Hi guys,(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Self-Dual Field Strength in complex coordinates

**Physics Forums | Science Articles, Homework Help, Discussion**