Twistor correspondence ($\mathrm{Gr}(2,{\mathbb{C}}^4)\leftarrow F_{1,2}({\mathbb{C}}^4)\to {\mathbb{CP}}^3$)

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.