An Explicit Construction of ${\mathbb{S}}^1$-Gerbes over the Stack $[G\mathrm{/}G]$

For a compact and connected Lie group $G$, we present an explicit construction of an ${\mathbb{S}}^1$-gerbe over the differentiable stack $[G\mathrm{/}G]$ in the framework of ${\mathbb{S}}^1$-central extensions of Lie groupoids. This gives a complete proof of the construction outlined earlier by Behrend–Xu–Zhang, together with an explicit proof of the differential-form identity stated there without proof. In particular, when $G$ is compact, simple, and simply connected, the Dixmier–Douady class of the resulting gerbe is the canonical generator of ${\mathrm{H}}_G^3(G,\mathbb{Z})$.