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A twist operator $\sigma$ is the operator that acts on the untwisted vacuum $|0\rangle$ to create a twisted vacuum $\sigma|0\rangle$. States belonging to the twisted sector of an orbifold are built on such twisted vacua (which are also in 1-1 correspondence with the fixed points of the orbifold action).

My question is given a specific twist operator $\sigma$, how can I possibly deduce the orbifold action that corresponds to it?

More specifically, let $$\sigma=\psi^1+i\psi^2$$ the operator that arises after bosonizing two real fermions. If the boundary conditions for the fermions are known, let's say $\psi^1\rightarrow \psi^1$ and $\psi^2\rightarrow -\psi^2$ then we have some information on $\sigma$. We should be able to interpret this $\sigma$ as the twist operator that generates a twisted sector of some orbifold, but what would the orbifold action in this case be?

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  • $\begingroup$ The twist operator that arises after bosonizing two real fermions? It's the other way round. Gamma-gamma pair production is where you start with untwisted vacuum and end up with twisted vacuum. Electron-positron annihilation reverses the process. $\endgroup$ – John Duffield Jan 11 '16 at 14:31
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For a toroidal orbifold, suppose you know the boundary condition in the k-twisted sector and the spin structure $\alpha, \beta \in \lbrace 0,1/2\rbrace $. In light cone gauge you have:

$\psi^{j}(\sigma^0, \sigma^1+2\pi)=- e^{2\pi i \alpha }e^{2\pi i k v_j }\psi^{j}(\sigma^0, \sigma^1)$

where:

$v_j=a_j/N$

in which $a_j$ are some integer, fixed by the possible crystallographic actions. From this boundary condition you can get N of the $Z_N$ orbifold.

Check "Basics concepts of String Theory", (Lust, Theisen, Bluemenhagen), pag.306

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