Julio Marny Hoff da Silva Aspectos Físicos e Algébricos de Espinores Escuros Guaratinguetá 2016 Julio Marny Hoff da Silva Aspectos Físicos e Algébricos de Espinores Escuros Texto Sistemático Crítico Apresen- tado à Faculdade de Engenharia da Universidade Estadual Paulista “Jú- lio de Mesquita Filho”- Câmpus de Guaratinguetá para o Concurso de Livre-Docência. Guaratinguetá, setembro de 2016 2016 Hoff da Silva, Julio Marny Aspectos Físicos e Algébricos de Espinores Escuros XXXXX páginas Texto Sistemático Crítico - Faculdade de Engenha- ria - Campus de Guaratinguetá - Universidade Estadual Paulista “Júlio de Mesquita Filho”. Departamento de Física e Química. I. Universidade Estadual Paulista “Júlio de Mesquita Filho”. Faculdade de Engenharia - Campus de Guaratin- guetá. Departamento de Física e Química. Dedicatória Para Ane, pois estaremos sempre sob a mesma lua... Agradecimentos Gostaria de expressar minha sincera gratidão • aos colegas e funcionários do Departamento de Física e Química, cuja cola- boração e aprendizado são de grande valia; • em particular aos colaboradores do grupo de partículas e campos, por toda a possibilidade de aprendizado que me proporcionam; • ao amigo Roldão da Rocha pelo exemplo de resiliência; • ao professor Dharam Vir Ahluwalia; • ao CNPq; • aos diversos estudantes (de graduação, pós-graduação, e orientados) cujos questionamentos e vontade de aprender nos impelem sempre em frente; • aos contribuintes do Estado de São Paulo. Julio Marny Hoff da Silva Texto Sistemático Crítico Apresentado à Faculdade de engenharia da Universidade Estadual Paulista - Campus de Guaratinguetá para o Concurso de Livre-Docência Guaratinguetá, setembro de 2016 . Texto Sistemático Sobre Nossas Contribuições aos "Aspectos Físicos e Algébricos de Espinores Escuros" Elaborado por Julio Marny Hoff da Silva APRESENTAÇÃO Optei pela apresentação de uma análise crítica de alguns trabalhos levados a termo após o ingresso como Docente do Departamento de Física e Química da Faculdade de Engenharia de Guaratinguetá - UNESP, em subsituição à tradicional tese de Livre-Docência. Essa opção na sistemática de apresentação se coaduna com um duplo as- pecto: apresentar criticamente as contribuições realizadas na área e consolidar perspectivas de continuidade do trabalho exposto. Sumário 1 Introdução 1 2 Aspectos Formais 7 2.1 Espinores Exóticos . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Álgebra Espinorial . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3 Física dos Espinores Escuros 67 3.1 Sinais no LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Radiação Hawking com espinores Elko . . . . . . . . . . . . . . . 93 4 Cosmologia com Espinores Elko 99 4.1 Aspectos Cosmológicos . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2 Modelo Sigma não-linear para Espinores Escuros . . . . . . . . . . 134 5 Considerações Finais 143 vii Capítulo 1 Introdução Nada menos propositado, talvez, do que uma citação no intróito de um texto formal. Havendo ainda o agravo do texto pretender conter uma parcela de crítica, a boa norma de uso do cálamo sempre nos remete a uma certa dose de parcimônia. Ainda assim, dada a relevância do autor da citação, comecemos nossa introdução apreciando a opinião de Sir Michael Atiyah no que concerne aos objetos centrais dessa tese: No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of -1 took centuries, the same might be true of spinors. Em certo sentido, abordaremos ao longo dos trabalhos explicitados nesse texto diversas facetas do entendimento corrente de espinores do ponto de vista mate- mático e de sua relação com a física, especialmente no que tange ao estudo de candidatos à matéria escura. A fim de encerrar nessa introdução um contexto 1 Capítulo 1. Introdução 2 explicativo do que se segue, recorremos novamente à frase supracitada, desta vez ponto a ponto. No one fully understand spinors. O começo desalentador da afirmação guarda também ampla perspectiva de tra- balho. Em física, em linhas gerais, espinores foram introduzidos na Mecânica Quântica como certas funções de onda descrevendo, via acoplamentos específivos no termo potencial da equação de Schrödinger, um grau de liberdade interno, o spin. A evolução conceitual que marca a incorporação da dinâmica relativística e o conceito de campo à Mecânica Quântica, culminando com a Teoria Quântica de Campos, levou também a refinamentos do conceito de espinor em física. O operador de campo quântico espinorial utilizado na descrição do setor de maté- ria do Modelo Padrão carrega consigo coeficientes de expansão espinoriais cuja propriedade definidora flerta com um viés matemático bem estabelecido para espinores: entidades que carregam a representação dos grupos de rotação em um espaço de dimensão finita. No caso específico dos coeficentes de expansão, estamos nos referindo às rotações que compõe o grupo de Lorentz. Na década de sessenta do século precedente, para nos atermos a um deter- minado recorte histórico, a Teoria Quântica de Campos passou por ampla for- malização, colocando suas bases teóricas em um patamar sólido. Entretanto, no que diz respeito ao uso de espinores, uma possível fenda investigativa se abre se os utilizamos como objetos que carregam representações de certos subgrupos do grupo de Lorentz, tais como os que erigem a chamada Very Special Relativity. É nesse sentido que contextualizamos essa primeira parte da citação, fazendo-lhe coro: a descrição de campos fermiônicos cujos coeficientes de expansão apresen- 3 tam modificações sutis, porém importantes, dos conceitos usuais, tem levado a um amplo campo de novas perspectivas. Tais novos campos fermiônicos apresen- tam (ou são construídos para que apresentem) propriedades que lhes faculta sua aplicação na descrição da matéria escura. No trajeto formal de desenvolvimento da teoria de campos incorporando esses novos campos há, acreditamos, muito a ser compreendido e formalizado. Their algebra is formally understood but their general significance is mysterious. Do ponto de vista algébrico, há uma classificação espinorial com forte apelo físico. Sabemos que em física de altas energias, um férmion sozinho (descrito por um espi- nor) não é passível de detecção. Estados acessíveis experimentalmente são aqueles formados pelos bilineares covariantes. O entendimento concreto de espinores como elementos (fibras) de um fibrado principal contendo transformações de 𝑆𝐿(2,C) e com características bem definidas do espaço de representação, possibilitou a execução de um programa algébrico profundo na década de oitenta, levando à categorização sistemática dos espinores de acordo com os valores assumidos pelos seus bilineares. Sem dúvida, no que é concernente a esse trabalho, podemos endossar a asserção de que a álgebra espinorial é bem entendida. No entanto, ainda há espaço para mistério: essa mesma formalização algébrica revela a existência de uma classe de espinores cuja contrapartida física ainda não foi completamente explorada. De fato, e vamos nos remeter a esse ponto em momento oportuno nessa tese, há até então apenas um sistema físico conhecido cujo setor espinorial recai sobre essa classe fugidia. Capítulo 1. Introdução 4 In some sense they describe the “square root” of geometry · · · Um programa investigativo levado a termo por Eliè Cartan mostra uma faceta bastante intrigante relacionada aos espinores. De fato, espinores podem ser enten- didos como objetos pré-geométricos, no sentido de que pontos do espaço-tempo podem ser escritos como composições de espinores. Nesse contexto eles seriam a “raíz quadrada da geometria”, objetos mais fundamentais do que pontos que compõe o espaço pseudo-euclidiano. Mais do que curioso, tal resultado permite uma abordagem interessante para a existência dos chamados espinores exóticos. A existência de espinores exóticos está vinculada à não trivialidade da vari- edade de base na qual a teoria toma forma. Por exemplo, a dinâmica espinorial em um espaço de Minkowski não simplesmente conexo abre a possibilidade de existência de outras estruturas espinoriais. Tais estruturas possuem dinâmica es- sencialmente usual, porém com uma correção de origem topológica, mas que pode ser entendida como um acoplamento adicional com um campo vetorial externo. É precisamente aqui que entra novamente a relevância dos espinores escuros. Uma propriedade essencial para um candidato a matéria escura é a impossibilidade de acoplamento com campos usuais do Modelo Padrão. Logo, diferentemente do que acontece com espinores usuais, o estudo de espinores escuros exóticos traz informação genuína sobre a topologia do sistema. Para que possamos abordar com propriedade o que foi dito nessa curta ex- posição, separamos nossas contribuições em duas frentes de trabalho, a saber: uma voltada às propriedades algébricas dos espinores escuros, e outra enfocando o entendimento das propriedades físicas dos mesmos. Nesse último caso, fazemos ainda outra ramificação separando nossas contribuições em duas subáreas: teoria 5 geral de partículas e campos e cosmologia. Percorreremos essa trajetória em dez trabalhos que tipificam a atuação e contribuição que construímos (e continuamos a construir) nessa área. Ainda que uma tal divisão seja imperfeita, pois vários dos trabalhos apresentados transitam entre essas diversas classificações, ela será útil na categorização geral. O texto está organizado da seguinte maneira: no Capítulo 1 expomos nossas contribuições acerca de espinores escuros e exóticos com mais foco nos aspectos algébricos para o estabelecimento dos mesmos. Essa exposição será composta de quatro trabalhos cujo mote central é formal. O Capítulo 2 será destinado a contribuições voltadas a teoria de partículas. Serão três trabalhos, com propostas de estudos específicos de sinais em aceleradores e uma aplicação, com o cálculo da radiação Hawking. O Capítulo 3 fica ao encargo de estudos de alguns efeitos dos espinores escuros em Cosmologia. Veremos também três trabalhos, sendo dois eminentemente de cunho cosmológico e outro onde propomos um modelo sigma para tais férmions, estudando alguma aplicação em cosmologia. Uma vez que parte desse texto pretende ter um aspecto crítico, reservamos o capítulo final para tal fim. Capítulo 1. Introdução 6 Capítulo 2 Aspectos Formais Em uma perspectiva bastante abarcativa e ambiciosa de trabalho, o estudo desde a álgebra espinorial até a extração de algum possível observável físico se mostra robusto. Gostaríamos de iniciar esse programa nos remetendo neste Capítulo à fundamentação algébrica do nosso estudo. Aqui veremos aspectos relevantes de espinores exóticos escuros, bem como a apreciação de um sistema físico levando a um tipo de espinor jamais utilizado. Finalizaremos estudando novos possíveis espinores escuros do ponto de vista algébrico. Cada um desses tópicos será exposto em uma seção deste capítulo, que contará sempre com um texto introdutório discutindo aspectos de relevância do trabalho. Começamos com o que podemos entender como uma introdução às demais seções: um trabalho curto contendo uma descrição de vários temas a que este capítulo diz respeito. 7 Physics Letters B 718 (2013) 1519–1523 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Unfolding physics from the algebraic classification of spinor fields J.M. Hoff da Silva a,∗, Roldão da Rocha b a Departamento de Física e Química, Universidade Estadual Paulista, Av. Dr. Ariberto Pereira da Cunha, 333, Guaratinguetá, SP, Brazil b Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09210-170, Santo André, SP, Brazil a r t i c l e i n f o a b s t r a c t Article history: Received 11 September 2012 Received in revised form 10 December 2012 Accepted 11 December 2012 Available online 19 December 2012 Editor: S. Dodelson After reviewing the Lounesto spinor field classification, according to the bilinear covariants associated to a spinor field, we call attention and unravel some prominent features involving unexpected properties about spinor fields under such classification. In particular, we pithily focus on the new aspects — as well as current concrete possibilities. They mainly arise when we deal with some non-standard spinor fields concerning, in particular, their applications in physics. © 2012 Elsevier B.V. All rights reserved. 1. Introduction From the classical point of view, the definition of spinors is based upon irreducible representations of the group Spin+(p,q), where p + q = n is the spacetime dimension. Due to the imme- diate physical interest, mainly the Minkowski spacetime R1,3 has being regarded since the 1920s. On the another hand, the repre- sentation space associated to an irreducible regular representation in a Clifford algebra is a minimal left ideal. Its elements are the so-called algebraic spinors. Another possible definition of a spinor, which is denominated operatorial, can be introduced from another representation — distinct of the regular representation — of a Clif- ford algebra, using the representation space associated to the even subalgebra. This definition is equivalent to the classical and alge- braic ones, in particular in the cases of great interest for physical applications. The classical definition of spinor is the customary approach in several superb textbooks in physics, e.g., [1]. There is no damage in asserting that, in Minkowski spacetime, classi- cal spinors are irreducible representations of the Lorentz group Spin+(1,3) � SL(2,C). Notwithstanding, this paradigm severely restricts the analysis to the usual Dirac, Weyl, and Majorana spinors. A new possibility involving the spinor fields classification was introduced by Lounesto [2], as a palpable paradigm shift. It is based upon the bilinear covariants and their underlying multivec- tor structure. In particular, this classification evinces the existence of a new type of spinor field, the so-called flag-dipole spinor fields. Furthermore, it additionally presents another class of spinor fields * Corresponding author. E-mail addresses: hoff@feg.unesp.br, hoff@ift.unesp.br (J.M. Hoff da Silva), roldao.rocha@ufabc.edu.br (R. da Rocha). (the flagpoles) that accommodates Elko spinor fields, which are prime candidates to the dark matter description [3]. They gener- alize Majorana spinor fields. As it is well known, any spin-half spinor field, that potentially describes the dark matter, respects the symmetries of the Poincaré group in the sense of Weinberg, if it is an element of a standard Wigner class of representations of the Poincaré group. As it will be reported, Elko spinor fields do not belong to the standard Wigner class. Among a significant amount of unexpected and interesting properties, it was recently demonstrated that the topological exotic spacetime structure can be probed uniquely by Elko spinor fields: they are, hence, suitable to investigate the eventual non-trivial topology of the universe [4]. By such exoticness, dynamical constraints converted into a dark spinor mass generation mechanism, with the encrypted VSR sym- metries holding as well. The aim of this work is to report some of the recent advances in this field of research, calling special attention to the interest- ing features associated to the new spinor fields appearing in the Lounesto’s classification. In order to accomplish that, we organize this work as follows: in the next section we review the formal and necessary aspects regarding the Lounesto spinor classification. In Section 3, we explore some of the odd and captivating aspects as- sociated to Elko and flag-dipole spinor fields. In the final section we conclude. 2. Classifying spinor fields We start this section reviewing some indispensable preliminary concepts. For a deeper approach see, e.g., [5]. Consider the tensor algebra T (V ) = ⊕∞ i=0 T i(V ), where V is a finite n-dimensional real vector space. Henceforth V is regarded as being the tangent space on a point on a manifold. Let Λk(V ) denote the antisymmetric k-tensors space, indeed the k-forms vector space. In this way 0370-2693/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2012.12.026 8 1520 J.M. Hoff da Silva, R. da Rocha / Physics Letters B 718 (2013) 1519–1523 Λ(V ) = ⊕n k=0 Λk(V ) is the space of the differential forms over V . For any ψ ∈ Λ(V ), the reversion is defined by ψ̃ = (−1)[k/2]ψ (the integer part of m is denoted by [m]), which is an anti- automorphism in Λ(V ). Moreover, ψ̂ = (−1)kψ denotes the graded involution, also called main automorphism. It is possi- ble to use the metric g : V ∗ × V ∗ → R extended to the k-forms space, in order to define the left and right contractions. Hence, for ψ = ∧p i=1 ui ≡ u1 ∧ · · · ∧ up and φ = ∧r j=1 vr , with ui,v j ∈ V ∗ , the extension of g to Λ(V ) reads g(ψ,φ) = det(g(ui,v j)) for p = r, and zero otherwise. Now one defines the left contraction by g(ψ � ϕ,χ) = g(ϕ, ψ̃ ∧ χ), for ψ,ϕ,χ ∈ Λ(V ). (1) For v ∈ V , the Leibniz rule for the contraction is v � (ψ ∧ ϕ) = (v � ψ) ∧ ϕ + ψ̂ ∧ (v � ϕ) (2) respectively. The Clifford product between v ∈ V and χ ∈ Λ(V ) is vχ = w ∧ χ + v � χ and the pair (Λ(V ), g), endowed with the Clifford product, is denoted by Cl(V , g) (Clp,q is a notation that shall be reserved to the Clifford algebra when V � Rp,q). In order to properly revisit the bilinear covariants let us fix the gamma matrices notation. All the formalism in representa- tion independent, and hence we use hereon the Weyl (or chiral) representation of γ μ: γ0 = γ 0 = ( O I I O ) , γk = −γ k = ( O σk −σk O ) , where I = ( 1 0 0 1 ) , O = ( 0 0 0 0 ) and the σi are the Pauli matrices. Moreover γ 5 = iγ 0γ 1γ 2γ 3. All the spinor fields in this work are placed in the Minkowski spacetime (M � R1,3, η, D, τ ,↑), where η = diag(1,−1,−1,−1) is a metric which has a compatible (Levi- Civita) connection D associated. Besides, M has spacetime orienta- tion induced by the volume element τ as well as time orientation denoted by ↑. We denote by {xμ} global coordinates, in terms of which an inertial frame — a section of the frame bundle PSO1,3(M) — reads eμ = ∂/∂xμ . At this point we recall that classical spinor fields are sections of the vector bundle PSpin1,3 ×C2, where the specific representation of SL(2,C) � Spin1,3 in C2 is implicit. In this framework, the bilinear covariants associated to a spinor field ψ ∈ PSpin1,3 ×C2 are sections of Λ(T M) into the Clifford bundle of multiform fields, given by σ = ψ†γ0ψ, J = Jμθμ = ψ†γ0γμψθμ, S = Sμνθμν = 1 2 ψ†γ0iγμνψθμ ∧ θν, K = Kμθμ = ψ†γ0iγ0123γμψθμ, ω = −ψ†γ0γ0123ψ, (3) where {θμ} is the dual basis of {eμ}. The bilinear covariants obey quadratic equations, the so-called Fierz–Pauli–Kofink identities [2] J � K = 0, J2 = ω2 + σ 2, J ∧ K = −(ω + σγ0123)S, K2 = −J2, (4) which are particularly interesting in what follows. The Fierz aggre- gate Z is defined by Z = σ + J + iS − iγ0123K + γ0123ω. (5) Eqs. (3) may be recast in terms of Z , yielding Z 2 = 4σ Z , Zγμ Z = 4 Jμ Z , Z iγμν Z = 4Sμν Z , Zγ0123 Z = −4ωZ , Z iγ0123γμ Z = 4Kμ Z . (6) Therefore, it is possible to categorize different spinor fields by different Z ’s, or similarly by distinct bilinear covariants. The Lounesto spinor field classification provides the following spinor field classes [2]: 1) σ = 0, ω = 0; 4) σ = 0 = ω, K = 0, S = 0; 2) σ = 0, ω = 0; 5) σ = 0 = ω, K = 0, S = 0; 3) σ = 0, ω = 0; 6) σ = 0 = ω, K = 0, S = 0. The first three classes are composed by Dirac spinor fields and it is implicit that in this case J,K,S = 0. In particular, for a Dirac spinor fields describing an electron, J is a future-oriented timelike current vector providing the current of probability; S is the distri- bution of intrinsic angular momentum, and the spacelike vector K is associated to the direction of the electron spin. A Majorana spinor field belongs to the class (5), while Weyl spinor fields are in the class (6). Type-(4) spinor fields are the so- called flag-dipole spinor fields. Furthermore, if ψ is a typical Dirac spinor field and ζ is an arbitrary spinor field such that ζ †γ0 = 0, ψ is herewith proportional to Zζ , where Z is given by Eq. (5). Before delving deeper into the investigation of some interest- ing outputs in this approach, let us first emphasize that there are no other possible classes for the spinor fields based on dif- ferent bilinear covariants. In fact, when σ = 0 and/or ω = 0, it implies that S = 0 and K = 0 — note that J 0 > 0 and hence J does not equal zero. Besides, the constraint ω = 0 = σ implies that Z = J(1 + i(s + hγ0123)), where (s + hγ0123) 2 = −1, s is a space- like vector, and h a real number given by h = ±√ 1 + s2. In this vein J(s + hγ0123) = S + Kγ0123. It is useful to provide further fea- tures of type-(4) spinor fields. For flag-dipole spinor fields, Eq. (5) gives Z = J + i J s − ihγ0123 J, where s = ‖s‖. It implies forthwith that (1 + is − ihγ0123)Z = 0, and taking into account that J2 = 0 for type-(4) spinor fields, Z is shown to be Clifford multivector satis- fying Z 2 = 0. Such spinor fields were widely investigated in [15] in a more topological geometric context, as well as some interesting applications. The bilinear covariant S in (3) is given by S = J ∧ s. For type-(4) spinor fields the real coefficient satisfies h = 0. Lounesto shows that either J2 = 0 or (s − ihγ0123) 2 = −1. The helicity h relates K and J by K = h J. The definition of helicity h in terms of bi- linear covariants precedes and implies the definition of helicity in quantum mechanics, as well the equivalent relation for anti- particles [6]. Such approach further prov ides a straightforward form for the Hamiltonian describing the one-layer superconductor graphene, given by Tr(γ 5Kγ 0) [6]. 3. Peculiar features Roughly speaking, the framework of Lounesto’s classification al- lows a twofold approach: on the one hand it is possible to study and classify new spinor fields recently discovered in the literature. Moreover, their geometric content can be explored and it sheds new light in the investigation on their physical content. We shall deal with this aspect in the following two subsections. On the an- other hand, it permits the exploration of genuinely different spinor fields, without any physical counterpart. We delve into this issue in the third subsection. 3.1. Elko spinor fields and its properties Imagine a mass dimension one spinor field with 1/2 spin, obeying the Klein–Gordon, but not the Dirac field equations. En- dowed with such predicates, it is indeed possible to call that spinor field as strange. In what follows, however, we shall argue that the strangeness of such spinor, the so-called Elko spinor, is far from pejorative. 9 J.M. Hoff da Silva, R. da Rocha / Physics Letters B 718 (2013) 1519–1523 1521 Elko spinor fields are eigenspinors of the charge conjuga- tion operator with eigenvalues ±1. The plus [minus] sign stands for self-conjugate [anti-self-conjugate] spinors λS (p) [λA(p)]. Elko spinor fields arise from the equation of helicity (σ · p̂)φ±(0) = ±φ±(0) [3]. The four spinor fields are given by λ S/A {∓,±}(p) = √ E + m 2m ( 1 ∓ p E + m ) λ S/A {∓,±}(0), (7) where λ S/A {∓,±}(0) = ( ±iΘ[φ±(0)]∗ φ±(0) ) . The operator Θ denotes the Wigner’s spin-1/2 time reversal operator. As Θ[φ±(0)]∗ and φ±(0) present opposite helicities, Elko cannot be an eigenspinor field of the helicity operator, and indeed carries both helicities. In order to guarantee an invariant real norm, as well as positive definite norm for two Elko spinor fields, and negative definite norm for the other two, the Elko dual is given by [3] ¬ λ S/A {∓,±}(p) = ±i [ λ S/A {±,∓}(p) ]† γ 0. (8) It is useful to choose iΘ = σ2, as in [3], in such a way that it is possible to express λ(p) = ( σ2φ∗ L (p) φL(p) ) . The dual is defined in such way that the product (λ S/A {∓,±})†ζλ S/A {±,∓} remains invariant un- der Lorentz transformations. This requirement implies ζ = ±iγ 0 for the Elko case, since it belongs to the right ⊕ left representation space [7]. Endowed with a new dual, Elko respects different or- thonormality relations, which engenders non-standard spin sums. Following this reasoning it is possible to envisage the Elko non- locality (see [7] for the details). Denoting by Λ(x, t) the quantum field constructed out of Elko spinor fields as the expansion coeffi- cients and Π(x, t) its conjugate momentum, although the follow- ing property { Λ(x, t),Λ ( x′, t )} = 0 = { Π(x, t),Π ( x′, t )} (9) holds, an unexpected anti-commutation relation is elicited [3]: { Λ(x, t),Π ( x′, t )} = i ∫ d3 p (2π)2 1 2m eip·(x−x′)2m [ 1 + G(p) ] . (10) Here 1 stands for the identity matrix and G(p) = γ 5γμnμ is a factor arising from the spin sums. The vector nμ = (0,n) defines some preferential direction [3], where n = 1 sin θ dp̂ dφ . It was recently demonstrated [9], by explicitly calculation, that the integration over the second term of Eq, (10) equals zero. This is a crucial point, since this term decides the locality structure of the quan- tum field. The mass dimension one related to such spinor fields severely suppresses the possible couplings to other fields of the standard model. In fact, if we keep in mind power counting arguments, Elko spinor fields may interact — in a perturbative renormalizable way — with itself and with a scalar (Higgs) field. Obviously, the former type of interaction means an unsuppressed quartic self interaction. At this point it is important to remark that this feature (quartic self interaction) is present in the dark matter characteristics ob- servations [10]. Therefore Elko spinor fields seems to perform an adequate fermionic dark matter candidate. It is worth notice that the appearance of the G(p) function in the spin sums, however, shall not be underestimated. Its presence turns out to be impossible to conciliate Elko quantum field to the full Lorentz group. Nevertheless, Elko fields are, in fact, a spinor representation under the SIM(2) avatar [11] of Very Special Rela- tivity (VSR) [12]. The group SIM(2) is the largest possible subgroup of VSR which is necessary to define a quantum theory when parity symmetry is violated. Hence, understanding Elko as a dark matter prime candidate, it may signalize that in the dark matter sector the Lorentz group may not be the underlying relevant group. Indeed, using the Lounesto framework previously outlined, Elko are classi- fied as type-(5) spinor fields, a generalization of Majorana spinor fields carrying both helicities [13]. As mentioned in the Introduc- tion, Lounesto classification goes beyond the standard classification by irreducible representations of the Lorentz group Spin+(1,3). From this perspective, it is quite conceivable that the quantum fields, constructed out from expansion coefficients which do not belong to Lorentz representation, do not respect Lorentz symme- tries themselves. 3.2. The usefulness of topologically exotic terms Among an extended inventory of relevant new physical possi- bilities arising from the use of the non-standard spinor fields, we can branch the role of Elko spinor fields as a detector of exotic spacetime structures [4]. If the base manifold M upon which the theory is built is simply connected, then the first homotopy group π1(M) is well known to be trivial. In this case, supposing that M satisfies the assumptions in the Geroch theorem [14], there exists merely one possible spin structure. Consequently, the spin-Dirac operator in the formalism is the standard one. Notwithstanding, when non-trivial topologies on M are regarded, there is a non- trivial line bundle on M . The set of line bundles and the set of inequivalent spin structures are labeled by elements of the co- homology group H1(M,Z2) — the group of the homomorphisms of π1(M) into Z2. In this regard, there are several globally dif- ferent spin structures arising from the different (and inequivalent) patches of the local coverings. The spin-Dirac operator has in this case an additional term, essentially a one-form field, that reflects the non-trivial topology. Spinor fields associated to these inequiv- alent spin structures are called exotic spinor fields. Let us make those considerations more precise. Throughout this section we denote by Spin1,3 and SO1,3 the components of such groups connected to the identity, for the sake of concise- ness. Given the fundamental map, in fact a two-fold covering re- lating the orthonormal coframe bundle and the spinor bundle1 s : PSpin1,3(M) → PSO1,3 (M), a spin structure on M is a princi- pal fiber bundle πs : PSpin1,3(M) → M satisfying: (i) π(s(p)) = πs(p) for every point p of PSpin1,3 (M), where π is the pro- jection of PSO+(1,3)(M) on M , and (ii) s(pφ) = s(p)Adφ . Here given φ ∈ Spin1,3(M), we have Adφ(κ) = φκφ−1, for all κ ∈ Cl1,3. A spin structure P := (PSpin1,3(M), s) exists solely when the second Stiefel–Whitney class satisfies specific criteria. To our presentation, however, it is remarkable that if H1(M,Z2) is not trivial, then the spin structure is not uniquely defined. Two spin structures, say P and P̃ , are said to be equivalent if there exists a map χ : P → P̃ compatible with s and s̃; they are said to be inequivalent oth- erwise. Given an arbitrary spinor field ψ ∈ sec PSpin1,3 (M) × C4, where “sec” means “section of”, to each element of the non-trivial H1(M,Z2) one can associate a Dirac operator ∇ . This construc- tion provides an one-to-one correspondence between elements of H1(M,Z2) and inequivalent spin structures (for more details see [8,4,14]). A crucial difference between the exotic and the standard spinor field is the action of the Dirac operator on exotic spinor fields. In a non-trivial topology scenario, the Dirac operator changes by an ad- ditional one-form field, which is a manifestation of the non-trivial 1 Let PSO1,3 (M) denote the orthonormal coframe bundle, that always exist on spin manifolds. Sections of PSO1,3 (M) are orthonormal coframes, and sections of PSpin1,3 (M) are also orthonormal coframes such that although two coframes dif- fering by a 2π rotation are distinct, two coframes differing by a 4π rotation are identified. 10 1522 J.M. Hoff da Silva, R. da Rocha / Physics Letters B 718 (2013) 1519–1523 topology. The exotic structure endows the Dirac operator with an additional term given by a−1(x)da(x), where x ∈ M and d denotes the exterior derivative operator. The term 1 2iπ a−1(x)da(x) is real, closed, and defines an integer Cěch cohomology class [16]. Using the relation between the Cěch and the de Rham cohomologies, it follows that∮ 1 2iπ a−1(x)da(x) ∈ Z. (11) When Dirac spinor fields are regarded, the exotic term can be ab- sorbed into a new shifted potential A �→ A + 1 2iπ a−1(x)da(x): the exotic term may be understood as an external electromagnetic po- tential that is summed to the physical electromagnetic potential, which plays the role of a disguise for the exotic term. In this way the exotic spacetime structures cannot be probed by Dirac spinor fields, which perceive the exotic term as an effective electromag- netic potential. From the perspective of Elko spinor fields, however, the situ- ation changes drastically. The reason is that the spinor field dis- cussed in the previous section is an eigenspinor of the charge conjugation operator. Therefore it does not carry local U (1) charge of the standard type. Hence, any type of extra term present in the Dirac operator cannot be absorbed into the electromagnetic poten- tial. As it is extensively discussed in [14], the exotic term may be expressed as a(x)√ 2π = exp (iθ(x)) ∈ U (1). It yields 1 2π a−1(x)da(x) = exp (−iθ(x) )( iγ μ∇μθ(x) ) exp ( iθ(x) ) = iγ μ∂μθ(x). (12) Now, making the conceivable exigency that the exotic Dirac op- erator must be considered the square root of the Klein–Gordon operator, we have2 [ iγ μ(∇μ + ∂μθ) ± m ][ iγ ν(∇ν + ∂νθ) ∓ m ] λ = ( gμν∇μ∇ν + m2)λ = 0. (13) Therefore, the corresponding Klein–Gordon equation for the exotic Elko spinor field reads (� + m2 + gμν∇μ∇νθ + ∂μθ∇μ + ∂μθ∂μθ ) λ = 0. (14) Finally, in order to have the Klein–Gordon propagator for the exotic Elko, as in the standard one, it follows from Eq. (14) that (�θ(x) + ∂μθ(x)∇μ + ∂μθ(x)∂μθ(x) ) λ = 0. (15) The result encoded in Eq. (15) makes Elko spinor field a very use- ful tool to explore unusual topologies in many contexts. Indeed Eq. (15) asserts that the Elko spinor structure constrains the exotic term related to the non-trivial spacetime topology. The possibil- ity of extracting information from the subjacent topology without using any additional (sometimes ill defined) shifted potentials is, in fact, quite attractive. Eq. (15) further encompasses the relation- ship between gravitational sources induced by exotic topologies. Recently the combined action of a spinor field coupled to the grav- itational field was obtained in [17]. Furthermore, Eq. (15) complies with the differential-topological restrictions on the spacetime for the evolution of our Universe. The differential-geometric descrip- tion of matter by differential structures of spacetime might leads to a unifying model of matter, dark matter and dark energy. In- deed, by taking into account exotic differential structures, it may be the source of the observed anomalies without modifying the 2 Hereon we are not going to specify the different Elko types, which simplify the content of indexes in Eq. (13). Again, for a complete discussion, see [4]. Einstein equations or introducing unusual types of matter, as a vast resource of possible explanations for recently observed surprising astrophysical data at the cosmological scale, merely provided by differential topology [17]. Furthermore, such exoticness induces a dynamical mass which is embedded in the VSR framework [18]. It is accomplished by identifying the VSR preferential direction with a dynamical depen- dence on the kink solution of a λφ4 theory, for a scalar field φ. The exotic term ∂μθ is chosen to be vμφ, where vμ provides a prefer- ential direction, an inherent preferred axis — along which Elko is local. This is solely one among various possible scenarios, using ex- otic couplings among dark spinor fields and scalar field topological solutions [18]. 3.3. The appearance of new spinors In the specific context of f (R)-cosmology, it was recently re- ported a solution for the Dirac equation with torsion, considering Bianchi type-I cosmological models [19]. The gravitational dynam- ics of the theory may be described by the metric and its compat- ible connection, or alternatively by the tetrad field and the spin- connection as well. The equations of motion are f ′(R)Rρσ − 1 2 f (R)gρσ = Σρσ , 1 2 ( ∂ f ′(R) ∂xα + Sαγ γ )( δα σ δ β ρ − δα ρ δ β σ ) + Sρσ β = f ′(R)Tρσ β, where Rρσ is the Ricci tensor and Tρσ β stands for the torsion ten- sor. The quantities σρσ and Sρσ α are the stress–energy and spin tensors of the matter fields. The energy–momentum tensor is given by Σρσ . The idea is to couple f (R)-gravity to spinor fields and to a spinless perfect fluid. These spinor fields are shown not to be Dirac spinor fields [20]. In addition the second equation of mo- tion assents the existence of torsion even in the absence of spinor fields. Implementing all the necessary constraints, it is possible to show that the spinor solutions reads ψ1 = 1√ 2τ ⎛ ⎜⎝ √ A − B cos ζ1eiθ1 0 0√ A + B sin ζ2eiθ2 ⎞ ⎟⎠ , (16) ψ2 = 1√ 2τ ⎛ ⎜⎜⎝ 0√ A + B cos ζ1eiϑ1 √ A − B sin ζ2eiϑ2 0 ⎞ ⎟⎟⎠ , (17) where A and B are constants, the angular functions have time dependence, and τ is defined as the product of the scale fac- tors appearing in the Bianchi type-I model (not relevant to our purposes). The point to be stressed is that, after a tedious cal- culation, the bilinear covariants associated to ψ1 and ψ2 classify the spinor fields (16) as type-(4): legitimate flag-dipole spinor fields that are obtained when the Dirac equation with torsion is regarded in the f (R)-cosmological scenario [21]. It is the first time, up to our knowledge, that a physical solution corresponds to a type-(4) spinor.3 Eq. (16) evinces a physical manifestation of 3 This fact is more remarkable than it may sound. Several spinor solutions are of the form presented in (16). Notwithstanding, after all, the class under Lounesto’s classification appears to be other than type-(4). For instance, on p. 65 of [22] it is possible to find similar structured spinor fields. Twenty pages of calculations led the authors to the very exciting conclusion that they belong to the type-(4) set. After some ponderation, however, we were brought back to the Earth: professor Leite Lopes’ book was not written using the Weyl representation! 11 J.M. Hoff da Silva, R. da Rocha / Physics Letters B 718 (2013) 1519–1523 1523 type-(4), or flag-dipole, spinor fields according to Lounesto’s clas- sification. We finalize this section by pointing out a provocative interpre- tation of the type-(4) spinor fields as manifested via Eq. (16). There is no quantum field constructed out yet with type-(4) spinor fields and it is certainly an interesting branch of research. In view of the analysis of Section 3.1, such a quantum field shall not respect Lorentz symmetry. From this perspective, it would be the darkest possible candidate to dark matter. Being more conservative, with- out making any reference to its possible quantum field, type-(4) spinor fields, as it appears, are also quite provocative. Usually, gen- eralizations of General Relativity are studied to give account of cosmological problems, without appealing to the existence of dark matter, for instance. Nevertheless, as we have mentioned, type-(4) spinor fields appeared only in a (double) generalization of General Relativity. Moreover, the presence of torsion in an f (R) gravity is crucial to the functional form of these spinor fields as explicit in (16). Hence, type-(4) spinor fields, an essentially dark spinor (we restrain to say dark matter), comes up in a far from usual gravitational theory, which is commonly investigated to preclude the necessity of “dark” objects. 4. Final remarks A plethora of open questions still haunts (in particular) the- oretical physicists. The non-standard spinor fields — both under Lounesto as well as Wigner classification — are an evidently use- ful alternative to pave the road to solve some questions, mainly in field theory and cosmology/gravitation. It brings some nice and unexpected properties, like the existence of fermions with mass dimension one and a subtle Lorentz symmetry breaking, for instance. Facing such paradigm shift seems to upheaval what we know already about field theory and the elementary parti- cles description, which were restricted to Dirac, Majorana and Weyl spinor fields heretofore, in Minkowski spacetime. As we have shown, flag-dipole type-(4) spinor fields are physical solutions of the Dirac equation with torsion in the context of f (R)-cosmology. Furthermore, Elko spinor fields representing type-(5), abreast of Majorana spinor fields, are evinced to be prime candidates to de- scribe dark matter. We moreover have introduced the exotic dark spinor fields, which dynamics constraints both the spacetime met- ric structure and the non-trivial topology of the universe. In par- ticular, it brings exotic couplings among dark spinor fields and scalar field topological solutions. The topics here introduced are merely the tip of the iceberg, and there are more useful proper- ties on spinor fields (and their application in physics) still to be explored. Acknowledgements The authors would like to thanks Prof. José Abdalla Helayël- Neto for the continuous motivation. R. da Rocha is grateful to Con- selho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grants 476580/2010-2 and 304862/2009-6 for financial support. References [1] S. Weinberg, The Quantum Theory of Fields. Vol. 1: Foundations, Cambridge Univ. Press, Cambridge, UK, 1995. [2] P. Lounesto, Clifford Algebras and Spinors, 2nd ed., Cambridge Univ. Press, Cam- bridge, 2002. [3] D.V. Ahluwalia, D. Grumiller, JCAP 0507 (2005) 012, arXiv:hep-th/0412080; D.V. Ahluwalia, D. Grumiller, Phys. Rev. D 72 (2005) 067701, arXiv:hep-th/ 0410192. [4] R. da Rocha, A.E. Bernardini, J.M. Hoff da Silva, JHEP 1104 (2011) 110, arXiv: 1103.4759 [hep-th]. [5] R.A. Mosna, W.A. Rodrigues Jr., J. Math. Phys. 45 (2004) 2945, arXiv:math-ph/ 0212033. [6] C.G. Boehmer, L. Corpe, J. Phys. A: Math. Theor. 45 (2012) 205206, arXiv: 1204.0135 [math-ph]. [7] D.V. Ahluwalia, C.-Y. Lee, D. Schritt, T.F. Watson, Phys. Lett. B 687 (2010) 248, arXiv:0804.1854 [hep-th]. [8] T. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics, vol. 25, AMS, Providence, USA, 2000. [9] E. Capelas de Oliveira, W.A. Rodrigues Jr., arXiv:1210.7207 [math-ph]. [10] D.N. Spergel, P.J. Steinhardt, Phys. Rev. Lett. 84 (2000) 3760; P.K. Samal, R. Saha, P. Jain, J.P. Ralston, Mon. Not. Roy. Astron. Soc. 396 (2009) 511, arXiv:0811.1639 [astro-ph]; M. Frommert, T.A. Ensslin, Mon. Not. Roy. Astron. Soc. 403 (2010) 1739, arXiv: 0908.0453 [astro-ph]. [11] D.V. Ahluwalia, S.P. Horvath, JHEP 1011 (2010) 078, arXiv:1008.0436 [hep-ph]. [12] A.G. Cohen, S.L. Glashow, Phys. Rev. Lett. 97 (2006) 021601, arXiv:hep-ph/ 0601236. [13] R. da Rocha, W.A. Rodrigues Jr., Mod. Phys. Lett. A 21 (2006) 65, arXiv:math-ph/ 0506075. [14] R.P. Geroch, J. Math. Phys. 9 (1968) 1739; R.P. Geroch, J. Math. Phys. 11 (1970) 343. [15] R. da Rocha, J.M. Hoff da Silva, Adv. Appl. Clifford Algebras 20 (2010) 847, arXiv:0811.2717 [math-ph]. [16] S.J. Avis, C.J. Isham, Nucl. Phys. B 156 (1979) 441. [17] T. Asselmeyer-Maluga, C.H. Brans, Gen. Rel. Grav. 34 (2002) 1767, arXiv:gr-qc/ 0110043; C.H. Brans, Class. Quant. Grav. 11 (1994) 1785, arXiv:gr-qc/9404003; C.H. Brans, J. Math. Phys. 35 (1994) 5494, arXiv:gr-qc/9405010. [18] A.E. Bernardini, R. da Rocha, Phys. Lett. B 717 (2012) 238, arXiv:1203.1049 [hep-th]. [19] S. Vignolo, L. Fabbri, R. Cianci, J. Math. Phys. 52 (2011) 112502, arXiv:1106.0414 [gr-qc]. [20] L. Fabbri, S. Vignolo, Class. Quant. Grav. 28 (2011) 125002, arXiv:1012.1270 [gr- qc]. [21] R. da Rocha, R.T. Cavalcanti, J.A. Silva-Neto, L. Fabbri, J.M. Hoff da Silva, in preparation. [22] J. Leite Lopes, Introducción a la Electrodinámica Cuántica, Ed. Trillas, México, 1977. 12 13 2.1. Espinores Exóticos 2.1 Espinores Exóticos Um dos tópicos abordados no trabalho introdutório diz respeito à relevância do estudo de espinores escuros exóticos. Sobre espinores escuros, e suas propriedades definidoras por assim dizer, falaremos detidamente nos capítulos ulteriores. Assim, uma vez que a formulação para a compreensão das estruturas exóticas faz-se a mesma para todos os espinores vamos nos ater a apresentá-la, chamando a atenção, posteriormente, às peculiariedades dos espinores escuros exóticos. No trabalho que reproduzimos nesta seção há uma abordagem bastante formal e sólida da adequação de espinores exóticos ao caso de espinores escuros, com um posterior vislumbre de aplicação em cosmologia. Gostaríamos aqui, entretanto, de aproveitar o ensejo para realizar uma introdução menos precisa, porém mais intuitiva, da noção de espinor exótico. Também, devido ao fato dos termos de exoticidade espinorial serem pouco vistos na literatura corrente, estenderemo-nos um pouco mais na sua apresentação. Comecemos por trabalhar o conceito de espinor do ponto de vista puramente geométrico. Devido ao caráter pseudo-euclideano do espaço-tempo sabemos haver vetores tipo-luz que, sob atuação da métrica, levam ao conceito de cone-de-luz1. Considerando uma intersecção de um dado hiperplano (𝑇1 =1,𝑋,𝑌,𝑍) (com 𝑇1 = 1 por simplicidade) com o cone de luz, temos como resultado uma casca esférica de raio unitário, a esfera de Riemann (ver figura 2.1). Em seguida, consideremos um mapa injetivo associando a cada ponto na esfera um dado ponto em um plano complexo que intersepta a esfera em 𝑍 = 0. Essa é a chamada projeção estereográ- fica. Nessa projeção, as coordenadas (𝑋,𝑌,𝑍) na esfera podem ser descritas por 1A abordagem que descreveremos é válida para qualquer tipo de vetor, mas para os tipo-luz ela é mais simples. Capítulo 2. Aspectos Formais 14 Figura 2.1: Origem da esfera de Riemann. um número complexo 𝛽 = 𝑋 ′ + 𝑖𝑌 ′. A figura 2.2 mostra como podemos construir o mapeamento a partir dos triângulos 𝑃 ′𝐶𝑁 e 𝑃𝐵𝑁 de modo que 𝛽 = 𝑋 − 𝑖𝑌 1 − 𝑍 . Entretanto para que se possa descrever o polo norte (𝛽 = ∞) é conveniente se associar aos pontos da esfera não apenas um único complexo, mas um par (𝜂,𝜉) tal que 𝛽 = 𝜉/𝜂. Desse modo, o polo norte é obtido pela coordenada (︃ 𝜉 𝜂 )︃ = (︃ 1 0 )︃ . Figura 2.2: Construção do mapeamento. O ponto de vista formal que queremos apreciar nesta seção pode agora ser anunciado: espinores são, de fato, as coordenadas projetivas da projeção estereo- 15 2.1. Espinores Exóticos gráfica de uma seção do cone de luz (com a ressalva da nota de rodapé da página 13) no plano complexo. É uma questão de simples álgebra agora se ver que 𝑋 = 𝜉𝜂 + 𝜂𝜉 𝜉𝜉 + 𝜂𝜂 , 𝑌 = 𝜉𝜂 − 𝜂𝜉 𝑖(𝜉𝜉 + 𝜂𝜂) , 𝑍 = 𝜉𝜉 − 𝜂𝜂 𝜉𝜉 + 𝜂𝜂 , onde a barra indica conjugação. Nota-se agora o caráter especial de “raíz quadrada da geometria” atribuído aos espinores. A concepção padrão de entendimento de espinores como elementos que carregam representações irredutíveis do grupo de Lorentz pode ser obtida da análise acima como se segue: considere um ponto dado por (1,𝑋,𝑌,𝑍)(𝜉𝜉− 𝜂𝜂)/ √ 2. Uma transformação de 𝑆𝐿(2,C) nas coordenadas do ponto (equivalentemente, em (𝜉, 𝜂)) deixa invariante o determinante det ⎡⎣(︃𝜉 𝜂 )︃ (𝜉 𝜂) ⎤⎦, que, traduzido em termos das coordenadas do espaço-tempo, nada mais é do que a métrica de Minkowski. Voltando ao tópico central de espinores exóticos, notemos que a existência de “buracos” no espaço-tempo (levando a uma topologia não-trivial) inviabiliza a concepção usual de espinores (ver Figura 2.3). A topologia não-trivial é refletida Figura 2.3: Visualização da topologia não-trivial na estrutura espinorial. (dentre outros efeitos) por um primeiro grupo de homotopia não-trivial do espaço- Capítulo 2. Aspectos Formais 16 tempo 𝜋1(𝑀). Por outro lado o grupo de homomorfismos de 𝜋1(𝑀) em Z2 rotula os diferentes (e não equivalentes) cobrimentos locais necessários para se contornar a região com o “buraco”. Esse grupo de homomorfismos é o primeiro grupo de cohomologia do espaço-tempo, e sua não trivialidade é herdada do fato de 𝜋1(𝑀) ser não-trivial. Assim a topologia não-trivial dá origem a cobrimentos inequiva- lentes, que por sua vez levam a projeções estereográficas também inequivalentes e, portanto, espinores diferentes surgem. Dá-se assim origem aos espinores exóticos. Por fim, enfatizamos que a única diferença na dinâmica de ambos os espinores se dá na conexão relacionada ao espinor exótico, que deve levar em conta a topologia não-trivial. J H E P 0 4 ( 2 0 1 1 ) 1 1 0 Published for SISSA by Springer Received: March 25, 2011 Accepted: April 13, 2011 Published: April 26, 2011 Exotic dark spinor fields Roldão da Rocha,a Alex E. Bernardinib and J. M. Hoff da Silvac aCentro de Matemática, Computação e Cognição, Universidade Federal do ABC Rua Santa Adélia, 166 09210-170, Santo André, SP, Brazil bDepartamento de F́ısica, Universidade Federal de São Carlos PO Box 676, 13565-905, São Carlos, SP, Brazil cUNESP — Campus de Guaratinguetá — DFQ Av. Dr. Ariberto Pereira da Cunha, 333 12516-410, Guaratinguetá-SP, Brazil. E-mail: roldao.rocha@ufabc.edu.br, alexeb@ufscar.br, hoff@feg.unesp.br; hoff@ift.unesp.br Abstract: Exotic dark spinor fields are introduced and investigated in the context of inequivalent spin structures on arbitrary curved spacetimes, which induces an additional term on the associated Dirac operator, related to a Čech cohomology class. For the most kinds of spinor fields, any exotic term in the Dirac operator can be absorbed and encoded as a shift of the electromagnetic vector potential representing an element of the cohomology group H1(M,Z2). The possibility of concealing such an exotic term does not exist in case of dark (ELKO) spinor fields, as they cannot carry electromagnetic charge, so that the full topological analysis must be evaluated. Since exotic dark spinor fields also satisfy Klein- Gordon propagators, the dynamical constraints related to the exotic term in the Dirac equation can be explicitly calculated. It forthwith implies that the non-trivial topology associated to the spacetime can drastically engender — from the dynamics of dark spinor fields — constraints in the spacetime metric structure. Meanwhile, such constraints may be alleviated, at the cost of constraining the exotic spacetime topology. Besides being prime candidates to the dark matter problem, dark spinor fields are shown to be potential candidates to probe non-trivial topologies in spacetime, as well as probe the spacetime metric structure. Keywords: Cosmology of Theories beyond the SM, Differential and Algebraic Geometry, Topological Field Theories ArXiv ePrint: 1103.4759 c© SISSA 2011 doi:10.1007/JHEP04(2011)110 17 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 Contents 1 Introduction 1 2 Preliminaries: exotic spin structures 4 3 Dark (ELKO) spinor fields 7 4 ELKO dynamics in the exotic spin structure 10 5 Concluding remarks and outlook 16 A Clifford bundles 17 B Principal bundles and associated vector bundles 18 1 Introduction ELKO — Eigenspinoren des Ladungskonjugationsoperators — spinor fields1 describe a non- standard Wigner class of fermions, for which charge conjugation and parity are commuting operators, rather than anticommuting ones [1–4]. They support two types of dispersion relations, accomplish dual-helicity eigenspinors of the spin-1/2 charge conjugation operator, and carry mass dimension one, besides having non-local properties. At low-energy limits, ELKO behaves as a representation of the Lorentz group through the setup of a preferred frame related to its wave equation [3–6]. Ahluwalia-Khalilova and Grumiller embedded ELKO [1] into the quantum field theory, from which large applications in cosmology and gravity can be outlined. The corresponding ELKO Lagrangian neither predicts interactions with Standard Model (SM) fields nor shows coupling with gauge fields. Otherwise, exotic interactions with the Higgs boson can somehow be depicted in order to endow such spinor fields to be prime candidates to describe dark matter [7]. In particular, observational aspects on such a possibility has been proposed at LHC: dark (ELKO) spinor fields can be observed, at center of mass energy around 7 TeV and total luminosity from 1 fb−1 to 10 fb−1, indicating that the number of events is large enough to motivate a detailed analysis about ELKO particle at high energy experiments [8]. In addition, the embedding of dark spinor fields into the SM [10, 11] was introduced. ELKO spinor fields dominant interaction via the gravitational field makes them naturally dark, and recently [12–14] dark spinor fields were investigated in a cosmological setting, where interesting solutions and also models where the spinor is coupled conformally to gravity are provided. Some additional applications of ELKO spinor fields to cosmology 1ELKO is the German acronym for eigenspinors of the charge conjugation operator. – 1 – 18 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 can be seen, e.g., in [15–27]. In particular, possible applications of ELKO spinor fields to more general f(R) gravitational theories were accomplished in [28], and supersymmetric models concerning ELKO were introduced in [29]. The main aim of our manuscript is to investigate dark spinor fields in spacetimes with non-trivial topologies, in order to clarify how the dynamics of such dark spinor fields can induce constraints on the metric spacetime structure, as well as in the non-trivial topology itself. Physical applications of non-trivial topologies on spacetime, including thermodynamics, superconductivity, and condensed matter have been extensively explored in the last years. For instance, the quantum theory of fields propagating on a manifold M not simply connected was investigated in [30]. The existence of a nontrivial line bundle on a manifold M , whose sections may be regarded as a generalization of the concept of a scalar field, is inherent in multiply connected manifolds, which in addition can imply in the existence of inequivalent spin structures. It is well known that the set of real line bundles on M and the set of inequivalent spin structures are both labeled by elements of the cohomology group H1(M,Z2) — the group of homomorphisms of the fundamental group π1(M) into Z2. Namely, there are many globally different spin structures which arise from inequivalent patchings of the local double coverings, see, e.g., [31]. The use of generalized spin structures has been discussed in [32–36], in particular the ones considered by [37] for the case where the original fiber bundle has no spin structure at all. Whereas the manifold M may have no spinor bundle — when it does not satisfy Geroch theorem hypotheses [38, 39] — it may have also many of them, which are split into equivalence classes — called spin structure. The formal aspects about the inequivalent spin structures in spacetime in terms of the different possible spin connections [30, 36, 40, 41] have been explored and reveal prominent physical applications. In an arbitrary spacetime that admits spin structure we delve into the problem of how to select a particular spin structure and its corresponding Dirac equation. In [30] it is discussed in what content the quantum field theory associated with a spinor field must involve some kind of “average” of all the spin connections.2 On simply connected spacetimes, the associated fundamental group satisfies π1(M) = 0, therefore there is only one spin structure. Although compact simply connected 4- dimensional manifolds admitting a spin structure can be classified in terms of the Euler and Pontryagin numbers [42], multiply connected spacetimes are devoid in general of such a classification. As argued in [30, 40, 41], since Nature seems to use all mutually consistent degrees of freedom in a physical system, the Feynman path integral formalism, for instance, should also include multiply connected manifolds — which is among other prominent mo- tivations to investigate quantum field theory on such spacetimes, which in addition are used in quantum gravity at both cosmological and Planck length scales [42, 43]. Multiply connected manifolds are also elicited in the theory of superconductivity. In fact, in [36] the inequivalent generalized spin structures are investigated in order to explain Cooper pairing phenomena in superconductors. In addition, such manifolds are used in the instan- ton compactification [44–46] of 4-dimensional Euclidean spacetimes and also in t’ Hooft’s 2As it must be summed over all topological sectors in instanton physics [30]. – 2 – 19 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 treatment of confinement. In [47, 48], the finite-temperature stress-energy-momentum for a conformally coupled massive scalar field in multiply connected spaces was calculated and cogently investigated from a thermodynamic viewpoint. Also, in [49] and [50–54] covariant Casimir calculations were performed for the massless scalar field in several flat multiply connected spaces, expressing the stress expectation values as the coincidence limit of a bilinear operator acting on the Feynman propagator for the manifold, motivating the in- troduction of a robust mathematical approach [55, 56]. Finally, in [36] exotic spinor fields provides pure geometrical explanation of the charge dependence on the quantized flux and also the Joseph current in superconductivity. To summarize, one of the main outstanding exotic spinor fields features is that they must be taken into account and employed in a variety of problems, wherein standard spinor fields cannot. For instance, when the vacuum polarization tensor of spinor electrodynamics is calculated [57, 58], it was found that the two types of spinor fields — standard and exotic — generate different vacuum polarization effects, which are physically inequivalent. When the effect of the vacuum polarization upon photon propagation is considered, it is shown that standard spinor fields give rise to non causal photon propagation, whereas exotic spinor fields do not. Even when the vacuum energy for a free spinor field is calculated, it is found that the exotic configuration gives rise to a vacuum state of lower energy than the standard one. These prominent features make exotic spinor fields as a broad audience candidate for concrete physical problems. We shall address to the question about such inequivalent spin structures and their consequences to the coupled system of Dirac equations satisfied by the four types of dark spinor fields. The physical assumption that — under the exotic spin structure — the exotic dark spinor fields satisfy the Klein-Gordon propagator brings up some constraint on the metric spacetime structure, as well as in the exotic topology, both arbitrary a priori. The characterization of dark (ELKO) spinor fields, and its inherent analysis is obtained through the natural introduction — topologically impelled — of an exotic term in the Dirac operator that, contrary to the case of the Dirac spinor field, cannot be absorbed in any external electromagnetic vector field. For Dirac fields, such term can be concealed and encoded as a shift of the electromagnetic potential.3 Therefore, besides addressing feasible aspect to the dark matter problem, dark spinor fields are also useful to probe non-trivial topological properties in spacetime. The manuscript is organized as follows. After briefly presenting some algebraic prelimi- naries in section 2 regarding inequivalent spin structures, in section 3, the ELKO properties are introduced, together with the bilinear covariants that completely characterize a spinor field through the Fierz identities. In section 4 the exotic structure is introduced and the corresponding implications on the behaviour of ELKO are depicted. Dark spinor dynamics not only constrains the possibilities for the exotic topology but also induces constraints in the spacetime geometry through the exotic topology coming from the dynamics of dark spinor fields. We prove that it brings up some subtle consequences on the spacetime geom- etry. A brief summary of important useful results throughout the manuscript is described in the appendices A and B, and we draw our conclusions in section 5. 3Representing an element of the cohomology group H1(M,Z2). – 3 – 20 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 2 Preliminaries: exotic spin structures In this section we review some results concerning general inequivalent spin structures in spinor bundles. For more details see the appendix. One denotes by (M,g,∇, τg , ↑) the spacetime structure [59, 60]: M denotes a 4- dimensional manifold — which we shall assume as a compact, paracompact, pseudo- Riemannian manifold which is both space and time orientable and which admits spinor fields — g is the metric, ∇ denotes the connection associated to g, τg defines a spacetime orientation and ↑ refers a time orientation. As usual T ∗M [TM ] denotes the cotangent [tangent] bundle over M , F (M) denotes the principal bundle of frames, and PSOe 1,3 (M) de- notes the orthonormal coframe bundle. Such bundles do exist on spin manifolds. Sections of PSOe 1,3 (M) are orthonormal coframes, and sections of PSpine 1,3 (M) are also orthonormal coframes such that although two coframes differing by a 2π rotation are distinct, two coframes differing by a 4π rotation are identified. A spin structure on M consists of a principal fiber bundle πs : PSpine 1,3 (M) → M , with group Spine 1,3, and the fundamental map — indeed a two-fold covering s : PSpine 1,3 (M) → PSOe 1,3 (M), satisfying the following conditions: 1. π(s(p)) = πs(p), ∀p ∈ PSpine 1,3 (M); π is the projection map of PSOe 1,3 (M) on M . 2. s(pφ) = s(p)Adφ, ∀p ∈ PSpine 1,3 (M) and Ad : Spine1,3 → Aut(Cℓ1,3), Adφ : Ξ 7→ φΞφ−1 ∈ Cℓ1,3 [59]. Namely, the following diagram PSpine 1,3 (M) s - PSOe 1,3 (M) M � π π s - commutes. The conditions for existence of a spin structure in a general manifold are discussed in [61, 62]. It is well known that a spin structure (PSpine 1,3 (M), s) exists if and only if the second Stiefel-Whitney class associated to M satisfies certain properties. If H1(M,Z2) is not trivial, the spin structure is not uniquely defined,4 and all the other inequivalent spin structures can be provided from (PSpine 1,3 (M), s). 4Up to trivial bundle isomorphisms. – 4 – 21 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 Two spin structures P :=(PSpine 1,3 (M), s) and P̊ := ((P̊Spine 1,3 (M), s̊) are said to be equivalent if there exists a Spine 1,3-equivariant map ζ : P → P̊ compatible with s and s̊: P ζ - P̊ PSOe 1,3 (M) � s̊s - Now we briefly review some few definitions necessary to introduce exotic spinor fields. Spinor fields are sections of vector bundles associated with the principal bundle of spinor coframes. A complex spinor bundle for M is a vector bundle Sc(M) = PSpine 1,3 (M)×µc Mc, where Mc is a complex left module for C⊗Cℓ1,3 ≃ M(4,C), and where µc is a representation of Spine 1,3 in End(Mc) given by left multiplication by elements of Spine 1,3. When Mc = C4 and µc the D(1/2,0) ⊕ D(0,1/2) representation of Spine 1,3 ≃ SL(2,C) in End(C4), we immediately recognize the usual definition of the covariant spinor bundle of M as given, e.g., in [61] and [62]. Classical spinor fields5 carrying a D(1/2,0) ⊕D(0,1/2), or D(1/2,0), or D(0,1/2) represen- tation of SL(2,C) are sections of the vector bundle PSpine 1,3 (M) ×ρ C4, where ρ stands for the D(1/2,0) ⊕ D(0,1/2) (or D(1/2,0), or D(0,1/2)) representation of SL(2,C) in C4. Other important spinor fields, like Weyl spinor fields are obtained by imposing some constraints on the sections of PSpine 1,3 (M) ×ρ C4, see, e.g., [70, 71] for details. Two spin structures (PSpine 1,3 (M), s) and (P̊Spine 1,3 (M), s̊) are respectively described by the maps hjk and h̊jk from Ui∩Uj to Spine 1,3, both satisfying eq. (A.1), and also the property ς ◦hjk = ajk = ς ◦ h̊jk. The following diagram illustrates such relations, summarizing what should be emphasized heretofore: Ui ∩ Uj ⊂ M hij - Spine 1,3 SOe 1,3 � ς a ij - PSpine 1,3 (M) ⊂ - ˚Spin e 1,3 h̊ij ? ς - - P̊Spine 1,3 (M) ? ζ - PSOe 1,3 (M) ? ∩ id - PSOe 1,3 (M) s ? s̊ - 5Quantum spinor fields are operator valued distributions, as well known. It is not necessary to introduce quantum fields in order to know the algebraic classification of ELKO spinor fields. – 5 – 22 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 Here another identical copy of Spine 1,3 is denoted by ˚Spin e 1,3, in order to become clearer the analysis about the inequivalent spin structures. Now one defines a map cjk by the relation hij(x) = h̊ij(x)cij such that cij : Ui ∩Uj → ker ς = Z2 →֒ Spine 1,3, satisfying cij ◦ cjk = cik. Such a map cij defines an 1-dimensional real bundle denoted [72]. Given the irreducible representation ρ : Cℓ1,3 → M(4,C) in PSpine 1,3 (M) ×ρ C4, as the map cij(x) is an element of Z2, it follows that ρ(cij(x)) = ±1, since ρ is faithful. When ρ is restricted to Spine 1,3, it is called Dirac representation. We assume as in [30, 36, 37, 40, 41] that there is a set of functions ξi : Ui → C such that ‖ξi(x)‖ = 1, namely ξi(x) ∈ U(1), and ξi(x)(ξj(x)) −1 = ρ(cij(x)) = ±1. (2.1) In the case where the second integral cohomology H2(M,Z2) has no 2-torsion, such func- tions always do exist [30, 36, 40, 41, 72], and ξ2i (x) = ξ2j (x), x ∈ Ui ∩Uj. Consequently the local functions ξi define a unique unimodular function ξ : M → C such that for all x ∈ Ui it follows that ξ(x) = ξ2i (x). Given now an arbitrary spinor field ψ ∈ sec PSpine 1,3 (M) ×ρ C4, to each element of H1(M,Z2), associate a covariant derivative ∇. This construction provides indeed a one- to-one correspondence between elements of H1(M,Z2) and inequivalent spin structures. A local component ψi : Ui → C4 of a spinor field in PSpine 1,3 (M) ×ρ C4 is the unique function such that ρ(ℓi, ψi(x)) = ψ(x), given local sections ℓi : Ui → (PSpine 1,3 (M), s), we have the transition law ψi(x) = ρ(hij(x))ψj(x), where x ∈ Ui ∩ Uj. A system of local sections ℓ̊i : Ui → P̊Spine 1,3 (M) can be constructed from the standard ones ℓi in such a way that s ◦ ℓi = s̊ ◦ ℓ̊i, as presented in the following diagram: Ui P̊Spine 1,3 (M) s̊- � ℓ̊ i PSOe 1,3 (M) mi ? �s PSpine 1,3 (M) ℓ i - It enables the (exotic) local spinor field components to present the respective transition property ψ̊j(x) = ρ(̊hij) = ρ(hij(x))ρ(cij(x))ψ̊i(x), where x ∈ Ui ∩ Uj . (2.2) From eq. (2.1) it follows that ρ(ξi) = ρ(cij(x))ρ(ξj) and if we compare it with eq. (2.2), it is clear that ρ(ξi)ψ̊i transforms as the local component ψi of PSpine 1,3 (M) ×ρ C4, which subsequently induces a bundle map f : P̊Spine 1,3 (M) ×ρ C4 → PSpine 1,3 (M) ×ρ C4 ψ̊i 7→ f(ψ̊i) := ρ(ξi)ψ̊i = ψi (2.3) – 6 – 23 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 such that ∇̊Xf(ψ̊) = f(∇X ψ̊) + 1 2 (Xy(ξ−1dξ))f(ψ̊) (2.4) holds for all sections ψ ∈ PSpine 1,3 (M) ×ρ C4 and all vector fields X. Details on how to derive eq. (2.4) are comprehensively given in, e.g., [30, 36, 37, 40, 41, 72, 73]. 3 Dark (ELKO) spinor fields This section is devoted to a brief review of the bilinear covariants through the programme introduced in [70, 71, 74]. The spinor fields classification is provided by a brief review of [10, 11, 74–76]. Given a spinor field ψ ∈ secPSpine 1,3 (M)×ρ C4, the bilinear covariants are the following sections of Λ(TM) = ⊕4 r=0 Λr(TM) →֒ Cℓ(M,g) [59, 77, 78]: σ = ψ†γ0ψ, J = Jµe µ = ψ†γ0γµψe µ, S = Sµνe µν = 1 2 ψ†γ0iγµνψe µ ∧ eν , K = ψ†γ0iγ0123γµψe µ, ω = −ψ†γ0γ0123ψ, (3.1) with σ, ω ∈ sec Λ0(TM), J,K ∈ sec Λ1(TM) and S ∈ sec Λ2(TM). In the formulæ ap- pearing in eq. (3.1) the set {γµ} refers to the Dirac matrices in chiral representation (see eq. (3.3)). Also, {1, eµ, eµeν , eµeνeρ, e0e1e2e3}, where µ, ν, ρ = 0, 1, 2, 3, and µ < ν < ρ is a basis for Cℓ(M,g), and {14, γµ, γµγν , γµγνγρ, γ0γ1γ2γ3} is a basis for M(4,C). In addi- tion, these bases satisfy the respective Clifford algebra relations [70] γµγν + γνγµ = 2gµν14 and eµeν + eνeµ = 2gµν , where 14 is the identity matrix. When there is no opportunity for confusion we shall omit the 14 identity matrix in our formulæ. For the orthonormal covector fields eµ and eν , µ 6= ν, their Clifford product eµeν is equal to the exterior product of those vectors, i.e., eµeν = eµ ∧eν = eµν . Also, for µ 6= ν 6= ρ, eµνρ = eµeνeρ, etc. More details on our notations, if needed, can be found in [59, 60]. In Minkowski spacetime, the case of the electron is described by Dirac spinor fields (classes 1, 2 and 3 below), J is a future-oriented timelike current vector which gives the current of probability, and J2 = JµJ µ > 0. Furthermore, for the case of Dirac spinor fields, the bivector S is associated with the distribution of intrinsic angular momentum, and the spacelike vector K is associated with the direction of the electron spin. For a detailed discussion concerning such entities, their relationships and physical interpretation, and generalizations, see, e.g., [70, 71, 77–79]. The bilinear covariants satisfy the Fierz identities6 [70, 71, 77–79] J2 = ω2 + σ2, K2 = −J2, JxK = 0, J ∧ K = −(ω + σγ0123)S. A spinor field such that not both ω and σ are null is said to be regular. When ω = 0 = σ, a spinor field is said to be singular, and in this case the Fierz identities are in 6 Given the spacetime metric g, it is possible to extend g to the exterior bundle Λ(TM). Given ψ = u1∧ · · ·∧uk and φ = v1∧· · ·∧vl, for ui, vj ∈ sec TM , one defines g(ψ,φ) = det(g(ui, vj)) if k = l and g(ψ,φ) = 0 if k 6= l. Given ψ, φ, ξ ∈ Λ(TM), the left contraction is defined implicitly by g(ψyφ, ξ) = g(φ, ψ̃ ∧ ξ). – 7 – 24 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 general replaced by the more general conditions [79] Z2 =4σZ, ZγµZ=4JµZ, ZiγµνZ=4SµνZ, Ziγ0123γµZ=4KµZ, Zγ0123Z=−4ωZ, where Z = σ + J + iS + iKγ0123 + ωγ0123. Lounesto spinor field classification is given by the following spinor field classes [70, 71], where in the first three classes it is implicit that J, K, S 6= 0: 1) σ 6= 0, ω 6= 0. 2) σ 6= 0, ω = 0. 3) σ = 0, ω 6= 0. 4) σ = 0 = ω, K 6= 0, S 6= 0. 5) σ = 0 = ω, K = 0, S 6= 0. 6) σ = 0 = ω, K 6= 0, S = 0. The current density J is always non-zero. Type 1, 2 and 3 spinor fields are denominated Dirac spinor fields for spin-1/2 particles and type 4, 5, and 6 are respectively called flag- dipole, flagpole and Weyl spinor fields. Majorana spinor fields are a particular case of a type 5 spinor field. It is worthwhile to point out a peculiar feature of types 4, 5 and 6 spinor fields: although J is always non-zero, J2 = −K2 = 0. It shall be seen below that the bilinear covariants related to an ELKO spinor field, satisfy σ = 0 = ω, K = 0, S 6= 0 and J2 = 0. Since Lounesto proved that there are no other classes based on distinctions among bilinear covariants, ELKO spinor fields must belong to one of the disjoint six classes. In [74] it is shown that ELKO spinor fields are indeed in class 5 above. Some properties of dark (ELKO) spinor fields,7 as introduced in [1, 2, 7] can be now briefly reviewed. An ELKO Ψ corresponding to a plane wave with momentum p = (p0,p) can be written, without loss of generality, as Ψ(p) = λ(p)e±ip·x where λ(p) = ( iΘφ∗(p) φ(p) ) , (3.2) and given the rotation generators J, the Wigner’s spin-1/2 time reversal operator Θ satisfies ΘJΘ−1 = −J∗. Hereon, as in [1], the Weyl representation of γµ is used γ0 =γ0 = ( O I I O ) , −γk=γk = ( O −σk σk O ) , γ5 =−iγ0γ1γ2γ3 =−iγ0123 = ( I O O −I ) , (3.3) 7Hereon throughout the text the term dark spinor field and ELKO are alternatively used having the same meaning, since ELKO is a candidate to describe dark matter, as comprehensively proposed, derived, and investigated in, e.g., [1–4, 7, 8, 10–14, 19–24, 74, 75, 80–82]. We choose the acronym ELKO to denote field theoretical and more formal properties of such a spinor field, whereas the naming dark spinor fields shall be used hereon alternatively to ELKO, in order to present and investigate the potentially cosmological applications as well as its usefulness as an attempt to the dark matter problem. – 8 – 25 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 where I = ( 1 0 0 1 ) , O = ( 0 0 0 0 ) , σ1 = ( 0 1 1 0 ) , σ2 = ( 0 −i i 0 ) , σ3 = ( 1 0 0 −1 ) . The σi are the Pauli matrices. ELKO spinor fields are eigenspinors of the charge conjugation operator C Cλ(p) = ±λ(p), for C = ( O iΘ −iΘ O ) K. The operator K C-conjugates 2-component spinor fields appearing on the right. The plus sign stands for self-conjugate spinor fields, λS(p), while the minus yields anti self-conjugate spinor fields, λA(p). Explicitly, the complete form of ELKO can be found by solving the equation of helicity (σ · p̂)φ± = ±φ± in the rest frame and subsequently make a boost, to recover the result for any p [1]. Here p̂ := p/‖p‖. The boosted four spinor fields are λ S/A {∓,±}(p)= √ E +m 2m ( 1 ∓ p E +m ) λ S/A {∓,±}(0), where λ S/A {∓,±}(0)= ( ±iΘ[φ±(0)]∗ φ±(0) ) . (3.4) One should notice that, since Θ[φ±(0)]∗ and φ±(0) have opposite helicities, ELKO cannot be an eigenspinor field of the helicity operator. The ELKO dual is given by [1] ¬S/A λ {∓,±}(p) = ±i [ λ S/A {±,∓}(p) ]† γ0. (3.5) Now let one denotes the eigenspinors of the Dirac operator for particles and antipar- ticles respectively by u±(p) and v±(p). The subindex ± regards the eigenvalues of the helicity operator (σ · p̂). The parity operator acts as Pu±(p) = +u±(p), Pv±(p) = − v±(p), which implies that P 2 = I in this case. The action of C on these spinors is given by C(u±1/2(p)) = ∓v∓(p), C(v±1/2(p)) = ±u∓1/2(p). (3.6) which implies that {C,P} = 0. On the another hand the parity operator P acts on ELKO by PλS ∓,±(p) = ± i λA ±,∓(p) , PλA ∓,±(p) = ∓ i λS ±,∓(p), (3.7) and it follows that [C,P ] = 0. Denoting [1] for Dirac spinor fields u+(p) = d1, u−(p) = d2, v+(p) = d3 v−(p) = d4, and for the ELKO λS {−,+}(p) = e1, λS {+,−}(p) = e2, λA {−,+}(p) = e3, λA {+,−}(p) = e4, it is possible to write ELKO as [80] ei = 4∑ j=1 Ωijdj , i = 1, 2, 3, 4, where Ωij = { + (1/2m) dj eiI, for j = 1, 2 , − (1/2m) dj eiI, for j = 3, 4 . (3.8) – 9 – 26 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 In matrix form, Ω reads Ω = 1 2   I −iI −I −iI iI I iI −I I iI −I iI −iI I −iI −I   = 1 2 ( B −B∗ B∗ −B ) ⊗ I, (3.9) where B := (I + σ2), Such results show that ELKO can be expressed somehow as a linear combination of the Dirac particle and antiparticle spinor fields. It reinforces the Lounesto theorems, showing that classes of spinor fields under Lounesto spinor fields classification are not preserved by sum (for details see [70, 71, 83]). In order to obtain the ELKO evolution, a prescription where the momentum is written in terms of the covariant derivative as pµ 7→ i∇µ is regarded. As one shall see in the following section, such a prescription is convenient when one considers the coordinate representation λS/A(x) = λS/A(p) exp ( εS/A ipµx µ ) . However, that is not the only way to prescribe the ELKO evolution. The momentum can also be replaced with the derivatives times the γ5 matrix as performed, for instance, in the investigation of ELKO auto interactions when one considers the ELKO field interacting with its own spin density via contorsional auto interactions [84]. 4 ELKO dynamics in the exotic spin structure In spacetimes with non-trivial topology it is well known that there is an additional degree of freedom for fermionic particles [85]. Albeit in the classical level it might be naively suggested that exotic spinor fields describe different particles, the breakthrough idea pro- posed is that, in the quantum framework, a new partition function which is the sum over all possibilities must be taken into account. See [30, 85] and references therein for more details. In this section it is thoroughly shown that dark spinor fields are a natural probe of the non-trivial topology and also provide, from their inherent dynamics, constraints either in the spacetime metric structure or in its topology, or in both. Essentially, exotic spinor fields are parallel transported like standard spinor fields, but an outstanding property distinguishes both kinds of spinor fields: the covariant derivative acting on these exotic spinor fields changes by an additional one-form field that is manifes- tation of the non-trivial topology, as it was shown in section 2. The exotic structure endows the Dirac operator with an additional term ξ−1(x)dξ(x), x ∈ M , where d : sec Λ0(TM) → sec Λ1(TM) denotes the exterior derivative operator. The term 1 2πiξ −1(x)dξ(x) is real and closed, but not exact, and defines an integer cohomology class in the Čech sense [30, 36, 37, 40, 41]. Using the relation between Čech and de Rham cohomologies, the integral of 1 2πiξ −1(x)dξ(x) around any closed curve is an integer. In the context of the exotic Dirac equation, the electromagnetic vector potential A term is affected by the transformation A 7→ A + 1 2πiξ −1dξ, which exactly corresponds to the addition of another electromagnetic potential, when Dirac spinor fields are taken into account. In such case the exotic term may be then absorbed in an external electromagnetic potential, representing an element of H1(M,Z2) [36, 40, 41, 73, 86]. Namely, in this case the interaction is encoded as a shift in the vector potential. – 10 – 27 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 The importance to analyze dark spinor fields in this context is that this possibility is not present if ELKO spinor fields are employed, as they cannot carry electromagnetic charge and the full topological treatment is appropriate in this case [73]. In addition to the ELKO spinor fields λ(x) — that was indeed defined as sections in the bundle PSpine 1,3 (M) ×ρ C4, in section 2 — one can get a second type of ELKO λ̊(x), which can be described by sections in the inequivalent spin structure-induced spinor bundle P̊Spine 1,3 (M) ×ρ C4, (4.1) with a variation of the covariant derivative, given by [36] ∇̊X λ̊(x) = ∇X λ̊(x) − 1 2 [ Xy ( ξ−1(x)dξ(x) )] λ̊(x), (4.2) where X denotes a vector field in M . In general, the exotic term in eq. (4.2) is assumed — in order to be an integer of a Čech cohomology class — to be indeed 1 2πi ( ξ−1(x)dξ(x) ) [30, 36, 37, 40, 41]. We henceforth redefine ξ(x) 7→ ξ(x)/ √ 2π in such a way that the exotic Dirac operator can be written as (see eq. (2.4)) iγµ∇̊µ = iγµ∇µ + ξ−1(x)dξ(x). (4.3) The exotic Dirac equation is given by (iγµ∇µ + (ξ−1(x) dξ(x)) −mI)ψ(x) = 0, where ψ denotes a Dirac spinor field. The exotic Dirac spinor fields are annihilated by ( iγµ∇µ + (ξ−1(x) dξ(x)) ±mI ) { For particles: ( iγµ∇µ + (ξ−1(x) dξ(x)) −mI ) u(x) = 0 , For antiparticles: ( iγµ∇µ + (ξ−1(x) dξ(x)) +mI ) v(x) = 0 . (4.4) Hereon we denote ξ−1(x) dξ(x) by a(x) in order to shorten all formulæ notations. Now it is straightforward to show that ELKO can not be eigenspinors of the exotic Dirac operator iγµ∇µ + a(x). Indeed, denoting e :=   e1 e2 e3 e4   , d :=   d1 d2 d3 d4   , and Γ := I ⊗ (iγµ∇µ + a(x)), eq. (3.8) becomes e = Ωd. Using [Γ,Ω] = 0 yields Γe = ΩΓd. Eqs. (4.4) imply Γd = mγ5 ⊗ I d and then Γe = Ω ( mγ5 ⊗ I ) Ω−1e. An explicit evaluation of µ := Ω ( mγ5 ⊗ I ) Ω−1 reveals µ = m ( σ2 O O −σ2 ) ⊗ I . – 11 – 28 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 Thus, making the direct product explicit again, finally one reaches the result   iγµ∇µ+a(x) O O O O iγµ∇µ+a(x) O O O O iγµ∇µ+a(x) O O O O iγµ∇µ+a(x)     λ̊S {−,+} λ̊S {+,−} λ̊A {−,+} λ̊A {+,−}   −imI   −̊λS {+,−} λ̊S {−,+} λ̊A {+,−} −̊λA {−,+}   =0 (4.5) which establishes that (iγµ∇µ + a(x) ±mI) do not annihilate the ELKO (dark) spinor fields. The antisymmetric symbol defined as ε {−,+} {+,−} := −1, the above equations reduces to ( (iγµ∇µ+a(x))δβα+mIεβα ) λ̊S β(x) = 0, ( (iγµ∇µ+a(x))δβα−mIεβα ) λ̊A β (x) = 0, (4.6) which are the inherent counterparts of eqs. (4.4). The term of δβα is iγµ∇µ + a(x), and the existence of εβα in the mass term forbids ELKO spinor fields to be eigenspinors of the iγµ∇µ + a(x) operator. Namely, the mass terms carry opposite signs and consequently ELKO cannot be annihilated by (iγµ∇µ + a(x) ±mI), because the term εβα in eq. (4.6), which implies that ǫS = −1 and ǫA = +1. Furthermore, as comprehensively discussed in, e.g., [36, 87], we can express ξ(x) = exp(iθ(x)) ∈ U(1), x ∈ M . The exotic spin structure term in this way reads ξ−1(x)dξ(x) = exp(−iθ(x))(iγµ∇µθ(x)) exp(iθ(x)) = iγµ∂µθ(x). (4.7) From eq. (4.7), eqs. (4.6) are written as ( (iγµ∇µ + iγµ∂µθ)δ β α ±mIεβα ) λ̊ S/A β (x) = 0 . (4.8) The exotic Dirac operator iγµ∇µ + iγµ∂µθ −mI, annihilates each of the four exotic Dirac spinor fields u±(x) and v±(x), but as the wave operator in (4.8) couples the {±,∓} degrees of freedom such exotic Dirac operator does not annihilate ELKO. Much has been extensively discussed about the subtle differences between Majorana and ELKO spinor fields, see e. g., [74]. Both in the Lounesto spinor field classification are type-(5) spinor fields, satisfying (3). We now shall discuss whether the exotic Dirac operator can be considered as a square root of the Klein–Gordon operator – in the sense that (iγµ∇µ + iγµ∂µθ −mI)(iγµ∇µ + iγµ∂µθ +mI) = (gµν∇µ∇ν +m2)I. This feature must remain true for the ELKO and its exotic partner: ((iγµ∇µ+iγµ∂µθ)δ β α±mIεβα)((iγµ∇µ+iγµ∂µθ)δ β α ∓mIεβα)=(gµν∇µ∇ν+m2)I δβα, (4.9) since the introduction of an exotic spin structure does not modify the Klein–Gordon prop- agator fulfillment by dark spinor fields. The corresponding Klein-Gordon equation is given by (� +m2 + gµν∇µ∇νθ + ∂µθ∇µ + ∂µθ∂µθ)̊λ(x) S/A {±,∓} = 0, (4.10) where � denotes the square of the spin-Dirac operator, that can be related to the Laplace- Beltrami operator by the Lichnerowicz formula [88–90]. In order that the Klein-Gordon – 12 – 29 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 propagator for the exotic ELKO remains the same as the standard Klein-Gordon propagator for the ELKO spinor field, from eq. (4.10) it follows that (�θ + ∂µθ∇µ + ∂µθ∂µθ)̊λ S/A {±,∓}(x) = 0. (4.11) Explicitly, for consistency with the standard formalism it can be written that λ̊(x) = ( σ2φ ∗(x) φ(x) ) , where φ(x) = ( α(x) β(x) ) , α(x), β(x) ∈ C, (4.12) implying that ( �θ + i∂µθ∇µ − i∂µθ∂µθ 0 0 �θ + i∂µθ∇µ − i∂µθ∂µθ )( β − iβ∗ α+ iα∗ ) = ( 0 0 ) . (4.13) Still, the carrier of the representation space can be written as λ̊ S/A {±,∓}(x) = ( (β1 ∓ β2) exp(∓iπ4 ) (α1 ± α2) exp(±iπ4 ) ) , β = β1 + iβ2, α = α1 + iα2. (4.14) Note that the condition in eq. (4.13) is independent of the function θ(x) in the case where Im(α) = −Re(α) and Im(β) = Re(β), by eq. (4.14). As this condition is too stringent, since we want to analyze the function θ(x) for an arbitrary ELKO and not for such so particular case, we demand the most general condition given by eq. (4.11) when arbitrary exotic dark spinor fields are taken into account, since the general case must be formulated without restricting the theory on any particular case as in eq. (4.14). Our analysis hereon sheds new light on the character of the function θ — that is a priori arbitrary — that defines the exotic topology. Furthermore, it delves into the way how the exotic topology can be constrained to the spacetime metric structure, via the dynamics of exotic ELKO spinor fields. Since eq. (4.11) holds for every exotic dark spinor field λ̊ S/A {±,∓}(x), in particular let us analyze the solutions of eq. (4.11) applied to, for instance, λ̊S{−,+}(x). We omit hereon the argument “(x)” for simplicity. Using the expression8 ∇µλ̊ S/A {∓,±} = ∂µλ̊ S/A {∓,±} − 1 4 Γµρσγ ργσλ̊ S/A {∓,±}, (4.15) 8Assume a metric compatible covariant derivative operator, we emphasize here that the connection is not required to be symmetric, and it is decomposed into the Christoffel symbol and the contortion tensor. – 13 – 30 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 for such case, after some calculation9 — denoting x0 = t — it follows that (�θ) λ̊S/A{∓,±} + (∂0θ) [ ∂0λ̊ S/A {∓,±} − 1 4 ( (Γ000 − Γ011 − Γ022 − Γ033)̊λ S/A {∓,±} + iΓ001λ̊ A/S {±,∓} +Γ002λ̊ S/A {±,∓} ∓ Γ003λ̊ S/A {∓,±} ± iΓ012λ̊ A/S {∓,±} + iΓ013λ̊ A/S {±,∓} ∓ Γ023λ̊ S/A {±,∓} )] −g00(∂0θ) 2λ̊ S/A {∓,±} = 0 (4.16) The equation above couples again all the four exotic spinor fields λ̊ S/A {±,∓}, in the case of spacetimes which the associated connection are non zero. As proposed in, e. g., [12–14, 21, 22], it is possible for cosmological applications, to assume that the dark spinor fields depend only on the time variable t via a matter field κ(t) compatible with homogeneity and isotropy [22] and acts as the only dynamical cosmological variable, in such a way that λ̊ S/A {±,∓}(x) can be explicitly written as λ̊ A/S {−,+}(x) = κ(t)χ A/S {−,+}, λ̊ A/S {+,−}(x) = κ(t) ζ A/S {+,−}, (4.17) where ζS/A and χS/A are linearly independent constant spinor fields given by [22] χS {−,+} =   0 i 1 0   , χA {−,+} =   0 −i 1 0   , ζS{+,−} =   1 0 0 −i   , ζA{+,−} = −   1 0 0 i   . (4.18) The matter field κ(t) was introduced and satisfies a first order ordinary differential equation in time derivative, involving the time component of the total energy-momentum tensor Σtt, the Planck mass, and the Hubble constant. In the limits proposed in [22] we can write κ̇ κ = −1 3 √ 1 3M2 Pl Σtt + O(κ4). (4.19) where M−2 Pl = 8πG is the coupling constant. The last term in the right hand side of the equation above is in ref. [22] kept apart, and such an approximation gives robust cosmological results in full compliance with the references therein. The most general case shall be considered still in this section. Therefore we can write κ(t) = exp(at), where a is the constant given in the equation above. Using now eqs. (4.17) and (4.18), and considering each one of the four exotic dark 9Here we consider the torsionless connection. For the torsion case it must be written ∇µλ = ∂µλ − 1 4 Γµρσ[γµ, γσ]λ+ 1 4 Kµρσγ ργσλ, where Kµρσ are the contorsion tensor coefficients. Such case is used in the analysis of dark spinor fields in Cosmology in [19, 20, 22], and it is important to remark that in the presence of torsion an additional dynamical term appears [91]. Such more general formalism for while is unnecessary here, since our main aim now is to verify that exotic dark spinor fields dynamics indeed can constraint the metric spacetime structure. By now, we just call some attention to the fact that the torsion fields may act in order to cancel the connection effects in the constraint equation. – 14 – 31 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 spinor λ S/A {−,+} components in eq. (4.16), we have the system    (−iΓ001 + Γ002 − iΓ013 − Γ023)∂0θ = 0 i�θ+∂0θ ( i− 1 4 (iΓ000−iΓ011−iΓ022−iΓ033+Γ012−iΓ003) ) −i(∂0θ) 2 = 0 �θ+∂0θ ( 1− 1 4 (Γ000−Γ011−Γ022−Γ033+iΓ012−Γ003) ) −(∂0θ) 2 = 0 (Γ001 − iΓ002 + Γ013 + iΓ023)∂0θ = 0. (4.20) The first and fourth equations above together imply that (∂0θ)Γ012 = 0, (4.21) what means that if θ is time dependent, it necessarily means that Γ012 = 0. Otherwise, in the case where θ does not depends on time, it implies that ∂0θ = 0, and then we obtain the Laplace equation for θ ∇2θ = 0. (4.22) It is worthwhile to note, by passing, that eq. (4.16) in spacetimes where the connection symbols above are zero — in the Minkowski space with Cartesian coordinates, for instance — is reduced to ( �θ + (∂0θ)a− (∂0θ) 2 ) λ̊ S/A {∓,±} = 0, (4.23) and in this way the dark spinor field dynamics imposes constraints only on the topological sector determined by θ, and there is no coupling among the four exotic spinor fields. On the another hand, the second and the third equations in the system above together imply that �θ + (∂0θ) ( 1 − 1 2 (Γ000 − Γ011 − Γ022 − Γ033 − Γ003) ) − (∂0θ) 2 = 0, (4.24) what means that if θ = θ(t), so necessarily 4 − (Γ000 − Γ011 − Γ022 − Γ033 − Γ003) = ∂0θ. Otherwise, again eq. (4.22) holds. Now, using eqs. (4.17) and (4.18) and considering each one of the four exotic dark spinor λ S/A {+,−} components in eq. (4.16), we have the same results as for the λ S/A {−,+}. It shows that the exotic topology induces constraints in the spacetime geometry, coming from the dynamics of dark spinor fields. Indeed it is the case in such an approach when the function θ(x) that generates the exotic structure — realized by eq. (4.7) — is most general, time dependent. As previously observed, eq. (4.15) was solved for λ̊ S/A {∓,±}(x), in order to illustrate the exotic dark spinor fields dynamics. It evinces the constraints either on the spacetime metric structure — given an arbitrary 1-form field in spacetime, manifestation of the exotic topology encrypted in the term θ(x) in eq. (4.7) — or on the exotic parameter θ(x). To the most general case, it is not necessary indeed to consider any particular case about κ(t), and the system (4.20) is written as    (−iΓ001 + Γ002 − iΓ013 − Γ023)∂0θ = 0 �θ−∂0θ ( 1 4 (Γ000−Γ011−Γ022−Γ033+iΓ012−Γ003) ) −(∂0θ) 2 = −(∂0θ) κ̇(t) κ(t) �θ+∂0θ ( −1 4 (Γ000−Γ011−Γ022−Γ033+iΓ012−Γ003) ) −(∂0θ) 2 = −(∂0θ) κ̇(t) κ(t) (Γ001 − iΓ002 + Γ013 + iΓ023)∂0θ = 0. (4.25) – 15 – 32 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 The analysis that evinces the constraints among topological and geometrical terms is simi- lar, except for the term −(∂0θ) κ̇(t) κ(t) on the right hand side of the second and third equations above. For instance, in the case previously analyzed, all terms in eq. (4.19) are given by κ̇ κ = − √ 1/M2 pl 4 √ 3 ( 8 + 3κ4/M4 pl 12 − κ4/M4 pl )√ 4 − κ4/M4 pl, (4.26) and cogently the exotic dark spinor fields dynamics constraints the spacetime topology, or the spacetime metric structure, or both, whatever the form of κ(t), and also even for the most general dark spinor fields λ̊ S/A {∓,±}, predicted by eq. (4.11). 5 Concluding remarks and outlook Given an a priori arbitrary manifold M with non-trivial topology, and using the fact that the inequivalent spin structures give rise to the exotic term endowing the Dirac operator — in our analysis, the exotic term in the Dirac operator (evinced when an arbitrary inequivalent spin structure is taken into account) — we have shown that the exotic dark spinor fields dynamics indeed can constraint the metric spacetime structure. Such constraints can be mitigated for some particular choices of the exotic term θ in (4.7) — but in the most general case both the spacetime metric structure and the non-trivial topology are constrained by the exotic dark spinor field dynamics. Much has been discussed the about equations constraining the dynamics and the spinor structures, and some questions were addressed about the validity of Klein-Gordon propaga- tor globally, but not locally, for dark spinor fields [12–14]. The formalism here introduced is promising to derive and provide open questions on the dark spinor fields models structures and their subsequent application in cosmology — in particular the dark matter problem. Eq. (4.17) is successful to decouple topological terms evinced by the exotic θ function and the geometrical terms given by the connection symbols, in some particular cases an- alyzed from eq. (4.20) on. As in such situations the connection symbols are constrained, it also induces constraints among Christoffel symbols and contorsion tensor components, in the case where torsion is taken into account in the covariant derivative. In addition, eq. (4.16) is the most general coupling between topological and geometrical terms when no particular exotic dark spinor field is considered. Besides analyzing the exotic dark spinor fields elicited from a non-trivial topology endowed manifold, such additional term in the Dirac operator may be useful to solve some open questions addressed in the current literature [1, 2, 5–7, 9, 12–14, 21–23, 25, 26]. In certain sense, that idea that is in the background of the theoretical tools through which one set the quoted constraints can be identified with the problem of constraints in gauge theories known as the Velo-Zwanziger problem [92]. In the context of such theories, to avoid algebraic inconsistencies originated from a kind of exotic interactions, one sets the constraints independently of the equations of motion. Furthermore, in [93] it is used a similar prescription to introduce a more suitable Dirac operator, that can be also related to the one introduced in [94]. In some cases, the Lagrangian device by itself does not provide – 16 – 33 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 satisfactory wave equations [93], a problem that is given an adequate interpretation, and we expect to have overcome. Acknowledgments R. da Rocha is grateful to Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico (CNPq) grants 472903/2008-0 and 304862/2009-6 for financial support. A. E. B. would like to thank the financial support FAPESP 2008/50671-0 and CNPq grant 300233/2010-8. A Clifford bundles One thus introduces the Clifford bundle of differential forms Cℓ(M,g), which is a vector bundle associated with PSpine 1,3 (M) [90, 95]. Sections of the Clifford bundles are sums of non-homogeneous differential forms, called Clifford fields. Remember that Cℓ(M,g) = PSOe 1,3 (M) ×Ad′ Cℓ1,3, where Cℓ1,3 ≃ M(2,H) is the spacetime algebra and H denotes the quaternions. Details of the bundle structure are as follows: 1. Let πc : Cℓ(M,g) → M be the canonical projection of Cℓ(M,g) and let ∪i∈IUi be an open simple covering of M , together with a set of transition functions aij : Ui ∩Uj → SOe 1,3 such that aij ◦ ajk = aik in Ui ∩ Uj ∩ Uk and ajj = id. There are trivialization mappings ψi : π−1 c (Ui) → Ui×Cℓ1,3 of the form ψi(p) = (πc(p), ψi,x(p)) = (x, ψi,x(p)). If x ∈ Ui ∩ Uj and p ∈ π−1 c (x), then ψi,x(p) = hij(x)ψj,x(p), for hij(x) ∈ Aut(Cℓ1,3), where hij : Ui ∩ Uj → Aut(Cℓ1,3) are the transition map- pings of Cℓ(M,g). Since every automorphism of Cℓ1,3 is inner, then hij(x)ψj,x(p) = aij(x)ψi,x(p)aij(x) −1 for some aij(x) ∈ Cℓ⋆1,3, the group of invertible elements of Cℓ1,3. In particular, a spin structure (PSpine 1,3 (M), s) on M is precisely comprised by the system of transition functions hij : Ui ∩ Uj → Spine 1,3 such that ς ◦ hij = aij, hij ◦ hjk = hik, hii = id, (A.1) where ς is defined in eq. (A.2). 2. Since Cℓ⋆1,3 acts naturally on Cℓ1,3 as an algebra automorphism through its adjoint representation, the group SOe 1,3 has a natural extension in the Clifford algebra Cℓ1,3. A set of lifts of the transition functions in Cℓ(M,g) is a set of elements {aij} ⊂ Cℓ⋆1,3 such that, if10 Ad : φ 7→ Adφ Adφ(Ξ) = φΞφ−1, ∀Ξ ∈ Cℓ1,3, then Adaij = hij in all intersections. 10It is well known that Spine 1,3 = {φ ∈ Cℓ01,3 : φφ̃ = 1} ≃ SL(2,C) is the universal covering group of the restricted Lorentz group SOe 1,3. Notice that Cℓ01,3 ≃ Cℓ3,0 ≃ M(2,C), the even subalgebra of Cℓ1,3 is the Pauli algebra. – 17 – 34 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 3. The application Ad|Spine 1,3 defines a group homomorphism ς : Spine 1,3 → SOe 1,3, which is onto and ker ς = Z2. (A.2) Then Ad±1 = identity, and Ad : Spine 1,3 → Aut(Cℓ1,3) descends to a representation of SOe 1,3. Let us call Ad′ this representation, i.e., Ad′ : SOe 1,3 → Aut(Cℓ1,3). Then Ad′ ς(φ)Ξ = AdφΞ = φΞφ−1. 4. It is clear that the structure group of the Clifford bundle Cℓ(M,g) is reducible from Aut(Cℓ1,3) to SOe 1,3. The transition maps of the principal bundle PSOe 1,3 (M) can thus be — through Ad′ — taken as transition maps for the Clifford bundle. It follows that [63–65] Cℓ(M,g) = PSOe 1,3 (M) ×Ad′ Cℓ1,3, i.e., the Clifford bundle is a vector bundle associated with the principal bundle PSOe 1,3 (M) of orthonormal coframes. B Principal bundles and associated vector bundles In this section it is reviewed the main definitions and concepts of the theory of principal bundles and their associated vector bundles, which is needed to introduce the Clifford and spin-Clifford bundles used in this paper. Propositions are in general presented without proofs, which can be found, e.g., in [90, 96, 97]. A fiber bundle on a manifold M with Lie group G is denoted by (E,M, π,G, F ). E is a topological space called the total space of the bundle, π : E → M is a continuous surjective map, called the canonical projection, and F is the typical fiber. The following conditions must be satisfied: a) π−1(x), the fiber over x, is homeomorphic to F . b) Let {Ui, i ∈ I}, where I is an index set, be a covering of M , such that: b1) Locally a fiber bundle E is trivial, namely it is diffeomorphic to a product bundle π−1(Ui) ≃ Ui × F for all i ∈ I. b2) The diffeomorphisms Φi : π−1(Ui) → Ui × F have the form Φi(p) = (π(p), φi,x(p)), φi|π−1(x) ≡ φi,x : π−1(x) → F is onto (B.1) The collection {(Ui,Φi)}, i ∈ I, are said to be a family of local trivializations for E. b3) The group G acts on the typical fiber. Considering x ∈ Ui∩Uj , then, φj,x◦φ−1 i,x : F → F must coincide with the action of an element of G, for all x ∈ Ui ∩ Uj and i, j ∈ I. b4) One calls transition functions of the bundle the continuous induced mappings aij : Ui ∩ Uj → G, where aij(x) = φi,x ◦ φ−1 j,x. (B.2) – 18 – 35 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 For consistence of the theory the transition functions must satisfy the cocycle condi- tion aij(x)ajk(x) = aik(x). The 5-tuple (P,M, π,G, F ≡ G) ≡ (P,M, π,G) is called a principal fiber bundle (PFB) if all the conditions about fiber bundles are fulfilled and, moreover, if there is a right action of G on elements p ∈ P , such that: a) the mapping (defining the right action) P ×G ∋ (p, g) 7→ pg ∈ P is continuous. b) given g, g′ ∈ G and ∀p ∈ P , (pg)g′ = p(gg′). c) ∀x ∈ M,π−1(x) is invariant under the action of G: each element of p ∈ π−1(x) is mapped into pg ∈ π−1(x), i.e., it is mapped into an element of the same fiber. d) G acts free and transitively on each fiber π−1(x), which means that all elements within π−1(x) are obtained by the action of all the elements of G on any given element of the fiber π−1(x). This condition is, of course, necessary for the identification of the typical fiber with G. A bundle (E,M, π, G = GL(n,K), V ), where K = R or C, and V is an n-dimensional vector space over K is called a vector bundle. A vector bundle (E,M, π,G, F ) denoted E = P ×ρF is said to be associated to a PFB bundle (P,M, π,G) by the linear representation ρ : G → GL(V ) — which is called the carrier space of the representation — if its transition functions are the images under ρ of the corresponding transition functions of the PFB (P,M, π,G). This precisely means the following: consider the following local trivializations of P and E respectively Φi :π−1(Ui) → Ui ×G, Ξi : π−1 1 (Ui) → Ui × F, (B.3) Ξi(q) = (π1(q), χi(q)) = (x, χi(q)), χi|π−1 1 (x) ≡ χi,x : π−1 1 (x) → F, (B.4) where π1 : P ×ρ F → M is the projection of the bundle associated to (P,M, π,G). Then, for all x ∈ Ui ∩ Uj , i, j ∈ I, it follows that χj,x ◦ χ−1 i,x = ρ(φj,x ◦ φ−1 i,x ). (B.5) In addition, the fibers π−1(x) are vector spaces isomorphic to the representation space V . Let (E,M, π,G, F ) be a fiber bundle and U ⊂ M an open set. A local section of the fiber bundle (E,M, π,G, F ) on U is a mapping s : U → E such that π ◦ s = IdU , (B.6) If U = M s is said to be a global section. There is a relation between sections and local trivializations for principal bundles. Indeed, each local section s (on Ui ⊂ M) for a principal bundle (P,M, π,G) determines a local trivialization Φi : π−1(U) → U × G, of P by setting Φ−1 i (x, g) = s(x)g = pg = Rgp. Conversely, Φi determines s since s(x) = Φ−1 i (x, e). (B.7) – 19 – 36 J H E P 0 4 ( 2 0 1 1 ) 1 1 0 A principal bundle is trivial if and only if it has a global cross section. A vector bundle is trivial if and only if its associated principal bundle is trivial. Any fiber bundle (E,M, π,G, F ) such that M is a paracompact manifold and the fiber F is a vector space admits a cross section. Then, any vector bundle associated to a trivial principal bundle has non-zero global sections. Note however that a vector bundle may admit a non-zero global section even if it is not trivial. Indeed, as shown in the main text, any Clifford bundle possesses a global identity section, and some spin-Clifford bundles admits also identity sections once a trivialization is given. References [1] D.V. Ahluwalia and D. G