The non-linear dual gravity equation of motion in eleven dimensions

We derive the non-linear dual graviton equation of motion in eleven dimensions in the context of E theory.


Introduction
The classic paper of Montonen and Olive [1] suggested that spontaneously broken SU(2) Yang-Mills theory with a triplet Higgs possessed a duality symmetry that transformed the electric particles into the magnetic particles. Duality symmetries also play an important role in supergravity theories but in these theories one finds that the duality transformations often exchange form fields of different rank instead of the vector potentials in the case considered by Montonen and Olive. The non-linear realisation of E 11 ⊗ s l 1 is a theory that contains all the maximal supergravity theories, depending on the decomposition of E 11 that one takes [2,3]. Contained in this symmetry are the E 11−D (Cremmer-Julia) symmetry [4] of the maximally supergravity theories in D dimensions as well as the Sl(2,R) symmetry [5] of IIB supergravity. However, the E 11 symmetry also contains symmetries that transform the fields of different spin into each other [6]. For example, in eleven dimensions the symmetry transforms the graviton field into the three form which also transforms into the six form field etc. The non-linear realisation of E 11 ⊗ s l 1 contains an infinite number of fields quite a few of these correspond to the infinitely many different ways of describing the some on-shell state [7,8]. For example the on-shell states described by the three form can also be described by a six form field but also by the field A a 1 ...a 9 ,b 1 b 2 b 3 and indeed any such field that has any number of antisymmetric blocks of nine indices. The E 11 symmetry contains not only the known duality transformations but also an infinite number of new duality symmetries that transform these duality equivalent field descriptions into each other.
The non-linear realisation of E 11 ⊗ s l 1 contains the usual field of gravity which satisfies Einstein equation in the presence of the three form field. This statement holds if one keeps only the lowest level coordinates of spacetime which are just those of our familiar spacetime. The non-linear realisation of E 11 ⊗ s l 1 also contains the field of dual gravity. A field that described dual gravity was first proposed by Curtright in five dimensions, it was of the form h a 1 a 2 ,b [9] and it was later proposed that the field h a 1 ...a D−3 ,b described dual gravity in D dimensions [10]. In reference [6] the equation of motion of this field was found and it was shown to describe the degrees of freedom of gravity in D dimensions at the linearised level.
To find the non-linear equation obeyed by the dual graviton has proved much more difficult. Indeed a no go theorem has been proved [11]. However, discussions of duality in the context of gravity require such an equation to exist. Since the non-linear realisation of E 11 ⊗ s l 1 contains the field of dual gravity one should be able to deduce its equation of motion from the non-linear realisation. In particular in eleven dimensions one finds that the E 11 variation of the six form equation of motion contains the dual graviton equation of motion. The correct linearised dual gravity equation of motion was derived in this way at the linearised level in reference [16] while the dual graviton equation at the non-linear level was derived in reference [12]. This latter derivation did not appear to lead to a unique dual gravity equation of motion but one that was ambiguous up to certain types of terms which were albeit it of a very restricted type. In reference [12] this ambiguity was apparently resolved by insisting on the additional requirement of diffeomorphism invariance. Such additional requirements were not used in any other E 11 papers when deriving the equations of motion This paper also found the duality equation that is first order in spacetime derivatives which relates the gravity field to the dual gravity field.
It was apparent from reference [12] how the no go theorem of reference [11] is circumvented. Although the field h a 1 ...a 8 ,b on its own does correctly describe gravity at the linearised level this is not the case at the non-linear level. However, the dual gravity equation of motion that follows from the non-linear realisation of E 11 ⊗ s l 1 contains both the usual gravity field as well as the dual gravity field. This is to be expected. It is well known that the non-linear equation of motion of the six form must also contains the three form while the duality related equation of motion of the three form just contains the three form field.
Recently the non-linear realisation the semi-direct product of A +++ 1 and it first fundamental (vector) representation l 1 , denoted as A +++ 1 ⊗ s l 1 , was constructed [13]. This theory contains the graviton h ab , at level zero, and the dual gravitonh ab =h (ab) at the next level as well as higher level fields. The gravity and dual gravity equations of motion as well as the duality relation that relates the two fields were found. The dual graviton equation of motion was essentially unique provided we demanded that it had the same index structure as the dual graviton field, that is, it was symmetric in its two indices. Of course this is not so much a demand as a necessity. The results of this paper made it clear that while many of the results in reference [12] were correct the equation of motion of the dual gravity was not correct.
In this paper we revisit the calculations of reference [12] and consider again the E 11 variation of the six form equation of motion. We demand that the dual graviton equation E a 1 ...a 8 ,b carries the same index structure as the dual graviton field h a 1 ...a 8 ,b , that is, it obeys the condition E [a 1 ...a 8 ,b] = 0. The dual graviton equation is then essentially unique without the need for any extraneous additional requirements.

The calculation
The construction of the non-linear realisation of E 11 ⊗ s l 1 in eleven dimensions has been much studied and so we will not repeat it here. The reader is referred to references [2,3] and [12], the review of reference [14] and the book of reference [15] for the details. We will now just recall our essential starting points from reference [12]. The Cartan forms, up to level three, are given by [2,3] where the vierbein is given in terms of the field h a b by e µ a ≡ (e h ) µ a They are inert under rigid E 11 ⊗ s l 1 transformations but transform under I c (E 11 ) transformations as [2,3] The above formulae are true when the Cartan forms are written as forms, for example G a b = dz Π G Π,a b . We will convert their first world volume index into a tangent index by using the formula G A,α = E A Π G Π,α where E Π A is the vierbein on the spacetime encoded in the non-linear realisation. Under the I c (E 11 ) transformations this first index on the Cartn forms transforms as Thus the Cartan forms G A,α transform under equation (2.6) on their first (l 1 ) index and their E 11 indices transform as in equations (2.2) to (2.5).
In the papers in references [2,3,12] etc we have found the equations of motion only to lowest order in derivatives of the spacetime coordinates, that is, just with derivatives with respect to the usual coordinates of spacetime. However, according to equation (2.6) Cartan forms with level one derivatives can transform into Cartan forms with derivatives with respect to the level zero coordinates. Consequently to find the equations of motion to lowest order in the spacetime derivatives we need the equations that we are varying to contain the required derivatives with respect to the level one coordinates. As in the previous papers we denote the former quantities by just capital letters and the later by calligraphic letters. In reference [12] the six form equation of motion was found by varying the three form equation of motion, its precise form is given bŷ We have put a hat on the symbol for the six form equation as this will not be our final result. Its I c (E 11 ) variation can be written as [12] δÊ a 1 ...
where E a,bc is the gravity-dual gravity duality relation derived, for example, in reference [12] to be where the spin connection is given by and E µ 1 µ 2 µ 3 is the equation of motion of the three form gauge field which was given in references [6,2,3]. The dual graviton equation was found by reading off the coefficient of the parameter Λ c 1 c 2 c 3 in equation (2.8) and it was found to bê where + . . . mean terms that are constructed from the Cartan forms and are of the generic form f G b,• where f is any function of the fields of the non-linear realisation and G b,• is a Cartan form with • being any E 11 index. We refer to such terms as l 1 terms. The precise indices on this expression are not shown but they are to be arranged so that they are those of the dual gravity equation of motion.
The reason for this ambiguity is that one can add to the six from equation of motion of equation (2.7) terms of the form f G c 1 c 2 ,• where the c 1 c 2 indices correspond to a level one derivative with respect to the coordinate x c 1 c 2 . Using equation (1.6) we find that this varies into the expression 6f Λ c 1 c 2 b G b,• . Looking at the variation of the six form equation (2.8) we see that this corresponds to the ambiguity in the dual gravity equation (2.11). This ambiguity reflects the fact that we compute the equation of motion to only lowest level in the spacetime derivatives and in order to achieve this we must include terms in the object being varied that contain terms that have derivatives with respect to the level one coordinates.
An attempt to resolve this ambiguity was made in reference [12] by demanding diffeomorphism invariance. This assumed that the dual graviton field transformed under a general coordinate transformation as a standard tensor as indicated by its indices. However, this is not the case. The dual graviton is not just any matter field but it describes gravity in its own way which is different to that due to the usual graviton. As a result it is to be expected that it does not behave like any other matter field. The derivation of equation (2.11) is correct if, as was stated in the paper, one takes it to be subject to the ambiguity discussed above. However, the attempt to resolve the ambiguity by using standard diffeomorphism invariance was not correct and as a result the last term added to equation (2.11) in reference [12] was not correct.
Our task in this paper is to resolve the ambiguity and so derive the correct equation of motion for the dual graviton in eleven dimensions. The dual graviton equation must belong to the same representation of GL(11) as the dual graviton field. As h [a 1 ...a 8 ,b] = 0 the dual graviton equation must satisfy the condition Equation (2.11) of reference [12] does not satisfy this condition but we will, in this paper, use the ambiguity mentioned above to make it symmetric. A priori, it is far from clear that this will work but we will find that it does. We will refer to quantities that we add to the dual graviton equation that satisfy the condition of equation (2.12) as being symmetric. The first step is to rewrite equation (2.11) such that it involves the dual graviton Cartan form with tangent space indices, it becomeŝ We have written the dual graviton equation as a sum of two parts, the first of which and G 1,1,1 denote the dual gravity and gravity Cartan forms respectively. As above we will adopted the convention in this paper that the a 1 . . . a 8 indices contained in any equation are always completely antisymmetrised.
We begin by processing the G 1,1,1 G 1,8,1 terms which can be written aŝ ,c (2.14) where A1, A2, A3 and A4 are the terms in the first bracket in the order in which they occur.
The terms in the second bracket in equation (2.14) contain Cartan forms whose first index is a b which is contracted with the parameter Λ c 1 c 2 b in the variation of the six form equation (2.8). Such terms are l 1 terms and can, as we explained above, be removed from the dual graviton equation by adding terms to the six form equation of motion. The final term in the first bracket, the term A4, can be written as The first term in this equation is obviously symmetric as G c,[a 1 ...a 8 ,b] = 0 as the dual graviton field and its corresponding generator satisfy this irreducibility condition. We reserve the use of {} brackets to denote quantities that are symmetric as this will make it easier to keep track of them. The second term in this equation is also symmetric as the expression in the bracket can also be written as We are placing all l 1 terms in square brackets so that it will be easier to keep track of them.
The third term in the first bracket, the term A3, in equation (2.14) can also be written in a very similar way, namely The first two terms are symmetric and the last term is an l 1 term. The first and second terms in equation (2.14), that is, the terms A1 and A2, are not symmetric and we will return to them later. We will now analyse the ∂G 1,8,1 terms which are contained inÊ (1)a 1 ...a 8 ,b in equation (2.13). We can write these terms aŝ (det e) The first and third terms in equation (2.18) are obviously symmetric. The second terms is also symmetric for the same reason as outlined in equation (2.16). However, the remaining three terms are not symmetric.
Let us consider the fourth term which we can rewrite as (2.19) For a term to be an l 1 term it must, when multiplied by the parameter Λ c 1 c 2 b that arises in the variation of the six form equation (2.8), contain a factor of the form Λ c 1 c 2 b G b,• . The first term in the first line is an l 1 term and so can be removed from the dual gravity equation. However, the second term in the first line is not of this form as there is a derivative in between the parameter Λ c 1 c 2 b and the Cartan form G b,• . As a result we have rewritten the expression in the second line. Here the first term is an l 1 terms and so is the third term. If we multiply this term by the parameter Λ c 1 c 2 b we find the factor e ρ 1 c 1 e ρ 1 c 2 Λ ρ 1 ρ 2 ν and using the fact that the parameter with upper world indices is a constant we can take it past the derivative to find that that the term is indeed an l 1 term. Left over from the term of equation (2.19) is the second term which can be written as In this equation we find two l 1 terms and one term, denoted as M11, which will be needed later.
The last term in equation (2.18) can not be analysed in this way and we now use the fact that it can be further evaluated using the Maurer-Cartan equations of E 11 for the dual graviton Cartan form which we now derive . The Cartan forms of E 11 are contained in the expression V = g −1 E dg E where g E is the group element of E 11 . It obviously obeys the usual Maurer-Cartan equation dV + V ∧ V = 0. The precise expression for V in terms of the Cartan forms has been discussed in, for example, in references [2] and [12]. Using the E 11 algebra one can show that the last term in equation ( The terms in the first bracket are symmetric and the terms in the second line can be removed as they are l 1 terms. The third, fourth and fifth terms in equation (2.22) can be evaluated as follows M 3+M 4+M 5 = − 4 · 7 9 · 9 6G a 1 ,a 2 e G a 3 ,a 4 ...a 8 bec, c −G a 1 ,b e G a 2 ,a 3 ...a 8 ec, c +G a 1 ,c e G a 2 ,a 3 ...a 8 eb, The terms in curly brackets are symmetric and the remaining terms are l 1 terms that must be removed. (2.28) Where the first term is symmetric and the last term is an l 1 that can be removed. We note that the term The only terms we have not processed so far are the terms A1 and A2 of equation (2.14) and the term M10 of equation (2.20). We find that the A2 + M 10 can be written as In this equation, the first term we will be needed later, the middle term is a symmetric term and the last term is an l 1 term that we will remove.
The first term, A1, of equation (2.14) can be written as We can reformulate the first term in equation (2.30) to be given by where the gravity-dual gravity relation is given in equation (2.9). The last term is in fact symmetric.
After some work the second term (A.1.2) in equation (2.30) can be rewritten as Examining equation (2.32) we see that the first term cancels the first term in equation (2.29). The two terms in curly brackets are symmetric and the terms in the final two bracket are l 1 terms which can be removed. It may not be immediately apparent to the reader that the second term in curly brackets really is symmetric. This becomes obvious if one uses the irreducibility of the dual gravity Cartan form and in particular the identity G a 1 ,ca 2 ...a 8 ,e − G a 1 ,ea 2 ...a 8 ,c = 7G a 1 ,cea 2 ...a 7 ,a 8 (2.33) In addition to the above terms there are terms which we can add to the dual graviton equation of motion which are symmetric but at the same time are l 1 terms. Such terms must contain a b index as the first index on one of the two Cartan forms and an a 1 as the first index on the other Cartan. To saturate the remaining eleven indices we also need two summed over indices. The possible terms are

The dual gravity equation of motion
To find the dual graviton equation we just need to collect up all the symmetric parts given in the curly brackets in the previous section. To make the expression self contained we will write the so far surpressed antisymmetry on the a indices and drop the curly brackets. The dual graviton equation is given by (det e)  As we have discussed varying the six form equation of motion we can find the dual gravity equation of motion. However, this equation is only determined up to the presence of certain terms. In this paper we have resolved this ambiguity by demanding that the dual graviton equation have the same symmetries as the dual graviton field. The l 1 terms we have added, or subtracted, to the dual gravity equation in order to make it symmetric are contained in the terms in the square brackets given in the previous section. To complete the calculation we have to list the changes to the six form equation (2.6) that result to these terms in the dual graviton equation through the variation of the six form equation of motion given in equation (2.8). The l 1 terms contain derivatives with respect to the level one derivatives and so they do not change the parts of the six form equation that contain only derivative with respect to the usual coordinates of spacetime, that is, the part we are familiar with.
Let us give an example, the final term in equation (2.15) is such an l 1 term, and this occurs in the first term of the right-hand side of the variation of the six form equation Thus term will be removed in the dual graviton equation of motion by adding to six form equation equation (2.7) a term with level two derivatives whose variation under (2.6) is equal to this term with the opposite sign. The result is that we must add to the six form equation of motion (2.7) the term −2G d,e e G c 1 c 2 , a 1 ...a 6 c 1 c 2 ,d . 3) The coefficient −2 = −432 · 1 4·9 · 1 6 where the one over six comes from the variation in equation (2.6).
A similar procedure holds for every l 1 term that arose throughout section 2.The resulting l 1 extension of the six form equation of motion (2.7) is given by where we have definedĜ τ ,µ ν = (∂ τ e ρ b )e b ν ,Ĝ τ,µ 1 µ 2 µ 3 = ∂ τ A µ 1 µ 2 µ 3 , In these definitions we have removed the (det e) which describes just gravity in eleven dimensions. As such to obtain the dual gravity equation contained in this theory one just has to set to zero the three form and six form gauge fields in equation (3.1). It would be interesting to find the diffeomorphism and dual gravity gauge transformations that leave equation (3.1) invariant. From this one could understand the geometry that describes a dually symmetric theory of gravity. It would be interesting to carry out the I c (E 11 ) variation of the dual gravity equation (3.1) to find the non-linear level four equation of the non-linear realisation. This would also resolve if the terms of equation (2.34) are present or not.