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\begin{document}

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\section*{bundle gerbe}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Interpretation}{Interpretation}\dotfill \pageref*{Interpretation} \linebreak
\noindent\hyperlink{AsGroupoidExtension}{As a groupoid extension}\dotfill \pageref*{AsGroupoidExtension} \linebreak
\noindent\hyperlink{AsTotalSpace}{As the total space of a principal 2-bundle}\dotfill \pageref*{AsTotalSpace} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{equivariant_bundle_gerbes_over_the_point}{Equivariant bundle gerbes over the point}\dotfill \pageref*{equivariant_bundle_gerbes_over_the_point} \linebreak
\noindent\hyperlink{tautological_bundle_gerbe}{Tautological bundle gerbe}\dotfill \pageref*{tautological_bundle_gerbe} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{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 .

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 \emph{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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{bundle gerbe} over a [[smooth manifold]] $X$ is

\begin{itemize}%
\item a [[surjective submersion]]

\begin{displaymath}
\itexarray{
    Y
    \\
    \downarrow^{\mathrlap{\pi}}
    \\
    X
    }
\end{displaymath}

\item together with a $U(1)$-[[principal bundle]]

\begin{displaymath}
\itexarray{
    L
    \\
    \downarrow^{\mathrlap{p}}
    \\
    Y \times_X Y
      }
\end{displaymath}
over the [[fiber product]] of $Y$ with itself, i.e.

\begin{displaymath}
\itexarray{
    L
    \\
    \downarrow^{\mathrlap{p}}
    \\
    Y \times_X Y &\stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}}& Y
    \\
    && \downarrow^{\mathrlap{\pi}}
    \\
    && 
    X
    }
   \,,
\end{displaymath}

\item an [[isomorphism]]

\begin{displaymath}
\mu : 
   \pi_{12}^*L \otimes
   \pi_{23}^*L 
   \to 
   \pi_{13}^* L
\end{displaymath}
of $U(1)$-bundles on $Y \times_X Y \times_X Y$


\item such that this satisfies the evident associativity condition on $Y\times_X Y \times_X Y \times_X Y$.



\end{itemize}
Here $\pi_{12}, \pi_{23}, \pi_{13}$ are the three maps

\begin{displaymath}
Y^{[3]} \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow}
   Y^{[2]}
\end{displaymath}
in the [[Cech nerve]] of $Y \to X$.

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

\hypertarget{Interpretation}{}\subsection*{{Interpretation}}\label{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

\begin{itemize}%
\item \hyperlink{AsGroupoidExtension}{As a groupoid extension}

\end{itemize}
and then describe how this is the total space of a principal 2-bundle in

\begin{itemize}%
\item \hyperlink{AsTotalSpace}{As the total space of a 2-bundle}.

\end{itemize}
\hypertarget{AsGroupoidExtension}{}\subsubsection*{{As a groupoid extension}}\label{AsGroupoidExtension}

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

\begin{displaymath}
C(Y) := \left(
    Y \times_X Y \stackrel{\to}{\to} Y
  \right)
\end{displaymath}
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{displaymath}
\itexarray{
     C(Y)
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     X
  }
  \,.
\end{displaymath}
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]]

\begin{displaymath}
\mathbf{B}U(1) \to P_{(Y,L,\mu)} \to X
  \,.
\end{displaymath}
This Lie groupoid is the groupoid whose space of morphisms is the total space $L$ of the $U(1)$-bundle

\begin{displaymath}
P_{(Y,L,\mu)} = 
  \left(
    L \stackrel{\overset{\pi_1 \circ p}{\to}}{\underset{\pi_2 \circ p}{\to}} Y
  \right)
\end{displaymath}
with composition given by the composite

\begin{displaymath}
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
  \,.
\end{displaymath}
\hypertarget{AsTotalSpace}{}\subsubsection*{{As the total space of a principal 2-bundle}}\label{AsTotalSpace}

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]].)

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

\begin{displaymath}
\itexarray{
     P_{(Y,L,\mu)} &\to& *
     \\
     \downarrow &\swArrow_{\simeq}& \downarrow
     \\
     X &\stackrel{g}{\to}& \mathbf{B}^2 U(1)
  }
\end{displaymath}
of a [[cocycle]] $[g] \in H(X,\mathbf{B}^2 U(1)) \simeq H^3(X,\mathbb{Z})$.

\end{prop}
In fact a somewhat stronger statement is true, as shown in the following proof.

\begin{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

\begin{displaymath}
\mu : Y \times_X Y \times_X Y  \to U(1)
\end{displaymath}
which in turn we may identify with a smooth 2-[[anafunctor]]

\begin{displaymath}
\itexarray{
    C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1)
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
  \,.
\end{displaymath}
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 omega-groupoid|strict 2-groupoip]]s. Then using an evident cartoon-notation we have

\begin{displaymath}
\mathbf{B}^2 U(1)
  = 
  \left\{
    \itexarray{
       & \nearrow \searrow
       \\
       \bullet &\Downarrow^{\mathrlap{c \in U(1)}}& \bullet
       \\
       & \searrow \nearrow
    }
  \right\}
\end{displaymath}
and $\mathbf{E}\mathbf{B}U(1)$ is the 2-groupoid whose \emph{morphisms} are diagrams

\begin{displaymath}
\itexarray{
      && \bullet
      \\
      & \nearrow &\swArrow_{c}& \searrow
     \\
     \bullet &&\to&& \bullet
  }
\end{displaymath}
in $\mathbf{B}^2 U(1)$ with composition given by horizontal [[pasting]]

\begin{displaymath}
\itexarray{
     &&& \bullet
     \\
     & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow
    \\
    \bullet &\to&& \bullet &&\to& \bullet
  }
\end{displaymath}
and 2-morphisms are paper-cup diagrams

\begin{displaymath}
\itexarray{
      && \bullet
      \\
      & \nearrow &\swArrow_{c}& \searrow
     \\
     \bullet &&\to&& \bullet
     \\
     & \searrow &\swArrow_{k}& \swarrow
     \\
     && \bullet
  }
  \;\;\;\;\;
  =
  \;\;\;\;\;
  \itexarray{
      && \bullet
      \\
      & \nearrow &\swArrow_{c k}& \searrow
     \\
     \bullet &&\to&& \bullet
  }
  \,.
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    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
  }
\end{displaymath}
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{displaymath}
\itexarray{
      && \bullet
      \\
      & \nearrow &\swArrow_{c}& \searrow
     \\
     \bullet &&\to&& \bullet
     \\
     \\
     (x,i) &&\to&& (x,j)
  }
\end{displaymath}
hence form a trivial $U(1)$-bundle over the morphisms of $C(U)$, and the 2-morphims are pairs consisting of 2-morphisms

\begin{displaymath}
\itexarray{
     && (x,j)
     \\
     & \nearrow &\swArrow& \searrow
     \\
     (x,i) &&\to&& (x,k)
  }
\end{displaymath}
in $C(U)$ and paper-cup diagrams of the form

\begin{displaymath}
\itexarray{
     &&& \bullet
     \\
     & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow
    \\
    \bullet &\to&& \bullet &&\to& \bullet
    \\
    & \searrow &&\swArrow_{\mu_{i j k}(x)}&&& \swarrow
  }
  \;\;\;\;
  =
  \;\;\;\;
  \itexarray{
      && \bullet
      \\
      & \nearrow &\swArrow_{c_1 c_2 \mu_{i j k}(x)}& \searrow
     \\
     \bullet &&\to&& \bullet
   }
\end{displaymath}
in $\mathbf{B}^2 U(1)$, which exhibits indeed the composition operation in $P_{(Y,L,\mu)}$.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{equivariant_bundle_gerbes_over_the_point}{}\subsubsection*{{Equivariant bundle gerbes over the point}}\label{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]]

\begin{displaymath}
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
\end{displaymath}
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 $c$.

\hypertarget{tautological_bundle_gerbe}{}\subsubsection*{{Tautological bundle gerbe}}\label{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]]

\begin{displaymath}
H : T X \to b^2 \mathbb{R}
  \,.
\end{displaymath}
By a slight variant of  we obtain from this a bundle gerbe on $X$ by the following construction

\begin{itemize}%
\item pick any point $x_0 \in X$;


\item let $Y = P_* X$ be the based smooth [[path space]] of $X$;


\item 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 \emph{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

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

\item 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.



\end{itemize}
This produces a bundle gerbe whose class in $H^3(X,\mathbb{Z})$ has $[H]$ as its image in [[de Rham cohomology]].

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[centrally extended groupoid]]


\item [[line 2-bundle]], [[holomorphic line 2-bundle]]


\item [[transgression of bundle gerbes]]


\item [[connection on a bundle gerbe]]


\item [[discrete torsion]]



\end{itemize}
and

\begin{itemize}%
\item [[principal bundle]] / [[torsor]] / [[associated bundle]]


\item [[principal 2-bundle]] / [[gerbe]] / \textbf{bundle gerbe}


\item [[principal 3-bundle]] / [[bundle 2-gerbe]]


\item [[principal ∞-bundle]] / [[associated ∞-bundle]]



\end{itemize}
especially

\begin{itemize}%
\item [[circle n-bundle with connection]], [[ordinary differential cohomology]].

\end{itemize}
For applications in [[string theory]] see also

\begin{itemize}%
\item [[B-field]], [[WZW model]].

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion of \emph{bundle gerbe} as such was introduced in

\begin{itemize}%
\item [[Michael Murray]], \emph{Bundle gerbes}, J.Lond.Math.Soc. \textbf{54} (1996) pp403-416, arXiv:\href{http://arxiv.org/abs/dg-ga/9407015}{dg-ga/9407015}.

\end{itemize}
Early texts also include

\begin{itemize}%
\item [[David Chatterjee]], \emph{On Gerbs}, 1998 (\href{https://people.maths.ox.ac.uk/hitchin/files/StudentsTheses/chatterjee.pdf}{pdf})

\end{itemize}
(notice that the title here suppresses one ``e'' intentionally);

\begin{itemize}%
\item [[Michael Murray]], [[Danny Stevenson]], \emph{Bundle gerbes: stable isomorphism and local theory}, J.Lond.Math.Soc. \textbf{62} (2000) 925-937 arXiv:\href{https://arxiv.org/abs/math/9908135}{math/9908135}

\end{itemize}
A general picture of bundle $n$-gerbes (with connection) as [[circle n-bundle with connection|circle (n+1)-bundles with connection]] classified by [[Deligne cohomology]] is in

\begin{itemize}%
\item [[Pawel Gajer]], \emph{Geometry of Deligne cohomology} Invent. Math., 127(1):155-207 (1997) (\href{http://arxiv.org/abs/alg-geom/9601025}{arXiv})

\end{itemize}
Reviews are in

\begin{itemize}%
\item [[Nigel Hitchin]], \emph{What is\ldots{}a gerbe?}, Notices of the AMS \textbf{50} no. 2 (2003) pp 218-219 \href{http://www.ams.org/notices/200302/what-is.pdf}{pdf}


\item [[Michael Murray]], \emph{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:\href{http://dx.doi.org/10.1093/acprof:oso/9780199534920.001.0001}{10.1093/acprof:oso/9780199534920.001.0001}, arXiv:\href{https://arxiv.org/abs/0712.1651}{0712.1651}



\end{itemize}
[[!redirects bundle gerbes]]



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