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\section*{higher 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{higher_abelian_chernsimons_theory}{Higher abelian Chern-Simons theory.}\dotfill \pageref*{higher_abelian_chernsimons_theory} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{formulation_in_chernsimons_theory}{Formulation in $\infty$-Chern-Simons theory}\dotfill \pageref*{formulation_in_chernsimons_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{holographic_relation_to_dimensional_theory}{Holographic relation to $4k+2$-dimensional theory}\dotfill \pageref*{holographic_relation_to_dimensional_theory} \linebreak
\noindent\hyperlink{BackgroundCharge}{Background charges and square root action functionals}\dotfill \pageref*{BackgroundCharge} \linebreak
\noindent\hyperlink{higherdimensional_nonabelian_cs_theory}{Higher-dimensional non-abelian CS theory}\dotfill \pageref*{higherdimensional_nonabelian_cs_theory} \linebreak
\noindent\hyperlink{1d_chernsimons_theory}{1d Chern-Simons theory}\dotfill \pageref*{1d_chernsimons_theory} \linebreak
\noindent\hyperlink{2d_chernsimons_theory}{2d Chern-Simons theory}\dotfill \pageref*{2d_chernsimons_theory} \linebreak
\noindent\hyperlink{3d_chernsimons_theory}{3d Chern-Simons theory}\dotfill \pageref*{3d_chernsimons_theory} \linebreak
\noindent\hyperlink{4d_chernsimons_theory}{4d Chern-Simons theory}\dotfill \pageref*{4d_chernsimons_theory} \linebreak
\noindent\hyperlink{5d_chernsimons_theory}{5d Chern-Simons theory}\dotfill \pageref*{5d_chernsimons_theory} \linebreak
\noindent\hyperlink{6d_chernsimons_theory}{6d Chern-Simons theory}\dotfill \pageref*{6d_chernsimons_theory} \linebreak
\noindent\hyperlink{7d_chernsimons_theory}{7d Chern-Simons theory}\dotfill \pageref*{7d_chernsimons_theory} \linebreak
\noindent\hyperlink{11d_chernsimons_theory}{11d Chern-Simons theory}\dotfill \pageref*{11d_chernsimons_theory} \linebreak
\noindent\hyperlink{infinitedimensional_chernsimons_theory}{Infinite-dimensional Chern-Simons theory}\dotfill \pageref*{infinitedimensional_chernsimons_theory} \linebreak
\noindent\hyperlink{aksz_models}{AKSZ $\sigma$-models}\dotfill \pageref*{aksz_models} \linebreak
\noindent\hyperlink{string_field_theory}{String field theory}\dotfill \pageref*{string_field_theory} \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{formulation_in_differential_cohomology}{Formulation in differential cohomology}\dotfill \pageref*{formulation_in_differential_cohomology} \linebreak
\noindent\hyperlink{relation_to_selfdual_theories}{Relation to self-dual theories}\dotfill \pageref*{relation_to_selfdual_theories} \linebreak
\noindent\hyperlink{higher_chernsimons_supergravity}{Higher Chern-Simons (super)gravity}\dotfill \pageref*{higher_chernsimons_supergravity} \linebreak
\noindent\hyperlink{ReferencesInvariants}{Higher Chern-Simons invariants}\dotfill \pageref*{ReferencesInvariants} \linebreak
\noindent\hyperlink{formulation_in_chernsimons_theory_2}{Formulation in $\infty$-Chern-Simons theory}\dotfill \pageref*{formulation_in_chernsimons_theory_2} \linebreak
\noindent\hyperlink{boundary_theories}{Boundary theories}\dotfill \pageref*{boundary_theories} \linebreak


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

There are two kinds of higher dimensional generalizations of ordinary 3-dimensional [[Chern-Simons theory]] that are often called ``higher dimensional Chern-Simons theory'' in the literature. Both are special cases of [[schreiber:infinity-Chern-Simons theory]].

Recall that for $\mathfrak{g}$ a [[Lie algebra]] (not necessarily abelian) with non-generate binary [[invariant polynomial]] $\langle -,-\rangle$, the corresponding [[schreiber:infinity-Chern-Simons theory]] [[QFT]] is ordinary [[Chern-Simons theory]] in dimension 3.

But also every other [[invariant polynomial]] $\langle-,-,\cdots,-\rangle$ on $\mathfrak{g}$ induces an [[schreiber:infinity-Chern-Simons theory]], now in higher dimension.

Moreover, every [[line Lie n-algebra]] $b^n \mathbb{R}$ carries a canonical invariant polynomial. The [[schreiber:infinity-Chern-Simons theory]] associated with that is often called \emph{abelian higher dimensional CS theory} .

(\ldots{})

More general ``higher''-generalization of Chern-Simons theory to [[schreiber:infinity-Chern-Simons theory]] allow $\mathfrak{g}$ to be a (nonabelian) [[Lie 2-algebra]] or more generally a (nonabelian) [[L-infinity algebra]] or fully generally a [[L-infinity algebroid]].

\hypertarget{higher_abelian_chernsimons_theory}{}\subsection*{{Higher abelian Chern-Simons theory.}}\label{higher_abelian_chernsimons_theory}

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

The definition of higher abelian Chern-Simons theory is simple \emph{locally} when certain global [[cohomology|cohomological]] effects can be ignored. We first give the simple local definition and then the full global definition.

Let $k \in \mathbb{N}$ be a [[natural number]], let $d = 4 k + 3$ and let $\Sigma$ be a [[compact space|compact]] [[smooth manifold]] of [[dimension]] $d$.

Then the \emph{simple version} of abelian $d$-dimensional Chern-Simons theory is defined as follows.

\begin{itemize}%
\item the [[configuration space]] is the [[space]] of [[differential form]]s on $\Sigma$ of degree $2k+1$

\begin{displaymath}
Conf_{simpl} = \Omega^{2k+1}(\Sigma)
  \,,
\end{displaymath}

\item the [[Lagrangian]] is

\begin{displaymath}
L : B \mapsto B \wedge d_{dR} B
  \,,
\end{displaymath}

\item and the [[action functional]]

\begin{displaymath}
S : \Omega^{2k+1}(\Sigma) \to \mathbb{R}
\end{displaymath}
is therefore

\begin{displaymath}
S : B \mapsto \int_\Sigma B \wedge d_{dR}B
  \,.
\end{displaymath}


\end{itemize}
Notice that generally for an $n$-form $B$ on a closed $(2n+1)$-dimensional manifold $\Sigma$ we have

\begin{displaymath}
\int_\Sigma B \wedge d_{dR} B
  =
  (-1)^{1+n + n(1+n)}
  \int_\Sigma B \wedge d_{dR} B
\end{displaymath}
by first using integration by parts and then switching the order of the wedge factors. Therefore this kind of action vanishes identically when $deg B$ is even. This is the reason for the above assumption that $deg B = 2k+1$ for $k \in \mathbb{N}$ and hence that the Chern-Simons theory is in dimension $4k+3$.

In the full theory instead the configuration space is

\begin{displaymath}
Conf = \mathbf{H}_{diff}^{2k+2}(\Sigma)
  \,,
\end{displaymath}
the space of [[circle n-bundle with connection|circle (2k+1)-bundles with connection]] (given by [[cocycle]]s in degree $2k+2$ [[ordinary differential cohomology]]). This contains the above simplified configuration space as the subspace of $(2k+1)$-connections whose underlying circle $(2k+1)$-bundle is trivial.

The action functional is given by

\begin{displaymath}
S : \hat B \mapsto \int_\Sigma \hat B \cup \hat B
  \,,
\end{displaymath}
where now the integral is [[fiber integration in ordinary differential cohomology]] and in the integrand we have the [[cup product in ordinary differential cohomology]] of differential cocycles.

(See for instance (\hyperlink{GT}{GT, section 4.1}), (\hyperlink{FMS}{FMS, (1.28)})).

\hypertarget{formulation_in_chernsimons_theory}{}\subsubsection*{{Formulation in $\infty$-Chern-Simons theory}}\label{formulation_in_chernsimons_theory}

We discuss how the above definition arises as a special case of the general notion of [[schreiber:infinity-Chern-Simons theory]].

These theories are defined by

\begin{itemize}%
\item an [[L-infinity algebroid]] $\mathfrak{a}$;


\item equipped with an [[invariant polynomial]] $\langle \rangle$.



\end{itemize}
The abelian higher dimensional Chern-Simons theories in dimension $4k+3$ are the special case of this general situation where

\begin{itemize}%
\item $\mathfrak{a} = b^{2k+1}\mathbb{R}$ is the [[line Lie n-algebra|line Lie (2k+1)-algebra]], the $(2k+1)$-fold [[delooping]] of the abelian [[Lie algebra]] $\mathbb{R}$;


\item $\langle - \rangle$ is the canonical quadratic [[invariant polynomial]] on this.



\end{itemize}
(\ldots{})

See (\hyperlink{FRS}{FRS, 4.1.4}).

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

Higher dimensional abelian Chern-Simons theories appear automatically as components of systems of higher [[supergravity]], for instance in [[11-dimensional supergravity]] (they are automatically induced by the requirement of [[local supersymmetry]] in these higher dimensional supergravity theories).

\begin{itemize}%
\item see at \emph{[[M5-brane]]} the section \emph{\href{http://ncatlab.org/nlab/show/M5-brane#7dCSDual}{Conformal blocks and 7d Chern-Simons dual}}.

\end{itemize}
\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

\hypertarget{holographic_relation_to_dimensional_theory}{}\paragraph*{{Holographic relation to $4k+2$-dimensional theory}}\label{holographic_relation_to_dimensional_theory}

Higher Chern-Simons theory in dimension $4k+3$ is related by a [[holographic principle]] to [[self-dual higher gauge theory]] in dimension $4k+2$ (at least in the abelian case).

\begin{itemize}%
\item $(k=0)$: ordinary 3-dimensional [[Chern-Simons theory]] is related to a [[string]] [[sigma-model]] on its boundary;


\item $(k=1)$: 7-dimensional Chern-Simons theory is related to a [[fivebrane]] model on its boundary;


\item $(k=2)$: 11-dimensional Chern-Simons theory is related to a parts of a [[type II string theory]] on its bounday (or that of the space-filling 9-[[brane]], if one wishes) (\hyperlink{BelovMoore}{BelovMoore})



\end{itemize}
\hypertarget{BackgroundCharge}{}\paragraph*{{Background charges and square root action functionals}}\label{BackgroundCharge}

The [[supergravity C-field]] is an example of a general phenomenon of higher abelian Chern-Simons QFTs in the presence of \emph{background charge}. This phenomenon was originally noticed in (\hyperlink{Witten96}{Witten}) and then made precise in (\hyperlink{HopkinsSinger05}{HopkinsSinger 05}). The holographic dual of this phenomenon is that of self-dual higher gauge theories, which for the supergravity $C$-field is the 2-form theory on the [[M5-brane]] -- see there for a discussion of this example. Here we discuss this effect generally, for higher abelian Chern-Simons theory in arbitrary dimension $4k+3$.

Fix some natural number $k \in \mathbb{N}$ and an [[orientation|oriented]] [[manifold]] ([[compact topological space|compact]] [[manifold with boundary|with boundary]]) $X$ of [[dimension]] $4 k + 3$. The gauge equivalence class of a $(2k+1)$-form gauge field $\hat G$ on $X$ is an element in the [[ordinary differential cohomology]] group $\hat H^{2k+2}(X)$. The [[cup product]] $\hat G \cup \hat G \in \hat H^{4k+4}(X)$ of this with itself has a natural [[higher holonomy]] over $X$, denoted

\begin{displaymath}
\exp(i S (-)) : \hat H^{2k+2}(X) \to U(1)
\end{displaymath}
\begin{displaymath}
\hat G \mapsto \exp(i \int_X \hat G \cup \hat G)
  \,.
\end{displaymath}
This is the exponentiated [[action functional]] for bare $(4k+3)$-dimensional abelian Chern-Simons theory, as discussed above.

Observe now that the above action functional may be regarded as a \emph{[[quadratic form]]} on the [[cohomology group]] $\hat H^{2k+2}(X)$. The corresponding [[bilinear form]] is the (``[[secondary characteristic class|secondary]]'', since $X$ is of dimension $4k+3$ instead of $4k+4$) \emph{[[intersection pairing]]}

\begin{displaymath}
\langle -,-\rangle : \hat H^{2k+2}(X) \times \hat H^{2k+2}(X) \to U(1)
\end{displaymath}
\begin{displaymath}
(\hat a_1 , \hat a_2) \mapsto \exp(i \int_X \hat a_1 \cup \hat a_2 )
  \,.
\end{displaymath}
But note that from $\exp(i S(-))$ we do \emph{not} obtain a \emph{[[quadratic refinement]]}\newline of the pairing. A quadratic refinement is, by definition, a function

\begin{displaymath}
q : \hat H^{2k+2}(X) \to U(1)
\end{displaymath}
(not necessarily homogenous of degree 2 as $\exp(i S(-))$ is), such that the [[intersection pairing]] is reobtained from it by the polarization formula

\begin{displaymath}
\langle \hat a_1, \hat a_2\rangle 
    = 
  q(\hat a_1 + \hat a_2)
  q(\hat a_1)^{-1}
  q(\hat a_2)^{-1}
  q(0)
  \,.
\end{displaymath}
If we took $q := \exp(i S(-))$, then the above formula would yield not $\langle -,-\rangle$, but the square $\langle -,-\rangle^2$, given by (the exponentiation of) \emph{twice} the integral.

The observation in (\hyperlink{Witten96}{Witten96}) was that for the correct [[holographic principle|holographic]] physics, we need instead an action functional which is indeed a genuine [[quadratic refinement]] of the [[intersection pairing]].

But since the [[differential cohomology|differential classes]] in $\hat H^{2k+2}(X)$ refine \emph{[[integral cohomology]]}, we cannot in general simply divide by 2 and pass from $\exp( i \int_X \hat G \cup \hat G)$ to $\exp( i  \int_X \frac{1}{2} \hat G \cup \hat G)$. The integrand in the latter expression does not make sense in general in differential cohomology. If one tried to write it out in the ``obvious'' local formulas one would find that it is a functional on fields which is not [[gauge invariance|gauge invariant]]. The analog of this fact is familiar from nonabelian $G$-[[Chern-Simons theory]] with [[simply connected space|simply connected]] $G$, where also the theory is consistent only at interger \emph{levels}. The ``level'' here is nothing but the underlying integral class $G \cup G$.

Therefore the only way to obtain a [[square root]] of the [[quadratic form]] $\exp(i S(-))$ is to \emph{shift its origin}. Here we think of the analogy with a quadratic form $q : x \mapsto x^2$ on the [[real numbers]] (a parabola in the plane). Replacing this by $q^{\lambda} : x \mapsto x^2 + \lambda x$ for some real number $\lambda$ means keeping the shape of the form, but shifting its [[minimum]] from 0 to $-\frac{1}{2}\lambda$. If we think of this as the potential term for a scalar field $x$ previously with rotation-symmetric dynamics about $x = 0$, then the new potential exhibits [[spontaneous symmetry breaking]]: its ground state is now at $x = -\frac{1}{2}\lambda$ (and has [[energy]] $-\frac{1}{4}\lambda^2$ there). We may say that there is a \emph{background field} or \emph{background [[charge]]} that pushes the field out of its free equilibrium.

To lift this reasoning to our action quadratic form $\exp(i S(-))$ on differential cocycles, we need a differential class $\hat \lambda \in H^{2k+2}(X)$ such that for every $\hat a \in H^{2k+2}(X)$ the composite class

\begin{displaymath}
\hat a \cup \hat a + \hat a \cup \hat \lambda
  \in 
  H^{4k+4}(X)
\end{displaymath}
is even, hence is divisible by 2. Because then we could define a shifted action functional

\begin{displaymath}
\exp(i S^\lambda(-)) : \hat a \mapsto \exp(i \int_X \frac{1}{2}(\hat a \cup \hat a + \hat a \cup \hat \lambda))
  \,,
\end{displaymath}
where now the fraction $\frac{1}{2}$ in the integrand does make sense. One directly sees that if this exists, then this shifted action is indeed now a [[quadratic refinement]] of the [[intersection pairing]]

\begin{displaymath}
\exp(i S^\lambda(\hat a + \hat b))
  \exp(i S^\lambda(\hat a))^{-1}
  \exp(i S^\lambda(\hat b))^{-1}
  \exp(i S^\lambda(0)
  = 
  \exp(i \int_X \hat a \cup \hat b))
  \,.
\end{displaymath}
The condition on the existence of $\hat \lambda$ here means equivalently that the image of the underlying integral class in cohomology with coefficients in $\mathbb{Z}_2$ vanishes:

\begin{displaymath}
(a)_{\mathbb{Z}_2} \cup (a)_{\mathbb{Z}_2} 
     + 
   (a)_{\mathbb{Z}_2} \cup (\lambda)_{\mathbb{Z}_2}  
   = 
   0 \in H^{4k+4}(X, \mathbb{Z}_2)
  \,,
\end{displaymath}
Precisely such a class $(\lambda)_{\mathbb{Z}_2}$ does uniquely exist on every oriented manifold. It is called the \emph{[[Wu class]]} $\nu_{2k+2} \in H^{2k+2}(X,\mathbb{Z}_2)$, and may be \emph{defined} by this condition. Moreover, if $X$ is a [[spin structure|Spin-manifold]], then every second [[Wu class]], $\nu_{4k}$, has a pre-image in [[integral cohomology]], hence $\lambda$ does exist as required above

\begin{displaymath}
(\lambda)_{\mathbb{Z}_2}
    = 
  \nu_{2k+2}
  \,.
\end{displaymath}
It is given by [[polynomials]] in the [[Pontryagin classes]] of $X$. For instance the degree-4 Wu class (for $k = 1$) is refined by the [[first fractional Pontryagin class]] $\frac{1}{2}p_1$

\begin{displaymath}
(\frac{1}{2}p_1)_{\mathbb{Z}_2} = \nu_4
  \,.
\end{displaymath}
This was the original observation in \hyperlink{Witten96}{Witten96, around (3.3)}.

Notice that the [[equations of motion]] of the shifted action $\exp(i S^\lambda(\hat a))$ are no longer $F_a = 0$, but are now $F_a = - \frac{1}{2}F_\lambda$. Comparing this to the [[Maxwell equations]], we see that $-\frac{1}{2}\hat \lambda$ here plays the role of a \emph{background [[charge]]} (or rather, of the \emph{background [[current]]} that underlies a background charge). We therefore think of $\exp(i S^\lambda(-))$ as the exponentiated action functional for \emph{higher dimensional abelian Chern-Simons theory with background charge $-\frac{1}{2}\lambda$}.

This of course only makes sense if $X$ is such that $\lambda$ is further divisible by 2, which we will assume now. In (\hyperlink{HopkinsSinger05}{Hopkins-Singer 05}) is discussed a way to make sense of the further division in general if one passes to a certain notion of twisted differential cohomology. One can also adopt a different perspective and interpret the condition that $\frac{1}{2}p_1$ is further divisible by 2 precisely as a $\mathrm{String}^{2a}$-structure (\href{http://ncatlab.org/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures}{SSS3}). This is a higher analog of a [[Spin{\tt \symbol{94}}c structure]].

With respect to the shifted action functional it makes sense to introduce the shifted field

\begin{displaymath}
\hat G  := \hat a  + \frac{1}{2}\lambda
  \,.
\end{displaymath}
This is simply a re-parameterization such that the Chern-Simons equations of motion again look homogenous, namely $FieldStrength(\hat G) = 0$. In terms of this shifted field the action $\exp(i S^\lambda(\hat a))$ from above equivalently reads

\begin{displaymath}
\exp(i S^\lambda(\hat G)) = 
  \exp(
    i \int_X \frac{1}{2}(\hat G \cup \hat G - (\frac{1}{2}\hat \lambda)^2)
  )
  \,.
\end{displaymath}
This is the form of the action functional first given as (\hyperlink{Witten96}{Witten96 (3.6)}) in for the case $k = 1$.

In the language of [[twisted differential c-structures]], we may summarize this sitation as follows:

in order for the action functional of higher abelian Chern-Simons theory to be correctly divisible, the images of the fields in $\mathbb{Z}_2$-cohomology need to form a [[twisted Wu structure]]. Therefore the fields themselves need to constitute a \emph{twisted $\lambda$-structure}. For $k = 1$ this is a [[twisted differential string structure|twisted string structure]] and explains for instance the quantization condition on the [[supergravity C-field]] in [[11-dimensional supergravity]]. For that case see also the corresponding discussion at \emph{[[M5-brane]]}.

\hypertarget{higherdimensional_nonabelian_cs_theory}{}\subsection*{{Higher-dimensional non-abelian CS theory}}\label{higherdimensional_nonabelian_cs_theory}

Chern-Simons actions for [[Lie algebra]]s $\mathfrak{g}$ but with higher-degree [[invariant polynomial]]s have in particular received attention for $\mathfrak{g} = \mathfrak{siso}$ the [[super Poincare Lie algebra]]. In this case these action functionals can be regarded as defining higher [[Chern-Simons supergravity]].

\hypertarget{1d_chernsimons_theory}{}\subsubsection*{{1d Chern-Simons theory}}\label{1d_chernsimons_theory}

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

\end{itemize}
\hypertarget{2d_chernsimons_theory}{}\subsubsection*{{2d Chern-Simons theory}}\label{2d_chernsimons_theory}

\begin{itemize}%
\item [[2d Chern-Simons theory]]

\begin{itemize}%
\item [[Poisson sigma-model]]

\end{itemize}


\end{itemize}
\hypertarget{3d_chernsimons_theory}{}\subsubsection*{{3d Chern-Simons theory}}\label{3d_chernsimons_theory}

\begin{itemize}%
\item [[3d Chern-Simons theory]]

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


\item [[Dijkgraaf-Witten model]]


\item [[Courant sigma-model]]



\end{itemize}


\end{itemize}
\hypertarget{4d_chernsimons_theory}{}\subsubsection*{{4d Chern-Simons theory}}\label{4d_chernsimons_theory}

\begin{itemize}%
\item [[4d Chern-Simons theory]]

\begin{itemize}%
\item [[Yetter model]]

\end{itemize}


\end{itemize}
\hypertarget{5d_chernsimons_theory}{}\subsubsection*{{5d Chern-Simons theory}}\label{5d_chernsimons_theory}

\begin{itemize}%
\item [[5d Chern-Simons theory]]

\end{itemize}
\hypertarget{6d_chernsimons_theory}{}\subsubsection*{{6d Chern-Simons theory}}\label{6d_chernsimons_theory}

\begin{itemize}%
\item [[6d Chern-Simons theory]]

\end{itemize}
\hypertarget{7d_chernsimons_theory}{}\subsubsection*{{7d Chern-Simons theory}}\label{7d_chernsimons_theory}

\begin{itemize}%
\item [[7d Chern-Simons theory]]

\end{itemize}
\hypertarget{11d_chernsimons_theory}{}\subsubsection*{{11d Chern-Simons theory}}\label{11d_chernsimons_theory}

\begin{itemize}%
\item [[11d Chern-Simons theory]]

\end{itemize}
\hypertarget{infinitedimensional_chernsimons_theory}{}\subsubsection*{{Infinite-dimensional Chern-Simons theory}}\label{infinitedimensional_chernsimons_theory}

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

\end{itemize}
\hypertarget{aksz_models}{}\subsubsection*{{AKSZ $\sigma$-models}}\label{aksz_models}

\begin{itemize}%
\item [[AKSZ sigma-model]]

\end{itemize}
\hypertarget{string_field_theory}{}\subsubsection*{{String field theory}}\label{string_field_theory}

\begin{itemize}%
\item [[string field theory]]

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

\begin{itemize}%
\item [[Wu classes]]


\item [[schreiber:infinity-Chern-Simons theory]]

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


\item \textbf{higher dimensional Chern-Simons theory}

\begin{itemize}%
\item [[Chern-Simons gravity]]

\end{itemize}

\item [[AKSZ sigma-model]]

\begin{itemize}%
\item [[Poisson sigma model]]

\begin{itemize}%
\item [[A-model]], [[B-model]]

\end{itemize}

\item [[Courant sigma-model]]

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

\end{itemize}


\end{itemize}

\item [[1-dimensional Chern-Simons theory]]



\end{itemize}

\item [[higher dimensional WZW models]]



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

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

For higher dimensional non-abelian Chern-Simons theory see for instance

\begin{itemize}%
\item M\'a{}ximo Ba\~n{}ados, \emph{Higher dimensional Chern-Simons theories and black holes} (\href{http://worldscibooks.com/etextbook/4388/4388_chap01.pdf}{pdf})


\item M\'a{}ximo Ba\~n{}ados, Luis Garay, [[Marc Henneaux]], \emph{Existence of local degrees of freedom for higher dimensional pure Chern-Simons theories} Phys. Rev. D 53, R593--R596 (1996) (\href{http://prd.aps.org/pdf/PRD/v53/i2/pR593_1}{pdf})


\item M\'a{}ximo Ba\~n{}ados, Luis Garay, [[Marc Henneaux]], \emph{The dynamical structure of higher dimensional Chern-Simons theory} (\href{http://arxiv.org/abs/hep-th/9605159}{arXiv:hep-th/9605159})


\item G Giachetta, L Mangiarotti, G Sardanashvily, \emph{Noether conservation laws in higher-dimensional Chern-Simons theory} Modern Physics Letters A Volume 18(2003) (\href{http://gnsardan.appfarm.ru/a84.pdf}{pdf})



\end{itemize}
\hypertarget{formulation_in_differential_cohomology}{}\subsubsection*{{Formulation in differential cohomology}}\label{formulation_in_differential_cohomology}

For the formulation of abelian Chern-Simons theory by fiber integration over cup products in [[ordinary differential cohomology]] see

\begin{itemize}%
\item Enore Guadagnini, Frank Thuillier, \emph{Deligne-Beilinson cohomology and abelian links invariants} SIGMA 4 (2008), 078, 30 (\href{http://arxiv.org/abs/0801.1445}{arXiv:0801.1445})


\item R. Floreanini, R. Percacci, \emph{Higher dimensional abelian Chern-Simons theory} Physics Letters B Volume 224, Issue 3, 29 (1989)


\item [[Daniel Freed]], [[Gregory Moore]], [[Graeme Segal]], \emph{The Uncertainty of Fluxes} (\href{http://arxiv.org/abs/hep-th/0605198}{arXiv:hep-th/0605198})



\end{itemize}
\hypertarget{relation_to_selfdual_theories}{}\subsubsection*{{Relation to self-dual theories}}\label{relation_to_selfdual_theories}

The idea of describing [[self-dual higher gauge theory]] by abelian Chern-Simons theory in one dimension higher originates in

\begin{itemize}%
\item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory} Journal of Geometry and Physics, Volume 22, Issue 2, May 1997, Pages 103-133 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234})

\end{itemize}
(there for the [[6d (2,0)-susy QFT]] on the [[fivebrane]]) and

\begin{itemize}%
\item [[Edward Witten]], \emph{Duality Relations Among Topological Effects In String Theory} (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086})

\end{itemize}
Motivated by this the [[differential cohomology]] of self-dual fields had been discussed in

\begin{itemize}%
\item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology, and M-Theory]]}, 2005

\end{itemize}
More discussion of the general principle is in

\begin{itemize}%
\item Dmitriy Belov, [[Greg Moore]], \emph{Holographic Action for the Self-Dual Field} (\href{http://arxiv.org/abs/hep-th/0605038}{arXiv:hep-th/0605038})

\end{itemize}
The application of this to the description of [[type II string theory]] in 10-dimensions to [[schreiber:infinity-Chern-Simons theory|11-dimensional Chern-Simons theory]] is in the followup

\begin{itemize}%
\item Dmitriy Belov, [[Greg Moore]], \emph{Type II Actions from 11-Dimensional Chern-Simons Theories} (\href{http://arxiv.org/abs/hep-th/0611020}{arXiv:hep-th/0611020})

\end{itemize}
\hypertarget{higher_chernsimons_supergravity}{}\subsubsection*{{Higher Chern-Simons (super)gravity}}\label{higher_chernsimons_supergravity}

There are various discussions identifying or conjecturing higher dimensional Chern-Simons theories as parts of or related to [[gravity]] and [[supergravity]].

An original articles includes

\begin{itemize}%
\item M\'a{}ximo Ba\~n{}ados, Ricardo Troncoso, Jorge Zanelli, \emph{Higher dimensional Chern-Simons supergravity} Phys. Rev. D 54, 2605--2611 (1996) (\href{http://cdsweb.cern.ch/record/293524/files/9601003.pdf}{pdf})

\end{itemize}
An introduction and survey is in

\begin{itemize}%
\item Jorge Zanelli, \emph{Lecture notes on Chern-Simons (super-)gravities} (\href{http://arxiv.org/abs/hep-th/0502193}{arXiv:hep-th/0502193})

\end{itemize}
For more references see at

\begin{itemize}%
\item \emph{[[Chern-Simons gravity]]} .

\end{itemize}
\hypertarget{ReferencesInvariants}{}\subsubsection*{{Higher Chern-Simons invariants}}\label{ReferencesInvariants}

\begin{itemize}%
\item F. Thuillier, \emph{Deligne-Beilinson cohomology and abelian link invariants: torsion case} J. Math. Phys. 50, 122301 (2009)


\item B. Broda, \emph{Higher-dimensional Chern-Simons theory and link invariants} Physics Letters B Volume 280, Issues 3-4, 30 (1992) Pages 213-218


\item L. Gallot, E. Pilon, F. Thuillier, \emph{Higher dimensional abelian Chern-Simons theories and their link invariants} (\href{http://arxiv.org/abs/1207.1270}{arXiv:1207.1270})



\end{itemize}
\hypertarget{formulation_in_chernsimons_theory_2}{}\subsubsection*{{Formulation in $\infty$-Chern-Simons theory}}\label{formulation_in_chernsimons_theory_2}

Below (5.11) of (\hyperlink{HopkinsSinger05}{Hopkins-Singer 05}).

Section 4.1.4 of

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]]}

\end{itemize}
Section 4.4 of

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

\end{itemize}
\hypertarget{boundary_theories}{}\subsubsection*{{Boundary theories}}\label{boundary_theories}

Boundary [[higher dimensional WZW models]] for nonabelian [[higher dimensional Chern-Simons theory]] are discussed in

\begin{itemize}%
\item J. Gegenberg , G. Kunstatter, \emph{Boundary Dynamics of Higher Dimensional Chern-Simons Gravity} (\href{http://arxiv.org/abs/hep-th/0010020}{arXiv:hep-th/0010020})


\item J. Gegenberg , G. Kunstatter, \emph{Boundary Dynamics of Higher Dimensional AdS Spacetime} (\href{http://arxiv.org/abs/hep-th/9905228}{arXiv:http://arxiv.org/abs/hep-th/9905228})



\end{itemize}
[[!redirects higher Chern-Simons theory]] [[!redirects higher Chern-Simons theories]]

[[!redirects higher dimensional Chern-Simons theories]]



\end{document}