Constant Spinors Research Articles

The translational μ-τ reflection symmetry of Majorana neutrinos

The present neutrino oscillation data allow [Formula: see text] (or [Formula: see text]) for the neutrino mass spectrum and support [Formula: see text] and [Formula: see text] as two good approximations for the PMNS lepton flavor mixing matrix [Formula: see text]. We show that these intriguing possibilities can be a very natural consequence of the translational[Formula: see text]-[Formula: see text] reflection symmetry — the effective Majorana neutrino mass term keeps invariant under the transformations [Formula: see text], [Formula: see text] and [Formula: see text] (for [Formula: see text] or 3), where [Formula: see text] is the charge conjugation of a constant spinor field [Formula: see text]. By extending such a working flavor symmetry to the canonical seesaw mechanism at a superhigh-energy scale, we calculate its soft breaking effects at the electroweak scale by using the one-loop renormalization-group equations.

On the Wess-Zumino Model: a supersymmetric field theory

We consider the free massless Wess-Zumino Model in 4D which describes a supersymmetric field theory that is invariant under the rigid or global supersymmetry transformations where the transformation parameter ϵ (or ϵ ) is a constant Grassmann spinor. We quantize the theory using the Hamiltonian and path integral formulations.

A translational flavor symmetry in the mass terms of Dirac and Majorana fermions

Requiring the effective mass term for a category of fundamental Dirac or Majorana fermions of the same electric charge to be invariant under the translational transformations ψ αL(R) → ψ αL(R) + n α z ψL(R) in the flavor space, where n α and z ψL(R) stand respectively for the flavor-dependent complex numbers and a constant spinor field anticommuting with the fermion fields, we show that n α can be identified as the elements U αi in the ith column of the unitary matrix U used to diagonalize the corresponding Hermitian or symmetric fermion mass matrix M ψ , and m i = 0 holds accordingly. We find that the reverse is also true. Now that the mass spectra of charged leptons, up- and down-type quarks are all strongly hierarchical and current experimental data allow the lightest neutrino to be massless, we argue that the zero mass limit for the first-family fermions and the translational flavor symmetry behind it should be a natural starting point for building viable fermion mass models.

Supersymmetric spindles

In the context of holography, we analyse aspects of supersymmetric geometries based on two-dimensional orbifolds known as spindles. By analysing spinc spinors on a spindle with an azimuthal rotation symmetry we show that under rather general conditions there are just two possibilities, called the ‘twist’ and the ‘anti-twist’, which are determined by the quantized magnetic flux through the spindle. A special case of the twist is the standard topological twist which is associated with constant and chiral spinors. We construct solutions of D = 5 and D = 4 STU gauged supergravity theories that are dual to D3-branes and M2-branes wrapping spindles, respectively, which realize both the anti-twist, as seen before, but also the twist. For the wrapped D3-brane solutions we reproduce the central charge of the gravity solution from the dual field theory by analysing the anomaly polynomial of mathcal{N} = 4 SYM theory. We also discuss M5-branes wrapped on spindles both from a gravity and a field theory point of view.

Nothing really matters

We study non-perturbative instabilities of AdS spacetime in General Relativity with a cosmological constant in arbitrary dimensions. In this simple setup we explicitly construct a class of gravitational instantons generalizing Witten’s bubble of nothing. We calculate the corresponding Euclidean action and show that its change is finite. The expansion of these bubbles is described by a lower-dimensional de Sitter geometry within a non- compact foliation of the background spacetime. Moreover we discuss the existence of covariantly constant spinors as a possible topological obstruction for such decays to occur. This mechanism is further connected to the stability of supersymmetric vacua in string theory.

Supersymmetric 3D model for gravity with SU(2) gauge symmetry, mass generation and effective cosmological constant

A Chern–Simons system in 2+1 dimensions invariant under local Lorentz rotations, SU(2) gauge transformations, and local supersymmetry (SUSY) transformations is proposed. The field content is that of (2+1)-gravity plus an SU(2) gauge field, a spin-1/2 fermion charged with respect to SU(2) and a trivial free abelian gauge field. A peculiarity of the model is the absence of gravitini, although it includes gravity and SUSY. Likewise, no gauginos are present. All the parameters involved in the system are either protected by gauge invariance or emerge as integration constants. An effective mass and effective cosmological constant emerge by spontaneous breaking of local scaling invariance. The vacuum sector is defined by configurations with locally flat Lorentz and SU(2) connections sporting nontrivial global charges. Three-dimensional Lorentz-flat geometries are spacetimes of locally constant negative—or zero—Riemann curvature, which include Minkowski space, AdS3, BTZ black holes, and point particles. These solutions admit different numbers of globally defined, covariantly constant spinors and are therefore good candidates for stable ground states. The fermionic sector in this system could describe the dynamics of electrons in graphene in the long wavelength limit near the Dirac points, with the spin degree of freedom of the electrons represented by the SU(2) label. If this is the case, the SU(2) gauge field would produce a spin–spin interaction giving rise to strong correlation of electron pairs.

On the Mechanism of Fermion-Boson Transformation

We study a fermion-boson transformation. Our approach is based on the 3 × 3 equations which are subequations of both the Dirac and Duffin-Kemmer-Petiau equations and thus provide a link between these equations. We show that solutions of the free Dirac equation can be converted to solutions of spin- 0 Duffin-Kemmer-Petiau equation and vice versa. Mechanism of this transition assumes existence of a constant spinor.

Witten Spinors on Nonspin Manifolds

Motivated by Witten’s spinor proof of the positive mass theorem, we analyze asymptotically constant harmonic spinors on complete asymptotically flat nonspin manifolds with nonnegative scalar curvature.

Conformally-coupled ELKO Spinor and Scalar-tensor Gravity

We suggest that the theory with conformal coupling of ELKO dark spinor fields to gravity belongs to a class of scalar-tensor theories of gravity. When the dark spinor can be decomposed into a homogeneous real scalar field and a constant spinor field in Robertson-Walker spacetime, we study cosmology by considering the renormalizable potential of the field. With the energy density and the pressure of the spinor calculated, we find solutions for a(t) in the case where the scalar field is assumed to take a special form. We discuss the possibility that both the inflaton- and the dark energy-dominated eras of our universe can be described by using the dark spinor.

Witten spinors on maximal, conformally flat hypersurfaces

The boundary conditions that exclude zeros of the solutions of the Witten equation (and hence guarantee the existence of a 3-frame satisfying the so-called special orthonormal frame gauge conditions) are investigated. We determine the general form of the conformally invariant boundary conditions for the Witten equation, and find the boundary conditions that characterize the constant and the conformally constant spinor fields among the solutions of the Witten equations on compact domains in extrinsically and intrinsically flat, and on maximal, intrinsically globally conformally flat spacelike hypersurfaces, respectively. We also provide a number of exact solutions of the Witten equation with various boundary conditions (both at infinity and on inner or outer boundaries) that single out nowhere vanishing spinor fields on the flat, non-extreme Reissner–Nordström and Brill–Lindquist data sets. Our examples show that there is an interplay between the boundary conditions, the global topology of the hypersurface and the existence/non-existence of zeros of the solutions of the Witten equation.

A new proposal for the picture changing operators in the minimal pure spinor formalism

Using a new proposal for the "picture lowering" operators, we compute the tree level scattering amplitude in the minimal pure spinor formalism by performing the integration over the pure spinor space as a multidimensional Cauchy-type integral. The amplitude will be written in terms of the projective pure spinor variables, which turns out to be useful to relate rigorously the minimal and non-minimal versions of the pure spinor formalism. The natural language for relating these formalisms is the Cech-Dolbeault isomorphism. Moreover, the Dolbeault cocycle corresponding to the tree-level scattering amplitude must be evaluated in SO(10)/SU(5) instead of the whole pure spinor space, which means that the origin is removed from this space. Also, the Cech-Dolbeault language plays a key role for proving the invariance of the scattering amplitude under BRST, Lorentz and supersymmetry transformations, as well as the decoupling of unphysical states. We also relate the Green's function for the massless scalar field in ten dimensions to the tree-level scattering amplitude and comment about the scattering amplitude at higher orders. In contrast with the traditional picture lowering operators, with our new proposal the tree level scattering amplitude is independent of the constant spinors introduced to define them and the BRST exact terms decouple without integrating over these constant spinors.

Gravitational radiation and angular momentum flux from a slowly rotating dynamical black hole

A four-dimensional asymptotic expansion scheme is used to study the next-order effects of the nonlinearity near a spinning dynamical black hole. The angular-momentum flux and energy flux formula are then obtained by constructing the reference frame in terms of the compatible constant spinors and the compatibility of the coupling leading-order Newman-Penrose equations. By using the slow rotation and small-tide approximation for a spinning black hole, the horizon cross-section we chose is spherical symmetric. It turns out the flux formula is rather simple and can be compared with the known results. Directly from the energy flux formula of the slow-rotating dynamical horizon, we find that the physically reasonable condition on requiring the positivity of the gravitational energy flux yields that the shear will monotonically decrease with time. Thus a slow-rotating dynamical horizon will asymptotically approach an isolated horizon during late time.

Cosmological Einstein–Maxwell instantons and Euclidean supersymmetry: anti-self-dual solutions

We classify super-symmetric solutions of the minimal N = 2 gauged Euclidean supergravity in four dimensions. The solutions with an anti-self-dual Maxwell field give rise to anti-self-dual Einstein metrics given in terms of solutions to the SU(∞) Toda equation and more general three-dimensional Einstein–Weyl structures. Euclidean Kastor–Traschen metrics are also characterized by the existence of a certain supercovariantly constant spinor.

Gravitational radiations of generic isolated horizons and nonrotating dynamical horizons from asymptotic expansions

Instead of using a three-dimensional analysis on quasilocal horizons, we adopt a four-dimensional asymptotic expansion analysis to study the next order contributions from the nonlinearity of general relativity. From the similarity between null infinity and horizons, the proper reference frames are chosen from the compatible constant spinors for an observer to measure the energy-momentum and flux near quasilocal horizons. In particular, we focus on the similarity of Bondi-Sachs gravitational radiation for the quasilocal horizons and compare our results with Ashtekar-Kirshnan flux formula. The quasilocal energy-momentum and flux of generic isolated horizons and nonrotating dynamical horizons are discussed in this paper.

Negative spectrum of the2+1black hole

In ($2+1$)-dimensional gravity with negative cosmological constant, the states in the negative energy range, between anti-de Sitter (AdS) ($M=\ensuremath{-}1$) and the so-called Ba\~nados-Teitelboim-Zanelli black hole ($M\ensuremath{\ge}0$), correspond to topological defects with angular deficit $0<\ensuremath{\alpha}<2\ensuremath{\pi}$. These defects are produced by (static or spinning) 0-branes which, in the extreme case $M\ensuremath{\ell}=\ensuremath{-}|J|$, admit globally-defined covariantly constant spinors. Thus, these branes correspond to Bogomol'nyi-Prasad-Sommerfield solitons and are stable ground state candidates for the corresponding supersymmetric extension of $2+1$ AdS gravity. These branes constitute external currents that couple in a gauge-invariant way to three-dimensional AdS gravity.

Generalised G2-structures and type IIB superstrings

The recent mathematical literature introduces generalised geometries which are defined by a reduction from the structure group SO(d,d) of the vector bundle Td⊕Td* to a special subgroup. In this article we show that compactification of IIB superstring vacua on 7-manifolds with two covariantly constant spinors leads to a generalised G2-structure associated with a reduction from SO(7,7) to G2 × G2. We also consider compactifications on 6-manifolds where analogously we obtain a generalised SU(3)-structure associated with SU(3) × SU(3), and show how these relate to generalised G2-structures.

Supersymmetry and Lorentzian holonomy in various dimensions

We present a systematic method for constructing manifolds with Lorentzian holonomy group that are non-static supersymmetric vacua admitting covariantly constant light-like spinors. It is based on the metric of their Riemannian counterparts and the realization that, when certain conditions are satisfied, it is possible to promote constant moduli parameters into arbitrary functions of the light-cone time. Besides the general formalism, we present in detail several examples in various dimensions.

PP-waves, [formula omitted]-theory and fluxes

We study a new type of warped compactifications of M -theory on eight-manifolds for which nowhere vanishing covariantly constant spinors of indefinite chirality on the internal manifold can be found. We derive the constraints on the fluxes and the warp factor following from supersymmetry and the equations of motion. We show, that the lift of type IIB pp-waves to M -theory is a special type of solution of this general class of models. We comment on the relation between the type IIB version of such compactifications as a dual description of the Polchinski–Strassler solution describing a four-dimensional confining gauge theory.

New cohomogeneity one metrics with Spin(7) holonomy

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A 8 , is complete and non-singular on R 8 . The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S 4, and are denoted by B 8 + , B 8 − and B 8 . The metrics on B 8 + and B 8 − occur in families with a non-trivial parameter. The metric on B 8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A 8 . The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3 . We construct the covariantly constant spinor and calibrating 4-form. We also obtain an L 2-normalisable harmonic 4-form for the A 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold.

Positivity of energy in Einstein–Maxwell axion–dilaton gravity

Avoiding the problem of the existence of asymptotically constant spinors satisfying certain differential equations on a non-compact hypersurface, we presented the proof of positivity of the Arnowitt–Deser–Misner and Bondi energy in Einstein–Maxwell axion–dilaton gravity. In our approach, spinor fields defining the energy need only be defined near infinity, satisfying propagation equations there.