Twistor Spinors Research Articles

Higher-degree Dirac currents of twistor and Killing spinors in supergravity theories

We show that higher degree Dirac currents of twistor and Killing spinors correspond to the hidden symmetries of the background spacetime which are generalizations of conformal Killing and Killing vector fields respectively. They are the generalizations of one-form Dirac currents to higher degrees which are used in constructing the bosonic supercharges in supergravity theories. In the case of Killing spinors, we find that the equations satisfied by the higher degree Dirac currents are related to Maxwell-like and Duffin–Kemmer–Petiau equations. Correspondence between the Dirac currents and harmonic forms for parallel and pure spinor cases is determined. We also analyze the supergravity twistor and Killing spinor cases in ten and eleven-dimensional supergravity theories and find that although different inner product classes induce different involutions on spinors, the higher degree Dirac currents still correspond to the hidden symmetries of the spacetime.

Lower bounds for the eigenvalues of the Spin c Dirac operator on submanifolds

We prove lower bounds for the eigenvalues of the Spinc Dirac operator on submanifolds. These estimates are expressed in terms of extrinsic and intrinsic quantities. We also give estimates involving the energy–momentum tensor as well as conformal bounds. The limiting cases of these estimates give rise to particular spinor fields, called generalized twisted Killing spinors, which are also studied.

Reducible conformal holonomy in any metric signature and application to twistor spinors in low dimension

We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally isotropic subspace of arbitrary dimension, there is, w.r.t. a local metric in the conformal class defined off a singular set, a parallel, totally isotropic distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant isotropic lines and planes and closes a gap in the understanding of the geometric meaning of reducibly acting conformal holonomy groups. We show how this result naturally applies to the classification of geometries admitting twistor spinors described in terms of parallel spin tractors using conformal spin tractor calculus. As an example we obtain together with already known results about generic distributions in dimensions 5 and 6 a complete geometric description of local geometries admitting real twistor spinors in signatures (3,2) and (3,3). In contrast to the generic case where generic geometric distributions play an important role, the underlying geometries in the non-generic case without zeroes turn out to admit integrable distributions.

The zero set of a twistor spinor in any metric signature

Using tractor methods, we exhibit the local structure of the zero set of a twistor spinor in any metric signature. It is given as the image under the exponential map of a distinguished totally lightlike subspace. Based on these results we are able to construct starting from a conformal structure admitting a twistor spinor with zero a projective structure on its zero set in a natural way. The construction is presented both explicitly, using a fixed metric in the conformal class, and more abstractly on the level of associated parabolic Cartan geometries. Finally, we use known results about pseudo-Riemannian extensions of 2-dimensional projective structures to \((2,2)\)-ASD conformal structures to find first examples of non-conformally flat, non Riemannian manifolds admitting a twistor spinor with nontrivial zero set.

Conformal superalgebras via tractor calculus

We use the manifestly conformally invariant description of a Lorentzian conformal structure in terms of a parabolic Cartan geometry in order to introduce a superalgebra structure on the space of twistor spinors and normal conformal vector fields formulated in purely algebraic terms on parallel sections in tractor bundles. Via a fixed metric in the conformal class, one reproduces a conformal superalgebra structure that has been considered in the literature before. The tractor approach, however, makes clear that the failure of this object to be a Lie superalgebra in certain cases is due to purely algebraic identities on the spinor module and to special properties of the conformal holonomy representation. Moreover, it naturally generalizes to higher signatures. This yields new formulas for constructing new twistor spinors and higher order normal conformal Killing forms out of existing ones, generalizing the well-known spinorial Lie derivative. Moreover, we derive restrictions on the possible dimension of the space of twistor spinors in any metric signature.

Rigid supersymmetry, conformal coupling and twistor spinors

We investigate the relationship between conformal and spin structure on lorentzian manifolds and see how their compatibility influences the formulation of rigid supersymmetric field theories. In dimensions three, four, six and ten, we show that if the Dirac current associated with a generic spinor defines a null conformal Killing vector then the spinor must obey a twistor equation with respect to a certain connection with torsion. Of the theories we consider, those with classical superconformal symmetry in Minkowski space can be reformulated as rigid supersymmetric theories on any lorentzian manifold admitting twistor spinors. In dimensions six and ten, we also describe rigid supersymmetric gauge theories on bosonic minimally supersymmetric supergravity backgrounds.

Superconformal quantum field theory in curved spacetime

By conformally coupling vector and hyper multiplets in Minkowski space, we obtain a class of field theories with extended rigid conformal supersymmetry on any Lorentzian 4-manifold admitting twistor spinors. We construct the conformal symmetry superalgebras which describe classical symmetries of these theories and derive an appropriate BRST operator in curved spacetime. In the process, we elucidate the general framework of cohomological algebra which underpins the construction. We then consider the corresponding perturbative quantum field theories. In particular, we examine the conditions necessary for conformal supersymmetries to be preserved at the quantum level, i.e. when the BRST operator commutes with the perturbatively defined S-matrix, which ensures superconformal invariance of amplitudes. To this end, we prescribe a renormalization scheme for time-ordered products that enter the perturbative S-matrix and show that such products obey certain Ward identities in curved spacetime. These identities allow us to recast the problem in terms of the cohomology of the BRST operator. Through a careful analysis of this cohomology, and of the renormalization group in curved spacetime, we establish precise criteria which ensure that all conformal supersymmetries are preserved at the quantum level. As a by-product, we provide a rigorous proof that the beta-function for such theories is one-loop exact. We also briefly discuss the construction of chiral rings and the role of non-perturbative effects in curved spacetime.

Conformal symmetry superalgebras

We show how the rigid conformal supersymmetries associated with a certain class of pseudo-Riemannian spin manifolds define a Lie superalgebra. The even part of this superalgebra contains conformal isometries and constant R-symmetries. The odd part is generated by twistor spinors valued in a particular R-symmetry representation. We prove that any manifold which admits a conformal symmetry superalgebra of this type must generically have dimension less than 7. Moreover, in dimensions 3, 4, 5 and 6, we provide the generic data from which the conformal symmetry superalgebra is prescribed. For conformally flat metrics in these dimensions, and compact R-symmetry, we identify each of the associated conformal symmetry superalgebras with one of the conformal superalgebras classified by Nahm. We also describe several examples for Lorentzian metrics that are not conformally flat.

Eigenvalue estimates for generalized Dirac operators on Sasakian manifolds

We consider a two-parameter generalization \(D_{ab}\) of the Riemann Dirac operator \(D\) on a closed Sasakian spin manifold, focusing attention on eigenvalue estimates for \(D_{ab}\). We investigate a Sasakian version of twistor spinors and Killing spinors, applying it to establish a new connection deformation technique that is adapted to fit with the Sasakian structure. Using the technique and the fact that there are two types of eigenspinors of \(D_{ab}\), we prove several eigenvalue estimates for \(D_{ab}\) which improve Friedrich’s estimate (Friedrich, Math Nachr 97, 117–146, 1980).

The functional equation for the twisted spinor L-series of genus 2

We prove the functional equation for the twisted spinor L-series of a cuspidal, holomorphic Siegel eigenform for the full modular group of genus 2. It follows from a more general functional equation, valid for Rankin convolutions of paramodular cuspforms. A non-vanishing result for Fourier-Jacabi coefficients of the eigenforms in question is the central pillar of the deduction of the former from the latter functional equation.

Skew Killing spinors

Abstract We study the existence of a skew Killing spinor on 2- and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.

Coupling solutions of BGG-equations in conformal spin geometry

BGG-equations are geometric overdetermined systems of partial differential equations (PDEs) on parabolic geometries. Normal solutions of BGG-equations are particularly interesting, and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of BGG-equations to produce new ones. Employing a suitable calculus for conformal spin structures, this yields explicit coupling formulas and conditions between almost Einstein scales, conformal Killing forms, and twistor spinors. Finally, we discuss a class of generic twistor spinors that provides an invariant decomposition of conformal Killing fields.

The length function of a twistor spinor with zero

We examine the Taylor expansion of the length function of a twistor spinor with zero on a Riemannian orbifold around its zero and study it on the Eguchi–Hanson orbifold. This expansion is written in some conformal normal coordinates (CNC) around the zero up to order 7. In the example of the Eguchi–Hanson orbifold, CNC are found explicitly. We use the expansion in computing the mass (a generalization of ADM–mass) of the asymptotically locally Euclidean coordinate system, which is constructed from a conformal normal coordinate system around the zero of a twistor spinor on a Riemannian spin orbifold admitting isolated singularities.

The twistor spinors of generic 2- and 3-distributions

Generic distributions on 5- and 6-manifolds give rise to conformal structures that were discovered by P. Nurowski resp. R. Bryant. We describe both as Fefferman-type constructions and show that for orientable distributions one obtains conformal spin structures. The resulting conformal spin geometries are then characterized by their conformal holonomy and equivalently by the existence of a twistor spinor which satisfies a genericity condition. Moreover, we show that given such a twistor spinor we can decompose a conformal Killing field of the structure. We obtain explicit formulas relating conformal Killing fields, almost Einstein structures and twistor spinors.

Invariant Bilinear Pairing for Twistor Spinors

It is proved in [5] that one can manufacture (conformal) Killing forms from (conformal) twistor spinors on any Riemannian Spin-manifold. We shall prove that this procedure can be realized in the framework of a cup product on twistor spinors in the Bernstein-Gelfand-Gelfand sequence associated to conformal geometry induced from the Riemannian structure.

About Twistor Spinors with Zero in Lorentzian Geometry

We describe the local conformal geometry of a Lorentzian spin manifold $(M,g)$ admitting a twistor spinor $\phi$ with zero. Moreover, we describe the shape of the zero set of $\phi$. If $\phi$ has isolated zeros then the metric $g$ is locally conformally equivalent to a static monopole. In the other case the zero set consists of null geodesic(s) and $g$ is locally conformally equivalent to a Brinkmann metric. Our arguments utilise tractor calculus in an essential way. The Dirac current of $\phi$, which is a conformal Killing vector field, plays an important role for our discussion as well.

Holomorphic superspace

We give a twisted holomorphic superspace description for the super-Yang-Mills theory, using holomorphic and antiholomorphic decompositions of twisted spinors. We consider the case of the N=1 super-Yang-Mills theory in four dimensions. We solve the constraints in two different manners, without and with a prepotential. This might have further application for an holomorphic superspace description of N=1,d=10 theory. We also explain how the N=1 and N=2 holomorphic superspaces are related.

Killing spinor initial data sets

A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is made using the space-spinor formalism. Conditions on initial data sets for the Einstein vacuum equations are given so that their developments contain solutions to the twistor and/or Killing equations. These lead to the notions of twistor and Killing spinor initial data. These notions are used to obtain a characterisation of initial data sets whose developments are of Petrov type N or D.

CR-tractors and the Fefferman space

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non-degenerate CR manifold of hypersurface type. In particular, we construct and treat the basic objects that relate the natural bundles and natural operators on the two spaces. This is illustrated with several applications: We prove that a number of conformally invariant overdetermined systems, including Killing form equations and the equations for twistor spinors, admit non-trivial solutions on any Fefferman space. We show that, on a Fefferman space, the space of infinitesimal conformal isometries naturally decomposes into a direct sum of subspaces, which admit an interpretation as solutions of certain CR invariant PDE's. Finally we explicitly analyse the relation between tractor calculus on a CR manifold and the complexified conformal tractor calculus on its Fefferman space, thus obtaining a powerful computational tool for working with the Fefferman construction.

Causal conformal vector fields, and singularities of twistor spinors

In this paper, we study the geometry around the singularity of a twistor spinor, on a Lorentz manifold (M, g) of dimension greater or equal to three, endowed with a spin structure. Using the dynamical properties of conformal vector fields, we prove that the geometry has to be conformally flat on some open subset of any neighbourhood of the singularity. As a consequence, any analytic Lorentz manifold, admitting a twistor spinor with at least one zero has to be conformally flat.