Hi, I was working through a Twistor paper and it was explaining the link between holomorphic vector bundles and anti self dual gauges and it had an equation like this, for electro-magnetism.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\lambda^a \lambda^b(\frac{\partial A_{b\dot{b}}}{\partial x^{a\dot{a}}}-\frac{\partial A_{a\dot{a}}}{\partial x^{b\dot{b}}})=\lambda^a \lambda^b\epsilon_{\dot{a}\dot{b}}\bar{\Phi_{ab}}=0[/tex] I can derive this equation, but what bothers me about it is the spinors. Namely, that [tex]\lambda^a \lambda^b\epsilon_{\dot{a}\dot{b}}[/tex] always equals zero so doesn't imply that [tex]\frac{\partial A_{b\dot{b}}}{\partial x^{a\dot{a}}}-\frac{\partial A_{a\dot{a}}}{\partial x^{b\dot{b}}}=\epsilon_{\dot{a}\dot{b}}\bar{\Phi_{ab}}=0[/tex] Even though this is equation is true. What I"m saying is the first equation is true regardless if the gauge is anti selfdual because of the nullity of the spinor norm. So why do they define [tex]\bar{\partial}=\lambda^a\frac{\partial}{x^{a\dot{a}}}[/tex]

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# Twistor question the link holomorphic vector bundles and anti self dual guage

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