Twistor theory

Twistor theory encodes the conformal geometry of four-dimensional spacetime using incidence geometry in a complex vector space. In the flat correspondence, a spacetime point is a two-plane in a four-dimensional twistor space, and hence a projective line in projective twistor space. Null separation becomes intersection of these lines. Choosing a spacetime signature is an additional reality condition on the complex geometry.

The flag manifold of a line $\ell$ contained in a two-plane $W$ has projections $(\ell ,W)\mapsto W$ and $(\ell ,W)\mapsto \ell$. Their fibers are respectively ${\mathbb{CP}}^1$ and ${\mathbb{CP}}^2$. This is the flat twistor correspondence: complexified compactified Minkowski space is the Grassmannian of two-planes in twistor space, while projective twistor space is its projective space of lines.

A holomorphic vector bundle on an appropriate twistor domain, trivial on its space-time incidence lines, determines a complex anti-self-dual gauge gauge connection. Split its patching matrix on each line and apply the tangent vector fields ${\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$. They annihilate the unsplit twistor patching matrix, so the two logarithmic derivatives of the factors agree and define a global degree-one function of $\pi$. It is consequently ${\pi }^{A^{\prime }}{A}_{A{A}^{\prime }}$. The inverse splitting bundle frame is parallel along each alpha-plane; commuting these parallel equations proves that the primed gauge curvature two-component spinor vanishes. Reality conditions recover unitary Euclidean gauge fields; Euclidean instanton bundles need triviality on the real twistor lines, not on every possible complex line.

On a twistor line use ${\pi }_{0^{\prime }}=\zeta ,{\pi }_{1^{\prime }}=1$ and ${\omega }^A=x^{A{0}^{\prime }}\zeta +x^{A{1}^{\prime }}$. For a degree-zero holomorphic patching logarithm, its Laurent series splits as $h_0=-{\sum }_{n\ge 0}{f}_n{\zeta }^n$, $h_1={\sum }_{n<0}{f}_n{\zeta }^n$. The twistor incidence relation implies ${\partial }_{A{0}^{\prime }}{f}_n={\partial }_{A{1}^{\prime }}{f}_{n-1}$. Hence $(-{\partial }_{A{0}^{\prime }}+\zeta {\partial }_{A{1}^{\prime }})h_{\alpha }={\partial }_{A{1}^{\prime }}{f}_{-1}$ on both patches. The gauge potential is $A_{A{0}^{\prime }}=-{\partial }_{A{1}^{\prime }}{f}_{-1}$, $A_{A{1}^{\prime }}=0$. Exponentiating the two additive factors produces the linewise holomorphic splitting of $e^f$.

The Abelian Ward splitting by Laurent series gives this gauge potential through the Cauchy integral formula, with $f_{,B}=\partial f/\partial {\omega }^B$ evaluated on the twistor line. The integrand has total projective degree zero. Holding the contour fixed locally and differentiating gives $F_{A{A}^{\prime }B{B}^{\prime }}={ϵ}_{A^{\prime }{B}^{\prime }}(2\pi i{)}^{-1}{\oint }_{\Gamma }{f}_{,AB}{\pi }_{D^{\prime }}d{\pi }^{D^{\prime }}$, using ${\pi }_{A^{\prime }}{\iota }_{B^{\prime }}-{\pi }_{B^{\prime }}{\iota }_{A^{\prime }}={ϵ}_{A^{\prime }{B}^{\prime }}({\iota }_{C^{\prime }}{\pi }^{C^{\prime }})$. Its unprimed two-component spinor is symmetric, so the primed gauge curvature vanishes and the field obeys the anti-self-dual Maxwell equations. These signs use ${ϵ}_{0^{\prime }{1}^{\prime }}=1$, ${ϵ}^{0^{\prime }{1}^{\prime }}=-1$ and raising by left multiplication; other consistent raising conventions can reverse both displayed overall signs.

On a twistor domain containing the lines $L_x$ for $x$ in a space-time domain $U$, the Ward construction uses a holomorphic vector bundle $E$ with $E{|}_{L_x}\cong {\mathcal{O}}^{\oplus r}$ for every $x\in U$. A degree-zero restriction is not sufficient: $\mathcal{O}(1)\oplus \mathcal{O}(-1)$ has degree zero but is not holomorphically trivial. A trivial restriction has $r$ independent global holomorphic sections; their evaluations give a linewise bundle frame and hence a factorization of its patching matrix. Nearby factors can be chosen holomorphically because $H^1({\mathbb{CP}}^1,\mathcal{O}{)}^{r^2}=0$. Lines on which the splitting type is nontrivial are called jumping lines and obstruct this regular reconstruction there. For a globally regular field on affine complexified Minkowski space the condition is needed on all its incidence lines; on a smaller domain only those lines are required.

A bundle trivial on a twistor line has a global holomorphic bundle frame there. Comparing it with the given two patch bundle frames produces invertible matrices $H,\tilde{H}$, holomorphic on their respective affine patches, with $F=\tilde{H}{H}^{-1}$. The obstruction to varying such a bundle frame locally with space-time is $H^1({\mathbb{CP}}^1,\mathcal{O}{)}^{n^2}=0$, so the splitting factors can be chosen holomorphically in a neighbourhood. If two splittings differ by $H\mapsto Hg$ and $\tilde{H}\mapsto \tilde{H}g$, then $g$ is holomorphic on the compact complex projective line and hence independent of its projective coordinate. It is the space-time gauge freedom.

An alpha-plane in the compact twistor correspondence is the set of two-planes containing a fixed line $\ell$ in twistor space. Its affine portion has tangent vectors $u^A{\pi }^{A^{\prime }}$ for fixed primed spinor $\pi$, so any two tangent vectors are orthogonal and null. Two distinct compact alpha-planes meet at the unique two-plane spanned by their defining lines; that point can lie at conformal infinity.

The projective line associated to a two-plane $W_x$ representing a point of complexified compactified Minkowski space. Two distinct twistor lines intersect precisely when their spacetime points are null separated. On the Lorentzian real slice, the line lies entirely in projective null twistor space and parametrizes the null geodesics through the point.

The four-complex-dimensional Grassmannian of two-planes in twistor space. Its Plücker embedding is the Klein quadric. An affine chart consists of graphs $W_x=\{(ix\pi ,\pi )\}$. A Lorentzian reality structure on twistor space or an Euclidean reality structure on twistor space selects different four-real-dimensional compact spacetimes inside this same complex manifold.

In flat twistor theory, a twistor is a pair $Z=({\omega }^A,{\pi }_{A^{\prime }})$ of two-component spinors. The twistor incidence relation with a spacetime point is ${\omega }^A=i{x}^{A{A}^{\prime }}{\pi }_{A^{\prime }}$. The nonzero vector $Z$ and its projective class play different roles: the latter belongs to projective twistor space.

Choose an antilinear quaternionic structure $J$ on ${\mathbb{C}}^4$. It induces a fixed-point-free involution on projective twistor space. A real Euclidean spacetime point is a $J$-invariant complex two-plane, equivalently a quaternionic line in ${\mathbb{H}}^2$. These form ${\mathbb{HP}}^1\cong S^4$, the conformal compactification of Euclidean space. Every projective twistor lies on the unique invariant line $\mathbb{P}{\mathrm{span}}_{\mathbb{C}}(Z,JZ)$, giving a fibration ${\mathbb{CP}}^3\to S^4$ with fiber ${\mathbb{CP}}^1$.

For $Z=(\omega ,\pi )$ and $W=(\eta ,\rho )$ the displayed Hermitian form has signature $(2,2)$. It is zero on every pair of vectors in $W_x=\{(ix\pi ,\pi )\}$ when $x$ is Hermitian, that is, Lorentzian real. Conversely a transverse totally isotropic two-plane is such a graph with $x$ Hermitian. This gives the Lorentzian reality structure on twistor space.

Real compactified Minkowski points are the two-dimensional totally isotropic subspaces for the signature-$(2,2)$ twistor Hermitian form. In a graph chart this is the usual Hermitian reality condition on $x^{A{A}^{\prime }}$. Their twistor lines lie in projective null twistor space. On the Klein quadric the same real form is the projectivized null cone of a real quadratic form of signature $(2,4)$.

A nonzero twistor null for the twistor Hermitian form. Its projective class determines a real null geodesic in compactified Minkowski space. For nonzero $\pi$, its affine real solutions of the twistor incidence relation have the form $x=x_0+r{\overline{\pi }}^A{\pi }^{A^{\prime }}$, $r\in \mathbb{R}$. Twistors with $\pi =0$ encode generators at infinity rather than finite affine solutions.

The five-real-dimensional space of projective null twistors. A projective line lies entirely in it precisely when its corresponding two-plane is totally isotropic for the twistor Hermitian form. Such lines represent real points of Minkowski conformal compactification. Two independent null twistors lie on one such line exactly when $h(X,Y)=0$.

For a fixed affine spacetime point $x$, this linear relation cuts out a two-dimensional subspace $W_x=\{(ix\pi ,\pi )\}$ of twistor space. Projectivizing gives its twistor line. For a fixed projective twistor with $\pi \ne 0$, its solutions are $x=x_0+u^A{\pi }^{A^{\prime }}$, a totally null affine alpha-plane. The compact description replaces affine solutions by the two-planes containing the fixed twistor line in the vector space.

The projective space of one-dimensional complex subspaces of twistor space. A point corresponds to an alpha-plane in complexified compactified Minkowski space; a projective line corresponds to a spacetime point.

For affine complexified Minkowski space, the relevant twistor domain isIncidence can be written ${\omega }^A=x^{A{A}^{\prime }}{\pi }_{A^{\prime }}$, absorbing the conventional factor $i$ into the definition of $x$. Each space-time point gives a twistor line $L_x\cong {\mathbb{CP}}^1$. On charts ${\pi }_{0^{\prime }}\ne 0$ and ${\pi }_{1^{\prime }}\ne 0$, the fibre coordinates transform by $\omega /{\pi }_{1^{\prime }}=(\omega /{\pi }_{0^{\prime }})/\lambda$, with $\lambda ={\pi }_{1^{\prime }}/{\pi }_{0^{\prime }}$, giving the two $\mathcal{O}(1)$ summands. The full projective twistor space corresponds to complexified compactified Minkowski space.