Spinorial Approach Research Articles

A Note on Invariant Description of SU(2)–Structures in Dimension 5

We develop an invariant approach to SU(2)–structures on spin 5–manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to SU(2). Moreover, we show how to induce quaternionic structure on the contact distribution of the considered SU(2)–structure. We show the invariance of certain components of the covariant derivative ∇φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla \\varphi $$\\end{document}, where φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} is any spinor field defining SU(2)–structure. This shows, as expected, that (at least some of) the intrinsic torsion modules can be derived invariantly with the spinorial approach. We conclude with the explicit description of the intrinsic torsion and the characteristic connection.

Spinorial Representation of Submanifolds in a Product of Space Forms

We present a method giving a spinorial characterization of an immersion into a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory for such target spaces. We also study special cases: we recover previously known results concerning immersions in mathbb {S}^2times mathbb {R} and we obtain new spinorial characterizations of immersions in mathbb {S}^2times mathbb {R}^2 and in mathbb {H}^2times mathbb {R}. We then study the theory of H=1/2 surfaces in mathbb {H}^2times mathbb {R} using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of H=1/2 surfaces in mathbb {R}^{1,2}.

Nontrivial Topology Dynamical Corrections and the Magnetic Monopole-like Effect in Minkowski Spacetime

We investigate two physical systems within a spacetime region affected by the nontrivial topology. The set-up for our analysis is a Minkowski metric perturbed by elements reflecting the topological nontriviality. These elements arise when exploring Cartan’s spinorial approach along with the exotic spinors counterpart. This evinced nontrivial topology corrections in the free particle dynamics and charged particles coupled to an external electromagnetic field. As a complement, we show the appearance of a magnetic monopole-like effect.

Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3‑dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth ‘spacetime harmonic functions’ in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved.

Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds

We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n+1. The problem of finding solutions with minimum energy has been resolved for all dimensions except for dimension 5 (n=2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach.

Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen–Yau minimal hypersurface technique and Witten’s spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten’s argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen–Yau minimal hypersurfaces.

Spinorial formulation of the GW -BSE equations and spin properties of excitons in two-dimensional transition metal dichalcogenides

In many paradigmatic materials, like Transition Metal Dichalcogenides, the role played by the spin degrees of freedom is as important as the one played by the electron-electron interaction. Thus an accurate treatment of the two effects and of their interaction is necessary for an accurate and predictive study of the optical and electronic properties of these materials. Despite the GW-BSE approach correctly accounts for electronic correlations the spin-orbit coupling effect is often neglected or treated perturbatively. Recently spinorial formulations of GW-BSE have become available in different flavours in material-science codes. Still an accurate validation and comparison of different approaches is missing. In this work we go through the derivation of non collinear GW-BSE approach. The scheme is applied to transition metal dichalcogenides comparing perturbative and full spinorial approach. Our calculations reveal that dark-bright exciton splittings are generally improved when the spin orbit coupling is included non perturbatively. The exchange-driven intravalley mixing between the A and B exciton is found to be extremely important in the case of MoSe$_2$. We finally define the excitonic spin and use it to sharply analyze the spinorial properties of Transition Metal Dichalcogenides excitonic states.

New spinorial approach to mass inequalities for black holes in general relativity

A new spinorial strategy for the construction of geometric inequalities involving the Arnowitt-Deser-Misner mass of black hole systems in general relativity is presented. This approach is based on a second order elliptic equation (the approximate twistor equation) for a valence 1 Weyl spinor. This has the advantage over other spinorial approaches to the construction of geometric inequalities based on the Sen-Witten-Dirac equation that it allows us to specify boundary conditions for the two components of the spinor. This greater control on the boundary data has the potential of giving rise to new geometric inequalities involving the mass. In particular, it is shown that the mass is bounded from below by an integral functional over a marginally outer trapped surface (MOTS) which depends on a freely specifiable valence 1 spinor. From this main inequality, by choosing the free data in an appropriate way, one obtains a new nontrivial bounds of the mass in terms of the inner expansion of the MOTS. The analysis makes use of a new formalism for the $1+1+2$ decomposition of spinorial equations.

Notes on G2: The Lie algebra and the Lie group

These notes have been prepared for the Workshop on “(Non)-existence of complex structures on S6”, celebrated in Marburg in March, 2017. The material is not intended to be original. It contains a survey about the smallest of the exceptional Lie groups: G2, its definition and different characterizations as well as its relationship to the spheres S6 and S7. With the exception of the summary of the Killing–Cartan classification, this survey is self-contained, and all the proofs are provided. Although these proofs are well-known, they are scattered, some of them are difficult to find, and others require stronger background, while we will usually stick to linear algebra arguments. The approach is algebraical, working at the Lie algebra level most often. We analyze the complex Lie algebra (and group) of type G2 as well as the two real Lie algebras of type G2, the split and the compact one. The octonion algebra will play its role, but it is not the starting point. Also, both the 3-forms approach and the spinorial approach are viewed and connected. Special emphasis is put on relating all the viewpoints by providing precise models.

The birth of E8 out of the spinors of the icosahedron.

E8 is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional (8D) space very different from the space we inhabit; for instance, the Lie group E8 features heavily in 10D superstring theory. Contrary to that point of view, here we show that the E8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3D geometry. The 240 roots of E8 arise in the 8D Clifford algebra of 3D space as a double cover of the 120 elements of the icosahedral group, generated by the root system H3. As a by-product, by restricting to even products of root vectors (spinors) in the 4D even subalgebra of the Clifford algebra, one can show that each 3D root system induces a root system in 4D, which turn out to also be exactly the exceptional 4D root systems. The spinorial point of view explains their existence as well as their unusual automorphism groups. This spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions, which opens up a novel interpretation of these phenomena in terms of spinorial geometry.

Spinors and the Weyl tensor classification in six dimensions

A spinorial approach to six-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the four-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the Goldberg-Sachs theorem. We work in complexified spaces, showing that the results for any signature can be obtained by taking the desired real slice.

UNIFIED ELECTROWEAK MODEL BASED ON SPACE–TIME AND ISOSPIN SYMMETRIES

We propose a spinorial approach to the unified electroweak interactions, in which no use is made of spontaneous symmetry breakdown. No scalar particles are needed in order to break the symmetry. No reference is made to gauge symmetry. Our approach stresses the role of space–time and isospin symmetries in the build up of the electroweak model. Internal degrees of freedom, such as isospin, are incorporated in the theory by using spinors carrying isospin indices. All vector bosons are described by a rank 2 field in the spinorial and the isospinorial indices. Leptons are accomodated in a rank 1 spinor field and in a rank 2 isospin field as well. The dynamical variables of the theory are the chiral and isochiral components of these fields.

MAGNETIC MONOPOLES AND CHIRAL ASYMMETRY

The asymmetry, between electric (E) and magnetic (H) fields of Maxwell's equation is here analyzed by using the concept of chirality. The chiral spinorial approach sets the stage for the construction of a more general theory of spin-1 particles than usual electrodynamics. Chiral components of a rank-2 spinor field are taken as the dynamic variables of the theory. A rank-2 spinor accommodates another particle (the magnetic photon). This new particle emerges naturally from chiral invariance arguments. The nonexistence, in nature, of such a particle is the reason for the nonexistence of monopoles and the asymmetry in Maxwell's equation. The existence of magnetic monopoles would restore the symmetry of Maxwell's equation. We establish, in this way, at a very formal level, the connection between magnetic monopoles and chiral asymmetry.

Chirality in the context of spin 32 particles

In this paper we apply the chiral spinorial approach to the description of spin 32 particles. Chiral components of rank 3 spinor fields are considered to be the dynamical variables of the theory. The free Lagrangian is built from the same principles as in the case of spin 1 and spin 2 particles: chiral symmetry at the free field level and chiral symmetry breakdown at the interaction level. We show how the chiral spinorial approach provides an unambiguous Lagrangian approach for massless spin 32 particles. This approach provides a fairly rich set of effective Lagrangians for the interaction of spin 32 particles.

CHIRAL ASYMMETRY AND GAUGE INVARIANCE

In this letter we address the question on whether gauge invariance can be derived from general basic principles. We show that, within the chiral spinorial approach developed by us, gauge invariance can be seen as a result of the left–right asymmetry of nature. This is valid for Abelian and non-Abelian gauge theories. In order to select the right couplings of the chiral components, we just have to require masslessness of the photon (or gluon) and renormalizability of the theory.

Physical symmetry groups and associated bundles in field theory

In his paper Some geometrical consequences of physical symmetries [8], J. Rzewuski describes in some detail invariant submanifolds of the linear representation space C 4m for the physical symmetry group: SU(2,2) x SU( m), and its subgroup P x SU( m). First we intend to give a geometric version of this paper using homogeneous spaces and our spinorial approach. In the second place we give examples of some concrete orbits by means of spinor structures considered in the modern scope and some plausible physical consequences.

Electromagnetic and gravitational Hertz potentials

With generality of complex relativity, the classical theory of the electromagnetic Hertz potentials is outlined in terms of spinors and forms. Particularly interesting are D (0,1) and D (1,0) null Hertz potentials. Then, a new spinorial approach to heavens (ℋ spaces) is proposed, which reveals in their structure the presence of the left null gravitational Hertz potential [of the type D (0,2)]. The relevant hints which follow from our results and concern the structure of the most general solutions of the Einstein vacuum equations (type G⊗G), are discussed, in particular on the level of the linearized theory.

Killing vectors in plane HH spaces

Employing the spinorial approach to the structure of hyperheavens, we integrate completely the Killing vector equations for plane (case I) hyperheavens, reducing them to one master equation of an extremely plausible form. (In this process, optimally simple gauges are demonstrated for each Petrov type.) The mechanism of generating Ernst potentials by Killing vectors is then investigated, and explicit forms are given. Also, some interesting preliminary study of Killing spinors of type D (0,k) in heavens is presented.

A spinorial approach to Palatini variational principles

The condition ▪ μ g α ▪=0 postulated in spinor analysis is shown to be deducible from a variational principle of the Palatini type. A generalized action principle which yields the Einstein equations, the Maxwell equations and the algebraic relations among derivatives of the metric quantities and connection quantities is studied. An action which leads to the Einstein equations with the energy-momentum tensor of a scalar massless field is investigated.