Non-Abelian U-duality for membrane

T-duality of string theory can be extended to the Poisson-Lie T-duality when the target space has a generalized isometry group given by a Drinfel'd double. In M-theory, T-duality is understood as a subgroup of U-duality, but the non-Abelian extension of U-duality is still a mystery. In this paper, we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal{E}_n$ algebra. This provides a natural setup to study non-Abelian U-duality because the $\mathcal{E}_n$ algebra has been proposed as a U-duality extension of the Drinfel'd double. We show that the standard treatment of Abelian U-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian U-duality still exists in the non-Abelian extension.


Introduction
Abelian T -duality is a symmetry of string theory when the target space has D commuting Killing vector fields. This T -duality can be extended to the Poisson-Lie (PL) T -duality [1,2] when the target geometry has a certain symmetry generated by the Lie algebra of the Drinfel'd double. For the PL T -duality, the usual Killing vector fields are not necessary and we can consider the extended T -duality in a more general class of target spaces. Similar to Abelian Tduality, the PL T -duality is a symmetry of the supergravity equations of motion (see e.g. [3]), and it can generate various supergravity solutions (see e.g. [2,[4][5][6][7][8][9][10][11][12][13][14][15][16]). To be more precise, some dual geometries do not solve the standard supergravity equations but rather the modified ones, known as the generalized supergravity equations [17,18]. However, as shown in [19,20], the generalized supergravity equations can be derived from double field theory (DFT) [21][22][23][24][25], which is a T -duality-manifest formulation of supergravity. Thus, now the PL T -duality has been understood as the symmetry of DFT [14,[26][27][28][29]. Recently, various aspects of the PL T -duality, in particular, its relation to the Yang-Baxter deformation [7,[30][31][32][33]  According to recent developments in the U -duality-manifest formulation, known as the exceptional field theory (EFT) [34][35][36][37][38], the U -duality symmetry in supergravity has been clearly understood. EFT also exhibits the duality between M-theory and type IIB supergravity, and it also provides a useful framework to study various non-geometric backgrounds or non-trivial compactifications. Recently, by using EFT, a U -duality extension of the PL T -duality has been discussed in [39,40] for n ≤ 4, where the Drinfel'd double is realized as a subgroup of the proposed E n group. There still remain many things to be clarified, but it has been expected that this E n group is the symmetry underlying the non-Abelian extension of U -duality. In contrast to the success in supergravity, U -duality symmetry in membrane theory remains to be mysterious. In the case of string theory, the equations of motion in a flat space have been successfully expressed in a T -duality-covariant form [41]. By closely following this approach, the equations of motion for a membrane have been expressed in a U -duality-covariant form in [42] (see also [43,44]). However, as it has been pointed out in [45], a certain integrability is broken under a general U -duality transformation, and it has been concluded that only a subgroup of Abelian U -duality is the symmetry of the membrane equations of motion. Only when the dimension of the target spacetime is n = 3 (where the membrane is spacefilling and called the topological membrane), the full SL(2) × SL(3) U -duality group can be consistently performed [45] (see also [46][47][48] for related earlier works).
In this paper, focusing on the successful case n = 3, we investigate non-Abelian U -duality in membrane theory. Our main results are as follows. In the T -duality-covariant formulation of string theory, the displacement dx m (σ) ( ) and it can be locally express as P I = dx I .
This paper studies its extension to the case where the target space is curved. By requiring the target space to have a symmetry of the E n algebra, we show that the generalized displacement the structure constants of the E n algebra. The MC equation does not depend on the choice of the generators T A in the E n algebra, and it is manifestly covariant under the change of . This arbitrariness in the choice of generators is what we call non-Abelian U -duality. It naturally unifies the PL T -duality and Abelian U -duality.
For clarity, we here note several subtleties. In string theory, the number of the equations of motion for the scalar fields x m is D, and the number of non-trivial MC equation is also D. However, the situation is completely the same as Abelian U -duality. Our result is a natural non-Abelian extension of the standard Abelian U -duality, and the fact that the generalized displacement P A satisfies the MC equation is non-trivial.
The structure of this paper is as follows. In section 2, we review Abelian T -duality and U -duality. The famous issue in U -duality is also reviewed. In section 3, we discuss the non-Abelian extensions. The PL T -duality in string theory is reviewed in section 3.1. In section 3.2, we discuss non-Abelian U -duality in membrane theory. Section 4 is devoted to the summary and discussion. Technical details are given in Appendices.

Abelian T -duality and U -duality
In this section, we review Abelian T -duality in string theory. We also review Abelian U -duality in membrane theory and explain a notorious issue specific to U -duality.

Abelian T -duality in string theory
In order to perform Abelian T -duality, the target space needs to have Abelian Killing vectors.
Thus, we here consider string theory in a D-dimensional flat space, where the supergravity fields are constant. The string equations of motion can be expressed as We also have a trivially conserved current, and Abelian T -duality can be understood as a permutation of the equations of motion and the Bianchi identities [41], If we introduce the generalized metric we find that the 1-form fields J M (σ) satisfy the self-duality relation, In order to keep this relation under the rotation (2.3), H M N also should be transformed as This shows that, under the equations of motion, we can locally express the 1-form fields as The scalar fields x M (σ) are interpreted as the embedding functions into a 2D-dimensional doubled space and P M = dx M is interpreted as the generalized displacement.

Abelian U-duality in membrane theory
In [42], the same idea has been applied to membrane theory in a flat space. By following [42,43], we consider the dynamics of a membrane in an n-dimensional Lorentzian spacetime (n ≤ 4).
Similar to the string case, the equations of motion are expressed as where i, j = 1, . . . , n and * is the Hodge star operator associated with the induced metric The trivially conserved current, known as the topological current is defined as J ij ≡ dx i ∧ dx j and we consider a combination, (2.14) Definitions: In order to see that the matrix C I J is restricted to the E n U -duality group, let us make several definitions. The generalized metric in EFT, 1 which is a U -duality-covariant combination of supergravity fields, is defined as where g i 1 i 2 , j 1 j 2 ≡ g i 1 [j 1 g j 2 ]i 2 and the inverse is denoted as We also introduce the U -duality-invariant tensor η IJ;K , where K denotes the index for the socalled R 2 -representation of the E n group that can be decomposed as (η IJ;K ) = (η IJ;k , ) .
For n ≤ 4, they are explicitly defined as [49] jp] . As discussed in [50], in order to formulate membrane theory in a U -duality-covariant manner, it is important to introduce the charge vector for a membrane We then define a 1-form, which we call the η-form, as The membrane charge vector q I transforms covariantly under the U -duality transformation (2.15), and accordingly, the η-form also transforms covariantly as When we consider the M2-brane (without any M5-charge induced), the charge vector should be chosen as [50] and then the η-form becomes We also define a matrix which corresponds to the matrix (2.4) defined in string theory, although here it is a 1-form.
E n U -duality: By using the above definitions, we find the self-duality relation, Similar to the string case, we introduce 1-form fields P I (σ) as Then, from the relation (2.25), we obtain the self-duality relation for P I , This corresponds to the relation (2.10) in string theory. Since J I and H IJ transform as under U -duality transformations, the 1-form P I (σ) should be transformed as The U -duality-covariant equations of motion dJ I = 0 can be also expressed as An issue specific to U -duality: So far, everything is parallel to the string case. However, as it has been pointed out in [45], the transformation (2.29) generally causes an issue. Here, we explain the issue by following the presentation given in [50]. By using the self-duality relation (2.27), the equations of motion (2.30) are equivalent to Unlike the string case, H I J is not invertible and they are not equivalent to dP I = 0 . To be more precisely, the equations of motion are weaker than the (Abelian) MC equation dP I = 0 . Indeed, in [45], an explicit solution of membrane theory where the equations of motion (2.31) are satisfied but dP I = 0 . By the definition of P I given in Eq. (2.26), the first component P i trivially satisfies dP i = 0, but for the second component P i 1 i 2 , dP i 1 i 2 = 0 is not ensured.
Let us suppose that we have a solution x i (σ) satisfying dP i 1 i 2 (σ) = 0 . Under a particular U -duality transformation, 2 The GL(n) matrix contained in the En group has the form |det(Λi j )| , but since we are using the generalized tensors with the effective weight 0, the determinant factor is dropped out.
with Λ i j and c ijk constants, the (constant) supergravity fields are transformed as At the same time, the 1-form fields are transformed as This shows that is a solution of membrane theory in the dual geometry, and the (geometric) U -duality (2.32) always maps a solution to the dual solution. On the other hand, a serious problem happens if we consider the (non-geometric) Ω-transformation, After the Ω-transformation, we obtain By assumption, we have dP i 1 i 2 = 0, and thus dP ′i (σ) = 0 . This shows that we cannot parameterize the dualized 1-form field P ′I (σ) as because the integrability dP ′i = d 2 x ′i (σ) = 0 is now violated.
In general, E n U -duality transformations (for n ≤ 4) are generated by the geometric transformations (2.32) and the Ω-transformation (2.36), but only the former preserve the integrability. Thus, it is concluded in [45] that only the geometric subgroup of U -duality is the (classical) symmetry of membrane theory. A resolution has been discussed in [50], but even in their approach, it is impossible to realize the full MC equation dP I (σ) = 0. Therefore, unlike the string case, we cannot express the 1-form as In summary, the point is that the equations of motion H I J ∧ dP J = 0 are weaker than the MC equation dP I = 0 and we cannot realize P I (σ) = dx I (σ) even under the equations of motion. Accordingly, unlike the string case, we cannot interpret that the membrane is fluctuating in an extended spacetime with coordinates x I .
An exceptional case where n = 3: As discussed in [45], the case n = 3 is exceptional.
There, the membrane is called the topological membrane because it is non-dynamical. Indeed, by using the identity the equations of motion (2.13) are identically satisfied. Moreover, as it is clear from is also identically satisfied, and at least locally, we can express the 1-form as Here, x I describes the embedding of the membrane into the 6-dimensional extended space.
We can freely rotate a given solution x I (σ) as . Therefore, these coincide only when n = 3 (see [47] for a similar discussion). For n ≥ 4, we cannot expect to obtain the full components of dP ij = 0 . If any component of the MC equation is not satisfied, we obtain dP ′i = 0 after a certain Ω-transformation, and the integrability is broken.
Of course, as it is discussed well in EFT, at the level of supergravity, the Lagrangian or the equations of motion have the E n U -duality symmetry for an arbitrary n ≤ 8 (or perhaps n ≤ 11). The issue arises only when we try to realize the symmetry in membrane theory. A membrane is only a member of the supersymmetric branes, which form a U -duality multiplet.
In order to realize the full U -duality symmetry, we will need to formulate a brane theory which describes all of the supersymmetric branes in a unified manner (see section 4 for more discussion on such formulation). At present, such a formulation has not been found, and we can realize the U -duality symmetry only for the topological membrane. Accordingly, as we discuss below, we can realize non-Abelian U -duality only for the topological membrane.

Non-Abelian T -/U -duality
In this section, we study the non-Abelian extension of U -duality.

PL T -duality in string theory
Before studying non-Abelian U -duality, we review the PL T -duality in string theory [1,2].
PL T -dualizability: In order to perform the PL T -duality, the target geometry is required to satisfy the differential equations [1, 2] As discussed in [1,2], Eq. (3.1) suggests that f ab c andf ab c can be identified with the structure constants of the Lie algebra of the Drinfel'd double, This is sometimes expressed as [T A , T B ] = F AB C T C by denoting the set of generators as In addition, an ad-invariant 3 bilinear form is defined for the generators, We denote a subgroup G generated by {T a } as the physical subgroup while a subgroupG generated by {T a } as the dual group. If we assume that the target space is a group manifold of G and identify the vector fields v m a with the left-invariant vector fields, we can solve the differential equation (3.1) as follows [1]: Here,Ê ≡ (Ê ab ) is a constant matrix and several quantities are defined as follows. For a group element g(x) ∈ G, we define a matrix M A B (x) as From the structure of the algebra (3.3), the matrix M A B (x) can be generally parameterized as follows by using two matrices a a b (x) and Π ab (x) = −Π ba (x): The left-and right-invariant 1-forms are denoted as Here,Ĥ AB is a constant matrix associated withÊ ab ≡ (ĝ +B) ab (ĝ ab ≡Ê (ab) ,B ab ≡Ê [ab] ) and the coordinate dependence is contained only in the twist matrix E M A (x), We note that the inverse of the twist matrix, denoted as E A M , is known as the generalized frame fields and, in fact, they satisfy the relation [26] where£ denotes the generalized Lie derivative in DFT. Then, we can show that the generalized metric satisfies the equation which shows that the target space has the symmetry of the Drinfel'd double.
Now, we rewrite the equations of motion (3.2) into a T -duality-manifest form. Similar to the Abelian case, we define 1-form fields which reduce to Eq. (2.8) in the Abelian case (where ℓ a m = δ a m and v m a = δ m a ). For convenience, we also define By further acting the adjoint action, we define Eq. (3.2) suggests that, under the equations of motion, J can be identified with the rightinvariant 1-formr ≡ dgg −1 associated with a dual group elementg(x), and we obtain This shows that the 1-form field P(σ) is the right-invariant 1-form on the Drinfel'd double, which satisfies Similar to the Abelian case, the 1-form fields are subjected to the self-duality relation, and only D components are independent. Thus, the 2D MC equation PL T -duality: The PL T -duality (or the PL T -plurality [6]) is a symmetry under redefinitions of the generators Under the redefinition, the structure constants are transformed as By requiring that the redefined algebra is also a Lie algebra of the Drinfel'd double, the metric η AB [i.e. the bilinear form (3.4)] must be preserved Namely, the constant matrix C A B should be an element of the O(D, D) group. After the redefinition, we introduce new group elements g ′ (σ) andg ′ (σ), such that gg = l = l ′ = g ′g′ is satisfied. Then, we obtain P(σ) = dl l −1 = dl ′ l ′−1 ≡ P ′ (σ), or equivalently,  rotation, we again provide parameterizations of group elements, such as g ′ (x ′ ) = e x ′a T ′ a and g ′ (x ′ ) = ex ′ a T ′a , and then obtain the dual geometry In order to relate the two geometries, we require Then, in principle, we can find the relation between the two coordinates, Using this relation, we can map a string solution in the original geometry to the dual solution.
Another easier method is as follows. From a given solution x m (σ), we can easily compute the 1-form field P(σ). Expanding P by means of the redefined generators T ′ A , we obtain P ′A . Then, we can compute P ′A = M ′ A B P ′A , whose first component is P ′a = ℓ ′a . Solving the differential equation P ′a T ′ a = g ′−1 (x ′ ) dg ′ (x ′ ), we can find x ′m (σ) . Either way, we can map a solution x m (σ) to a new solution x ′m (σ) of the dual sigma model.

Non-Abelian U-duality
Here, we study the U -duality extension of the PL T -duality in membrane theory. We note that our analysis is restricted to n ≤ 4 .
The E n algebra: The PL T -duality is based on the Lie algebra of the Drinfel'd double, and similarly, non-Abelian U -duality will be based on a new algebra that extends the Lie algebra of the Drinfel'd double. Such an algebra has been recently proposed in [39,40] and we call it the E n algebra by following [39]. For n ≤ 4, the algebra is given by . For simplicity, we denote the algebra as Since the first two indices of the structure constants are not antisymmetric (i.e. F AB C = −F BA C ), this is a Leibniz algebra rather than a Lie algebra. The Leibniz identity, requires that the structure constants should satisfy [39] Similar to the Drinfel'd double, a U -duality-invariant metric has been defined as has the same matrix elements as Eq. (2.18).
Target space with the E n symmetry: Similar to the case of the PL T -duality, where the target geometry is expressed as (3.10), we can construct a target geometry as [39] H by using the E n algebra. Here,Ĥ AB is a constant matrix and E I A has the form The right-invariant 1-form r a i and its dual e i a has been defined by using a physical group element g(x), which we parameterize as g(x) = e x a Ta . In addition, similar to Eqs. (3.6) and (3.7), the tri-vector Π abc is defined as and we can show that the target geometry has the generalized isometry group, which is generated by the E n algebrâ If we introduce the dual metricg ij and Ω ijk through the non-geometric parameterization of the generalized metric (see for example [51][52][53]) where we have parameterized the constant matrixĤ AB as whereΩ abc andĝ ab are constants that are assumed to satisfy Then, we find that the dual fields satisfy [39] £ vagij = 0 , The target space constructed in this way is the setup to discuss non-Abelian U -duality.
In order to identify the standard supergravity fields, we make the following identification between the standard fields (g ij , C ijk ) and the dual fields (g ij , Ω ijk ) [51,52] (see also [42]): where the density factors are needed in order to remove the weight of the generalized metric.
From this relation, the standard supergravity fields (for n ≤ 4) are obtained as follows: where K −1 ≡ 1 + 1 3! Ω ijk Ω ijk and the indices of C ijk and Ω ijk are raised or lowered by the metric g ij and the dual metricg ij , respectively.
In terms of the standard fields, the relations (3.46) read 4 where ℓ a ≡ g ab ℓ b , and curved indices i, j of g ij and C ijk have been converted to the indices a, b by using v i a (e.g.
. We also note that the metric G ab ≡ e i a e j b g ij also satisfies Indeed, in n = 3, g ij ∝g ij and Eq. (3.50) is trivial. In n = 4, we can parameterize Ω ijk as Ω ijk =ε ijkl Ω l ε 0123 = 1 √ |g| (see Appendix A) and then we obtain Then, we obtain This shows the desired relation in n = 4, Thus, both in n = 3 and n = 4 (with f ab a = 0), G ab is an invariant metric, and we have

54)
Membrane theory: Now, let us consider membrane theory. In a general curved spacetime, the equations of motion for the scalar fields x i (σ) become where ǫ 012 = 1 . By contracting the free index i with a set of vector fields v i a , we obtain where we have defined In the target geometry given by Eqs. (3.37) and (3.48), by choosing the vector fields v i a as the left-invariant vector fields, the equations of motion become Note also that Eq. (3.58) reduces to the equations of motion (2.13) in the Abelian case, where f a bcd = 0 and ℓ a = δ a i dx i . For the manifest U -duality, we define a combination similar to the Abelian case (2.14). Similar to Eq. (2.26), we also define the Hodge dual of the 2-form field J A through where Then, we find that the 1-form fields have the form Then, we obtain Similar to Eq. (3.19), this satisfies the self-duality relation where we have defined the η-form as Equations of motion: By using an identity, the equations of motion (3.58) can be expressed as where we have used dℓ b = − 1 2 f cd b ℓ c ∧ ℓ d . We then consider a projection, To be more precise, when n = 3 we can show the equivalence without any assumption, but when n = 4 we need to assume f ab a = 0 . Then, the equations of motion (3.70) imply In fact, as we show in Appendix C, the relation (3.71), which is suggested by the equations of motion, is equivalent to the MC equation of the E n algebra, where F BC A are the structure constants of the E n algebra. Thus, in n = 3, the generalized displacement P A satisfies the MC equation, which generalizes the Abelian one given in Eq. (2.42). In n = 4, P A does not satisfy the MC equation similar to the Abelian case, and we cannot perform the full U -duality transformation.
Non-Abelian U -duality: In n = 3, non-Abelian U -duality is realized as a redefinition of the E n generators, where C A B is an element of the U -duality group SL(2) × SL (3). Under the redefinition, the structure constants are transformed as In order to keep the MC 1-form P invariant, the components should be transformed as The η-form is also transformed covariantly and by further transforming the constant matrix aŝ we can in principle determine the dual solution x ′i (σ) .

Discussion
In this paper, we have studied membrane theory in a curved background (3.37), which has the symmetry of the E n algebra. Similar to the case of Abelian U -duality, we can show that the generalized displacement P A satisfies the MC equation of the E n algebra only when n = 3.
Both the MC equation and the self-duality relation for P A are manifestly covariant under non-Abelian U -duality (3.73) (which is a redefinition of the E n generators) and we have naturally extended the standard story of Abelian U -duality to the non-Abelian setup. In n = 4, we face the difficulty already known in the Abelian case, and P A do not satisfy the MC equation even under the equations of motion.
In addition to the membrane, M-theory contains the M5-brane as well (see [50,54,55] for M5-brane theory in U -duality-covariant approaches). Again in the M5-brane theory, the equations of motion will not generally provide the MC equation. The only exceptional case will be n = 6, where the M5-brane becomes space-filling. There, the generalized displacement is extended as P A = P a , (see for example [50]), and the number of the non-trivial components of the MC equations is n! 2! (n−2)! + n! 5! (n−5)! , corresponding to P a 1 a 2 and P a 1 ···a 5 . The dynamical fields on the M5-brane are x i and the 2-form gauge field A αβ (α, β = 0, . . . , 5) and, naively, the number of the equations of motion coincides with that of the non-trivial MC equations when n = 6. 5 Thus, we expect that U -duality symmetry in the M5-brane theory can be realized for n = 6. For n > 6, the number of the equations of motion is smaller and the full MC equation will not be reproduced. In order to examine this possibility, it is important to construct the E n algebra for n = 6 or higher.
As it is well-known, when the (self-dual) field strength on the M5-brane is non-vanishing, the M2-brane is induced on the M5-brane. Then, the dynamics of the induced M2-brane will be described by the 2-form gauge fields A αβ on the M5-brane. For example, in section 6 of [56], the gauge fields are dualized to the embedding functions x i of the M2-brane, and the membrane action has been reproduced from the M5-brane action. Then, it is interesting to consider the following possibility. As we discussed in this paper, in n = 6, membrane theory does not have the E 6 U -duality symmetry. However, if the E 6 U -duality symmetry is realized in the topological (or space-filling) M5-brane theory, it is interesting to interpret the topological M5brane theory as the E 6 -covariant membrane theory. Since the M5-brane is space-filling x i will be non-dynamical, and only the gauge fields A αβ are dynamical, which describe the fluctuation of the membrane. If P A satisfies the MC equation, we can perform non-Abelian U -duality.
This approach may resolve the issue of U -duality in membrane theory. Moreover, in the approach of [50,55], gauge fields on the worldvolume are introduced as the diffeomorphism parameters along the dual direction in the extended spacetime. In other words, the gauge fields are interpreted as the fluctuation along the dual directions in the extended spacetime.
Since the number of diffeomorphism parameters along the dual direction is always the same as the number of the non-trivial components of the MC equations, naively we can expect that the MC equation is realized under the equations of motion even for higher n. For example, in n = 8, it will be impossible to realize the E 8 duality symmetry in M5-brane theory. However, there, the Kaluza-Klein monopole (KKM) is space-filling, and its worldvolume theory may have the E 8 U -duality symmetry. If so, it may be possible to regard the topological KKM theory as the E 8 M5-brane theory. We hope to work on this in the future.

Acknowledgments
The work by Y.S. is supported by JSPS Grant-in-Aids for Scientific Research (C) 18K13540 and (B) 18H01214.

A Formulas in n = 4
In n = 4, relations between the non-geometric fields and the standard fields are given as where the indices of C ijk and Ω ijk are raised or lowered with the metric g ij andg ij , respectively.

(B.2)
In n = 3, the tri-vector Ω ijk should be proportional toε ijk and we define From Eq. (3.48), we obtain the standard supergravity fields as where we have defined Using £ vaε ijk = 0 (which follows from £ vagij = 0), Eq. This allows us to simplify P ab as and we obtain Now, let us rewrite each term on the right-hand side. The first term is which is an identity in n = 3 .

C Maurer-Cartan equation for the E n algebra
In this appendix, for n ≤ 4, we show that given in Eq. (3.71) is equivalent to the U -duality-manifest MC equation By using the explicit form of F BC A , The MC equation (C.2) is equivalent to As we show later, the former follows from the latter. Thus, in the following, we show that the latter is equivalent to Eq. (C.1). To this end, we employ the following identities [39]: a a e a −1  where we have definedP a ≡ a b a P b ,Π abc ≡ a d a a e b a f c Π def . (C.14)