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On BPS preons, generalized holonomies and D=11 supergravities

We develop the BPS preon conjecture to analyze the supersymmetric solutions of D=11 supergravity. By relating the notions of Killing spinors and BPS preons, we develop a moving G-frame method (G=GL(32,R), SL(32,R) or Sp(32,R)) to analyze their associated generalized holonomies. As a first application we derive here the equations determining the generalized holonomies of k/32 supersymmetric solutions and, in particular, those solving the necessary conditions for the existence of BPS preonic (31/32) solutions of the standard D=11 supergravity. We also show that there exist elementary preonic solutions, i.e. solutions preserving 31 out of 32 supersymmetries in a Chern--Simons type supergravity. We present as well a family of worldvolume actions describing the motion of pointlike and extended BPS preons in the background of a D'Auria-Fre type OSp(1|32)-related supergravity model. We discuss the possible implications for M-theory.


I. INTRODUCTION
A complete, algebraic classification of M-theory BPS states, based on the number k of supersymmetries preserved by a given BPS state, has been given in [1]. BPS states preserving k out of 32 supersymmetries are denoted ν = k/32 states; k = 32 corresponds to fully supersymmetric vacua. The observation [1] (see also Sec. II) that BPS states that break 32 − k supersymmetries can be treated as composites of those preserving all but one supersymmetries, suggests that the k = 31 states might be considered as fundamental constituents of M-theory. These ν = 31/32 BPS states were accordingly named BPS preons [1].
Interestingly enough, the notion of BPS preons may be extended to arbitrary dimensions, including D = 4. In this case, a point-like BPS preon may be identified with a tower of massless higher spin fields of all possible helicities (see [2,3], [4] and refs. therein).
In this paper we develop the notion of BPS preons to analyze the various supersymmetric solutions of the supergravity equations. A ν = k/32 supersymmetric solution corresponds to a BPS state composed of n = (32−k) BPS preons. The corresponding (32 − k) bosonic spinors λ α r (r = 1, . . . , n) are orthogonal to the k Killing spinors ǫ J α characterizing the ν = k/32 solution, Thus, BPS preonic spinors and Killing spinors provide an alternative (dual) characterization of a ν-supersymmetric solution; either one can be used and, for solutions with supernumerary supersymmetries [12,13], the characterization provided by BPS preons is a more economic one. Moreover, the use of both BPS preonic (λ α r ) and Killing (ǫ J α ) spinors allows us to develop (in Sec. II) a moving G-frame method, which may be useful in the search for new supersymmetric solutions of CJS supergravity.
We apply this moving G-frame method (in Sec. III, G = SL(32, R)) to studying the generalized holonomies of CJS supergravity and discuss the basic equations characterizing the still hypothetical BPS preonic solutions (Sec. IV B, G = GL(32, R), SL (32, R) or Sp(32, R)).
Although no definite answer to the question of the existence of BPS preonic solutions for the standard CJS supergravity is given here, we do show (in Sec. IV A) that ν = 31/32 supersymmetric preonic configurations solve the equations in the Chern-Simons (CS) supergravity case [14] (for a review, see [15]) i.e., that CS supergravity does have preonic solutions.
Using the recent results [16,17] on a gauge-fixed form of the action for dynamical supergravity interacting with dynamical superbrane sources (see Sec. IV C), we also propose (in Sec. IV D) a D = 11 worldvolume action for BPS preons in the background of a D'Auria-Fré OSp(1|32)-related supergravity [18] (see also [19]), a model allowing for an economic 'embedding' of the standard D = 11 CJS supergravity.
A. Equations of D = 11 supergravity The purely bosonic limit of the 'free' CJS supergravity equations is given by which include the Einstein equations with a contribution from the energy-momentum tensor of the antisymmetric gauge field A µνρ (x) only, Eq. (2), as well as the free equations (3) and Bianchi identities (4) for the gauge field in curved D = 11 spacetime. Here a, b, c = 0, . . . , 9, 10 = µ, ν, ρ, It is also assumed that the torsion and the gravitino vanish, where ω Lb a is the Lorentz connection. Such equations possess nonsingular pp-wave solutions with supernumerary supersymmetries [12,13].
As the supergravity multiplet is the only one without higher spin fields in D = 11, no usual field-theoretical (spacetime) matter contribution to the r.h.s.'s of Eqs.
The ν = 16/32 supersymmetric M2-brane solution of D = 11 supergravity (see [24,25] and refs. therein) possesses a singularity on the (2 + 1)-dimensional worldvolume surface W 2+1 ⊂ M 11 , x µ =x µ (ξ) ≡x µ (τ, σ 1 , σ 2 ) (a caret indicates a function of the worldvolume coordinates). In other words, it solves the Einstein field equation E ab = T ab − 1 9 η ab T c c with a singular energymomentum tensor density T ab ∝ δ 3 (x −x(ξ)) [42]. The gauge field equation also possesses a singular contribution J 8 in the r.h.s., G 8 = J 8 , similar to that of the electric current to the r.h.s. of Maxwell equations. In this sense, the M2-brane carries a supergravity counterpart of the electric charge in Maxwell electrodynamics (see [26] for a discussion).
The other basic ν = 16/32 supersymmetric solution of the CJS supergravity, the M5-brane (see [24,25,27] and refs. therein), is a counterpart of Dirac monopole, i.e. of the magnetically charged particle. It is characterized by a modification of the Bianchi identities (Eq. (4)) with the analogue of a magnetic current in the r.h.s., dF 4 = J 5 .
Notice, however, that one should not exclude the possible existence of a brane solution [i.e. solutions of E ab = T ab − 1 9 η ab T c c with a T ab ∝ δ p+1 (x−x(ξ))] with supernumerary supersymmetries, although certainly these solutions would describe quite unusual branes. The reason why the 'standard' brane solutions (like M-waves, M2 and M5-branes in D = 11) always break 1/2 of the supersymmetry is that their κ-symmetry projector (the bosonic part of which is identical to the projector defining the preserved supersymmetries [28,29]) has the form (1 −Γ) with trΓ = 0,Γ 2 = I. However, worldvolume actions for branes with a different form for the κ-symmetry projector are known [4,30,31] although in an enlarged superspace (see [32]). A question arises, whether such actions may be written in usual spacetime or superspace.
As a partial answer to this question, we present here a D = 11 spacetime action for a BPS preon in the background of a D'Auria-Fré type supergravity [18]. The experience provided by the usual D = 11 M-branes and D = 10 D-branes together with the analysis [16,17] of the partial preservation of local supersymmetry by the purely bosonic limit of the super-p-brane action suggests that the existence of such a preonic action implies that ν = 31/32 solitons should exist in a D'Auria-Fré type model.
In CJS supergravity, as well as in type IIB D = 10 supergravity, the gauge symmetry is restricted to SO(1, 10), and ω in Eq. (9) is not a true connection. However, the selfconsistency (integrability) condition for (8), DDǫ J α = 0 has the suggestive form [6,8], in terms of the generalized curvature Eqs. (8), (10) formally possess a GL(32, R) gauge invariance with ω transforming as a GL(32, R) connection. However, GL(32, R) is not a gauge invariance of CJS supergravity, hence the name 'generalized' connection and curvature for ω and R. Notice that, in contrast, a rigid GL(k, R) transformation acting on the index I results in a redefinition of the Killing spinors ǫ I α i.e., in replacing the Killing spinors by independent linear combinations with constant coefficients. This is clearly allowed and GL(k, R) can be treated as a rigid symmetry of the system of k Killing spinors characterizing the ν = k/32 supersymmetric solution of any model.
A true connection takes values in the Lie algebra G of the structure group G of the principal fiber bundle. The curvature may take values in a smaller subalgebra H ⊂ G which is associated with a proper subgroup H ⊂ G of the structure group, the holonomy group. For generalized connections ω one may accordingly introduce the notion of generalized holonomy group H ⊂ G [6] (see also [7,8,10]), such that R β α ∈ H (while ω β α ∈ G). In this light, the necessary condition for the existence of k Killing spinors, Eq. (10), can be treated as a restriction on the generalized holonomy group H [6,7,10,11,13].
It has been shown that for both CJS D = 11 supergravity [7] and for type IIB supergravity [11], the generalized holonomy group H is a subgroup of SL(32, R), H ⊂ SL(32, R). As the generalized connections are clearly traceless in both cases, one also has G ⊂ SL(32, R).
A full expression for the generalized curvature R α β corresponding to purely bosonic solutions of CJS supergravity may be found e.g., in [9,33] (see Appendix B of [9] and refs. therein; there, η ab and F correspond to −η ab , −2F ). For our purposes here it is sufficient to note that this R α β obeys where E ab , G 8 are the r.h.s's of the Einstein and the gauge field equations as defined in Eqs. (2), (3) and i a is defined by i a e b = δ b a so that i.e., for Ω p = 1 p! e ap ∧. . .∧e a1 Ω a1...ap , in particular i a R α β = e b R abα β . The equality (12) implies that the set of the free bosonic (Eq. (7)) equations for the CJS supergravity, Eqs. (2), (3), (4), is equivalent to the following simple equation for the generalized curvature of Eq.

A. Killing spinors and BPS states
A BPS state |BP S , k described by a solitonic solution preserving k supersymmetries is, schematically, one satisfying where Q α are the supersymmetry generators obeying so that P αβ = P βα . The generalized momentum P αβ may be decomposed e.g. in the basis of D = 11 Spin(1, 10) (32 × 32) Dirac matrices, giving then rise to the standard D = 11 momentum P µ and to the tensorial 'central' charge generators Z µν , Z µ1...µ5 of the M-algebra {Q α , Q β } = P αβ . These generators may be identified as topological charges [34] related to the M2-and M5-branes (as well as to the M9-brane and KK7-brane of M-theory [35]; see [29] for the rôle of the worldvolume gauge fields of the M5-brane, and [32,36] for D-branes).

B. BPS preons as constituents
A BPS preon [1] state |BP S , 31 ≡ |λ is a state characterized by a single bosonic spinor λ α , (hence the notation |λ ) or as a state preserving all supersymmetries but one (hence the notation |BP S, 31 ) [1]. The bosonic spinor parameters ǫ I α corresponding to the supersymmetries preserved by a BPS preon |λ , are 'orthogonal' to the bosonic spinor λ α that labels it, (see below). Identifying ǫ I α with 31 Killing spinors satisfying (8), one finds that (20) expresses the fact that these Killing spinors are orthogonal to the single bosonic spinor λ α characterizing a hypothetical BPS preonic solution.
For BPS states |BP S, k preserving k ≤ 30 supersymmetries (15) one may introduce n = 32 − k bosonic spinors λ α r corresponding to the n broken supersymmetries. They may be treated as characterizing n BPS preons |λ r , r = 1, . . . , n, out of which the corresponding k/32 BPS state is composed [1].
To make this transparent, let us consider the eigenvalue matrix p αβ of the generalized momentum operator P αβ corresponding to the BPS state |BP S, k (which is usually assumed to be an eigenstate of the generalized momentum, i.e. having definite energy and definite brane charges), αβ may be diagonalized by a general linear transformation i.e., with p (γ)(δ) = diag(. . .). Moreover, as g α (γ) ∈ GL(n, R), this diagonal matrix can be put in the form where the number of nonvanishing elements, all +1 or −1, is equal to n = rank(p (k) αβ ). However, the usual assumptions for the supersymmetric quantum mechanics describing BPS states do not allow for negative eigenvalues of P αβ = {Q α , Q β } [p 11 = −1, e.g., would imply (Q 1 ) 2 |BP S, k = −|BP S, k , contradicting positivity]. Thus, only positive eigenvalues are allowed and Substituting (23) into (21), one arrives at or, equivalently, denoting g α One sees using Eq. (16) that, if the preserved supersymmetries correspond to the generators which, immediately implies Eq. (25) may be looked at as a manifestation of the composite structure of the ν = k/32 BPS state |BP S, k , where |λ 1 , . . ., |λ n are BPS elementary, preonic states characterized by the spinors λ α 1 , . . ., λ α n , respectively.

C. Moving G-frame
When a BPS state |k is realized as a solitonic solution of supergravity, it is characterized by k Killing spinors ǫ J β (x) or by the n = 32 − k bosonic spinors λ α r (x) associated with the n BPS preonic components of the state |BP S, k . The Killing spinors and the preonic spinors are orthogonal, and, hence, may be completed to obtain bases in the spaces of spinors with upper and with lower indices by introducing n = 32 − k spinors w r α and k spinors u α L satisfying w s α λ α r = δ r s , w s α u α Either of these two dual bases defines a generalized moving G-frame described by the nondegenerate matrices where (α) = (s, J) = (1, . . . , 32 − k; J = 1, . . . , k). Indeed, g −1 is equivalent to Eqs. (30) and (29), while provides the unity decomposition or completeness relation in terms of these dual bases.
One may consider the dual basis g −1

(β)
α to be constructed from the bosonic spinors in g α (β) by solving Eq. (32) or g −1 g = I (Eqs. (30) and (29)). Alternatively, one may think of w r α and u α J as being constructed from ǫ J α and λ α r through a solution of the same constraints. In this sense the generalized moving G-frame (31) is constructed from k Killing spinors ǫ J α characterizing the supersymmetries preserved by a BPS state (realized as a solution of the supergravity equations) and from the n = 32 − k bosonic spinors λ α r characterizing the BPS preons from which the BPS state is composed.
Although many of the considerations below are general, we shall be mainly interested here in the cases G = SL(32, R) and G = Sp(32, R).
Clearly, in D = 11, the charge conjugation matrix C αβ = −C βα allows us to express the dual basis g −1 in terms of the original one g or vice versa. In particular, in the preonic k = 31 case one finds that, as λ α C αβ λ β ≡ 0, λ α = C αβ λ β has to be expressed as λ α = λ I ǫ I α , for some coefficients λ I , I = 1, . . . , 31. In general (as e.g., in CJS supergravity with nonvanishing F 4 ), the charge conjugation matrix is not 'covariantly constant', DC αβ = −2ω [αβ] = 0. This relates the coefficients λ I = λ α u α I to the antisymmetric (non-symplectic) part of the generalized connection, [43]. In OSp(1|32)-related models ω [αβ] = 0 and A = 0, hence λ I is constant and we may set λ I = δ I 31 using the global transformations of GL(31, R), which is a rigid symmetry of the system of Killing spinors . This allows us to identify λ α itself with one of the Killing spinors i = 1, . . . , 30 .
On the other hand, one might reverse the argument and ask for the structure of a theory allowing for ν = k/32 supersymmetric solutions. This question is especially interesting for the case of BPS preonic and ν = 30/32 solutions as, for the moment, such solutions are unknown in the standard D = 11 CJS and D = 10 type II supergravities.

B. Generalized holonomy for BPS preons and for ν = k/32 supersymmetric solutions
The simplest application of the moving G-frame construction above is to find an explicit form for the general solution of Eq. (10), which expresses the necessary conditions for the existence of k Killing spinors. As the Killing spinor equation (35) implies Eqs. (36), (37), one may solve instead the selfconsistency conditions for these equations, Using the unity decomposition (32), which implies R α β = R α γ λ γ r w r β +R α γ u γ I ǫ I β , one finds the following expression for the generalized curvature where For k = 31, corresponding to the case of a BPS preon, Eq. (50) simplifies to Eqs. (50) and (53) imply R α β = λ α r (· · · ) r β and, thus, clearly solve Eq. (10), ǫ I β R β α = 0. The conditions G ⊂ SL(32, R) and hence H ⊂ SL(32, R), R α α = 0 (which is always the case for bosonic solutions of 'free' CJS and type IIB supergravities [7,11] Finally, for G ⊂ Sp(32, R) ω [αβ] = 0, the holonomy group H ⊂ Sp(32, R), R αβ := C αγ R γ β = R (αβ) , and Eq. (54) reduces to where only one arbitrary one-form B appears [to obtain (55) one has to keep in mind that ǫ I α = (ǫ i α , C αβ λ β ), I = (i, 31), Eq. The first observation is that the generalized curvature allowing for a BPS preonic (k = 31 supersymmetric) configuration for the case of H ⊂ SL(32, R) holonomy, Eq. (54), is nilpotent As a result it solves the purely bosonic equations of a Chern-Simons supergravity [see [14,15], although our statement may be related to a more general version of a hypothetical Chern-Simons-like supergravity], Clearly, the same is true for H ⊂ Sp(32, R) ⊂ SL(32, R), where R is given by Eq. (55). Thus, there exist BPS preonic solutions in CS supergravity theories, including OSp(1|32)-type ones.

B. Searching for preonic solutions of 'free' bosonic CJS equations
We go back now to the question of whether BPS ν = 31/32 (preonic) solutions exist for the standard CJS supergravity [5]. As it was noted in Sec. I A, this problem can be addressed step by step, beginning by studying the existence of preonic solutions for 'free' bosonic CJS equations. To this aim it is useful to observe [9] that these equations may be collected in a compact expression for the generalized curvature, Eq. (14). The generalized curvature of a BPS preonic configuration satisfies Eq. (54), and thus it solves the 'free' bosonic CJS supergravity equations (14) if Actually, Eq. (54) in Eq. (14) gives λ α i a dB I ǫ I γ Γ a γ β = 0. However, as λ α = 0, this is equivalent to Eq. (59).
Eq. (59) contains a summed I = 1, . . . , 31 index and, as a result, it is not easy to handle. It would be much easier to deal with the expression Γ a α γ i a R γ β which, with Eq. (54) is equal to Γ a α γ λ γ i a dB J ǫ J β . Indeed, (Γ a λ) α i a dB J ǫ J β = 0, for instance, would imply (Γ a λ) α i a dB J = 0 which may be shown to have only trivial solutions. However, Γ a α γ i a R γ β = 0 in general for a solution of the 'free' bosonic CJS equations (Eq. (14)), and O(F F ) denotes the terms of second order in F c1c2c3c4 , Eq. (54) then implies that for a hypothetical preonic solution of the 'free' bosonic CJS equations, the gauge field strength F abcd should be nonvanishing (otherwise dB J = 0 and R α β = 0, see above) and satisfy Using (30), Eqs. (63) split into a set of restrictions for F abcd , and equations for dB I , Eq. (63) or, equivalently, Eqs. (64),(65) are the equations to be satisfied by a CJS preonic configuration. Note that if a nontrivial solution of the above equations with some F abcd = 0 and some dB I = 0 is found, one would have then to check in particular that such a solution satisfies ddB I = 0 and D [e F abcd] = 0.
On the other hand, if the general solution of the above equation turned out to be trivial, dB I = 0, this would imply R α β = 0 and, thus, a trivial generalized holonomy group, H = 1. However, this is the necessary condition for fully supersymmetric, k = 32, solutions [8]. Hence a trivial solution for Eqs. (64), (65) would indicate that a solution preserving 31 supersymmetries possesses all 32 ones (thus corresponding to a fully supersymmetric vacuum) and, hence, that there are no preonic, ν = 31/32 solutions of the free bosonic CJS supergravity equations (2), (3), (4), (6) and (7).
If this happened to be the case, one would have to study the existence of preonic solutions for the CJS supergravity equations with nontrivial right hand sides. These could be produced by corrections of higher-order in curvature [20,21,23] (a counterpart of the string α ′ corrections in D = 10 [22]) and by the presence of sources (from some exotic p-branes).

C. On brane solutions and worldvolume actions
As far as supersymmetric p-brane solutions of supergravity equations are concerned, one notices that for most of the known ν = 1/2 supersymmetric solutions (ν = 16/32 in D = 11 and D = 10 type II cases) there also exist worldvolume actions in the corresponding (D = 11 or D = 10 type II) superspaces possessing 16 κ-symmetries, exactly the number of supersymmetries preserved by the supergravity solitonic solutions. The κsymmetry-preserved supersymmetry correspondence was further discussed and extended for the case of ν < 1/2 multi-brane solutions in [28,29].
In this perspective one may expect that if preonic ν = 31/32 supersymmetric solutions of the CJS equations with a source do exist, a worldvolume action possessing 31 κ-symmetries should also exist in a curved D = 11 superspace. For a moment no such actions are known in the standard D = 11 superspace, but they are known in a superspace enlarged with additional tensorial 'central' charge coordinates [30,31]. One might expect that the rôle of these additional tensorial coordinates could be taken over by the tensorial fields of D = 11 supergravity. But this would imply that the corresponding action does not exist in the flat standard D = 11 superspace as it would require a contribution from the above additional field degrees of freedom (replacing the tensorial coordinate ones corresponding to spirit of [32]). This lack of a clear flat standard superspace limit hampers the way towards a hypothetical worldvolume action for a BPS preon in the usual curved D = 11 superspace.
Nevertheless, a shortcut in the search for such an action may be provided by the recent observation [17] that the superfield description of the dynamical supergravitysuperbrane interacting system, described by the sum of the superfield action for supergravity (still unknown for D = 10, 11) and the super-p-brane action, is gauge equivalent to the much simpler dynamical system described by the sum of the spacetime, component action for supergravity and the action for the purely bosonic limit of the super-p-brane. This bosonic p-brane action carries the memory of being the bosonic limit of a superp-brane by still possessing 1/2 of the spacetime local supersymmetries [16]; this preservation of local supersymmetry reflects the κ-symmetry of the original super-pbrane action.
Thus the κ-symmetric worldvolume actions for super-p-branes have a clear spacetime counterpart: the purely bosonic actions in spacetime possessing a part of local spacetime supersymmetry of a 'free' supergravity theory. This fact, although explicitly discussed for the standard, ν = 1/2 superbranes in [17], is general since it follows from symmetry considerations only and thus it applies to any superbrane, including a hypothetical preonic one. The number of supersymmetries possessed by this bosonic brane action coincides with the number of κ-symmetries of the parent super-p-brane action. Moreover, these supersymmetries are extracted by a projector which may be identified with the bosonic limit of the κsymmetry projector for the superbrane. With this guideline in mind one may simplify, in a first stage, the search for a worldvolume action for a BPS preon in standard supergravity (or in a model minimally extending the standard supergravity) by discussing the bosonic limit that such a hypothetical action should have.
The question that remains to be settled is the meaning of the symmetric spin-tensor one-form e αβ with the local supersymmetry transformation rule (66) in D = 11 supergravity. The contraction of e αβ with Γ a , one finds that e αβ also contains the antisymmetric tensor one-forms B ab 1 (x) = dx µ B ab µ (x) and B a1...a5 µ (x). Such fields, moreover, with exactly the same supersymmetry transformation rules, are involved in the D = 11 supergravity model by D'Auria and Fré [18].
Thus the action (68) can be teated as a worldline action for a BPS preon in the background of the D'Auria-Fré, OSp(1|32)-related 'gauge' supergravity model. This might be regarded as an indirect indication of the existence of BPS preonic ν = 31/32 solutions in D'Auria-Fré D = 11 supergravity.
The possibility of having preonic actions in the standard CJS supergravity requires additional study.

V. DISCUSSION AND OUTLOOK
In this paper we have studied the rôle of the BPS preon notion [1] in the analysis of supersymmetric solutions of D = 11 supergravity. This notion suggests the moving G-frame method, which we propose as a new tool in the search for supersymmetric solutions of D = 11 and D = 10 supergravity. We used this method here to make a step towards answering whether the standard CJS supergravity [5] possesses a solution preserving 31 supersymmetries, a solution that would correspond to a BPS preon state. Although this question has not been settled for the CJS supergravity case, we have shown in our framework that preonic, ν = 31/32 solutions do exist in a Chern-Simons type D = 11 supergravity [14].
Although the main search for preonic solutions concerns the 'free' bosonic CJS supergravity equations, one should not exclude other possibilities, both inside and outside the CJS standard supergravity framework. When, e.g., super-p-brane sources are included, the Einstein equation (2), and possibly the gauge field equations (3) and even the Bianchi identities (4), acquire r.h.s's. In this case (see Eq. (12)), a r.h.s. also appears in Eq. (14) and the situation would have to be reconsidered. Another source of modification of the CJS supergravity equations might be due to 'radiative' corrections of higher order in curvature. Such modified equations might also allow for preonic solutions not present in the unmodified ones. If it were found that only the inclusion of these higherorder curvature terms allows for preonic BPS solutions, this would indicate that BPS preons cannot be seen in a classical low energy approximation of M-theory and, hence, that they are intrinsically quantum objects.
The special rôle of BPS preons in the algebraic classification of all the M-theory BPS states [1] allows us to conjecture that they are elementary (quark-like) necessary ingredients of any model providing a more complete description of M-theory. In such a framework, if the standard supergravity did not contain ν = 31/32 solutions, neither in its 'free' form, nor in the presence of a super-pbrane source, this might just indicate the need for a wider framework for an effective description of M-theory. Such an approach could include Chern-Simons supergravities [14] and/or the use of larger, extended superspaces (see [32,37] and refs. therein), in particular with additional tensorial coordinates (also relevant in the description of massless higher-spin theories [2,3]). In this perspective our observation that the BPS preonic configurations do solve the bosonic equations of Chern-Simons supergravity models looks interesting.
Note added. We mention that it might be interesting to look at the role of vectors and higher order tensors that may be constructed from the preonic spinors λ α r , in analogy with the use of the Killing vectors K a IJ = ǫ I Γ a ǫ J and higher order bilinears ǫ I Γ a1···as ǫ J made in Refs. [9,39,40,41]. supported by the research grants BFM2002-03681, BFM2002-02000 from the Ministerio de Ciencia y Tecnología and from EU FEDER funds, by the grant N 383 of the Ukrainian State Fund for Fundamental Research, the INTAS Research Project N 2000-254, and by the Junta de Castilla y León grant A085-02. M.P. and O.V. wish to thank the Ministerio de Educación, Cultura y Deporte and the Generalitat Valenciana, respectively, for their FPU and FPI research grants.