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\begin{document}

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\section*{1-dimensional Chern-Simons theory}

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

\hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory}

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

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{for_the_first_chern_class}{For the first Chern class}\dotfill \pageref*{for_the_first_chern_class} \linebreak
\noindent\hyperlink{for_a_group_character_on_a_coadjoint_orbit}{For a group character, on a coadjoint orbit}\dotfill \pageref*{for_a_group_character_on_a_coadjoint_orbit} \linebreak
\noindent\hyperlink{for_a_symplectic_lie_0algebroid}{For a symplectic Lie 0-algebroid}\dotfill \pageref*{for_a_symplectic_lie_0algebroid} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{for_the_first_chern_class_2}{For the first Chern class}\dotfill \pageref*{for_the_first_chern_class_2} \linebreak
\noindent\hyperlink{for_a_symplectic_lie_0algebroid_2}{For a symplectic Lie 0-algebroid}\dotfill \pageref*{for_a_symplectic_lie_0algebroid_2} \linebreak


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

By the general mechanism of [[schreiber:∞-Chern-Simons theory]], every [[invariant polynomial]] of total degree 2 induces a 1-dimensional Chern-Simons-like theory.

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

\hypertarget{for_the_first_chern_class}{}\subsubsection*{{For the first Chern class}}\label{for_the_first_chern_class}

By the general mechanism of [[schreiber:∞-Chern-Simons theory]] there is a Chern-Simons action functional associated to the first [[Chern class]], or rather to the corresponding [[invariant polynomial]], which is simply the [[trace]] map on the [[unitary Lie algebra]]

\begin{displaymath}
tr : \mathfrak{u}(n) \to \mathbb{R}
  \,.
\end{displaymath}
This yields an [[action functional]] for a 1-dimensional [[QFT]] as follows:

The [[configuration space]] over a 1-dimensional $\Sigma$ is the [[groupoid of Lie algebra valued 1-forms]] $\Omega^1(\Sigma, \mathfrak{u})$. After identifying $\Sigma \subset \mathbb{R}$ this may be identified with the space of $\mathfrak{u}(n)$-valued functions.

The action functional is simply the [[trace]] operation

\begin{displaymath}
S_{CS}(\phi) = \int_\Sigma tr(\phi)
  \,.
\end{displaymath}
Degenerate as this situation is, it can be useful to regard the [[trace]] as a Chern-Simons action functional.

\begin{itemize}%
\item Arguments for a role in large $N$ gauge theory are in (\hyperlink{Nair}{Nair 06}).


\item The \emph{[[spectral action]]} is of this form.



\end{itemize}
\hypertarget{for_a_group_character_on_a_coadjoint_orbit}{}\subsubsection*{{For a group character, on a coadjoint orbit}}\label{for_a_group_character_on_a_coadjoint_orbit}

For $G$ a suitable [[Lie group]] (compact, semi-simple and simply connected) the [[Wilson loops]] of $G$-[[principal connections]] are equivalently the [[partition functions]] of a 1-dimensional Chern-Simons theory.

This appears famously in the formulation of [[Chern-Simons theory]] \href{Chern-Simons+theory#WithWilsonLineObservables}{with Wilson lines}. More detailes are at \emph{[[orbit method]]}.

\hypertarget{for_a_symplectic_lie_0algebroid}{}\subsubsection*{{For a symplectic Lie 0-algebroid}}\label{for_a_symplectic_lie_0algebroid}

A [[symplectic manifold]] regarded as a [[symplectic Lie n-algebroid]] with $n = 0$ induces a 1d Chern-Simons theory whose [[Chern-Simons form]] is a Liouville form of the symplectic form.

This case is discussed in \ldots{}

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

\begin{itemize}%
\item [[higher dimensional Chern-Simons theory]]

\begin{itemize}%
\item \textbf{1d Chern-Simons theory}


\item [[2d Chern-Simons theory]]


\item [[3d Chern-Simons theory]]


\item [[4d Chern-Simons theory]]


\item [[5d Chern-Simons theory]]


\item [[6d Chern-Simons theory]]


\item [[7d Chern-Simons theory]]


\item [[AKSZ sigma-models]]


\item [[string field theory]]


\item [[infinite-dimensional Chern-Simons theory]]



\end{itemize}


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

\hypertarget{for_the_first_chern_class_2}{}\subsubsection*{{For the first Chern class}}\label{for_the_first_chern_class_2}

A discussion of 1d CS theory in the context of large $N$-gauge theory is in

\begin{itemize}%
\item V.P. Nair, \emph{The Matrix Chern-Simons One-form as a Universal Chern-Simons Theory} Nucl.Phys.B750:289-320,2006 (\href{http://arxiv.org/abs/hep-th/0605007}{arXiv:hep-th/0605007})

\end{itemize}
An exposition of this theory formulated via an [[extended Lagrangian]] in [[higher geometric quantization]] is in section 1 of

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:A higher stacky perspective on Chern-Simons theory]]}

\end{itemize}
Further discussion is in section 5.7 of

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

\end{itemize}
\hypertarget{for_a_symplectic_lie_0algebroid_2}{}\subsubsection*{{For a symplectic Lie 0-algebroid}}\label{for_a_symplectic_lie_0algebroid_2}

A 1d Chern-Simons theory with target a [[cotangent bundle]] is discussed in

\begin{itemize}%
\item [[Ryan Grady]], [[Owen Gwilliam]], \emph{One-dimensional Chern-Simons theory and the $\hat{A}$ genus} (\href{http://arxiv.org/abs/1110.3533}{arXiv:1110.3533})

\end{itemize}
[[!redirects 1-dimensional Chern-Simons theories]]

[[!redirects 1d Chern-Simons theory]] [[!redirects 1d Chern-Simons theories]]



\end{document}