So I am following this script here: https://arxiv.org/abs/1712.02196
I am already stuck at chapter 1.3: I understand for the three cases Lorentzian, Euclidean and Split case that the coordinates need to be what they are written.
But what I don't get is how it should induce a complex conjugation. E.g. it says that the fact $x^{\alpha \dot{\alpha}}$ should stay invariant under $\dagger$ implies that the natural notion of conjugation on spinor space is sending left chiral spinors to right chiral representation with its components complex conjugated.
Or that in the euclidean case, the naturally induced conjugation on spinor space is to switch the components of the spinor and multiplying by a - in the first component.
How does he arrive at this conclusion given the "reality conditions" (i.e. what conditions the a priori complex components of $x^{\alpha \dot{\alpha}}$ must fulfill to be on the desired slice)?
