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

TL;DR

This paper explicitly constructs an $ ext{S}^1$-gerbe over the stack $[G/G]$ for a compact Lie group $G$, providing a detailed proof and differential-form identities, with applications to the Dixmier--Douady class.

Contribution

It offers a complete explicit construction of $ ext{S}^1$-gerbes over $[G/G]$ and clarifies the associated differential-form identities, extending prior work by Behrend--Xu--Zhang.

Findings

Constructed an explicit $ ext{S}^1$-gerbe over $[G/G]$ for compact Lie groups.

Proved the differential-form identity associated with the gerbe.

Identified the Dixmier--Douady class as the canonical generator in specific cases.

Abstract

For a compact and connected Lie group $G$, we present an explicit construction of an $\mathbb{S}^1$-gerbe over the differentiable stack $[G/G]$ 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 ${\rm H}^3_G(G,\mathbb Z)$.