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The insertion of an orientifold $O_{p}$-plane in a string theory setup, results to the following action on the string coordinates $X^{\mu}(\tau,\sigma)$ (leave aside the Chan-Paton factors for now):$$\Omega_{p}X^{i}(\tau,\sigma)\Omega^{-1}_{p}=X^{i}(\tau,\pi-\sigma)$$ and $$\Omega_{p}X^{\alpha}(\tau,\sigma)\Omega^{-1}_{p}=-X^{\alpha}(\tau,\pi-\sigma),$$ where "$i$" are the directions parallel to the orientifold plane and "$\alpha$" the transverse ones.

Now, consider the Green-Schwartz superstring, where we have spacetime supersymmetry. In this case except for the common bosonic $X^{\mu}(\tau,\sigma)$ we have also the fermionic fields: $$\Theta^{I}(\tau,\sigma),\quad \bar{\Theta}^{I}(\tau,\sigma)$$

I am not sure about the action of $\Omega_{p}$ on these fermionic fields. My first thought is that they transform like the bosonic ones but unfortunately I cannot justify it. Is there any justification for the transformation rule or any relevant literature?

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  • $\begingroup$ No, they don't transform as the bosonic fields, they do so up to a sign. The easiest way to answer your question is to work with space-time fields (ie representations of the little group $SO(8)$). The details can be found in section 6.3 of the book by Becker, Becker and Schwarz, or at the end of section 10.3 in the book by Polchinski. $\endgroup$
    – Antoine
    Commented Oct 24, 2016 at 9:21
  • $\begingroup$ thank you so much for your reply, I will carefully check it now $\endgroup$
    – Jordan
    Commented Oct 24, 2016 at 9:26

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