SelfDual Field Strength in complex coordinatesby earth2 Tags: complex, coordinates, field, selfdual, strength 

#1
Apr2512, 05:27 AM

P: 86

Hi guys,
I have to brush up my knowledge about selfdual 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 selfduality 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_1i x_2[/tex] and [tex]\sqrt{2}z=x_3+i x_3 \quad \sqrt{2}\bar{z}=x_3i 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 selfduality 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 



#2
Apr2512, 10:30 AM

Sci Advisor
Thanks
P: 3,846

The selfduality condition says F_{12} =  F_{34}, F_{13} = F_{24}, F_{14} =  F_{23}. So for example (ignoring √2's)
F_{yz} = F_{13} i F_{23} i F_{14}  F_{24} ≡ 0 



#3
Apr2512, 12:24 PM

P: 86

Ah cool, i didn't know that i could just plug in numbers back :) Nice, thank you!



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