nLab
bundle gerbe

Idea

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

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

• a surjective submersion

  $$\begin{array}{c}Y\\ {\downarrow }^{\pi }\\ X\end{array}$$\array{ Y \\ \downarrow^{\mathrlap{\pi}} \\ X }

• together with a $U(1)$-principal bundle

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

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

  $$\begin{array}{c}L\\ {\downarrow }^p\\ Y{\times }_{X}Y& \stackrel{\stackrel{{\pi }_1}{\to }}{\underset{{\pi }_2}{\to }}& Y\\ & & {\downarrow }^{\pi }\\ & & X\end{array},$$\array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y &\stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}}& Y \\ && \downarrow^{\mathrlap{\pi}} \\ && X } \,,

• an isomorphism

  $$\mu :{\pi }_{12}^{*}L\otimes {\pi }_{23}^{*}L\to {\pi }_{13}^{*}L$$\mu : \pi_{12}^*L \otimes \pi_{23}^*L \to \pi_{13}^* L

  of $U(1)$-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

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

• As a groupoid extension

and then describe how this is the total space of a principal 2-bundle in

• As the total space of a 2-bundle.

As a groupoid extension

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

$$C(Y):=\left(Y{\times }_{X}Y\overrightarrow{\to }Y\right)$$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)

$$\begin{array}{c}C(Y)\\ {\downarrow }^{\simeq }\\ X\end{array}.$$\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 $BU(1)$-extension of $C(Y)$, exhibiting a fiber sequence

$$BU(1)\to P_{(Y,L,\mu )}\to X.$$\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

$$P_{(Y,L,\mu )}=\left(L\stackrel{\stackrel{{\pi }_1\circ p}{\to }}{\underset{{\pi }_2\circ p}{\to }}Y\right)$$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

$$L{\times }_{s,t}L\stackrel{\simeq }{\to }{\pi }_{12}^{*}L\times {\pi }_{23}^3*L\stackrel{}{\to }{\pi }_{12}^{*}L\otimes {\pi }_{23}^3*L\stackrel{\mu }{\to }{\pi }_{13}^{*}L\to L.$$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 $BU(1)$-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 $X\to BG$ 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 $P_{(Y,L,\mu )}$ defined by a bundle gerbe is in ∞LieGrpd the (∞,1)-pullback

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

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

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

$$\mu :Y{\times }_{X}Y{\times }_{X}Y\to U(1)$$\mu : Y \times_X Y \times_X Y \to U(1)

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

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

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

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

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

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

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

$$\begin{array}{cccc}& & & •\\ & \swarrow & {⇙}_{c_1}& \downarrow & {⇙}_{c_2}& \searrow \\ •& \to & & •& & \to & •\end{array}$$\array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet }

and 2-morphisms are paper-cup diagrams

$$\begin{array}{ccc}& & •\\ & \nearrow & {⇙}_c& \searrow \\ •& & \to & & •\\ & \searrow & {⇙}_k& \swarrow \\ & & •\end{array}=\begin{array}{ccc}& & •\\ & \nearrow & {⇙}_{ck}& \searrow \\ •& & \to & & •\end{array}.$$\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 $EBU(1)$ 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

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

$$\begin{array}{ccc}& & •\\ & \nearrow & {⇙}_c& \searrow \\ •& & \to & & •\\ \\ (x,i)& & \to & & (x,j)\end{array}$$\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

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

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

$$\begin{array}{cccc}& & & •\\ & \swarrow & {⇙}_{c_1}& \downarrow & {⇙}_{c_2}& \searrow \\ •& \to & & •& & \to & •\\ & \searrow & & {⇙}_{{\mu }_{ijk}(x)}& & & \swarrow \end{array}=\begin{array}{ccc}& & •\\ & \nearrow & {⇙}_{c_1{c}_2{\mu }_{ijk}(x)}& \searrow \\ •& & \to & & •\end{array}$$\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 $B^{2}U(1)$, 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 $c:BG\to B^{2}A$, the corresponding fiber sequence

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

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

$$H:TX\to b^2ℝ.$$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 $\varphi :D^3\to X$ 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

  $$c_2{c}_1^{-1}={\int }_{D^3}{\varphi }^{*}H\in ℝ/\mathbb{Z}.$$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.

Related entries

• centrally extended groupoid

• transgression of bundle gerbes

• connection on a bundle gerbe

and

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle

References

The notion of bundle gerbe as such was introduced in

• Michael Murray, Bundle gerbes (arXiv:dg-ga/9407015)

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)