On a ℤ2n-Graded Version of Supersymmetry

We extend the notion of super-Minkowski space-time to include Z 2 n -graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst themselves. The mathematical framework we employ is the recently developed category of Z 2 n -manifolds understood as locally ringed spaces. The formalism we present resembles N -extended superspace (in the presence of central charges), but with some subtle differences due to the exotic nature of the grading employed.

Radiative corrections in 2 + 1 dimensional supergravity

Supergravity in 2 + 1 dimensions has a set of first-class constraints that result in two bosonic and one fermionic gauge invariances. When one uses Faddeev–Popov quantization, these gauge invariances result in four fermionic scalar ghosts and two bosonic Majorana spinor ghosts. The BRST invariance of the effective Lagrangian is found. As an example of a radiative correction, we compute the phase of the one-loop effective action in the presence of a background spin connection, and show that it vanishes. This indicates that unlike a spinor coupled to a gauge field in 2 + 1 dimensions, there is no dynamical generation of a topological mass in this model. An additional example of how a BRST invariant effective action can arise in a gauge theory is provided in Appendix B where the BRST effective action for the classical Palatini action in 1 + 1 dimensions is examined.

Majorana spinor from the point of view of geometric algebra

Majorana spinors are constructed in terms of the multivectors of relativistic Cl1,3 algebra. Such spinors are found to be multiplied by primitive idempotents which drastically change spinor properties. Running electronic waves are used to solve the real Dirac–Majorana equation transformed to Cl1,3 algebra. From the analysis of the solution it is concluded that free Majorana particles do not exist, because relativistic Cl1,3 algebra requires the massive Majorana particle to move with light velocity.

The Dynamics of the Wave Packet in the Majorana Equation

In the Majorana equation for particles with arbitrary spin, the wave packet is due not only to the uncertainty affecting position and momentum but also to the infinite components with decreasing mass forming the Majorana spinor. In this paper we prove that such components contribute to increase the spreading of the wave packet. Moreover, as occurs in the time propagation of the Dirac wave packet, also in that of Majorana the Zitterbewegung takes place, but it shows a peculiar fine structure. Finally, the group velocity always remains subluminal and the contributions due to the infinite components decrease progressively increasing the spin.

NEUTRAL MASSIVE SPIN ½ PARTICLES EMISSION IN A RINDLER SPACE

The Unruh effect for the rate of emission and absorption of neutral massive Majorana spinor particles — plausible constituents of the Dark Matter — in a Rindler space–time is thoroughly investigated. The corresponding Bogoliubov coefficients are explicitly calculated and the consistency with Fermi–Dirac statistics and the Pauli principle is actually verified.

Canonical analysis of a system with fermionic gauge symmetry

A non-abelian gauge field with a topological action is coupled to a spin-3/2 Majorana spinor. The symmetries of this model are analyzed using the Dirac constraint formalism. These symmetries include a fermionic symmetry and the algebra of these symmetries closes; it is not the algebra of supergravity. The action is invariant without the need to introduce auxiliary fields.

KALUZA–KLEIN GAUGE AND MINIMAL INTEGRABLE EXTENSION OF OSp(4|6)/(SO(1, 3) × U(3)) SIGMA-MODEL

Basing upon experience from performing double-dimensional reduction of the D = 11 supermembrane on AdS 4 × S7 background to Type IIA superstring on AdS 4 × ℂℙ3 we introduce Kaluza–Klein (partial) κ-symmetry gauge as a vanishing condition of the contributions to the D = 11 supervielbein components tangent to D = 10 space–time that are proportional to the differential of the coordinate parametrizing compact 11th space–time dimension identified with the supermembrane worldvolume compact dimension. For AdS 4 × S7 supermembrane Kaluza–Klein gauge removes half Grassmann coordinates associated with eight space–time supersymmetries, broken by the AdS 4 × ℂℙ3 superbackground, by imposing D = 3 (anti-)Majorana condition on them. The consideration relies on the realization of osp(4|8) isometry superalgebra of the AdS 4 × S7 superbackground as [Formula: see text] superconformal algebra. Requiring further vanishing of the D = 10 dilaton leaves in the sector of broken supersymmetries just two Grassmann coordinates organized into D = 3 (anti-)Majorana spinor that defines minimal SL(2, ℝ)-covariant extension of the OSp(4|6)/(SO(1, 3) × U(3)) sigma-model. Among four possibilities of such a minimal extension we consider in detail one, that corresponds to picking out D = 3 Majorana coordinate related to broken Poincaré supersymmetry, and show that the AdS 4 × ℂℙ3 superstring equations of motion in this partial κ-symmetry gauge are integrable. Also the relation between the OSp(4|6)/(SO(1, 3) × U(3)) sigma-model and the AdS 4 × ℂℙ3 superstring is revisited.

ELKO SPINOR FIELDS: LAGRANGIANS FOR GRAVITY DERIVED FROM SUPERGRAVITY

Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong — together with Majorana spinor fields — to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. Using the mapping from ELKO spinor fields to the three classes Dirac spinor fields, it is shown that the Einstein–Hilbert, the Einstein–Palatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), as the prime Lagrangian for supergravity. The Holst action is related to the Ashtekar's quantum gravity formulation. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. Furthermore we consider the necessary and sufficient conditions to map Dirac spinor fields (DSFs) to ELKO, in order to naturally extend the Standard Model to spinor fields possessing mass dimension one. As ELKO is a prime candidate to describe dark matter and can be obtained from the DSFs, via a mapping explicitly constructed that does not preserve spinor field classes, we prove that — in particular — the Einstein–Hilbert, Einstein–Palatini, and Holst actions can be derived from the QSL, as a fundamental Lagrangian for supergravity, via ELKO spinor fields. The geometric meaning of the mass dimension-transmuting operator — leading ELKO Lagrangian into the Dirac Lagrangian — is also pointed out, together with its relationship to the instanton Hopf fibration.

SUPERSYMMETRIC LORENTZ INVARIANT DEFORMATIONS OF SUPERSPACES

Lorentz invariant supersymmetric deformations of superspaces based on Moyal star product parametrized by Majorana spinor λa and Ramond Grassmannian vector [Formula: see text] in the spinor realization42 are proposed. The map of supergravity background into composite supercoordinates: [Formula: see text] valid up to the second order corrections in deformation parameter h and transforming the background dependent Lorentz noninvariant (anti)commutators of supercoordinates into their invariant Moyal brackets is revealed. We found one of the deformations to depend on the axial vector [Formula: see text] and to vanish for the θ components with the same chiralities. The deformations in the (super)twistor picture are discussed.

WHERE ARE ELKO SPINOR FIELDS IN LOUNESTO SPINOR FIELD CLASSIFICATION?

This paper proves that from the algebraic point of view ELKO spinor fields belong together with Majorana spinor fields to a wider class, the so-called flagpole spinor fields, corresponding to the class 5, according to Lounesto spinor field classification. We show moreover that algebraic constraints imply that any class 5 spinor field is such that the 2-component spinor fields entering its structure have opposite helicities. The proof of our statement is based on Lounesto general classification of all spinor fields, according to the relations and values taken by their associated bilinear covariants, and can eventually shed some new light on the algebraic investigations concerning dark matter.