hat acts nontrivially on the base, acts nontrivially on a rank 8 bundle, that subgroup of the gauge group is locally duplicating the effect of one of the ten-dimensional left-moving GSO projections. If one starts with a Spin$(32)/{\mathbb Z}_2$ string, then the dual looks locally like an $E_8 \times E_8$ string.

In this section, we will describe[^12] a precise duality relating such Spin$(32)/{\mathbb Z}_2$ compactifications to ordinary $E_8 \times E_8$ compactifications, and discuss some examples.

First, let us consider the easiest case. If the ${\mathbb Z}_2$ gerbe is trivial, the result is automatic: the worldsheet left-moving GSO projection is duplicated exactly, not just locally. When the gerbe is nontrivial, one must think a little harder to find a precise duality.

We propose[^13] a duality to heterotic $E_8 \times E_8$ strings as follows. To set conventions, suppose our stack $\mathfrak{X} = [X/\tilde{G}]$, where $$1 \: \longrightarrow \: {\mathbb Z}_2 \: \longrightarrow \:
\tilde{G} \: \longrightarrow \: G \: \longrightarrow \: 1$$ and ${\mathbb Z}_2$ acts trivially on $X$. Suppose furthermore that ${\cal E}$ is a bundle on $\mathfrak{X}$, [*i.e.*]{} a $\tilde{G}$-equivariant bundle on $X$, whose embedding into $E_8$ is via the standard worldsheet fermionic construction, in which left-moving fermions are in the fundamental representation of the structure group. Suppose that the ${\mathbb Z}_2$ acts nontrivially on ${\cal E}$.

In general ${\cal E}$ will not admit a $G$-equivariant structure. Nevertheless, at least in the special case that the ${\mathbb Z}_2$ is central in $\tilde{G}$, the bundles ${\cal E}^* \otimes {\cal E}$ and $\wedge^{\rm even} {\cal E}$ will admit a $G$-equivariant structure, and so can be defined on $[X/G]$. Moreover, it was observed in [@dist-greene] that, for the ‘typical’ worldsheet embeddings of $SU(n)$ in $E_8$ (including the present one), massless spectra of heterotic compactifications on smooth spaces can be defined solely in terms of sheaf cohomology with coefficients 