(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
arXiv:dg-ga/9407015.
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, Simon Salamon, OUP, 2010. doi:10.1093/acprof:oso/9780199534920.001.0001, arXiv:0712.1651
Last revised on March 22, 2019 at 10:18:41. See the history of this page for a list of all contributions to it.