l examine properties of those classes.

The first class we consider involves the special case that the gauge bundle is a pullback from the base. This is equivalent to the statement that the subgroup $G$ of the gauge group that acts trivially on the base, also acts trivially on the fibers of the gauge bundle.

In this case, we will argue that, at least for banded gerbes, the heterotic (0,2) SCFT factorizes – it is equivalent to a heterotic string on a disjoint union of spaces with bundles, following essentially the same mechanism as in (2,2) strings.

Review of (2,2) decomposition conjecture {#sect:decomp-22review}
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As was reviewed earlier in section \[sect:rev-gerbes\], gauge theories in which a subgroup of the gauge group acts trivially on massless matter break cluster decomposition. However, it was argued in [@summ] that such theories are equivalent to tensor products / disjoint unions of cluster-decomposition-obeying theories. For example, a gauged nonlinear sigma model of this form is equivalent to a nonlinear sigma model on a disjoint union of ordinary spaces. The latter violates cluster decomposition, but does so in an obviously trivial fashion, and so there is no essential difficulty with the quantum field theory.

For (2,2) supersymmetric gauged nonlinear sigma models in two dimensions, this was encapsulated in [@summ] in the “decomposition conjecture,” which we shall generalize to heterotic strings. To make this paper self-contained, we take a moment here to review the statement of the decomposition conjecture.

Suppose we have a $K$-gerbe over $[X/H]$, defined by the quotient $[X/G]$ where $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \:
H \: \longrightarrow \: 1.$$ Let $\hat{K}$ denote the set of irreducible representations of $K$. There is a natural action of $H$ on $\hat{K}$, defined as follows: given $h \in H$ and $\rho \in \hat{K}$, pick a lift $\tilde{h} \in G$ of $h$, and define $h \cdot \rho$ by, $$(h \cdot \rho)(g) \: \equ