= Abelian anti-self-dual connections from harmonic scalar potentials
{c}
{title2=$A=\tfrac12(\bar\partial f-\partial f),\quad f_{w\bar w}+f_{z\bar z}=0$}
Locally a <U(1) connection> whose <gauge curvature> has no $(2,0)$ or $(0,2)$ part can be written $A=\partial u+\bar\partial v$ by the <Dolbeault-Poincaré lemma>. Unitarity permits $v=-\bar u$. Writing $u=p+iq$, a <gauge transformation> removes $i\,dq$ and leaves $A=\partial p-\bar\partial p$. Setting $f=-2p$ gives the displayed <unitary> form, whose <gauge curvature> is $\partial\bar\partial f$. Its remaining <anti-self-duality of gauge curvature> condition is the scalar <Laplace equation>. Alternatively a complex <gauge transformation> by $e^{-u}$ gives $A=\bar\partial(v-u)$; this second gauge generally does not preserve unitarity.
// Destination: complex-geometry.bigb
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