 us turn to an example which does not have such a realization, but which is relevant to (0,2) GLSMs. Take $\mathfrak{X} = {\mathbb P}^4_{[1,1,1,2,2]}$, with bundle $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \:
\oplus_a {\cal O}(n_a) \: \stackrel{F_a}{\longrightarrow} \: {\cal O}(m)
\: \longrightarrow \: 0$$ where $\det {\cal E}^* \cong K_{\mathfrak{X}}$: $$\sum n_a \: - \: m \: = \: 7,$$ and second Chern classes match: $$\sum n_a^2 \: - \: m^2 \: = \: 11.$$ This is not Calabi-Yau, so it would not be directly useful for a string compactification, but can help illuminate some general aspects. This stack has a ${\mathbb P}^1$ of ${\mathbb Z}_2$ orbifolds, so the inertia stack has the form $$I_{\mathfrak{X}} \: = \: \mathfrak{X} \amalg 
{\mathbb P}^1_{[2,2]}.$$ On the nontrivial component ${\mathbb P}^1_{[2,2]}$, call it $\alpha$, we can work out the decomposition of the gauge bundle. Suppose, for example, that $m$ is odd. For any given $a$, if $n_a$ is even, then $F_a$ is odd, so $F_a = 0$; if $n_a$ is even on the other hand, there is no constraint on $F_a$. In this case, we can decompose $$q^* {\cal E} |_{\alpha} \: = \: {\cal E}_+ \oplus {\cal E}_-,$$ where ${\cal E}_+$ is invariant, ${\cal E}_-$ anti-invariant under ${\mathbb Z}_2$, and specifically $$\begin{aligned}
{\cal E}_+ & = & \oplus {\cal O}(n_a \, {\rm even}),
\\
{\cal E}_- & = & {\rm ker}\left( \oplus {\cal O}(n_a \, {\rm odd}) \:
\longrightarrow \: {\cal O}(m) \right).\end{aligned}$$ A closely related decomposition exists for $m$ even.

Now that we have illuminated the definitions, let us return to our description of the general procedure for spectrum computation. At this point, the computation of spectra becomes more or less identical to that in an ordinary global orbifold by a finite group, if we think of $\alpha$ as denoting a twisted sector. We will walk through the details, as there are a few important subtleties for general cases not usually discussed in the literature, especially regarding Fock vacua, but the rest of the computation i