Bosonic D=11 supergravity from a generalized Chern-Simons action
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Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 923 (2017) 633–652 www.elsevier.com/locate/nuclphysb Bosonic D=11 supergravity from a generalized Chern–Simons action D. Camarero a, J.A. de Azcárraga b,∗, J.M. Izquierdo a aDepartamento de Física Teórica, Universidad de Valladolid, 47011-Valladolid, Spain bDepartamento de Física Teórica and IFIC (CSIC-UVEG), 46100-Burjassot, Valencia, Spain Received 3 July 2017; received in revised form 7 August 2017; accepted 22 August 2017 Available online 31 August 2017 Editor: Leonardo Rastelli Abstract 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). ©2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction It is known [1–4] that various D=3 (super)gravities are actually Chern–Simons (CS) theories based on Lie superalgebras. Although supergravities in D>3, Dodd, do not have a true CS nature, it has been argued that certain CS theories may be related to supergravities for odd D>3 dimensions. These CS theories have been generically called ‘CS supergravities’ [5–7] (see [8] for further references). *Corresponding author. E-mail address: [email protected].es (J.A. de Azcárraga). http://dx.doi.org/10.1016/j.nuclphysb.2017.08.015 0550-3213/©2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.
634 D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 CS actions are constructed (see e.g. [9]) as follows. Let Ai, Fi(i=1, ..., dim G) be the Maurer–Cartan (MC) gauge fields and curvatures associated with a Lie algebra Gin a certain basis. Then, the 2-form (the exterior product symbol ∧will be omitted throughout) H=ki1...1Fi1...Fi,(1.1) where ki1...1are the coordinates of a symmetric invariant tensor of order , is closed and gauge invariant. Since a gauge free differential algebra is contractible, His also exact, H=dB, and the potential Bdefines a Chern–Simons (2 −1)-form, which is gauge invariant up to an exterior differential. Then, the CS action is given by the integral ICS = M2−1 B(1.2) over a (2 −1)-dimensional manifold M2−1; it is gauge invariant up to non-trivial topological situations ignored in this paper. The possible connection between CS supergravity and the actual supergravities for D>3 suggested in refs. [10–13] (see [14] for another connection in D=11 based on the comparison of the linearized models) is best analyzed by expressing the gauge fields and curvatures associated with the superalgebra Gin terms of supermatrices Aand F, with oneand two-form fields entries respectively. This is the case for D=3 and G=osp(p|2) ⊕osp(q|2), for D=5 and G= su(1|2, 2)and for D=11 and G=osp(1|32)(or osp(1|32) ⊕osp(1|32)). His typically of the form H=Tr(F)where Tr denotes the supertrace, although other non-primitive, closed gauge invariant forms will be considered below. Depending on the case, the MC one-form gauge fields of the superalgebras may, or may not, correspond to the fields of D-dimensional supergravities. In the second and almost general case, the association between ‘CS supergravities’ and the standard supergravities in Ddimensions fails. Let us show this by summarizing the D=3, 5 and 11 cases. We use mostly plus metric throughout. 1.1. The D=3case Let us first consider the simplest algebra G=osp(1|2) ⊕sp(2)(i.e. p=1, q=0 above). The osp(1|2)and sp(2)gauge fields, denoted Aand Arespectively, can be written in matrix form as A=fξ ¯ ξ0,f=faγa; A= f, f= faγa,(1.3) where ξis a two-component Grassmann odd Majorana spinor form and γaare the 2 ×2D=2 gamma matrices. Note that osp(1|2)alone would not provide enough fields for D=3 supergravity and that all fields fa, ξand fain (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 ωa, ea, and ψobtained from f, ξand fare then defined by fa=ωa+λea, fa=ωa,ξ=λ1 2ψ, (1.4) so that they have the right dimensions [ωa] =L0, [ea] =L1and [ψ] =L1 2to be identified with the fields of D=3, N=1 supergravity in the first order formulation.
D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 635 The action is constructed starting from the closed, invariant polynomial four-form H(f, f,ξ;α) =Tr(F2)+αTr( F2), (1.5) where αis a dimensionless constant and F=dA+A2=df +f2+ξ¯ ξdξ+fξ d¯ ξ+¯ ξf 0, F=d f+ f2.(1.6) Inserting (1.4) into (1.5) and collecting the terms in equal powers of λgives H(ω,e,ψ;λ,α) =H0+λH1+λ2H2+λ3H3,(1.7) where H0=H0(ω, α) only since H(ω, e, ψ; λ, α) is dimensionless and H1,2,3= H1,2,3(α). We note in passing that this re-scaling in λis 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 dimensionpreserving in general1(see [16,17] for details). By construction, the above two-form Hand the associated CS action are osp(1|2) ⊕sp(2) gauge-invariant. In particular, the local supersymmetry transformations under the odd dimensionless gauge parameter ηthat corresponds to the gauge field ξare, written in terms of =λ−1 2η, [] =L1/2, δea=ψγa, δψ=D +λeaγa, δωa=0,(1.8) where D=d−ωaγais the Lorentz covariant derivative. Since ωais supersymmetry invariant, so is H0which only contains this field. Thus, the action obtained from H(ω, e, ψ; λ, α) −H0(ω, α) is invariant under the local supersymmetry transformations (1.8), and provides the first order formulation of (1, 0)D=3AdS supergravity. Moreover, the leading λterm in H−H0, H1, 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 H0which is not invariant under the gauge transformation that cannot be ignored and the linear term in λdoes not yield Poincaré supergravity. In this case, a proper Poincaré limit may still be taken by enlarging Gas Gto osp+(p|2) ⊕osp−(q|2) ⊕so(p) ⊕so(q), and adding to Hthe two invariant so(p) and so(q)-valued four-forms [4,18]. 1.2. The D=5case The next simplest case is D=5. The smallest real superalgebra that contains the AdS5one so(4, 2) ∼su(2, 2)is the 24-dimensional G=su(1|2, 2). A su(1|2, 2)-valued form can be written in the form 1It is terminologically unfortunate that algebras of different dimensions are sometimes said to be related by so-called ‘generalized’. Inönü–Wigner contractions. There are, of course, generalizations of the original I–W contraction procedure with respect to a subalgebra, but these are also dimension-preserving, as it corresponds to the mathematical idea of contraction (see e.g. [17]).
636 D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 A=fξ i¯ ξ4if0,f=if0+faγa+1 4fabγab ;F=dA+A2,(1.9) where γa, a=0, ..., 4are 4 ×4 gamma matrices, ξis a four-component spinor form and ¯ ξits adjoint. Let us introduce again λ, [λ] =L−1, and new fields ea, φ, ωab and ψ, with dimensions 1, 1, 0 and 1/2 respectively, through the scalings f0=λφ, fa=λea, fab =ωab, ξ=λ1 2ψ. We now express the 16 real bosonic fields 1(φ) +5(e) +10(ω) and the 4 complex fermionic ones ψ (8 real) associated with the supergroup parameters in the form f=iλφ +λeaγa+1 4ωabγab ,ξ=λ1 2ψ. (1.10) Using these expressions in Fand H=Tr(F3)and collecting the different powers in λwe obtain H(φ,e,ω,ψ)=H0+H1λ+H2λ2+H3λ3+H4λ4+H5λ5,(1.11) where H0=H0(ω) and Hi, =1, ...5, depend on the gauge fields ea, φ, ωab and ψ. The term H3in λ3has the right dimension [H3] =LD−2=L3for a D=5 action. Therefore, it makes sense comparing the CS action obtained from H3with that of simple D=5 supergravity which, in the first order formulation, has the same spacetime fields content; including also the terms proportional to λ4and λ5and retaining only the last three terms would lead (removing a common λ3factor) to an action with a ‘cosmological constant’ term in λ2coming from H5(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)gauge transformations in terms of the rescaled fields depend on λ, δφ=−1 4¯ψ −¯ ψ δea=−i 4¯γaψ−¯ ψγa δωab =iλ 2¯γabψ−¯ ψγab δψ=d +1 4ωabγab+λ−3iφ +eaγa, (1.12) the individual terms are not invariant separately. The leading H0term 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 H3. In fact, it is easily seen that the H3term in (1.11) does not lead to D=5 supergravity. The quickest way to see it is by noticing that this H3term coming from the su(1|4)based CS action is not gauge invariant under the one-dimensional subgroup of transformations ϕcorresponding to the field φ, δϕφ=dϕ, in contrast with the action of the D=5 supergravity. 1.3. The D=11 case: preliminary considerations 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 Agiven by A=fξ ¯ ξ0,f=faγa+1 4fabγab +fa1...a5γa1...a5,(1.13)
D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 637 where γaare the 32 ×32 gamma matrices and ξis 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 ea, ωab, ψαand the three-form field Aof 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 fa, ˜ fa, fab, ˜ fab, fa1...a5, fa1...a5, ξα, ξα, as linear combinations of new fields ea, Ba, ωab, Bab, Ba1...a5, Ba1...a5, ψα, ψα, with dimension Lexcept for [ψα] =L1 2, [ψα] =L3 2, [ωab] =L0 (and perhaps Ba1,...,a5and Ba) using the scale factor λ, [λ] =L−1. It was conjectured in [10] that the three-form field Acould be a composite of ea, Bab, Ba1...a5, ψα, ψαas explicitly considered in [20]. A closed, osp(1|32)gauge invariant twelve-form Hhas the general expression H=Tr(F6)+αTr(F2)Tr(F4)+βTr(F2)3 ,(1.14) where F =dA +A2. The corresponding form Hfor osp(1|32)is expressed similarly in terms of F=d A+ A2. Then, introducing H(λ) =H(λ) + H(λ)and collecting the different powers of λ we can write H(λ) =H(λ)+ H(λ)=H 0+···+H 9λ9+H 10λ10 +H 11λ11 .(1.15) It was conjectured [10] that the H 9term would depend on ωab, eaand ψα, with the remaining fields either included in A =A(ea, Bab, Ba1...a5, ψα, ψα)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 ea, Ba, ωab, Bab, Ba1...a5, Ba1...a5, ψαand ψαwith the following assumptions: (a) the action corresponding to H 9has the gauge symmetries of the above fields realized in the generic form δAi=dαi+...; (b) the fields Baand Ba1...a5, which do not enter in A, are also absent in H 9, so that we can ignore them; (c) the field equations of ωab can be used to eliminate the ωab; (d) the linearized field equations for the elfbein eaand the gauge one-form fields Bab and Ba1...a5have a structure similar to the eaequation of D=11 supergravity and (e) the linearized field equations for ψα and ψαare linearized Rarita–Schwinger equations. With these assumptions, the counting of on-shell degrees of freedom goes as follows: ea μ(ψα μ,ψ α μ)B ab μBa1...a5 μ 9·11 −55 32·8 2each 9 ·11 29·11 5,(1.16) i.e. there are 4697 bosonic and 256 fermionic degrees of freedom.2But D=11 supergravity has 44 +84 =128 bosonic and 128 fermionic degrees of freedom, so that for H 9to lead to CJS supergravity there should be 128 fermionic and 4569 bosonic extra hidden gauge symmetries. Secondly, there is no reason why the H 9term in the expansion (1.15) of the right dimension L9should correspond to a locally supersymmetric action. Besides, the local supersymmetry 2The vielbein ea μand Rarita–Schwinger ψα μfields in Ddimensions have, respectively, (D −2)D −D 2=1 2(D − 1)(D −2)−1 (after using local Lorentz invariance) and 1 2(D −3)2[D/2]on-shell degrees of freedom. Similarly, a p-form gauge field Aμ1...μphas D−2 pon-shell d.o.f.; the B’s above are one-form gauge fields with additional antisymmetric aindices.
638 D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 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 on-shell 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 9term in (1.15). A second problem is the three-form field Ain the action of CJS supergravity. For Ato be a composite field, A =A(ea, Bab, Ba1...a5, ψα, ψα), the supersymmetry algebra of the H 9term 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 Acomplicating 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 Fa1···a4which 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 Aand Fa1···a4and keeping only ea, ωab and ψα, supports this conclusion. One may consider adding separately an osp(1|32)-gauge invariant dimensionless three-form field Ato look for an action involving the fields of a single osp(1|32). The additional Ais inert under osp(1|32)gauge transformations and, under two-form gauge transformations , A transforms as δA =d; thus, the four-form dAis δ-gauge invariant. Then, the general gauge invariant twelve-form H(F, A)(cf. (1.14)) is given by H=Tr(F6)+αT r(F4)T r(F2)+βTr(F2)3+νTr(F4)dA +δTr(F2)2 dA+ρTr(F2)(dA)2+σ(dA)3,(1.17) where α, ..., σare dimensionless constants. An action with the right dimensions would correspond to the H9term in the expression above with A =λ3A, [A] =L3. However, this construction still would not explain the need for the auxiliary Fa1···a4fields. 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 fa=λeeis chosen, consistency of the contraction limit would require a new field, ea1···a5say, with fa1···a5=λBa1···a5, so that the osp(1|32)MC equations df a∝ab1···b5c1···c5fb1···b5fc1···c5+··· (1.18) have a well defined λ →0 limit. But, if this consistency condition is removed, we may now set fa1···a5=ωa1···a5, [ωa1···a5] =L0, (rather than fa1···a5=λBa1···a5, which implies [Ba1···a5] =L1). Indeed, it will be shown that the ωa1···a5fields play the role of the Fa1···a4(see below eq. (3.61)).
D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 639 Unfortunately, a calculation shows that the λ9term 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, ···, σin (1.17) such that the H9 term in Hresulting from the re-scalings fa=λea, fab =ωab, fa1···a5=ωa1...a5and A =λ3A lead to the equations of its bosonic sector. In particular, the ωa1...a5equation determines ωa1...a5 itself in terms of the coordinates of dA =(dA)a1···a4ea1···ea4, ωa1...a5∝(dA)[a1···a4ea5],(1.19) so that ωa1...a5plays 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 λdescribed above. This result also extends others in refs. [12,13] in which standard pure gravity with just ωab and ea, without the fields φin D=5 and Ain 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 λis 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 Appendix A. 2. The generalized sp(32)Chern–Simons action 2.1. sp(32)Cartan structure equations and gauge transformations In terms of its MC forms fαβ, α, β=1, ...32, the sp(32)algebra is defined by df αβ=−fαγ∧fγβ,df=−f2.(2.20) Using the symplectic metric Cαγ =−Cγα, fαβ is given by fαβ =Cαγ fγβ,f αβ =fβα .(2.21) Since fαβ 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 fαβ =faγa αβ +1 4fabγab αβ +fa1...a5γa1...a5 αβ .(2.22) 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 fto satisfy the sp(32) algebra MC equations. Let be the two-form matrix incorporating the curvatures. Then, =Df=df +f2,(2.23) where fcontains the one-form gauge fields, and d=f −f=[,f],(2.24)
640 D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 is the Bianchi identity D =d +[f, ] ≡0for the sp(32)connection f. As f, the curvature may be similarly expressed as αβ =aγa αβ +1 4abγab αβ +a1...a5γa1...a5 αβ .(2.25) The infinitesimal gauge transformations of f, are given by the standard expressions, δbf=db +fb−bf =db +[f,b],δ b=b −b =[,b],(2.26) where the zero-form matrix b=bαβcontains the gauge functions b=baγa+1 4babγab +ba1...a5γa1...a5.(2.27) 2.2. 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 three-form Ainert under sp(32)δ b gauge transformations and under δones. Thus, the most general twelve-form H(, A), closed and invariant under both δband δgauge transformations, may be written as H=Tr( 6)+αT r(4)T r(2)+βTr( 2)3+νTr(4)dA +δTr( 2)2 dA+ρTr(2)(dA)2+σ(dA)3,(2.28) where the bosonic has replaced Fin eq. (1.17), in which fermions were present. Then, the integral I= M11 B, dB =H, (2.29) 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 Tr( 6)and those in νand σwill 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. 2.3. Generalized CS action for the bosonic sector of D=11 supergravity Again, the component fields in the one-form f, the two-form and the three-form Afield are dimensionless. Dimensions are introduced by setting A=λ3A, [A]=L3,(2.30) f=λe aγa+1 4ωabγab +ωa1...a5γa1...a5,(2.31) where in (2.22) we set
D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 641 fa=λea,[ea]=L, fab =ωab ,[ωab]=L0,(2.32) fa1...a5=ωa1...a5,ωa1...a5=L0. With our mostly plus metric we use real gamma matrices such that γa1...a11 =a1...a11 . 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 ea, ωa1···a5and A. Different coefficients would lead to different values for the constants α, ..., σin (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 LD−2=L9. Thus, writing now H|ifor Hiand expressing the twelve-form Hin (2.28) and the eleven-form Bin powers of λ, we obtain H=H|0+λH|1+... , B=B|0+λB|1+... . (2.33) Then, H|i=dB|iallows us to write for the different IGCS|i=M11 B|i, IGCS =IGCS |0+λI GCS |1+... . (2.34) The physically relevant term is in λ9since [IGCS |9]=L9. Therefore, IGCS|9=M11 B|9. We are thus interested in H|9. Since Hcontains the sp(32)curvature two-forms a, ab, a1...a5of (2.25), we need their expressions in terms of ea, ωab, ωa1...a5. To simplify the calculations, we write =df +f2=0+λ1+λ22,(2.35) with fin (2.31) expressed as f=λe +ωL+ω5=λe +ω, (2.36) where e=eaγa, ωL=1 4ωabγab is the spin connection, ω5=ωa1...a5γa1...a5and ω=ωL+ω5. In this way, the sp(32)-valued curvature in (2.35) gives =d(λe +ω) +(λe +ω)(λe +ω) =dω +ω2+λ(de +ωe +eω) +λ2e2 ≡R(ω) +λT +λ22.(2.37) Thus, 0=R(ω) =dω+1 2[ω, ω], 1=T(e, ω) =de+[ω, e]and 2(e) =e2=1 2[e, e]. Notice that Tcontains a piece proportional to γaand another proportional to γa1...a5; similarly, the curvature R(ω) contains contributions proportional to γa, γab and γa1...a5, because it depends on both ωLand ω5. The previous equations tell us that to obtain the piece H|9that comes e.g. from Tr( 6), one has to consider all the contributions containing a number n0of Rfactors, n1 of Tand n2of 2in such a way that 1. n0+n1+n2=6 (there are 6 curvatures) 2. n1+2n2=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:
648 D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 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 λ9term in the λexpansion. 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=3supergravity 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 Aare deformations of an algebra which is the expansion osp(1|32)(2, 3) of osp(1|32)in the sense of [16,17]. Acknowledgements This work has been partially supported by the grants MTM2014-57129-C2-1-P from the MINECO (Spain), VA057U16 from the 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 A This Appendix provides details of some main text calculations. A.1. Solving for ω5in the ω5equation Let us solve (3.59) for ω5. Contracting the equation with a1...a6d1...d5we find, 9! 2a1...a5d1...d5a1...a5b1...b4cdωb4...b1a6 c+4ν a1.....a6b1...b4da1...a6d1...d5(dA)b1...b4=0. (A.79) Taking into account that a1...akb1...b11−ka1...akc1...c11−k=−k!δb1...b11−k c1...c11−k(A.80) for our signature choice, (− +...+), where δb1...bk a1...ak=σ∈skδb1 aσ(1)...δbk aσ(k), we obtain
D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 649 9! 25!δa6d1...d5 b1...b4cdωb4...b1a6 c+4ν6!δd1...d5 b1...b4d(dA)b1...b4=0.(A.81) Now, using δaa1...ak−1 b1...bk= k l=1 δa blδa1...ak b1... ˆ bl...bk (A.82) in the first term with a=a6, it follows that 9! 25!ωb4...b1cc+4ν6!(dA)b1...b4δd1...d5 b1...b4d−9! 25!δd1...d5 b1...b4cωb4...b1dc=0.(A.83) Now, contracting d5and din (A.83) we get ωb4...b1cc=−56 9!ν(dA) b1...b4(A.84) and, inserting this in (A.83), we find ω[d1...d4dd5]=−ν4·2 9!(dA)[d1...d4δd5] d.(A.85) We now use this equation to find ωd1...d4dd5without antisymmetrization. To this end, we use the following trick: first we make eq (A.85) more explicit, with d5interchanged with d, so that the antisymmetrization involves d1, ···d4and d, ωd1d2d3d4d5d−ωd1d2d3dd5d4 −ωd1d2dd5d4d3−ωd1dd3d4d5d2 −ωd1d2d3d4d5 d=−4·2 9!ν(dA)d1d2d3d4δd5 d(A.86) −(dA)dd2d3d4ηd5d1−(dA)d1dd3d4ηd5d2 −(dA)d1d2dd4ηd5d3−(dA)d1d2d3dηd5d4. Antisymmetrizing the indices d1...d5with weight one leads to ωd1...d5d−4·ω[d1...d4dd5]=−4·2 9!ν(dA) [d1d2d3d4δd5] d,(A.87) and using (A.85) in (A.87), we finally obtain ωd1...d5d=−40 9!ν(dA) [d1...d4δd5] d.(A.88) or, equivalently, (3.61). A.2. Calculation of the terms in (3.70) Defining the zero-form matrix dA=(dA)a1...a4γa1...a4, eq. (3.61) may be rewritten as ω5=−20 9!( dAe +e dA) (A.89) Inserting this relation into (3.70), we obtain
650 D. Camarero et al. / Nuclear Physics B 923 (2017) 633–652 30Tr(R Le9)=4820 9!ν2dATr( dAe7)(A.90) −54ν220 9!2 Tr(4 dA dAe11 +3 dAe dAe10 +4 dAe2 dAe9 +4 dAe3 dAe8+4 dAe4 dAe7+4 dAe5 dAe6). Let us now compute the terms in this equation. First, the trace on the l.h.s. is given by Tr(R Le9)=1 4Tr(R b1b2 La1a2γb1b2ea1ea2ea3...ea11 γa3...a11 ) =1 4Tr(γ b1b2γa3...a11 )Rb1b2 La1a2a1...a11 E =8b1b2a3...a11 a1...a11 Rb1b2 La1a2E =−8.9!δa1a2 b1b2Rb1b2 La1a2E =−16.9!RLE, (A.91) where Eis an 11-form defined by ea1...ea11 =a1...a11 E, and we have written Rb1b2= Rb1b2a1a2ea1ea2. The first term on the r.h.s. of (A.91) contains the form dATr( dAe7)=32(dA)b1...b4eb1...eb4a1...a11 (dA)a1...a4ea5...ea11 E =32 (dA)b1...b4(dA)a1...a4a1...a11 b1...b4a5...a11 E =−7!·32 (dA)b1...b4(dA)a1...a4δb1...b4 a1...a4E =−7!·4!·32 (dA)a1...a4(dA)a1...a4E, (A.92) where, as before, we have written dA =(dA)b1...b4eb1...eb4. The calculation of the remaining terms is slightly more complicated. These terms have the form Tr( dAek dAe11−k)=Tr( dAγa1...ak dAγak+1...a11 a1...a11 E) (A.93) =(−1)k(k−1) 2 k!Tr( dAγa1...ak dAγb1...bk)b1...bkak+1...a11 a1...a11 E =−(−1)k(k−1) 2(11 −k)!Tr( dAγa1...ak dAγa1...ak)E =−32(−1)k(k−1) 24!(11 −k)!Nk(dA)a1...a4(dA)a1...a4E, where we have used the property (3.52) and the numbers Nkin the equation are defined through γa1...ak dAγa1...ak=Nk dA . (A.94) These numbers may be computed using gamma matrix algebra; alternatively, they can be found in Ref. [29]. Their values are: N0=1, N1=3, N2=2, N3=66, N4=−144, N5=1680. Then, the second trace on the r.h.s. of (A.90) is given by −32 ·168 ·9! ·4!(dA)a1...a4(dA)a1...a4E. When this is taken into account, eq. (A.90) reads RL=16 ·7!·4! (9!)2γ2(dA)a1...a4(dA)a1...a4.(A.95)
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