= Solution
Represent affine <complexified Minkowski space> by complex $2\times2$ <matrices> $x^{AA'}$, with the conformal metric characterized by $\det(dx)$. Let an element of <twistor space> be $Z=(\omega^A,\pi_{A'})\in\mathbb C^4$, and identify nonzero twistors under $Z\sim cZ$, $c\ne0$. The compact <projective twistor space> is $\mathbb{CP}^3$. For affine space-time the corresponding open domain is the <affine projective twistor space>
$$
\boxed{PT_{\rm aff}=\{[\omega,\pi]:\pi\ne0\}
=\mathbb{CP}^3\setminus\mathbb{CP}^1_\infty
\cong\operatorname{Tot}(\mathcal O(1)\oplus\mathcal O(1)).}
$$
Use the <twistor incidence relation> in the convention $\omega^A=x^{AA'}\pi_{A'}$; the more usual factor $i$ can be absorbed into the complex coordinate $x$. A point $x$ determines a <twistor line>
$$
L_x=\{[x\pi,\pi]:[\pi]\in\mathbb{CP}^1\}\cong\mathbb{CP}^1.
$$
In the two projective charts, fibre coordinates $\omega/\pi_{0'}$ and $\omega/\pi_{1'}$ differ by division by $\lambda=\pi_{1'}/\pi_{0'}$, giving the two $\mathcal O(1)$ summands. Two distinct space-time points have intersecting lines precisely when $(x-y)\pi=0$ for some nonzero $\pi$, equivalently $\det(x-y)=0$; this recovers conformal null separation.
The <twistor correspondence> is the double fibration
$$
M_{\mathbb C}\,\xleftarrow{p}\,
\mathcal F=M_{\mathbb C}\times\mathbb{CP}^1
\ \xrightarrow{\ q\ }\ PT_{\rm aff},
\qquad q(x,[\pi])=[x\pi,\pi].
$$
For a fixed twistor, the incidence solutions form an affine <alpha-plane>, a two-dimensional totally null plane. Its tangent distribution on $\mathcal F$ is generated by
$$
V_A=\pi^{A'}\partial_{AA'},\qquad
\pi^{A'}=\epsilon^{A'B'}\pi_{B'}.
$$
These <vector fields> annihilate the incidence coordinates, since $\pi^{A'}\pi_{A'}=0$, and commute. This distribution is the geometric source of the linear system in the <Penrose-Ward correspondence>.
For the inverse construction, take a rank-$r$ <holomorphic vector bundle> $E$ on the twistor domain corresponding to a space-time open set $U$. The crucial hypothesis is
$$
\boxed{E|_{L_x}\cong\mathcal O^{\oplus r}\quad\text{for every }x\in U.}
$$
These are <line-trivial bundles in the Ward correspondence>. It is not enough to specify an arbitrary <holomorphic vector bundle>, nor just zero <First Chern class> on its lines: $\mathcal O(1)\oplus\mathcal O(-1)$ has degree zero but is nontrivial. A line with a nontrivial splitting is a jumping line and does not support the regular linewise construction there. Thus a globally nonsingular field on all affine complexified space-time requires triviality on all its relevant lines; a local construction uses only the lines over $U$.
Choose <holomorphic vector bundle> <bundle frames> on two twistor patches, and let $P$ be the invertible transition <matrix> on their overlap. Pull it back to $\mathcal F$. Because it comes from <affine projective twistor space>, it is constant along the $q$-fibres:
$$
V_A P(x,\pi)=0.
$$
Triviality on $L_x$ supplies a global <holomorphic> <bundle frame> on that line. Comparing this global <bundle frame> with the two local <bundle frames> produces a <holomorphic splitting of a twistor patching matrix>,
$$
P(x,\lambda)=\Psi_-(x,\lambda)^{-1}\Psi_+(x,\lambda),
$$
where $\Psi_+$ and $\Psi_-$ are <holomorphic> and invertible near the two affine patches of the <complex projective line>. The splitting can be chosen holomorphically in $x$ locally: the restrictions have $r$ independent global sections and $H^1(\mathbb{CP}^1,\mathcal O^{\oplus r})=0$, so these <bundle frames> extend through a neighborhood of the given line. This explains where regularity and linewise triviality enter the construction.
Differentiate the factorization using $V_AP=0$. Multiplication by the invertible factors gives
$$
(V_A\Psi_+)\Psi_+^{-1}
=(V_A\Psi_-)\Psi_-^{-1}.
$$
Define their common negative to be $\mathcal A_A$. It is regular on both projective patches and homogeneous of degree one in $\pi$, because $V_A$ has that degree while the <bundle frames> have degree zero. Consequently
$$
\mathcal A_A(x,\pi)=\pi^{A'}A_{AA'}(x).
$$
This last conclusion is elementary: on one affine chart its entries are entire in $\lambda$; regularity at infinity after dividing by the transition factor $\lambda$ allows at most linear growth, so each entry is affine in $\lambda$. Equivalently, a global section of $\mathcal O(1)$ is a linear homogeneous polynomial. We have thereby recovered a <gauge connection> $D_{AA'}=\partial_{AA'}+A_{AA'}$ on the space-time bundle whose fibre is $H^0(L_x,E|_{L_x})$.
The splitting factors now obey
$$
\boxed{\pi^{A'}D_{AA'}\Psi_\pm=0.}
$$
Their invertibility implies the compatibility conditions
$$
\pi^{A'}\pi^{B'}F_{AA'BB'}=0\quad\text{for every }[\pi].
$$
In the <spinor decomposition of gauge curvature>,
$$
F_{AA'BB'}=\epsilon_{AB}\Phi_{A'B'}+\epsilon_{A'B'}\Phi_{AB},
$$
the two spinors are symmetric. The antisymmetric primed factor vanishes when contracted with $\pi^{A'}\pi^{B'}$, leaving $\epsilon_{AB}\pi^{A'}\pi^{B'}\Phi_{A'B'}=0$. A quadratic polynomial that vanishes for every spinor has every coefficient zero, so $\Phi_{A'B'}=0$. With the orientation convention used here, this is exactly <anti-self-duality of gauge curvature>, hence the <ASDYM equations>. This establishes the field equation from the <holomorphic> data, rather than assuming it as an extra input.
The construction naturally gives a gauge class. If another linewise splitting is chosen, $\Psi'_\pm\Psi_\pm^{-1}$ agrees on the overlap and is <holomorphic> on the entire compact <complex projective line>; its entries are therefore independent of $\lambda$. Write this common <matrix> as $g(x)^{-1}$. Then
$$
A'_{AA'}=g^{-1}A_{AA'}g+g^{-1}\partial_{AA'}g,
$$
the ordinary <Yang-Mills gauge transformation>. Changing the original <holomorphic vector bundle> <bundle frames> or replacing $E$ by an isomorphic bundle produces the same space-time gauge class. A determinant trivialization or another <holomorphic> structure-group reduction gives the corresponding restriction from $GL(r,\mathbb C)$ to $SL(r,\mathbb C)$ or another complex <gauge group>.
For perspective, the forward direction starts from an ASD <gauge connection> and uses its flat <gauge covariant derivative> along each <alpha-plane> to descend parallel sections to <affine projective twistor space>. The reconstructed linear system is exactly that parallel-section system. Going back through it recovers the original line-trivial bundle, while changing space-time <bundle frames> amounts to <Yang-Mills gauge transformation>. Thus, locally and with the stated regularity and triviality conditions, \b[<holomorphic> line-trivial bundles determine solutions of the <ASDYM equations> uniquely up to gauge].
No unitary reality condition is imposed on <complexified Minkowski space> in this question. Obtaining real Euclidean <gauge fields> requires the compatible antiholomorphic twistor involution and a bundle reality structure; obtaining finite-action instantons requires further global and boundary conditions. Those are additional restrictions on the complex correspondence, not automatic consequences of having any <holomorphic vector bundle> on <projective twistor space>.
Back to article page
