# Self-Dual Field Strength in complex coordinates

by earth2
Tags: complex, coordinates, field, selfdual, strength
 P: 86 Hi guys, 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 $$2F_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$$. If I know go to complex coords defined by $$\sqrt{2}y=x_1+i x_2 \quad \sqrt{2}\bar{y}=x_1-i x_2$$ and $$\sqrt{2}z=x_3+i x_3 \quad \sqrt{2}\bar{z}=x_3-i x_4$$ the metric transforms to $$g_{y\bar{y}}=g_{\bar{y}{y}}=g_{z\bar{z}}=g_{\bar{z}{z}}=1$$. So far i've understood everything. But then Yang says it's easy to see that the self-duality condition becomes $$F_{yz}=0=F_{\bar{y}\bar{z}}$$ $$F_{y\bar{y}}=F_{z\bar{z}}$$ 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
 Sci Advisor P: 3,696 The self-duality condition says F12 = - F34, F13 = F24, F14 = - F23. So for example (ignoring √2's) Fyz = F13 -i F23 -i F14 - F24 ≡ 0
 P: 86 Ah cool, i didn't know that i could just plug in numbers back :) Nice, thank you!

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