Additional fermionic fields onto parallelizable 7-spheres

The geometric Fierz identities are here employed to generate new emergent fermionic fields on the parallelizable (curvatureless, torsionfull) 7-sphere ($S^7$). Employing recently found new classes of spinor fields on the $S^7$ spin bundle, new classes of fermionic fields are obtained from their bilinear covariants by a generalized reconstruction theorem, on the parallelizable $S^7$. Using a generalized non-associative product on the octonionic bundle on the parallelizable $S^7$, these new classes of algebraic spinor fields, lifted onto the parallelizable $S^7$, are shown to correctly transform under the Moufang loop generators on $S^7$.


I. INTRODUCTION
(Classical) spinor fields are well known to be elements in the carrier space of the Spin group irreducible representations on any given spacetime that admits a spin structure, namely, if the second Stiefel-Whitney class vanishes. Spinor fields are, in particular, employed for constructing the so called bilinear covariants, consisting of tensorial quadratic forms involving the spinors. The bilinear covariants were shown to be the homogeneous part of a multivector Fierz aggregate [1].
Particularizing for 4D the Minkowski spacetime case, spinor fields were classified with respect to their bilinear covariants in the Dirac-Clifford algebra, by the so called Lounesto's spinor field classification [2]. Further lattice generalizations in the context of quantum Clifford algebras were also studied [3]. The bilinear covariants are not independent, but constrained by the Fierz identities [2,4]. Reciprocally, given the bilinear covariants, their associated spinor fields can be re-obtained up to a phase, by the reconstruction theorem [6,7]. The Lounesto's classification is based upon the U(1) gauge symmetry of the first-order equations of motion that rule spinor fields in each spinor class. However, a more general classification has been proposed in Ref. [8] encompassing spinor multiplets as realizations of (non-Abelian) gauge fields. In this more general classification, composed flagpoles, dipoles, and flag-dipoles naturally descend within fourteen disjoint classes of spinor fields, under the gauge symmetry SU(2) × U (1). In this setup, the spinor fields in the standard Lounesto's classification were shown to be a limiting case, equivalent to Pauli singlets [8].
Further spinor representations were studied in Refs. [9,10], with also other proposals to construct the bilinear covariants for flagpole spinors [11]. An analogous classification in the framework of second quantization and a quantum reconstruction algorithm was also proposed, being the Feynman propagator extended for regular and singular spinor fields, in Ref. [12].
In any fixed spacetime dimension, n, and signature, (p, q), the very construction of the bilinear covariants depends on the existence of either real, or complex, or even quaternionic structures.
Hence, the existence of non null bilinear covariants can be impeded by the geometric Fierz identities. Despite the natural obstructions due to the existence of algebraic and geometric structures on a given spacetime dimension/signature, the Lounesto's spinor field classification on 4D Minkowski spacetime was successfully generalized to other spacetime dimensions and signatures, of relevance in their applications, as the emergence of fermionic fields in the respective spacetime compactifications. Spinor fields on the 7-sphere S 7 , as an Einstein space composing the compactification AdS 4 × S 7 , were studied in Ref. [1], where new spinor classes were derived. On the other hand, new spinor field classes in the compactification AdS 5 × S 5 were derived and investigated in Ref. [13], representing new recently obtained fermionic solutions in string theory. More precisely, Ref. [1] proposed new classes of spinor fields on S 7 , based on the geometric Fierz identities in Ref. [14].
The underlying structure of the geometric Fierz identities on S 7 was shown to sternly obstruct the amount of non null bilinear covariants were found on S 7 . Nevertheless, further three new emergent classes of fermionic fields on S 7 . From a more physical point of view, investigating these new classes of spinors S 7 may afford new fermionic solutions of first order equations of motion, that can play an important role on supergravity. In fact, one of the spontaneous compactification schemes on n = 11 supergravity can be implemented by the so called Freund-Rubin-Englert solution, obtained on a product manifold AdS 4 × S 7 [15]. As important as the standard S 7 , the so called parallelizable S 7 , a curvatureless manifold that has torsion, emerges when the antisymmetric gauge field strength in the Englert's solution excedes the Freund-Rubin one, being identified with the Cartan-Schouten torsion on the 7-sphere.
Our main aim here is to construct new fermionic fields on the parallelizable S 7 , that can be then obtained when new classes of S 7 spinor fields are lifted onto the parallelizable S 7 . This paper is organised as follows: in Sect. II, after briefly reviewing how the geometric Fierz identities are used to derive additional spinor field classes on S 7 , we propose a reconstruction procedure for obtaining the spinor fields, in these new classes, from the bilinear covariants and the geometric Fierz identities. Sect. III is then devoted to briefly review the parallelizable sphere, whose torsion is defined with respect to the non-associative X-product on the octonionic bundle. The geometric Fierz identities are used to derive the spinor field classes on S 7 , that are going to be lifted onto the parallelizable S 7 , whereon new fermionic fields can be then constructed through the introduction of a generalized octonionic law of transformation.

II. GEOMETRIC FIERZ IDENTITIES AND BILINEAR COVARIANTS
Let (M, g) be a manifold endowed with a metric tensor. The exterior bundle Ω(M ) = i=0 Ω i (M ) has endomorphisms that come from the tensor algebra quotient construction. Given a k-form field 1 a ∈ sec Ω k (M ), the grade involution,â = (−1) k a, is an automorphism; the reversion, a = (−1) [(k/2)] a, for [(k)] denoting the integer part of the degree k, is an antiautomorphism. These composition of these two morphisms define the conjugation, denoted byā. The Clifford bundle can be obtained by equipping the exterior bundle with the universal Clifford product u⋄a = u∧a+u a, for all 1-forms u ∈ sec Ω 1 (M ), where is the left contraction. 1 One calls a k-form field a section of an homogeneous space of the exterior bundle.
In Eqs. (3b) -(3f) below, we just denote the bilinear covariants that do not vanish: Classes 1, 2, and 3 consist of regular spinor fields, since not both the scalar and the pseudoscalar vanish. Classes 4, 5, and 6 realize singular spinor fields, where both σ and ω are null. Refs. [3, 16-18] illustrate a vast range of applications of these classes in quantum field theory and gravity. Ref.
[4] introduced more two exclusive classes into the Lounesto's classification, through a generallyrelativistic gauge classification, whereas the most general spinor field class in each spinor class was derived in Ref. [5] as a prominent computational tool for the reconstruction theorem.
The Fierz identities (2) are well known not to be valid for the case of singular spinors. In this case, based upon a Fierz aggregate, the Fierz identities (2) can be replaced by Fierz aggregates that are self-adjoint multivectors, γ 0 Zγ 0 = Z † , are better known as boomerangs [2].

2 √ῡ
Zυ e −iα Zυ, for an arbitrary phase α, such that −iα = ln 2 √ῡ ψῡZυ . In particular, any regular spinor can be reconstructed as [6,7] Heretofore spinor fields were approached without mentioning the spinor bundle. We denoted the Minkowski spacetime manifold by M ≃ R 1,3 . Since it is an affine space, being isomorphic to its own tangent spaces, a lot of important structures were hidden throughout the text, for simplicity.
Nevertheless, to approach spinor fields on higher dimensions, we should recall the spinor structures case of interest, consisting of the 7-sphere S 7 , its S spin bundle is, thus, equipped with the induced : S → S product, accordingly [14]. This composition just indicates the product between spinor fields, usually denoted by juxtaposition, when Minkowski spinor fields are regarded. Denoting by End(S) all the linear mappings from S to S and by "sec" any section of a bundle, a bilinear pairing B : sec S × sec S → R can define a bilinear mapping [1,14]. Indeed, given sections ψ, ψ on the spin bundle, a bilinear mapping B 0 : sec S × sec S → R, on the S 7 spinor bundle, reads [14] B for the real, ℜ ψ, and the imaginary, ℑ ψ, components of the spinor field ψ [14]. This bilinear mapping is the one that shall generalize the bilinear covariants (3b -3f) scalar components, that were constructed on R 1,3 to the 7-sphere. This can be implemented by the bilinear mapping on the S 7 spin bundle: To define new spinor classes on S 7 , when k is odd, the bilinear mapping B(ψ, γ τ 1 . . . γ τ k ψ) is not equal to zero [14]. Defining givenψ,ψ,Ψ ∈ sec S, then the (geometric) Fierz identities then read [14] A ψ|Ψ¯Aψ |Ψ = B(ψ, Ψ)A ψ|Ψ .
Given the structure D that defines the complex conjugate on S by D(ψ) = ℑ ψ, the elements A ψ|ψ are differential forms that can be always split into A ψ|ψ = D¯A 1 ψ|ψ + A 0 ψ|ψ [14], where The geometric Fierz identities then follow for S 7 [14]: These equations are the equivalent of Eqs. (2), for S 7 .
Moreover, the bilinear covariants on S 7 emulate the ones of Minkowski spacetime (3b -3f), by It is worth to emphasize that the bilinear covariants construction on R 1,3 are not obstructed by a dimensional accident. However, on S 7 (and also on other specific dimensions), the geometric Fierz identities (12a, 12b) severely obstruct the very existence of homogeneous bilinear covariants [14].
In fact, spinors on S 7 have the bilinear covariants φ k equal to zero, with the exceptions when k = 0 or k = 4 [1,14], namely, Hence, the complex bilinear covariants can be defined [1], yielding three (non-trivial) classes of spinor fields on the S 7 spin bundle [1], The Fierz aggregate (4) in the R 1,3 Minkowski spacetime can be now emulated for the 7-sphere.
In fact, the reconstruction theorem can be then employed for constructing the original spinor field as a section of the spin bundle, from the corresponding Fierz aggregatě that is simpler than its 4D Minkowski counterpart Fierz aggregate, defined in Eq. (4). Hence, when an arbitrary spinor ξ ∈ S 7 satisfies ξ † ( spinor ψ can be obtained from its Fierz aggregate (19), where

III. LIFTING NEW SPINOR FIELDS ON THE PARALLELIZABLE S 7
Heretofore, new classes of S 7 spinors were derived into the classes (18a -18c), whose representative spinor fields can be reconstructed by Eq. (20). These representative spinor fields are now aimed to be lifted onto the so called parallelizable S 7 , that can be regarded as the manifold of unit octonions. Among the parallelizable spheres, S 7 is the sole one that does not carry a Lie group structure, however a Moufang loop structure, instead. The (sub)bundle of octonionic sections on written as X = X 0 e 0 + 7 a=1 X a e a . Instead of the vector space R 8 , one can take the paravector space V 7 := R ⊕ R 0,7 endowed with the octonionic standard product • : V 7 × V 7 → V 7 . In fact, the scalar part, X 0 does correspond to the real part of an octonion, whereas the vector component, 7 a=1 X a e a , regards the imaginary part. In this case, the identity e 0 = 1 and an orthonormal basis {e a } 7 a=1 of V 7 ֒→ Cℓ 0,7 generate the octonion algebra [20]. The octonionic product can be emulated at the Clifford algebra Cℓ 0,7 as where ✵ = e 7 e 1 e 5 + e 6 e 7 e 4 + e 5 e 6 e 3 + e 4 e 5 e 2 + e 3 e 4 e 1 + e 2 e 3 e 7 + e 1 e 2 e 6 is a 3-form, and the juxtaposition denotes the Clifford product. The symbol χ 0⊕1 denotes the projection of a multivector χ ∈ Cℓ 0,7 onto its paravector components. For the underlying Lie algebra g 7 , the Lie bracket where A [ab] = 1 2 (A ab − A ba ), for any tensor A ab , and the Einstein's summation convention is used hereon. The (Clifford) conjugation of X = X 0 + X b e b ∈ O readsX = X 0 − X b e b , for X 0 , X a real coefficients. Given X ∈ S 7 , the X-product is defined by [22] A The expressions below are shown in, e.g., [22] ( As we dealed with bundles in the previous sections, the octonion bundle shall be employed, where T S 7 denotes the tangent bundle on S 7 , with fibers R ⊕ T X S 7 [21]. Hence, given A, B, C ∈ OS 7 , and the associator [A, Eq. (24) shows that the octonionic field X ∈ sec(OS 7 ) determines two endomorphisms of the . The quasi-alternativity of the • X -multiplication then follows as The X-product can be, thus, seen as the original octonionic product. In fact, there exists an orthogonal mapping T ∈ SO(R 0,7 ), such that the mapping ρ : , is an isomorphism, for all a ∈ R and v ∈ R 0,7 [29]. The reciprocal statement is up to now a conjecture. Besides, an orbit whose elements are isomorphic copies of O obtained out of any fixed copy of O is an orbifold S 7 /Z 2 = RP 7 , being diffeomorphic to SO(7)/G 2 . In fact, identifying two antipode points on S 7 yields A • −X B = A • X B. One of the most natural ways of obtaining a parallelizable S 7 is choosing two non-canonical connections on Spin(7)/G 2 [23].
Besides, the sphere S 7 plays a prominent role on the (quaternionic 2 ) Hopf fibration S 3 ֒→ S 7 p → S 4 , [25]. In this sense, S 7 can be realized as being the set {(q 1 , q 2 ) ∈ H 2 | q 1 2 + q 2 2 = 1}, where p : S 7 → S 4 maps the pair (q 1 , q 2 ) to q 1 /q 2 , an element in the projective line HP 1 ≈ S 4 . Thus, each fiber is represented by a torsor that is parametrized by quaternions of unit norm, defining S 3 . A construction of this Hopf algebra was also realized using regular spinors, being the most important realization with respect to the Lounesto's spinor field classification, in Refs. [16,25,26].
More generally speaking, without considering just the S 7 manifold, a n-manifold M is said to have the property of global parallelizability if there are n linearly independent vector fields defined on M . Thereupon, for each X ∈ M , one can linearly combine these fields to obtain an orthonormal basis for T X M . Given one of these bases, since vectors are linear combinations of such elements, their covariant derivative in different points can be taken in a natural way, which results in path independence for the parallel transport. In fact, it follows that whereD denotes the covariant derivative defined with respect to this parallel transport, whereas R µν denotes the curvature tensor. As usual,D = ∂ +Γ = D − T, where T denotes the parallelizing torsion andΓ is the parallelizing connection. Let e ν a indicate the vielbein, related to a noncoordinate basis, wherein roman letters indicate the indexes of the tangent spaces, accordingly. As D µ e ν a = 0, the covariant derivative of the vielbein yieldsD µ e ν a = −T µν a . Now, one can look at a manifold M , that for our case is S 7 , and consider the infinitesimal translations determined by the covariant derivatives. As may be seen, it is straightforward that these translations configure a closed algebra [22]: When the manifold is also a group manifold, it is evident that the parallelizing torsion does not depend on the point chosen and is, thus, only expressed by the structure constants. Nonetheless, S 7 must be carefully considered, for the torsion varies at each point on the manifold. This fact is intrinsically related to the non-associativity of O, as it can be seen in Ref. [22].
For a field X ∈ sec(OS 7 ), one can construct a parametrization of S 7 with respect to unitary octonionic fields X |X| ∈ sec(OS 7 ). The tangent space T X S 7 is spanned by the usual octonionic basis as {X • e i } 7 i=1 . Now, as introduced in Ref. [22], let us consider the infinitesimal operator δ A , where A ∈ sec(OS 7 ) is now a pure imaginary octonionic field, acting on X as δ A X = X • A. This transformation defines the parallel transport on the basis spanned by the choice of X. An explicit derivation can be realized [22] to find the commutator of the defined transformations: It can be shown that the parameterX is twice the negative of the parallelizing torsion [27]. Componentwise, presenting, thus, a Moufang loop (or Moufang quasigroup) structure in the second equation in (31). Therefore, one can see that the operator δ and the parallelizing covariant derivative are, in fact, in a 1-1 correspondence. Now, taking another field ζ ∈ O, with ζ |ζ| = Y ∈ sec(OS 7 ), over the same orientation given by the choice of X, and transforming it such that the relations in Eq. (31) are preserved, such properties preclude the straightforward ansatz δ A Y = Y • A [22]. The two regarded fields on S 7 must, thus, transform according to another rule, that may seem at a first glance, not the simplest choice. Ref. [22] derived the appropriate transformation rule for fermionic fields on S 7 , taking into account its underlying parallelizable torsion, as Now, the new classes of spinors on S 7 can be lifted onto the parallelizable S 7 . In fact, for it we need to remember the equivalence between the classical and the algebraic spinor fields. Going back to the 4D Minkowski spacetime, the standard Dirac spinor ψ was identified, e. g., in Ref. [2] as an element of the minimal left ideal (C ⊗ Cℓ 1,3 )f associated to the Dirac-Clifford algebra (C ⊗ Cℓ 1,3 ), generated by the primitive idempotent f = 1 4 (1 + γ 0 )(1 + iγ 1 γ 2 ) yielding ψ ∈ (C ⊗ Cℓ 1,3 )f is an algebraic spinor [2]. Hence, using the Dirac representation of the gamma matrices, the algebraic is equivalent to the classical spinor ψ = (ψ 1 , ψ 2 , ψ 3 , ψ 4 ) ⊺ ∈ C 4 . Now, this concept can be extended for the parallelizable S 7 , emulating the transformation is a primitive one. Hence, a spinor ψ ∈ S 7 has its algebraic version as the elementψf of the left ideal Cℓ 0,7 f , for some multivectorψ ∈ Cℓ 0,7 . This is accomplished just for introducing the S 7 spinor into the Clifford bundle itself, on S 7 . Now, to write the correct transformation of a fermionic field on the parallelizable S 7 , given an element of the vector space underlying Cℓ 0,7 , a non-associative product called the ξ-product was introduced in [19] as a natural generalization for the X-product. For homogeneous multivectors ξ = u 1 ∧ . . . ∧ u k ∈ sec Λ k (R 0,7 ) ֒→ sec Cℓ 0,7 , where {u p } k p=1 ⊂ sec T R 0,7 and A ∈ sec(OS 7 ), the products • and • are defined (and extended by linearity) by [19,29] • : sec(OS 7 ) × sec Λ k (R 0,7 ) → sec(OS 7 ) Hence, within the above constructions, the transformation of the reconstructed spinor field on S 7 from its bilinear covariants in Eq. (20), that is a representative of the new classes (18a -18c) of spinor fields on S 7 , can be defined as In this way, the previous new classes of S 7 spinors are lifted onto the parallelizable S 7 . This transformation is compatible to the ones defined in Ref. [30].

IV. CONCLUSIONS
We have managed to establish the reconstruction theorem for the new classes of spinor fields on S 7 using the generalized Fierz aggregate, for each recently found new class of spinor fields on the S 7 spin bundle according to their bilinear covariants. Besides, this categorization has enabled the construction of new fermionic fields on the parallelizable S 7 , promoting the new classes of classical spinor fields on S 7 to new classes of algebraic ones. Hence, the correct transformation of these elements, generating a Moufang loop structure on the parallelizable S 7 was derived. Aiming to this procedure, we briefly reviewed the parallelizability property on the parallelizable S 7 , wherein the parallel transport could be analyzed with respect to the torsion. Therein, the non-associativity of the octonionic bundle on S 7 was related to the torsion tensor on the parallelizable S 7 , as a function dependent on each point on S 7 , via the X-product. In this way, additional classes of fermionic (spinor) fields on the parallelizable S 7 have been constructed, according to the classes obtained heretofore, lifted from the S 7 spin bundle, with the right transformation under infinitesimal transformations. Our results, thus, generalize the ones in Ref. [22], also proposing new classes of fermionic fields that may play the role of the solutions in compactfications of supergravity.
The Clifford bundle of differential forms Cℓ(M, g) is a vector bundle associated with P Spin e 1,3 (M ), whose sections are sums of non-homogeneous differential forms. Hence Cℓ(M, g) ≃ P SO e 1,3 (M ) × Ad ′ Cℓ 1,3 is a bundle defined by: (1) Let π : Cℓ(M, g) → M be the canonical projection and let {U α } be an open covering of M .

Appendix B: The Radon-Hurwitz theorem
Let Cℓ p,q be the Clifford algebra associated to R p,q and {e i } (i = 1, . . . , n) an orthonormal basis of this quadratic space. A primitive idempotent of Cℓ p,q is given by f = 1 2 (1 + e I 1 ) · · · 1 2 (1 + e I k ), where {e I 1 , . . . , e I k } is a set of elements in Cℓ p,q that commute and such that (e Iα ) 2 = 1 for α = 1, . . . , k. It generates a group of order 2 k , where k = q − r q−p , and r j are the Radon-Hurwitz numbers defined by [28] with the recurrence relation r j+8 = r j + 4.