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

[Jim Kata]

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.

$$\lambda^a \lambda^b(\frac{\partial A_{b\dot{b}}}{\partial x^{a\dot{a}}}-\frac{\partial A_{a\dot{a}}}{\partial x^{b\dot{b}}})=\lambda^a \lambda^b\epsilon_{\dot{a}\dot{b}}\bar{\Phi_{ab}}=0$$ I can derive this equation, but what bothers me about it is the spinors. Namely, that $$\lambda^a \lambda^b\epsilon_{\dot{a}\dot{b}}$$ always equals zero so doesn't imply that $$\frac{\partial A_{b\dot{b}}}{\partial x^{a\dot{a}}}-\frac{\partial A_{a\dot{a}}}{\partial x^{b\dot{b}}}=\epsilon_{\dot{a}\dot{b}}\bar{\Phi_{ab}}=0$$ 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 $$\bar{\partial}=\lambda^a\frac{\partial}{x^{a\dot{a}}}$$

 

[sheaf]

I agree that

$$\lambda^a\lambda^b\epsilon_{ab} = 0$$

and

$$\lambda^{\dot{a}}\lambda^{\dot{b}}\epsilon_{\dot{a}\dot{b}} = 0$$

but there's no reason for

$$\lambda^{{a}}\lambda^{{b}}\epsilon_{\dot{a}\dot{b}} = 0$$

to be true is there ?

 

[Jim Kata]

Sorry, I'm a little toasty.

What i meant to write is:

$$\lambda^a \lambda^b(\frac{\partial A_{b\dot{b}}}{\partial x^{a\dot{a}}}-\frac{\partial A_{a\dot{a}}}{\partial x^{b\dot{b}}})=\lambda^a \lambda^b\epsilon_{ab}\bar{\Phi_{\dot{a}\dot{b}}}= 0$$

but $$\lambda^a \lambda^b\epsilon_{ab} =0$$ is always true

 

[sheaf]

So why do they define $$\bar{\partial}=\lambda^a\frac{\partial}{x^{a\dot{a}}}$$

I can't be sure without seeing the paper you're reading but I think the significance of the operator

$$\lambda^a\partial_{a\dot{a}}$$

usually comes about when you're establishing the correspondance between self and anti self dual zero rest mass fields and (equivalence classes of) functions on twistor space. The correspondance is established by using the fact that the spin bundle (with coordinates $(x^{a\dot{a}}, \lambda^a)$) is a bundle over spacetime when you fibre it one way, and a bundle over twistor space when you fibre it another way. Functions defined on the spin bundle i.e. functions $f(x^{a\dot{a}}, \lambda^a)$ reduce to functions on twistor space if they're constant along the twistor space fibration. To satisfy this, they have to be annihilated by the vector field $\lambda^a\partial_{a\dot{a}}$ on the spin bundle. If this is true, then they're not just any old functions $f(x^{a\dot{a}}, \lambda^a)$, but rather they're functions of the form $f(ix^{a\dot{a}}\lambda_{a}, \lambda^a)$