Spinors and diffeomorphisms

We discuss the action of diffeomorphisms on spinors on an oriented manifoldM. To do this, we first describe the action of the diffeomorphism groupD(M) on the set =H ¹ (M,Z ₂) of inequivalent spin structures and show that it is affine. We argue that in the presence of spinors the gauge group of gravity is a certain double cover ofD(M) which depends on the spin structure. We explicitly compute the action ofD(M) on whenM is a closed Riemann surface; is seen to consist of exactly two orbits, corresponding to even and odd spin structures.On leave of absence from I.F.T., University of Wrocaw, Poland     

Euclidean Majorana and Weyl Spinors

The complex form algebra of Schwinger functions of a Dirac field on a Euclidean R ^d with arbitrary dimension d is decomposed into the form algebras of Majorana spinors and of Weyl spinors. The existence of real form algebras is investigated. The reality condition leads to severe restrictions in the case of Majorana forms which do not agree with the results of classical field theory. For all real form algebras Euclidean spinors are constructed as elements of a measure space.     

Asymptotically Euclidean Manifolds and Twistor Spinors

We analyse two asymmetric discrete Painlevé equations, namely d-P_II and q-P_III which are known to be discrete forms of P_III and P_VI respectively. We show that both equations are self-dual. This means that the same equation governs the evolution along the discrete independent variable and the transformations under the action of the Schlesinger transforms along the parameters of the discrete Painlevé. A bilinear formulation of the self-dual equation is given as a system of nonautonomous Hirota–Miwa equations. Received: 15 February 1997 / Accepted: 19 June 1997     

Spinors without spin structure

A class of space-times without spin-structure is discussed. It is shown that these space-times admit a globally defined structure which is locally equivalent to a spin structure. Generalised spinors can be defined, locally indistinguishable from ordinary spinors, which suffice for defining particle fields.On leave of absence from Jesus College, Cambridge. Work supported by a fellowship under the European Programme of the Royal Society.     

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     

Spinors and torsion in general relativity

Conformal rescalings of spinors are considered, in which the factor Ω, inε _AB ?Ωε _AB, is allowed to be complex. It is argued that such rescalings naturally lead to the presence of torsion in the space-time derivative▽ _a. It is further shown that, in standard general relativity, a circularly polarized gravitational wave produces a (nonlocal) rotation effect along rays intersecting it similar to, and apparently consistent with, the local torsion of the Einstein-Cartan-Sciama-Kibble theory. The results of these deliberations are suggestive rather than conclusive.     

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.     

Matrix Integrals and the Geometry of Spinors

Abstract

We obtain the collection of symmetric and symplectic matrix integrals and the collection of Pfaffian tau-functions, recently described by Peng and Adler and van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector of a fermionic Fock space. This fermionic Fock space is the same space as one constructs to obtain the KP and 1-Toda lattice hierarchy.     

Parallel and Killing Spinors on Spinc Manifolds

We describe all simply connected Spin^c manifolds carrying parallel and real Killing spinors. In particular we show that every Sasakian manifold (not necessarily Einstein) carries a canonical Spin^c structure with Killing spinors. Received: 24 December 1996 / Accepted: 6 January 1997     

Spinors, Inflation, and Non-Singular Cyclic Cosmologies

We consider toy cosmological models in which a classical, homogeneous, spinor field provides a dominant or sub-dominant contribution to the energy-momentum tensor of a flat Friedmann-Robertson-Walker universe. We find that, if such a field were to exist, appropriate choices of the spinor self-interaction would generate a rich variety of behaviors, quite different from their widely studied scalar field counterparts. We first discuss solutions that incorporate a stage of cosmic inflation and estimate the primordial spectrum of density perturbations seeded during such a stage. Inflation driven by a spinor field turns out to be unappealing as it leads to a blue spectrum of perturbations and requires considerable fine-tuning of parameters. We next find that, for simple, quartic spinor self-interactions, non-singular cyclic cosmologies exist with reasonable parameter choices. These solutions might eventually be incorporated into a successful past- and future-eternal cosmological model free of singularities. In an Appendix, we discuss the classical treatment of spinors and argue that certain quantum systems might be approximated in terms of such fields.     

Spinors in quantum space with torsion

Dirac spinors are considered in quantized fiber-bundled spaces. It is shown that the spin connection has the same internal structure as in the Riemann-Cartan space as well as the quantized one. It is also assumed that the neutrino oscillation mechanism can be linked to the quantized and fibered character of the space at small distances.     

Invariant Spinors and their Use for Expansions of Scattering Amplitudes

A general covariant scheme is proposed for the kinematical analysis of arbitrary reactions. By this scheme the problem of finding invariant amplitudes and corresponding kinematical covariants can be solved systematically. The amplitudes are labelled by two invariant indices for each particle related to spin and parity. The method provides a tool for exploiting the symmetries of the M-function, including crossing symmetry and the Pauli principle. Amplitudes free from kinematical singularities can easily be constructed.     

Relation Between Rigid-Body Motion,Isotropic Cones,and Spinors

No Heading Spinors play a role in classical physics, since the fact that the rotation group is not simply connected has consequences for macroscopic bodies. Here we relate the orientations of a rigid body to two-dimensional complex spinors by combining O. Rodrigues parameterization of rotations with Cartans 2:1 identification of spinors with complex isotropic 3-vectors.     

Twistor Spinors with Zero on Lorentzian 5-Space

We present in this paper a C ¹-metric of Lorentzian signature (1,4) on an open neighbourhood of the origin in \(\mathbb{R}^{5}\), which admits a solution to the twistor equation for spinors with a unique isolated zero at the origin. The metric is not conformally flat in any neighbourhood of the origin and the associated conformal Killing vector to the twistor generates a one-parameter group of essential conformal transformations. The construction is based on the Eguchi-Hanson metric in dimension 4.