Higher dualisations of linearised gravity and the $A_1^{+++}$ algebra

The non-linear realisation based on $A_1^{+++}$ is known to describe gravity in terms of both the graviton and the dual graviton. We extend this analysis at the linearised level to find the equations of motion for the first higher dual description of gravity that it contains. We also give a systematic method for finding the additional fields beyond those in the non-linear realisation that are required to construct actions for all of the possible dual descriptions of gravity in the non-linear realisation. We show that these additional fields are closely correlated with the second fundamental representation of $A_1^{+++}\,$.


Introduction
It was shown that the conjectured [1,2] non-linear realisation of the semi-direct product E 11 ⋉ℓ 1 of E 11 with its vector representation contains the fields and the equations of motion of every maximal supergravity theory [3,4]. For a review, see [5]. As such, it contains the metric of gravity and the three form in eleven dimensions and there are very good reasons to believe that these are the only degrees of freedom that the non-linear realisation possesses [6,7]. However, the non-linear realisation contains an infinite number of fields, of which only a few are the usual fields of the maximal supergravity theories.
It was conjectured in [8] and proven in [9] that many of these remaining fields represent equivalent descriptions of the degrees of freedom of the maximal supergravity theories. For example, in E 11 at levels 1, 4, 7, . . . , we find the fields A a 1 a 2 a 3 , A a 1 ...a 9 ,b 1 b 2 b 3 , A a 1 ...a 9 ,b 1 ...b 9 ,c 1 c 2 c 3 , and so on, which are related by an infinite set of duality relations. This ensures that the only degrees of freedom are those which are usually contained in the first field, the three form.
However, any of these fields can be used to give an equivalent formulation of these degrees of freedom. At levels 2, 5, 8, . . . , the story is similar except that the the block of three indices a 1 a 2 a 3 in each field is replaced by a block of six indices a 1 . . . a 6 . Then, at levels 0, 3, 6, 9, . . . , we find fields associated with gravity. Indeed, at level zero, we find the usual description of gravity with the field h ab . At level three, we find the field A a 1 ...a 8 ,b which was proposed to provide a dual description of gravity, while at level six we have A a 1 ...a 9 ,b 1 ...b 8 ,c , at level nine we find A a 1 ...a 9 ,b 1 ...b 9 ,c 1 ...c 8 ,d , . . . , and so on. These fields also provide alternative descriptions of gravity and all the fields are related by a set of duality relations which ensure that the theory only propagates a single graviton. In fact, there are other fields in the non-linear realisation and some of these are required to account for the gauged supergravities.
It is useful to give an account of the history of the dual graviton field. It was first observed by Curtright that the field A a 1 a 2 ,b = A [a 1 a 2 ],b could describe pure gravity in five dimensions [10].
It was then proposed that the field A a 1 ...a D−3 ,b may describe pure gravity in D dimensions [11].
In order the show that the field A a 1 ...a 8 ,b at level three in the non-linear realisation of E 11 ⋉ℓ 1 did indeed describe gravity, a parent action in D dimensions was given in [1,12]. By first linearising the parent action of [1], then varying the result with respect to one field or the other and finally substituting inside the linearised parent action, we obtain either the Fierz-Pauli action in the form where local Lorentz invariance holds, i.e. in terms of the field h ab that is neither symmetric nor antisymmetric, or we obtain an action in terms of the dual field A a 1 ...a D−3 ,b . This result was fully explained and also extended to higher spin fields in reference [13]. As shown in [1,12], the parent action of [1] also led to duality relations between the two fields. In this way, it was clear that the dual graviton field A a 1 ...a D−3 ,b really did provide an equivalent formulation of gravity at the linearised level. Further connections were also established in [13] between [1], [10] and [11].
These developments are reviewed at the beginning of Section 3.
It was also conjectured that the non-linear realisation of the semi-direct product A +++ D−3 ⋉ ℓ 1 of the very-extended algebra A +++ D−3 with its vector representation, contains pure gravity in D dimensions [14]. Following early preparatory work in references [15] and [16], this was indeed shown to be the case in four and eleven dimensions [17] and [18] respectively. In four dimensions, at the lowest level, this non-linear realisation contains the usual field of gravity h ab . At higher levels -indicated by numbers in brackets after each field -in addition to other fields, it contains h a b (0) ; A (ab) (1) ; A a 1 a 2 ,(b 1 b 2 ) (2) ; A a 1 a 2 ,b 1 b 2 ,(c 1 c 2 ) (3) ; A a 1 a 2 ,b 1 b 2 ,c 1 c 2 ,(d 1 d 2 ) (4) ; . . . (1.1) where groups of indices are antisymmetric unless otherwise indicated by round brackets ( · · · ) in which case they are symmetric. We interpret these fields as being related to dual descriptions of gravity. The field at level one is called the dual graviton. We then find the first higher dual graviton at level two, the second higher dual graviton at level three, and so on. The equations of motion at the full non-linear level, as well as the duality relations, were found for h a b and A (ab) in four dimensions [17] and in eleven dimensions [18].
The non-linear realisations of E 11 ⋉ ℓ 1 and A +++ D−3 ⋉ ℓ 1 lead to an infinite number of duality relations which can then be used to derive the equations of motion of the fields. These field equations are constructed from fields that are irreducible representations of A D−1 and, as a result, they have more and more space-time derivatives for the fields at higher and higher levels. The equations of motion require only the fields in the non-linear realisation, and they correctly describe the relevant degrees of freedom. In [19] and [20], equations of this type which describe the irreducible representations of the Poincaré group were given precisely. As shown in [20] and reviewed in [21], one can also integrate these equations to find equations of motion that are second order in space-time derivatives provided that one makes a particular gauge choice that leads to the Labastida [22,23] gauge transformations for arbitrary mixed-symmetry fields where the gauge parameters obey trace constraints. In fact, the duality relations derived from the non-linear realisation only hold modulo gauge transformations which can, as a matter of principle, be deduced from the non-linear realisation. See, for example, [4] or the review [24].
However, one must also introduce extra fields in order to have duality relations that hold as equations of motion in the usual sense and not just as equivalence relations [25].
A parent action containing the fields A a 1 a 2 a 3 , A a 1 ... a 9 ,b 1 b 2 b 3 , . . . occurring at levels 1, 4, . . . in the E 11 ⋉ ℓ 1 non-linear realisation, also containing certain extra fields, was worked out in [25] along the lines of [9,26]. Depending on which field was eliminated, one found an action only in terms of one field or the other. In this way, the authors of [25] found an action for the latter field which we call the first higher dual of the three from. The higher level fields were also discussed in [25], as were the infinite chain of duality relations and analogous results for the six form. Hence, using parent actions, one could find the additional fields required in order to write down an action, or duality relations, for the higher dual fields.
A similar strategy had previously been suggested for pure gravity in [9]. The method of parent actions was used to produce, for the first time, an infinite number of higher dual action principles, thereby proving the conjecture established in [8] on the equivalent dual descriptions of gravity. These parent actions involve extra fields in comparison to those that appear in the non-linear realisation of A +++ 1 ⋉ ℓ 1 .
In this paper, we further pursue the approach set forth in [9] to higher dual descriptions of gravity, focusing on four space-time dimensions for the sake of concreteness. We provide an explicit procedure for constructing the parent actions that relate the different higher dual formulations of gravity. Using these parent actions, one can directly obtain action principles for each subsequent higher dual graviton. We find extra fields on top of those already in the non-linear realisation of A +++ 1 ⋉ ℓ 1 . These extra fields are required to formulate actions for the dual fields as well as the duality relations between dual fields at adjacent levels.
We will compare the type of additional fields required to form higher dual actions with those contained in the adjoint representation and the second fundamental representation, denoted l 2 , of A +++  Following earlier results [15,16], the non-linear realisation of the semi-direct product A +++ 1 ⋉ ℓ 1 of A +++ 1 with its vector representation was computed at low levels in [17]. This calculation will be reviewed in this section. The Dynkin diagram for A +++ While no complete description of the generators of any such Kac-Moody algebra exists, they can still be analysed by decomposing them with respect to certain subalgebras. Deleting node 4 from the Dynkin diagram of A +++ 1 allows us to analyse the algebra in terms of its decomposition into GL(4) [15]. The resulting generators can be classified in terms of a level which, in this case, is the number of up minus down GL(4) indices divided by two. The positive low level generators R α are given, alongside the level zero generator, by The number in brackets corresponds to the level of the generators and the subscripts enumerate the generators when there is more than one with the same index structure. Groups of indices are antisymmetric except when shown to be symmetrised using round brackets. generators can be found in [17].
The generators in the vector representation of the A +++ 1 are denoted by L A and, when decomposed into representations of GL(4), the low level generators found in [17] are given by where, as before, groups of indices are antisymmetric except for those in round brackets which are symmetric. Subscripts denote different generators when the multiplicity is greater than one.
Generators in the vector representation commute and their commutators with the generators of A +++ 1 are given in [17].
The construction of the equations of motion follows the same pattern as that for E 11 ⋉ ℓ 1 .
See [5,24] for reviews. For the non-linear realisation based on A +++ 1 ⋉ℓ 1 in [17], we start from the group element of A +++ 1 ⋉ ℓ 1 denoted by g = g L g A , where g A and g L are group elements that are constructed in terms of non-negative level generators of the adjoint and vector representations, respectively, of A +++ 1 . They take the form Therefore, the theory is populated by a set of fields A α which contains the graviton h a b , the dual graviton A ab , the first higher dual graviton A ab,cd , and so on. We see from the list of generators in (2.1) that we have the generator R a 1 a 2 ,b 1 b 2 ,c 1 c 2 ,(d 1 d 2 ) at level four which results in the second higher dual graviton A a 1 a 2 ,b 1 b 2 ,c 1 c 2 ,(d 1 d 2 ) . Indeed, the pattern continues so that one finds such fields at every level. This leads to an infinite tower of dual formulations of pure gravity with fields that depend on the generalised coordinates z A = {x a , z a , z abc , z ab,c , . . .} .
The non-linear realisation is invariant under rigid transformations g 0 ∈ A +++ . This means that generic group elements g = g L g A are invariant under g → g 0 g and g → gh , (2.5) where g 0 is a rigid (i.e. constant) group element and h is a local transformation which can be used to set the coefficients of all negative level generators in g A to zero [27]. The equations of motion are just those that are invariant under these transformations and, as for E 11 , they are essentially unique.
The dynamics of the non-linear realisation is often constructed using Maurer-Cartan forms Here, E Π A can be thought of as a vierbein on the generalised space-time. Its lowest component is the gravitational vierbein given by e µ a = (exp(h)) µ a . The G Π,α are the components of the Maurer-Cartan form where the index Π is a world-volume (derivative) index and α is an index in the adjoint representation.
The low level Maurer-Cartan forms in the A +++ 1 direction are given by They are found as the coefficients of K a b , R bc and R a 1 a 2 ,bc in the Maurer-Cartan form, where ∂ Π is the derivative with respect to the coordinates z Π . The dynamics is actually constructed where "· · · " corresponds to terms arising when higher contributions to the vierbein E Π A are taken into account. These contributions contain derivatives with respect to the higher level coordinates. Working with G A,α has the advantage that it only transforms under I c (A +++ 1 ) .
Rather than deriving the equations of motion, one can derive a set of duality relations from which the equations of motion can be deduced, as explained in [3,5,24]. The duality relations between low level fields at the lowest level of generalised space-time derivatives are [17] where ω a,b 1 b 2 is the usual expression for the spin connection in terms of the vierbein which is given in terms of the low level Maurer-Cartan forms by with G a,bc = e a µ e b ν ∂ µ e νc .
Equation (2.9) relates the graviton field h a b appearing at level zero to the dual graviton field A ab at level one, while equation (2.10) is a duality relation between A ab at level one and the first higher dual graviton field A a 1 a 2 ,bc appearing at level two.
By combining equations (2.9) and (2.10), one derives a duality relation between the graviton and the first higher dual graviton: The above duality relations only hold modulo certain gauge transformations, as indicated by the symbol "= ", so they really are equivalence relations. This is explained in [3,4,24,27].
In order to further manipulate the above duality relations, one needs to know what the gauge transformations are. We will obtain them in the next section. As explained in [1], the duality relation (2.9) may be turned into a usual equation by adding an antisymmetric component to the symmetric dual graviton A a 1 a 2 . This 2-form field will later be found inside the second fundamental representation of A +++
In what follows we are only concerned with the linearised theory and so we drop, in particular, the det e factors.

Gauge transformations
It was proposed in [28] that a theory constructed from a non-linear realisation of g +++ ⋉ ℓ 1 , where g +++ is any very-extended Kac-Moody algebra and ℓ 1 is its vector (first fundamental) representation, is invariant under a particular set of gauge transformations whose parameters are in a one-to-one correspondence with the spectrum of ℓ 1 . For the linearised theory where base and fiber indices are identified, these gauge transformations take the form In this equation, C −1 α,β is the inverse of the Cartan-Killing metric C α,β for g +++ . The matrix (D β ) E F is that for the vector representation and, in particular, it occurs in the commutator (2.14) In addition, we have used the partial derivative in the linearised theory ∂ F = ∂ ∂z F . The gauge parameters Λ A correspond to elements in the vector representation.
Hence, in order to evaluate the gauge transformations, we require the inverse Cartan-Killing matrix and the analogous matrix for ℓ 1 at the corresponding level. The gauge transformations for the graviton and the dual graviton in the non-linear realisation of A +++ 1 ⋉ ℓ 1 were computed in [17] and we now extend these previous results to the main object of study in this paper: the first higher dual graviton.
We begin with the computation of the Cartan-Killing form which is determined by requiring that it is invariant. For our current purposes this means that it should satisfy where (· , ·) is the symmetric non-degenerate bilinear form on A +++ 1 that generalises the Killing form for finite-dimensional semi-simple Lie algebras. One finds that . Taking the previous results from [17], we find that the Cartan-Killing metric up to the level of the first higher dual graviton is given by where the basis is ordered to match the scalar products of Note that the only non-zero entries of C α,β are found when the levels of α and β sum to zero.
The inverse Cartan-Killing metric is given by The vector representation appears in the commutators of (2.14) which were given in [17] at low levels. Omitting the commutators with K a b , they are given by To study the vector representation at the level of the first higher dual graviton, we need to compute certain commutators at higher levels. One finds that where the last expression should be taken so that is is anti-symmetric in a 1 and a 2 .
Using the inverse Cartan-Killing metric (2.18) and reading off the analogous matrix for ℓ 1 from equations (2.19)-(2.21), we find that the gauge transformations with gauge parameters are given, for the fields at low levels, by The parameters satisfy Λ a 1 a 2 a 3 = Λ (a 1 a 2 a 3 ) and Λ a 1 a 2 ,b = Λ [a 1 a 2 ],b with the irreducibility condition Λ [a 1 a 2 ,b] = 0. In these equations, we have not written the gauge transformations that involve derivatives with respect to the higher level coordinates.

Linearised equations of motion
The duality relations given in (2.9)-(2.12) only hold modulo certain transformations which arise from the gauge transformations for the fields involved in the duality relations. As we computed these in the previous section, we can now compute the resulting transformations up to which the duality relations hold. Having done this, we can then compute the equations of motion from the duality relations at the linearised level.
We first consider the duality relation between gravity and dual gravity in (2.9). Using the gauge transformation (2.24) we find that, at the linearised level, it takes the form In deriving this result, we have used local Lorentz symmetry to symmetrise the h ab field, and so we obtain the variation δh ab = ∂ (a ξ b) . Had we not done this Lorentz gauge fixing, then the first term on the right-hand-side of (2.26) would have been replaced by a Lorentz transformation.
We have removed the dot above the equals sign in (2.25) since it holds as a usual equation.
To find the equations of motion, we have to eliminate the gauge transformations from the duality relations by taking derivatives and, at the same time, eliminating one of the two fields involved. In the case of (2.25), we can eliminate the gauge parameter ξ b 1 b 2 by taking an exterior derivative of E a,b 1 b 2 which produces By contracting a 1 with b 1 , we find that the term involving the dual graviton vanishes due to the fact that we have anti-symmetrised derivatives and A ab is symmetric. Thus, we find that which is the equation of motion for linearised gravity.
We can also write (2.27) as and so it vanishes when we contract it with η a 1 c 1 . As a result, we find that which we recognise as the equation of motion for the dual graviton at the linearised level, which agrees with the results of [17] where the full non-linear equation of motion was found and its linearised version was also given.
We will now carry out the same procedure for the duality relation involving the dual graviton and the first higher dual graviton (2.10). Using the gauge transformations in (2.23) and (2.24), we find that the duality relation becomes By taking two derivatives, we find that the gauge parameters disappear. We obtain which can also be written as The first term vanishes if we sum over e 1 and b 1 and also e 2 and b 2 . From this, we obtain which is indeed the correct equation of motion for the first higher dual graviton in four spacetime dimensions [19].
Clearly, the duality equation (2.12) between the graviton and the first higher dual graviton will also lead to the same equations since it can be deduced from the above duality relations.
However, it is instructive to treat this in the same way. Using the gauge transformation (2.24), we find that this duality relation is given by The gauge parameter ξ a is then eliminated by taking a derivative as follows: Contracting a 1 and b 1 allows us to discard the first term as it is the equation of motion for linearised gravity. We are left with the equation Then, taking one more derivative, we can eliminate the last gauge parameter to arrive at This is the correct equation of motion for the first higher dual graviton at the linearised level that we have also found in (2.34).
The equations of motion (2.28), (2.30) and (2.34) for the graviton, the dual graviton, and the first higher dual graviton, with their respective symmetry types , , and , are tracelessness equations that may be written in the form where Tr ij denotes a trace over columns i and j in a given Young diagram, and where we have introduced the curvature tensors for each field. They are given explicitly by As we have explained above, the duality relations only hold modulo gauge transformations, although the equations of motion derived from them hold exactly. One of the points of this paper is to obtain the extra fields that are required to have duality relations that also hold exactly. We will find evidence that these extra fields are contained in the second fundamental representation of A +++ 1 , denoted ℓ 2 . The content of this representation can be deduced by enlarging the A +++ 1 algebra by attaching an additional node to the node labelled 2 in the A +++ 1 Dynkin diagram, and then by taking only the generators of this enlarged algebra that have level one with respect to this new node. One may then deduce the commutation relations between generators in the adjoint and ℓ 2 representations of A +++ 1 by using the fact that the level is preserved and that the Jacobi identities must hold. One can then add new fields corresponding to the ℓ 2 generators and deduce their A +++ 1 transformations from their commutation relations.
As the role of the new fields is to soak up the gauge transformations in the duality relations, the next step must be to propose their gauge transformations. This involves writing down the variation of the ℓ 2 fields in terms of the derivative, which belongs to ℓ 1 , acting on the gauge parameters that also belong to the ℓ 1 representation. This transformation can be deduced using level matching and group theory. Given these transformations, one can then finally obtain new duality relations which hold as exact equations, at least in principle, and in detail at low levels.
We leave this calculation to a future paper.

Higher dualisations of linearised gravity
In this section we give an action principle in four dimensions for the first higher dual graviton A ab,cd whose equations of motion and gauge transformations were obtained from the non-linear realisation of A +++ 1 ⋉ ℓ 1 in the previous section. We will only be concerned with free dynamics and we will build the action principle for the dual field A ab,cd ≡ A [ab],cd ≡ A ab,(cd) using the off-shell dualisation procedure proposed in [9]. In that paper, a field-theoretical interpretation was given for an infinite subset of E 11 generators that transform in the GL(11)-irreducible representations whose Young tableaux are given in column notation as In what follows, we first recall the basic ideas behind the parent action procedure to derive dual actions for linearised gravity in any dimension D , and then we will direct our attention to the four-dimensional case for which A +++ 1 is the relevant Kac-Moody algebra.

From the graviton to the dual graviton
Off-shell dualisation of linearised gravity h a b around D-dimensional Minkowski space-time was initiated in [1] and [12]. This was investigated further in [13] where the authors made contact with the Curtright action [10] and generalised this duality to higher-spin fields with spin s > 2 .
Although the analysis of [1] began with the fully non-linear Einstein-Hilbert action, it is only for its linearisation that one can make the dual graviton and all of its higher dual generalisations appear off-shell [9]. Following the original idea [1], consider the second order Einstein-Hilbert action based on the vielbein e µ a : Indeed, the field equation of Y ab;c can solved for Y ab;c in terms of Ω(e) which yields the parent action linearised around Minkowski spacetime, where e µ a = δ µ a + h µ a , reads where Ω ab,c (h) := 2 ∂ [a h b]c and the field h ab has no symmetry on its two indices. The equation of motion for h ab yields The Poincaré lemma implies that the dual field This new field is completely antisymmetric in its first D −3 indices but it has no definite GL(D) symmetry otherwise: Inserting this back into the linearisation of (3.6) produces a consistent quadratic action S[C] that describes linearised gravity by construction. Note that the field h ab acted as a Lagrange multiplier for the constraint (3.7). It is not an auxiliary field like Y a 1 ...a D−2 ;b is, but the dual action obtained by substituting (3.8) inside the parent action (3.6) is classically equivalent to the original linearised Einstein-Hilbert action. The reader might want to see [29] for more comments on this issue.
Until now, the dual field C a 1 ...a D−3 ;b as defined in (3.8)  However, one may check [12,13] that, after inserting (3.8) into (3.6), the resulting action S[C] is invariant under a shift symmetry inherited from the local Lorentz symmetry that we call the dual graviton [1,12,13]. In the antisymmetric convention for Young tableaux, the GL(11) irreducibility condition of the dual graviton is the over-antisymmetrisation identity It is important to stress the fact that the dynamics of linearised gravity around Minkowski space-time, as given by the variational principle based on the original Fierz-Pauli action, can equivalently be described from the dual action principle S[A a 1 ...a D−3 ,b ] given in [13]. The reason is that both the Fierz-Pauli action and the dual action appear upon elimination of different fields from the same parent action. Moreover, as explained in [13], the dual graviton in four dimensions is a symmetric field A ab = A (ab) and the dual action S[A ab ] reproduces the standard Fierz-Pauli action. In D = 4, one concludes that "Fierz-Pauli is dual to Fierz-Pauli" [13].
In the next part, we review the dualisation procedure first explained in [9], which takes the and produces a dual action featuring the first higher dual graviton c as well as an extra field that cannot be eliminated from the action. In four dimensions, the first higher dual graviton A a 1 a 2 ,bc corresponds to the A +++ 1 generator R a 1 a 2 ,(bc) at level 2. Therefore, this approach makes direct contact with the previous section where the non-linear realisation of A +++ 1 ⋉ ℓ 1 was reviewed. The extra fields that enter each higher dual action principle will then be shown to be closely correlated with the ℓ 2 representation of A +++ 1 .
Although they are not needed in order to write down self-duality equations, they are necessary for the off-shell formulation of various generations of higher dual graviton fields.
3.2 The first higher dual graviton in four dimensions Action principle. As explained in [13], around Minkowski spacetime of dimension D = 4, the dual graviton A a 1 ...a D−3 ,b is a symmetric rank-2 tensor A ab ≡ A (ab) and the dual action is just the Fierz-Pauli action given as follows, up to boundary terms that we neglect: We stress that the curl Ω ab,c (A) := 2 ∂ [a A b]c is not featured in this formulation of the Fierz-Pauli action. Instead, it features the full gradient G a;bc (A) := ∂ a A bc without any antisymmetrisation over indices. As proposed in [9], we define the following parent action S[G a;bc , D ab; cd ] : featuring the two independent fields G a;bc = G a;(bc) and D ab; (cd) . The latter of these two fields is defined up to a gauge transformation it is easy to see that the parent action (3.12) is invariant under the combined transformations On the one hand, one can vary the parent action (3.12) with respect to the GL(4)-reducible field D da;bc that acts as a Lagrange multiplier for the constraint ∂ [a G d];bc = 0 . This constraint is identically solved by for some symmetric tensor A bc . Substituting G a;bc (A) for G a;bc inside the parent action (3.12) reproduces the original Fierz-Pauli action (3.11).
On the other hand, in the parent action (3.12), the independent field G a;bc can be considered to be an auxiliary field. Its equation of motion c)e (3.18) can be solved algebraically to express G a;bc in terms of D ab; cd as follows: Upon substituting this expression for G a;bc into the parent action (3.12), we obtain the following alternative description of linearised gravity around four-dimensional Minkowski spacetime: This action is invariant under the gauge transformation (3.16). We emphasise that (3.20) describes the same free graviton dynamics as the Fierz-Pauli action (3.11). The reason is that both action principles arise from the same parent action S[G, D] when it is extremised with respect to one field or the other. Note that the spectrum of fields is in one-to-one correspondence with those that are obtained by taking the tensor product of a 2-form with a symmetric rank-2 tensor. This is depicted in terms of Young tableaux as follows: In what follows, we decompose the dual field D ab; cd into GL (4) (3.24) Hodge dualising X ab; cd and Z a; c on their lower indices produces the GL(4)-irreducible fields These fields satisfy GL(4) irreducibility conditions in the antisymmetric convention for Young tableaux. That is to say, they satisfy the over-antisymmetrisation identities:

This action is invariant under the gauge transformations
where the gauge parameters λ abc and µ ab,c are GL(4)-irreducible: , µ (ab,c) ≡ 0 . In terms of this equivalent representation for the gauge parameter, one has Notice that λ abc and m ab,c ∼ µ ab,c match the gauge parameters Λ a 1 a 2 a 3 and Λ a 1 a 2 ,b at level 2 in the ℓ 1 representation of A +++ 1 given in (2.22). Up to trivial gauge parameter redefinitions, the transformation law of A ab,cd with respect to m ab,c and λ abc fully agrees with (2.24).
Off-shell dualisation is a different approach to the A +++ is closely related to extra fields that appear during off-shell dualisation.
Therefore, we expect to obtain (3.33) and (3.34) from the non-linear realisation by modifying it in a suitable way that incorporates the ℓ 2 representation.
The gauge parameters λ abc and µ ab,c arise from the decomposition Θ abc; de = 2 ε abci (−λ dei + µ de,i ) , (3.35) so that the Θ abc; de part of the gauge transformation in (3.16) reads We stress that this dual action principle (3.28)-(3.30) describes equivalent dynamics to the Fierz-Pauli action principle. Namely, it propagates a single graviton in four-dimensional Minkowski spacetime. It is an alternative off-shell description of linearised gravity and we will further analyse this action principle in Section 4. In that section, in order to make contact with the Labastida formulation for a gauge field with the symmetries of the first higher-dual graviton, we need to change convention for Young tableau. We refer to Appendix A for this change of convention.
Note that the field content of the theory {A ab,cd , Z abc,d } is in one-to-one correspondence with the set of Young diagrams obtained in the tensor product This depicts the set of GL(4)-irreducible tensors that are contained in the reducible tensor D ab;cd := 1 2 ε abij D ij; cd . (3.38) In Section 4, we will build two gauge invariant curvature tensors that do not vanish on-shell.
Anticipating this (more technical) result, the curvature for the gauge field A ab,cd starts like where ". . ." is used to denote terms that involve the field Z abc,d and where it is understood that indices with the same letters are antisymmetrised. For example, . Then, in that same section, we show that the field equations for A ab,cd are equivalent to As demonstrated in [19,20], this form of field equation is precisely what one should have for a mixed-symmetric gauge field A ab,cd that propagates non-trivially in four dimensional Minkowski spacetime. It is of higher-derivative type for a gauge field with more than two columns in its Young tableau representation, but a partial gauge-fixing procedure was found in [20] that brings such higher-derivative field equations down to the two-derivative equations (for bosonic fields) postulated in [22,23]. with respect to its GL(4) subalgebra (see Table 2).

Field theoretical analysis at higher levels
After having discussed, in great detail, off-shell dualisation from the dual graviton A ab = A (ab) to the first higher dual graviton A ab,cd , we may now proceed to the next step in the off-shell dualisation procedure. Recall that the dualisation procedure at level one transformed our set of fields from a symmetric tensor A ab to the GL(4)-reducible field D ab; cd whose Hodge dual This demonstrates how to label Young tableaux either by the heights of their columns as in (cde,f ) ≡ 0 . This decomposition into GL(4)-irreducible components becomes where · · · denotes projection onto irreducible components. Indices a 1 a 2 are antisymmetric and indices c 1 c 2 c 3 are symmetric. The GL(4) irreducibility conditions are together with the tracelessness constraints Projecting onto the symmetry of the final four indices, we find with inverse formulas Previously, in order to dualise A ab off-shell, we decomposed D ab; cd into traceless components {X, Z} , whose Hodge duals are the irreducible components of the Hodge dual D ij;cd of D ab; cd .
In order to make contact with the E -theory literature expressed using fields and generators in the antisymmetric convention, we will do something similar to {X, Y, Z, W } . Hodge dualising all of them on their first blocks of indices creates GL(4)-irreducible fields in the symmetric convention, which may then be written in the antisymmetric convention with fields labelled {A , Y , Z, W } . The full calculation is given in Appendix A, from which we find Off-shell dualisation from the dual graviton to the first higher dual graviton is given by where D A denotes one round of off-shell dualisation applied only to the dual graviton at the previous level. At the next level, dualising only the first higher dual graviton A a 1 a 2 ,bc produces a new action D 2 A (S[A ab ]) = D A (S[A ab,cd , Z abc,d ]) with the following set of fields The pattern is starting to become clear now. Label groups of k symmetric and k antisymmetric indices by a(k) and a[k], respectively. For example, After dualising the (n − 1) th higher dual graviton A a 1 [2],a 2 [2], ... ,a n−1 [2],c (2) , the set of independent fields will contain the n th higher dual graviton , ... ,a n−1 [2],a n [2],c(2) ∼ a 1 1 a 2 1 · · · a n 1 c 1 c 2 a 1 2 a 2 2 · · · a n which generalises (3.43). Ultimately, at the n th level of higher dualisation, the action will be given in terms of the following set of independent fields: In parallel with the A (n) family of dual graviton fields, the Z (n) family of extra fields starts to appear at the first level of higher dualisation when Z abc,d enters the action, while the Y (n) and W (n) families both enter the action at the second level. With index structure explicit, we see that there is only A ab ∼ A (0) [1,1] at level zero, i.e. at the level of the usual dual graviton. , [4,1,1] , P [4,2] , Q (2) [3,3] , (3.76) or equivalently as , [4,2,1,1] , P , (3.83) or equivalently as Let's take inventory. At levels one and two, dualising every field at every level, we have whereas the third level is visualised as The third higher dual graviton which consists of the fields A (4) , Y (4) , Z (4) and W (4) . In addition to this, the extra fields at the third level of higher dualisation can also be dualised off-shell, and they produce the following fields at the fourth level of higher dualisation: and respective multiplicities in the order presented above (1 , 4 , 3 , 12 , 12 , 3 , 8 , 2 , 24 , 6 , 6 , 12 , 10 , 9 , 6) .
The number of families of fields will clearly continue to increase with further dualisation.
Finally, note that W (n) is the same as A (n−2) with four extra antisymmetric indices, so they are Hodge dual. Dualising every field at every stage but only keeping track of the A (n) and W (n) families, we know that off-shell dualisation of A (n) produces both A (n+1) and W (n+1) = * A (n−1) .
At the n th level of higher dualisation, we have one copy of A (n) and at least one copy of the k th Hodge dual A (n−2k) for every positive integer k such that 0 ≤ n − 2k ≤ n . When n is even, we find that our set of independent fields contains A (0) , A (2) , A (4) , . . . , A (n) as a subset.
Similarly, when n is odd, we find that it contains A (1) , A  with respect to its A 3 subalgebra. The structure of A +++ 1 is studied at each level by looking at the representation content (i.e. the weight space) of the A 3 subalgebra. Any generic A 3 weight can be expressed as

Contact with
where λ i is the i th fundamental weight of A 3 . This weight may also be written as λ = [p 1 , p 2 , p 3 ] and we may depict this weight (and its corresponding representation) by a Young diagram with p 3 columns of height 1, p 2 columns of height 2, and p 1 columns of height 3. With this, we have roots so that the new simple root α * is included.
In previous sections, we have found the extra fields appearing in the action principles and duality relations at low levels. Now we are finally ready to show, level-by-level, that off-shell dualisation produces a set of extra fields that is closely correlated with the ℓ 2 representation.
In particular, at the n th level of higher dualisation, we count fields that appear in the adjoint representation at level n + 1 and in the ℓ 2 representation at level n. This will then be compared against the set of extra fields that we obtain by off-shell dualising every field at every level.
In Table 3, the 'adj' and 'ℓ 2 ' columns contain the field multiplicities in the adjoint and ℓ 2 representations, respectively, and the 'total' column gives their sum. The 'maximal off-shell' column tells us how many of each field is found by off-shell dualising every field at every level.
Lastly, the 'net' column is the 'maximal off-shell' column minus the 'total' column. It tells us if we have too many, too few, or the right amount of fields with maximal off-shell dualisation.
From gravity to dual gravity. Recall the dualisation of gravity in Section 3.1 which gave us the dual graviton with GL(4)-irreducible symmetry type Y[D − 3, 1] and an extra (D − 2)-form which may be gauged away with a shift symmetry [1]. In four dimensions, the decomposition gives us a symmetric rank-2 field A ab and a 2-form field. The symmetric field is the familiar dual graviton a 0 := A (0) [1,1] in the adjoint representation of A +++ 1 at level 1 in Table 1, and the  [2] is found in the ℓ 2 representation at level 0 in Table 2.
In contrast to what happens later, we do not dualise the extra 2-form field as it can be shifted away.
The weight [0, 1, 2] in the adjoint representation at level 2 (see Table 1) corresponds to the first higher dual graviton field a 1 := A (1) [2,1,1] , and the weight [1, 0, 1] in the ℓ 2 representation at level 1 (see Table 2) corresponds to the extra field b := Z (1) [3,1] required to build a consistent dual action principle. So, at the first level of higher dualisation, we see the complete correspondence between the fields produced during off-shell dualisation and the generators of the adjoint and There is a perfect match at this level. In Table 3, we see that b appears zero times in the adjoint at level 2 and once at level 1 in ℓ 2 . It is also required exactly once at this off-shell dualisation at this level, hence the zero in the 'net' column. Note that, although they both have the same A 3 weight [1, 0, 1], the two fields b and d 8 in ℓ 2 at levels 1 and 3, respectively, are not compared or counted together since they appear at different levels.
In fact, d 8 should be viewed as a Y[4, 3, 1] field.
The second level of higher dualisation. Taking the first higher dual graviton A [3,2,1] and W (2) [4, 1,1] . This is the minimal off-shell dualisation of linearised gravity. In addition, we may dualise the extra field from the   Since it does not appear in the adjoint, c 1 appears only once in our tables. Moreover, c 1 appears precisely once in maximal off-shell dualisation at this level, so there is a perfect match for c 1 .
Moving onto the next field, we see in Table 3 that c 2 := Z (2) [3,2,1] appears once in the adjoint and once in the ℓ 2 , and it also appears twice in maximal off-shell dualisation: once from the dualisation of a 1 and once more from the dualisation of b . Another perfect match. The reader might like to check that c 4 := P (2) [4,2] gives yet another match. Unfortunately, we do not find a match for every extra field. For example, c 3 := W (2) [4,1,1] does not appear in the adjoint, and it appears once in the ℓ 2 . However, two of them are required for maximal off-shell dualisation. In other words, although ℓ 2 contains enough for the minimal off-shell description, we go slightly over when we dualise every field at every level. It is even more peculiar with c 5 := Q (2) [3,3] since it is not contained in the tables at all, yet it is needed for There is a perfect match for the GL(D) types of fields required but, as for multiplicities, we appear to be lacking a small number of fields in the tables. One possible solution could be to dualise some extra fields but not all of them. By carefully selecting which fields to dualise, this would provide an off-shell description of gravity that does not exceed the field content of the adjoint and ℓ 2 representations of A +++ 1 but nonetheless, within this restriction, as many fields as possible are dualised. This lies somewhere between the minimal and maximal off-shell dualisations, and we call it the optimal off-shell dualisation of linearised gravity.
The third level of higher dualisation. Moving onto the next level, we find that maximal off-shell dualisation produces the fields in (3.84) with Young tableaux (3.87). This set of fields contains the third higher dual graviton a 3 := A [2,2,2, 1,1] in the adjoint at level 4, and a number of extra fields that are obtained by dualising the set independent fields {a 2 , c 1 , . . . , c 5 } from the previous level. As before, looking at Table 3, we find that some fields are a perfect match and some are not. In fact, for almost all of the extra fields introduced at this level, we have a surplus of fields in the maximal off-shell description compared with the fields that are available from the adjoint and ℓ 2 representations. With the exception of the rogue scalar field d 9 := O (3) [4,4] , the Young tableaux for the extra fields perfectly match those of the spectrum of ℓ 2 at level 3. Rogue scalars like this one are found in the maximal off-shell description at all odd levels of higher dualisation greater than or equal to this level.
Optimal off-shell dualisation. In order to understand the differences between these higher dualisation schemes, it is useful to give examples at low levels. All three of them coincide at the first level of higher dualisation where the dual graviton A ab is dualised to give A ab,cd and so optimal off-shell dualisation is attained by choosing any one of these six fields not to dualise at the previous level. This is important: optimal off-shell dualisation is, in general, not unique.
Then again, we do not yet know what will happen at higher levels. It is possible that these various pathways to optimal off-shell dualisation may converge at higher levels. It would be interesting to draw the graph of optimal pathways at higher levels and to study its topology. root (0, 0, n, n + 1) whose squared length is equal to 2. That is, the n th higher dual graviton appears in the adjoint at level n + 1.
However, even at low levels, this will turn out not to produce the entire spectrum of ℓ 2 and, indeed, less than half of the spectrum of ℓ 2 at level 3 is found if we only dualise the A (n) fields.
The A +++ 1 algebra has been shown to contain the minimal off-shell dualisation of linearised gravity in four dimensions. Of course, extra fields may also be dualised off-shell, but dualising too many of them leads to field multiplicities that exceed those provided by the adjoint and Maximal off-shell dualisation contains too many fields, but it is quite interesting nonetheless. Despite some discrepancies in the multiplicities of Table 3, the correct Young tableaux shapes appear in this maximal scheme. More work is needed to fully understand the role of the rogue scalar O [4,4] at the third level of higher dualisation, and the other scalars at odd higher levels of dualisation. The situation at the fourth level of maximal off-shell dualisation is more severe with fields that have a surplus as high as +7. However, ignoring multiplicities, the Young tableaux at this level in the maximal off-shell description perfectly matches the spectrum of ℓ 2 at level 4.
In this section, we have identified a possible solution to the tricky problem of mismatched multiplicities at each level: optimal off-shell dualisation. In this scheme, one carefully chooses which extra fields to dualise so that, at each level of higher dualisation, the set of extra fields is contained in the relevant representations of A +++ 1 with multiplicities that do not exceed those in the 'total' column in Table 3. In this section, at the first level of higher dualisation, we show that the fields A ab,cd and Z abc,d may be repackaged into new fields: A ab and A ab,cd with the respective symmetry types of A (0) = A ab and A (1) = A ab,cd . We will show that the gauge transformation laws of these two fields are almost identical to those of the Fierz-Pauli field for A ab and the Labastida gauge field [23] with symmetry type Y(3, 1) = Y[2, 1, 1] for A ab,cd , with additional terms that entangle the two gauge transformation laws. In order to make contact with the Labastida formalism where mixed-symmetry fields are given in the symmetric convention for Young tableaux, we will also use this convention for the first higher dual graviton in this section. However, it should be noted that this convention is not used in the context of E -theory.

The graviton tower action at low levels
The equivalent formulation we will present for the action of the first higher dual graviton in terms of A ab,cd and A ab has the advantage of showing more explicitly the number of degrees of freedom through an on-shell duality relation between the gauge invariant curvature tensors of the two fields, as is usual in this context [11,19,34].
Change of variables. Recall that the fields appearing in the action (3.24) were X ab; ij and Z a; e , the latter being the Hodge dual of Z bcd,e , see (3.25 From the independent field variables X ab; ij and Z a; e we introduce the two-form field that transforms like while we recall from Section 3.2 that the field φ abc, d transforms like We define the Y(3, 1)-type gauge field φ abc,d := φ abc,d + 3 4 η (ab U c)d (4.4) that transforms like where λ abc := λ abc − 1 4 η (ab τ c) , µ ab,c := µ ab,c + 1 6 η ab τ c − η c(a τ b) . (4.6) The advantage of this change of variable is that the newly defined gauge parameters λ abc and µ ab,c have the same trace: This also implies that, among the three linearly independent gauge fields { φ abc,d , U ab , Z (a;b) }, We now express the action (3.24) in terms of the independent fields X ab; cd , U ab and f ab := Z (a;b) : Substituting this into the parent Lagrangian, we obtain the following dual Lagrangian which is given, up to a total derivative, by This action is invariant under δf ab = 1 4 ε a cde ∂ c µ bd,e + ε b cde ∂ c µ ad,e + ∂ (a ǫ b) − 1 4 η ab ∂ c ǫ c , (4.14) Finally, we combine the scalar field S with the traceless symmetric field f ab := Z (a;b) to get We therefore obtain a new action S[ A ab , A ab,cd ] which takes the form and X a 1 a 2 ; are understood throughout. This allows us to write the repackaged first higher dual graviton A ab,cd in a variety of useful ways. For example, it was convenient to write the above action in terms of A ab and X ab; cd . It is invariant under the following intertwined gauge transformations: Trivially, one would need to make use of (4.18) before checking gauge invariance under (4.21).
The λ abc and µ ab,c parts of the gauge transformations for φ abc,d coincide with the Labstida gauge transformations for a gauge field of type Y(3, 1). In particular, the two gauge parameters are constrained to have equal trace. The ǫ a part of the gauge transformations for the (traceful) symmetric rank-two tensor A ab corresponds to linearised diffeomorphisms. However, notice that A ab also transforms with the µ ab,c gauge parameter, and that φ abc,d transforms with the gauge parameter ǫ a . As we have seen in [30] for higher dualisation of gauge fields, we find that the action contains fields that resemble the original dual graviton A ab and the first higher dual graviton φ abc,d with entangled gauge transformation laws. By the construction of our dual action using the parent action procedure, we know that the on-shell degrees of freedom are only those of a single massless spin-2 field around four-dimensional Minkowksi spacetime.
Nevertheless, we will rederive this fact from the field equations. It is clear that the dual action is more than just the sum of the Fierz-Pauli and Labastida actions.
It is also important to remember that this repackaging approach seeks to drastically redefine our fields for reasons that will become clear towards the end of this section. As a result, gauge transformations (4.20) and (4.21) are not expected to resemble the gauge transformations for A ab and A ab,cd that were found in Section 2.2 and Section 3.2.
Field equations. The equations of motion for the fields A ab and φ abc,d are given by We find that the left-hand side of the field equation for A ab is related to the trace of K ma,nb : where K ab := η mn K am,bn and K := η ab K ab . Obviously, on-shell, we have K ab ≈ 0 which is to be compared with the field equation (2.28) in Section 2.3. This is analogous to the Ricci-flat equation in linearised gravity.
We also have the following gauge-invariant quantity with three derivatives: Therefore, on-shell, we find the following relation that will be instrumental in showing that the degrees of freedom are those of a single graviton: On-shell duality relation. We find that the gauge-invariant tensor G ab,cd; e is related to K ab,cd and the left-hand-sides of the equations of motion in the following way: These equations are important in several respects. The tensors K ab,cd and G abc,de,f g can be called the field strengths for the repackaged dual graviton and first higher dual graviton, resepctively. Indeed, they do not vanish on-shell and they are gauge invariant. This field equation completes those found in Section 2.3.
With these field equations, we have found a strong parallel with the analogous equations derived in Section 2.3. However, since no action principle was considered in that section, each field strength was a function of a single field. Instead, in the off-shell formulation found in the present section that requires the extra field Z abc,d to be repackaged, equations necessarily entangle both fields due to the nature of the gauge transformation laws.
We conjecture that this dualisation and repackaging procedure creates an increasingly tall tower of new repackaged dual gravitons whose gauge transformation laws are intertwined. In particular, for a given tower with highest level N, the field φ (n) at level n ≤ N should transform as a Labastida gauge field of symmetry type Y[2, . . . , 2, 1, 1] . Its gauge transformation law should contain terms that entangle it with the repackaged fields at every level lower than n .
Moreover, if n < N , then its gauge transformation law will also be entangled with that of the repackaged field at level n + 1 . It may even be possible to redefine fields so that the repackaged dual graviton at level n is entangled only with those at level n + 1 and n − 1 .
For the graviton tower action S[ A ab , A ab,cd ] in (4.17), the extra field Z abc,d was completely hidden by the specific field and gauge parameter redefinitions used to construct A ab . However, this may only be possible at low levels, so we cannot yet exclude the possibility that some extra fields may still be present in the graviton tower actions at higher levels.
The non-linear realisation contains an infinite number of dualisations of gravity. It consists of an infinite set of duality relations, the first of which involves only the graviton and the dual graviton. This relation was worked out at the full non-linear level in [17,18]. In Section 2, we have used the non-linear realisation to work out the linearised equations of motion for the first higher dual graviton.
While on the other hand, in [9] it was shown that pure linearised gravity could be described by any member of an infinite family of action principles, each involving more and more fields.
Some of these fields were shown in [9] to have a direct connection with the adjoint representation of the very-extended A +++ D−3 algebra, while other fields received no interpretation at that time. In the present paper, where we focus on D = 4 for the sake of concreteness, we showed that the aforementioned fields are all associated with generators in the ℓ 2 representation of A +++ 1 in the sense that there exist generators in ℓ 2 that have the same GL(4) types. We have carried out this match up to level four and, while there is a striking agreement at low levels, some of the multiplicities differ for the extra fields.
We also constructed, at the level of the first higher dual graviton, a new action principle featuring two fields A ab and A ab,cd with the GL(4) symmetry types Y[1, 1] and Y[2, 1, 1] of the dual graviton and the first higher dual graviton, respectively. The gauge transformations of these two fields are those of the dual graviton and the corresponding Labastida field, along with extra terms that entangle the two fields. Remarkably, the field equations can be obtained from a duality relation between the gauge invariant curvatures of these repackaged fields, which further demonstrates that our original action only propagates a single graviton.
That the field equations can be encapsulated in a set of duality relations is in full agreement with the method of obtaining the field equations in the non-linear realisation of A +++ 1 ⋉ ℓ 1 .
In a future work in preparation, we will extend our analysis to pure gravity in five dimensions where the relevant algebra is A +++ 2 . We will also consider pure gravity and the bosonic sector of maximal supergravity in eleven dimensions. It is well-known that the relevant Kac-Moody algebras for these theories are A +++ 8 and E 11 , respectively. There, we will also show how their ℓ 2 representations are related to the set of off-shell fields entering higher dual action principles.
It will be important to modify the coset space used to construct the non-linear realisation for these algebras in order to incorporate ℓ 2 . Consequently, this will account for the extra fields that were thought to be missing from E -theory until now.
Finally, it would be interesting to make a contact with [35] where the importance of the ℓ 2 representation of E 11 was noticed in a similar context. It is not yet clear to us that there is a connection since their equations of motion are obtained from the E 11 pseudo -Lagrangian by a variational principle supplemented by extra duality relations that are not derived by variation.
More specifically, variations with respect to constrained fields (which carry a section constraint index) vanish only when these extra duality relations are imposed. In contrast, our off-shell dualisation approach produces equations of motion and duality relations that are all obtained by varying dual actions. Nothing external needs to be imposed here. Another line of research is to investigate the possible non-linear extensions of the higher dual actions considered here.
B Representations of A +++ 1 at the next level