= Contour potential for an anti-self-dual Maxwell field
{title2=$A_{BB'}=\frac1{2\pi i}\oint_\Gamma\frac{\iota_{B'}}{\iota_{C'}\pi^{C'}}\,f_{,B}\,\pi_{D'}d\pi^{D'}$}
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 $F_{AA'BB'}=\epsilon_{A'B'}(2\pi i)^{-1}\oint_\Gamma f_{,AB}\,\pi_{D'}d\pi^{D'}$, using $\pi_{A'}\iota_{B'}-\pi_{B'}\iota_{A'}=\epsilon_{A'B'}(\iota_{C'}\pi^{C'})$. 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 $\epsilon_{0'1'}=1$, $\epsilon^{0'1'}=-1$ and raising by left multiplication; other consistent raising conventions can reverse both displayed overall signs.
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