Twistor Spinors Research Articles

Bessel periods and anticyclotomic 𝑝-adic spinor 𝐿-functions

We construct the anticyclotomic p p -adic L L -function that interpolates a square root of central values of twisted spinor L L -functions of a quadratic base change of a Siegel cusp form of genus 2 2 with respect to a paramodular group of square-free level.

Modified conformal extensions

We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive a complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show the vanishing of the Q-curvature.

First BGG operators on homogeneous conformal geometries

We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing–Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.

Geometric analysis of the Yang–Mills–Higgs–Dirac model

The harmonic sections of the Kaluza–Klein model can be seen as a variant of harmonic maps with additional gauge symmetry. Geometrically, they are realized as sections of a fiber bundle associated to a principal bundle with a connection. In this paper, we investigate geometric and analytic aspects of a model that combines the Kaluza–Klein model with the Yang–Mills action and a Dirac action for twisted spinors. In dimension two we show that weak solutions of the Euler–Lagrange system are smooth. For a sequence of approximate solutions on surfaces with uniformly bounded energies we obtain compactness modulo bubbles, namely, energy identities and the no-neck property hold.

Twistor spinors and extended conformal superalgebras

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing–Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing–Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing–Yano forms and the symmetry operators of twistor spinors. The even part of the extended conformal superalgebra corresponds to conformal Killing–Yano forms and the odd part consists of twistor spinors.

Spinorially Twisted Spin Structures. II: Twisted Pure Spinors, Special Riemannian Holonomy and Clifford Monopoles

We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special K\"ahler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also introduce the Clifford monopole equations as a natural geometric generalization of the Seiberg-Witten equations. We show that they restrict to the Seiberg-Witten equations in 4 dimensions, and that they admit non-trivial solutions on manifolds with special Riemannian holonomy.

Harmonic spinors from twistors and potential forms

Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators that transform gauged twistor spinors to gauged harmonic spinors are found. Symmetry operators of gauged harmonic spinors in terms of conformal Killing-Yano forms are obtained. Algebraic conditions to obtain solutions of the Seiberg-Witten equations are discussed.

Some examples of Dirac-harmonic maps

We discuss a method to construct Dirac-harmonic maps developed by Jost et al. (J Geom Phys 59(11):1512–1527, 2009). The method uses harmonic spinors and twistor spinors and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in Jost et al. (2009) were fulfilled. We show that for isometric immersions into space forms, these conditions are fulfilled only under special assumptions. In several cases, we show the existence of solutions.

Spin raising and lowering operators for Rarita-Schwinger fields

Spin raising and lowering operators for massless field equations constructed from twistor spinors are considered. Solutions of the spin-$\frac{3}{2}$ massless Rarita-Schwinger equation from source-free Maxwell fields and twistor spinors are constructed. It is shown that this construction requires Ricci-flat backgrounds due to the gauge invariance of the massless Rarita-Schwinger equation. Constraints to construct spin raising and lowering operators for Rarita-Schwinger fields are found. Symmetry operators for Rarita-Schwinger fields via twistor spinors are obtained.

Self-consistency of conformally coupled ABJM theory at the quantum level

We study the mathcal{N} = 6 superconformal Chern-Simons matter field theory (the ABJM theory) conformally coupled to a Lorentzian, curved background spacetime. To support rigid supersymmetry, such backgrounds have to admit twistor spinors. At the classical level, the symmetry of the theory can be described by a conformal symmetry superalgebra. We show that the full mathcal{N} = 6 superconformal algebra persists at the quantum level using the BV-BRST method.

A Projective-to-Conformal Fefferman-Type Construction

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.

A New $$\frac{1}{2}$$ 1 2 -Ricci Type Formula on the Spinor Bundle and Applications

Consider a Riemannian spin manifold $$(M^{n}, g)$$ $$(n\ge 3)$$ endowed with a non-trivial 3-form $$T\in \Lambda ^{3}T^{*}M$$ , such that $$\nabla ^{c}T=0$$ , where $$\nabla ^{c}:=\nabla ^{g}+\frac{1}{2}T$$ is the metric connection with skew-torsion T. In this note we introduce a generalized $$\frac{1}{2}$$ -Ricci type formula for the spinorial action of the Ricci endomorphism $${{\mathrm{Ric}}}^{s}(X)$$ , induced by the one-parameter family of metric connections $$\nabla ^{s}:=\nabla ^{g}+2sT$$ . This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schrodinger–Lichnerowicz formula related to the square of the Dirac operator $$D^{s}$$ , induced by $$\nabla ^{s}$$ , under the condition $$\nabla ^{c}T=0$$ . In the same case, we provide integrability conditions for $$\nabla ^{s}$$ -parallel spinors, $$\nabla ^{c}$$ -parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly Kahler manifolds and nearly parallel $$\hbox {G}_2$$ -manifolds, in dimensions 5, 6 and 7, respectively.

The ambient obstruction tensor and conformal holonomy

For a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures admitting twistor spinors or normal conformal Killing forms. As our main tool we introduce the notion of a conformal holonomy distribution and show that its integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurowski and Bryant.

Extended superalgebras from twistor and Killing spinors

The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and relevant symmetry operators of twistor spinors and Killing spinors are constructed from Killing–Yano (KY) and conformal Killing–Yano (CKY) forms in constant curvature and Einstein manifolds. The squaring map of spinors gives KY and CKY forms for Killing and twistor spinors, respectively. They constitute a graded Lie algebra structure in some special cases. By using the graded Lie algebra structure of KY and CKY forms, extended Killing and conformal superalgebras are constructed in constant curvature and Einstein manifolds.

Gauged twistor spinors and symmetry operators

We consider gauged twistor spinors which are supersymmetry generators of supersymmetric and superconformal field theories in curved backgrounds. We show that the spinor bilinears of gauged twistor spinors satisfy the gauged conformal Killing-Yano equation. We prove that the symmetry operators of the gauged twistor spinor equation can be constructed from ordinary conformal Killing-Yano forms in constant curvature backgrounds. This provides a way to obtain gauged twistor spinors from ordinary twistor spinors.

Twistor spinors and quasi-twistor spinors

The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on closed Riemannian spin manifolds. As a consequence, it is known that on a locally decomposable closed spin manifold with nonzero Ricci curvature, the space of twistor spinors is trivial. Some integrability condition for twistor spinors is also obtained.

Dirac operators on quasi-Hamiltonian G-spaces

We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian G-spaces.

Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension

We consider a class of smooth oriented Lorentzian manifolds in dimensions three and four which admit a nowhere vanishing conformal Killing vector and a closed two-form that is invariant under the Lie algebra of conformal Killing vectors. The invariant two-form is constrained in a particular way by the conformal geometry of the manifold. In three dimensions, the conformal Killing vector must be everywhere causal (or null if the invariant two-form vanishes identically). In four dimensions, the conformal Killing vector must be everywhere null and the invariant two-form vanishes identically if the geometry is everywhere of Petrov type N or O. To the conformal class of any such geometry, it is possible to assign a particular Lie superalgebra structure, called a conformal symmetry superalgebra. The even part of this superalgebra contains conformal Killing vectors and constant R-symmetries while the odd part contains (charged) twistor spinors. The largest possible dimension of a conformal symmetry superalgebra is realised only for geometries that are locally conformally flat. We determine precisely which non-trivial conformal classes of metrics admit a conformal symmetry superalgebra with the next largest possible dimension, and compute all the associated submaximal conformal symmetry superalgebras. In four dimensions, we also compute symmetry superalgebras for a class of Ricci-flat Lorentzian geometries not of Petrov type N or O which admit a null Killing vector.

Killing and twistor spinors with torsion

We study twistor spinors (with torsion) on Riemannian spin manifolds $$(M^{n}, g, T)$$ carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection $$\nabla ^{c}=\nabla ^{g}+\frac{1}{2}T$$ and under the condition $$\nabla ^{c}T=0$$ , we show that the twistor equation with torsion w.r.t. the family $$\nabla ^{s}=\nabla ^{g}+2sT$$ can be viewed as a parallelism condition under a suitable connection on the bundle $$\Sigma \oplus \Sigma $$ , where $$\Sigma $$ is the associated spinor bundle. Consequently, we prove that a twistor spinor with torsion has isolated zero points. Next we study a special class of twistor spinors with torsion, namely these which are T-eigenspinors and parallel under the characteristic connection; we show that the existence of such a spinor for some $$s\ne 1/4$$ implies that $$(M^{n}, g, T)$$ is both Einstein and $$\nabla ^{c}$$ -Einstein, in particular the equation $${{\mathrm{Ric}}}^{s}=\frac{{{\mathrm{Scal}}}^{s}}{n}g$$ holds for any $$s\in \mathbb {R}$$ . In fact, for $$\nabla ^{c}$$ -parallel spinors we provide a correspondence between the Killing spinor equation with torsion and the Riemannian Killing spinor equation. This allows us to describe 1-parameter families of non-trivial Killing spinors with torsion on nearly Kahler manifolds and nearly parallel $${{\mathrm{G}}}_2$$ -manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere $${{\mathrm{S}}}^{3}$$ . We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.

A new upper bound for the Dirac operators on hypersurfaces

We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schr\"odinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.