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Self-Dual Field Strength in complex coordinates

  1. Apr 25, 2012 #1
    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
     
  2. jcsd
  3. Apr 25, 2012 #2

    Bill_K

    User Avatar
    Science Advisor

    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
     
  4. Apr 25, 2012 #3
    Ah cool, i didn't know that i could just plug in numbers back :) Nice, thank you!
     
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