bundles

cohomology

# Contents

## Idea

A bundle gerbe is a special model for the total space Lie groupoid of a $BU\left(1\right)$-principal 2-bundle for $BU\left(1\right)$ the circle 2-group.

More generally, for $G$ a more general Lie 2-group (often taken to be the automorphism 2-group $G=\mathrm{AUT}\left(H\right)$ 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

• $\begin{array}{c}Y\\ {↓}^{\pi }\\ X\end{array}$\array{ Y \\ \downarrow^{\mathrlap{\pi}} \\ X }
• together with a $U\left(1\right)$-principal bundle

$\begin{array}{c}L\\ {↓}^{p}\\ Y{×}_{X}Y\end{array}$\array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y }

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

$\begin{array}{c}L\\ {↓}^{p}\\ Y{×}_{X}Y& \stackrel{\stackrel{{\pi }_{1}}{\to }}{\underset{{\pi }_{2}}{\to }}& Y\\ & & {↓}^{\pi }\\ & & X\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y &\stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}}& Y \\ && \downarrow^{\mathrlap{\pi}} \\ && X } \,,
• $\mu :{\pi }_{12}^{*}L\otimes {\pi }_{23}^{*}L\to {\pi }_{13}^{*}L$\mu : \pi_{12}^*L \otimes \pi_{23}^*L \to \pi_{13}^* L

of $U\left(1\right)$-bundles on $Y{×}_{X}Y{×}_{X}Y$

• such that this satisfies the evident associativity condition on $Y{×}_{X}Y{×}_{X}Y{×}_{X}Y$.

Here ${\pi }_{12},{\pi }_{23},{\pi }_{13}$ are the three maps

${Y}^{\left[3\right]}\stackrel{\stackrel{\to }{\to }}{\to }{Y}^{\left[2\right]}$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

$C\left(Y\right):=\left(Y{×}_{X}Y\stackrel{\to }{\to }Y\right)$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)

$\begin{array}{c}C\left(Y\right)\\ {↓}^{\simeq }\\ X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ C(Y) \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.

The data of a bundle gerbe $\left(Y,L,\mu \right)$ induces a Lie groupoid ${P}_{\left(Y,L,\mu \right)}$ which is a $BU\left(1\right)$-extension of $C\left(Y\right)$, exhibiting a fiber sequence

$BU\left(1\right)\to {P}_{\left(Y,L,\mu \right)}\to X\phantom{\rule{thinmathspace}{0ex}}.$\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 $U\left(1\right)$-bundle

${P}_{\left(Y,L,\mu \right)}=\left(L\stackrel{\stackrel{{\pi }_{1}\circ p}{\to }}{\underset{{\pi }_{2}\circ p}{\to }}Y\right)$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

$L{×}_{s,t}L\stackrel{\simeq }{\to }{\pi }_{12}^{*}L×{\pi }_{23}^{3}*L\stackrel{}{\to }{\pi }_{12}^{*}L\otimes {\pi }_{23}^{3}*L\stackrel{\mu }{\to }{\pi }_{13}^{*}L\to L\phantom{\rule{thinmathspace}{0ex}}.$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 $BU\left(1\right)$-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 $X\to BG$ 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 ${P}_{\left(Y,L,\mu \right)}$ defined by a bundle gerbe is in ∞LieGrpd the (∞,1)-pullback

$\begin{array}{ccc}{P}_{\left(Y,L,\mu \right)}& \to & *\\ ↓& {⇙}_{\simeq }& ↓\\ X& \stackrel{g}{\to }& {B}^{2}U\left(1\right)\end{array}$\array{ P_{(Y,L,\mu)} &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}^2 U(1) }

of a cocycle $\left[g\right]\in H\left(X,{B}^{2}U\left(1\right)\right)\simeq {H}^{3}\left(X,ℤ\right)$.

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 $U\left(1\right)$-bundle $L=Y{×}_{X}Y×U\left(1\right)$ by refining, if necessary, the surjective submersion $Y$ by a good open cover. In that case we may identify $\mu$ with a $U\left(1\right)$-valued function

$\mu :Y{×}_{X}Y{×}_{X}Y\to U\left(1\right)$\mu : Y \times_X Y \times_X Y \to U(1)

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

$\begin{array}{ccc}C\left(U\right)& \stackrel{\mu }{\to }& {B}^{2}U\left(1\right)\\ {↓}^{\simeq }\\ X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\infty ,1\right)$-pullbacks that defines principal $\infty$-bundles in terms of ordinary pullbacks of the universal $BU\left(1\right)$-principal 2-bundle $EBU\left(1\right)\to {B}^{2}U\left(1\right)$.

We may model all this in the case at hand in terms of strict 2-groupoips. Then using an evident cartoon-notation we have

${B}^{2}U\left(1\right)=\left\{\begin{array}{cc}& ↗↘\\ •& {⇓}^{c\in U\left(1\right)}& •\\ & ↘↗\end{array}\right\}$\mathbf{B}^2 U(1) = \left\{ \array{ & \nearrow \searrow \\ \bullet &\Downarrow^{\mathrlap{c \in U(1)}}& \bullet \\ & \searrow \nearrow } \right\}

and $EBU\left(1\right)$ is the 2-groupoid whose morphisms are diagrams

$\begin{array}{ccc}& & •\\ & ↗& {⇙}_{c}& ↘\\ •& & \to & & •\end{array}$\array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet }

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

$\begin{array}{cccc}& & & •\\ & ↙& {⇙}_{{c}_{1}}& ↓& {⇙}_{{c}_{2}}& ↘\\ •& \to & & •& & \to & •\end{array}$\array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet }

and 2-morphisms are paper-cup diagrams

$\begin{array}{ccc}& & •\\ & ↗& {⇙}_{c}& ↘\\ •& & \to & & •\\ & ↘& {⇙}_{k}& ↙\\ & & •\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccc}& & •\\ & ↗& {⇙}_{ck}& ↘\\ •& & \to & & •\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $EBU\left(1\right)$ is the Lie 2-groupoid with a single object, with $U\left(1\right)$ worth of 1-morphisms and unique 2-morphism between these.

From this we read of that

$\begin{array}{ccc}{P}_{\left(Y,L,\mu \right)}& \to & EBU\left(1\right)\\ ↓& & ↓\\ C\left(U\right)& \stackrel{\mu }{\to }& {B}^{2}U\left(1\right)\\ {↓}^{\simeq }\\ X\end{array}$\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

$\begin{array}{ccc}& & •\\ & ↗& {⇙}_{c}& ↘\\ •& & \to & & •\\ \\ \left(x,i\right)& & \to & & \left(x,j\right)\end{array}$\array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet \\ \\ (x,i) &&\to&& (x,j) }

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

$\begin{array}{ccc}& & \left(x,j\right)\\ & ↗& ⇙& ↘\\ \left(x,i\right)& & \to & & \left(x,k\right)\end{array}$\array{ && (x,j) \\ & \nearrow &\swArrow& \searrow \\ (x,i) &&\to&& (x,k) }

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

$\begin{array}{cccc}& & & •\\ & ↙& {⇙}_{{c}_{1}}& ↓& {⇙}_{{c}_{2}}& ↘\\ •& \to & & •& & \to & •\\ & ↘& & {⇙}_{{\mu }_{ijk}\left(x\right)}& & & ↙\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccc}& & •\\ & ↗& {⇙}_{{c}_{1}{c}_{2}{\mu }_{ijk}\left(x\right)}& ↘\\ •& & \to & & •\end{array}$\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 ${B}^{2}U\left(1\right)$, which exhibits indeed the composition operation in ${P}_{\left(Y,L,\mu \right)}$.

## Examples

### Equivariant bundle gerbes over the point

For $A\to \stackrel{^}{G}\to G$ 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 $c:BG\to {B}^{2}A$, the corresponding fiber sequence

$A\to \stackrel{^}{G}\to G\to BA\to B\stackrel{^}{G}\to BG\stackrel{c}{\to }{B}^{2}A$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 $B\stackrel{^}{G}$ as the bundle gerbe over $BG$ (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 $H\in {\Omega }^{3}\left(X{\right)}_{\mathrm{cl},\mathrm{int}}$ a degree 3 differential form with integral periods.

We may think of this a cocycle in ∞-Lie algebroid cohomology

$H:TX\to {b}^{2}ℝ\phantom{\rule{thinmathspace}{0ex}}.$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 $L\to Y{×}_{X}Y$ be the $U\left(1\right)$-bundle which over an element $\left({\gamma }_{1},{\gamma }_{2}\right)$ in $Y{×}_{X}Y$ – which is a loop in $X$ assigns the $U\left(1\right)$-torsor whose elements are equivalence class of pairs $\left(\Sigma ,c\right)$, where $\Sigma$ is a surface cobounding the loop and where $c\in U\left(1\right)$, and where the equivalence relation is so that for any 3-ball $\varphi :{D}^{3}\to X$ cobounding two such surfaces ${\Sigma }_{1}$ and ${\Sigma }_{2}$ we have that $\left({\Sigma }_{1},{c}_{1}\right)$ is equivalent to $\left({\Sigma }_{2},{c}_{2}\right)$ the difference of the labels differs by the integral of the 3-form

${c}_{2}{c}_{1}^{-1}={\int }_{{D}^{3}}{\varphi }^{*}H\in ℝ/ℤ\phantom{\rule{thinmathspace}{0ex}}.$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 ${H}^{3}\left(X,ℤ\right)$ has $\left[H\right]$ 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 October 30, 2013 09:57:06 by Urs Schreiber (145.116.130.141)