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

$Y^{[3]} \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow} Y^{[2]}$

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 bunde 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

## References

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

• Pawel Gajer, Geometry of Deligne cohomology Invent. Math., 127(1):155–207 (1997) (arXiv)

Revised on May 29, 2014 23:58:06 by Urs Schreiber (31.55.15.191)