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\begin{document}

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\section*{connection on a bundle gerbe}

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

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\begin{itemize}%
\item [[connection on a bundle]]


\item [[connection on a 2-bundle]] / [[connection on a gerbe]] / \textbf{connection on a bundle gerbe}


\item [[connection on a 3-bundle]] / [[connection on a bundle 2-gerbe]]


\item [[connection on an ∞-bundle]]



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{for_line_bundle_gerbes}{for line bundle gerbes}\dotfill \pageref*{for_line_bundle_gerbes} \linebreak
\noindent\hyperlink{for_principal_bundle_gerbes}{for principal bundle gerbes}\dotfill \pageref*{for_principal_bundle_gerbes} \linebreak
\noindent\hyperlink{surface_transport}{Surface transport}\dotfill \pageref*{surface_transport} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{connection on a [[bundle gerbe]]} is a slight variant of a [[Cech cohomology|Cech]]-realization of a degree 3 [[Deligne cohomology]] cocycle.

\begin{quote}%
old content, needs to be polished


\end{quote}
Like a [[connection on a bundle|connection]] on a locally trivialized bundle is encoded in a Lie algebra-valued connection $1$-form on $Y$, the connection on a bundle gerbe gives rise to a Lie-algebra valued $2$-form on $Y$ (this shift in degree is directly related to the step from second to third integral cohomology). This $2$-form is sometimes addressed as the \emph{curving $2$-form} of a bundle gerbe.

But there is more data necessary to describe a connection on a bundle gerbe. The details of the definition -- which is evident for line bundle gerbes but more involved for principal bundle gerbes -- can be naturally derived from a functorial concept of parallel surface transport, just like connection $1$-forms on bundles can be derived from parallel line transport.

\hypertarget{definitions}{}\subsubsection*{{Definitions}}\label{definitions}

\hypertarget{for_line_bundle_gerbes}{}\paragraph*{{for line bundle gerbes}}\label{for_line_bundle_gerbes}

A connection (also known as ``connection and curving'') on a line bundle gerbe

\begin{displaymath}
B \stackrel{p}{\to} Y^{[2]} \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X
\end{displaymath}
is

\begin{itemize}%
\item a 2-form on $Y$

\begin{displaymath}
B \in \Omega^2(Y)
\end{displaymath}

\item a connection $\nabla$ on the line bundle $B \to Y^{[2]}$


\item such that

\begin{displaymath}
\pi_1^*B \; -\;  p_2^*B \;=\; F_\nabla
\end{displaymath}

\item together with an extension of the bundle gerbe product $\mu$ to an isomorphism

\begin{displaymath}
\mu_\nabla \;:\; p_{12}^* (B,\nabla) \;\; \otimes p_{23}^* (B,\nabla) \;\to\; p_{13}^* (B,\nabla)
\end{displaymath}
of line bundles with connection.



\end{itemize}
Notice that this condition ensures that $d B$ is a $3$-form on $Y$ which agrees on double intersections

\begin{displaymath}
p_1^* d B \;\; = \;\; p_2^* d B
  \,.
\end{displaymath}
This means that $d B$ actually descends to a 3-form on $X$.

The \textbf{curvature} associated with the connection on a line bundle gerbe is the unique 3-form on $X$

\begin{displaymath}
H \in \Omega^3(X)
\end{displaymath}
such that

\begin{displaymath}
\pi^* H = d B
  \,.
\end{displaymath}
The deRham class $[H]$ of this 3-form is the image in real cohomology of the class in integral coholomology classifying the bundle gerbe.

\hypertarget{for_principal_bundle_gerbes}{}\paragraph*{{for principal bundle gerbes}}\label{for_principal_bundle_gerbes}

A connection on a $G$-principal bundle gerbe is

\begin{itemize}%
\item a $\mathrm{Lie}(G)$-valued 2-form on $Y$

\begin{displaymath}
B \in \Omega^2(Y,\mathrm{Lie}(G))
\end{displaymath}

\item together with a $\mathrm{Lie}(\mathrm{Aut}(G))$-valued 1-form on $Y$

\begin{displaymath}
A \in \Omega^1(Y,\mathrm{Lie}(\mathrm{Aut}(G)))
\end{displaymath}

\item and a certain twisted notion of connection on the $G$-bundle $B$


\item satisfying a couple of conditions that reduce to those described above in the case $G = U(1)$.



\end{itemize}
For the case that $F_{A} + \mathrm{ad} B = 0$, these conditions are nothing but a tetrahedron law on a 2-functor from 2-paths in $Y$ to the category $\Sigma(G\mathrm{BiTor})$. This is discussed in \href{http://arxiv.org/abs/math.DG/0511710}{math.DG/0511710}.

For the more general case a choice for these conditions that harmonizes with the conditions found for (proper) gerbes with connection by Breen \& Messing in \href{http://arxiv.org/abs/math.AG/0106083}{math.AG/0106083} has been given by Aschieri, Cantini \& Juro in\newline \href{http://arxiv.org/abs/hep-th/0312154}{hep-th/0312154}.

\hypertarget{surface_transport}{}\subsubsection*{{Surface transport}}\label{surface_transport}

From a line bundle gerbe with connection one obtains a notion of [[parallel transport]] along surfaces in a way that generalizes the procedure for locally trivialized fiber bundles with connection.

Recall that in the case of fiber bundles, the holonomy associated to a based loop $\gamma$ is obtained by

\begin{itemize}%
\item choosing a triangulation of the loop (i.e., a decomposition into intervals) such that each vertex sits in a double intersection $U_{ij}$ and such that each edge sits in a patch $U_i$


\item choosing for each edge a lift into $Y = \sqcup_i U_i$


\item choosing for each vertex a lift into $Y^{[2]} = \sqcup_{ij} U_i\cap U_j$


\item assigning to each edge lifted to $U_i$ the transport computed from the connection 1-form $a_i$


\item assigning to each vertex lifted to $U_i \cap U_j$ the value of the transition function $g_{ij}$ at that point


\item multiplying these data in the order given by $\gamma$ .



\end{itemize}
For bundle gerbes this generalizes to a procedure that assigns a triangulation to a closed surface, that lifts faces, edges, and vertices to single, double and triple intersections, respectively, and which assigns the exponentiated integrals of the 2-form over faces, of the connection 1-form over edges, and assigns the isomorphism $\mu_{ijk}$ to vertices.

For the abelian case (line bundle gerbes) this procedure has been first described in

\begin{itemize}%
\item K. Gawedzki \& N. Reis, \emph{WZW branes and Gerbes} (\href{http://arxiv.org/abs/hep-th/0205233}{arXiv})

\end{itemize}
based on

\begin{itemize}%
\item O. Alvarez, \emph{Topological quantization and cohomology} Commun. Math. Phys. 100 (1985), 279-309.

\end{itemize}
Further discussion can be found in

\begin{itemize}%
\item A. Carey, S. Johnson \& M. Murray, \emph{Holonomy on D-branes}, (\href{http://arxiv.org/abs/hep-th/0204199}{arXiv})

\end{itemize}
Gawedzki and Reis showed this way that the Wess-Zumino term in the WZW-model is nothing but the surface holonomy of a (line bundle) gerbe.

In terms of string physics this means that the string (the $2$-particle) couples to the Kalb-Ramond field -- which hence has to be interpreted as the connection (``and curving'') of a gerbe -- in a way that categorifies the coupling of the electromagnetically charged ($1$-)particle to a line bundle.

The necessity to interpret the Kalb--Ramond field as a connection on a gerbe was originally discussed in

\begin{itemize}%
\item D. Freed and E. Witten \emph{Anomalies in string theory with D-branes}, Asian J. Math. 3 (1999), 819-851 (\href{http://arxiv.org/abs/hep-th/9907189}{arXiv})

\end{itemize}
Underlying the Gawedzki--Reis formula is a general mechanism of transition of transport $2$-functors, described in

\begin{itemize}%
\item [[Urs Schreiber]], [[Konrad Waldorf]], \emph{Connections on non-abelian gerbes and their holonomy}, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (\href{http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html}{TAC}, \href{http://arxiv.org/abs/0808.1923}{arXiv:0808.1923}, )

\end{itemize}
and similarly in

\begin{itemize}%
\item Joao Faria Martins, Roger Picken, \emph{A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces} (\href{http://arxiv.org/abs/0808.3964}{arXiv})

\end{itemize}
This applies to more general situations than ordinary line bundle gerbes with connection.

The generalization to unoriented surfaces (hence to type I strings) was given in

\begin{itemize}%
\item K. Waldorf, C. Schweigert \&{} U. S., \emph{Unoriented WZW Models and Holonomy of Bundle Gerbes} (\href{http://arxiv.org/abs/hep-th/0512283}{arXiv})

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

(\ldots{})

[[!redirects connections on a bundle gerbes]] [[!redirects connections on bundle gerbess]]

[[!redirects bundle gerbe with connection]] [[!redirects bundle gerbes with connection]]



\end{document}