Resources for self-dual solutions to Maxwell's equation on Euclidean or pseudo-Euclidean space I am attempting to understand a question posed to me by an acquaintance, who asked me if I could refer him to literature on "self-dual solutions to Maxwell's equations on Euclidean space, or pseudo-Euclidean space, not Minkowski space (where there are none)" and he labeled them "instantons". I'm lost, so my questions: 


*

*aren't instantons solutions to equations of motion? Can you consider Maxwell's equations as equations of motion? 

*In what sense are solutions to such equations self-dual, or more simply, what concept of duality is meant here? 

*does anyone have any appropriate literature  to recommend to answer my acquaintance's question? 
Any indications on any or all of these questions would be greatly appreciated.
 A: On flat $R^4$ the only bounded solutions to $dF=0$, $F=*F$ are linear combinations of the three usual Kahler forms; this is going to be due to $\triangle F = 0$ and so $\triangle |F| \ge 0$ (by the Kato inequality), and the fact that a subharmonic function with 2-sided bounds is constant (a version of Liouville's theorem); then one finds out which solutions F have constant norm, and learns they are just the Kahler forms on $R^4$ (eg. using the identity $\frac12\triangle|F|^2=|\nabla F|^2+\left<\triangle{}F,F\right>$).
On a compact Riemannian 4-manifold, the solutions to $dF=0$, $F=*F$ are precisely given by a cohomology space, the self-dual de Rham space $H^+ \subseteq H^2$. On a non-compact Riemannian 4-manifold the question is harder, and I'm not sure if all bounded solutions to your friend's equations are known in general. Of course there are the $L^2$ de Rham spaces, but computing these is tough, and might or might not provide a satisfactory answer even still.
Needless to say, in 4-dimensional Riemannian geometry, solutions to $dF=0$ where $F$ is a 2-form always split into non-interacting self-dual and anti-self-dual parts, quite in contrast to the Lorentzian case.
