Bundle Gerbes

The notion of bundle gerbe is a higher categorical analogue of that of line bundle. Bundle gerbes over $X$ provide a geometric realisation for the Dixmier-Douady class $H^3(X,\mathbb{Z})$.

Definition

Given a subduction of diffeological spaces $\pi: Y\to X$, denote by $Y^{[n]}$ the $n$-fold iterated fiber product over $X$ equipped with the subspace diffeology.

Let $\text{pr}_{i_1…i_k}: Y^{[n]}\to Y^{[k]}$ be the smooth face maps of the simplicial diffeological space $Y^{[\cdot]}$ over the subduction groupoid $Y\times_X Y \rightrightarrows X$.

Definition(Bundle gerbe) . A (hermitean) bundle gerbe over a diffeological space $X$ is a subduction $\pi: Y\to X$, a hermitean line bundle $L\to Y^{[2]}=Y\times_X Y$, a unitary isomorphism (bundle gerbe multiplication) $\mu: \text{pr}_{12}^\ast L \otimes \text{pr}_{01}^\ast L \to \text{pr}_{02}^\ast L$ of line bundle over $Y^{[3]}$ which is associative over $Y^{[4]}$.

One can equivalently use principal $U(1)$-bundles instead of the hermitean line bundle. Just like in the case of line bundles, bundle gerbes can also be equipped with connections.

Definition(Connections on a hermitean bundle gerbe) . A connection on a hermitean bundle gerbe $(\pi:Y\to X,L,\mu)$ is a pair $(\nabla^L,B)$ where
  1. $\nabla^L$ is a hermitean connection on $L$,
  2. $B\in \Omega^2(Y)$ called curving,
such that
  1. the isomorphism $\mu$ is parallel w.r.t $\nabla^L$, and
  2. the curvature of $\nabla^L$ is equal to $(\text{pr}_1^\ast B - \pi_0^\ast B)$
where the second condition implies that $dB = \pi^\ast H$ descends to a unique closed 3-form $H \in H^3(X,\mathbb{Z})$, the curvature of the bundle gerbe connection $(\nabla^L, B)$.