Spinor Fields Classification in Arbitrary Dimensions and New Classes of Spinor Fields on 7-Manifolds
PPrepared for submission to JHEP
Spinor Fields Classification in Arbitrary Dimensionsand New Classes of Spinor Fields on 7-Manifolds
L. Bonora a K. P. S. de Brito b Roldão da Rocha a,c a International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy b CCNH, Universidade Federal do ABC 09210-580, Santo André, SP, Brazil c CMCC, Universidade Federal do ABC 09210-580, Santo André, SP, Brazil
E-mail: [email protected] , [email protected] , [email protected] Abstract:
A classification of spinor fields according to the associated bilinear covariantsis constructed in arbitrary dimensions and metric signatures, generalizing Lounesto’s 4Dspinor field classification. In such a generalized classification a basic role is played by thegeometric Fierz identities. In 4D Minkowski spacetime the standard bilinear covariants canbe either null or non-null – with the exception of the current density which is invariablydifferent from zero for physical reasons – and sweep all types of spinor fields, includingDirac, Weyl, Majorana and more generally flagpoles, flag-dipoles and dipole spinor fields.To obtain an analogous classification in higher dimensions we use the Fierz identities, whichconstrain the covariant bilinears in the spinor fields and force some of them to vanish. Ageneralized graded Fierz aggregate is moreover obtained in such a context simply fromthe completeness relation. We analyze the particular and important case of Riemannian7-manifolds, where the Majorana spinor fields turn out to have a quite special place. Inparticular, at variance with spinor fields in 4D Minkowski spacetime that are classified insix disjoint classes, spinors in Riemannian 7-manifolds are shown to be classified, accordingto the bilinear covariants: (a) in just one class, in the real case of Majorana spinors; (b)in four classes, in the most general case. Much like new classes of spinor fields in 4DMinkowski spacetime have been evincing new possibilities in physics, we think these newclasses of spinor fields in seven dimensions are, in particular, potential candidates for newsolutions in the compactification of supergravity on a seven-dimensional manifold and itsexotic versions. a r X i v : . [ h e p - t h ] N ov ontents Generalizing Fierz identities in 4D Minkowski spacetime has led to new interestingresults on spinor fields, with respect to the textbook ones, and unexpected applicationslikewise. It is therefore natural to try to do the same in higher dimensions. Fierz iden-tities for form-valued spinor bilinears were considered in arbitrary dimensions and metricsignatures [1, 2], using the geometric algebra, being also based in the developments [3, 4].The most general Fierz identities were further used to construct independent effective four-fermion interactions that contain spin-3/2 chiral fields [5]. Besides, the existence of twonatural bilinear forms on the space of spinors were shown in [6] to be related with elementsof the exterior algebra. All the Fierz identities can be reduced to a single equation usingthe extended Cartan map [7]. Reciprocally, spinors are reconstructed from Fierz identities[8] and quantum tomography for Dirac spinors were considered in this context as well [9].General Fierz identities were further used to find completeness and orthogonality relations[10, 11], where an equivalence between spinor and tensor representation of various quanti-ties have been also constructed. Moreover, Fierz identities are useful to calculate scatteringamplitudes [12], being thus employed to calculate electroweak interactions. The inverseproblem can be solved likewise by using the Fierz identities [13]. Various Fierz identitiesare further needed to show the invariance of the D = 11 , N = 1 supergravity action under alocal supersymmetry transformation, and the supercovariance of the fermion field equation[14]. In supersymmetric gauge theories, supersymmetry constraints imply the existence ofcertain Fierz identities for real Clifford algebras [15]. These identities hold merely for 2, 4,8 and 16 supercharges [16]. Fierz identities were also generalized for non-integer dimensionsin [17].The prominent relevance of Fierz identities can be moreover measured by their roleon the recent emergence of new kinds of spinor fields. The subject concerning those new– 1 –pinor fields and their applications has been widening, mainly since the middle of thelast decade. Fierz identities were used by Lounesto to classify spinor fields in Minkowskispacetime according to the bilinear covariants [18]. Indeed, Lounesto showed that spinorfields can be accommodated in six disjoint classes, that encompass all the spinor fields inMinkowski spacetime. The first three types of spinor fields in such classification are namedDirac spinor fields: this is actually a generalization, as it does not restrict to the standardDirac spinor field, which is an eigenspinor of the parity operator, providing hence furtherphysical solutions for the Dirac equation. Indeed such three classes of regular spinor fieldsappear as solutions of the Dirac equation in different contexts. They are characterizedby either the scalar or the pseudoscalar (or even both) bilinear covariants being nonzero.The other three classes of (singular) spinors are known as flag-dipole, flagpole and dipolespinor fields, and have both the scalar and pseudoscalar bilinear covariants vanishing. Thelatter classes contain, besides a rich geometric structure, spinor fields with new dynamics.Flagpole spinor fields have been recently considered in cosmology [19], being explored ascandidates for dark matter in various contexts [20–23], wherein Elko and Majorana spinorfields evince prominent roles [24]. Flag-dipole ones are typified for instance by recentlyfound new solutions of the Dirac equation in ESK gravities [25]. Dipole spinor fields includeWeyl spinor fields as their most known representative. All the spinor classes have beenlately thoroughly characterized [26]. A complete overview of this classification with furtherapplications in field theory and gravitation can be found in [27], being also further exploredin the context of black hole thermodynamics [28]. Indeed, black hole tunnelling methodswere studied for Elko spinor fields as special type of flagpoles [28], which play an essentialrole in constructing various theories of gravity naturally arising from supergravity [29, 30].Flagpoles spinor fields and Lounesto spinor field classification are moreover discussed inthe context of the instanton Hopf fibration [31], and experimental signatures of the type-5spinors in such a classification are related to the Higgs boson at LHC [32]. An up-to-dateoverview on a special class of such spinor fields can be found in [33] and references therein.Fierz identities make it possible to deduce a classification of spinors on the spinor bun-dle associated to manifolds of arbitrary dimensions, departing from the case proposed byLounesto, that holds solely for Minkowski spacetime [18]. The aim of the present paper istwo-fold: besides generalizing Lounesto’s spinor field classification for spacetimes of arbi-trary dimension and metric signatures and proposing a graded Fierz aggregate, we focusin particular on the noteworthy case of spinor fields on seven-dimensional manifolds, inparticular those on the 7-sphere. This is motivated by their wide physical applications,for instance, in D = 11 supergravity [34–36]. In fact, spontaneous compactifications of D = 11 supergravity [34] on Riemannian 7-manifolds are well-known to contain all the de-grees of freedom of the massless sector of gauged N = 8 supergravity theory [36]. Explicitforms of the Killing spinors can be also obtained in the supergravity theory that admits anAdS × S solutions. S spinors and the Ka ˇc -Moody algebra of S were also considered onthe parallelizable 7-sphere [37, 38].We use the Fierz identities constraining bilinear covariants constructed by spinor fieldsin arbitrary dimensions [1], that force some of the respective bilinear covariants to bezero and study the important case of seven dimensions, in particular classifying Majorana– 2 –pinor fields. New kinds of classes of spinor fields are hence obtained, which are hidden ifone considers only the real spin bundle on Riemannian 7-manifolds.This paper is organized as follows: in Section II, the bilinear covariants are used torevisit the classification of spinor fields in Minkowski spacetime, according to the Lounesto’sclassification prescription, and the Fierz aggregate and its related boomerang are defined.In Section III, the bilinear covariants associated to spinor fields in arbitrary dimensionsand metric signatures are introduced and all the possibilities for them are listed. In SectionIV, the geometric Fierz identities are employed and from the admissible pairings betweenspinor fields the number of classes in the spinor field classification are constrained. We studythe case of Majorana spinor fields on Riemannian 7-manifolds and define the graded Fierzaggregate as a particular case of the completeness condition. We conclude that Majoranaspinors in seven dimensions pertain to solely one class, according the bilinear covariants,as some of these bilinears are identically zero. Spinor fields on these 7-manifolds can beclassified in four classes, departing from the classification for the real spin bundle. Oneof the new classes encompasses the Majorana spinor fields and the others provide newcandidates for physical solutions, for instance, in supergravity. In order to fix the notation, consider an oriented manifold ( M, g ) , where the metric g hassignature ( p, q ) , and its associated tangent [cotangent] bundle T M [ T ∗ M ], having sectionsconsisting of n -dimensional ( n = p + q ) real vector spaces. Denoting sections of the exteriorbundle by sec (cid:86) ( T M ) , given a k -vector a ∈ sec (cid:86) k ( T M ) , the grade involution is defined by ˆ a = ( − k a and the reversion by ˜ a = ( − [[ k/ a , where [[ k ]] stands for the integral part of k . The conjugation is the composition of the two previous morphisms. Moreover, when g isextended from sec (cid:86) ( T M ) = sec T ∗ M to sec (cid:86) ( T M ) , and by considering a, b, c ∈ sec (cid:86) ( V ) ,the left [right] contraction can be defined by g ( a (cid:121) b, c ) = g ( b, ˜ a ∧ c ) [ g ( a (cid:120) b, c ) = g ( b, a ∧ ˜ c ) ]. The Clifford product between a vector field v ∈ sec (cid:86) ( T M ) and a multivector a ∈ sec (cid:86) ( T M ) is prescribed by v ◦ a = v ∧ a + v (cid:121) a . The dual Hodge operator (cid:63) : sec (cid:86) ( T M ) → sec (cid:86) ( T M ) is defined by a ∧ (cid:63)b = g ( a, b ) . The Grassmann algebra ( (cid:86) ( T M ) , g ) endowed withthe Clifford product is denoted by C(cid:96) p,q , the Clifford algebra associated with sec (cid:86) ( T M ) (cid:39) R p,q .When the Minkowski spacetime is considered, the set { e µ } represents sections of theframe bundle P SO e , ( M ) and { θ µ } is the dual basis { e µ } , namely, θ µ ( e µ ) = δ µν . Clas-sical spinor fields are objects in the carrier space associated to a ρ = D (1 / , ⊕ D (0 , / representation of the Lorentz group, and can be thought as being sections of the vec-tor bundle P Spin e , ( M ) × ρ C . Moreover, the classical spinor fields carrying either the D (0 , / or the D (1 / , representation of the Lorentz group are sections of the vector bundle P Spin e , ( M ) × ρ (cid:48) C , where ρ (cid:48) stands for either the D (1 / , or the D (0 , / representation ofthe Lorentz group. Given a spinor field ψ ∈ sec P Spin e , ( M ) × ρ C , the bilinear covariants– 3 –re the following sections of the exterior algebra bundle (cid:86) ( T M ) [18, 39, 40]: σ = ¯ ψψ , (2.1a) J = J µ θ µ = ¯ ψγ µ ψ θ µ , (2.1b) S = S µν θ µ ∧ θ ν = i ¯ ψγ µν ψ θ µ ∧ θ ν , (2.1c) K = K µ θ µ = i ¯ ψγ γ µ ψ θ µ , (2.1d) ω = − ¯ ψγ ψ , (2.1e)where ¯ ψ = ψ † γ , γ := γ γ γ γ and the set { , γ µ , γ µ γ ν , γ µ γ ν γ ρ , γ } ( µ < ν < ρ ) is a basisfor M (4 , C ) satisfying γ µ γ ν + γ ν γ µ = 2 η µν and the Clifford product is denoted here byjuxtaposition [40].The space-like 1-form K designates the spin direction, the 2-form S denotes the well-known intrinsic angular momentum, and the time-like 1-form J stands for the current ofprobability. The bilinear covariants satisfy the Fierz identities [18, 39] − ( ω + σγ ) S = J ∧ K , K + J = 0 = J (cid:120) K , J = ω + σ . (2.2)When ω = 0 = σ , a spinor field is said to be singular, and regular otherwise.Lounesto [18] classified spinor fields into six disjoint classes. In the classes (1), (2), and(3) beneath it is implicit that J , K and S are simultaneously different from zero, and inthe classes (4), (5), and (6) just J (cid:54) = 0 :1) ω (cid:54) = 0 , σ (cid:54) = 0 ω = σ = 0 , K (cid:54) = 0 , S (cid:54) = 0 ω = 0 , σ (cid:54) = 0 ω = σ = 0 , K = 0 , S (cid:54) = 0 ω (cid:54) = 0 , σ = 0 ω = σ = 0 , S = 0 , K (cid:54) = 0 Spinor fields of types-1, -2, and -3 are called Dirac spinor fields whilst spinor fields oftypes-4, -5, and -6 are flag-dipoles, flagpoles and dipole spinor fields, respectively [18]. Itis worthwhile to emphasize that the naming “Dirac spinors” in Lounesto’s classification iswider than the one adopted in textbooks, where Dirac spinors are eigenstates of the parityoperator. The first physical example of flag-dipole spinor fields has been found very recentlyin [25] as solutions of the Dirac equation in a f ( R ) background with torsion. Moreover,Majorana and Elko spinor fields reside in the class of spinors of type-5 [24], and Weyl spinorfields are a particular case of a type-6 dipole spinor fields [18], wherein further spinor fieldshave been scarcely scrutinized. It is also worthwhile to point out that in four and in sixdimensions pure spinors coincide with Weyl spinors due to an accident in these dimensions[18, 31], while there are quadratic constraints that pure spinors obey in higher dimensions[41]. In particular, the constraints in ten dimensions play an important role in Berkovits’approach to superstrings [42, 43].A multivector field [18] Z = ωγ + i K γ + i S + J + σ (2.3)– 4 –s called a Fierz aggregate when ω, S , K , J , σ fulfil the Fierz identities (2.2). Additionally,if γ Zγ = Z † , the Fierz aggregate is named a boomerang [18]. When singular spinor fieldsare scrutinized, the Fierz identities are replaced by the most assorted expressions [39]: Zγ µ Z = 4 J µ Z, Z = 4 σZ, iZγ µν Z = 4 S µν Z, − Zγ Z = 4 ωZ, iZγ γ µ Z = 4 K µ Z. (2.4) Going to arbitrary dimensions, one starts from the spin bundle S associated to amanifold ( M, g ) . A crucial role is played by the Kähler-Atiyah bundle ( sec (cid:86) ( T M ) , ◦ ),where the Clifford product shall be denoted by ◦ . The spin bundle S is defined upon theeven Kähler-Atiyah bundle ( ˚ (cid:86) ( T M ) , ◦ ) and has module structure specified by a morphism γ : ( (cid:86) ( T M ) , ◦ ) → (End( S ) , ¯ ) (for more details, see, e. g., [1, 2, 40, 44–46]). In addition,a direct sum decomposition S = S ⊕ S is provided by an idempotent endomorphism R ∈ Γ(End( S )) (here Γ(End( S )) denotes the space of smooth sections of End( S ) ) – which forsome dimensions and signatures is usually identified with the volume element γ n +1 [1]. Thesub-bundles S and S are determined by the eigenvalues ± of R, and the above directsum decomposition is said to be non-trivial if both S and S are different from zero. Thisis equivalent to saying that the Clifford algebras C (cid:96) p,q constructed on the cotangent bundleon each point of M are universal [47]. The restriction ˚ γ : sec ˚ (cid:86) ( T M ) → End( S ) is namedspin endomorphism if it commutes with γ ( ξ ) , for all ξ ∈ sec ˚ (cid:86) ( T M ) [1].Spin projectors are defined by Π ± = ( I ± R) , where I denotes the identity operatoron S , providing the direct sum S = S + ⊕ S − , where S ± = Π ± ( S ) . The sections of S ± are called [symplectic] Majorana-Weyl spinors when p − q ≡ [ p − q ≡ ],while the sections of S + are known as [symplectic] Majorana spinors when p − q ≡ [ p − q ≡ ]. Classical spinors S p,q of the Clifford bundle of M are well-known to beelements of the irreducible representation space of the component of the group Spin ( p, q ) connected to the identity, and their classification can be summarized as follows: p − q mod S p,q R [( n − / ⊕ R [( n − / R [( n − / C [( n − / H [( n − / − p − q mod S p,q H [( n − / − ⊕ H [( n − / − H [( n − / − C [( n − / R [( n − / Table I. Classical Spinors Classification Table – Real Case ( p + q = n ) – 5 –he ring of quaternions is denoted by H . For the complex case the classification iswell-known to be simpler: n = 2 k C k − ⊕ C k − n = 2 k + 1 C k Table II. Classical Spinors Classification Table – Complex Case
Now consider the complex structure J ∈ Γ(End( S )) , which in particular is given by J = ± γ n +1 = ± γ ¯ · · · ¯ γ n when p − q ≡ , [1], and an endomorphism D on thespin bundle that satisfies for all ξ ∈ sec (cid:86) ( T M ) the following expressions [1, 48] D ¯ D = ( − p − q I, D ¯ γ ( ξ ) = γ ( ˆ ξ ) ¯ D . (3.1)Such expressions are taken into account hereupon, being essential to define both the clas-sification of spinor fields and the graded Fierz aggregate as well.Starting from an orthonormal local coframe { e a } na =1 ⊂ P SO ep,q ( M ) , recall from [49]that a non-degenerate bilinear pairing B on the spin bundle S is named admissible if thefollowing requirements hold: a) B is either symmetric or skew-symmetric; b) if p − q ≡ , , , , then S + and S − are either isotropic or orthogonal with respect to B [1]; c) for any ξ ∈ sec (cid:86) ( T M ) one has the transpose relation γ ( ξ ) (cid:124) = γ (˜ ξ ) if and onlyif B ( γ ( ξ ) ψ, ψ (cid:48) ) = B ( ψ, γ (˜ ξ ) ψ (cid:48) ) , where ˜ ξ stands for either the usual reversion ˜ ξ , if B issymmetric, or the Clifford conjugation ¯ ξ otherwise.A more general pairing can be taken into account, by complexifying its restriction tothe real bundle S + . In fact, by adopting hereon the notation ψ, ψ (cid:48) ∈ Γ( S ) , where Γ( S ) denotes the space of smooth sections of the spin bundle S , the bilinear pairing β on S isobtained [1] β ( ψ, ψ (cid:48) ) = B (cid:16) (Re) ψ, (Re) ψ (cid:48) (cid:17) − B (cid:16) (Im) ψ, (Im) ψ (cid:48) (cid:17) + i (cid:104) B (cid:16) (Re) ψ, (Im) ψ (cid:48) (cid:17) + B (cid:16) (Im) ψ, (Re) ψ (cid:48) (cid:17)(cid:105) , (3.2)where (Re) ψ = ( ψ + D ( ψ )) and (Im) ψ = ( ψ − D ( ψ )) are the real and imaginary parts of ψ , respectively [1].For an ordered set of indexes ( α , . . . , α k ) , ≤ α k ≤ n = p + q , the notation e α ...α k = e α ∧ · · · ∧ e α k and γ α ...α k = γ α ¯ · · · ¯ γ α k shall be adopted hereupon, with analogousexpressions for their respective contravariant counterparts. The product “ ¯ ” may be alsodenoted by juxtaposition, being used explicitly solely when we want to emphasize theunderlying structure.In the previous section the usual spinor conjugate ¯ ψ = ψ † γ represents the spinor dualto ψ . In Minkowski spacetime it has this form in order to produce a Lorentz invariantquantity. In arbitrary dimensions it can have a much more general form, which plays aprominent role in the framework of bilinear forms. In fact, a spin-invariant product canalways be written as [47] β ( ψ, ψ (cid:48) ) = a − ˜ ψψ (cid:48) = ψ † a − ψ (cid:48) , – 6 –here ψ, ψ (cid:48) are spinor fields and a ∈ Γ(End( S )) . Since ˜ a = a and ˚˚ b = ab † a − , where ˚˚ indicates an arbitrary adjoint involution, then a † = a [47]. In this way, the spinorconjugation can be written more generally as ¯ ψ = a − ˜ ψ = ψ † a − , and we can use thisdefinition to write the most general bilinear on S : β k ( ψ, ψ (cid:48) ) = B ( ψ, γ α ...α k ψ (cid:48) ) = ¯ ψγ α ...α k ψ (cid:48) . (3.3)In the well-known case described in Section II, ψ is a spinor field in Minkowski spacetime,and usually ¯ ψ = ψ † γ .On the spin bundle S , bilinear covariants are thus more generally defined as follows,given ψ ∈ S : Ω = ¯ ψψ (3.4a) J α = ¯ ψγ α ψ (3.4b) S α α = ¯ ψγ α α ψ (3.4c)... S α ...α d +1 = ¯ ψγ α ...α d +1 ψ , d < k − (3.4d)... S α ...α k = ¯ ψγ α ...α k ψ (3.4e) K α ...α l − = ¯ ψγ α ...α l − γ n +1 ψ (3.4f)... K α ...α m = ¯ ψγ α ...α m γ n +1 ψ , m < l − (3.4g)... K α = ¯ ψγ α γ n +1 ψ (3.4h) Ω = ¯ ψγ n +1 ψ . (3.4i)These bilinear covariants are used to defined the corresponding Fierz aggregate Z = Ω + J µ γ µ + S µ µ γ µ µ + · · · + S µ ...µ p γ µ ...µ p ++ K α ...α q − γ α ...α q − γ n +1 + · · · + K µ γ µ γ n +1 + Ω γ n +1 , (3.5)that reduces to the standard Fierz aggregate (2.3) when n = 4 . Similarly to the case onMinkowski spacetime, when γ Z γ = Z † the above generalized Fierz aggregate will be calleda generalized boomerang.In the next section we shall see that a graded Fierz aggregate can be evinced exclusivelyfrom the completeness relation. In addition, most of such bilinear covariants that define itwill be shown to be null, as a consequence of constraints imposed by the geometric Fierzidentities [1]. It forces the number of spinor field classes to be reduced, being obstructed bythe geometric Fierz identities. We will see in the next section that the 4D Fierz identitiesfor regular spinor fields – given by (2.2) – and for any kind of spinor fields – provided by(2.4) – can be generalized to the geometric Fierz identities in arbitrary dimensions [1], andnew classes of Fermi fields on 7-manifolds can be evinced.– 7 – Where are Majorana Spinor Fields in the Generalized Spinor FieldClassification? New Classes of Spinor fields
When p − q ≡ , the endomorphism D is a real structure that defines thecomplex conjugate via D ( ψ ) = (Im) ψ , and can be identified to the spin endomorphism Rdiscussed in Section 3 [1]. The projectors P ± = ( I ± D ) are hence responsible to split aspinor field in real and imaginary components given by respectively by P ± ( ψ ) = ψ ± . Thereal vector bundles S ± ≡ P ± ( S ) are thus used to identify the spin bundle S = S + ⊕ S − tothe complexification of the real bundle S + of Majorana spinor fields [1]. In particular when n = 7 , other useful pairings can be defined from a basic admissible pairing, denoted hereonfor the sake of simplicity by B , as [1] B (cid:48) := B ¯ ( I ⊗ J ) , B (cid:48)(cid:48) := − B ¯ [ I ⊗ ( J ¯ D )] , B (cid:48)(cid:48)(cid:48) := B ¯ ( I ⊗ D ) . (4.1)We can use some of them to define the bilinears (3.2) and (3.3). However, might the higherdimensional analogues of Lounesto’s classes of spinor fields provide indeed a different spinorfields classification? We shall answer in an affirmative way this question and discuss latersuch new possibilities, showing that at least two choices in (4.1) are equivalent under Hodgeduality.When one chooses ψ to be a Majorana spinor, the non null bilinear pairings can bereduced through the fact that B ( ψ, γ α ...α k ψ ) = 0 except if k is even [1]. Given anyadmissible bilinear pairing B on S , the endomorphisms A ψ | ψ (cid:48) of the spin bundle S havebeen defined in [1]: A ψ | ψ ( ψ ) := B ( ψ, ψ ) ψ , for all ψ, ψ , ψ ∈ Γ( S ) , (4.2)and play an important role to determine the geometric Fierz identities, encrypted in theexpressions A ψ | ψ ¯ A ψ | ψ = B ( ψ , ψ ) A ψ | ψ , (4.3)as shall be briefly reviewed in the sequel [1].Consider now the completeness relation A ψ | ψ (cid:48) = (cid:96) n (cid:88) k k ! ( − k B ( ψ, γ α ...α k ψ (cid:48) ) e α ...α k , where either (cid:96) = 2 (cid:86) n (cid:87) , if p − q = 0 , , , or (cid:96) = 2 (cid:86) n (cid:87) +1 otherwise, where (cid:86) n (cid:87) ≡ n ( n − mod 2 . This expression mimics the Fierz aggregate (2.3). Moreover, every element in thespace Γ(End( S )) , in particular the A ψ | ψ (cid:48) , can be split uniquely as A ψ | ψ (cid:48) = D ¯ A ψ | ψ (cid:48) + A ψ | ψ (cid:48) [1], where A ψ | ψ (cid:48) = (cid:96) n (cid:88) k ( − k k ! B ( ψ, γ α ...α k ψ (cid:48) ) e α ...α k , (4.4a) A ψ | ψ (cid:48) = (cid:96) n (cid:88) k k ! ( − k + p − q ) B ( ψ, D ¯ γ α ...α k ψ (cid:48) ) e α ...α k . (4.4b)– 8 –he Fierz identities (2.2) – for regular spinor fields in Minkowski spacetime – or moregenerally (2.4) — for any kind of spinor fields in Minkowski spacetime – can be generalizedfor arbitrary dimensions, being hence provided by [1]: (cid:92) A ψ | ψ ◦ A ψ | ψ + A ψ | ψ ◦ A ψ | ψ = B ( ψ , ψ ) A ψ | ψ , (4.5a) A ψ | ψ ◦ A ψ | ψ + ( − p − q (cid:92) A ψ | ψ ◦ A ψ | ψ = B ( ψ , ψ ) A ψ | ψ . (4.5b)The notation for both elements in sec (cid:86) k ( T M ) A ,kψ | ψ (cid:48) = 1 k ! ( − k B ( ψ, γ α ...α k ψ (cid:48) ) e α ...α k , (4.6a) A ,kψ | ψ (cid:48) = 1 k ! ( − p − q + k B ( ψ, D ¯ γ α ...α k ψ (cid:48) ) e α ...α k , (4.6b)will be employed accordingly, in order to make it possible to write A λψ | ψ (cid:48) = (cid:96) n (cid:80) k A λ,kψ | ψ (cid:48) ,for λ = 0 , .As we are interested in determining the nature of Majorana spinor fields accordingto bilinear covariants in 7-manifolds, we focus in the particular, however important case[37] of n = p + q = 7 + 0 . The case of a Majorana spinor on a Riemannian 7-manifoldarises, for example, in the of N = 1 compactifications of M -theory on 7-manifolds [50–52],permitting a geometric characterization by means of the reduction of the structure groupof sec (cid:86) ( T M ) from the orthogonal group O(7) to the exceptional one G [1].Moreover, the expression ϕ k := | A ,kψ | ψ | = 1 k ! B ( ψ, γ α ...α k ψ ) e α ...α k (4.7)equals zero except if k is even. Together with the symmetry of B , this shows that theelement A ,k ≡ A ,kψ | ψ vanishes except if k = 0 , , , . Combining this with (4.7), it impliesthat, for ψ a Majorana spinor, the forms ϕ k equal zero except when k = 0 or k = 4 . Byregarding ψ normalized such that B ( ψ, ψ ) = 1 , it follows that A , = 1 and the followingbilinear can be defined [1]: ϕ = 14! B ( ψ, γ α α α α ψ ) e α α α α , (4.8)which are the components of the first generator A = (1+ ϕ ) of the so called Fierz algebrarepresented in (4.4a, 4.4b) [1], where A λ ≡ A λψ | ψ . Moreover, since A = A , then the Fierzidentities (4.5a, 4.5b) are concomitantly equivalent and imply that ( ϕ + 1) ◦ ( ϕ + 1) =8( ϕ + 1) [1].Now, recall that the product ∆ k : sec (cid:86) ( T M ) × sec (cid:86) ( T M ) → sec (cid:86) ( T M ) is definediteratively by χ ∆ k +1 ϑ = 1 k + 1 g ab ( e a (cid:121) χ ) ∆ k ( e b (cid:121) ϑ ) , χ, ϑ ∈ sec (cid:94) ( T M ) , (4.9)where g ab denotes the metric tensor coefficients. By fixing ∆ = ∧ as being the exteriorproduct, the next terms are for instance provided by χ ∆ ϑ = g ab ( e a (cid:121) χ ) ∧ ( e b (cid:121) ϑ ) ,χ ∆ ϑ = 12 g ab g cd [ e a (cid:121) ( e c (cid:121) χ )] ∧ [ e b (cid:121) ( e d (cid:121) ϑ )] . – 9 –hen the Clifford product is written as ϕ ◦ ϕ = || ϕ || − ϕ ∆ ϕ , the Fierz identitiescorrespond to the following conditions [1]: ϕ ∆ ϕ = − ϕ , || ϕ || = 7 . (4.10)They are the root to establish the spinor fields classification according to the bilinear co-variants.In the case here to be analyzed n = p + q = 7 , we already know that ϕ k = 0 except forthe values k ∈ { , , , } . In addition, due to the restriction B ( ψ, J ¯ γ α ...α k ψ ) = − B ( ψ, J ¯ D ¯ γ α ...α k ψ ) = 0 except it k = 2 j + 1 , k ∈ Z , (4.11)we obtain that unless k = 3 or k = 7 the form B (cid:48) ( ψ, γ α ...α k ψ ) vanishes. Hence, alternativelywe could have chosen any other of the bilinear pairings in (4.1) in order to define the bilinearcovariants. For instance, choosing B (cid:48) yields the following definition: ˇ ϕ k = 1 k ! B ( ψ, J ¯ γ α ...α k ψ ) e α ...α k ∈ sec k (cid:94) ( T M ) . (4.12)However, the Hodge duality (cid:63)ξ = ˜ ξJ , where J = γ n +1 , for our case n = 7 implies that thealternative homogeneous forms ˇ ϕ = 13! B ( ψ, J ¯ γ α α α ψ ) e α α α (4.13) ˇ ϕ = 17! B ( ψ, J ¯ γ α ...α ψ ) e α ...α (4.14)do not vanish likewise. A similar reasoning [1] implies that the other bilinear pairings in(4.1) contain no information besides the ones provided by (4.7). The Fierz identities takenow the form ˇ ϕ ∆ i ϕ = 0 ( i = 1 , , ˇ ϕ ∧ ϕ = || ˇ ϕ || γ . (4.15)Since ˇ ϕ = (cid:63)ϕ implies that (cid:107) ˇ ϕ (cid:107) = (cid:107) ϕ (cid:107) , Eq. (4.10) asserts that the 3-form ˇ ϕ is non nulllikewise, and a similar reasoning implies that also ˇ ϕ (cid:54) = 0 , as (cid:63) ϕ .Hence, only the bilinears ϕ = B ( ψ, ψ ) (4.16) ϕ = 14! B ( ψ, γ α α α α ψ ) e α α α α (4.17)are non null. Eq. (4.16) is the higher dimensional analogue of its Minkowski spacetimeversion provided by Eq.(2.1a). Thus the Fierz identities (4.10) imply, in particular, that ϕ (cid:54) = 0 and only one type of Majorana spinor results according to a generalized spinor fieldclassification: ϕ (cid:54) = 0 ∈ sec (cid:94) ( T M ) , ϕ (cid:54) = 0 ∈ sec (cid:94) ( T M ) . (4.18)– 10 –n fact, such class of Majorana spinor fields according to the bilinears in the Clifford bundle C (cid:96) , is provided by: ϕ (cid:54) = 0 , ϕ = 0 , ϕ = 0 , ϕ = 0 , ϕ (cid:54) = 0 , ϕ = 0 , ϕ = 0 , ϕ = 0 , (4.19)or equivalently, if Eq.(4.12) is taken into account, ˇ ϕ (cid:54) = 0 , ˇ ϕ = 0 , ˇ ϕ = 0 , ˇ ϕ = 0 , ˇ ϕ (cid:54) = 0 , ˇ ϕ = 0 , ˇ ϕ = 0 , ˇ ϕ = 0 , (4.20)where the Hodge duality ˇ ϕ k = (cid:63)ϕ − k is utilized.If we use the results from [1] and given ι ∈ Z , B ( ψ, γ α ...α k ψ ) = B (cid:0) (Re) ψ, γ α ...α k (Im) ψ (cid:1) + B (cid:0) (Im) ψ, γ α ...α k (Re) ψ (cid:1) , if k = 2 ιB (cid:0) (Re) ψ, ( J ¯ γ α ...α k ) (Re) ψ (cid:1) − B (cid:0) (Im) ψ, ( J ¯ γ α ...α k (cid:1) (Im) ψ ) , if k = 2 ι + 1 , and B ( ψ, J ¯ γ α ...α k ψ ) = − B (cid:0) (Re) ψ, γ α ...α k (Im) ψ (cid:1) + B (cid:0) (Im) ψ, γ α ...α k (Re) ψ (cid:1) , if k = 2 ιB (cid:0) (Re) ψ, ( J ¯ γ α ...α k ) (Re) ψ (cid:1) + B (cid:0) (Im) ψ, ( J ¯ γ α ...α k ) (Im) ψ (cid:1) , if k = 2 ι + 1 , Eq. (3.3) that describes the bilinear covariant coefficient of degree k can be thus generalized,in order to encompass the complex case, providing the higher degree generalization of (3.2): β k ( ψ, γ α ...α k ψ (cid:48) ) = B (cid:16) (Re) ψ, γ α ...α k (Re) ψ (cid:48) (cid:17) − B (cid:16) (Im) ψ, γ α ...α k (Im) ψ (cid:48) (cid:17) (4.21) + i (cid:104) B (cid:16) (Re) ψ, γ α ...α k (Im) ψ (cid:48) (cid:17) + B (cid:16) (Im) ψ, γ α ...α k (Re) ψ (cid:48) (cid:17)(cid:105) . By using the above results, the bilinear covariants can be now extended from the standardMajorana spinor fields in ψ ∈ Γ( S + ) to sections of Γ( S ) , by identifying now ϕ k := 1 k ! β k ( ψ, γ α ...α k ψ ) e α ...α k . (4.22)As both terms in the real part and also both terms in the imaginary part in (4.21) as wellcan cancel each other, in the complex version it is possible that the bilinears ϕ and ϕ can now be different from zero. Hence, four classes of spinor fields ψ ∈ Γ( S ) are found,concerning the classification in Riemannian 7-manifolds using the constraints above, as itis showed below: ϕ = 0 , ϕ = 0 , (4.23a) ϕ = 0 , ϕ (cid:54) = 0 , (4.23b) ϕ (cid:54) = 0 , ϕ = 0 , (4.23c) ϕ (cid:54) = 0 , ϕ (cid:54) = 0 . (4.23d)It is implicit in (4.23a-4.23d) that all other ϕ k = 0 for k = 1 , , , , , . Moreover, thespinor field classification according to the bilinears ˇ ϕ k , defined by substituting Eq.(4.12) in– 11 –4.21), is identical to the one provided by the ϕ k . The above class (4.23d) encompasses thesole spinor field class (4.19), and reduces to it when we restrict the field ψ to be an elementof the bundle Γ( S + ) , namely, a Majorana spinor field.Since B ( ψ, γ α ...α k ψ ) vanishes except when k ∈ { , , , } , the graded Fierz aggregate,that is defined by Z = (cid:96) n (cid:88) k ( − k B ( ψ, γ α ...α k ψ ) e α ...α k , (4.24)where the sum is ordered in k , has clearly the terms B ( ψ, γ α ...α k ψ ) = 0 for such values of k . The above expression coincides with its Minkowski spacetime version (2.3) provided byLounesto [18].After classifying the spinor fields in Riemannian 7-manifolds, they can be used fordefining one Lagrangian on S for matter fields. According to [16], terms in a Lagrangiandefined in this way depend on which realization is taken for the matter spinor fields. Theclassification of spinor fields in 7-manifolds can be very useful in order to study the behaviourof fields in AdS × S or, more generally, in AdS × M , where M denotes a 7-manifold. Spinor fields on a manifold ( M, g ) , with arbitrary dimension and arbitrary metric signa-ture have been classified according to the bilinear covariants. It encompasses the celebratedLounesto’s spinor field classification for Minkowski spacetime, generalizing it to arbitrarydimensions and metric signatures. The geometric Fierz identities [1] limit the amount ofclasses of spinor fields in such a generalized classification, which is explicitly analyzed forthe important case of Majorana spinor fields on Riemannian 7-manifolds. A generalizedgraded Fierz aggregate is also obtained in such a context simply from the completenessrelation, and we analyze the particular and prominent case of 7D. In this case, the higherthe spacetime dimension, the lesser the number of classes of spinor fields.Despite the generalizations regarding Fierz identities were known, analysis of the spinorfields themselves had been lacking up to the middle of the last decade. Since then newmodels had been proposed, as for instance a candidate for the field theory of some massdimension one fermions. A complete characterization and identification of these new spinorfields as elements of the Lounesto’s classes [24] have introduced in the literature new pos-sibilities, further investigated in e. g. [20, 21, 23–28, 30]. In fact, spinor fields in thesame class can present completely different dynamics. For instance, Elko spinor fields per-tain to the class 5 in Lounesto’s classification, and satisfy a coupled system of Dirac-likeequations, whereas Majorana spinor fields are also spinor fields in the class 5, but satisfythe well-known Majorana equations. Recently the first physical example of a flag-dipoleparticle, which is a type-4 spinor field in the Lounesto’s classification, has been found asthe solution of the Dirac equation with torsion in a f ( R ) background [25]. Type-6 spinorfields encompass for instance Weyl spinor fields, but the complete dynamics of all classesis nevertheless undetermined. The first important step toward a complete characterizationof all possible dynamics of spinor fields in Minkowski spacetime has been accomplished– 12 –y explicitly obtaining the most general type of spinor fields in each class of Lounesto’sclassification [26].Thus, the study of the Lounesto’s spinor fields classification has opened a huge path todiscover unexpected new physical features and to propose candidates for new particles inMinkowski spacetime [20–27, 30, 32]. With this motivation we have proposed a much moregeneral classification as well, encompassing pseudo-Riemannian spacetimes of arbitrary di-mensions and metric signatures. In particular, as the subject of S spinor fields is veryrich [34, 37], we investigated where are the Majorana spinor fields in such a classificationaccording to the bilinear covariants, and we concluded that the geometric Fierz identitiesobstruct the existence of more than one precise class, determined by (4.19), asserting thatfor instance that spinor fields studied in [34, 35, 37] reside in such class. In these papers, the3-form bilinear is the torsion tensor that works as a gauge potential. In the most generalcase that we analyzed, by taking spinor fields in the spin bundle over M , more three typesof spinor fields with potential new properties are achieved. As the singular spinor fields inLounesto’s classification were studied in exotic structures [20, 23], it is natural to relate thenew classes of spinor fields in 7-manifolds derived in (4.23a-4.23d) to their exotic version[53]; however this is beyond of the scope of the present paper. Acknowledgments
R. da Rocha is grateful for the CNPq grant No. 303027/2012-6 and No. 473326/2013-2 which has provided partial support, to CAPES for the
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