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When studying the CFT of a complex fermion $\Psi$ we know that if it's periodic, ie if $$\Psi(\sigma_1+2\pi,\sigma_2)=\Psi(\sigma_1,\sigma_2)$$ then there is a doubly degenerate Ramond vacuum which I denote $|\pm\rangle$.

The question is what happens if $\Psi$ is real? Is there a Ramond vacuum at all? Is it still doubly degenerate? Why/why not?

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