# Orbifold actions and twist operators

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?

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