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I have to brush up my knowledge about self-dual Yang Mills and I'm reading an ancient paper by Yang about it...and of course I'm stuck...although Yang writes 'it is easy to see that'...

Ok, so the self-duality condition of the YM field strength tensor is defined as

[tex] 2F_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}[/tex].

If I know go to complex coords defined by

[tex]\sqrt{2}y=x_1+i x_2 \quad \sqrt{2}\bar{y}=x_1-i x_2[/tex]

and

[tex]\sqrt{2}z=x_3+i x_3 \quad \sqrt{2}\bar{z}=x_3-i x_4[/tex]

the metric transforms to

[tex]g_{y\bar{y}}=g_{\bar{y}{y}}=g_{z\bar{z}}=g_{\bar{z}{z}}=1[/tex]. So far i've understood everything. But then Yang says it's easy to see that the self-duality condition becomes

[tex]F_{yz}=0=F_{\bar{y}\bar{z}}[/tex]

[tex]F_{y\bar{y}}=F_{z\bar{z}}[/tex]

The question know is: how do i see the last two equations? Does the epsilon tensor somehow transform if i go to these complex coords?

Cheers,

earth2

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# Self-Dual Field Strength in complex coordinates

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