New spinor classes on the Graf-Clifford algebra

Pinor and spinor fields are sections of the subbundles whose fibers are the representation spaces of the Clifford algebra of the forms, equipped with the Graf product. In this context, pinors and spinors are here considered and the geometric generalized Fierz identities provide the necessary framework to derive and construct new spinor classes on the space of smooth sections of the exterior bundle, endowed with the Graf product, for prominent specific signatures, whose applications are discussed.


Introduction
The Clifford algebras classification provides a relationship between supersymmetry and division algebras [1,2]. From the classical point of view, spinors can be defined as objects which carry an irreducible representation of the Spin group, which is the double covering of the special orthogonal group. Therefore spinors carry the spin 1/2 representation of the group of rotations in a quadratic space. On the other hand, the Spin group is naturally embedded into a Clifford algebra and an ulterior, equivalent, definition of spinor can be introduced, namely, the algebraic one. In fact, the representation space associated with an irreducible regular representation is a minimal left ideal related to the Clifford algebra [3]. An algebraic spinor is an element of a minimal left ideal in a Clifford algebra. The representation of the Clifford algebra obtained is called a spinor representation. Classical spinors can be classified with respect to their bilinear covariants, satisfying the generalized Fierz identities presented in Refs. [4,5]. This property has led to the well known Lounesto's classification of spinors in Minkowski spacetime R 1,3 into six classes [3], into regular and singular spinors [6][7][8][9][10]. Prominent features of singular spinors were studied in Ref. [11]. The spinors themselves can be reconstructed from their spinor bilinear covariants, by the reconstruction theorem [12], yielding a reciprocal spinor field classification [7,13].
The geometric Fierz identities are constraints on the bilinear covariants, yielding to classify spinors fields on any simply connected manifold with spin structure [5]. Recently, a classification of spinor fields on Lorentzian [14] and Riemannian [15] 7-manifolds has been constructed, emulating the Lounesto's classification. It introduces new classes of spinors and new fermionic solutions in supergravity, regarding, in particular, its AdS 5 × S 5 [16,22] and AdS 4 × S 7 [14] compactifications.
Having constructed new classes of spinors, implicitly using the Graf product and the geometric Fierz identities [5], for signatures (7,0), (6,1), and (1,4) [14][15][16], our main aim here is to derive new classes of spinors on a Lorentzian space with signature (1,2), employing this method, and also to construct pinors in signature (9,0). This last case is naturally linked to spinors in R 1,9 , as elements that carry the irreducible representation of the group Spin 1,9 , whose Lie algebra so (1,9) spin (1,9) is isomorphic to the Lorentz-like Lie algebra sl (2, O). Since the Fierz identities can be used to provide quite few realisations, we shall discuss here the last mod 8 possibility [5], by studying the occurrence of new pinor and spinor fields on manifolds of signatures (1,2) and (9,0). This last case is relevant in either the cone or the cylinder formalism, to study compactifications of M -theory with one supersymmetry, lifting the formalism on an 8-dimensional manifold onto a 9-dimensional conic or cylindric metric space, wherein the pin/spin bundle is a real vector bundle [23]. Hence, it is important to derive new classes of pinors in these signatures.
This paper is organized as follows: after presenting the fundamental properties of Clifford bundles in Sect. II, the Graf product is then introduced and studied in Sect. III, in the context of the Graf-Clifford algebra. In Sect. IV, pinors, spinors and the geometric Fierz identities are then constructed, from the Atiyah-Bott-Shapiro mod 8 periodicity. The main subalgebra is a pivotal ingredient for analysing the normal, the almost complex, and the quaternionic cases for the geometric Fierz identities, in the context of the Clifford bundles. In Sect. V, three non-trivial new classes of spinors on a 3-dimensional manifold of signature (1,2) are constructed, in a detailed and systematic analysis of the geometric Fierz identities. In Sect. VI, seven non-trivial pinor classes of Euclidean 9-manifolds are derived. Sect. VII is finally devoted for drawing the conclusions and perspectives.

Preliminaries
Let us regard the 2-uple (M, g) as a pseudo-Riemannian manifold with signature (p, q), the exterior bundle (T * M) and U an open subset on M. The k-forms are defined as local sections of the k-power exterior bundle k (T * M), for k = 0, . . . , dim M. The function 1 Γ ∈ C ∞ (M) denotes the unit element of Γ (U, (T * M)). The following suitable notations Γ ( k (T * M)) = Γ ( k ) and Γ ( (T * M)) = Γ ( ) shall be adopted, denoting the space of smooth sections of the exterior bundle. If dim M = n, given a set of indexes I n = {i 1 , . . . , i n }, let {e i | i ∈ I n } be a local frame to the tangent bundle T * M on U ⊂ M. A coframe for T * M is constituted by the set of covectorial sections {e i | i ∈ I} that are defined with respect to the dual relation e i (e j ) = δ i j 1 Γ . The metric tensor g * : Γ (U, T * M) × Γ (U, T * M) → R is such that g * (e i , e j ) =: g ij . The k-forms e I j = e i 1 ...i j = e i 1 ∧ e i 2 ∧ · · · ∧ e i j are elements of Γ (U, (T * M)) for j = 1, . . . , n. Therefore, a form f ∈ Γ (U, (T * M)) reads f = n k=1 f I k e I k , where f I k is a smooth function that constitutes a linear combination of element of I k . Since the fibers of the exterior bundle are Z 2 -graded, then there is a splitting (T * M) = + (T * M) ⊕ − (T * M), where + (T * M) is the even subbundle whose fibers are given by the grade involution # as being Besides, there is another Z 2 -grading on the sections of exterior bundle, given by the odd subspace Γ − ( ) := Γ ( − ) = k=odd Γ ( k ) = ker(# + Id Γ ( ) ) and by the even subbundle Γ + ( ) := Γ ( + ) = k=even Γ ( k ) = ker(# − Id Γ ( ) ). The volume form v = e 12...n is defined as an element of the space Γ ( n (T * M)).

The Graf product
The Graf product between a m-form f ∈ Γ ( m (T * M)) and a r-form g ∈ Γ ( r (T * M)), m ≤ r, [24][25][26][27] is defined as and the non-commutative relation where ∧ k denotes the contracted wedge product between f and g [25,28], iteratively constructed as: In Ref. [24] it was proved that the set of sections Γ ( (T * M)) is a Clifford algebra with respect to the Graf product, such algebra (Γ ( (T * M)), ) is named the Graf-Clifford algebra. Due to this definition, the volume element v satisfies [24] v remembering that p−q regards the signature of the pseudo-Riemannian metric that endows the M manifold.
For an arbitrary r-form f it implies that In addition, the volume element v is central in Γ ( (T * M)), with the condition that the dimension of M is odd whichever the grade of the form [24].
defining the upper truncation, P U (f ) = f U := n j=[ n 2 ]+1 f j I j e I j , and the lower truncation, In other words, Γ L = P L (Γ ( )) and Γ U = P U (Γ ( )). For a form f in the signatures p − q ≡ 8 0, 1, 4, 5, the Hodge operator action is the identity, up to a sign, f = ±f . It implies that ±(f L + f U ) = f L + f U . Now, as the f L , f U are in the set j=0 Γ ( j ) and, on the other hand, f U , f L are elements of n j=[ n 2 ]+1 Γ ( j ), then it is possible to assert that f U = ± f L and f L = ± f U . Hence, as The image of Γ L is not a subalgebra of Γ ( ) when it is equipped with the Graf product. Hence, it motivates the definition of the truncated Graf product ± : Γ ( ) → Γ L : f ± g = 2P L (P ± (f ) P ± (g)), ∀f, g ∈ Γ ( ), (3.12) which implies that P ± (f ± g) = P ± (f ) P ± (g). Hence, when n is odd and p − q ≡ 8 0, 1, 4, 5, the Eq. (3.12) implies that f ± g = 2P L (P ± (f g)) for all f, g ∈ Γ L , and consequently P ± (f ± g) = P ± (f g), i. e., P ± is a endomorphism. The isomorphism between the subalgebras (Γ L , ± ) and (Γ ± , ) was established in Ref. [24]. Thus, when n is odd and p − q ≡ 8 0, 1, 4, 5, the truncated subalgebra (Γ L , ± ) will be modeled by the subalgebra (Γ ± , ).

Pinors, spinors and Fierz identities
A way to approach pinors and spinors consists of respectively considering elements that carry the irreducible representation in C p,q or C + p,q , respectively [1]. In fact, unless n equals 3 or 7 modulo 8, the algebra C n is a real R, a complex (C), or a quaternionic (H) matrix algebra. Hence, it has a unique irreducible representation, known as the space of pinors.
Since the group Pin(n) is embedded into the algebra C n , the irreducible representations of C n then restrict to representations of Pin(n). Similarly, Spin(n) is embedded into the even subalgebra C + n of C n . Thus the irreducible representations of C + n restrict to irreducible representations of Spin(n), called spinors [1]. The Clifford algebra representations, coming from the Atiyah-Bott-Shapiro theorem, is comprised below [1].
Classification of the real Clifford algebras, for p + q = n.
The Clifford bundle over the cotangent bundle C (T * M) → M is defined by the disjoint union as follows whereas, the even Clifford subbundle C + (T * M) → M, whose fibers are even Clifford algebras, is defined as follows The pin bundle π p : P → M is the bundle whose fibers are the irreducible representation spaces of the fibers End P x , the pin bundle is given by: Thus, the algebra bundle morphism can be considered where End P is the bundle whose fibers are End P x for all x ∈ U ⊂ M. A section α : M → P is named a pinor field.
On the other hand, the spin bundle π s : S → M is the bundle constituted by the irreducible representation spaces of This means that the bundle S is given as follows where each S x comes from the mapping λ + Naturally, a spinor field is defined as being a section of S. Analogously to the pinor bundle case, the following algebra bundle morphism can be defined: The mappings λ x and λ + x are the pinor and spinor representations already defined. For Once the pin and spin bundles are established, consider the following mappings regarding the Graf-Clifford algebra: where "•" denotes the natural product of endomorphisms. Besides, the morphism Observe that the mapping λ Γ + is the application λ Γ restricted to Γ + ( ), then there exist an identification between Γ (P + ) (when the bundle P is decomposable) and Γ (S). In literature [23], the sections of P ± are named Majorana-Weyl spinors when p − q ≡ 8 0, sections of P ± are called symplectic Majorana-Weyl spinors when p − q ≡ 8 4, the sections of P + are named symplectic Majorana spinors when p − q ≡ 8 6 and sections of P + are called Majorana spinors when p − q ≡ 8 7.
Note that the image Im λ x is whithin the algebra End P x . By a direct consequence of the Schur's lemma, the algebra End P x is a division algebra, since λ x is irreducible.
The real algebra (End P x , •) is associative. Thus, by the Frobenius theorem for division algebras, each End P x is isomorphic either to R, C or H. Thereat, define the subset (4.10) Since End P x is associative, thus A x is a subalgebra. Since the centralizer is a division algebra, then A x is a division algebra, as End P x . Namely, A x is isomorphic to R, C or H. This subalgebra A x is very important to analyze the geometric Fierz identities into three different cases, according to the so called isomorphism type of A x . Therefore, this subalgebra will be called hereon the main subalgebra. With the results by Okubo [30], it is possible to conclude that each element f x ∈ A x can be written as: 1) The normal case: where I k R is the identity matrix of order k R × k R . This means that A x ∼ = R. This case occurs when p − q ≡ 8 0, 1, 2 and k R = 2 [ n 2 ] .
2) The almost complex case: 3) The quaternionic case: In this case, A x ∼ = H and it occurs with the condition that p − q ≡ 8 4, 5, 6 for k H = 2 [ n 2 ]−1 .
When A x ∼ = R (p−q ≡ 8 0, 1, 2), no additional structure is needed, since λ Γ is invertible [5,30]. If A x ∼ = C (p−q ≡ 8 3, 7) there exists an endomorphism J ∈ Cen End(Γ (P)) (λ Γ (Γ ( )) such that J 2 = −Id Γ (P) , in addition there is another endomorphism D ∈ End(Γ (P)) so ). An example of this sort of structure is J := ±λ Γ (v). In fact, If A x ∼ = H (p − q ≡ 8 4, 5, 6), the structure in End(Γ (P)) is given by H 0 = Id Γ (P) , H 1 , For each one of the isomorphism type of A x ∼ = F and α, β ∈ Γ (P), let E F α,β be the endomorphism E α,β on the respective division algebra F. The endomorphism E F α,β can be locally written as [23] for the normal case, for the almost complex case and for the quaternionic case, where I is an ordered index set and B is an admissible bilinear mapping.
As the inverse of λ Γ was defined into the three cases, consider the following forms in Locally, for an index set I k , they are respectively expressed as: The Fierz identities come from Eq. (4.19), being written according to each F: where X = E, E or E. Therefore, using the components defined in Eq. (4.29a), the Fierz identities in the normal case read [23,31]: Besides, in the almost complex case we have whereas the quaternionic case, for i = 1, 2, 3, reads

Spinor classes on signature (1, 2)
New classes of spinors have been found under the formalism of the Graf product and the Fierz identities, for signatures (7, 0), (6, 1), and (1, 4), respectively in Refs. [14][15][16]. In this section we will find three non-trivial new classes of spinor fields on 3-dimensional pseudo-Riemannian manifolds with signature (1, 2), employing this method. One of the main motivations to explore new classes of spinor fields in such dimension and signature is the existence of anyons in quantum field theory, interpolating between Fermi-Dirac and Bose-Einstein statistics [17]. Besides the formal aspects, one can find a vast variety of applications regarding (1, 2) spinor fields, playing a prominent role on the stage of condensed matter physics. Fermions quantum fields constructed upon these spinors can describe superconductors [18] and semimetals, with particular attention to the graphene [19][20][21].
In the next section, new classes of pinors on spaces of signature (9, 0) shall be explored.

Concluding remarks and outlook
In this work we implemented a systematic investigation of the normal, the almost complex, and the quaternionic cases for the geometric Fierz identities, in the context of the Clifford bundles. Using the geometric algebra approach of the Graf-Clifford, geometric Fierz identities were employed to reconstruct the form-valued spinor and pinor bilinear covariants in certain dimensions and signatures. New classes of spinors were constructed from the geometric Fierz identities for manifolds of signature (1,2) and new classes of pinors have been derived, in spaces of signature (9, 0), besides constructive formulae to iteratively implement these constructions. As in Ref. [23] the Kähler-Atiyah algebra was constructed on metric cones and cylinders on (9, 0), the lifting of generalized Killing equations and the geometric Fierz isomorphisms can also be used to study 8-manifolds. Beyond the scope of this paper, this last construction has a prominent link to the octonionic spinors and their classification in the context of the Lounesto's one [33][34][35], and may be valuable for finding other solutions in string theory. In fact, the new mass dimension one spinors found in Ref. [36] may be employed to construct pure spinor superstring ghosts in a curved heterotic background [37]. Acknowledgments R.L. is grateful to CAPES and R.dR. is grateful to CNPq (Grant No. 303293/2015-2), and to FAPESP (Grant No. 2017/18897-8), for partial financial support.

A Useful identities
Eqs. (6.13a -6.13j) were disposed to construct new classes of pinors on signature (9, 0). Here we provide a detailed proof for each one of them. The notation and definitions ar the same thereon.