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Fermion is left-handed spinor and anti-fermion is right-handed spinor. Firstly, you can see that $\xi = \chi ^{\dagger}$. From Sredniski, "... that hermitian conjugation swaps the two SU(2) Lie algebras...the hermitian conjugate of a field in the (2,1) representation should be a field in the (1,2) representation"... Thus $\chi$ is a right-handed fermion ...

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