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\section*{connection on a 2-bundle}

\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]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{on_trivial_2bundles}{On trivial 2-bundles}\dotfill \pageref*{on_trivial_2bundles} \linebreak
\noindent\hyperlink{differential_ech_cocycle_data}{Differential ech cocycle data}\dotfill \pageref*{differential_ech_cocycle_data} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{applications_to_physics}{Applications to physics}\dotfill \pageref*{applications_to_physics} \linebreak


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

The notion of \emph{connection on a 2-bundle} generalizes the notion of [[connection on a bundle]] from [[principal bundle]]s to [[principal 2-bundle]]s / [[gerbe]]s.

It comes with a notion of [[higher parallel transport|2-dimensional parallel transport]].

For an exposition of the concepts here see also at \emph{[[infinity-Chern-Weil theory introduction]]} the section \emph{\href{http://ncatlab.org/nlab/show/infinity-Chern-Weil%20theory%20introduction#ConnectionOn2Bundle}{Connections on principal 2-bundles}} .

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

For $G$ a [[Lie 2-group]], a \emph{connection} on a $G$-[[principal 2-bundle]] coming from a [[cocycle]] $g : X \to \mathbf{B}G$ is a lift of the cocycle to the [[2-groupoid of Lie 2-algebra valued forms]] $\mathbf{B}G_{conn}$

\begin{displaymath}
\itexarray{
     && \mathbf{B}G_{conn}
     \\
     & {}^{\mathllap{\nabla}}\nearrow & \downarrow
     \\
     X &\stackrel{g}{\to}& \mathbf{B}G
  }
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{on_trivial_2bundles}{}\subsubsection*{{On trivial 2-bundles}}\label{on_trivial_2bundles}

When the underlying [[principal 2-bundle]] over a [[smooth manifold]] $X$ is topologically trivial, then the connections on it are identified with [[Lie 2-algebra valued differential forms]] on $X$.

Recall from the discussion there what such form data looks like.

Let $\mathfrak{g}$ be some [[Lie 2-algebra]]. For instance for discussion of connections on $G$-[[gerbe]]s ($G$ a [[Lie group]]) this would be the [[automorphism 2-group|derivation Lie 2-algebra]] of the [[Lie algebra]] of $G$.

Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let

\begin{displaymath}
\{t^a\} \,, \;\;\; \{b^i\}
\end{displaymath}
be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general [[Chevalley-Eilenberg algebra]]

\begin{displaymath}
CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}
\end{displaymath}
with these generators.

We thus have

\begin{displaymath}
d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i
\end{displaymath}
\begin{displaymath}
d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j  
    - 
    r^i{}_{a b c} t^a \wedge t^b \wedge t^c
  \,,
\end{displaymath}
for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathfrak{g}_0$) and $\{\alpha^i{}_{a j}\}$ (the [[action]] of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the ``Jacobiator'' for the bracket on $\mathfrak{g}_0$).

These constants are subject to constraints (the weak [[Jacobi identity]] and its higher [[coherence law]]s) which are precisely equivalent to the condition

\begin{displaymath}
(d_{CE(\mathfrak{g})})^2 = 0
  \,.
\end{displaymath}
Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism

\begin{displaymath}
\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) 
   : 
  (A,B)
\end{displaymath}
from the [[Weil algebra]] $W(\mathfrak{g})$.

This is given by a 1-form

\begin{displaymath}
A \in \Omega^1(U, \mathfrak{g}_0)
\end{displaymath}
and a 2-form

\begin{displaymath}
B \in \Omega^2(U, \mathfrak{g}_1)
  \,.
\end{displaymath}
The [[curvature]] of this is $(\beta, H)$, where the 2-form component (``fake curvature'') is

\begin{displaymath}
\beta^a = d_{dR} A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c
  + r^{a}_{i} B^i
\end{displaymath}
and whose 3-form component is

\begin{displaymath}
H^i
     = 
   d_{dR} B^i
   +
   \alpha^i{}_{a j} A^a \wedge B^j
   +
   r^i{}_{a b c} A^a \wedge A^b \wedge A^c
  \,.
\end{displaymath}
\hypertarget{differential_ech_cocycle_data}{}\subsubsection*{{Differential ech cocycle data}}\label{differential_ech_cocycle_data}

We spell out the data of a connection on a 2-bundle over a [[smooth manifold]] $X$ with respect to a given [[open cover]] $\{U_i \to X\}$, following (\hyperlink{FSS}{FSS}, \hyperlink{SchreiberCohesive}{SchreiberCohesive})

(\ldots{})

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

\begin{itemize}%
\item A [[differential string structure]] (untwisted) is a 2-connection with coefficients in the [[string 2-group]] / [[string Lie 2-algebra]].


\item The [[worldvolume]] theory of the [[fivebrane]] is expected to be a [[6d (2,0)-supersymmetric QFT]] containing a [[self-dual higher gauge theory]] whose fields are 2-connections (see \emph{\href{http://ncatlab.org/nlab/show/NS5-brane#SelfDual2Connections}{Self-dual 2-connections}} there).


\item A connection on a [[twisted vector bundle]] is naturally a 2-connection.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

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


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

\begin{itemize}%
\item [[2-groupoid of Lie 2-algebra valued forms]]

\end{itemize}

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


\item [[connection on an ∞-bundle]]


\item [[parallel transport]], [[higher parallel transport]]


\item [[holonomy]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Connections on 2-bundles with vanishing 2-form curvature and arbitrary 3-form curvature are defined in terms of their [[higher parallel transport]] are discussed in

\begin{itemize}%
\item [[Urs Schreiber]], [[Konrad Waldorf]],

\emph{Smooth Functors and Differential Forms}, Homology, Homotopy Appl., 13(1), 143-203 (2011) (\href{http://arxiv.org/abs/0802.0663}{arXiv:0802.0663})

\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}
expanding on

\begin{itemize}%
\item [[John Baez]], [[Urs Schreiber]], \emph{Higher gauge theory} (\href{http://arxiv.org/abs/math/0511710}{arXiv:math/0511710}) (\href{http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos+--+references/#BaezSchreiber}{web})

\end{itemize}
Much further discussion and illustration and relation to [[tensor networks]] is in

\begin{itemize}%
\item [[Arthur Parzygnat]], \emph{Two-dimensional algebra in lattice gauge theory} (\href{https://arxiv.org/abs/1802.01139}{arXiv:1802.01139})

\end{itemize}
Examples of 2-connections with vanishing 2-form curvature obtained from [[geometric quantization]] are discusssed in

\begin{itemize}%
\item Olivier Brahic, \emph{On the infinitesimal Gauge Symmetries of closed forms} (\href{http://arxiv.org/abs/1010.2189}{arXiv})

\end{itemize}
The [[cocycle]] data for 2-connections with coeffcients in [[automorphism 2-group]]s but without restrictions on the 2-form curvature have been proposed in

\begin{itemize}%
\item [[Lawrence Breen]], [[William Messing]], \emph{Differential Geometry of Gerbes} Advances in Mathematics, Volume 198, Issue 2, 20 (2005) (\href{http://arxiv.org/abs/math/0106083}{arXiv:math/0106083})


\item [[Lawrence Breen]], \emph{Differential Geometry of Gerbes and Differential Forms} (\href{http://arxiv.org/abs/0802.1833}{arXiv:0802.1833})



\end{itemize}
and

\begin{itemize}%
\item Paolo Aschieri, Luigi Cantini, [[Branislav Jurco]], \emph{Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory} , Communications in Mathematical Physics Volume 254, Number 2 (2005) 367-400,(\href{http://arxiv.org/abs/hep-th/0312154}{arXiv:hep-th/0312154}).

\end{itemize}
A discussion of fully general local 2-connections is in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{$L_\infty$-connections} (\href{http://arxiv.org/abs/0801.3480}{arXiv:0801.3480})

\end{itemize}
and the globalization is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech cocycles for differential characteristic classes]]}

\end{itemize}
For a discussion of all this in a more comprehensive context see section xy of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
See also [[connection on an infinity-bundle]] for the general theory.

\hypertarget{applications_to_physics}{}\subsubsection*{{Applications to physics}}\label{applications_to_physics}

Nonabelian 2-connections appear for instance as [[orientifold]] [[B-fields]] in [[type II string theory]], as [[differential string structure]] in [[heterotic string theory]], and as fields in non-abelian [[7-dimensional Chern-Simons theory]]. See at these pages for references.

An appearance in 4-dimensional [[Yang-Mills theory]] and [[4d TQFT]] is reported in

\begin{itemize}%
\item [[Sergei Gukov]], [[Anton Kapustin]], \emph{[[Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories]]} (\href{http://arxiv.org/abs/1307.4793}{arXiv:1307.4793})

\end{itemize}
[[!redirects connections on a 2-bundle]] [[!redirects connections on 2-bundles]]

[[!redirects 2-connection]] [[!redirects 2-connections]]

[[!redirects principal 2-connection]] [[!redirects principal 2-connections]]

[[!redirects connection on a principal 2-bundle]]



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