From osp(1|32) ⊕ osp(1|32) to the M-Theory Superalgebra: a Contraction Procedure

ISSN 1063-7788, Physics of Atomic Nuclei, 2017, Vol. 80, No. 2, pp. 340–346. c Pleiades Publishing, Ltd., 2017. ELEMENTARY PARTICLES AND FIELDS Theory From osp(1|32) ⊕osp(1|32) osp(1|32) ⊕osp(1|32) osp(1|32) ⊕osp(1|32) to the M-Theory Superalgebra: a Contraction Procedure∗ J. J. Fern ´ andez**,J.M.Izquierdo ***,and M.A.delOlmo **** Departamento de F´ısica Te ´orica and IMUVA, Universidad de Valladolid, Spain Received May 18, 2016 Abstract—We show the impossibility to obtain the D’auria–Fr ´ e-type superalgebras that allow for an underlying gauge theoretical structure of D=11supergravity from the superalgebra osp(1|32)+⊕ osp(1|32)−, by means of a Weimar-Woods contraction. DOI: 10.1134/S1063778817020156 1. INTRODUCTION In the original paper where supergravity theory in D=11was introduced, Cremmer, Julia, and Scherk (CJS) [1] raised the question of the identification of its underlying gauge symmetry group. They conjectured that the theory could admit a geometrical interpretation in terms of the simple supergroup OSp(1|32). The evidence in favor of this suggestion was the fact that its graded Lie algebra osp(1|32) contains an internal o(8) subalgebra, which is also a subalgebra of the internal invariance group of a D=4reduction of the D=11model. However, the lack of understanding about how this connection could be realized, was caused because of the presence of a three-form field A3= Aμνρ(x)dxμ∧dxν∧dxρin the action found in [1]. While the graviton ea=ea μdxμ, the gravitino ψα= ψα μdxμand the spin connection wab =wab μdxμoneforms can be considered as the gauge fields of a Lie superalgebra, the antisymmetric Aμνρ(x)gauge field cannot be associated to a symmetry operator in an easy way. D’Auria and Fr ´ e [2], addressed this problem by looking at the free differential algebra (FDA) satisfied by the above forms in the absence of curvatures. The FDA formalism does not consist only of one-forms, so it is the natural extension of the Lie algebras, being particulary suitable to account the three-form field mentioned above. D’Auria and Fr ´ e’s idea was to express A3in terms of linear combinations of exterior products of one-forms, treated as gauge ∗The text was submitted by the authors in English. **E-mail: [email protected] ***E-mail: [email protected] ****E-mail: [email protected] fundamental fields belonging to a certain superalgebra which had to be found. For this to be possible, it was necessary to introduce a set of additional oneforms, two of them bosonic fields Bab =Bab μdxμ and Ba1...a5=Ba1...a5 μdxμ, and one extra fermionic contribution ψα=ψα μdxμ, which play a central role in the new algebra. Consequently, the composite nature of the three-form field A3required extending the underlying gauge group of D=11CJS supergravity into a new superalgebra with larger algebraic dimension (hereafter E(528|32+32)). Two superalgebras were obtained which allowed the decomposition of A3in that way. The question about how these superalgebras could be related to a simple supergroup was studied in [3], where the osp(1|32),aswellasthesu(32|1) and the conformal osp(1|64) superalgebras, were ruled out as algebras that could lead to the D’Auria–Fr ´ eonesbycontraction. Nevertheless, the semisimple superalgebra osp(1|32) ⊕osp(1|32) was not in the above list, but this is the algebra that was later considered by Ho ˘ rava [4] as a prospective candidate to construct a Chern–Simons (CS) M-theory group on a holographic scenario. The reason for this choice relies on the fact that the CS action must be parity invariant in the Ho˘rava–Witten construction [5], which is based on the properties of the heterotic string theory [6], and a single osp(1|32) superalgebra does not yield parity invariance [7–10]. Rather, parity invariance will require a non-minimal extension of the osp(1|32) superalgebra into an algebra with 64 supercharges. Another implication of Ho ˘rava’s suggestion is that D=11 CJS supergravity would be a low-energy limit of a CS theory based on osp(1|32) ⊕osp(1|32). 340 FROM osp(1|32) ⊕osp(1|32) TO THE M-THEORY 341 In [4, 11], 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 [11], where the superalgebras obtained, although with the same structure, did not coincide with those originally found by D’Auria and Fr ´ e. The question remained of interpreting this discrepancy. The two superalgebras found in [2] were shown to be two particular elements of an infinite set G(s)= E(528|32+32)(s)so(10,1) parametrized by one real parameter s[12, 13] (see Section 3). Moreover, all values of sexcept s=0, allow for the decomposition of the three-form field A3in terms of combinations of one-forms dual to the generators of the algebra. In fact, this particular case G(0) for which it is not possible make such decomposition can be obtained via an expansion procedure from osp(1|32).Anexpansion of a given Lie (super)algebra is obtained by a suitable rescaling of the Maurer–Cartan (MC) dual one-forms in terms of a parameter λ[14–16]. The resulting (super)algebra is expressed by the coefficients of each power in λin the resulting MC equations. In this way, superalgebras with an infinite number of generators are obtained, so this process does not preserve the dimension of the start handle algebra. Certain conditions can be imposed in order to ensure that, by cutting the expansion in λup to a finite power, the resulting equations are the MC equations of a finite (super)algebra [17]. 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. However, the inverse statement is obviously false: contractions are more general than expansions in the sense that the latter remember the structure of the original (super)algebra, whereas the former procedure leads to more possibilities. Nevertheless, taking two copies of the same algebra would reduce the freedom associated to contraction, and it is not clear whether a contraction of osp(1|32) ⊕osp(1|32) leads to an expansion G(0) or to the other class of the set G(s=0). For the above reasons, we have made a detailed computation of all possible contractions of osp(1|32) ⊕osp(1|32) leading to a superalgebra with the generic structure E(528|32+32)(s)⊕L(473)so(10,1),(1) where Lis an arbitrary superalgebra, not necessarily abelian, that mixes in a trivial way with the D’Auria– Fr ´ e superalgebra, and has to be present because the contraction procedure is dimension preserving, and the dimensions of osp(1|32) ⊕osp(1|32) and G(s)do not match. Our study states that it is only possible to obtain the expansion case (s=0)by contraction procedure from the osp(1|32) ⊕osp(1|32) superalgebra. Therefore, none of the Lie superalgebras, suitable for decomposing the three-form A3of D=11 supergravity in terms of MC one-forms, can be obtained by contraction from the direct sum of two osp(1|32) algebras. The paper is organized as follows: in Section 2 a brief review of osp(1|32) is presented. In Section 3 we show the main properties of the D’Auria–Fr ´ e family superalgebras. Lie algebra and Maurer–Cartan oneform languages are used in both sections. Section 4 is devoted to showing the main details in the contraction procedure from the two copies of osp(1|32). Finally, we collect some conclusions in the last section. 2. THE SUPERALGEBRA osp(1|32) The orthosymplectic supergroup OSp(1|32) defines the minimal grading of the symplectic Sp(32) bosonic group which, in turn, is the maximal group preserving the Majorana property of the SO(10,1) spinors. Since its algebraic counterpart osp(1|32) verifies the inclusion so(10,1) ⊂sp(32) ⊂osp(1|32), its bosonic generators Pa,J ab,Z a1...a5could be directly associated to the even symmetry operators of the D=11superPoincar ´ e bosonic extended superalgebra [18–20]. The orthosymplectic Lie algebra osp(1|32) can be defined, in a certain basis {Zαβ,Q γ}, by the following anticommutator and commutator relations: {Qα,Q β}=ηZαβ, [Zαβ,Q γ]=CαγQβ+CβγQα, [Zαβ,Z γδ]=CαγZβδ +CβγZαδ +CαδZβγ +CβδZαγ,(2) where Zαβ is a symmetric matrix in the spinorial indices (α, β, γ =1,...,32), which are raised and lowered by the 32 ×32 skewsymmetric charge conjugation matrix Cαβ. We point out that, in the first relation, the parameter ηmaytakethevalues±1,but these choices do not make any difference in the complex Lie algebra in contrast with the real case, where they determine two nonisomorphic superalgebras denoted by osp+(1|32),osp−(1|32), as it happens in the the case of osp(1|2) (see [21]). Decomposing Zαβ in the basis of the Spin(1,10) gamma-matrices, we can express it in terms of the usual tensorial generators Za,Zab,Za1...a5,as Zαβ =1 1! ·32Γa αβZa+1 2! ·32Γab αβZab +1 5! ·32Γa1...a5 αβ Za1...a5,(3) PHYSICS OF ATOMIC NUCLEI Vol. 80 No. 2 2017 342 FERN ´ ANDEZ et al. where the notation Γa1...an αβ =(Γ a1...anC−1)αβ,has been used to denote generically the antisymmetrized products of D=11Dirac matrix. Using this last relation (3) in (2), we obtain the commutator and anticommutator relations [22, 23] [Za,Z b]=1 8Jab, [Za,J b1b2]=1 4δa [b1δk b2]Zk, [Ja1a2,J b1b2]=1 2δ[a1 [k1δa2] [b1δk2] b2]Jk1k2, [Za,Z b1...b5] =i 8·5!c5...c1k1k2...k6δ[k1 aδk2 [b1...δk6] b5]Zc1...c5, [Ja1a2,Z b1...b5]=5 4δ[a1 [k1δa2] [b1δk2 b2...δk5] b5]Zk1k2...k5, [Za1...a5,Z b1...b5] =i 8δ[a1 [k1...δa5] k5δk6 [b1...δk10] b5]k1...k5k6...k10cZc +5i 4!δ[a1 [k1δa2 k2δa3 k3δa4 [b2δa5] b1δ[k4 b3δk5 b4δk6] b5] ×k1k2k3k4k5k6c5c4c3c2c1 ×Zc1c2c3c4c5+75δ[a1 [k1δa2 [b4...δa5] b1δk2] b5]Jk1k2, [Za,Q α]= 1 16(Γa)αβQβ, [Jab,Q α]=−1 16(Γab)αβQβ, [Za1...a5,Q α]= 1 16(Γa1...a5)αβQβ, {Qα,Q β}=Γ a αβZa+1 2!Γab αβJab +1 5!Γa1...a5 αβ Za1...a5,(4) where the square brackets in the r.h.s. denote antisymmetrization with height one. Remember that the bosonic generators Jab and Za1...a5are associated to even symmetry operators of the D=11 super- Poincar ´ e bosonic extended superalgebra. It is convenient to resort to a dual point of view to deal with Lie algebras, in agreement with the FDA context developed by D’Auria and Fr ´ e. If we use the dual algebra spanned by the MC one-forms Παβ,Πα dual to the algebraic generators Zαβ,Q α, which verify the identities Παβ(Zγδ)=2δ(α γδβ) δ≡δα γδβ δ+δβ γδα δ, Πα(Qβ)=δα β, the Eqs. (2) can be rewritten in a compact form by the MC close relations [24] dΠαβ =−(Παγ ∧Πγβ)−η(Πα∧Πβ),(5) dΠα=−Παγ ∧Πγ, which provide the 528 bosonic MC one-forms of the sp(32) algebra through the symmetric spin-tensor Παβ,andthe32 fermionic MC one-forms Πα. Expressing the spinorial symmetric one-form Παβ in terms of the MC one-forms Πa,Πab,Πa1...a5dual to the algebraic generators Za,Zab,Za1...a5respectively, the couple of MC equations (5) can be split as dΠa=−1 8Πb∧Πba−1 2Γa αβ πα∧πβ −i 16(5!)2ab1...b5c1...c5(Πb1...b5∧Πc1...c5), dΠab =−1 8Πa∧Πb−1 8Πac ∧Πcb −1 2Γab αβ πα∧πβ−1 4! ·8Πac1...c4∧Πc1...c4b, dπα=1 16(Γa)βαπβ∧Πa−1 2·16(Γab)βα ×πβ∧Πab+1 5! ·16(Γa1...a5)βαπβ∧Πa1...a5, dΠa1...a5=−i 5! ·8cb1...b5 a1...a5Πc∧Πb1...b5 −5 8Π[a1b∧Πba2...a5]−1 2Γa1...a5 αβ πα∧πβ −i 2·(4!)2a1...a5b1b2b3c1c2c3 ×Πb1...b5∧Πb5b4c1c2c3.(6) which provide the same information as its algebraic counterpart described by the (anti)commutator relations (4). 3. THE D’AURIA–FR ´ E SUPERALGEBRAS The two solutions found originally by D’Auria and Fr ´ e can be identified as two examples of an infinite family of superalgebras G(s)=E(528|32+32)(s) so(10,1), which solved in general the problem posed by them. All of these superalgebras contain a set of 528 bosonic and 32 + 32 = 64 fermionic generators, plus the Lorentz generators Jab,andaredefined through the (anti)commutator relations [Za,Q α]=τ2(s−1)(Γa)αβQ β, [Zab,Q α]=τ2(Γab)αβQ β, [Za1...a5,Q α]=τ2s 6! −1 5!(Γa1...a5)αβQ β, {Qα,Q β}=Γ a αβZa+1 2!Γab αβZab +1 5!Γa1...a5 αβ Za1...a5, Q α,·=0,(7) PHYSICS OF ATOMIC NUCLEI Vol. 80 No. 2 2017 FROM osp(1|32) ⊕osp(1|32) TO THE M-THEORY 343 where the fermionic generator Q αdoes not introduce a new grading in the extended algebra because of its central character. We point out that Zab and Za1...a5are the central generators of the M algebra [25, 26], so this set of superalgebras are just a class of fermionic central extensions of the M-theory superalgebra. This point of view has been used as a way to try to understand the underlying symmetry structure of the Mtheory from the knowledge of the gauge symmetry of D=11supergravity [12]. Intheaboveequationstherealparameterτ2is always different from zero and it can be included in the normalization of the additional central charged Q α,so it is thus inessential. Then, only one free parameter sremains which labels all the equivalent, but nonisomorphic, members belonging to this uniparameter family E(528|32+32)(s)[12]. Note that this factorization also includes the case when τ2→0and so s→ ∞,suchthatτ2·sremains finite. In particular, the two specific D’Auria–Fr ´ e solutions take the values E(3/2) and E(−1) under the above notation. In a similar way as in the previous section, introducing the MC one-forms Πa,Πab,Πa1...a5,πα, παdual to the algebraic generators Za,Zab,Za1...a5, Qα,Q α, respectively, the family of superalgebras E(528|32+32)(s)can be equivalently descibed by the MC equations dΠa=−1 2Γa αβ(πα∧πβ), dΠab =−1 2Γab αβ(πα∧πβ), dΠa1...a5=−1 2Γa1...a5 αβ (πα∧πβ), dπα=0, dπα=−τ2(s−1)(Γa)βα(Πa∧πβ) +1 2!(Γab)βα(Πab ∧πβ) +s 6! −1 5!Γa1...a5)βα(Πa1...a5∧πβ,(8) where the real parameter sis only involved in the last relation. In this parametrization, all the algebras in (7) and (8) can be used to write the three-form A3of D= 11 supergravity as a composite one, except of the case s=0. This particular value corresponds to the only superalgebra G(0) = E(528|32+32) (s=0) so(10,1) for which the Lorentz groupSO(10,1) can be enlarged to Sp(32), and is ruled out on the searching of the local symmetry of the D=11supergravity [12, 13, 15]. Hence, it appears that the real factor splays an important role in the study of the connection of the D=11supergravity with the osp(1|32) ⊕osp(1|32) superalgebra. 4. CONTRACTIONS OF osp(1|32) ⊕osp(1|32) In [4], the author tried to explore the possible relation of the M theory with Chern–Simons supergravities. He focused the attention on the fact that M theory is parity invariant. However, the Chern– Simons action based on the eleven dimensional antide Sitter group Osp(1|32) is not compatible with such invariance. Thus, in order to respect parity invariance, Horava pointed out that the gauge group will contain extra bosonic charges and an extra supercharge Q α, so that the complete set of bosonic and fermionic generators lead to an algebraic structure isomporphic to osp(1|32) ⊕osp(1|32). Morever, he suggested that this algebra contracts to the D’Auria–Fr ´ e superalgebra G(s=0), because in low-energy limit we have to recover the D=11CJS supergravity. In this section we study whether this is the case. At a first step, we are interested in writing the explicit relations of the osp(1|32) ⊕osp(1|32) superalgebra in a generic basis. In general, we consider a change of basis from the basis of generators {Xi} and ¯ Xiof the component Lie algebras Gand ¯ G,to anewone{Yi,¯ Yi}of G⊕¯ G. In our case, we have G=osp+(1|32) and ¯ G=osp−(1|32). Since these two superalgebras are actually two non-isomorphic real versions of the same complex algebra (see Section 2), we can take G=¯ G=osp+(1|32) by considering complex factors aj i,b j i,c j i,d j ion the mix process. Consequently, we can take two copies of (4) and consider linear combinations of the generators given by Yi=aj iXj+bj i¯ Xj, ¯ Yi=cj iXj+dj i¯ Xj.(9) This process can also be done applied to the Maurer– Cartan one-forms Πi,¯ Πidual to the algebraic generators Xi,¯ Xi. In fact, this is how we have done our calculations. Then, the linear combination (9) leads to a new set of Maurer–Cartan oneforms denoted ρ(n) +,ρ (n) −, which can be written generically as ρ(n) +=α(n)Π(n)+β(n)¯ Π(n), ρ(n) −=γ(n)Π(n)+δ(n)¯ Π(n),(10) PHYSICS OF ATOMIC NUCLEI Vol. 80 No. 2 2017 344 FERN ´ ANDEZ et al. where α(n);β(n);γ(n);δ(n)are a set of 16 complex scalars and n=(1,2,5,α)denotes the number of Lorentz indices for the bosonic one-forms ρa ±,ρ ab ±,ρ a1...a5 ±or the spinorial index for the fermionic one ρα ±≡ψα ±. We have to emphasize that the above linear combinations cannot be arbitrary in order to keep the Lorentz transformation law inside the osp(1|32) ⊕ osp(1|32). This means that we have to take combinations of pairs of one-forms Πaand ¯ Πa,Πab and ¯ Πab, etc. separately. Moreover, we must ensure that these linear combinations have to be invertible in order to really perform a change of basis, so det ⎛ ⎝α(n)β(n) γ(n)δ(n)⎞ ⎠=0. Thus, the osp(1|32) ⊕osp(1|32) superalgebra may be written explicity in terms of these complex scalar coefficients α(n);β(n);γ(n);δ(n), their inverse relations α (n);β (n);γ (n);δ (n)and the structure constants (6) (see appendix on reference [27]). Now let us perform a generalized Weimar-Woods contraction on these equations. Generalized, or Weimar-Woods [28, 29], contractions can be constructed as follows: let Gbe a Lie (super)algebra given, as a vector space, by the direct sum G=V0⊕V1⊕···⊕Vn,(11) and such that the (graded) commutators obey [Vp,V q]⊂ p+q  l=0 Vl.(12) In particular, V0is a subalgebra of G.Let{Xp,αp}, p=0,...,n,αp=1,...,dim Vp,beabasisofGrelative to the splitting (11), then expression (12) can be written explicitly as [Xp,αp,X q,βq]=Cr,γr p,αp;q,βqXr,γr, Cr,γr p,αp;q,βq=0 ∀r>p+q. If ωp,αpare the one-forms dual to the vector fields Xp,αp, i.e., ωp,αp(Xq,βq)=δp qδαp βq, the MC equations of Gare dωr,γr=−1 2 p+q≤r Cr,γr p,αpq,βqωp,αp∧ωq,βq.(13) It turns out that the same vector space (11), but now with modified MC equations given by (13) with the sum only extended to p+q=r,defines a new Lie (super)algebra Gc, known as the Weimar- Woods contracted/super)algebra relative to the splitting (11). This contracted algebra can be obtained by rescaling in terms of parameter λthe forms ωp,αpas ωp,αp→λpωp,αpin the starting MC equations, and then taking the limit λ→0. This is the procedure that we use in this paper. The case n=1corresponds to the original, ˙In ¨ on ¨ u–Wigner [30, 31], contractions. Given a starting algebra and a set of structure constants, in practice one has to study a system of equations of the rescaled exponents for which the contraction limit (λ→0)iswelldefined and reproduces the algebraic structure desired. However, in our case the starting structure constants of osp(1|32) ⊕osp(1|32) are written in terms of α(n);β(n);γ(n);δ(n)and α (n);β (n);γ (n);δ (n),so the set of exponents of λin the rescaling ρa +⇒λnρa +,ρ ab +⇒λpρab +, ρa1...a5 +⇒λrρa1...a5 +,ψ α +⇒λvψα +, ρa −⇒λmρa −,ρ ab −⇒λqρab −, ρa1...a5 −⇒λtρa1...a5 −,ψ α −⇒λwψα −,(14) arenottheonlycoefficients to be determined. Hence, we have to add a set of extra conditions, in terms of the coefficients of the linear combinations (10), that ensures that the structure constants after the contraction limit λ→0reproduce the structure (1). We list below the steps that we have followed: 1. First, we have performed the change of scale (14) on the MC equations of the osp(1|32) ⊕ osp(1|32) ([27]). The resulting MC equations could be rewritten by sums of terms with the following structure λE(m,n,p,q,r,t,u,w)C(α(n),β (n),γ (n),δ (n))(ρ±∧ρ±), where, apart from the exterior product of two oneforms, there is a power of λthat depends on the scaling factors of (14), and a coefficient (structure constant) that depends on the parameters of the linear combination (10) and their inverse relations. 2. We then have chosen the one-form ρab ±dual to the Lorentz generator Jab which fixes the tensorial transformation on the resulting algebras, as well as the fermionic one-form field associated to the central supercharge Q α. Without loss of generality, we have made the election ρab +as the boost generator and ψα as the central fermionic one. 3. Next, we have carried out a suitable election, via visual inspection, of the products of MC one-forms ρa +,ρ ab −,ρ a1...a5 +,ψαthat reproduce the D’Auria–Fr ´ e structure (8), plus the Lorentz transformations. On the other hand, the remaining MC one-forms ρa −,ρ a1...a5 −belonging to the bosonic algebra Lare identified by exclusion procedure. PHYSICS OF ATOMIC NUCLEI Vol. 80 No. 2 2017 FROM osp(1|32) ⊕osp(1|32) TO THE M-THEORY 345 4. The above step fixes the exponents Ewhich have to be vanished, as well as the values of the structure constants Cin terms of the real parameter s. Moreover, the study of the compatibility in terms of the scaling factors (14) for all the 112 powers associated to the 112 terms of the MC equations of the osp(1|32) ⊕osp(1|32), leads to: (a) terms with E=0that should not appear in the limit, so we have to impose the vanishing of their associated structure constant C=0, (b) terms whose powers are negative E<0,so we have to ensure that their algebraic coefficient also vanishes C=0, consistently with the Weimar- Woods approach. 5. Finally, we have imposed that the linear combination in (10) is invertible. These steps lead to a system of equations and inequations in terms of the complex scalar coefficients α(n);β(n);γ(n);δ(n), their inverse matrix relations α (n);β (n);γ (n);δ (n)and the real parameter s,performed by fifteen conditions C(α(n),β (n),γ (n)δ(n))=0, associated to the third step, six equations C(α(n),β (n),γ (n),δ (n))=0 for the fourth step, and four inequalities needed to express the admissible basic changes of the fifth step. The results of this problem rely on heavy algebraic manipulations which have been performed by using a symbolic manipulation program (Mathematica). The resulting system of equations and inequations have asolutiononlywhens=0, i.e., the expansion case of osp(1|32) for which the three-form of D=11su- pergravity A3cannot 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. 5. CONCLUSIONS The main result of this paper is the proof that it is not possible to obtain by generalized Weimar-Woods contraction from osp+(1|32) ⊕osp−(1|32) any of the algebras found in [12], which allow a gauge group interpretation of the three-form field in the sense of [2]. In other words, D=11 supergravity cannot be connected with the semi-simple supergroup OSp+(1|32) ⊗OSp−(1|32) by trivializing the threeform field A3. But we cannot claim that the conjecture made in [4], according to which D=11supergravity can be obtained as a low-energy limit of a Chern–Simons theory based on osp+(1|32) ⊕osp−(1|32),isincorrect. This is due to the fact that there is no reason why the λqterm in the expansion of the CS action should be invariant under the contracted algebra. 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