Eleven-dimensional CJS supergravity and the D Auria Fré group

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1 Physics Letters B ) Eleven-dimensional CJS supergravity and the D Auria Fré group A. Pérez P. Minning P. Salgado Departamento de Física Universidad de Concepción Chile Received 8 November 2007; received in revised form 9 January 2008; accepted 0 January 2008 Available online 9 January 2008 Editor: M. Cvetič Abstract An action for eleven-dimensional standard supergravity genuinely invariant under the D Auria Fré group is proposed. The construction of the action is carried out through a generalization of the Stelle West Grignani Nardelli formalism. The action is genuinely invariant under this group without the need to impose auxiliary conditions such as a torsion-free condition Published by Elsevier B.V. Open access under CC BY license.. Introduction The problem of the underlying symmetry of eleven-dimensional supergravity was considered by Cremmer Julia Scherk CJS) in the original paper []. They considered its possible association with a gauge theory and suggested that the gauge group could be related to OSp32/). This problem was addressed by D Auria and Fré [2] where the search for the supergroup of D = supergravity was formulated as a search for a composite structure of its three-form A 3. D Auria and Fré expressed this three form A 3 in terms of the graviton e a x) the gravitino ψ α x) two bosonic one-forms B abx) = dxμ Bμ ab x) Babcde x) = dx μ Bμ abcde x) and a fermionic one-form η α x) = dx μ η μα x). This composite structure suggests [2] a possible underlying symmetry of the D = supergravity. The new fields B ab B abcde η α ) may be treated as gauge fields associated with new antisymmetric generators Z ab = Z ba Z abcde = Z [abcde] and a new fermionic generator Q α which extends the Poincare superalgebra in which Q α P a and M ab correspond to the gravitino field ψ α x) the graviton field e a x) and the spin connection ω ab x). Two possible superalgebras allowing a composite nature of A 3 were found in Ref. [2]. InRefs.[34] these algebras were called D Auria Fré superalgebras. These superalgebras are related to the orthosimplectic supergroup OSp32/). Indeed both D Auria Fré superalgebras and M-algebra can be obtained via an expansion procedure [56] from OSp32/) algebra which was considered by Cremmer Julia and Scherk as a possible gauge group for CJS supergravity. It is the purpose of this Letter to show that the supersymmetric generalization of the Stelle West Grignani Nardelli SWGN) formalism [78] permits constructing an eleven-dimensional standard supergravity off-shell invariant under the D Auria Fré group. This Letter is organized as follows: In Section 2 we shall review some aspects of CJS supergravity. In Section 3 we shall review the principal features of the general non-linear realizations formalism which is used to obtain a non-linear realization of the D Auria Fré superalgebra. An action for eleven-dimensional CJS supergravity genuinely invariant under the D Auria Fré superalgebra is constructed in Section 4. Section 5 concludes the work with a look forward to find some relations between CJS supergravity and Chern Simons/transgression supergravity. We will follow the notation and conventions of Ref. [9]. * Corresponding author. address: pasalgad@udec.cl P. Salgado) Published by Elsevier B.V. doi:0.06/j.physletb Open access under CC BY license.

2 408 A. Pérez et al. / Physics Letters B ) The action for the standard eleven-dimensional supergravity In this section we review some aspects of CJS supergravity. 2.. CJS supergravity The fields of the first order CJS supergravity are the -form vielbein e a theso0 ) -form spin-connection ω ab the Rarita Schwinger -form ψ and the three-form A = /3!)A μνρ dx μ dx ν dx ρ and its associated field strength F = /4!) F μνρσ dx μ dx ν dx ρ dx σ. The supergravity action in first order form was originally obtained in Ref. [2]. It also is given by Eq. 78) of Ref. [9] which is invariant under the supersymmetry transformations δ ψ = ˆD ε := D ε + 44 εγ abcde 8Γ bcd η ea )e a F bcde δe a = i εγ a ψ ) 2) δa = i 2 εγ 2)ψ 3) as long as we requires the vanishing of the supertorsion ˆT a := 0 and the fulfillment of the equation F := da i 4 ψγ ab ψe a e b. The introduction of these conditions in Eq. 78) of Ref. [9] leads to the action found by Cremmer Julia and Scherk where the spin connection ω ab and the 4-form F coincide with the spin connection ˆω ab and with the field ˆF used in Ref. []. In the context of the.5 formalism the action is on-shell invariant under the supersymmetry transformation given by ) 2) and 3). This means that the connection and the 4-form F are no longer independent variables. Rather their variations are given in terms of δe a δψ and δa and differ from those dictated by group theory CJS supergravity with a composite three-form gauge field The general form of the D Auria Fré superalgebras is given by Eqs..2).3).4) of Ref. [3]. The so-called minimal D Auria Fré algebra is obtained when s = 6. In this case the anti)commutation relations takes the form [3] [J ab J cd ]= iη bc J ad + η ad J bc η bd J ac η ac J bd ) [J ab P c ]= iη bc P a η ac P b ) 4) 5) [Q α J ab ]= i 2 Γ ab) β α Q β [ ] Q i α J ab = 2 Γ ab) β α Q β [J ab Z cd ]= iη bc Z ad + η ad Z bc η bd Z ac η ac Z bd ) [ Jab Z a ]...a 5 = 2!5!iδ [a [a Z a 2...a 5 ] b] {Q α Q β }= Γ a) αβ P a + i Γ a ) a 2 αβ Z a a 2 + Γ a...a 5 )αβ Z a...a 5 [P a Q α ]= 0γ Γ a ) β α Q β [Z a a 2 Q α ]=iγ Γ a a 2 ) β α Q β where Z a a 2 and Z a...a 5 are the central generators of the M-algebra which transforms as a tensor under Lorentz rotations and Q is a new central generator which transforms as a spinor under Lorentz rotations. Q is not a new generator of supersymmetry because of its central character. Using this algebra the gauge connection 6) 7) 8) 9) 0) ) 2) h = ie a P a + i 2 ωab J ab + ψq+ ηq + 2 Ba a 2 Z a a 2 + 5! Ba...a 5 Z a...a 5 3) the gauge parameter λ = iρ a P a i 2 κab J ab εq τq 2 ρab Z ab 5! ρa...a 5 Z a...a 5 4) and δh = dλ [h λ] we obtain that e a ω ab ψ B ab and η transform under the D Auria Fré group as: δe a = Dρ a + i εγ a ψ δω ab = 0 δψ = Dε 5) 6) 7)

3 A. Pérez et al. / Physics Letters B ) δη = Dτ + 0iγ e a Γ a ε iγ 2 Ba a 2 Γ a a 2 ε 0iγ ρ a Γ a ψ + iγ 2 ρab Γ ab ψ δb ab = Dρ ab 2i εγ ab ψ δb a...a 5 = Dρ a...a 5 5! εγ a...a 5 ψ. In Ref. [2] it was shown that it is possible to find a Lie group that reproduces the Free Differential Algebra described by Eqs. 3.5) of Ref. [2] provided that the three-form A 3 has a composite nature given by [3] A = 48 B abe a e b 3. Non-linear realization formalism B abb b cb ca + 2 8γ e a ηγ a ψ 240 8γ B ab ηγ ab ψ. A group G can be realized linearly or non-linearly. A linear realization is a rule which assigns to each element g G a linear operator which acts on a vector space representation space) such that the group product is preserved by the linear operator which defines the representation. It is also possible to define non-linear realizations of a group which correspond to maps of a Manifold M in itself characterized by an element g 0 of a Lie group G 8) 9) 20) 2) x = fg 0 ; x) which satisfy the following properties 22) x = fi G ; x) f g 2 ; [ fg x) ]) = fg 2 g ; x) i.e. the identity and the group product are preserved. In general the transformations x x are not linear. However in the case that the manifold M is a vector space and the function f is linear we obtain a linear realization of the Lie group. The problem of the characterization of all possible non-linear realizations of a Lie group was solved in Refs. [0]. 3.. Non-linear realization of Lie groups Following Refs. [0 2] we consider a Lie super)group G with n parameters and a subgroup H with n d parameters which is called stability group. Let us call {V i } n d i= the generators of H and {A l} d l= the remaining generators. Thus the generators of G are {V i A l }. Since H is a subgroup of G we have that the generators V i define a Lie subalgebra i.e. [VV] V. We assume that the remaining generators {A l } d l= can be chosen so that they form a representation of H. In other words the commutator [V i A l ] should be a linear combination of A l alone i.e. [VA] A. A group element g G can be represented uniquely) in the form g = e ξ l A l h where h is an element of H.Theξ l parametrize the coset space G/H. We do not specify here the parametrization of h. One can define the effect of a group element g 0 on the coset space by 23) 24) 25) or g 0 g = g 0 e ξ l A l h ) = e ξ l A l h 26) g 0 e ξ l A l = e ξ l A l h where ξ = ξ g 0 ξ) h = h h h = h g 0 ξ). If g 0 is infinitesimal 27) implies e ξ l A l g 0 )e ξ l A l e ξ l A l δe ξ l A l = h. The right-hand side of 3) is a generator of H. 27) 28) 29) 30) 3)

4 40 A. Pérez et al. / Physics Letters B ) Let us first consider the case in which g 0 = h 0 H. Then 27) gives e ξ l A l = h 0 e ξ l A l h 0. Since the A l form a representation of H this implies h = h 0 ; h = h 0 h. The transformation from ξ to ξ given by 32) is linear. On the other hand consider now g 0 = e ξ l 0 A l. In this case 27) becomes e ξ l 0 A l e ξ l A l = e ξ l A l h. This is a non-linear inhomogeneous transformation on ξ. The infinitesimal form 3) becomes 32) 33) 34) 35) e ξ l A l ξ i 0 A ie ξ j A j e ξ l A l δe ξ i A i = h. The left-hand side of this equation can be evaluated using the algebra of the group. Since the results must be a generator of H one must set equal to zero the coefficient of A l. In this way one finds an equation from which δξ i can be calculated Non-linear gauge fields The construction of a Lagrangian invariant under coordinate-dependent group transformations requires the introduction of a set of gauge fields a = a i μ A i dx μ ρ = ρ i μ V i dx μ p = p l μ A l dx μ v = v i μ V i dx μ associated respectively with the generators V i and A l. Hence ρ + a is the usual linear connection for the gauge group G and the corresponding covariant derivative is given by: 36) D = d + fρ+ a) and its transformation law under g G is g : ρ + a) ρ + a ) = [gρ + a)g f ] dg)g 37) 38) where f is a constant which as it turns out gives the strength of the universal coupling of the gauge fields to all other fields. We now consider the Lie algebra valued differential form [0]: e ξ l A l [ d + fρ+ a) ] e ξ l A l = p + v. The transformation laws for the forms pξdξ) and vξdξ) are easily obtained. In fact using 34) 35) one finds [3] 39) p = h ph ) v = h vh ) + h dh ). Eq. 40) shows that the differential forms pξdξ) are transformed linearly by a group element of the form 34). The transformation law is the same as by an element of H except that now this group element h ξ 0 ξ) is a function of the variable ξ. Therefore any expression constructed with pξdξ) which is invariant under the subgroup H will be automatically invariant under the entire group G Non-linear realizations of the D Auria Fré algebras Following Refs. [783 7] we consider as group G a group generated by the D Auria Fré algebra and as subgroup L to the Lorentz group. If we denote as ɛs) the Lie algebra generated by the generators {Q Z a...a 5 Z a a 2 QP a } then the D Auria Fré can be expressed as a semidirect addition of ɛs) and the Lorentz algebra so 0). Thus we have the necessary conditions to non-linearly realize a non-compact group. Following the usual procedure we associate a parameter with each generator of the coset space G/L: 40) 4) χ α Q; θ α Q ; ξ a P a ; λ ab Z ab ; λ a...a 5 Z a...a 5. It is interesting to note that the D Auria Fré algebras have the following set of subalgebras 42) L ={J ab } H ={P a J ab }

5 A. Pérez et al. / Physics Letters B ) B ={Z a a 2 P a J ab } D ={Z a...a 5 Z a a 2 P a J ab } C ={Q Z a...a 5 Z a a 2 P a J ab } G ={Q Q Z a...a 5 Z a a 2 P a J ab } which are contained one in another. This permits us to write Eq. 27) in the form e θ β Q β e 5! λa...a 5 Z a...a 5 e 2 λab Z ab e iξa P a = e χ α Q α e θ β Q β e 5! λ a...a 5 Z a...a 5 e 2 λ ab Z ab e iξ a P a l. Multiplying 44) by e iξa P a we have e θ β Q β e 5! λa...a 5 Z a...a 5 e 2 λab Z ab = e χ α Q α e θ β Q β e 5! λ a...a 5 Z a...a 5 e 2 λ ab Z ab h h e iξa P a l h = h e iξa P a with h l. Multiplying 45) by e 2 λab Z ab we have e θ β Q β e 5! λa...a 5 Z a...a 5 = e χ α Q α e θ β Q β e 5! λ a...a 5 Z a...a 5 b b e 2 λab Z ab = e 2 λ ab Z ab h h e iξa P a l b = b e 2 λab Z ab with b = e 2 λ ab Z ab h. Multiplying 48) by e 5! λa...a 5 Z a...a 5 we have e θ β Q β = e χ α Q α e θ β Q β d d e 5! λa...a 5 Z a...a 5 = e 5! λ a...a 5 Z a...a 5 b b e 2 λab Z ab = e 2 λ ab Z ab h h e iξa P a l d = d e 5! λa...a 5 Z a...a 5 with d = e 5! λ a...a 5 Z a...a 5 b. Multiplying 52) by e θ β Q β we have 43) 44) 45) 46) 47) 48) 49) 50) 5) 52) 53) 54) 55) 56) = e χ α Q α c c e θ β Q β = e θ β Q β d d e 5! λa...a 5 Z a...a 5 = e 5! λ a...a 5 Z a...a 5 b b e 2 λab Z ab = e 2 λ ab Z ab h h e iξa P a l c = c e θ β Q β with c = e θ β Q β d where the multiplication is done by the right side and where c C b B h H l L. In a word we have = e χ α Q α c c e θ β Q β = e θ β Q β d d e 5! λa...a 5 Z a...a 5 = e 5! λ a...a 5 Z a...a 5 b b e 2 λab Z ab = e 2 λ ab Z ab h h e iξa P a l. This decomposition is possible due to the fact that we obtain a succession of subalgebras L H B D C G in proportion as we insert new generators. The order in which we make such decompositions is rigid because if for example we introduce Q before another generator then the sets L H B D C G should lose their characteristic of subalgebras: it is the characteristic of subalgebras which allows the mentioned decomposition. 57) 58) 59) 60) 6) 62) 63) 64) 65) 66) 67)

6 42 A. Pérez et al. / Physics Letters B ) CJS gauge invariant supergravity 4.. Transformation law of the three-form A From 2) and 5) 9) we find that the transformation law of the three-form A under supersymmetry in its minimal form) is given by δ susy A = i [ ] 4 εγ abψe a e b + d e a ηγ a ε B ab ηγ ab ε ˆT a ηγ a ε) 2 8γ 240 8γ 2 8γ where γ R ab ηγ ab ε) + 2 8γ e a σγ a ε 240 8γ B ab σγ ab ε 68) R ab = DB ab + i ψγ ab ψ σ = Dη + 0iγ e a Γ a ψ i 2 γ B ab Γ ab ψ. 69) 70) From 68) and from δ susy e a = i εγ a ψ δ susy ω ab = 0 δ susy ψ = Dε it is direct to see that the action for CJS supergravity is not invariant under supersymmetry. Clearly this is due to the presence of curvature terms en δa and to the absence of the supercovariant derivative in δψ. This result is valid in general for any element of the family of the D Auria Fré algebras Transformation law of the parameters The infinitesimal form of Eqs. 63) 67) is given by e χq g 0 )e χq e χq δe χq = c e θq c )e θq e θq δe θq = d e 5! λa...a 5 Z a...a 5 d )e 5! λa...a 5 Z a...a 5 e 5! λa...a 5 Z a...a 5 δe 5! λa...a 5 Z a...a 5 = b e 2 λab Z ab b )e 2 λcd Z cd e 2 λab Z ab δe 2 λcd Z cd = h e iξa P a h )e iξb P b e iξa P a δe iξb P b = l where 7) 72) 73) 74) 75) c = c χετρ ab ρ a...a 5 ρ a κ ab) d = d χθετρ ab ρ a...a 5 ρ a κ ab) b = b χθλ a...a 5 ετρ ab ρ a...a 5 ρ a κ ab) h = h χθλ a...a 5 λ ab ετρ ab ρ a...a 5 ρ a κ ab) l = l χθλ a...a 5 λ ab ρ a ετρ ab ρ a...a 5 ρ a κ ab). 76) From 7) 75) it is possible to obtain the transformation law of the parameters under the action of an element g 0 of the group. For an element g 0 = εq τq 5! ρa...a 5 Z a...a 5 2 ρab Z ab iρ a P a 77) such transformation laws are given by δχ = ε δθ = τ γ εγ a χ ) Γ a χ + 3 γ εγ ab χ ) Γ ab χ + i 2 γ ρ ab Γ ab χ 0iγ ρ a Γ a χ δλ a...a 5 = ρ a...a 5 5! a εγ...a 5 χ ) 2 δλ ab = ρ ab i εγ ab χ δξ a = ρ a + i 2 εγ a χ. 78) 79) 80) 8) 82)

7 A. Pérez et al. / Physics Letters B ) CJS supergravity action invariant under the D Auria Fré group From 39) we can see that if h = ie a P a + i 2 ωab J ab + ψq+ ηq + 2 Bab Z ab + 5! Ba...a 5 Z a...a 5 is the linear gauge connection then the non-linear gauge connection is given by iv a P a + i 2 W ab J ab + ΨQ+ ˆηQ + 2 ˆB ab Z ab + 5! ˆB a...a 5 Z a...a 5 = e iξa P a e 2 λab Z ab e 5! λa...a 5 Z a...a 5 e θq e χq [ d + ie a P a + i 2 ωab J ab + ψq+ ηq + 2 Bab Z ab + 5! Ba...a 5 Z a...a 5 ]e χq e θq e 5! λa...a 5 Z a...a 5 e 2 λab Z ab e iξa P a. 83) Using 83) we find that the non-linear gauge fields are given by V a = e a Dξ a i χγ a ψ ) i 2 D χγ a χ ) 84) W ab = ω ab Ψ = ψ Dχ ˆB ab = B ab Dλ ab + 2i χγ ab ψ ) + i D χγ ab χ ) ˆB a...a 5 = B a...a 5 Dλ a...a 5 + 5! χγ a...a 5 ψ ) + 5! D χγ a...a 5 χ ) 2 ˆη = η Dθ 0iγ ξ a Γ a Dχ + iγ 2 λab Γ ab Dχ 5 3 γ D χγ a χ ) Γ a χ γ D χγ ab χ ) Γ ab χ 0iγ e a Γ a χ 6 + 0iγ ξ a Γ a ψ iγ 2 λab Γ ab ψ 5γ χγ a ψ ) Γ a χ γ χγ ab ψ ) Γ ab χ + iγ 2 2 Bab Γ ab χ. The non-linear three-form  is then given by  = 48 ˆB ab V a V b ˆB ab ˆB b c ˆB ca 2 8γ V a ˆηΓ a Ψ γ ˆB ab ˆηΓ ab Ψ. Within the supersymmetric extension of the non-linear realization formalism the action for CJS supergravity can be rewritten as S = 4 ˆR ab ˆΣ ab + i 2 Ψ Γ ˆ 8) DΨ + i T a i ) 8 4 ΨΓ a Ψ V a Ψ Γ ˆ 6) Ψ 85) 86) 87) 88) 89) 90) where 2 ˆF ˆF + ˆF + ˆb)d â) + 2â ˆb 3ÂdÂd ˆΣ a...a r := D r)! ɛ a...a r a r+...a D V a r+...v a D ˆ Γ n) := n! Γ a...a n V a...v a n â := i 4 Ψ ˆ Γ 2) Ψ ˆb := i 4 Ψ ˆ Γ 5) Ψ 9) 92) 93) 94) 95) with ˆT a = dv a + W a bv b i 2 ΨΓ a Ψ ˆR ab = dw ab + W a cw cb DΨ = dψ + 4 W ab Γ ab Ψ ˆF = dâ i 8 ΨΓ ab ΨV a V b. 96) 97) 98) 99) This action is off-shell gauge invariant under the D Auria Fré group.

8 44 A. Pérez et al. / Physics Letters B ) Comments and possible developments We have shown in this work that the successful formalism of Stelle West [7] and Grignani-Nardelli [8] used to construct an action for 3 + )-dimensional gravity genuinely invariant under the Poincaré group can be generalized to the case of eleven-dimensional standard supergravity. The main result of this Letter is that we have shown that in order to construct an eleven-dimensional standard supergravity invariant under the minimal D Auria Fré superalgebra it is necessary to use the SWGN formalism. This means using the fields vielbein V a spin connection W ab gravitino Ψ ˆB ab ˆB abcde which involve in their definitions the fields ξ a χ θ α λ ab λ abcde. This fields can be interpreted as some kind of auxiliary fields which permits the superalgebra to close off shell without the need to impose conditions such as ˆT a = 0. From the action 9) and Eqs. 84) 89) we can see that once the gauge ξ a = χ = θ α = λ ab = λ abcde = 0 is chosen we obtain that the Lagrangian of the action 9) takes the form of Eq. 78) of Ref. [9]. Several aspects deserve consideration and many possible developments can be worked out. A still unsolved problem is to find a relation between Chern Simons/transgression supergravity and eleven-dimensional CJS supergravity work in progress). Acknowledgements The authors wish to thank F. Izaurieta E. Rodriguez and J. Crisostomo for enlightening discussions. P.S. wishes to thank J.A. de Azcarraga and M.A. Lledó for their warm hospitality at the Universidad de Valencia where this work was started. One of the authors A.P.) wishes to thank Consejo Nacional de Ciencia y Tecnología CONICYT) for financial support. He is also grateful to Departamento de Física Universidad de Concepción for the atmosphere. P.S. was supported by FONDECYT Grant # and by Universidad de Concepción through Semilla Grants S and S. References [] E. Cremmer B. Julia J. Scherk Phys. Lett. B ) 409. [2] R. D Auria P. Fré Nucl. Phys. B ) 0. [3] I.A. Bandos J.A. de Azcarraga M. Picón O. Varela Ann. Phys ) 238. [4] I.A. Bandos J.A. de Azcarraga M. Picón O. Varela Phys. Lett. B ) 45. [5] J.A. de Azcarraga J.M. Izquierdo M. Picón O. Varela Nucl. Phys. B ) 85. [6] F. Izaurieta E. Rodriguez P. Salgado J. Math. Phys ) [7] K.S. Stelle P.C. West Phys. Rev. D 2 980) 466. [8] G. Grignani G. Nardelli Phys. Rev. D ) 279. [9] B. Julia S. Silva JHEP ) 026. [0] S. Coleman J. Wess B. Zumino Phys. Rev ) [] C.G. Callan S. Coleman J. Wess B. Zumino Phys. Rev ) [2] D.V. Volkov Sov. J. Part. Nucl ) 3. [3] B. Zumino Nucl. Phys. B ) 89. [4] P. Salgado M. Cataldo S. del Campo Phys. Rev. D ) ; P. Salgado M. Cataldo S. del Campo Phys. Rev. D ) 02403; P. Salgado M. Cataldo S. del Campo Phys. Rev. D ) [5] P. Salgado F. Izaurieta E. Rodriguez Phys. Lett. B ) 283; P. Salgado F. Izaurieta E. Rodriguez Eur. Phys. J. C ) 429. [6] F. Izaurieta E. Rodriguez P. Salgado Phys. Lett. B ) 397. [7] P. Salgado G. Rubilar J. Crisóstomo Eur. Phys. J. C ) 587.