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I'm trying to get my head around integrability in twistor theory, but am struggling with interpreting the concept of self-duality on complexified spaces.

One can complexify Minkowski space to $\mathbb{C}^4$ with coordinates $(z_1,z_2,z_3,z_4)$ and metric $ds^2=2(dz^2dz^4-dz^1dz^3)$. I have see it written that the two-forms

$$dz^1\wedge dz^2, \ dz^1\wedge dz^3-dz^2\wedge dz^4,\ dz^3\wedge dz^4$$

form a basis for the self-dual two-forms on $\mathbb{C}^4$. I can see that this agrees with the definition of self-duality in the standard Euclidean slice.

But what does it mean in general for these complex forms to be self-dual? Just applying the definition of the Hodge star to these forms blindly doesn't suggest that they are self-dual at all! Is there some other notion of self-dual in complexified spaces, or does the statement simply mean that they reduce to self-dual forms in some real slice?

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My best guess is that the source you are quoting writes $\mathbb{C}^4$ for $\mathbb{R}^4\otimes_\mathbb{R} \mathbb{C}$? And $*$ is meant to be the complex linear extension of the Hodge operator on $\mathbb{R}^4$. Can you include the source? – Willie Wong Mar 8 '13 at 13:20
Dunajski - Solitons, Instantons and Twistors is the source. So should I interpret $*(x+iy)$ as $*x + i*y$ then? – Edward Hughes Mar 9 '13 at 9:49

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