Gerbes give a nice way to group together bundle data on a smooth manifold, but gerbes also naturally define degree two cohomology. Thus the idea of using gerbes in differential geometry is to have a nice language that relates geometric concepts such as connections and curvature to cohomological classifications.

In addition, one can use the analogies above that are made precise with gerbes to define other new concepts such as 3-curvature and “local” fiberwise connections.

Definitions

Let $X$ be a smooth manifold. A Dixmier-Douady sheaf of groupoids over $X$ is a $\underline{\mathbb{C}}_X^*$- gerbe on $X$ where $\underline{\mathbb{C}}_X^*$ is the sheaf of smooth $\mathbb{C}^*$-valued functions (not to be confused with the constant sheaf $\mathbb{C}_X^*$).

We define $\mathbb{Z}(1)$ to be the term in the exponential sequence on $X$: $0\to \mathbb{Z}(1)\to \underline{\mathbb{C}}_X \to \underline{\mathbb{C}}_X^*\to 0$.

Preliminaries

Taking the associated sequence in cohomology to the exponential sequence gives us an isomorphism $H^2(X, \underline{\mathbb{C}}_X^*)\overset\sim\to H^3(X, \mathbb{Z}(1))$.

We have a canonical isomorphism between the group of equivalence classes of Dixmier-Douady sheaves of groupoids over $X$ (basically by the definition of $\mathcal{A}$-gerbe) and $H^2(X, \underline{\mathbb{C}}_X^*)\simeq H^3(X, \mathbb{Z}(1))$.

Geometric Interpretation of $H^3(X, \mathbb{Z}(1))$

Idea: Classes in $H^3(X, \mathbb{Z}(1))$ correspond to principal $G$- bundles over $X$ where $G$ is the projective linear group of a separable Hilbert space, namely $C^\infty (\mathbb{T})$.

Matt: Actually, a slight issue has arisen. Most of the things I thought would go here actually already appear in other places even though they aren’t grouped as coming from the same idea.

For instance, bundle gerbe contains the geometric interpretation of $H^3(X, \mathbb{Z}(1))$. Also, 3-curvature and fiber-wise connections occur at connection on a bundle gerbe. Although I think there is still a lot to say, I’m not convinced that “gerbe (in differential geometry)” is necessary anymore…

References

Jean-Luc Brylinski Loop Spaces, Characteristic Classes, and Geometric Quantization

I. Moerdijk, Introduction to the language of stacks and gerbes (arXiv)