Spontaneous symmetry breaking origin for the difference between time and space

We suggest that the difference between time and space is due to spontaneous symmetry breaking. In a theory with spinors the signature of the metric is related to the signature of the Lorentz group. We discuss a higher symmetry that contains pseudo-orthogonal groups with an arbitrary signature as subgroups. The fundamental asymmetry between time and space can then result as a property of the ground state rather than being put into the formulation of the theory a priori. We show how the complex structure of quantum field theory as well as gravitational field equations arise from spinor gravity--a fundamental spinor theory without a metric.     

Nonmetricity and torsion induced by dilaton gravity in two dimension

We develop a theory in which there are couplings amongst Dirac spinor, dilaton and non-Riemannian gravity and explore the nature of connection-induced dilaton couplings to gravity and Dirac spinor when the theory is reformulated in terms of the Levi-Civita connection. After presenting some exact solutions without spinors, we investigate the minimal spinor couplings to the model and in conclusion we cannot find any nontrivial dilaton couplings to spinor.     

Pure spinor integration from the collating formula

We use the technique developed by Becchi and Imbimbo to construct a well-defined BRST-invariant path integral formulation of pure spinor amplitudes. The space of pure spinors can be viewed from the algebraic geometry point of view as a collection of open sets where the constraints can be solved and a set of free and independent variables can be defined. On the intersections of those open sets, the functional measure jumps and one has to add boundary terms to construct a well-defined path integral. The result is the definition of the pure spinor integration measure constructed in terms of differential forms on each single patch.     

The Character of Pure Spinors

The character of holomorphic functions on the space of pure spinors in 10, 11 and 12 dimensions is calculated. From this character formula, we derive in a manifestly covariant way various central charges which appear in the pure spinor formalism for the superstring. We also derive in a simple way the zero momentum cohomology of the pure spinor BRST operator for the D=10 and D=11 superparticle Mathematics Subject Classifications (2000): 81T30, 83E30, 83E50     

Classification of the Weyl curvature spinors of neutral metrics in four dimensions

Neutral geometry is of increasing interest. As with Riemannian and Lorentzian geometry, spinors can be expected to provide a valuable tool in neutral geometry. For a neutral metric in four dimensions, the classification of the Weyl curvature spinors by the pattern of principal spinors each admits is given. For each Weyl curvature spinor, there are nine nontrivial types. This classification is then related to the classification, given previously by the author, of a Weyl curvature spinor when regarded as a curvature endomorphism (four types). These results are the neutral analogues of well known and fundamental results in Lorentzian geometry, but display the peculiarities of neutral geometry. One can expect these results to be an essential ingredient in a full understanding of neutral geometry in four dimensions.     

Pure spinors and quadric Grassmanians

If, following E. Cartan, the simplest spinors (pure) are conceived as equivalent to isotropic (or null or optical) polarized planes in complex spaces, then the most natural tensors generated (bilinearly) by the simplest spinors are isotropic vectors rather than ordinary linear ones. The conjecture that spinors are fundamental would then imply that non-linear geometry of isotropic elements should be more elementary in general than the linear one; and the relevance of optical geometry (optical flags, optical groups) on space-time manifolds for the explanation of optical phenomenology in the frame of general relativity [5] could already constitute a first confirmation of this conjecture.Only 2- and 4-component spinors build up linear spinor spaces while 8, 16, 32,...component pure spinors, instead, are subject to covariant (quadratic) constraint equations and build up non-linear sets isomorphic, up to a sign, to quadric Grassmanians and, for neutral and conformal spaces, to Lie groups.The possible relevance of such pure spinor properties for physics is conjectured and exemplified.     

Geometric significance of the spinor Lie derivative. I

In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant.     

Algebraic spinors and directed random walks in the McKane-Parisi-Sourlas theorem

We present the Dirac propagator as a random walk on anS ^D–1 sphere for Majorana spinors, even spinor space, Dirac spinors, and Chevalley-Crumeyrolle spinors built from Minkowski space. We propose the Dirac propagator constructed from Chevalley-Crumeyrolle spinors as the generators of a Markov process such that McKane-Parisi-Sourlas theorem can be applied to calculate the expectation values for functions of local times.     

The role of singular spinor fields in a torsional gravity,Lorentz-violating,framework

In this work, we consider a generalization of quantum electrodynamics including Lorentz violation and torsional-gravity, in the context of general spinor fields as classified in the Lounesto scheme. Singular spinor fields will be shown to be less sensitive to the Lorentz violation, as far as couplings between the spinor bilinear covariants and torsion are regarded. In addition, we prove that flagpole spinor fields do not admit minimal coupling to the torsion. In general, mass dimension four couplings are deeply affected when singular—flagpoles—spinors are considered, instead of the usual Dirac spinors. We also construct a mapping between spinors in the covariant framework and spinors in Lorentz symmetry breaking scenarios, showing how one may transliterate spinors of different classes between the two cases. Specific examples concerning the mapping of Dirac spinor fields in Lorentz violating scenarios into flagpole and flag-dipole spinors with full Lorentz invariance (including the cases of Weyl and Majorana spinors) are worked out.     

On “Conformal spinor geometry”: An attempt to “Understand” internal symmetry

The natural homomorphism of pure spinors corresponding to a given Clifford algebraC _2n to polarized isotropicn-planes of complex Euclidean spaceE _2n ^c is taken as a starting point for the construction of a geometry called spinor geometry where pure spinors are the only elements out of which all tensors have to be constructed (analytically as bilinear polynomials of the components of a pure spinor).C ₄ andC ₆ spinor geometry are analyzed, but it seems that C₈ spinor geometry is necessary to construct Minkowski spaceM ^3,1.C ₆ spinor field equations give rise in Minkowski space to a pair of Dirac equations (for conformal semispinors) presenting ansu(2) internal symmetry algebra. Mass is generated by breaking spontaneously the originalO(4,2) symmetry of the spinor equation.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Killing spinors on Lorentzian manifolds

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.     

SO(r+2, r) pure spinors

Chevalley gave a comprehensive treatment of pure spinors for Clifford algebras whose quadratic form has maximal index. We here show how the notion of pure spinor can be extended to the real Clifford algebras associated with quadratic forms with r+2 positive and r negative eigenvalues.     

Asymptotically covariantly constant spinor hair on black holes

It is demonstrated, in a straightforward approach, that a static black hole has no spinor hair if we assume that the fields vanish at infinity. We extend this to the case of asymptotically covariantly constant spinors (Witten spinors) which approach a constant value at spatial infinity. We show that with the dominant energy condition and in absence of external spinor sources, Witten spinors must vanish everywhere in the region exterior to a stationary black hole with an horizon in an asymptotically flat space-time.     

Supersymmetry in Lorentzian Curved Spaces and Holography

We consider superconformal and supersymmetric field theories on four-dimensional Lorentzian curved space-times, and their five-dimensional holographic duals. As in the Euclidean signature case, preserved supersymmetry for a superconformal theory is equivalent to the existence of a charged conformal Killing spinor. Differently from the Euclidean case, we show that the existence of such spinors is equivalent to the existence of a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, this vector field is further restricted to be Killing. We demonstrate how these results agree with the existing classification of supersymmetric solutions of minimal gauged supergravity in five dimensions.     

The twistor equation on Riemannian manifolds

It is shown that a twistor spinor on a Riemannian manifold defines a conformal deformation to an Einstein manifold. Twistor spinors on 4-manifolds are considered. A characterization of the hyperbolic space is given. Moreover the solutions of the twistor equation on warped products Mⁿ × , where Mⁿ is an Einstein manifold, are described.     

Spinorial charges and their role in the fusion of internal and space-time symmetries

The advent of supersymmetry immediately led to speculations that a non-trivial mixing of internal and space-time symmetries could be achieved within its framework. In fact, the well-known no-go theorems do not apply to the supersymmetry algebra due to the presence, in the latter, of (anticommuting) spinorial charges. However, not until the recent work of Haag, Lopuszanski and Sohnius did a clearcut picture emerge as to how the aforementioned nontrivial mixing can take place. Most notably, the presence of the conformal algebra within the supersymmetry algebra turns out to be vital. We solidify the findings of Haag et al. through an explicit construction which uses as underlying space the pseudo-Euclidean space E(4,2), i.e. the space for which the conformal group is the group of rotations, and which employs as main tools the spinors associated with the space E(4,2). We follow the algebro-geometric approach of Cartan in order to understand both the introduction and the properties of these spinors. In this manner, we gain many insights regarding the mathematical foundations of supersymmetry. Thus, we fully understand the emergence of the anticommutator, rather than the commutator, among spinor charges as a natural algebraic consequence and not as an a priori given fact. In addition, we clearly see how an (internal) unitary symmetry group can make its appearance within the supersymmetry scheme and verify, via our explicit construction, the results of Haag et al.     

Geometric significance of the spinor Lie derivative. II

The formulas for the Lie covariant differentiation of spinors are deduced from an algebraic viewpoint. The Lie covariant derivative of the spinor connection is calculated, and is given a geometric meaning. A theorem about the Lie covariant derivative of an operator in spin space that was stated in Part I of this work is discussed.     

Materializing superghosts

The off-shell Batalin-Vilkovysky (BV) realization has been constructed for N = 1, d = 10 super-Yang-Mills theory with seven auxiliary fields. This becomes possible due to the materialized ghost phenomenon. Namely, supersymmetry ghosts are coordinates on a manifold B of ten-dimensional spinors with the pure spinors cut out. Auxiliary fields are sections of a bundle over B, and supersymmetry transformations are nonlinear in ghosts. By integrating out the auxiliary fields, we obtain an on-shell supersymmetric BV action with quadratic terms in the antifields. Exactly this on-shell BV action was obtained in our previous paper after integration out of auxiliary fields in the framework of a pure spinor superfield formalism. The text was submitted by the authors in English.     

SO（d,d;Z) Transformation Property for Gauge Fluxes and Ramond-Ramond Fields in Noncommutative Geometry

In this paper we study the spinor constructions of gauge fluxes and Ramond-Ramond fields on noncommutative tori T^d up to d=6. In which the spinor and conjugate spinor are distinguished and dual bases are also introduced.So that we can express the Chern-Simons Lagrangian in toroidal compactification as a product of spinors.