Bosonic D=11 supergravity from a generalized Chern-Simons action

It is shown that the action of the bosonic sector of D=11 supergravity may be obtained by means of a suitable scaling of the originally dimensionless fields of a generalized Chern-Simons action. This follows from the eleven-form CS-potential of the most general linear combination of closed, gauge invariant twelve-forms involving the sp(32)-valued two-form curvatures supplemented by a three-form field. In this construction, the role of the skewsymmetric four-index auxiliary function needed for the first order formulation of $D=11$ supergravity is played by the gauge field associated with the five Lorentz indices generator of the bosonic sp(32) subalgebra of osp(1|32).


Introduction
can be written in matrix form as where ξ is a two-component Grassmann odd Majorana spinor form and γ a are the 2 × 2 D = 2 gamma matrices. Note that osp(1|2) alone would not provide enough fields for D = 3 supergravity and that all fields f a , ξ and f a in (1.3) are necessarily dimensionless; to define 'physical' one-form fields, we introduce a scale parameter λ, [λ] = L −1 . We use geometrized units for which c = 1 = G, so that all the quantities have physical dimensions in terms of powers of length; with them, the dimensions of an action in D-dimensional spacetime is L (D−2) . The new fields ω a , e a , and ψ obtained from f , ξ and f are then defined by f a = ω a + λe a , f a = ω a , ξ = λ where H 0 = H 0 (ω, α) only since H(ω, e, ψ; λ, α) is dimensionless and H 1,2,3 = H 1,2,3 (α). We note in passing that this re-scaling in λ is the starting point of the (super)Lie algebra expansions procedure, introduced in [15] and considered in general in [16], by which new (super)algebras may be obtained from a given one. Note that, unlike in the contraction of algebras, where the dimensions of the original algebra and that of the contracted one are necessarily equal, the dimension of the expanded algebra is usually higher since the expansion process is not dimension-preserving in general 1 (see [16,17] for details). By construction, the above two-form H and the associated CS action are osp(1|2) ⊕ sp(2) gauge-invariant. In particular, the local supersymmetry transformations under the odd dimensionless gauge parameter η that corresponds to the gauge field ξ are, written in terms of ǫ = λ − 1 2 η, [ǫ] = L 1/2 , δ ǫ e a = ψγ a ǫ , δ ǫ ψ = Dǫ + λe a γ a ǫ , δ ǫ ω a = 0 , (1.8) where D = d − ω a γ a is the Lorentz covariant derivative. Since ω a is supersymmetry invariant, so is H 0 which only contains this field. Thus, the action obtained from H(ω, e, ψ; λ, α) − H 0 (ω, α) is invariant under the local supersymmetry transformations (1.8), and provides the first order formulation of (1, 0) D=3 AdS supergravity. Moreover, the leading λ term in H − H 0 , H 1 , is also invariant under the transformations (1.8) for λ = 0, and hence provides the action for D = 3 Poincaré supergravity; this will not be the case for higher D. Also, as noted in [3], in the general (p, q) case the action contains a term that comes from H 0 which is not invariant under the ǫ gauge transformation that cannot be ignored and the linear term in λ does not yield Poincaré supergravity. In this case, a proper Poincaré limit may still be taken by enlarging G as G to osp + (p|2) ⊕ osp − (q|2) ⊕ so(p) ⊕ so(q), and adding to H the two invariant so(p) and so(q)-valued four-forms [18] [4].
The term H 3 in λ 3 has the right dimension [H 3 ] = L D−2 = L 3 for a D=5 action. Therefore, it makes sense comparing the CS action obtained from H 3 with that of simple D = 5 supergravity which, in the first order formulation, has the same spacetime fields content; including also the terms proportional to λ 4 and λ 5 and retaining only the last three terms would lead (removing a common λ 3 factor) to an action with a 'cosmological constant' term in λ 2 coming from H 5 (as it would be similarly the case taking the higher order terms in D = 3 [1]). However, here there is no reason why local supersymmetry should be preserved by selecting any group of terms in (1.11): since the su(1|4) ǫ gauge transformations in terms of the rescaled fields depend on λ, the individual terms are not invariant separately. The leading H 0 term will be invariant under the above gauge algebra for λ = 0, but this will not be the case for the other terms including the one with the correct dimension H 3 . In fact, it is easily seen that the H 3 term in (1.11) does not lead to D = 5 supergravity. The quickest way to see it is by noticing that this H 3 term coming from the su(1|4) based CS action is not gauge invariant under the one-dimensional subgroup of transformations ϕ corresponding to the field φ, δ ϕ φ = dϕ, in contrast with the action of the D = 5 supergravity.
The D=11 AdS algebra so(2, 10) is contained in sp(32), which is of dimension (32+1)·16. The relevant superalgebra in this case would be, in principle, the smallest one that contains sp(32), namely osp(1|32), of dimension 528+32=560. A convenient way of describing its elements is provided by the osp(1|32)-valued one-form gauge field supermatrix A given by where γ a are the 32 × 32 gamma matrices and ξ is a 32-component Majorana spinor one-form. Clearly, the osp(1|32)-valued one-forms in (1.13) cannot be identified with the one-form fields e a , ω ab , ψ α and the three-form field A of Cremmer-Julia-Scherk (CJS) D=11 supergravity [19]. One could think of using two copies [10] osp(1|32), osp(1|32), to write the gauge fields f a ,f a , f ab ,f ab , f a 1 ...a 5 , f a 1 ...a 5 , ξ α , ξ α , as linear combinations of new fields e a , B a , ω ab , B ab , B a 1 ...a 5 , B ′ a 1 ...a 5 , ψ α , ψ ′ α , with dimension L except for [ψ α ] = L 1 2 , [ψ ′ α ] = L 3 2 , [ω ab ] = L 0 (and perhaps B ′ a 1 ,...,a 5 and B a ) using the scale factor λ, [λ] = L −1 . It was conjectured in [10] that the threeform field A could be a composite of e a , B ab , B a 1 ...a 5 , ψ α , ψ ′ α as explicitly considered in [20]. A closed, osp(1|32) gauge invariant twelve-form H has the general expression It was conjectured [10] that the H ′ 9 term would depend on ω ab , e a and ψ α , with the remaining fields either included in A = A(e a , B ab , B a 1 ...a 5 , ψ α , ψ ′ α ) or absent, and that it would also be invariant under local supersymmetry. However, this has not been verified, and there are arguments against this being the case. First, the bosonic and fermionic on-shell degrees of freedom do not match unless there is a large hidden extra gauge symmetry. To be more precise, let us consider Horava's choice of osp(1|32) ⊕ osp(1|32) and possible gauge action depending on e a , B a , ω ab , B ab , B a 1 ...a 5 , B ′ a 1 ...a 5 , ψ α and ψ ′ α with the following assumptions: (a) the action corresponding to H ′ 9 has the gauge symmetries of the above fields realized in the generic form δA i = dα i + . . . ; (b) the fields B a and B ′ a 1 ...a 5 , which do not enter in A, are also absent in H ′ 9 , so that we can ignore them; (c) the field equations of ω ab can be used to eliminate the ω ab ; (d) the linearized field equations for the elfbein e a and the gauge one-form fields B ab and B a 1 ...a 5 have a structure similar to the e a equation of D = 11 supergravity and (e) the linearized field equations for ψ α and ψ ′ α are linearized Rarita-Schwinger equations. With these assumptions, the counting of on-shell degrees of freedom goes as follows: Secondly, there is no reason why the H ′ 9 term in the expansion (1.15) of the right dimension L 9 should correspond to a locally supersymmetric action. Besides, the local supersymmetry transformations of D = 11 supergravity are not osp(1|32) gauge transformations, but rather local superspace transformations of the component fields the commutators of which close onshell only (see, for instance, [21]). It is thus unclear how the osp(1|32) gauge transformations could lead to these local superspace transformations after selecting the H ′ 9 term in (1.15). A second problem is the three-form field A in the action of CJS supergravity. For A to be a composite field, A = A(e a , B ab , B a 1 ...a 5 , ψ α , ψ ′ α ), the supersymmetry algebra of the H ′ 9 term in (1.15) would have to be related with the algebra defined by the MC equations including the one-form gauge fields appearing in the expression of a composite A. A natural candidate for a supersymmetry algebra would be a contraction of osp(1|32) ⊕ osp(1|32) but, as shown in [22], there is no way of obtaining by contraction the algebras given in [20,23] that allow for a one-forms decomposition of the CJS supergravity three-form field A.
As we have seen, already in the D = 5 case where there is no A complicating matters, the CS action does not lead to D = 5 supergravity. So it is hard to imagine why moving to D = 11 would improve the situation so that supersymmetry is preserved after selecting the proper H ′ 9 term in the expansion (1.15). Further, if there were such a mechanism, working only in D = 11 and ensuring local supersymmetry after taking a non-leading term, it would presumably also apply to the H ′ 10 and H ′ 11 terms in (1.15); again, this would yield a D = 11 supergravity with a cosmological constant, which has been shown not to exist [24].
The D = 11 case is more convoluted than the D=5 one not only due to the three-form field A, but also because of the auxiliary zero-form field F a 1 ···a 4 which has to be added in the first order formulation of D = 11 supergravity, which is the one that would naturally appear from a CS action. But even if these difficulties were overcome, the D=5 case already tells us that the resulting action would not be locally supersymmetric. In fact, an attempt made in [13] using just one osp(1|32) algebra, ignoring A and F a 1 ···a 4 and keeping only e a , ω ab and ψ α , supports this conclusion.
One may consider adding separately an osp(1|32)-gauge invariant dimensionless three-form field A to look for an action involving the fields of a single osp(1|32). The additional A is inert under osp(1|32) gauge transformations and, under two-form gauge transformations Λ, A transforms as δ Λ A = dΛ; thus, the four-form dA is δ Λ -gauge invariant. Then, the general gauge invariant twelve-form H(F, A) (cf. (1.14)) is given by where α, ..., σ are dimensionless constants. An action with the right dimensions would correspond to the H 9 term in the expression above with A = λ 3 A, [A] = L 3 . However, this construction still would not explain the need for the auxiliary F a 1 ···a 4 fields. In fact, one of the results of this paper is that, since contractions do not appear to play a role in the present problem, the field re-scalings need not being those that allow for a consistent λ → 0 limit. Once f a = λe e is chosen, consistency of the contraction limit would require a new field, e a 1 ···a 5 say, with f a 1 ···a 5 = λB a 1 ···a 5 , so that the osp(1|32) MC equations have a well defined λ → 0 limit. But, if this consistency condition is removed, we may now set f a 1 ···a 5 = ω a 1 ···a 5 , [ω a 1 ···a 5 ] = L 0 , (rather than f a 1 ···a 5 = λB a 1 ···a 5 , which implies [B a 1 ···a 5 ] = L 1 ). Indeed, it will be shown that the ω a 1 ···a 5 fields play the role of the F a 1 ···a 4 (see below eq. (3.61)). Unfortunately, a calculation shows that the λ 9 term in the expansion of this new, generalized CS action is not D = 11 supergravity (in particular, the fermion equation will not correspond to the spinor equation for CJS supergravity). This was to be expected since, again, there is no reason for this term to be invariant under supersymmetry gauge transformations. Nevertheless, we will show below that our construction for the fields associated with the bosonic part of a osp(1|32), supplemented by the three-form A, does work for the bosonic sector of D = 11 supergravity. In other words, there are constants α, · · · , σ in (1.17) such that the H 9 term in H resulting from the re-scalings f a = λe a , f ab = ω ab , f a 1 ···a 5 = ω a 1 ...a 5 and A = λ 3 A lead to the equations of its bosonic sector. In particular, the ω a 1 ...a 5 equation determines ω a 1 ...a 5 itself in terms of the coordinates of dA = (dA) a 1 ···a 4 e a 1 · · · e a 4 , so that ω a 1 ...a 5 plays the role of the auxiliary zero-forms of D = 11 supergravity. In this way, the fact that the D = 11 supergravity action contains a generalized 'CS term' for the field A, the eleven-form A dA dA, is incorporated into the full bosonic action through the sum of powers of λ described above. This result also extends others in refs. [12,13] in which standard pure gravity with just ω ab and e a , without the fields φ in D = 5 and A in D = 11, is derived from a CS action in these odd dimensions. The plan of the paper is as follows. The 'generalized CS action' is defined in Sec. 2, where its expression in powers of the scale factor λ is given. Then, we study in Sec. 3 the field equations of the model and compare them with those of the bosonic sector of supergravity. We end with some conclusions and further comments. Some calculations are relegated to an Appendix. In terms of its MC forms f α β , α, β = 1, . . . 32, the sp(32) algebra is defined by Using the symplectic metric C αγ = −C γα , f αβ is given by Since f αβ is a 32 × 32 symmetric matrix, it can be expanded in the basis of (αβ)-symmetric matrices given by 'weight one' antisymmetrized products of D=11 Dirac matrices as The 1/4 factor is introduced to obtain the usual relation between the spin connection and its curvature (eq. (3.71)) as well as the definition of the torsion (eq. (3.45)). Gauge curvatures are introduced by moving from the MC equations (zero curvature) to the Cartan structure ones, in which the sp(32) curvatures express the failure of f to satisfy the sp(32) algebra MC equations. Let Ω be the two-form matrix incorporating the curvatures. Then, where f contains the one-form gauge fields, and is the Bianchi identity DΩ = dΩ + [f, Ω] ≡ 0 for the sp(32) connection f . As f , the curvature Ω may be similarly expressed as The infinitesimal gauge transformations of f, Ω are given by the standard expressions, where the zero-form matrix b = b α β contains the gauge functions (2.27)

Generic expression for a CS-type action
Since the bosonic sector of D = 11 supergravity contains the three-form field A, we add it explicitly to the one-form sp(32) fields by introducing the threeform A inert under sp(32) δ b gauge transformations and under δ Λ ones. Thus, the most general twelve-form H(Ω, A), closed and invariant under both δ b and δ Λ gauge transformations, may be written as where the bosonic Ω has replaced F in eq. (1.17), in which fermions were present. Then, the integral may be used to obtain a CS-type action.
Our task now is to extract from eq. (2.28) the physically relevant terms (it will turn out that only the first term T r(Ω 6 ) and those in ν and σ will contribute) and to fix their corresponding coefficients so that the resulting action determines the equations of motion for the bosonic sector of supergravity. Because of the presence of the three-form A, this action will be referred to as the generalized CS action for the bosonic sector of D=11 supergravity.

Generalized CS action for the bosonic sector of D=11 supergravity
Again, the component fields in the one-form f , the two-form Ω and the three-form A field are dimensionless. Dimensions are introduced by setting where in (2.22) we set [ω a 1 ...a 5 ] = L 0 .
With our mostly plus metric we use real gamma matrices such that γ a 1 ...a 11 = ǫ a 1 ...a 11 . Besides the 1/4 factor in (2.31) that was fixed in (2.22), there is no special reason for the factors accompanying the fields e a , ω a 1 ···a 5 and A. Different coefficients would lead to different values for the constants α, . . . , σ in (2.28) after requiring that the action corresponds to the bosonic sector of supergravity. Thus, these constants depend on the way the fields are introduced and will not affect the final result. Keeping this in mind, we now look for the relevant terms and their coefficients for the particular choices in (2.31), (2.30). An action for D=11 gravity has dimensions L D−2 =L 9 . Thus, writing now H| i for H i and expressing the twelve-form H in (2.28) and the eleven-form B in powers of λ, we obtain Then, H| i = dB| i allows us to write for the different I GCS | i = M 11 B| i , The physically relevant term is in λ 9 since [I GCS | 9 ] = L 9 . Therefore, I GCS | 9 = M 11 B| 9 . We are thus interested in H| 9 . Since H contains the sp(32) curvature two-forms Ω a , Ω ab , Ω a 1 ...a 5 of (2.25), we need their expressions in terms of e a , ω ab , ω a 1 ...a 5 . To simplify the calculations, we write where e = e a γ a , ω L = 1 4 ω ab γ ab is the spin connection, ω 5 = ω a 1 ...a 5 γ a 1 ...a 5 and ω = ω L + ω 5 . In this way, the sp(32)-valued curvature in (2.35) gives and Ω 2 (e) = e 2 = 1 2 [e, e]. Notice that T contains a piece proportional to γ a and another proportional to γ a 1 ...a 5 ; similarly, the curvature R(ω) contains contributions proportional to γ a , γ ab and γ a 1 ...a 5 , because it depends on both ω L and ω 5 . The previous equations tell us that to obtain the piece H| 9 that comes e.g. from T r(Ω 6 ), one has to consider all the contributions containing a number n 0 of R factors, n 1 of T and n 2 of Ω 2 in such a way that 1. n 0 + n 1 + n 2 = 6 (there are 6 curvatures) 2. n 1 + 2n 2 = 9 , where the first condition guarantees that the order of the forms is twelve and the second one that their length dimension is nine. The only two solutions are: • n 2 = 4, n 1 = 1, n 0 = 1 , or • n 2 = 3, n 1 = 3, n 0 = 0 Thus, the R, T, Ω 2 contributions are of the form T r(Ω 6 )| 9 = T r(W(Ω 4 2 , T, R)) + T r(W(Ω 3 2 , T 3 , R 0 )) , where e.g. W(Ω 4 2 , T, R) is the sum of all nine-form 'words' that can obtained out of four Ω 2 , one T and one R. This would give us the piece T r(Ω 6 )| 9 of H| 9 . We could now add to (2.38) the contributions to H| 9 coming from the other terms in (2.28), to find an 11-form B| 9 with dB| 9 = H| 9 , and compare with the action of the bosonic sector of D = 11 supergravity. Instead, we will obtain directly the field equations for the action M B 9 from the original, unexpanded H twelve-form.

Field equations
The field equations for I GCS can be obtained directly from H in a way similar to that used in [16]. To find them, we use the following fact (see [25]): let i f α β , i Ω α β , i A and i dA be the inner derivations associated with the fields and curvatures of the algebra with respect to f , Ω, A and dA, defined by where E(f ) is a ten-form, and by where E(A) is an eight-form. We have to extract now from the above the equations for e, ω (ω L and ω 5 ) and A for the action I GCS | 9 . Proceeding as in [17], where the equations for the dimensionful fields were derived from those for the dimensionless ones by selecting the appropriate powers of λ, they are given by E(e) = (E(f )| 9−1=8 )| γ [1] , E(ω) = E(f )| 9 , E(ω L ) = (E(f )| 9 ) γ [2] , E(ω 5 ) = (E(f )| 9 ) γ [5] [2,5] refer to the contributions proportional to the antisymmetrization of two and five D=11 gamma matrices respectively. Eqs. (3.42) constitute the complete set of equations of our bosonic model.

Field equation for ω
We need to know the contributions of all terms in equation (3.40), namely all the contributions containing n 2 factors Ω 2 , n 0 factors R and n 1 factors T in such a way that the order of the form is 10 and its dimension L 9 . Then, we find that the ω equation is given by the ten-form expression where the first term comes from the first one in eq. (3.40) and the other comes from the ν term. Since ω = ω L + ω 5 , eq. (3.43) contains two different contributions, one proportional to γ a 1 a 2 from the first term that gives the equation for ω L , and another proportional to γ a 1 ...a 5 that comes from both terms and gives the equation for ω 5 . We consider them now. The first ten-form in To see how the γ a 1 a 2 and γ a 1 ...a 5 contributions come out, note the identity i.e., all terms in the sum (3.48) add up, and the first term appears k times. The same pattern exists when two matrices γ a 1 ...a k , γ a 1 .....as are multiplied, but now there are contributions with all possible number of contractions. The e 8 T L terms have the structure γ [8] · γ [1] (again, the superscripts indicate the number of γ's in the skewsymmetric products). This gives, schematically, γ [8] · γ [1] ∼ γ [9] + γ [7] (3.50) where there are no contractions in γ [9] and one in γ [7] . The γ [7] contribution will cancel because only the matrices symmetric in all indices contribute (γ [1,2,5,6,9,10] are symmetric; 1, γ [3,4,7,8] skewsymmetric). Thus, only the e 8 T L terms appear in the ω L equation since γ a 1 ...a 9 ∝ ǫ a 1 ...a 9 ab γ ab . (3.51) In general, since with our metric signature we can choose γ a 1 ...a 11 = ǫ a 1 ...a 11 , we have On the other hand, the terms e 9 ω 5 coming from (3.47) are, again schematically, of the form γ [9] · γ [5] ∼ γ [14] + γ [12] + γ [10] + γ [8] + γ [6] + γ [4] , (3.53) The γ [10] ∼ γ [1] contribution vanishes because there is no ω a , i.e. there is no equation of dimension L 9 with a single Lorentz index. The only symmetric γ is γ [6] . So the e 9 ω 5 terms only appear in the ω 5 equation. The ω L equations are then (E(f )| 9 ) γ [2] = E(ω L ) ∝ e a 1 ...e a 8 (T L ) a 9 γ a 1 ...a 9 = 0 . This equation implies T L = 0, which, as usual, can be used to express ω abµ in terms of e a µ and its derivatives.

Equation for
This equation has contributions from the two terms in (3.43). One is given by its second term 4ν(dA)e 6 which, due to e 6 , is proportional to γ [6] , and the other is the contribution with four contractions from the terms with nine e and one ω 5 from W(e 9 , ω 5 ), which is also proportional to γ [6] ∼ γ [5] , contained in the first one, 6 W(Ω 4 2 , T ). A long calculation shows that this second contribution is given by  b 1 a 6 = ω b 4 ...b 1 a 6 c e c . We now write the products of ten e's above as for some ten-form E d . Then, factoring out this form in eq. (3.58) and γ a 1 ...a 6 , we find where [ ] indicates weight one antisymmetrization in a 1 ...a 6 . It is shown in the Appendix (sec. 5.1) that the solution is Hence, ω d 1 ...d 5 may be expressed in terms of the coordinates of dA so that, as anticipated, ω 5 plays a role analogous to that of the auxiliary zero-forms F a 1 ···a 4 of the first order formulation of D = 11 supergravity, where F ∝ dA.

Field equation for A
The sum of the contributions to the field equation where again D is the ω L covariant derivative; we see that there is no contribution from the δ and ρ terms. In the e a basis, this gives Now we can introduce the eight-form ..e b 8 , and use it to rewrite the factors with eight one-forms e a . If the E d 1 d 2 d 3 are then factorized, we obtain 4 ν 32 · 6! δ a 7 ...
(3.63) Using the expression (3.60) for ω b 2 a 7 ...a 11 in terms of the components of dA, the r.h.s of (3.63) reads In this way, the final expression for the A equation of the motion is found to be Note that this equation has the form required to reproduce the equations of D = 11 supergravity in the absence of fermions (see [19,21,26]).

Field equation for e
We need to know the contributions of all terms in eq. (3.40) again, but now we have to find (E(f )| 8 )| γ [1] = E(e) instead of E(f )| 9 in eq. (3.42). Collecting all the possible contributions as explained before, we find that they all come from the first and the ν term in eq. (3.40), where, again, | γ [1] selects the contribution accompanying a single gamma matrix γ a , or equivalently, a ten indices gamma matrix, γ a 1 ...a 10 . In particular we need the contributions coming from the term 6W(Ω 3 2 , T 2 )| γ [1] +6W(Ω 4 2 , R)| γ [1] in (3.65), but this is a very tedious calculation. Instead, it is more convenient to take advantage of the fact that the symmetry of the stress-energy tensor forces its terms to be the result of contracting three or four indices among two dA µνρσ (in the dx µ basis), namely (dA) µνρ α (dA) µνρβ and (dA) µνρσ (dA) µνρσ g αβ . Hence, Einstein's equations have the form with P, Q yet to be determined. With the sign for the curvature tensor as in [27], R(Γ) and R(ω L ) are related through the elfbein postulate by R(Γ) = 2R(ω L ). The P , Q constants are now determined using that the covariant derivative of the Einstein tensor is zero, ∇ µ R(Γ) µν − 1 2 g µν R(Γ) = 0. Then, the r.h.s of (3.66) must vanish when the supergravity field equation for the A field (equivalent to our (eq. (3.64))), where the proportionality factor is unimportant here, and , are used. Indeed, the covariant derivative of the r.h.s of eq. (3.66) may be written using (3.68) as a linear combination of (dA) ρσλτ ∇ ν (dA) ρσλτ and (dA) ρσλν ∇ µ (dA) ρσλµ . This last contribution vanishes due to eq. (3.67). Hence, the first contribution also has to vanish and, since it includes a factor (P + 8Q), it follows that P/Q = −8 (see, e.g., [28]). Thus, we only need now the overall factor. To fix it, we take the trace of eq. (3.66) to find the Ricci scalar We still need the value of P for our action. If we compute the trace of the E(e) = 0 (eq. (3.65)) times e a γ a , we obtain 0 = 6 T r(9ω 5 eω 5 e 8 + 9ω 2 5 e 9 + 9ω 5 e 2 ω 5 e 7 + 9ω 5 e 3 ω 5 e 6 (3.70) + 9ω 5 e 4 ω 5 e 5 ) + 30 4 T r(R L e 9 ) + 4ν(dA) 6T r(ω 5 e 6 ) , where the curvature R L is This expression leads to an equation for the Ricci scalar R(ω L ) ab ab that has the advantage that the different contributions are easier to compute. A calculation (Appendix, eq. (5.95)) shows that ν in our action is related to P by P = 12 · 32 · 4! · 7! (9!) 2 ν 2 .

The generalized CS action for the bosonic sector of D = 11 supergravity
Having found the field equations from our action, we now fix the remaining constants in (2.28) so that the equations of bosonic D = 11 supergravity follow from I GCS as stated. First, the D=11 supergravity equation for the e field is, after taking the trace (see e.g. [27]), Comparing with (3.69) we find P = 1 12 , which in eq. (3.72) then gives Secondly, the D=11 supergravity equation for A is Thus, the needed values of ν and σ in (2.28) are now fixed; the terms in α, β, δ, ρ do not appear once the relevant H 9 term is selected. Note that it is possible to obtain ν from (3.74) because its r.h.s. is positive. Summarizing, the generalized CS action for the bosonic sector of D=11 supergravity is obtained from

Conclusions
We have shown that the bosonic sector of D = 11 supergravity may be obtained from a generalized CS action based on the one-form gauge fields of the sp(32) subalgebra of osp(1|32) supplemented with a dimensionless threeform field A. The need for A could not have been guessed without having in mind D = 11 supergravity: the presence of fermions requires A by simply counting the degrees of freedom of the D = 11 supermultiplet. Further, we have also shown (see (3.60)) that the role of the auxiliary zero-form fields F a 1 ...a 4 that appear in the first-order version of D = 11 supergravity [26] is played by specific gauge fields associated with sp(32).
The values of the constants that determine our generalized CS bosonic action were obtained by requiring that the equations it leads to are those of the bosonic sector of D = 11 supergravity. It turns out that only three terms in eq. (2.28) are actually needed, the first one and those in ν and σ, since the others do not appear in the bosonic equations obtained from the λ 9 term in the λ expansion. The other terms and their constants would appear when including fermions, eq. (1.17), but nevertheless (Sec. 1.3) this will not lead to D=11 supergravity. Hence, there is no generalized CS action based on osp(1|32) with the addition of the three-form field leading to CJS supergravity. Therefore, although D = 3 supergravity may be described by a CS action, we conclude that this is not so in larger, odd spacetime dimensions.
It was already conjectured in the original paper [19] that osp(1|32) would provide the lead for a geometric interpretation of D=11 supergravity. The main obstacle to relate its field contents to the geometric MC fields of a superalgebra in the search for a possible CS action is the appearance of the three-form field A. As mentioned, it is possible to retain only one-form fields by assuming a composite nature for A [20] and then using a superalgebra that incorporates the one-form MC components of A. In fact, there is a whole family of superalgebras related to osp(1|32) that do just this [23] (another family of algebras structure has recently been shown to exist for N=2, D=7 supergravity [30]).
Summarizing, we have shown that although there is no CS action for CJS supergravity, its bosonic sector may be described by a generalized CS action in the sense of Sec. 2.2. But, if we insist in including fermions, we conclude that the only geometric way of relating CJS supergravity to the osp(1|32) superalgebra requires assuming the mentioned composite nature for A [20,23]. Even so, the connection with osp(1|32) is rather subtle [23]: the family of algebras that trivialize the three-form A are deformations of an algebra which is the expansion osp(1|32)(2, 3) of osp(1|32) in the sense of [16,17].
Junta de Castilla y León and by the Spanish Centro de Excelencia Severo Ochoa Programme (IFIC-SEV-2014-0398). Correspondence with R. D'Auria is also appreciated.

Appendix
This Appendix provides details of some main text calculations.   without antisymmetrization. To this end, we use the following trick: first we make eq (5.85) more explicit, with d 5 interchanged with d, so that the antisymmetrization involves d 1 , · · · d 4 and d, Antisymmetrizing the indices d 1 ...d 5 with weight one leads to