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Let $G$ be a Lie group, and let $\underline{G}$ denote the sheaf of smooth $G$-valued maps, i.e. for a smooth manifold $M$ we have $G(M) = C^\infty(M,G)$. What are conditions on $G$ that imply that …

Yes, it is exact, and there is in fact a canonical 2-form on each conjugacy class whose derivative is your $\Omega$. This was an important observation when studying D-branes in WZW models, see, e.g. h …

Here is one way to see it: gerbes on $M$ form a bigroupoid (= bicategory all of whose morphisms and 2-morphisms are invertible). In particular, if $\mathcal{G}$ is a gerbe over $M$, and $\mathcal{I}_ …

One prerequisite to any differential-geometric point of view of course is that $G$ is a Lie group rather than just a topological group. Then, one option is to pass from manifolds to Lie groupoids: If …

A nice point of view is to consider principal bundles with structure group $B\mathbb{C}^\times$. One can probably take any abelian Lie group instead of $\mathbb{C}^\times$. Principal $B\mathbb{C}^\tim …