Membrane Duality Revisited

Just as string T-duality originates from transforming field equations into Bianchi identities on the string worldsheet, so it has been suggested that M-theory U-dualities originate from transforming field equations into Bianchi identities on the membrane worldvolume. However, this encounters a problem unless the target space has dimension $D = p + 1$. We identify the problem to be the nonintegrability of the U-duality transformation assigned to the pull-back map. Just as a double geometry renders manifest the $O(D,D)$ string T-duality, here we show in the case of the M2-brane in $D = 3$ that a generalised geometry renders manifest the $SL(3) \times SL(2)$ U-duality. In the case of M2-brane in $D=4$, with and without extra target space coordinates, we show that only the ${\rm GL}(4,R)\ltimes R^4$ subgroup of the expected $SL(5,R)$ U-duality symmetry is realised.


Strings, T-duality and Double Geometry
Some time ago [1], it was pointed out that strings moving in an D-dimensional space M D with coordinates X µ (τ, σ), background metric g µν (X) and 2-form B µν (X), could usefully be described by a doubled geometry with 2D-dimensional coordinates and doubled metric 1 (1. 2) The motivation was twofold; worldsheet and spacetime:

Worldsheet
In the case when M D is the D-torus T D , this renders manifest the O(D, D) T-duality by combining worldsheet field equations and Bianchi identities via the constraint where Ω M N = 0 δ µ β δ α ν 0 , (1.4) and γ ij is the worldsheet metric.

Spacetime
In the case when M D is a generic manifold, the 2D-dimensional diffeomorphisms with parameter ξ M = (ξ µ , λ α ) suggest a way of unifying 2 D-dimensional diffeomorphisms and 2-form gauge invariance After all, G M N is just the Kaluza-Klein metric with spacetime metric g µν , gauge field A µ a and internal metric g ab where the "gauge field" is B µα and the "internal" metric is g αβ . If this programme were successful one would expect the SL(D)/SO(D) coset of general relativity to be promoted to an O(D, D)/(SO(D) × SO(D)), as conjectured in [3,5].
In summary, the worldsheet goal of rendering manifest the string T-duality O(D, D) by doubling the coordinates was achieved successfully in [1] and a T-dual worldsheet action using the doubled coordinates was then constructed in [6]. However, there were missing ingredients in the spacetime approach: the generalized diffeomorphisms were subsequently supplied in [4,7] δG M N = ξ P ∂ P G M N + (∂ M ξ P − ∂ P ξ M )G P N + (∂ N ξ P − ∂ P ξ N )G M P , (1.8) and the section condition subsequently supplied in [8] Ω M N ∂ M ∂ N = 0 .
(1.9) (The need for the section condition has, however, been called into question [9,10].) Once these ingredients were included, it was possible also to build a generalised spacetime action for G M N . This activity came to be known as "Double Field Theory." For further developments and variations on this doubled geometry theme, in addition to those already cited, including "Generalised geometry" and the E 11 approach see, for example, [8,9, Branes, U-duality and M-theory Following [1], it was pointed out [43][44][45] that membranes moving in a (D ≤ 4)-dimensional space M D with coordinates X µ (τ, σ, ρ), background metric g µν (X) and 3-form B µνρ (X) could usefully be described by a geometry with [D + D(D − 1)/2]-dimensional coordinates Z M = (X µ , Y ρσ ) (1.10) and generalized metric where g αβγδ = 1 2 (g αγ g βδ − g αδ g βγ ) . (1.12) Once again, the motivation was twofold; worldvolume and spacetime:

Worldvolume
In the case when M D is the D-torus T D , the hope was to render manifest the M-theory U-dualities (using modern parlance) by combining worldvolume field equations and Bianchi identities. For example, the U-duality would be SL(5, R) in the case D = 4.
The restriction to D ≤ 4 arises because, just as the usual coordinates X µ correspond to momentum in the supersymmetry algebra, so the extra coordinates Y µν correspond to the M2 central charge. But for D ≥ 5, this is not enough, as shown in Table 2 in [43]. There is also the M5 central charge with corresponding coordinates Y µνρστ , which first appears in D = 5. In Appendix A, we illustrate the emergence of extra coordinates from central charges in the M-theory algebra for general D. For example, in the D = 7 case X µ , Y µν ,Ỹ µν ∼ ǫ µνρστ λκ Y ρστ λκ andX µ form a 56 of the U-duality symmetry E 7(7) .

Spacetime
If this programme were successful, one would expect the SL(D)/SO(D) of general relativity to be promoted not merely to O(D, D)/(SO(D)×SO(D)) but to E 8 /SO (16), with possible infinite-dimensional extensions involving E 9 , E 10 and E 11 as conjectured in [5,14,46]. Once again, however, the generalised diffeomorphisms, section conditions and U-invariant actions came later. This activity has become known as "Exceptional Field Theory." For subsequent developments and variations on generalized geometry in M-theory and U-duality see, for example, [17, 22-24, 26, 28-30, 47-59], where the 5brane and other extended objects were incorporated, as required for D > 4. The E 11 approach [14] goes further with infinitely many coordinates of which those associated with the M-theory central charges are but a subset.
In summary, in contrast with strings where both the worldsheet and spacetime approaches have been successful, the brane worldvolume approach seems problematical and, with the exception of [22,29], recent developments have tended to focus on the spacetime approach where the extra coordinates (1.10) and generalised metric (1.11) have proved valuable. In fact, the worldvolume approach has been questioned by Percacci and Sezgin [60], by Sen [61], and by Lukas and Ovrut [62]. They suggest that it works only for target space dimensions D = p + 1. In this case, the D!/((D − p)!p!) wrapping modes on a D-torus (D ≥ p + 1) and the D Kaluza-Klein modes are equal in number as in the case of a string. Sen argues that this equality is a requirement. If so, the D = 3 U-duality SL(3) × SL(2) might be expected, but the D = 4 U-duality SL(5), would not.
In any event, the need to include coordinates corresponding to central charges in the M-theory algebra exposes a major difference between U-duality in M-theory and Tduality in string theory. In string theory, T-duality takes strings into strings, but in M-theory U-duality mixes up p-branes with different p. It seems unlikely, therefore, that the M2-brane worldvolume alone is sufficient. Somehow the totality of p-brane worldvolumes must conspire to give the full U-duality. This remains an unsolved problem.
Finally we note that the purpose of extra coordinates in both string and M-theory is to render the T and U dualities manifest. If one is content with non-manifest Tduality, one may invoke the Gaillard-Zumino (GZ) approach [63], as was done in [64]. The GZ approach to U duality is discussed below.

This Paper
This paper is devoted purely to the worldvolume approach. We shall show: • There is a problem with SL(5, R), which manifests itself both in the GZ approach (which doesn't introduce extra coordinates), as well as in approaches in which extra coordinates are introduced [43,65]. In the GZ approach, as well as the approach of [43], we shall show that the obstacle to the realisation of SL(5, R) symmetry is the nonintegrability of the transformation rule for the pull-back map. In the approach of [65], we shall show that the proposed manifestly SL(5, R)-invariant equation for a membrane in a target based on generalized geometry does not support linearized fluctuations about a Poincaré invariant vacuum solution.
• In the case of topological membranes, the SL(2, R) symmetry is known but we shall formulate it in a double geometry setting.
• We shall rederive the result that the membrane in d + 3 dimensions has a Heisenberg subgroup of the SL(2, R) symmetry [60], by making use of simple integrability considerations.

Topological Membranes
Although p-branes in p + 1 dimensions carry no dynamical degrees of freedom [66], they are nevertheless of considerable interest. In the present context, they provide us with a setting in which we can get a handle on duality symmetries that transform field equations and Bianchi identities into each other. Everything we will do here applies to topological p-branes for general p, but for simplicity in notation as well as our special interest in M 2branes, we shall focus on topological membranes. The standard action for the closed membrane is where in our conventions ε 012 = +1. For the topological membrane we take µ = 0, 1, 2 3 The SL(3) is manifest. For simplicity, we shall take the metric tensor g µν and 3-form potential B µνρ in the 3-dimensional target space to be constant 4 . In this case, using the algebraic field equation γ ij = ∂ i X µ ∂ j X ν g µν , we have the identity Therefore, letting B µνρ = √ −gε µνρ B, the action can be written as We shall, however, use the form (2.1) of the action below, motivated by the fact that this form will make it easier to compare with what happens in the case of the non-topological membrane. The resulting equations are There is also a conserved topological current: Because we are considering a target space that is three dimensional, we can make the definitions Noting that ε ijk ε µνρ ∂ j X ν ∂ k X ρ = −2(det ∂X)γ ij g µν ∂ j X ν and that det ∂X = |γ| 1/2 |g| −1/2 , we can write P i µ and J i µ as follows From these equations we find Note that this equation readily follows from the form of the action given in (2.3). We may now consider linear GL(2, R) = SL(2, R) × R transformations of the form 5 (2.10) 4 This is an important assumption. Otherwise the modification in equation (2.4) will obstruct the sought after duality symmetry. The spacetime background here can be viewed as being a subsector a time-like dimensional reduction of the bosonic sector of D = 11 supergravity down to 8 dimensions, where only the fields (gµν , Bµνρ) are kept, and the 8 dimensional Euclidean coordinates are taken to be constants. In a spacelike dimensional reduction, the signature of the metric gµν would be Euclidean, and we would take the 8 dimensional spacetime coordinates to be constant. All of our considerations apply for this case as well. 5 It is understood that there also exists the trivial SL(3, R) × R 3 symmetry realised as δP i µ = Rµ ν + S (µ) δ ν µ P i ν and δJ i µ = Rµ ν + S (µ) δ ν µ J i ν , where Rµ ν are real traceless matrices and S (µ) are the real scaling parameters.
We see that (2.9) is left invariant provided that |g| and B transform such that This can be seen from (2.9), noting that it implies C1l = P J −1 . It represents the infinitesimal form of a fractional linear transformation of the real variable C, and gives a representation of the algebra SL(2, R) × R. The fact that this symmetry acts on a combination of g and B is a consequence of the fact that the target spacetime is Lorentzian, as noted in [60].
To make the SL(2, R) symmetry manifest, we introduce a doubled system of coordinates Z aµ , with a = 1, 2, such that X µ = −Z 1µ and This doubling of coordinates is in accordance with the generalized target space geometry recently studied in [69] for maximal supergravity in eight dimensions. Using these definitions, it follows that (2.9) can be written as where transforming by conjugation under SL(2, R). Note that det G ab = −1, such that the product (det G ab )(det g µν ) = 1. Denoting the inverse of this metric by G ab , it transforms under infinitesimal SL(2, R) transformations as δG ab = Λ a c G cb + Λ b c G ac , with Λ as given in (2.10). Written out, this gives Using these rules, the transformation of −|g| 1/2 (1 + B) indeed gives the result (2.12). Demanding manifest SL(2, R) invariance thus fixes the separate variation of g and B under SL(2, R), not just its combination (2.11).
It is important to note that the SL(2, R) transformation under which (∂ i Z 1µ , ∂ i Z 2µ ) forms a doublet is embedded into (2.10) with a field-dependent scale transformation with parameter as can be determined from (2.13).
Turning to the key equation (2.14), it can be written as P ab ∂ i Z bµ = 0, where P ab = 1 2 (G ab − ǫ ab ) is a projector, and it amounts to Acting with ∇ i does not yield the field equation ∂ i √ −γγ ij g µν ∂ j X ν = 0. However, the latter is identically satisfied for the topological membrane, since recalling that the target space metric is constant. It is instructive to perform a double dimensional reduction [70] of (2.14). To this end, we letĝμν where g µν , φ and B are constants. Note that It follows that (2.14), or equivalently (2.18), holds for the topological string, where the indices now run over two values, namely, i = 0, 1 and µ = 0, 1. Settingî = 2 andμ = 2 in (2.18) fixes Z 22 , giving it a linear dependence on the coordinate σ 2 . Comparing (2.14) for the topological string with (1.3), they are in fact, contrary to appearance, the same equation, with the identification x µ = −Z 1µ and y µ = −ε µν Z 2ν . This can be seen by where we have used the formula for the determinant of ∂ j Z 1ν in the second equation. From this we conclude showing the equivalence of (2.14) and (1.3), up to a relative sign which can be attributed to convention choices.

Membrane in D = 8
The M2-brane action in D = 8 can be obtained from the M2-brane action in D = 11 by dimensional reduction on 3-torus. The bosonic sector of such a reduction has been studied in [62] where SL(3, R) × SL(2, R) symmetry could not be established. In a different approach aiming at a direct construction of an M2-brane action in D = 8 which couples to all the bosonic fields of the maximal supergravity theory in which the SL(3, R) × SL(2, R) symmetry is built in manifestly has been proposed [71]. However, the condition of SL(2, R) symmetry puts nonlinear constraints on the field which have been solved only in a fashion that exhibits a two parameter subgroup of SL(2, R) as a symmetry. More specifically, the bosonic sector of maximal D = 8 supergravity has the fields where the seven scalars parametrize the coset (SL(3, R)/SO (3)) × (SL(2, R)/SO (2)), the vector fields transform as (3,2) of SL(3, R)× SL(2, R). The field strength of the 3-form field is combined with the dual field strength for a doublet of SL(2, R). The gauge invariance of the pullbacks of the field strengths of the 1, 2, 3-form potentials requires the introduction of worldvolume fundamental potentials, resulting in the field strengths h r = db r − B r + · · · , g m = dc m − C m + · · · and f mr = dφ mr − A mr , where the underlined fields are the pullback of the target space forms. The action proposed in [71] then takes the form where λ is a Lagrangian multiplier field, G rs is SL(2, R) matrix parametrized in terms of the SL(2, R)/SO(2) coset scalars and Φ is a function of the field strengths (f mr , g m ). Duality relations for these field strengths are imposed by hand in addition to the field equations that follow from the action to ensure the correct number of propagation degrees of freedom, namely the 5 scalars coming from X µ and 3 scalars φ m1 . The resulting field equations have not lent themselves to a solution in general, however, and a special solution discussed in [71] breaks SL(2, R) symmetry. Our approach here is instead to consider a membrane propagating in D = 8 dimensions and coupled to the target space metric and 3-form potential only, and to study the duality symmetry of the standard membrane action. The background can be viewed as the truncated version of the maximal supergravity. In fact, all considerations below apply equally well to p-branes in d + p dimensions propagating in the background of a metric and p + 1 potential. We thus consider an 8 + 3 dimensional space-time with coordinates Xμ = (x µ , y α ) , µ = 0, . . . , 7 , α = 1, 2, 3 , with x 0 being in the time direction. We take the background geometry to have the form where g (3) ≡ det g αβ andḡ µν ,ḡ αβ are assumed to be SL(2, R) invariant, and κ is an exponent to be determined. We take the only nonvanishing component of the 3-form to be B αβγ , and assume that all the target space background fields to depend on x µ only. Thus, the action is In maximal supergravity theory in D = 8, the fields (g αβ , B αβγ ) contain five scalars that parametrise the coset SL(3, R)/SO(3) and two scalar parametrising the coset SL(2, R)/SO (2). In addition to these fields and the metric g µν there are two triplet of vectors, g µα and B µαβ , that transform as (3,2) under three 2-forms B µνα that transform as (3, 1) under the Uduality group SL(3, R) × SL(2, R). We are neglecting the latter fields below, with the expectation that they would not effect the realisation of SL(3, R) × SL(2, R) U-duality symmetry at the level of duality rotations on the worldvolume of the membrane in D = 8, should such symmetry exist at all 6 . Turning the action (3.4), it implies that the induced worldvolume metric is given by 5) and the field equations are There is also a conserved topological current: such that, defining we have where V ij is the inverse of We can combine these equations as where In [60], the self-consistency of (3.12) was studied in detail, and it was shown that a two-parameter subgroup of SL(2, R) can be realized. In doing so, the complicated transformation rule for the induced metric, which is no longer a scaling transformation we saw in the case of topological membrane, was taken into account in [60] 7 .
Here, we shall avoid this complication and show that this same result can be derived more simply from the integrability of the transformation rule for ∂ i y α . Assuming that P i α and J i α transform as in (2.10), we can compute the transformation of ∂ i y α by using (3.8) and (2.10), finding (3.14) As discussed above, the integrability of this variation is in general not guaranteed, even on-shell. Clearly the a transformation is always integrable, and so the remaining question is whether the c transformation in (3.14) is integrable. In order to test this on-shell, it will suffice to consider a particular membrane solution [72], for which gμν = ημν and we take Thus we have the induced metric giving It is straightforward to verify that this is a solution of the equations of motion. We see that in this background Let us consider the transformations δ∂ 1 y 2 and δ∂ 2 y 2 under c, which from (3.15), (3.14) and (3.18) will therefore give δ∂ 1 y 2 = 0 , δ∂ 2 y 2 = cβ h . Checking the integrability, we see from the first equation that ∂ 2 δ∂ 1 y 2 = 0, whereas from the second equation ∂ 1 δ∂ 2 y 2 = cβ ∂h ∂σ 1 = 0. This example is therefore sufficient to show that the proposed transformation for δ∂ i y α under the c transformation, subject only to the use of the membrane equations of motion, is not integrable. Choosing κ = − 1 6 , equations (3.12) transform properly under the remaining SL(2, R) transformations provided the background parameters g and B transform according to (2.16). As a consequence, the transformation of the induced metric takes the form Equations (2.9) can be combined as The presence of the matrix X i j shows that the duality symmetry of this equation is the Heisenberg group with the underlying algebra parametrized as in (3.20).

Membrane in D = 4
In this section we shall study the action (2.1) for D = 4 target spacetime with Lorentzian signature. In this action (2.1), the target space fields can be interpreted as the bosonic sector of N = 1, D = 4 supergravity in which the cosmological constant is dualised to a 3-form potential [73][74][75]. Supermembrane action in this setting exists [76,77] and it has been studied in detail in [75]. The spacetime background can also be viewed as being a subsector a time-like dimensional reduction of the bosonic sector of D = 11 supergravity down to 7 dimensions, where only the fields (g µν , B µνρ ) are kept, and the 7 dimensional Euclidean coordinates are taken to be constants. In a spacelike dimensional reduction, the signature of the metric g µν would be Euclidean, and again, we would take the 7 dimensional spacetime coordinates to be constant. Our considerations apply to this case as well but we shall adhere to the Lorentzian signature for concreteness. We shall consider two approaches to the problem of duality rotations in this theory. In the first approach, due to Gaillard and Zumino [63], there is no need to introduce any extra coordinates. Rather, one examines the consistency of the duality rotations, since the definition of the conjugate momentum field associated with the worldvolume scalar fields involves the topological current whose conservation is the Bianchi identity. In a second approach considered in [43][44][45], one introduces extra coordinates in order to try to achieve a manifest realization of the duality symmetry.

Gaillard-Zumino approach
The equations of motion for a membrane in a four-dimensional target space with coordinates X µ , (µ = 0, ..., 3) can be written as and It is important to note that (4.3) implies the relation 8 It will also be useful to note the relation The question then is what is the largest set of transformations that transforms P i µ and J iµν into each other in a consistent manner, such that the system of field equations ∂ i P i µ = 0 and the Bianchi identity ∂ i J iµν = 0 remain invariant.
Our task is to check whether the equations of motion are invariant under SL(5, R) transformations which can be parametrised as where a µ ν is traceless. It acts on a 5-plet of SL(5, R) as δV M = −Λ P M V P . Thus, defining the components of a triplet of second-rank antisymmetric tensor of K i M N as K i µ5 := P i µ and where δ µν ρσ = 1 2 δ µ ρ δ ν σ − δ ν ρ δ µ σ . Next, assembling (g µν , B µνρ ) into a symmetric SL(5, R) matrix G M N , where M, N = 1, ..., 5, with identifications G µν = g −2/5 g µν , G µ5 = G 5µ = 1 3! g −2/5 g µα ε αβγδ B βγδ , where indices are lowered on the parameters c µνρ using the metric g µν , and we have defined c · B ≡ c αβγ B αβγ . Next, the variation of F iµ can be found from (4.2) by using the variations (4.7), (4.9) and (4.10). The result is In deriving this variation, one makes use of the identity B λ αβ c αβσ B σνρ = 1 3 c · BB λ νρ , which can be proven by writing c µνρ and B µνρ in terms of dual vector fields. Finally, from the variations above, one can also determine the variation of γ ij by using (4.5), finding Now we turn to a key test for the above transformation rules, which is the requirement that the transformations of J i µν , F i µ , and γ ij , must be compatible with equation (4.4). The question of whether this nontrivial condition holds was raised in [60]. In fact, as we shall show here, it does actually hold. Firstly, the invariance of (4.4) under the a-and b-dependent transformations is manifest. The nontrivial check is the invariance under the c-dependent transformations, which requires that (4.14) The first three terms sum up to zero since J [σµ i c ναβ] = 0 identically. The remaining three terms also sum up to zero, upon using the fact that γ ij ∂ i X µ ∂ j X ν = g µν − n µ n ν , where n µ is normal to the brane in the target space, i.e. ∂ i X µ g µν n ν = 0. There remains, however, the condition that the variation of ∂ i X µ , which follows from the first equation in (4.3), must be curl-free. Using (4.11) and (4.12) in the first equation in (4.3) we find that Thus the integrability condition amounts to Using the field equation ∇ i ∂ i X µ = 0, this equation can be simplified to read As we did earlier, we can most conveniently check this equation by considering a particular membrane solution [72], namely which solves the equations of motion (4.1). If suffices to consider the integrability condition (4.16) for i = 0 and µ = 0. It is then immediately evident that the first term on the right-hand side of (4.16) gives zero, whereas the second term gives a non-vanishing result that is proportional to the parameter c 0 12 . Thus the integrability condition is not satisfied, and so the proposed SL(5, R) transformation of ∂ i X µ is not valid. The subgroup that is consistent with the curl-free condition is therefore the semi-direct product GL(4, R) ⋉ R 4 , generated by a, a µ ρ , and b µνρ .

Introduction of Extra Coordinates
In seeking a manifestly realised SL(5, R) symmetry, six extra coordinates were introduced in [43], such that together with the four coordinates X µ of spacetime they form a 10-plet of SL(5, R). The extra coordinates are antisymmetric tensorial, and are denoted by Y µν . The field equations and Bianchi identities of the membrane were cast into a manifestly SL(5, R)covariant form. However, the equations satisfied by the extended system are problematic, and this can be seen as follows. Setting B µνρ = 0 and taking g µν = η µν for simplicity, these equations take the form [43] Taking the curl of the second equation gives the integrability condition where we have used the field equation ∇ i ∂ i X µ = 0 to simplify the result. This integrability condition, which implies second-order differential constraints over and above the field equations, therefore poses a problem with the desired SL(5, R) duality-symmetric system of membrane equations. Note also in the GZ approach as well as the approach in which extra coordinates are introduced, the obstacle to the sought-after SL(5, R) symmetry is the nonvanishing of the expression V µν i defined in (4.18).
A different proposal has been made in [65], where the original coordinates are embedded into an SL(5, R) 10-plet Z M N via The following SL(5) covariant equation was proposed in [65]: Here c is an arbitrary constant, and the induced SL(5)-invariant metric is given by which resemble the original membrane equations of motion, except that the worldvolume metric γ ij is now replaced by the SL(5)-invariant metric Γ ij . The curl of (4.23) also implies the integrability equation Unlike the previous proposal discussed above, this does not yield an immediate contradiction, since it involves the original X µ as well as the new Y µν coordinates. Indeed, taking B µνρ = 0 and g µν = η µν , we see that the equation (4.23) gives Apart from the fact that the induced worldvolume metrics are different, we see that the second equation above contains an extra term, in comparison to that given in (4.20). Consequently, its integrability condition will indeed mix X µ and Y µν , thereby avoiding an immediate conflict. However, to test whether the system described by (4.23) makes sense, we should also examine the spectrum of small fluctuations around a vacuum solution that respects the worldvolume Poincaré symmetry. Such a background can be taken to be given by where λ is an arbitrary constant, η M N is the SO(p, q) ⊂ SL(5, R) invariant tensor, and σ i M N specifies the embedding of SO(2, 1) into SO(p, q) ⊂ SL(5, R), for which we choose the canonical normalization Owing to the scale invariance of (4.23), the parameter λ drops out, and for convenience we choose it so that we may identify the worldvolume metric (4.24) with η ij = diag(−, +, +): Equation (4.23) then yields Denoting the spin of the representation σ i by j (assuming the representation carries a single spin), we have the relation which together with (4.31) determines Defining the fluctuations around this background as we fix the gauge freedom of worldvolume diffeomorphisms by imposing The expansion of equation (4.23) to linear order in the fluctuations then gives  For case A), using an explicit spin-2 representation for the generators σ i implies that the invariant tensor η M N is of signature (2,3), and one may verify that equation (4.23) reduces to for the components φ ijk ≡ (σ (i σ j σ k) ) M N φ M N surviving the gauge condition (4.36). This shows that around this background the fluctuations do not admit any non-trivial dynamics. Similarly, in case B) equations (4.23) restrict the fluctuations to which again kills all dynamics for the fluctuation components surviving the gauge condition (4.36). Finally, in case C) the background is most conveniently given in terms of the 't Hooft symbols which again kills all dynamics for the fluctuation components surviving the gauge condition.

Conclusions
Using the p-brane worldvolume approach to U-dualities, we have confirmed that there is a problem when D = p + 1, focusing on p = 2 in D = 4 where the expected SL(5, R) fails to materialise, and p = 2 in D = 3, which we refer to as the topological membrane, where the expected SL(3) × SL(2) does arise. In the case of the topological membrane, we have introduced extra coordinates to make the U-duality symmetry manifest. The features we have found for D = 4 are the same whether we use the approach where extra coordinates are introduced in order to make U-dualities manifest [43,65], or in the Gaillard-Zumino approach where the symmetries are not manifest. In the latter approach, as well as that of [43], we have shown that the SL(5, R) U-duality fails due to the nonintegrability of the transformation rule of the pull-back map. In the approach of [65] where a manifestly SL(5, R)-invariant equation is proposed for a membrane in a target based on generalized geometry, we have shown that these equations do not support linearized fluctuations about a Poincaré invariant vacuum solution. These problems extend to the worldvolume treatment of Berman and Perry [22], and also to the approach in Hatsuda et al. [29], which reformulates the diffeomorphism constraints for an M2-brane coupled to a supergravity background in D = 4 in an SL(5, R)-covariant form. In both cases the problem is that they use transformation rules that are not integrable, for the reasons we have explained above More specifically, in [29], the SL(5, R) transformations of the time components J 0 and P 0 are used to assert the SL(5, R) invariance of the Hamiltonian constraint, which is a quadratic form in these variables. By worldvolume Lorentz symmetry, however, also the space components must transform in the same way, which is equivalent to our (4.7). Then, our discussion leading to the nonintegrability of the resulting transformation rule for ∂ i X µ continues to be an obstacle for SL(5, R) invariance. We expect this will also appear in the case of SO(5, 5) symmetry of M5 branes that has been proposed in [30].
Going beyond D = 5 only exacerbates the problem, since the U-duality multiplets involve the M5-brane charges as well the M2-brane charges and the momentum, as shown in the Appendix. One possible approach to this problem may be along the lines studied in [71] for the case of M2-brane in D = 8. In this approach, firstly one keeps all the target space fields arising in the dimensional reduction of D = 11 supergravity down to d dimensions. Next, one ensures the gauge invariance of the pull-backs of all the target space form fields by introducing appropriate worldvolume potentials, in the same way the worldvolume vector fields are introduced in D-brane actions. Then, one imposes suitable duality equations that exhibit the expected U-duality symmetry group manifestly, while maintaining the correct number of degrees of freedom. The challenge in this approach is to resolve the resulting highly nonlinear constraint equations in a way that maintains the U-duality symmetry. So far, these equations have been solved for a restricted class of supergravity background such that only the two parameter Heisenberg subgroup of SL(2, R) we encountered in our treatment of membrane in D = 8 has been realised [71].
The E n(n) symmetry of D = 11 supergravity compactified on the torus T n also arises if ten-dimensional type IIA or IIB supergravity is compactified on the torus T n−1 . An alternative approach to finding a worldvolume realisation of these U-duality symmetries could be to look at string theories in (n − 1) dimensions rather than the membrane in n dimensions. Ideas along these lines have been pursued in [78][79][80][81].

A Supergravity Compactifications
Recall that the 11=dimensional M-theory superalgebra is given by [82] : which is, algebraically, the maximum possible number since the LHS is a symmetric 32 × 32 matrix. The spatial components of the momentum P 1 and the central charges Z 2 , Z 5 are associated with the plane wave W1, the 2-brane M2 and the 5-brane M5; the temporal components are associated with their duals, the KK-monopole K6, and objects we can call K9 and W10.
After dimensional reduction to n dimensions, the 528 charges will form representations of SO(n − 1, 1) × SO(D) as in (A.3). However, the charges carried by the waves, branes and monopoles do not fall into representations of SO(n − 1, 1) × SO(D) because they discriminate between the temporal and spatial components. For example, writing M = (0, I) with I = 1, 2, ...10, the 45 SO(10) M5 charges are given by Z IJ and the 10 K9 charges by Z 0J or equivalentlyZ IJKLM N OP Q .