Self-Dual Field Strength in complex coordinates


by earth2
Tags: complex, coordinates, field, selfdual, strength
earth2
earth2 is offline
#1
Apr25-12, 05:27 AM
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

[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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Bill_K
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#2
Apr25-12, 10:30 AM
Sci Advisor
Thanks
Bill_K's Avatar
P: 3,861
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
earth2
earth2 is offline
#3
Apr25-12, 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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