Contents

bundles

cohomology

# Contents

## Idea

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.

## Definition

A bundle gerbe over a smooth manifold $X$ is

• $\array{ Y \\ \downarrow^{\mathrlap{\pi}} \\ X }$
• together with a $U(1)$-principal bundle

$\array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y }$

over the fiber product of $Y$ with itself, i.e.

$\array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y &\stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}}& Y \\ && \downarrow^{\mathrlap{\pi}} \\ && X } \,,$
• $\mu : \pi_{12}^*L \otimes \pi_{23}^*L \to \pi_{13}^* L$

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

$Y^{} \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow} Y^{}$

in the Cech nerve of $Y \to X$.

In a nonabelian bundle gerbe the bundle $L$ is generalized to a bibundle.

## Interpretation

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

### As a groupoid extension

Give a surjective submersion $\pi : Y \to X$, write

$C(Y) := \left( Y \times_X Y \stackrel{\to}{\to} Y \right)$

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)

$\array{ C(Y) \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.$

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

$\mathbf{B}U(1) \to P_{(Y,L,\mu)} \to X \,.$

This Lie groupoid is the groupoid whose space of morphisms is the total space $L$ of the $U(1)$-bundle

$P_{(Y,L,\mu)} = \left( L \stackrel{\overset{\pi_1 \circ p}{\to}}{\underset{\pi_2 \circ p}{\to}} Y \right)$

with composition given by the composite

$L \times_{s,t} L \stackrel{\simeq}{\to} \pi_{12}^* L \times \pi_{23}^3* L \stackrel{}{\to} \pi_{12}^* L \otimes \pi_{23}^3* L \stackrel{\mu}{\to} \pi_{13}^* L \to L \,.$

### As the total space of a principal 2-bundle

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.)

###### Proposition

The Lie groupoid $P_{(Y,L,\mu)}$ defined by a bundle gerbe is in ?LieGrpd? the (∞,1)-pullback

$\array{ P_{(Y,L,\mu)} &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}^2 U(1) }$

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.

###### 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

$\mu : Y \times_X Y \times_X Y \to U(1)$

which in turn we may identify with a smooth 2-anafunctor

$\array{ C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.$

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

$\mathbf{B}^2 U(1) = \left\{ \array{ & \nearrow \searrow \\ \bullet &\Downarrow^{\mathrlap{c \in U(1)}}& \bullet \\ & \searrow \nearrow } \right\}$

and $\mathbf{E}\mathbf{B}U(1)$ is the 2-groupoid whose morphisms are diagrams

$\array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet }$

in $\mathbf{B}^2 U(1)$ with composition given by horizontal pasting

$\array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet }$

and 2-morphisms are paper-cup diagrams

$\array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet \\ & \searrow &\swArrow_{k}& \swarrow \\ && \bullet } \;\;\;\;\; = \;\;\;\;\; \array{ && \bullet \\ & \nearrow &\swArrow_{c k}& \searrow \\ \bullet &&\to&& \bullet } \,.$

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

$\array{ P_{(Y,L,\mu)} &\to& \mathbf{E} \mathbf{B}U(1) \\ \downarrow && \downarrow \\ C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ X }$

is indeed a pullback square (in the category of simplicial presheaves over CartSp). The morphisms of the pullback Lie groupoid are pairs of diagrams

$\array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet \\ \\ (x,i) &&\to&& (x,j) }$

hence form a trivial $U(1)$-bundle over the morphisms of $C(U)$, and the 2-morphims are pairs consisting of 2-morphisms

$\array{ && (x,j) \\ & \nearrow &\swArrow& \searrow \\ (x,i) &&\to&& (x,k) }$

in $C(U)$ and paper-cup diagrams of the form

$\array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet \\ & \searrow &&\swArrow_{\mu_{i j k}(x)}&&& \swarrow } \;\;\;\; = \;\;\;\; \array{ && \bullet \\ & \nearrow &\swArrow_{c_1 c_2 \mu_{i j k}(x)}& \searrow \\ \bullet &&\to&& \bullet }$

in $\mathbf{B}^2 U(1)$, which exhibits indeed the composition operation in $P_{(Y,L,\mu)}$.

## Examples

### Equivariant bundle gerbes over the point

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

$A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to} \mathbf{B}^2 A$

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 $c$.

### Tautological bundle gerbe

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

$H : T X \to b^2 \mathbb{R} \,.$

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

$c_2 c_1^{-1} = \int_{D^3} \phi^* H \in \mathbb{R}/\mathbb{Z} \,.$
• 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

especially

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

With emphasis on surface holonomy:

Bundle gerbes over Lie groups (cf. the basic bundle gerbe):

On equivariant bundle gerbes?:

with a restrictive notion of equivariance (“strong” equivariance):

with a general notion of higher equivariance: