Generalised vielbeins and non-linear realisations

We briefly review why the non-linear realisation of the semi-direct product of a group with one of its representations leads to a field theory defined on a generalised space-time equipped with a generalised vielbein. We give formulae, which only involve matrix multiplication, for the generalised vielbein, the Cartan forms and their transformations. We consider the generalised space-time introduced in 2003 in the context of the non-linear realisation of the semi-direct product of E(11) and its first fundamental representation. For this latter theory we give explicit expressions for the generalised vielbein up to and including the levels associated with the dual graviton in four, five and eleven dimensions and for the IIB theory in ten dimensions. We also compute the generalised vielbein, up to the analogous level, for the non-linear realisation of the semi-direct product of very extended SL(2) with its first fundamental representation, which is a theory associated with gravity in four dimensions.


Introduction
We begin by giving a very brief review of the general theory of non-linear realisations. While some aspects of this are very well known, the non-linear realisations that involve a group whose generators are associated with space-time are less well known. In particular we will make it clear why the non-linear realisations which lead to space-times automatically encode a generalised vielbein.
A non-linear realisation of a group G with local subgroup H is constructed from a group element g ∈ G which is subject to the transformations g → g 0 g, for g 0 ∈ G and also g → gh, for h ∈ H (1.1) where g 0 is a rigid transformation and h a local transformation. The meaning of rigid and local will be discussed below. The non-linear realisation is an action, or set of equations of motion, that are invariant under the transformations of equation (1.1). The dynamics is usually constructed from the Cartan forms V which are inert under the rigid g 0 transformations but transform under the h transformations as Before explaining the particular type of non-linear realisation that will be discussed in this paper it will be instructive to briefly discuss the three types of non-linear realisation.

Type 1
Non-linear realisations were first introduced to understand the scattering of pions and it was through this work that it became understood that symmetry was to play a crucial role in particle physics [1]. The theoretical underpinning of this method was set out in the classic papers of reference [2]. This work involved a group G which contained generators that were internal, that is, not associated with space-time. Space-time was introduced in an adhoc manner by taking the group element g, and so the parameters it contains, to depend on the chosen space-time, which in the application at that time, was just four dimensional Minkowski space-time. As a result, the parameters of the group element g became the fields of the theory defined on the chosen space-time. The rigid transformation g 0 is a constant group element, while the local transformation h is taken to depend on the chosen space-time and so can be used to gauge away parts of g. The Cartan forms can be written as where H i are the generators of H and T I the remaining generators of G. When the group is such that the commutators between generators of H with T I lead again to the generators T I (the reductive case), the forms P ≡ P I T I transform homogeneously and can be used to construct an invariant action which is just the space-time integral of T rP 2 . The dynamics of the pions were found to be very well described, in the limit of small pion mass, by the non-linear realisation of SU (2) ⊗ SU (2) with respect to its diagonal subgroup SU (2).

Type 2
A non-linear realisation at the other extreme is one where all the generators of the group G are associated with "space-time". In this case we have a simple coset space, often written as G H , which has been studied for a very long time, at least in the mathematics literature. In this case the H transformations enforce the usual equivalence relation that ensures that the group elements of G are regarded as equivalent if they belong to the same coset. Modulo this relation the parameters in the group element label the points in the coset, which for the application that physicists have in mind are the points in space-time. Thus in this case we have no fields.
For these non-linear realisations the Cartan forms can be written as where H i are the generators of H and l A the remaining generators of G. The objects E Π A define a preferred basis of one forms dx Π E Π A at every point of the coset which are just those swept out out, using the natural action of the group on the coset, from a basis of one forms at the origin of the coset. As a result we can interpret the objects E Π A as vielbeins on the coset space. The objects ω Πi can be thought of as the spin-connection on the coset. One can easily verify, using equation (1.1) that both of these objects transform as vielbeins and spin connections should on their A and i indices respectively under the local H transformations.
The classic example of such a non-linear realisation is to take G to be the Poincare group, which can be written as the semi-direct product of the Lorentz group, SO(1, D −1), with a set of generators in its vector representation, denoted l SO(1,D−1) , with the local subgroup H being the Lorentz group. We denote this semi-direct product by SO(1, D − 1) ⊗ s l SO(1,D−1) . Another example is superspace which is the non-linear realisation of the super Poincare group with the local subgroup being the Lorentz group [3].

Type 3
The final type of non-linear realisation is built from a group G that has some generators that are associated with space-time and some that are not. For simplicity, and as it is the case we wish to consider in this paper, we will take the group to be of a semi-direct product structure, that is, of the form G = G 1 ⊗ s l 1 where G 1 is a Lie group and l 1 is one of its representations. We denote the generators of G 1 by R α and those of the l 1 representation by l A . The Lie algebra for this group can be written in the form The Jacobi identity, implies that the generators l A belong to representation of G 1 and so the matrices in the above equation obey the matrix identity The commutators of the l A generators must be consistent with the Jacobi identities and we will take them, for simplicity, to commute.
The group element g ∈ G is constructed form the generators l A and R α and can be written in the form g = g l g A ≡ e x A l A e A α (x)R α (1.8) The parameters of the l A generators can be interpreted as the coordinates of a generalised space-time while the parameters of the R α generators are taken to depend on the coordinates of space-time and are fields defined on the generalised space-time. Rigid in this case means that the group element g 0 does not depend on the generalised space-time and so its parameters are constants. The local subalgebra H of G used in the non-linear realisation is a subalgebra of G 1 and local H transformations have a group element h which does depend on space-time. This transformation can be used to gauge away some of the fields in g. However, once this has been done we have to carry out compensating H transformation to preserve the form of the group element g under a rigid g 0 transformation. This last type of non-linear realisation is a hybrid of the first two types; if we take no l 1 generators then it is of type one while if we take no generators of the type R α then it is of type two. The Cartan forms belong to the Lie algebra of G and so can be written as were V l contains the generators of l 1 and V A the generators of G 1 and as such we can write them in the form We can interpret the objects E Π A as the vielbein on the generalised space-time. One of the early examples of this type of non-linear realisation was to take G = GL(D) ⊗ s l GL(D) where l GL(D) is the vector representation of SL(D), or equivalently its first fundamental representation [4,5]. This non-linear realisation gives, with a judicious choice of a few undetermined constants, Einstein's theory of gravity [4,5]. A more recent example, and the one of interest to us here, is to take G to be the semi-direct product of E 11 and its first fundamental representation l 1 , denoted E 11 ⊗ s l 1 [6]. This is a special case of non-linear realisations constructed from the groups G = G +++ ⊗ s l 1 where G +++ is the very extension of any finite dimensional semi-simple Lie algebra and l 1 is the first fundamental representation of G +++ . We note that E 11 = E +++ 8 . The non-linear realisations A +++ D−3 ⊗ s l 1 [6] and D +++ D−2 ⊗ s l 1 [8] are conjectured to be the low energy effective actions for gravity and the closed bosonic string in D dimensions respectively. A more detailed review of non-linear realisation can be found in [9].
In this paper we will consider non linear realisation of the last type that is the nonlinear realisation of G = G 1 ⊗ s l 1 . In section two we derive expressions for the generalised vielbein, Cartan forms and their transformations that require no more than matrix multiplication. In section three we consider the non-linear realisation E 11 ⊗ s l 1 and compute the generalised vielbeins in eleven, five and four dimensions and the IIB theory in ten dimensions up to levels three, four, two and five respectively. In section four we give the initial steps in the construction of the non-linear realisation of the A +++ 1 ⊗ s l 1 and compute the generalised vielbein up to level two. This later theory is conjectured to be the complete low energy effective action for four dimensional gravity. In appendix A we recall, up to the level associated with the dual graviton, the E 11 ⊗ s l 1 algebra in the decompositions appropriate to eleven and four dimensions and for five dimensions and the IIB theory in ten dimensions we give these algebras for the first time.

Formulae for the generalised vielbein and Cartan forms
In this section we consider the non-linear realisation of the semi-direct product of a group G 1 with one of its representations l 1 which we denote by G 1 ⊗ s l 1 and so we are discussing the case of type three of the section one. In this section the l 1 representation can be any representation and not just the first fundamental representation. We will take the generators of the l 1 representation to commute. It is straightforward to modify the discussion to the case when the generators of the l 1 representation do not commute, but form a group.
The generators of the group G 1 in the non-linear realisation are usually taken to be abstract objects, but if we take them to be in the l 1 representation then it is straightforward to derive expressions, that involve no more than matrix multiplication, for the generalised vielbein, their transformations, and the Cartan forms. These are well known for the nonlinear realisation of GL(D) ⊗ s l GL(D) [4] and were recently given [10] for the generalised vielbein for E 11 ⊗ s l 1 .
The generalised vielbein is defined in equation (1.10) and it is straightforward to evaluate using equation (1.6) to find that it is given by A and the expression on the right-hand side is evaluated by expanding the exponential and using matrix multiplication. Taking the generators of the G 1 algebra to be in the l 1 representations in the expression for the Cartan forms of equation (1.10) we find that and so Using the expression for the vielbein of equation (2.1) we find that where we have used the identity The action of the rigid transformation g 0 ∈ G +++ , which can be written in the form can also be given in explicit form. As the generators l A form a representation of G +++ under this transformation, equation (1.1) implies that Using equation (1.6) the first equation is found to imply the coordinate change While the change in the vielbein can be found by considering and as a result , or equivalently in matrix notation e A ′ = e a·D e A (2.9) We note that dx Π E Π A is inert under rigid g 0 transformations as it should be. It is often useful not to parameterise the group element g A by a single exponential but by a product of exponentials. In this case one just replaces the above matrix expressions by the corresponding products, for example, if let set g A = e A 1 ·R . . . e A n ·R then the vielbein takes the form.
where A 1 = A 1 · (D) Π A and there are analogous expressions for the above formulae. To proceed further we will need the Cartan Involution I c which can be taken to act on the generators of E 11 as I c (R α ) = −R −α . In fact we have in previous papers taken a plus sign for some of the involutions, but this can be undone by redefining the negative generators. The Cartan involution acts on the l 1 representation to give another representation denoted byl A as I c (l A ) = −J −1 ABl B for a suitable matrix J AB . Acting on the commutator of equation (1.6) with the Cartan involution we find that For the case of E 11 ⊗ s l 1 , the l 1 representation is a lowest weight representation with lowest weight state P 1 whilel 1 is a highest weight representation with highest weight stateP 1 where P a , a = 1, . . . , D are the usual space-time translation generators and P a , a = 1, . . . , D.
We take the local subalgebra in the G 1 ⊗ l 1 non-linear realisation to be the Cartan involution subgroup of G 1 which consists of group elements which obey I c (h) = h. Following similar arguments one finds that the local h = e b α (R α −R −α ) transformation of the generalised vielbein is given by It is sometimes useful to construct the dynamics not from the Cartan forms, but from the object M = g A I c (g −1 A ) which transforms as M → M ′ = g 0 M I c (g −1 0 ). We note that I c (M −1 ) = M and so M can be written in the form M = e φ α (R α +R −α ) which confirms that M is a group element that belongs to the coset. The matrix representation of M is given by where the matrixÃ = A α D −α . The transformation of M can be written, in matrix form, as M → M ′ = e a α D α M e a α D −α 3 Explicit computation of the E 11 generalised vielbein at low levels In this section we will consider the non-linear realisation of E 11 ⊗ s l 1 with the local subgroup being the Cartan involution invariant subalgebra of E 11 ; the analogue of the maximal compact subalgebra. This non-linear realisation has been conjectured to be the low energy effective action describing strings and branes [6,11]. The representations of E 11 can be studied by decomposing them into representations of a finite-dimensional Lie algebras, obtained by removing one node from the Dynkin diagram of E 11 . The Dynkin diagram of E 11 is given by The theories with different number of space-time dimensions emerge when computes the non-linear realisation of the E 11 ⊗ s l 1 algebra when decomposed into the algebras that follow by removing the different possible nodes [12][13][14]. In this paper we are interested in four particular cases: removing node 11 leads to GL (11) algebra that corresponds to 11-dimensional theory, removing node 9 results in 10-dimensional type IIB theory with GL (10) × SL (2, R) algebra, removing node 5 leads to GL(5) × E 6 algebra that describes 5-dimensional theory, and, finally, removing node 4 leads to GL(4) × E 7 algebra that corresponds to the 4-dimensional theory. The fields and coordinates in D dimensions can be classified by a level that is given by the number of down minus up SL(D) indices except that one adds one for the coordinates and divides the results by three in eleven dimensions and two for the ten dimensional IIB theory.
The l 1 representation decomposed in the way suitable to D dimensions leads to a generalised space-time that contains at level zero the usual coordinates x a and at level one coordinates that are scalars under the Lorentz group but transform as the 10, 16, 27, 56 and 248 ⊕ 1 of SL(5), SO(5,5), E 6 . E 7 and E 8 for D equal to seven, six, five, four and three dimensions respectively [16,17]. The corresponding generalised vielbeins have been partially constructed at low levels for these generalised space-times using the E 11 ⊗ s l 1 non-linear realisation. One of the first examples was the construction of the generalised vielbein for the five dimensional theory up to level two [15] which, in conjunction with the corresponding generalised space-time, was used to find all maximally supersymmetric gauged supergravities. In the four dimensional theory reference [18] computed the 56 by 56 vielbein that arises in the space of the level one coordinates [18]. The full generalised vielbein up to and including level one in the four dimensional theory was given in [19]. The generalised vielbein, but restricted to the space of the level one coordinates, was also subsequently computed in [20] in dimensions four up to seven inclusive. The eleven dimensional generalised vielbein was computed up to level two in [21]. A metric that appeared in the duality invariant first quantised actions studied in reference [26] was used in reference [25] to discuss theories formulated on a seven dimensional space-time. However, we note that this generalised space-time is just the part of l 1 representation of E 11 at level one in seven dimensions [6,16,17] and the vielbein, or equivalently the metric, is a truncation of the vielbeins found earlier in the context of E 11 papers.
Siegel theory [22], sometimes called doubled field theory, was developed in 1993. This was motivated by string theory and it consists of a theory with the same massless fields as appear in the NS-NS sector of the superstring, but defined in a 20-dimensional spacetime that transformed in the vector representation of O (10,10). A generalised vielbein defined on this space-time, was found in reference [22], it played an important part in the construction of Siegel theory. The Virasoro operators appeared in construction and they were to contain a corresponding metric which agreed with that found when reducing string theory on a torus. This theory was subsequently formulated as the non-linear realisation of E 11 ⊗ s l 1 in ten dimensions at level zero [23]. The extension of this theory to include the R-R sector is just the level one contribution and it was first found in reference [24]. The generalised vielbein computed from this later viewpoint agrees with that found earlier.
In this section we calculate the generalised vielbein in eleven, five and four dimensions and also the for the ten dimensional IIB theory at much higher levels using the E 11 ⊗ s l 1 non-linear realisation.

D = 11
The eleven dimensional theory is obtained by deleting node 11 from the Dynkin diagram of E 11 . and decomposing the E 11 ⊗ s l 1 into representations of GL (11) [11]. In this section we will restrict ourselves with level 3 calculations. The non-negative level generators of the E 11 are K a b ; R a 1 a 2 a 3 ; R a 1 ... a 6 ; R a 1 ... a 8 , b .
The negative level generators are The l 1 representation contains the generators [6] P a ; Z a 1 a 2 ; Z a 1 ... a 5 ; Z a 1 ... a 8 , Z a 1 ... a 7 , b . (3.1. 3) The group element g = g l g A can be parametrised in the following way: 1.4) where we have introduced the generalised coordinates [6] x a ; x a 1 a 2 ; x a 1 ... a 5 ; x a 1 ... a 8 , x a 1 ... a 7 , b . (3.1.5) We have used the local subalgebra to gauge away part of the g A group element and we have the, by now well known, fields of the E 11 ⊗ s l 1 non-linear realisation up to level three, namely, the graviton, the three and six form gauge fields and the dual graviton [11]. The corresponding generalised tangent space structure is obvious and the tangent space group is I c (E 11 ) which at lowest level is just the Lorentz group and at higher levels has an algebra can be found in reference [6] and also the book of reference [8].
The generalised vielbein is defined in equation (1.10) and, while one can straightforwardly compute it using the commutators of appendix A.1, we will find it using the matrix expression of equation (2.10), which in eleven dimensions takes the form We begin with the level zero matrix which is given by the expression from which we conclude that and I is the identity matrix.
It then follows that from which we conclude, using the commutators of appendix A.1, that Proceeding in a similar way we find that and To compute the generalised vielbein we just need to evaluate the matrix expression of equation (3.1.6), being careful to evaluate the unusual index sets, we find that where the symbols in the first line of this matrix are given by the symbols in the second line are given by 1.18) and, finally, the symbols in the third line are given by There are two ways of obtaining a ten-dimensional theory: removing node 10 leads to type IIA supergravity theory, while removing node 9 leads to type IIB supergravity [12]. We will be interested in the latter. The corresponding Dynkin diagram is In this case all the generators fall into representations of GL (10) × SL (2, R). The nonnegative level generators of adjoint representation up to level 5 are [12,27] The negative level generators are The l 1 representation generators up to level five are We note that some of the l 1 generators at level five have multiplicity two. Although we have listed the generators up to level five we will only use the generators up to level four, that is, we will work just up to level four in what follows. Lower case Greek indexes correspond to the fundamental representation of SL (2, R) (α, β, γ, ... = 1, 2). Tensors that have multiple Greek indexes are assumed to be symmetric in these indices. The general group element g = g l g A , up to level 4, can be written as

2.4)
Where we have introduced the generalised coordinates The tangent space group is I c (E 11 ) which at level zero is SO(1, 9) × SO(2). It is very straightforward to compute at higher levels.
In this section we are going to calculate the generalised vielbein using its definition in equation (1.10) rather than the matrix method of section two. In this approach the generalised vielbein is computed by conjugating the l 1 generators with the E 11 group element. We recall that Using the algebra from Appendix A.2 we can perform this conjugation for the D = 10 case. Conjugation with level 0 group element gives (3.2.8) In the above equation and what follows we denote world, rather than tangent, SL (2) indices with a dot, that isα, . . .. Conjugating with positive level generators can be obtained by Taylor-expanding the exponents and truncating the series by level 4. For level one E 11 generator we have (3.2.9) For level 2 E 11 generator: For level 3 E 11 generator: and, finally, for level E 11 4 generators: Using all these results we find, from equation (3.2.5), that the generalised vielbein is given by (3.2.13) In the above world indices, that is, µ, . . . orα, . . . arise, as vielbeins acting on objects with tangent indices, for example . . The symbols in the first line of the above matrix are given by in the second line are 2.15) in the third line are 2.16) and finally in the fourth line are The five dimensional theory is obtained by deleting node 5 from the E 11 Dynkin diagram, given below, to find the algebra GL(5) × E 6 and decomposing the E 11 ⊗ s l 1 algebra into representations of this algebra [15].
In this decomposition the positive, including zero, level generators of the E 11 algebra are The l 1 representation decomposes to give the generators [15] P a , Z N , Z a N , Z a 1 a 2 , α , Z a 1 a 2 , Z a 1 a 2 ,bN , Z a 1 a 2 a 3 N 1 N 2 , . . . An arbitrary group element can be parametrised in the following way:

3.4)
We find that the five dimensional theory has a generalised space-time that has the coordinates x a , x N , x a N , x a 1 a 2 ,α , x ab , . . . (3.3.5) and the fields h a b , ϕ α , A aM , A a 1 a 2  N , A a 1 a 2 a 3 , α , A a 1 a 2 , b , . . .

3.8)
We will compute the generalised vielbein up to level three. We begin by considering the level zero part and noting that dx

3.10)
from which we conclude that It then follows that e A 0 eÃ 0 = (det e) 3.14) and A dot over an index means that it is a world rather than a tangent index. We now compute A 1 in a similar way by considering 3.16) from which we conclude, using the commutators of appendix A.3, that Proceeding in a similar way we find that It is now just a matter of matrix multiplication, albeit with unusual index sets, to find the generalised vielbein using equation (3.3.7), the result is

3.22)
in the second line

3.23)
and in the third line

D = 4
The four dimensional theory is obtained by deleting node 4 from the Dynkin diagram and so decomposing the E 11 algebra into representations of GL (4)×E 7 [19]. However, it is easier to work with SL (8) subalgebra of E 7 , instead of E 7 itself; the E 7 representations can be reconstructed if needed. In this case all the generators belong to different representations of GL (4) × SL (8). In this section we are going to calculate the Cartan form up to level 2. The positive (and zero) level generators of E 11 are K a b , R I J , R I 1 ...I 4 ; R aI 1 I 2 , R a I 1 I 2 ;K (ab) , R a 1 a 2 I J , R a 1 a 2 I 1 ...I 4 . (3.4.1) The negative level generators are R aI 1 I 2 , R a I 1 I 2 ;K (ab) , R a 1 a 2 I J , R a 1 a 2 I 1 ...I 4 . (3.4. 2) The l 1 representation generators are P a ; Z I 1 I 2 , Z I 1 I 2 ; Z a , Z aI J , Z aI 1 ...I 4 . The parametrisation of an arbitrary level 2 group element is of the form To calculate the generalised vielbein we used the definition of equation (1.10), which was the same technique as was used in section (3.2) for the ten dimensional IIB theory. Conjugation of any l 1 generator with group element that contains the K a b and R I J generators gives the following: We place a dot on a SL(8) index to denote that it is a world, rather than a tangent, index. Conjugation with R I 1 ...I 4 generator gives exp − ϕ I 1 ...I 4 R I 1 ...I 4 P a , Z I 1 I 2 , Z I 1 I 2 , Z a , Z aI J , Z aI 1 ...I 4 exp ϕ I 1 ...I 4 R I 1 ...
where the β-matrices that mix level 1 elements are defined as δ I I 1 ϕ JI 2 I 3 I 4 ε I 1 ...
(3.4.11) Conjugation with level 1 and level 2 elements is performed by Taylor-expanding the exponents. The generalised vielbein is The quantities in the above matrix which have world indices are given in terms of quantities with all tangent indices by which form the generalised vielbein on the coset space of the non-linear realisation of E 7 ⊗ s l 56 with local subgroup SU (8), and in addition With these definitions the symbols in the first line of the matrix are given by , (3.4.13) and the second line by (3.4.14) 4 The non-linear realisation of A +++ 1 and its generalised vielbein As we have mentioned the non-linear realisations of the semi-direct product of very extended A 1 , denoted A +++ 1 with its their first fundamental representation, denoted l 1 is conjectured to lead to the complete low energy effective action for four dimensional gravity [7]. The Dynkin diagram for the Kac-Moody algebra A +++ which corresponds to the Cartan matrix The four dimensional theory appears when we delete node four, as indicated in the above diagram, to leave the algebra GL (4). Decomposing A +++ 1 into this subalgebra we find that the positive level generators of to level 2 are given by algebra can be constructed in the usual way, see reference [8] for a review of this process in the context of E 11 . The commutators for the listed generators preserve the level and must obey the Jacobi identities, as such one proceeds level by level writing down the most general right-hand side for each commutator and then tests the Jacobi identities level by level. The generators belong to representations of GL (4) and so their commutators with the generators K a b are The level 2 (−2) two commutators must give on the right-hand side the unique level 2 (−2) generators and can be written in the form where the normalisation of the level 2 (−2) generators are fixed by these relations. The reader may verify that the right-hand side of these commutators do indeed have the symmetries of the generators which occur in the left-hand side using the constraints on the generators given below equation (4.2). The commutators between the positive and negative level generators are given by where δ (ab) d . The relation of the above generators to the Chevalley generators of A +++ 1 is given by (4.8) One can verify that the satisfy the defining relations were A ab is the Cartan matrix of A +++ 1 given in equation (4.1). The Cartan involution acts on the generators of A +++ 1 as follows The reader may verify that it leaves invariant the above commutators.
We pause here to review how the above construction of the A +++ 1 algebra was carried out as this can act as an illustration of how to construct any Kac-Moody algebra from a knowledge of the generators. We have first written down the commutators of the known generators of equations (4.2) and (4.3) which are consistent with the level, SL(4) algebra, the Cartan involution and the symmetries of the indices on the generators. Strictly we should have included arbitrary constants on the right-hand sides of these commutators, that is, two constants in first of equations (4.7) and one constant in the last two of the equations (4.7), which are related by the action of the Cartan involution. The Jacobi identity [[R ab , R cd ], R ef ] + . . . = 0 then gives one relation between these three constants.
We have then consider the Chevalley relations which by definition must satisfy the relations of equation (4.11). Those for the first three nodes, that is, E a , F a and H a , a = 1, 2, 3 are just those for the subalgebra A 3 and are given in equation (4.8-10). The Chevalley generators E 4 must be constructed out of the level one generators R ab . It must also commute with F 1 , F 2 and F 3 and as a result it must, up to scale, be R 44 . We can choose it to be E 4 = R 44 . Similarly, or using the Cartan involution, we find that F 4 = R 44 . The Chevalley generator H 4 must be a sum of the K a a generators and finding the correct relations with E 1 , . . . , E 4 we find it is as given in equation (4.8). Finally, we impose that [E 4 , F a ] = 2H 4 = [R 44 , R 44 ] which using the first of equations (4.7) fixes the two constants we should have introduced in this relations to be as they are given.
The l 1 representation generators up to level two are given by P a ; Z a ; Z abc , Z ab,c , (4.13) where Z abc = Z (abc) , Z ab,c = Z [ab],c and Z [ab,c] = 0. Their commutators with the level 0 generators of GL(4) are given by (4.14) The commutators of the level one A +++ 1 generators with the l 1 generators must increase their level by one and they can be chosen to be of the form Using the Jacobi identities, the commutator of P a with the level 2 generator of A +++ 1 is found to be The commutators with level-lowering generators are given by The very first relation reflects the fact that the l 1 representation is a lowest weight representation.
Having constructed the A +++ 1 ⊗ s l 1 algebra up to level two we can construction its non-linear realisation. The group element g = g l g A can, up to level two, be written in the form g l = exp x a P a + y a Z a + x abc Z abc + x ab, c Z ab, c , We find that we have introduced the fields h a b ; A ab ; A ab,cd (4.19) where A ab = A (ab) ; A ab,cd = A [ab],cd = A ab,(cd) , and the coordinates where x abc = x (abc) , x ab,c = x [ab],c ,. The field h a b is the usual graviton while the field A ab is the dual graviton. Analogously the coordinates x a are the usual coordinates of space-time while the coordinates y a are the dual coordinates.
This non-linear realisation is a good arena in which to discuss the dual graviton and the resulting dynamics will be discussed elsewhere. Here we will content ourselves with calculating the generalised vielbein up to level two. We will use the definition of equation (1.10) which involves conjugating the l 1 generators with g A using the above algebra. Conjugation with level 0 group element gives    As we are only computing up to level two, that is, up to the l 1 elements Z abc and Z ab,c the order in which we calculate the action of the group elements on the l 1 generators is irrelevant. Combining these results together we find that the generalised vielbein up to level two is given by where the symbols in the first line are given by while the symbols in the second line are given by (4.27)

Conclusion
In this paper we have reviewed how to construct the generalised vielbein associated with the generalised space-time that arises in the non-linear realisation of E 11 ⊗ s l 1 . We find the generalised vielbein up to, and including, the level containing the dual graviton in eleven, five and four dimensions as well as for the ten dimensional IIB theory. To find these results one requires E 11 ⊗ s l 1 algebra up to the level concerned. These algebras were previously known in eleven and four dimensions and in this paper we have also found them in five dimensions and for the ten dimensional IIB theory, the explicit formulae being given in appendix A.
In a recent paper the gauge transformations of the fields in the E 11 ⊗ s l 1 non-linear realisation were proposed [29]. These are formulated in terms of the generalised vielbein and the results for this object given in this paper will prove useful for finding the explicit gauge transformations.

Acknowledgment
We would like to thank Nikolay Gromov for discussions and the SFTC for support from Consolidated grant number ST/J002798/1.

Appendix A
For convenience we give in this appendix the E 11 ⊗ s l 1 algebra appropriate to four, five and eleven dimensions and also for the IIB ten dimensional theory.

A.1 D = 11 algebra
In this appendix we repeat, for convenience, the E 11 ⊗ s l 1 algebra decomposed into representations of GL (11) [11]. The commutators of the E 11 generators with the generators of K a b are given by The commutators between the positive and negative level generators are given by The commutators of the GL(11) generators with those of the l 1 representation are given by [6] [ The commutators of the positive root generators of E 11 with l 1 generators are given by While the commutators of the l 1 generators with the level minus one E 11 generators are given by [R a 1 a 2 a 3 , P a ] = 0, [R a 1 a 2 a 3 , A.2 D = 10 algebra In this appendix we give the commutators of E 11 ⊗ s l 1 algebra, decomposed into representations of GL (10) ⊗ SL (2, R). Parts of this algebra for the form generators were given in references [12] and [27]. The l 1 multiplet and their commutators with the E 11 generators are given for the first time in this paper as are many of the commutators of the E 11 algebra that involve the negative level generators. The commutators of the E 11 generators with the SL(10) generators K a b are The commutators of the E 11 generators with the SL (2, R) generators R αβ are The commutators of the positive level E 11 generators are given by (A.2.4) To find the commutators between positive and negative level generators we need to utilize Jacobi identities. These commutators up to level 3 are given by with levels ±3: and, finally, the commutators of level ±4 generators between themselves are The action of the Cartan involution on the adjoint generators is given by We now consider the commutators of the E 11 generators with those of the l 1 representation. The members of the l 1 representation are most easily found using the Nutma programme Simplie [30]. The commutators of the l 1 representation generators with the level 0 E 11 generators, that is the SL(11) generators, are given by The commutators with the SL (2) generators R αβ are The commutators with level one E 11 generators can be taken as The commutators with other positive-level generators can be found using the Jacobi identities to be given by The commutators with level −1 E 11 generators are given by while the commutators with level −2 generators are and, finally, with level −4 generators The E 11 algebra for the generators decomposed into representations of GL(5) ⊗ E 6 is given below. This algebra for SL (5) and the form generators, up to level four, can be found in references [15,28] and [10], which also include some useful identities. Here we compute the full E 11 algebra up to level four and its commutators with the l 1 representation up to level 3. These include, in particular, the generators associated with the dual graviton and were given in equation (3.3.1) and (3.3.2). By construction the generators of E 11 are in representations of SL(5) and this determined their commutators with K a b to be given by b R |a|a 2 a 3 ], cN + δ c b R a 1 a 2 a 3 , aN , [K a b , R a 1 a 2 a 3 , cN ] = − 3 δ a [a 1 R |b|a 2 a 3 ], cN − δ a c R a 1 a 2 a 3 , bN . The commutation relation of any generator with R α is determined by the representation of E 6 that this generator belongs to: R α , R a 1 a 2 a 3 , β = f αβ γ R a 1 a 2 a 3 , γ , R α , R a 1 a 2 a 3 β = f αβ γ R a 1 a 2 a 3 γ , R α , R a 1 a 2 , b = 0, [R α , R a 1 a 2 , b ] = 0,  The commutators between the positive and negative level generators of E 11 up to level 4 are given by We now give the commutators between the generators of E 11 and those of the l 1 representation given in (3.3.3) up to level 3. The commutation relations between the later and the generators of GL(5) are given by The elements of the l 1 representation at a given level can be introduced into the algebra by taking the commutators of suitable E 11 generators of the same level with P a , namely [R a 1 a 2 a 3 , α , P a ] = 3 δ [a 1 a Z a 2 a 3 ], α , R a 1 a 2 , b , P a = − 2 δ b a Z [a 1 a 2 ] − 2 δ [a 1 a Z |b|a 2 ] , The commutators of the remaining positive level generators of E 11 with the l 1 generators is determined by the Jacobi identities and they are found to be given by In this appendix we give the E 11 ⊗ s l 1 algebra decomposed into representations of GL (4) × SL (8) that corresponds to four-dimensional theory [19]. This latter reference contains a few typographical errors in the commutators which are corrected here. We will first give the commutation relations of level 0 generators with the rest of E 11 algebra. The Finally, the commutators of level − 2 with the level 1 E 11 generators are