Self-Dual Field Strength in complex coordinates

[earth2]

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

 

[Bill_K]

The self-duality condition says F₁₂ = - F₃₄, F₁₃ = F₂₄, F₁₄ = - F₂₃. So for example (ignoring √2's)

F_yz = F₁₃ -i F₂₃ -i F₁₄ - F₂₄ ≡ 0

 

[earth2]

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