Projective Twistor Space Research Articles

A twistorial description of the IKKT-matrix model

We consider the fuzzy 4-sphere {S}_N^4 as a background in the IKKT matrix model, and explore the relation between {S}_N^4 and fuzzy twistor space in the semi-classical limit. A novel description for the IKKT-matrix model in terms of spinorial indices is given, which is reminiscent of mathcal{N} = 4 super-symmetric Yang-Mills (SYM) in 4d. On fuzzy twistor space, the interactions of the IKKT model are of gravitational type. The higher-spin (HS) gauge theory emerging in this limit from the IKKT model, denoted as HS-IKKT, on fuzzy twistor space is shown to be a higher-spin extension of mathcal{N} = 4 SYM, with vertices that have more than two derivatives. We obtain its (Euclidean) spacetime action using the Penrose transform. Although this is a gravitational theory, it shares many features with the higher-spin extensions of Yang-Mills in 4d flat space obtained in [1, 2]. The tree-level amplitudes of the HS-IKKT are studied in the semi-classical flat limit. The self-dual gauge sector of the IKKT model is obtained by dropping some parts of the cubic- and the quartic interactions, which is shown to reduce to a mathcal{BF} -type action on commutative deformed projective twistor space.

Two-dimensional twistor manifolds and Teukolsky operators

The Teukolsky equations are currently the leading approach for analysing stability of linear massless fields propagating in rotating black holes. It has recently been shown that the geometry of these equations can be understood in terms of a connection constructed from the conformal and complex structure of Petrov type D spaces. Since the study of linear massless fields by a combination of conformal, complex and spinor methods is a distinctive feature of twistor theory, and since versions of the twistor equation have recently been shown to appear in the Teukolsky equations, this raises the question of whether there are deeper twistor structures underlying this geometry. In this work we show that all these geometric structures can be understood naturally by considering a {\em 2-dimensional} twistor manifold, whereas in twistor theory the standard (projective) twistor space is 3-dimensional.

Ambitwistors, oscillators and massless fields on AdS5

Positive energy unitary irreducible representations of $SU(2,2)$ can be constructed with the aid of bosonic oscillators in (anti)fundamental representation of $SU(2)_L\times SU(2)_R$ that are closely related to Penrose twistors. Starting with the correspondence between the doubleton representations, homogeneous functions on projective twistor space and on-shell $SL(2,\mathbb C)$ generalized Weyl curvature spinors and their low-spin counterparts, we study in the similar way the correspondence between the massless representations, homogeneous functions on ambitwistor space and, via the Penrose transform, with the gauge fields on Minkowski boundary of $AdS_5$. The possibilities of reconstructing massless fields on $AdS_5$ and some applications are also discussed.

Stringlike structures in the real and complex Kerr-Schild geometry

Four-dimensional Kerr-Schild (KS) geometry displays remarkable relationships with quantum world and theory of superstrings. In particular, the Kerr-Newman (KN) solution has gyromagnetic ratio g = 2, as that of the Dirac electron and represents a consistent background for gravitational and electromagnetic field of the electron. As a consequence of very big spin/mass ratio, black hole horizons disappear, exposing the naked Kerr singular ring. We consider four-decade history of development of this structure which took finally the form of a point-string-membrane-bubble complex which is reminiscent of the enhancon model of string/M-theory. A complex string obtained in the complex structure of the Kerr geometry gives an extra dimension to the world-sheet of the real Kerr string, forming a membrane by analogue with the string/M-theory unification. By analysis of the orientifold parity of the complex Kerr string, we obtain that the determined by the Kerr theorem principal null congruence of the Kerr geometry is described by a quartic equation in projective twistor space CP3, and therefore, it creates the known Calabi- Yau twofold (K3 surface) in twistor space of the 4d KS geometry. We connect it with N=2 superstring which has (complex) critical dimension two and may be embedded into complex KS geometry.

Stringlike structures in Kerr-Schild geometry: The N=2 string, twistors, and the Calabi-Yau twofold

The four-dimensional Kerr-Schild geometry contains two stringy structures. The first is the closed string formed by the Kerr singular ring, and the second is an open complex string obtained in the complex structure of the Kerr-Schild geometry. The real and complex Kerr strings together form a membrane source of the over-rotating Kerr-Newman solution without a horizon, a = J/m ≫ m. It was also recently found that the principal null congruence of the Kerr geometry is determined by the Kerr theorem as a quartic in the projective twistor space, which corresponds to an embedding of the Calabi-Yau twofold into the bulk of the Kerr geometry. We describe this embedding in detail and show that the four sheets of the twistorial K3 surface represent an analytic extension of the Kerr congruence created by antipodal involution.

(Pre-)Hilbert spaces in twistor quantization

In twistor theory, the canonical quantization procedure, called twistor quantization, is performed with the twistor operators represented as $\hat{Z}^{A}=Z^{A}\;\!(\in \mathbb{C})$ẐA=ZA(∈C) and $\hat{\bar{Z}}_{A} =-\partial /\partial Z^{A}$Z̄̂A=−∂/∂ZA. However, it has not been clarified what kind of function spaces this representation is valid in. In the present paper, we intend to find appropriate (pre-)Hilbert spaces in which the above representation is realized as an adjoint pair of operators. To this end, we define an inner product for the helicity eigenfunctions by an integral over the product space of the circular space S1 and the upper half of projective twistor space. Using this inner product, we define a Hilbert space in some particular case and indefinite-metric pre-Hilbert spaces in other particular cases, showing that the above-mentioned representation is valid in these spaces. It is also shown that only the Penrose transform in the first particular case yields positive-frequency massless fields without singularities, while the Penrose transforms in the other particular cases yield positive-frequency massless fields with singularities.

Quantum Instantons with Classical Moduli Spaces

We introduce a quantum Minkowski space-time based on the quantum group SU(2)q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective twistor space.

Twistor geometry of light rays

The geometry of real and complex light rays (mainly in Minkowski space) is studied using twistor methods. The properties of weak and strong incidence between rays are examined, some apparent anomalies arising for neighbouring rays being explained. Various apparently different interpretations of a complex surface in projective twistor space are given (one coming from the Kerr theorem describing shear-free congruences in Minkowski space). The relations between them are analysed in terms of the caustics of shear-free congruences, null hypersurfaces and a twistor description of spacelike or timelike 2-surfaces. The situation for curved spacetime is also considered. The relation between ray geometry in Minkowski space and a 6-quadric of signature (++++----), with its triality properties, is explored.

The linear system for self-dual gauge fields in a spacetime of signature 0

The overdetermined linear system for the self-dual Yang—Mills (SDYM) equations is examined in a flat four-dimensional space whose metric has signature 0. There are three different domains for the system, and correspondingly three (essentially) different solutions to the linear system for a given gauge field. If the gauge potential is real analytic, two of the solutions patch together to give a holomorphic function in an annular region of projective twistor space. Conversely, an arbitrary holomorphic GL( n, C )-valued function in such a domain can be uniquely factored (on the real lines) to give a solution to SDYM with gauge group U( n). The set of all real analytic u( n)-valued gauge fields can thus be parametrized by the points of a certain double coset space.

The causal geometry of twistor space

The problem of classifying the nature of the vector connecting a pair of points in Minkowski space is examined within the twistor theoretic framework. Two approaches are considered, one algebraic and the other geometric. The latter of the two is studied in some detail, providing some insight into the relation between the causal structure of Minkowski space and the geometry of projective twistor space.

A twistor approach to the regularization of divergences

One object of the twistor programme, as developed principally by R. Penrose, is the production of a manifestly finite theory of scattering in quantum field theory. Earlier work has shown that progress towards this goal is obstructed even at the first-order level, by the appearance of an infrared divergence in the standard theory. New studies in many-dimensional contour integration now suggest a simple but very powerful modification to this branch of twistor theory, in which the full (as opposed to the projective) twistor space plays an essential role. In this modified theory there arise natural contour-integral expressions with the effect of eliminating the infrared divergence previously noted, and replacing it by a finite quantity. This regularization can be specified by using a formalism of ‘inhomogeneous twistor diagrams’. The interpretation of this new formalism is not yet wholly clear, but the inhomogeneity can be seen as a means of relinquishing the concept of space-time point, while preserving light-cone structure. It therefore suggests a quite fresh approach to the divergences of quantum field theory.

A CP 5 calculus for space-time fields

Compactified Minkowski space can be embedded in projective five-space CP 5 (homogeneous coordinates X i , i = 0, …, 5) as a four dimensional quadric hypersurface given by Ω ijX iX j = 0. Projective twistor space (homogeneous coordinates Z α, α = 0, …, 3) arises via the Klein representation as the space of two-planes lying on this quadric. These two facts of projective geometry form the basis for the construction of a global space-time calculus which makes use of the coordinates X i↔X αβ(=-X βα) to represent spinor and tensor fields in a manifestly conformally covariant form. This calculus can be regarded as a synthesis of work on conformal geometry by Veblen, Dirac, and others, with the theory of twistors developed by Penrose. We provide here a systematic review of the basic framework: the underlying projective geometry; the calculus of tensor fields; the characterization of spinors as twistor-valued fields ψ α( X) which satisfy a geometrical condition ( ψ α X αβ = 0 on Ω ); and the introduction of the conformally invariant Laplacian operator ∇ 2 = Ω ij ∂ 2/∂ X i ∂ X j . In addition a number of subsidiary topics are discussed which illustrate the general scheme, including: the breaking of conformal symmetry to Poincaré symmetry; a derivation of the zero rest mass equations for all helicities; and a new and manifestly conformally covariant form of the twistor contour integral formulae for massless fields.

Scattering theory and the geometry of multitwistor spaces

Existing results which show the zero rest mass field equations to be encoded in the geometry of projective twistor space are extended, and it is shown that the geometries of spaces of more than one twistor contain information concerning the scattering of such fields. Some general constructions which describe spacetime interactions in terms of cohomology groups on subvarieties in twistor space are obtained and are used to construct a purely twistorial description of spacetime propagators and of first order ϕ 4 {\phi ^4} scattering. Spacetime expressions concerning these processes are derived from their twistor counterparts, and a physical interpretation is given for the twistor constructions.

Non-self-dual nonlinear gravitons

Penrose has given a twistor description of all self-dual complex Riemannian space-times. We modify his construction to characterize all complex Riemannian space-times and all complex teleparallel space-times. This construction may be useful in finding non-self-dual solutions to the gravitational field equations (Einstein's or otherwise) without or with sources. It may also lead to a nonperturbative method for computing path integrals. Whereas Penrose shows that a self-dual space-time may be specified by a deformation of projective twistor space (the set of α planes in complex Minkowski space), we find that a Riemannian or teleparallel space-time may be described by a deformation of projective ambitwistor space (the set of null geodesics in complex Minkowski space).

Structure of asymptotic twistor space

We show that asymptotic projective twistor space 𝒫𝒯+ is an Einstein–Kähler manifold of positive curvature. We then use the Chern–Moser theory of hypersurfaces in complex manifolds to show that the Kähler curvature of 𝒫𝒯+ closely related to the CR curvature of its boundary. We also give a proof that the Kähler potential function defining the boundary satisfies the complex Monge–Ampere equations.

Hyperfunction solutions of the zero-rest-mass field equations

In this paper it is shown how the Penrose transform maps tangential hyperfunction Dolbeault groups with coefficients in a power of the hyperplane section bundle on the hyperquadric of null twistors in projective twistor space isomorphically to all hyperfunction solutions of the massless field equations of nonnegative helicity on compactified Minkowski space. This is an extension of the Penrose transform which generated real-analytic solutions of the same field equations on the same space (cf. Eastwood, M., Penrose, R., Wells, R.O., [10]). In additions, one obtains the result that each hyperfunction solution of the massless field equations of nonnegative helicity is the sum of massless fields of positive and negative frequency, a generalization of the usual Fourier decomposition for solutions with appropriate growth conditions.

Self-dual gauge fields

Self-dual solutions of massless Yang-Mills gauge fields are constructed from a two-parameter family of elements $G({x}^{\ensuremath{\mu}},\ensuremath{\zeta},\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}})$ of a gauge group G. The construction uses the existence of a one-parameter family of anti-self-dual totally null two-surfaces through each point of complex Minkowski space. These surfaces allow one to introduce coordinates on the projective twistor space PT. Then $G({x}^{\ensuremath{\mu}},\ensuremath{\zeta},\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}})$ is used to construct a fiber bundle over PT giving Ward's encoding of the gauge field in the transition functions.

Some remarks on the nonlinear graviton

We give a new description of unprimed spinors for the nonlinear graviton and use this to prove that the nonlinear graviton is an anti-self-dual solution to Einstein's equations. We also provide a proof of the stability of the normal bundles to the projective lines in curved projective twistor space.

Projective geometry, Lagrangian subspaces, and twistor theory

The use of projective geometry for the characterization of Lagrangian subspaces and maps among them is of particular interest for the symplectic manifold that is twistor space. We raise some conjectures on how these should be interpreted on the space-time manifold by making use of the structure of projective twistor space.