aly-free (0,2) GLSM describing a bundle ${\cal E}'$, say, $$0 \: \longrightarrow \: {\cal E}' \: \longrightarrow \:
\oplus_a {\cal O}(n_a) \: \stackrel{F}{\longrightarrow} \:
\oplus_i {\cal O}(m_i) \: \longrightarrow \: 0,$$ over a hypersurface in a weighted projective space ${\mathbb P}^d_{w_0, \cdots, w_d}[w_0 + \cdots + w_d]$. Now, build a new (0,2) GLSM constructed from the one above by multiplying all gauge charges by an integer $k > 0$. The result is a bundle ${\cal E}$, $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \:
\oplus_a {\cal O}(k n_a) \: \stackrel{F}{\longrightarrow} \:
\oplus_i {\cal O}(k m_i) \: \longrightarrow \: 0,$$ over a hypersurface in a weighted projective stack ${\mathbb P}^d_{[k w_0, \cdots, k w_d]}[k(w_0 + \cdots + w_d)]$. The bundle map $F$ and hypersurface polynomial are unchanged. If one now goes to the Landau-Ginzburg point of this theory and computes the massless spectrum, it is trivial to see that the spectrum will consist of $k$ copies of the spectrum of the original theory, in agreement with the prediction of the decomposition conjecture.

The analogue of the decomposition conjecture for nonbanded gerbes is not currently known. It is tempting to speculate that it should be the statement that a heterotic string on $([X/G], {\cal E})$ is the same as a heterotic strings on $( [(X \times \hat{K})/H],
{\cal E})$, where (as in the (2,2) case) $\hat{K}$ is the set of irreducible representations of $K$, and we extend ${\cal E}$ trivially from $[X/H]$ to $[(X \times \hat{K})/H]$. However, on the (2,2) locus, the special case that ${\cal E} = TX$ with its natural $G$-equivariant structure, ${\cal E}$ reinterpreted as an $H$-equivariant bundle and extended trivially over $\hat{K}$ does not in general[^11] define the tangent bundle of $[(X \times \hat{K})/H]$, and so this would not reduce correctly to (2,2) decomposition.

Class II: Dualities {#sect:het-gsomods}
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Basic proposal
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In the special case of a heterotic string in which a ${\mathbb Z}_2$ t