A question about the antiperiodic conditions in string theory We know that for both bosonic and fermionic strings, there are possibly antiperiodic boundary conditions:
$$X^\mu(\tau,\sigma+2\pi)=-X^\mu(\tau,\sigma); \tag 1\\$$
$$\Psi(\tau,\sigma+2\pi)=-\Psi(\tau,\sigma). \tag2 $$
Eq.(1) can be interpreted as a string moving in a orbifold while Eq.(2) is simply the Neveu-Schwarz sector for fermionic strings. But I really find it uncomfortable to accept such anti-periodic conditions. For Eq.(1), is the orbifold physically real? For Eq.(2), how can the field (though fermionic) on the string be double-valued?
 A: The question involves orbifolds in general that map a string according to a discrete group $\Gamma$ as $X^i~\rightarrow~\theta^{ij}X^j~+~x^i$, for the indices $i,~j~>~3$ on the compactified manifold. The string or particle propagates on a space $M^4\times C$ in twisted theory. The space $C$, a Calabi-Yau space, is of the form $\mathbb R^6/S$, such that $S$ is a space group, similar to solid state physics, and that $\tau~=~S/\Gamma$ has the symmetry of a torus and is a twist that defines the orbifold on a torus $\mathbb T^6/\tau$.
The twists in the $6$-dimensional space induce $\psi(\sigma~+~2\pi)~=~\gamma\psi(\sigma)$, for $\gamma$ an element of the discrete group $\Gamma$. The complex valued string coordinates $Z^j~=~(X^{2j}~+~X^{2j+1})/\sqrt 2$ for $j~=~\{2,~3,~4\}$ satisfy the periodicity conditions
$$
Z^j(\sigma~+~2\pi)~=~e^{2\pi i(\phi_j+\theta}Z^j(\sigma).
$$
Here the phase term $\theta~=~0$ in the Ramond sector and $\theta~=~1/2$ in the Neveu-Schwarz sector. This induces a phase shift on the field as
$$
\psi^j(\sigma~+~2\pi)~=~e^{2\pi i(\phi_j+\theta)}\psi^j(\sigma),
$$
so that with compactification and reduction to $4$ dimensions the twisting theory of the string on an orbifold induces this phase shift.
This a simple look at the situation. In the NS sector the phase shift for $\phi_j~=~0$ is $\psi^j(\sigma~+~2\pi)$ $=~e^{\pi i}\psi^j(\sigma)$ $=~-\psi^j(\sigma)$.  This twisting is a form of $T$-duality and the Moebius or linear fractional transformation of a string which relates the mode number to the winding number of the string on the compactified manifold, a Calabi-Yau space or D-brane. 
