Contractions from $osp(1|32) \oplus osp(1|32)$ to the M-theory superalgebra extended by additional fermionic generators

We study here the generalized Weimar-Woods contractions of the superalgebra $osp(1|32) \oplus osp(1|32)$ in order to obtain a suitable algebra that could describe the gauge group of $D=11$ supergravity. The contracted superalgebras are assumed to be given in terms of fermionic extensions of the M-theory superalgebra. We show that the only superalgebra of this type obtained by contraction is the only one for which the three-form of $D=11$ supergravity cannot be trivialized. Therefore, $D=11$ supergravity cannot be connected in this way with a contraction of $osp(1|32) \oplus osp(1|32)$.


Introduction
In their original work on D = 11 supergravity, Cremmer, Julia and Scherck [1] conjectured that the theory could admit a geometrical interpretation in terms of the simple supergroup OSp(1|32). The evidence in favour of this conjecture was the fact that OSp(1|32) contains an SO (8) subgroup, which is also a subgroup of the internal invariance group of a D = 4 reduction of the D = 11 model. However, the presence in the D = 11 supergravity field content of a three-form field A 3 = A µνρ (x) dx µ ∧ dx ν ∧ dx ρ , apart from the elfbein e a = e a µ dx µ , the Rarita-Schwinger field ψ α = ψ α µ dx µ and the spin connection w ab = w ab µ dx µ one-forms, makes it difficult to identify the group theoretical structure of the model. D'Auria and Fré [2], addressed this problem by looking at the free differential algebra (FDA) satisfied by the above forms in the absence of curvatures. This FDA does not consist only of one-forms, so it cannot be interpreted as the Maurer-Cartan equations of a certain Lie superalgebra. D'Auria and Fré's idea was to express the three-form A 3 in terms of linear combinations of exterior products of one-forms for a certain superalgebra, which had to be found. Two superalgebras were obtained which allowed the decomposition of A 3 in such a way. For this, it was necessary to consider superalgebras that, ignoring the spin(10, 1) algebra, are fermionic central extensions of the M-theory superalgebra [3,4], with anticommutators: (1.1) An obvious question was whether these two superalgebras could be related to osp(1|32) or, more generally, to a simple superalgebra. This was considered in [5], where not only osp(1|32), but also osp(1|64) and su(32|1), were ruled out as algebras that could lead to the D'Auria-Fré ones by contraction.
In [6,7], it was assumed that the supersymmetry group in the low-energy limit had to be a contraction of osp(1|32) ⊕ osp(1|32). The contraction problem was considered in [7], where the superalgebras obtained, although with the same structure, did not coincide with those originally found by D'Auria and Fré. The question remained of interpreting this discrepancy.
The two superalgebras found in [2] were shown to be just two examples of an infinite set of them [8,9], which solved in general the problem posed by D'Auria and Fré. In [8], it was shown that all superalgebras with structure G(s) = E (528|32+32) (s) ⋊ so(10, 1), where E (528|32+32) (s) is a fermionic central extension of the M-Theory superalgebra and ⋊ indicates semidirect product of algebras, are actually parametrized by a real parameter s, and that for all values of s except one, say s = 0 (see later), it was possible to interpret the three-form field A 3 in terms of one-forms dual to the generators of the algebra. This particular value of s, for which it is not possible to decompose A 3 , corresponds to the only superalgebra G(0) = E (528|32+32) (s=0) ⋊ so(10, 1) for which the Lorentz group SO(10, 1) can be enlarged to Sp(32). Moreover, it is given by an expansion of osp(1|32). Under another name, Lie algebra expansions were used for the first time by Hatsuda and Sakaguchi [10] in order to relate the Wess-Zumino terms of the adS-type superstrings with the Poincaré ones. The expansion method, studied in full generality in [11], consists of expanding the Maurer-Cartan dual one-forms of an initial superalgebra in terms of a parameter λ, and then identifying the coefficients of each power in λ in the resulting Maurer-Cartan equations * . In this way, superalgebras with an infinite number of generators are obtained. Certain conditions can be imposed in order to ensure that, by cutting the expansion in λ up to a finite power, the resulting equations are the MC equations of a finite (super)algebra [11]. It turns out that E (528|32+32) (s=0) ⋊ so(10, 1) corresponds to the expanded superalgebra osp(1|32)(2, 3, 2) (see [8,9,11] for the notation).
In all the examples obtained so far, the resulting expansions can be viewed as extensions followed by contractions, and this will presumably be true in general. The inverse statement is obviously false, because an arbitrary extension plus a contraction does not have to be an expansion since expansions remember the structure of the original, unexpanded algebra. So, it makes sense to find out whether the contraction of osp(1|32) ⊕ osp(1|32) leads to an expansion of osp(1|32) or to the other (s = 0) superalgebras in the class of fermionic extensions of the M-theory superalgebras, G(s = 0). We have performed a systematic calculation of all possible contractions of osp(1|32) ⊕ osp(1|32) leading to a superalgebra with the genertic structure where L is an arbitrary superalgebra that has to be present because the contraction procedure does not change the dimension of the superalgebra and the dimensions of osp(1|32)⊕osp(1|32) and those of the G(s) do not match. Indeed, osp(1|32)⊕osp(1|32) has dimension 1120 = 2× 32×33 2 + 32 = 2× 11 + 11 2 + 11 5 + 32 , whereas E (528|32+32) (s)⋊ so(10, 1) have dimension 647 = 528 + 32 + 32 + 11 2 , so the dimension of the bosonic Lie algebra L is equal to 473.
The main result of this paper is that it is possible to obtain by contraction only the fermionic extension of the M-theory superalgebra given by an expansion, i.e. the case s = 0 in (1.2). In other words, none of the Lie superalgebras, suitable for decomposing the three-form A 3 of D = 11 supergravity in terms of MC one-forms, can be obtained by contraction of the mixture of two osp(1|32) algebras.
The plan of the paper is as follows: in section 2 a review of osp(1|32) and the fermionic extensions of the M-theory algebras is presented. Section 3 explains the procedure used to obtain the contractions of osp(1|32)⊕osp(1|32) and contains the statement of the main result of this paper. Finally, section 4 is devoted to the conclusions.
Using the relation (2.3) in (2.1), we obtain the commutators and anti-commutators relations [16,17] [Z a , where the antisymmetrisation in the r.h.s., denoted by the square brackets, is such that the overall weight is 1. Correspondingly, the analogue to equation (2.3) for Π αβ in (2.2) is given by the Maurer-Cartan equations (2.5) We will use, in what follows, eqs. (2.5) rather than the equivalent Lie superalgebra commutators of eqs. (2.4). The algebras that we are going to obtain are related with G(s) = E (528|32+32) (s) ⋊so(10, 1) [8,9]. They are determined by the generators Q α , Q ′ α , Z a , Z ab , Z a 1 ...a 5 , plus the Lorentz generators J ab , and their commutators and anti-commutators can be given in the form plus the obvious ones involving the Lorentz generators given the Lorentz character of the algebra. Note that in eqs. (2.6) the real parameter τ 2 is always different from zero and it can be absorbed in the definition of Q ′ β , so that only one free real parameter s remains. This factorization of the algebra also includes the case when τ 2 → 0 and so s → ∞, such that the product τ 2 · s remains finite.

Contractions of osp(1|32) ⊕ osp(1|32)
Generalized, or Weimar-Woods [18,19], contractions can be constructed as follows: let G be a Lie (super)algebra given, as a vector space, by the direct sum and such that the (graded) commutators obey In particular, V 0 is a subalgebra of G. Let { X p, αp }, p = 0, . . . , n, α p = 1, . . . , dim V p , be a basis of G relative to the splitting (3.1). Then, the structure constants C r, γr p, αp q, βq vanish for r > p + q. If ω p, αp are the one-forms dual to the vector fields X p, αp , ω p, αp (X q, βq ) = δ p q δ αp βq , the MC equations of G are then dω r, γr = − 1 2 p+q≤r C r, γr p, αp q, βq ω p, αp ∧ ω q, βq . It turns out that the same vector space (3.1), but now with modified MC equations given by (3.3) with the sum only extended to p + q = r, defines a new Lie (super)algebra G c , known as the Weimar Woods contracted /super)algebra relative to the splitting (3.1). This contracted algebra can be obtained by re-scaling in terms of a parameter λ the forms ω p, αp as ω p, αp → λ p ω p, αp in the starting MC equations, and then taking the limit λ → 0. This is the procedure that we use in this paper. The case n = 1 corresponds to the original,İnönü-Wigner [20,21], contractions.
Contractions are dimension preserving. What it is not always realized is that the contraction of a direct sum of two Lie (super)-algebras G ⊕Ḡ can be different from the direct sum of the contractions of G andḠ, i.e., For instance, the non-trivially extended Galilei algebra may be obtained as a contraction of the trivial extension of the Poincaré algebra by u(1) [22,23], and the superalgebra of D = 3 (p, q)-Poincaré supergravity as a contraction of osp + (p|2)⊕osp − (q|2)⊕so(p)⊕so(q) [24]. This situation happens when the contraction is performed relative to a basis that is a linear combination of generators in G andḠ, and the inverse of this linear combination is not defined in the contraction limit, so it cannot be undone after the contraction.
In our case we have that G = osp + (1|32) andḠ = osp − (1|32). Let {X i } and X i be bases of generators of the Lie algebras G andḠ, respectively. We will considerer a new basis and then perform a generalized Weimar-Woods contraction [18,19] by rescaling in a consistent way the set of generators {Y i ,Ȳ i } so that the contraction limit (when the scaling parameter goes to zero) is well defined. Alternatively, this can also be done with the Maurer-Cartan forms dual to the generators. In fact, this is how we have done our calculations. Since osp + (1|32) and osp − (1|32) are actually two non-isomorphic real versions of the same complex algebra, we can take G =Ḡ = osp + (1|32) by considering complex coefficients A j i , B j i , C j i , D j i . So, we shall take two copies of (2.3), (2.4) and (2.5) (or (2.6), (2.7)) and consider complex linear combinations of the generators, and then look for contractions that would correspond to a real superalgebra. The linear combinations will not be arbitrary, because the Lorentz character of the components of Z αβ in (3.4) has to be preserved. This means that we will take combinations of Z a and Z ′ a , Z ab and Z ′ ab , etc. separately, with scalar coefficients. In terms of the Maurer-Cartan one-forms, we write generically ρ where n = (1, 2, 5, α) denotes the number of Lorentz indices for the bosonic one-forms ρ a ± , ρ ab ± , ρ a 1 ...a 5 ± or the spinorial index for the fermionic one ρ α ± ≡ ψ α ± . We must ensure that the linear combinations have to be invertible, so that we really perform a change of basis.
Then, we pose the problem of finding the exponents of λ used to do the following rescaling ρ a + ⇒ λ n ρ a + , ρ ab 8) and the coefficients of (3.6) with the condition (3.7), such that the generalized Weimar-Woods contraction limit of the Maurer-Cartan equations leads to the structure (1.2).
We have performed the calculations by using a symbolic manipulation programme (Mathematica). The problem to solve becomes an algebraic system in the complex coefficients α (n) ; β (n) ; γ (n) ; δ (n) , those of the matrices inverse to the matrix in (3.7) (see Appendix) and in the real parameter s. More explicitly, after the redefinition (3.6) and the change of scale (3.8), the r.h.s of the resulting Maurer-Cartan equations will be given by sums of terms with the following structure λ E(m,n,p,q,r,t,u,w) C(α (n) , β (n) , γ (n) , δ (n) ) (ρ ± ∧ ρ ± ), i.e., apart from the exterior product of two one-forms, there is a power of λ that depends on the scaling factors of (3.8), and a coefficient (structure constant) that depends on the parameters of the linear combination (3.6). We have first imposed the condition that some of these terms reproduce the ones in (2.7), which fixes the values of their C ′ s in terms of the real parameter s and also implies that their corresponding exponents E vanish. As a result, some of the remaining exponents are negative so their coefficients C have to vanish consistently with the Weimar-Woods approach. Additionally, there are also terms that should not appear in the limit λ → 0, which means that for them either E > 0 or C = 0. Finally, as already mentioned, the linear combinations in (3.6) have to be invertible.
This process leads to a system of equations and inequations that turns out to have a solution only when s = 0. We remind that this value corresponds in our parametrization to the expansion of osp(1|32) for which the three-form of D = 11 supergravity A 3 cannot be written in terms of Maurer-Cartan one-forms. We have also considered the case s → ∞ and checked that there is no solution. We do not include here the detailed expressions of the explicit computing, but they are available from the authors upon request.

Conclusions
We have shown in this paper that is not possible to obtain by generalized Weimar-Woods contraction from osp + (1|32) ⊕ osp − (1|32) none of the algebras found in [8], which allow a gauge group interpretation of the three-form field in the sense of [2]. Hence, we can conclude that D = 11 supergravity cannot be connected with the semi-simple supergroup OSp + (1|32) ⊗ OSp − (1|32) by trivializing the three-form field A 3 .
This result, however, does not necessarily mean that the conjecture made in [6], according to which D = 11 supergravity can be obtained as a low-energy limit of a Chern-Simons theory based on osp + (1|32) ⊕osp − (1|32), is incorrect. This is so because although the low-energy limit corresponds to a rescaling of the gauge fields in terms of a parameter λ of the type that appears when a Weimar-Woods contraction is performed, only the leading or next to leading terms in the resulting expansion of the Chern-Simons action in powers of λ have the symmetries of the contracted algebra. However, the term that would correspond to supergravity is neither the leading nor the next to leading term, hence it is unclear how the problem of checking the connection of Chern-Simons supergravity with the ordinary one would be related to the algebra contractions.

Appendix
In this appendix we show the Maurer-Cartan equations of the osp + (1|32) ⊕ osp − (1|32) superalgebra. These relations are written in terms of the complex scalar coefficients (3.6) used to determine the linear combinations between the rescaled one-forms of the two osp + (1|32) algebras ; and the structure constants (2.5). These explicit relations are 1).-For one-index tensors To obtain the explicit form of dρ a − is enough to change in the previous expression of dρ a + the A and B for C and D, respectively, i.e.