 $G$-equivariantizable structure, which means that for every $g \in G$, we need a lift $\tilde{g}: {\cal E} \rightarrow {\cal E}$ such that $$\xymatrix{
{\cal E} \ar[r]^{\tilde{g}} \ar[d] & {\cal E} \ar[d] \\
X \ar[r]^{g} & X
}$$ and also such that the lifts obey the group law: $\tilde{g} \circ
\tilde{h} = \widetilde{gh}$.

We need such an equivariant structure on the bundle ${\cal E}$ for the following two reasons:

-   In the worldsheet theory, such an equivariant structure enables us to define a group action on the worldsheet fermions/bosons describing the bundle, such that summing over twisted sectors in the orbifold yields an honest projection operator onto $G$-invariant states.

-   In the low-energy supergravity, if ${\cal E}$ does not have an equivariant structure, then even if $G$ acts freely, on the quotient $X/G$ the bundle ${\cal E}$ will descend to a ‘twisted’ bundle, not an honest bundle, whose transition functions $g_{\alpha \beta}$ obey $$g_{\alpha \beta} g_{\beta \gamma} g_{\gamma \alpha} \: = \: h_{\alpha
    \beta \gamma} I$$ on triple overlaps, and whose gauge field $A$ obeys $$A_{\beta} \: = \: g_{\alpha \beta} A_{\alpha} g_{\alpha \beta}^{-1}
    \: + \: g_{\alpha \beta}^{-1} d g_{\alpha \beta} \: + \:
    \Lambda_{\alpha \beta} I$$ across intersections, for some affine translation $\Lambda_{\alpha \beta}$. As ten-dimensional super-Yang-Mills only describes honest bundles and ordinary gauge transformations, the structure above cannot be used to define a consistent string compactification.

However, there is a workaround. If the bundle ${\cal E}$ is invariant (meaning, its characteristic classes are invariant under the group action), but not equivariant, then we can find a larger group $\tilde{G}$, an extension of $G$ by a trivially-acting subgroup, such that ${\cal E}$ does admit a $\tilde{G}$-equivariant structure, and then take a $G'$ orbifold. This is precisely an example of a heterotic string on a gerbe, in this case a gerbe over $[X/G]$.

First, let us review some generalities on th