exceptional Drinfeld algebra. These admit a notion of “maximally isotropic subalgebras” and we show how to define a generalised Scherk-Schwarz truncation on the associated group manifold to such a subalgebra. This allows us to define a notion of Poisson-Lie U-duality. Moreover, the closure conditions of the exceptional Drinfeld algebra define natural analogues of the cocycle and co-Jacobi conditions arising in Drinfeld double. We show that upon making a further coboundary restriction to the cocycle that an M-theoretic extension of Yang-Baxter deformations arise. We remark on the application of this construction as a solution-generating technique within supergravity. emanuel.malek@aei.mpg.de 2 d.c.thompson@swansea.ac.uk


Introduction
Abelian T-duality asserts in its simplest form that closed strings do not distinguish between circular target space geometries of radius R and 1 R measured in units of the string length scale. This result is a crucial thread of a wider tapestry of U-dualities that are expected to be displayed M-theory whose target spaces contain toroidal directions. This article will consider some generalised notions of Abelian T-duality and show how they are upgraded to analogous concepts within U-duality.
Non-Abelian T-duality [1] is a proposed dualisation of closed-string non-linear sigma-models (NLSM) whose target space admits the action of non-Abelian isometry group. Whilst the status of Non-Abelian T-duality in terms of the string genus expansion remains unclear, recent evidence [2] at two-loops provides confidence that the duality could remain robust to quantum (α ′ ) corrections on the worldsheet. What is absolutely clear is that in the context of holography at large N , where both string genus and α ′ corrections are suppressed, Non-Abelian T-duality can be a powerful solution generating technique as advocated first in [3][4][5][6] (see [7] for a review and further references) .
More radically Poisson-Lie (PL) T-duality [8,9] dispenses with the requirement of isometry of a target space but does assume some underlying algebraic structure given by a Drinfeld double. Here again, though the quantum corrections are far from understood beyond one-loop in α ′ , there have been a number of significant developments. By exploiting the close connection of PL T-duality and the classical Yang-Baxter equation, this has given rise to wide classes of new integrable NLSMs called η-model or Yang-Baxter sigma-models [10,11]. Most notably when applied to the AdS 5 × S 5 superstring this has lead to a marginal deformation that is expected encode a quantum-group deformation, with q ∈ R, of N = 4 Super-Yang-Mills gauge theory [12,13]. Parallel to this has been the realisation with the λ-model [14,15] of an analogous quantum-group deformation with q a root-of-unity; the η-and λ-models are related (at least in simplest bosonic setting where it has been spelt out explicitly) by a PL duality transformation combined with some analytic continuation [16][17][18]. Alongside η and λ-integrable deformations, non-commutative β-deformations realised by TsT transformations can also be thought of as a Yang-Baxter sigma-model [19][20][21].
Critical to us will be that the most natural understanding of Poisson-Lie T-duality and its associated target spacetimes is provided by the tools and techniques of Double Field Theory (DFT) [22] and generalised geometry [23][24][25]. In essence, Poisson-Lie models, i.e. models where Poisson-Lie duality can act, arise as a generalised Scherk-Schwarz reduction [26] in which all non-trivial coordinate dependance is encoded in a twist matrix, or generalised frame field, [27,28]. Through the Courant bracket, these generalised frame fields realise the algebra of  [29], thus far an algebraically robust description has been lacking. It is this that we address in the current article in the context of exceptional field theory (ExFT)/exceptional generalised geometry [30][31][32][33][34][35][36], the M-theoretic analogues to Double Field Theory/generalised geometry. Given the importance of generalised Scherk-Schwarz reductions in the DFT realisation of Poisson-Lie T-duality, it is natural to use the analogous ExFT structure to build a notion of Poisson-Lie U-duality. This is our goal in this paper.
In ExFT, a splitting is made of 11 − d "external" directions and d "internal" directions such that all field content and gauge symmetries are repackaged into representations of the exceptional Lie algebras e d,(d) . Our focus will be case of d = 4 such that the duality group is SL(5) and we will restrict our attention to the "internal" directions in which the Poisson-Lie U-duality will act.
We begin with a review of Poisson-Lie T-duality and its realisation in Double Field Theory in 2, before introducing the natural analogue of a Drinfeld Double, which for want of a better name we call an Exceptional Drinfeld Algebra in section 3. In section 4 we will show how a generalised frame can be introduced that reproduces the algebra of the Exceptional Drinfeld Algebra, allowing us to define a U-duality notion of Poisson-Lie duality. Finally, in sections 5 and 6 we develop methods for constructing an Exceptional Drinfeld Algebra starting from a Lie Group G, and thereby also suggest a U-dual version of Yang-Baxter deformations. Finally, we present some examples in section 7 and conclude in section 8 with a brief outlook.
Note added: While finalising this manuscript, the paper [37] appeared which proposes a Uduality extension of Drinfeld Doubles and has some overlap with our sections 3 and 4 in the unimodular case.

Review of Poisson-Lie T-duality
In this section we provide a brief recap of Poisson-Lie T-duality. We will flip the conventional exposition by starting with algebraic considerations to eventually arrive at an associated NLSM describing the NS sector of a closed string; this will serve as a road map for what follows.
Central to the construction is a (classical) Drinfeld double: an even-dimensional real algebra d with generators T A obeying [T A , T B ] = iF AB C T C equipped with a symmetric split-signature ad-invariant pairing η(·, ·) such that d admits at least one decomposition d = g ⊕g with g and g sub-algebras that are maximally isotropic with respect to η. Letting t a (t a ) be generators for The Jacobi identity for d imposes a compatibility constraint, which is better understood as demanding that δ(t a ) =f bc a t b ⊗ t c defines a one-cocycle on g valued in g ∧ g which, as a trivial consequence of the Jacobi identity forg, also obeys a co-Jacobi identity. Equivalent we may then speak of (g, δ) as defining a Lie bi-algebra.
The exponential of a Lie bi-algebra is a Poisson-Lie group, that is a Lie-group manifold G equipped with a Poisson bi-vector compatible with the group composition law and obeying the Schouten identity. Equivalent to this Poisson bi-vector is a one-cocyle on G valued in g ∧ g denoted by Π = Π ab g t a ⊗ t b which is constructed from the adjoint action of g ∈ G on d as follows: As a consequence of this definition, Π enjoys some useful properties including: in which we have introduced the left-invariant one-forms l = il a t a = g −1 dg. The dual vector fields to l a will be denoted v a . It will be useful in what follows to build from Π ab a second set of vector fields π a ≡ Π ab g v b . A modest calculation, appealing to the properties of eq. (2.4), shows that these objects realise the algebra of d given in eq. (2.1) under the Lie derivative We now upgrade this discussion to generalised geometry which concerns the generalised tangent bundle E given locally as T G ⊕ T ⋆ G. On E there is a generalised Lie (Dorfman) derivative which acts on two sections In particular we can introduce two sets of generalised vectors, which we package together by defining a set of generalised frame fields E A = (E a ,Ẽ a ) that under the Dorfman derivative furnish the algebra of the double d, We may now perform a generalised Scherk-Schwarz reduction of type II supergravity on G.
For this, introduce a set of We can use the generalised frame field to translate this constant generalised metric to one From this "curved" generalised metric, we can extract the metric tensor and B-field E ij = G ij +B ij on G which match the target space geometry of the Poisson-Lie σ-model with action [9] in which the left invariant one-forms on G have been pulled back to the worldsheet and ± signify light-cone coordinates. The would-be-Noether currents, associated to the G action generated by the vector fields v a are not conserved but enjoy a remarkable non-commutative conservation law, that is often called a Poisson-Lie symmetry.
We could of course swap the role of g andg in the entire discussion above constructing Π ab ,π a ,ṽ a ,l a as well as generalised frame fieldsẼ A . This results in a dual σ-model,S, defined onG = expg that is canonically equivalent to the first [9,38]. A set of Buscher rules for such a dualisation is easily formulated in the generalised geometry by starting with a curved space generalised metric H, undressing the frame fields E A to return to the flat space generalised metricH, performing an O(d, d) action that implements the swapping (T a ,T a ) ↔ (T a , T a ) and then re-dressing with the the generalised framesẼ A . The RG equations governing the two sigma models S andS are equivalent at one-loop and can be formulated in terms of a renormalisation toH. Should the target space of the original theory define a solution of the appropriate (super)gravity (or part thereof) under normal circumstances so too will the dual, and hence this procedure defines a solution generating technique. This is called Poisson-Lie T-duality.

Exceptional Drinfeld Algebra
In the ExFT approach to eleven-dimensional supergravity a split is made into a d dimensional "internal" space M and 11 − d "external" directions but importantly no restriction is made on the coordinate dependence of any fields and no truncation is assumed from the outset. This splitting enables a rewriting of the variables of supergravity in a way that makes manifest the E d(d) U-duality symmetry. In order to get efficiently to the core issues, our focus in this paper will be exclusively on the internal directions and we shall ignore both dependance of internal fields on external coordinates and external fields entirely -these extra modes will be spectators as far as the Poisson-Lie U-duality is concerned. Our approach is essentially a specialisation of the general construction of gauged supergravities via ExFT and the inclusions of these spectator modes is well discussed in the literature.
On the internal space the bosonic field content, namely the components of the metric tensor and three-form and dual six-form potentials, are packaged in to a generalised metric M AB where the index A runs over a representation R 1 of the duality group. As it is most amenable for direct calculation we will henceforth consider the case of d = 4 for which E d(d) ∼ = SL (5) and Let us now construct the analogue of a Drinfeld double which, for want of inspiration, we call an "Exceptional Drinfeld Algebra". This is a (n ≤ 10)-dimensional subalgebra of sl(5) ⊕ R + which we write in a 10-dimensional SL(5) covariant manner, such that the generators are represented by with T AB = T [AB] and A, B = 1, . . . , 5. However, as we will will make explicit later, typically n < 10 so the generators T AB are not all linearly independent. Since d is a subalgebra of sl(5) ⊕ R + , we can introduce the sl(5)-invariant ǫ ABCDE which is left invariant up to scalings. The algebra of d is a Leibniz algebra given by where in analogy to 7-d gauged maximal SUGRA, the structure constants, also called the embedding tensor, are given by Here S AB is symmetric, τ AB is antisymmetric and T ABC D = 1 2 ǫ ABCEF Z EF ,D is traceless, i.e. Z [AB,C] = 0. This is, in general, not a Lie algebra because the structure constants need not be anti-symmetric under the interchange of the two sets of lower indices. Moreover, instead of the Jacobi identity, closure of g requires the quadratic constraint (also known as the Leibniz identity) We require that d contains a four-dimensional subalgebra g (corresponding group G) whose generators satisfy This condition plays the analogue of the maximally isotropic condition of Drinfeld doubles. We can always label these generators as t a ≡ T a5 with a = 1 . . . 4 and the remaining 6 generators byt ab ≡ 1 2 ǫ abcd T cd . Requiring closure under the adjoint action of g ∈ G on d implies that Moreover, since d ⊂ sl(5) ⊕ R + , this adjoint action must lie inside SL(5) × R + , i.e.
Finally, since g ⊂ d is a subalgebra, we must have is the adjoint action of G on g, and Ξ g and (λ g ) a are related to the R + action and the action of G ont ab , respectively.
Equations (3.7) and (3.9) imply the following properties for (λ g ) a : • λ a vanishes at the identity of G (λ e ) a = 0 . (3.11) • (λ g ) a inherits a group composition rule from that of the adjoint action which is reminiscent of the composition law obeyed by the Poisson-Lie bi-vector.
In the following, we will often drop the subscript g on Ξ g and (λ g ) a for simplicity.
The structure constants of the algebra d can now be related to λ a using where l = i l a t a = g −1 dg denotes the Maurer-Carten one-form as in section 2. We first use that g is a subalgebra of d with structure constants to identify We then find from (3.13) where we defined the derivative of λ a as dλ a = λ a,b l b .  18) and thus various relations between the structure constants (3.4), Ξ g , ad g and (λ g ) a which will be important for us in the following.

The Frame Fields and Embedding Tensor
The next step is to furnish the algebra eq. (3.3) via the generalised Lie derivative acting on a set of generalised frame fields. By analogy with the Poisson-Lie case of eq. (2.7) we will construct the generalised frame fields out of objects on d, in particular λ abc , as follows where λ abc = ǫ abcd λ d , α is a function on G and v a and l a are the left-invariant vector fields and one-forms on G.
Taken together these define generalised frame fields in the 10, ie. E AB = (E a , 1 2 ǫ abcdẼ cd ), that should obey with F AB,CD EF given in (3.4).
In addition the frame fields must define SL(5)×R + group elements and so can be decomposed in terms of objects in the5 representation: where under the generalised Lie-derivative ∆ has weight 1 2 . It is often easier to work with the E a , which are given by and must obey where θ AB,C D are the structure constants (embedding tensor) in the fundamental representation, given as We will now show that, with the λ a defined in section 3, via equation (3.9), and an appropriate function α that we will specify shortly, the E AB of (4.1) satisfy the differential conditions (4.7) Recall from (3.11) that λ a vanishes at the identity of G. Since the expressions (4.7) define structure constants, we can evaluate them at the identity where λ a = 0. This implies the following relation between the structure constants.
Therefore, to perform a Poisson-Lie U-duality, we must impose the further restriction on the algebra d that its structure constants (3.4) must satisfy (4.8).
With the structure constants related by (4.8), we can derive the following three equations 2 Here EA I has the natural weight 3 5 .
from the adjoint-invariance condition of the structure constants (3.18).
where K a = I a + 1 2 τ a5 . These relations together with (3.16) and for the structure constants related by (4.8) imply that the λ a introduced in (3.9) and for some constant α 0 , satisfy precisely the right differential conditions, i.e.

Poisson-Lie U-dualities
When an exceptional Drinfeld algebra admits two different subalgebras g andg satisfying the "maximal isotropy conditions" (3.6), we can perform a Poisson-Lie U-duality as follows. Firstly, note that the generators of g andg are necessarily related by some SL(5) transformation T A B .
Next, for both g andg, we can introduce Ξ, λ a and Ξ ′ , λ ′ a , respectively as shown in 3. We then construct the frame fields E A and E ′ A realising the algebra of the exceptional Drinfeld algebra based on g,g, respectively using our results in 4. Finally, we can perform a generalised Scherk-Schwarz reduction of 11-dimensional supergravity on G using the Ansatz where MĀB are the scalar fields of 7-dimensional maximal gauged supergravity and are thus constant on the internal space. We now perform a SL(5) transformation on the fields of the 7-dimensional supergravity, i.e. M ′ĀB = TĀCTBDMĀB , (4.13) and lift M ′ to 11-dimensional supergravity using the frame fields E ′ AB . Thus the dual background is described by the fields encoded in the generalised metric (4.14)

The quadratic constraint
If we are given a Drinfeld extension, as defined in section 3 with structure constants related by 4.8, we can use the results of the previous section to immediately construct the frame fields associated to it, and thus to perform a Poisson-Lie U-duality. However, in practice, we typically want to know when a given algebra g can be enlarged into a Drinfeld extension, and, in particular, how to define λ a and α and thus the frame fields (4.1) given g, so that we can perform a Poisson-Lie U-duality. To answer this question, we must first study the closure conditions of d, i.e. the quadratic constraints (3.5), to understand what conditions g imposes on the structure constants of d.
The quadratic constraints (3.5) imply that the irreducible representations of F AB,C D must • The compatibility conditions involving the g action on the remaining embedding tensor components, S ab and τ ab : in which we recall K a = I a + 1 2 τ a5 .
• A dual Jacobi condition, involving only S ab and τ ab and not the structure constants of g:

Dual structure constants
It is useful to introduce the "dual structure constants" These define a map:f : g −→ Λ 3 g, given bỹ Viewed this wayf defines a Λ 3 g * -valued Lie algebra 1-cochain.
Now the compatibility conditions (5.3) are equivalent to

Cocycle condition
The first equation in (5.7) is particularly interesting. Let us first focus on the case I a = τ a5 . Then (5.9) implies thatf abc d must be a Lie-algebra 1-cocycle, i.e.f must be closed under the Lie algebra differential The usual Lie algebra differential is nilpotent d 2 = 0, and thus the cocycle condition is solved by a coboundaryf = dm , (5.11) for some m ∈ Λ 3 g ⋆ , where dm(x) ≡ ad x m , ∀ x ∈ g . (5.12) Now let us turn to I a = τ a5 . Now, the modified cocycle condition (5.9) can be expressed in terms of the operator However, the operator d ′ is still nilpotent. Let for some m ∈ Λ 3 g ⋆ . Then it is easy to show that d ′2 = 0. Therefore, the modified cocycle condition (5.9) can be solved by a coboundarỹ for any m ∈ Λ 3 g ⋆ .

Coboundary Ansatz
As discussed above, the quadratic constraints lead to compatibility conditions between the structure constants of g andf abc d (5.7), which include the modified cocycle condition (5.9). Let us now focus on coboundary solutions to these equations, i.e. solutions of the form (5.16 In terms of the irreps τ ab and S ab , the Ansatz (6.1) is Now the first equation of (5.7) is automatically solved and the remaining compatibility conditions impose the following restrictions on m a : Now let us construct the object λ a satisfying the first-order constraints (4.11) with S ab and τ ab given as in (6.2), as well as the group composition properties (3.11) and (3.12). It is given by which clearly satisfies (3.11) and (3.12). Moreover, we can readily calculate the Lie derivative 6) and find that at the identity where λ a = 0 as required by (6.2).
Now we note that (4.11) is satisfied precisely when(4.9) holds. However, as shown in section 3, these equations follow from adjoint-invariance of the structure constants, and thus from integrating the quadratic constraints (3.5) over G. Since we have already satisfied the quadratic constraints (3.5) and ensured that λ a transforms the right way on G, our Ansatz (6.5) satisfies (4.9) and thus (4.11).
Finally, for a given group G, we can integrate dΞ = 1 2 l a τ a5 Ξ to find the function Ξ appearing via α, see equation (4.10), in the frame fields (4.1).
Given the analogy with Poisson-Lie T-duality, it is tempting to speculate that the matrix m abc defined above is related to an M-theoretic analogue of Yang-Baxter deformations. In particular, one may want to view m : g −→ Λ 2 g ⋆ and defining an associated 3-bracket. The appropriate notion of closure of the 3-bracket should then be (5.8). There should then be a simple condition involving two powers of m and one of the structure constants of g that ensures closure. This would be the analogue of the classical Yang-Baxter equation. We leave a detailed study of this question for future work.

Some examples
Let us now consider some examples of the co-boundary case.
The most obvious example here is g = su(2) + u(1) in which we choose T 4 be the u(1) Let us consider g = (II) + u(1) where (II) is the Bianchi II or Heisenberg-Weyl algebra for which the only non-vanishing commutator is We can choose a parameterisation of (II) as and a corresponding group element (in which the overall phase accounts for the U (1) factor) {0, 0, 0, k 4 } {0, 0, 0, 0} × − − − (7.1) Table 1: Solutions for g = (II)+u(1) in which × indicates a non-zero embedding tensor component The function Ξ g = e 1 3 θk 4 −xk 2 −zk 3 which enters into the co-boundary ansatz for λ is compatible with the group multiplication law and the differential constraint dΞ g = 1 2 l c τ a5 Ξ g . The one forms and dual vectors are given as The various constraint equations discussed above admit a set of solutions, illustrated in table 1, leading to constant embedding tensor components.

Discussion and outlook
This work opens a number of interesting lines.
As indicated in section 4, by a procedure of undressing, SL(5) action and redressing, one has a map between two generalised metrics giving rise to same lower dimensional supergravity theory and one should like to exploit this as a solution generating technique. An exciting opportunity is to explore pragmatic usages of this technique in constructing new holographic supergravity solutions, a programme which becomes richer as the dimensionality of the internal space is increased.
A immediate mechanical task is to scan through the four-dimensional Lie-algebras (see e.g. [39]) and classify all the Exceptional Drinfeld Algebras d ⊂ SL(5) × R + thereby providing an exceptional analogue of the classification of six-dimensional Drinfeld doubles [40]. The framework proposed above should admit a ready generalisation to the other exceptional generalised geometries based on e n,(n) , at least for n ≤ 6, though the details should be worked out. New features occur at larger n: at n = 6 a new object λ abcdef would enter and modify the structure; at n = 7 a mixed symmetry object λ a 1 ...a 7 ,a 8 must be considered; at n = 8 the generalised Lie derivative alone doesn't close and compensating additional shift symmetries must be incorporated. Beyond n = 8 one is dealing with infinite dimensional algebras which would be exciting to investigate in this context. Beyond SL (5), when five-branes should be considered this is likely to be more involved.
The Poisson-Lie scenario described in section 2 can be generalised in two important ways.
First, the assumption of a Drinfeld double can be relaxed to only require a single isotropic subgroup, this set-up captures WZW models as well as their integrable λ-deformations. Second, one can take a reduction of the Poisson-Lie model to "dressing cosets" which allows for target spaces that are cosets rather than groups (required to define e.g. integrable Yang-Baxter type deformations of AdS 5 × S 5 ). One would hope to situate both of these generalisations within the exception generalised geometry setting.
Here we have seen that how the components of the embedding tensor are realised as encoding structure constants of an exceptional Drinfeld algebra. One can thus reduce on the geometry we construct to a lower dimensional gauged supergravity theory. Here the resultant theories obtained after Scherk-Schwartz reduction will be maximally supersymmetric, and so a natural question related to the preceding paragraph, is to obtain the half-maximal or lower supersymmetric analogue construction exploiting the ideas in [41,42]. The converse question is interesting; under what circumstances can a lower dimensional gauged supergravity theory be uplifted to an exceptional Drinfeld algebra?