(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A bundle gerbe is a special model for the total space Lie groupoid of a $\mathbf{B}U(1)$-principal 2-bundle for $\mathbf{B}U(1)$ the circle 2-group.
More generally, for $G$ a more general Lie 2-group (often taken to be the automorphism 2-group $G = AUT(H)$ of a Lie group $H$), a nonabelian bundle gerbe for $G$ is a model for the total space groupoid of a $G$-principal 2-bundle.
The definition of bundle gerbe is not in fact a special case (nor a generalization) of the definition of gerbe, even though there are equivalences relating both concepts.
A bundle gerbe’* over a smooth manifold $X$ is
together with a $U(1)$-principal bundle
over the fiber product of $Y$ with itself, i.e.
an isomorphism
of $U(1)$-bundles on $Y \times_X Y \times_X Y$
such that this satisfies the evident associativity condition on $Y\times_X Y \times_X Y \times_X Y$.
Here $\pi_{12}, \pi_{23}, \pi_{13}$ are the three maps
in the Cech nerve of $Y \to X$.
In a nonabelian bundle gerbe the bundle $L$ is generalized to a bibundle.
A bundle gerbe may be understood as a specific model for the total space Lie groupoid of a principal 2-bundle.
We first describe this Lie groupoid in
and then describe how this is the total space of a principal 2-bundle in
Give a surjective submersion $\pi : Y \to X$, write
for the corresponding Cech groupoid. Notice that this is a resolution of the smooth manifold $X$ itself, in that the canonical projection is a weak equivalence (see infinity-Lie groupoid for details)
The data of a bundle gerbe $(Y,L,\mu)$ induces a Lie groupoid $P_{(Y,L,\mu)}$ which is a $\mathbf{B}U(1)$-extension of $C(Y)$, exhibiting a fiber sequence
This Lie groupoid is the groupoid whose space of morphisms is the total space $L$ of the $U(1)$-bundle
with composition given by the composite
We discuss how a bundle gerbe, regarded as a groupoid, is the total space of a $\mathbf{B}U(1)$-principal 2-bundles.
Recall from the discussion at principal infinity-bundle that the total $G$ 2-bundle space $P \to X$ classified by a cocycle $X \to \mathbf{B} G$ is simply the homotopy fiber of that cocycle. This we compute now.
(For more along these lines see infinity-Chern-Weil theory introduction. For the analogous nonabelian case see also nonabelian bundle gerbe.)
The Lie groupoid $P_{(Y,L,\mu)}$ defined by a bundle gerbe is in ∞LieGrpd the (∞,1)-pullback
of a cocycle $[g] \in H(X,\mathbf{B}^2 U(1)) \simeq H^3(X,\mathbb{Z})$.
In fact a somewhat stronger statement is true, as shown in the following proof.
We can assume without restriction that the bundle $L$ in the data of the bundle gerbe is actually the trivial $U(1)$-bundle $L = Y \times_X Y \times U(1)$ by refining, if necessary, the surjective submersion $Y$ by a good open cover. In that case we may identify $\mu$ with a $U(1)$-valued function
which in turn we may identify with a smooth 2-anafunctor
From here on the computation is a special case of the general theory of groupoid cohomology and the extensions classified by it.
Then recall from universal principal infinity-bundle that we model the $(\infty,1)$-pullbacks that defines principal $\infty$-bundles in terms of ordinary pullbacks of the universal $\mathbf{B}U(1)$-principal 2-bundle $\mathbf{E}\mathbf{B}U(1) \to \mathbf{B}^2 U(1)$.
We may model all this in the case at hand in terms of strict 2-groupoips. Then using an evident cartoon-notation we have
and $\mathbf{E}\mathbf{B}U(1)$ is the 2-groupoid whose morphisms are diagrams
in $\mathbf{B}^2 U(1)$ with composition given by horizontal pasting
and 2-morphisms are paper-cup diagrams
So $\mathbf{E}\mathbf{B}U(1)$ is the Lie 2-groupoid with a single object, with $U(1)$ worth of 1-morphisms and unique 2-morphism between these.
From this we read of that
is indeed a pullback square (in the category of simplicial presheaves over CartSp). The morphisms of the pullback Lie groupoid are pairs of diagrams
hence form a trivial $U(1)$-bundle over the morphisms of $C(U)$, and the 2-morphims are pairs consisting of 2-morphisms
in $C(U)$ and paper-cup diagrams of the form
in $\mathbf{B}^2 U(1)$, which exhibits indeed the composition operation in $P_{(Y,L,\mu)}$.
For $A \to \hat G \to G$ a group extension by an abelian group $G$ classified by a 2-cocycle $c$ in group cohomology, which we may think of as a 2-functopr $c : \mathbf{B}\mathbf{G} \to \mathbf{B}^2 A$, the corresponding fiber sequence
exhibits $\mathbf{B}\hat G$ as the bundle gerbe over $\mathbf{B}G$ (in equivariant cohomology of the point, if you wish) with Dixmier-Douady class $c$.
Let $X$ be a simply connected smooth manifold and $H \in \Omega^3(X)_{cl, int}$ a degree 3 differential form with integral periods.
We may think of this a cocycle in ∞-Lie algebroid cohomology
By a slight variant of Lie integration of oo-Lie algebroid cocycles we obtain from this a bundle gerbe on $X$ by the following construction
pick any point $x_0 \in X$;
let $Y = P_* X$ be the based smooth path space of $X$;
let $L \to Y \times_X Y$ be the $U(1)$-bundle which over an element $(\gamma_1,\gamma_2)$ in $Y \times_X Y$ – which is a loop in $X$ assigns the $U(1)$-torsor whose elements are equivalence class of pairs $(\Sigma,c)$, where $\Sigma$ is a surface cobounding the loop and where $c \in U(1)$, and where the equivalence relation is so that for any 3-ball $\phi : D^3 \to X$ cobounding two such surfaces $\Sigma_1$ and $\Sigma_2$ we have that $(\Sigma_1,c_1)$ is equivalent to $(\Sigma_2, c_2)$ the difference of the labels differs by the integral of the 3-form
the composition operation $\pi_{12}^* L \otimes \pi_{23}^* L \to \pi_{13}^* L$ is loop-wise the evident operation that on loops removes from a figure-8 the inner bit and whch is group multiplication of the labels.
This produces a bundle gerbe whose class in $H^3(X,\mathbb{Z})$ has $[H]$ as its image in de Rham cohomology.
and
principal 2-bundle / gerbe / bundle gerbe
especially
For applications in string theory see also
The notion of bundle gerbe as such was introduced in
Early texts also include
(notice that the title here suppresses one “e” intentionally);
A general picture of bundle $n$-gerbes (with connection) as circle (n+1)-bundles with connection classified by Deligne cohomology is in
Reviews are in
Nigel Hitchin, What is…a gerbe?, Notices of the AMS 50 no. 2 (2003) pp 218-219 pdf
Michael Murray, An Introduction to Bundle Gerbes, In: The Many Facets of Geometry, A Tribute to Nigel Hitchin, Edited by Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon, OUP, 2010. doi:10.1093/acprof:oso/9780199534920.001.0001, arXiv:0712.1651