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\begin{document}

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\section*{Chern-Simons element}

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

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

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

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

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - 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{uniqueness}{Uniqueness}\dotfill \pageref*{uniqueness} \linebreak
\noindent\hyperlink{CanonicalChernSimonsElement}{Canonical $\infty$-Chern-Simons elements}\dotfill \pageref*{CanonicalChernSimonsElement} \linebreak
\noindent\hyperlink{OriginAndRelatedConcepts}{Origin and relation to other concepts}\dotfill \pageref*{OriginAndRelatedConcepts} \linebreak
\noindent\hyperlink{PresentationForChernWeil}{As presentations for the $\infty$-Chern-Weil homomorphism}\dotfill \pageref*{PresentationForChernWeil} \linebreak
\noindent\hyperlink{ChernSimonsForms}{Chern-Simons forms}\dotfill \pageref*{ChernSimonsForms} \linebreak
\noindent\hyperlink{chernsimons_action_functionals}{Chern-Simons action functionals}\dotfill \pageref*{chernsimons_action_functionals} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{StandardCS}{On semisimple Lie algebra-- Standard Chern-Simons action functional}\dotfill \pageref*{StandardCS} \linebreak
\noindent\hyperlink{higher_csforms_on_semisimple_lie_algebras}{Higher CS-forms on semisimple Lie algebras}\dotfill \pageref*{higher_csforms_on_semisimple_lie_algebras} \linebreak
\noindent\hyperlink{action_functional_for_chernsimons_supergravity}{Action functional for Chern-Simons (super-)gravity}\dotfill \pageref*{action_functional_for_chernsimons_supergravity} \linebreak
\noindent\hyperlink{fractional_secondary_pontryagin_classes}{Fractional secondary Pontryagin classes}\dotfill \pageref*{fractional_secondary_pontryagin_classes} \linebreak
\noindent\hyperlink{BF}{On strict Lie 2-algebras -- BF-theory action functional}\dotfill \pageref*{BF} \linebreak
\noindent\hyperlink{Symplectic}{On symplectic $\infty$-Lie algebroids -- The AKSZ Lagrangian}\dotfill \pageref*{Symplectic} \linebreak
\noindent\hyperlink{higher_phase_space_hamiltonian_and_lagrangian_mechanics}{Higher phase space ---Hamiltonian and Lagrangian mechanics}\dotfill \pageref*{higher_phase_space_hamiltonian_and_lagrangian_mechanics} \linebreak
\noindent\hyperlink{on_a_symplectic_manifold__the_topological_particle}{On a symplectic manifold -- The topological particle}\dotfill \pageref*{on_a_symplectic_manifold__the_topological_particle} \linebreak
\noindent\hyperlink{on_a_poisson_lie_algebroid__the_poisson_model}{On a Poisson Lie algebroid -- The Poisson $\sigma$-model}\dotfill \pageref*{on_a_poisson_lie_algebroid__the_poisson_model} \linebreak
\noindent\hyperlink{on_higher_extensions_of_the_super_poincare_lie_algebra__supergravity}{On higher extensions of the super Poincare Lie algebra -- supergravity}\dotfill \pageref*{on_higher_extensions_of_the_super_poincare_lie_algebra__supergravity} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{Chern-Simons element} on an [[L-∞ algebroid]] (named after [[Shiing-shen Chern]] and [[James Simons]] who considered this for [[semisimple Lie algebras]]) is an element of its [[Weil algebra]] that exhibits a [[transgression]] between an [[∞-Lie algebroid cocycle]] and an [[invariant polynomial]].

It is construct that arises in the presentation of the [[∞-Chern-Weil homomorphism]] by an [[∞-anafunctor]] of [[simplicial presheaves]].

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

We discuss [[∞-Lie algebra]]s and [[∞-Lie algebroid]]s $\mathfrak{a}$ of [[finite type]] in terms of their [[Chevalley-Eilenberg algebra]]s $CE(\mathfrak{a})$. For $\infty$-Lie algebras these are objects in the [[category]] [[dgAlg]] of [[dg-algebra]]s (over a given ground [[field]]). For $\infty$-Lie algebroids these are dg-algebras equipped with a lift of the degree-0 algebra to an algebra over a given [[Fermat theory]] $T$ and such that the [[differential]] is a $T$-[[derivation]] in this degree. (See [[∞-Lie algebroid]] for details). We shall write in the following $dgAlg$ also for the category of dg-algebras with this extra structure and leave the [[Fermat theory]] $T$ implicit.

\begin{defn}
\label{}\hypertarget{}{}
For $\mathfrak{g}$ an [[∞-Lie algebra]] or more generally [[∞-Lie algebroid]], $\mu \in CE(\mathfrak{g})$ a [[∞-Lie algebra cocycle]] (a closed element of the [[Chevalley-Eilenberg algebra]]) and $\langle - \rangle \in W(\mathfrak{g})$ an [[invariant polynomial]], a \textbf{Chern-Simons element} exhibiting the \emph{[[transgression]]} between the two is an element

\begin{displaymath}
cs \in W(\mathfrak{g})
\end{displaymath}
such that

\begin{enumerate}%
\item we have $d_{W(\mathfrak{g})} cs = \langle -\rangle$


\item and $cs|_{CE(\mathfrak{g})} = \mu$



\end{enumerate}
where the restriction is along the canonical morphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Notice that a degree-$n$ [[∞-Lie algebroid cocycle]] $\mu$ is equivalently a morphism

\begin{displaymath}
CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R}) : \mu
\end{displaymath}
and an [[invariant polynomial]] of degree $n+1$ is equivalently a morphism

\begin{displaymath}
inv(\mathfrak{g}) \leftarrow inv(b^{n-1}\mathbb{R}) = 
  CE(b^n \mathbb{R}) : \langle - \rangle
\end{displaymath}
in [[dgAlg]].

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
A \textbf{Chern-Simons element} $cs$ an $\mathfrak{a}$ witnessing the [[transgression]] of $\langle - \rangle$ to $\mu$ is equivalently a morphism

\begin{displaymath}
W(\mathfrak{g}) \leftarrow W(b^{n-1} \mathbb{R})
  : cs
\end{displaymath}
such that we have a [[commuting diagram]] in [[dgAlg]]

\begin{equation}
\itexarray{
    CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1} \mathbb{R})
    &&&
    cocycle
    \\
    \uparrow && \uparrow^{\mathrlap{i^*}}
    \\
    W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1} \mathbb{R})
    &&&
    Chern-Simons\;element
    \\
    \uparrow && \uparrow
    \\
    inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}&
    inv(b^{n-1} \mathbb{R})
    &&&
    invariant\;polynomial
  }
  \,,
\label{TheDiagram}\end{equation}
where the vertical morphisms are the canonical ones.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
If we think of

\begin{itemize}%
\item $W(\mathfrak{g})$ as differential forms on the total space of the universal $G$-bundles;


\item $CE(\mathfrak{g})$ as differential forms on the fiber


\item $inv(\mathfrak{g})$ as differential forms on the base space



\end{itemize}
then the abov expresses the classical notion of transgression of forms from the fiber to the base of a fibe bundle (for instance \hyperlink{Borel}{Borel, section 9}).

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{uniqueness}{}\subsubsection*{{Uniqueness}}\label{uniqueness}

\begin{pop}
\label{}\hypertarget{}{}
For a given transgressive cocycle $\mu$ and transgressing invariant polynomial $\langle - \rangle$ the set of Chern-Simons elements witnessing the [[transgression]] is a [[torsor]] (based over the point and) over the additive [[group]]

\begin{displaymath}
\{\omega \in W(\mathfrak{g}) | d_{W(\mathfrak{g})} \omega = 0, i^* \omega = 0\}
\end{displaymath}
of Chern-Simons elements for vanishing cocycle and vanishing invariant polynomial.

\end{pop}
\hypertarget{CanonicalChernSimonsElement}{}\subsubsection*{{Canonical $\infty$-Chern-Simons elements}}\label{CanonicalChernSimonsElement}

Since the [[Weil algebra]] of an [[L-∞ algebra]] has trivial cohomolgy in positive degree, every [[invariant polynomial]] $\langle -,\cdots, -\rangle$ has a Chern-Simons element and there is a standard formula for it.

\begin{pop}
\label{}\hypertarget{}{}
Let $\mathfrak{g}$ be an [[L-∞ algebra]] with $k$-ary brackets $[-,\cdots, -]_k : \mathfrak{g}^{\otimes k} \to \mathfrak{g}$ and equipped with a quadratic [[invariant polynomial]] $\langle -,-\rangle$.

A Chern-Simons element for $\langle-,-\rangle$ is given by the formula

\begin{displaymath}
cs(A) = 
  \langle A, d_{dR} A\rangle
  + 
  \sum_{k = 1}^\infty
  \frac{2}{(k+1)!}
  \langle A, [A,\cdots,A]_k\rangle
  \,,
\end{displaymath}
where $A : W(\mathfrak{g}) \to \Omega^\bullet(\Sigma)$ is any $\mathfrak{g}$-[[infinity-Lie algebroid-valued differential form|valued form]]

\end{pop}
\begin{proof}
There is a canonical contracting [[homotopy]]

\begin{displaymath}
\tau : W(\mathfrak{g}) \to W(\mathfrak{g})
\end{displaymath}
satisfying $[d_W, \tau] = Id$

and the above element is

\begin{displaymath}
cs = \tau \langle -,-\rangle
  \,.
\end{displaymath}
To see this, let $\{t_a\}$ be a [[basis]] and $\{t^a\}$ the dual basis. Then the differential of the [[Chevalley-Eilenberg algebra]] can be written

\begin{displaymath}
d_{CE(\mathfrak{g})} t^a 
  =
  -
  \sum_{k = 1}^\infty
   \frac{1}{k!}
   [t_{a_1}, t_{a_2}, \cdots, t_{a_k}]^a
   \,\,
   t^{a_1} \wedge t^{a_2}\wedge \cdots t^{a_k}
  \,,
\end{displaymath}
where

\begin{displaymath}
[-,-, \cdots, -] : \mathfrak{g}^{\otimes_k} \to \mathfrak{g}
\end{displaymath}
is the corresponding $k$-ary bracket.

Write

\begin{displaymath}
P_{a b} := \langle t_a , t_b\rangle
  \,,
\end{displaymath}
for the components of the [[invariant polynomial]] in this basis.

Then the claim is that

\begin{displaymath}
cs 
  = 
  2 P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b
  + 
  \sum_{k = 1}^\infty
  C_{a b_1, \cdots, b_k}
   \,\,
  t^a \wedge t^{b_1} \wedge \cdots \wedge t^{b_k}
  \,,
\end{displaymath}
where the coefficients are

\begin{displaymath}
C_{a b_1, \cdots, b_k}
  :=
  \frac{1}{(k+1)!}
  (P_{a b} [t_{b_1}, \cdots, t_{b_k}]^b)
\end{displaymath}
Write $F(\mathfrak{g})$ for the [[free construction|free]] [[dg-algebra]] on the [[graded vector space]] $\mathfrak{g}^*$. In terms of the above basis this is generated from $\{t^a, \mathbf{d}t^a\}$. As discussed at [[Weil algebra]], there is a dg-algebra [[isomorphism]]

\begin{displaymath}
F(\mathfrak{g}) \stackrel{\simeq}{\to} W(\mathfrak{g})
\end{displaymath}
given by sending $t^a \mapsto t^a$ and $\mathbf{d}t^a \mapsto d_{CE} t^a + r^a$.

Let $h : F(\mathfrak{g}) \to F(\mathfrak{g})$ be the [[derivation]] which on generators is defined by

\begin{displaymath}
h : t^a \mapsto 0
\end{displaymath}
\begin{displaymath}
h : \mathbf{d}t^a \mapsto t^a
  \,.
\end{displaymath}
Notice that this is \emph{not} the homotopy that exhibits the triviality of $Id_{F(\mathfrak{g}^*)}$, rather that homotopy is $\frac{1}{L} h$, where $L$ is the word length operator for element in $F(\mathfrak{g}^*)$ in terms of the generators $\{t^a , \mathbf{d}t^a\}$.

Therefore the homotopy $\tau$ is the composite top morphism in the diagram

\begin{displaymath}
\itexarray{
    W(\mathfrak{g}) &\stackrel{\tau}{\to}& W(\mathfrak{g})
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \uparrow^{\mathrlap{\simeq}}
    \\
    F(\mathfrak{g}) &\stackrel{\frac{1}{L} h}{\to}& F(\mathfrak{g})
  }
  \,.
\end{displaymath}
Unwinding this, we find

\begin{displaymath}
\begin{aligned}
    cs & := \tau \left( P_{a b} r^{a} \wedge r^b \right)
    \\
    & =
     P_{a b} \tau
    \left(
      d_{W(\mathfrak{g})} t^a 
      + 
      \sum_{k = 1}^\infty
      [t_{a_1}, \cdots, t_{a_k}]^a
      t^{a_1} \wedge \dots t^{a_k}
    \right)
    \wedge
    \left(
      d_{W(\mathfrak{g})} t^b 
      + 
      \sum_{k = 1}^\infty
      [t_{b_1}, \cdots, t_{b_k}]^b
      t^{b_1} \wedge \dots t^{b_k}
    \right)
    \\
    & =
    P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b
    +
    \sum_{k = 1}^\infty
    \frac{2}{k! (k+1) }
    P_{a b} [t_{b_1}, \cdots t_{b_k}]^b t^{b_1} \wedge 
    \cdots \wedge t^{b_k} 
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{example}
\label{}\hypertarget{}{}
We consider the ordinary Chern-Simons element as an example of this formula: let $\mathfrak{g}$ be a [[semisimple Lie algebra]] and $\langle -,-\rangle$ the [[Killing form]] [[invariant polynomial]]. Then the above computation gives

\begin{displaymath}
\begin{aligned} 
    cs & = \tau \left( P_{a b} r^a \wedge r^b \right)
    \\
    & = 
    P_{a b}\tau
    \left(d_W t^a + \frac{1}{2}C^a{}_{a_1 a_2}t^{a_1} \wedge t^{a_2} \right)
    \wedge
    \left(d_W t^b + \frac{1}{2}C^b{}_{b_1 b_2}t^{b_1} \wedge t^{b_2} \right)
    \\
    & =
    P_{a b} t^a \wedge d_W t^b
    +
    \frac{2}{2! 3} P_{a b} t^a \wedge C^b_{b_1 b_2} t^{b_1} \wedge t^{b_2}
    \\
    & =
    P_{a b} t^a \wedge d_W t^b
    +
    \frac{1}{3} C_{a b c} t^a \wedge t^{b} \wedge t^{c}
  \end{aligned}
  \,.
\end{displaymath}
\end{example}
\hypertarget{OriginAndRelatedConcepts}{}\subsection*{{Origin and relation to other concepts}}\label{OriginAndRelatedConcepts}

We discuss the general abstract structures of which Chern-Simons elements are presentations and how they are related to other structures.

The term \emph{Chern-Simons element} alludes to the term \emph{[[Chern-Simons form]]} and [[Chern-Simons theory]]. In the following we explain the relation.

\hypertarget{PresentationForChernWeil}{}\subsubsection*{{As presentations for the $\infty$-Chern-Weil homomorphism}}\label{PresentationForChernWeil}

We explain here briefly how Chern-Simons elements provide a \emph{presentation} of a generalization of the [[Chern-Weil homomorphism]] -- the [[∞-Chern-Weil homomorphism]] in [[cohesive (∞,1)-topos]] theory -- in the sense in which [[(∞,1)-topos]]es have [[presentable (∞,1)-category|presentations]] by a [[model structure on simplicial presheaves]].

To warm up, we start with considering a traditional setup of [[Lie groupoid]] theory. Recall that for $G$ a [[Lie group]], we may form its [[delooping]] [[Lie groupoid]] dnoted $*//G$ or $\mathbf{B}G$. Then with $X$ any [[smooth manifold]], we have that the [[groupoid]] of [[morphism]]s of [[Lie groupoid]]s $X \to \mathbf{B}G$ is equivalent to that of $G$-[[principal bundle]]s on $X$:

\begin{displaymath}
SmoothGrpd(X, \mathbf{B}G) \simeq G Bund(X)
  \,.
\end{displaymath}
Here we are thinking of [[Lie groupoid]]s as [[differentiable stack]]s, hence as [[object]]s in the [[(2,1)-topos]]

\begin{displaymath}
SmoothGrpd := Sh_{(2,1)}(SmoothMfd)
\end{displaymath}
of [[stack]]s/[[(2,1)-sheaves]] on the [[site]] [[SmoothMfd]] (equivalently on its [[small category|small]] [[dense subsite]] [[CartSp]] of [[Cartesian space]]s). (This is discussed in detail at \emph{[[principal bundle]]} ).

There is a \emph{differential refinement} of the [[Lie groupoid]] $\mathbf{B}G$, to the smooth groupoid

\begin{displaymath}
\mathbf{B}G_{conn}
   :=
  SmoothGrpd(\mathbf{P}_1(-), \mathbf{B}G)
  \,,
\end{displaymath}
where $\mathbf{P}_1(X)$ is the [[path groupoid]] of $X$. This is the [[(2,1)-sheaf]] given by the [[(∞,1)-sheafification|(2,1)-sheafification]] of the assignment that sends a [[smooth manifold]] $U$ to the [[groupoid of Lie algebra-valued 1-forms]] on $U$.

There is a corresponding [[natural equivalence]]

\begin{displaymath}
SmoothGrpd(X, \mathbf{B}G_{conn}) \simeq G Bund_{conn}(X)
\end{displaymath}
of morphisms into $\mathbf{B}G_{conn}$ with the groupoid of $G$-[[principal bundle]]s with [[connection on a bundle|with connection]] on $X$. (This is described in detail at \emph{[[connection on a bundle]]} ).

In particular if $G = U(1)$ is the [[circle group]], a morphism $X\to \mathbf{B}U(1)_{conn}$ is a [[circle n-bundle with connection|circle bundle with connection]]. This

This allows already to consider a simple case of a [[characteristic class]] and its refinement to a [[differential characteristic class]]: Let $U$ be the [[unitary group]]. There is a canonical morphism of [[Lie groupoid]]s $\mathbf{c}_1 : \mathbf{B}U \to \mathbf{B}U(1)$ given by the [[determinant]]. This -- or rather its image in [[cohomology]]

\begin{displaymath}
\mathbf{c}_1 : SmoothGrpd(- ,\mathbf{B}U) \to SmoothGrpd(-, \mathbf{B} U(1))
\end{displaymath}
is a smooth representative of the [[characteristic class]] called the \emph{first [[Chern class]]} . Its differential refinement is the evident morphism

\begin{displaymath}
\hat \mathbf{c}_1 : \mathbf{B}U_{conn} \to \mathbf{B}U(1)_{conn}
\end{displaymath}
that sends a $\mathfrak{u}$-[[Lie algebra valued 1-form|valued differential form]] to the [[trace]] of its Lie algebra value. Postcomposition with this is the refined [[Chern-Weil homomorphism]]

\begin{displaymath}
\itexarray{
    SmoothGrpd(X, \mathbf{B}U)_{conn} 
      &\stackrel{\hat \mathbf{c}_1}{\to}&
    SmoothGrpd(X, \mathbf{B}U(1)_{conn})
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
    \\
    U Bund_\nabla(X) &\to& U(1) Bund_\nabla(X)
  }
\end{displaymath}
with values in circle bundles with connection, hence in degree-2 [[ordinary differential cohomology]].

It is this kind of construction on [[Lie groupoid]]s that we now want to generalize to a notion of [[smooth ∞-groupoid]]s, to see that Chern-Simons elements are a means to constructi morphisms akind to the differential first Chern-class $\hat \mathbf{c}_1$.

A general abstract context for [[higher geometry]] equipped with [[differential cohomology]] is a [[cohesive (∞,1)-topos]] $\mathbf{H}$ of [[∞-groupoid]]s equipped with \emph{cohesive structure} , such as [[Smooth∞Grpd|smooth cohesive structure]].

An example for such is the [[∞-stack]]-analog of the [[stack]]-[[(2,1)-topos]] over [[SmoothMfd]]: the [[(∞,1)-sheaf (∞,1)-topos|∞-stack (∞,1)-topos]] [[Smooth∞Grpd]] $:= \hat Sh_{(\infty,1)}(SmoothMfd)$.

In that context we have for instance all the higher [[delooping]]s of $U(1)$: the 

\begin{displaymath}
\mathbf{B}^n U(1) \in Smooth\infty Grpd
  \,.
\end{displaymath}
This is such that the evident generalizations of the above classification statements hold: we have that morphisms $X \to \mathbf{B}^n U(1)$ form an [[n-groupoid]]

\begin{displaymath}
Smooth\infty Grpd(X, \mathbf{B}^n U(1))
   \simeq
  U(1) (n-1)Bund(X)
\end{displaymath}
equivalent to that of [[circle n-bundle with connection|circle n-bundles]]/$(n-1)$-[[bundle gerbe]]s on $X$.

If here $X = \mathbf{B}G$ is again the [[delooping]] of a [[Lie group]], this means that now also the higher [[characteristic class]]es are represented by morphisms

\begin{displaymath}
\mathbf{B}G \to \mathbf{B}^n U(1)
  \,.
\end{displaymath}
For instance for $G = Spin$ the [[spin group]], the first fractional [[Pontryagin class]] has a smooth incarnation given by a morphism of the form

\begin{displaymath}
\frac{1}{2}\mathbf{p}_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)
\end{displaymath}
corresponding under the above equivalence to the ordinary [[Chern-Simons circle 3-bundle]] on $\mathbf{B}G$.

Every [[cohesive (∞,1)-topos]] comes canonically and essentially uniquely equipped with  that we need for the discussion of a refinement of this to [[differential characteristic class]]es:

There is an [[adjoint (∞,1)-functor|endo-(∞,1)-adjunction]]

\begin{displaymath}
(\mathbf{\Pi} \dashv \mathbf{\flat}) : Smooth\infty Grpd \to 
  Smooth \infty Grpd
\end{displaymath}
where

\begin{itemize}%
\item $\mathbf{\Pi}(X)$ is the  of a [[smooth ∞-groupoid]] $X$;


\item $\mathbf{\flat}\mathbf{B}G$ is the coefficient object for  on $G$-[[principal ∞-bundle]]s.



\end{itemize}
A [[morphism]] $\mathbf{\Pi}(X) \to \mathbf{B}^n U(1)$ encodes the flat [[higher parallel transport]] of a flat [[circle n-bundle with connection]], and we have that the [[n-groupoid]] of morphisms

\begin{displaymath}
Smooth \infty Grpd(\mathbf{\Pi}(X), \mathbf{B}^n U(1))
  \simeq
  U(1) n Bund_{\nabla_{flat}}(X)
\end{displaymath}
is that of flat [[circle n-bundles with connection]]/ (n-1)-[[connection on a bundle gerbe|bundle gerbes with connection]].

We observe that a \emph{trivial} circle $n$-bundle with connection is equivalently just a globally defined [[differential form|differential n-form]]. Therefore if we define the modified [[adjoint (∞,1)-functor|(∞,1)-adjunction]]

\begin{displaymath}
(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})
  : 
   */Smooth\infty Grpd
    \stackrel{\leftarrow}{\to}
   Smooth\infty Grpd
\end{displaymath}
by forming the [[(∞,1)-pullback]]

\begin{displaymath}
\mathbf{\flat}_{dR}\mathbf{B}^n U(1) 
  := * \prod_{\mathbf{B}^n U(1)} \mathbf{\flat} \mathbf{B}^n U(1)
  \,,
\end{displaymath}
which is the coefficient object for \emph{trivial} principal $\infty$-bundles equipped with flat $\infty$-connection, one finds (discussed in detail ) that morphisms $X \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ correspond to trivial circle bundle with connection, hence to cocycles in [[de Rham cohomology]] of $X$;

\begin{displaymath}
\pi_0 Smooth \infty Grpd(X, \mathbf{\flat}_{dR} \mathbf{B}^n U(1))
  = 
  \left\{
    \itexarray{
      H_{dR}^n(X) & n \geq 2
      \\
      \Omega^1_{cl}(X) & n = 1
    }
  \right.
  \,.
\end{displaymath}
This now allows us to construct differential refinements:

one can show (detailed discussion is ) that there are canonical cocycles

\begin{displaymath}
curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}
\end{displaymath}
in the degree $(n+1)$-[[de Rham cohomology]] of $\mathbf{B}^n U(1)$: these are the universal [[curvature characteristic form]]s on $\mathbf{B}^n U(1)$.

Then for $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ any smooth [[characteristic class]], the corresponding (unrefined) [[differential characteristic class]] is simply the composite

\begin{displaymath}
\mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to}
   \mathbf{B}^n U(1)
   \stackrel{curv}{\to}
  \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)
  \,.
\end{displaymath}
The (unrefined) [[∞-Chern-Weil homomorphism]] is postcomposition with this morphism:

\begin{displaymath}
(\mathbf{c}_{dR})_* : H^1(X,G) \to H_{dR}^{n+1}(X)
  \,.
\end{displaymath}
This is finally where the Chern-Simons elements come in:

Chern-Simons elements are a means to \emph{present} the composite morphism $\mathbf{c}_{dR}$ of [[smooth ∞-groupoid]]s by an [[∞-anafunctor]] between smooth [[Kan complex]]es.

This presentation we describe in the next section.

(In fact a bit more is true: the serve to present the refinement of $\mathbf{c}_{dR}$ to a morphism $\hat \mathbf{c}$ with values in [[ordinary differential cohomology]]. This we come to further below.)

\hypertarget{ChernSimonsForms}{}\subsubsection*{{Chern-Simons forms}}\label{ChernSimonsForms}

We explain now how Chern-Simons elements arise as a presentation of a [[differential characteristic class]] $\mathbf{c}_{dR}$ by a [[span]] of [[simplicial presheaves]].

At the heart of the presentation of differenial characteristic classes by morphisms of simplicial presheaves is a differential refinement of the [[Lie integration]] of [[L-∞ algebra]]s and [[∞-Lie algebroid]]: for $\mathfrak{g}$ an ordinary [[Lie algebra]], one finds that the 3-[[coskeleton]] of the simplicial presheaf that assigns flat [[vertical differential form|vertical]] $\mathfrak{g}$-[[Lie algebra valued 1-form]]s

\begin{displaymath}
\exp(\mathfrak{g}) : 
  (U,[k])
  \mapsto
  \left\{
     \itexarray{
       \Omega^\bullet_{si, vert}(U \times \Delta^k) 
        &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
     }
  \right\}
\end{displaymath}
is the [[delooping]] of the [[simply connected]] [[Lie group]] $G$ integrating $\mathfrak{g}$

\begin{displaymath}
\mathbf{cosk}_3 \exp(\mathfrak{g}) \stackrel{\simeq}{\to}
  \mathbf{B}G
  \,.
\end{displaymath}
Similarly the [[Lie integration]] of the [[line Lie n-algebra]] $b^{n-1}\mathbb{R}$

\begin{displaymath}
\exp(b^{n-1}\mathbb{R})
  : 
  (U,[k])
  \mapsto
  \left\{
     \itexarray{
       \Omega^\bullet_{si,vert}(U \times \Delta^k)
       &\leftarrow&
       CE(b^{n-1}\mathbb{R})
     }
  \right\}
\end{displaymath}
is the $n$-fold [[delooping]] of $\mathbb{R}$:

\begin{displaymath}
\exp(b^{n-1}\mathbb{R})  \stackrel{\simeq}{\to}
  \mathbf{B}^n \mathbb{R}
  \,.
\end{displaymath}
Moreover, for $\mu : \mathfrak{g} \to b^{n-1}\mathbb{R}$ a degree-$n$ [[cocycle]] in [[Lie algebra cohomology]], simple postcomoposition gives its image under [[Lie integration]]

\begin{displaymath}
\exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{n-1}\mathbb{R})
  \,.
\end{displaymath}
Under [[coskeleton|coskeletization]] on the left this carves out the [[period]]s of $\mu$ as a lattice in $\mathbb{R}$, which typically is the [[integer]]s, so that this descends to degree $n$-cocycle in [[Lie group cohomology]] with coefficients in $U(1) \simeq \mathbb{R}/\mathbb{Z}$

\begin{displaymath}
\exp(\mu) : \mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z}
  \,.
\end{displaymath}
(See [[Lie group cohomology]] and [[smooth ∞-groupoid]] for discussion of the refined notion of Lie group cohomology arising here.)

The differential refinement of these construction is based on the following fact (discussed in detail )

\begin{enumerate}%
\item the object $\mathbf{B}^n U(1) \in$ [[Smooth∞Grpd]] is equivalently presented by a quotient of the presheaf of [[Kan complex]]es given by

\begin{displaymath}
\mathbf{B}^n U(1)_{diff} : 
  (U \in SmoothMfd, [k] \in \Delta)
   \mapsto
  \left\{
    \itexarray{
      \Omega^\bullet_{si,vert}(U \times \Delta^k)
      &\leftarrow& CE(b^{n-1} \mathbb{R})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet_{si}(U \times \Delta^k)
      &\stackrel{\omega}{\leftarrow}&
      W(b^{n-1} \mathbb{R})
    }
  \right\}
  \,,
\end{displaymath}
where on the right we have the set of horizontal morphisms in [[dgAlg]] that make a [[commuting diagram]] with the canonical vertical morphisms as indicated.

\begin{displaymath}
\itexarray{
    \mathbf{B}^n U(1)_{diff}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}^n U(1)
  }
  \,.
\end{displaymath}
(We may think of a morphism of simplicial presheaves $X \to \mathbf{B}^n U(1)_{diff}$ as a [[circle n-bundle with connection|circle n-bundle]]/$(n-1)$-bundle gerbe equipped with a \emph{[[pseudo-connection]]} . )

Notice that the bottom morphism here encodes precisely a degree-$n$ [[differential form]] $\omega$ on $U \times \Delta^k$,


\item The morphism $curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$ is presented on this by the [[∞-anafunctor]]

\begin{displaymath}
\itexarray{
    \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}^n U(1)
  }
\end{displaymath}
by the map that sends such a form $\omega$ to its [[curvature]] $d \omega$. If the [[pseudo-connection]]s that we are dealing with are genuine connections the [[curvature]] is a basic form down on $U$ and this means diagrammatically that it forms the [[pasting]] composite

\begin{displaymath}
\itexarray{
    \Omega^\bullet_{si,vert}(U \times \Delta^k)
    &\leftarrow& CE(b^{n-1} \mathbb{R})
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet_{si}(U \times \Delta^k)
    &\stackrel{\omega}{\leftarrow}&
    W(b^{n-1} \mathbb{R})
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U)
    &\stackrel{d \omega}{\leftarrow}&
    inv(b^{n-1} \mathbb{R})
  }
\end{displaymath}
and then picks out the bottom horizontal morphism.



\end{enumerate}
Therefore our task of presenting $\mathbf{c}_{dR}$ amounts to computing the composition of [[∞-anafunctor]]s

\begin{displaymath}
\itexarray{
    && \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim}
    \\
    && \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}G &\stackrel{\exp(\mu)}{\to}& \mathbf{B}^n U(1)
  }
\end{displaymath}
To do se we need to complete componentwise to [[commuting diagram]]s. To this end we first complete the assignment of $\exp(\mathfrak{g})$ to a diagram

\begin{equation}
(U,[k])
  \mapsto
  \left\{
  \itexarray{
    \Omega^\bullet(U \times \Delta^k)_{vert}
    &\stackrel{A_{vert}}{\leftarrow}&
    CE(\mathfrak{g})
    &&& transition\;function
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U \times \Delta^k)
    &\stackrel{A}{\leftarrow}&
    W(\mathfrak{g})
    &&&
    connection
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U)
    &\stackrel{\langle F_A \rangle}{\leftarrow}&
    inv(\mathfrak{g})
    &&&
    curvature\;characteristic\;form
  }
  
  \right\}
  \,.
\label{ConnectionDiagram}\end{equation}
Here in the middle row an unrestricted $\mathfrak{g}$-[[∞-Lie algebra valued differential form]] appears, which is the local $\mathfrak{g}$-[[connection on an ∞-bundle|∞-connection]]. And in the lower row all its [[curvature characteristic forms]] appear, obtained by evaluating the [[curvature]] $F_A$ in the [[invariant polynomial]]s on $\mathfrak{g}$.

This is such that a choice of Chern-Simons element witnessing the [[transgression]] of an [[invariant polynomial]] to $\mu$ allows to refine $\exp(\mu)$ to

\begin{equation}
\cdots
  \stackrel{\exp(\mu)_{conn}}{\mapsto}
  \left\{
  \itexarray{
    \Omega^\bullet(U \times \Delta^k)_{vert}
    &\stackrel{A_{vert}}{\leftarrow}&
    CE(\mathfrak{g})
    &\stackrel{\mu}{\leftarrow}&
    CE(b^{n-1}\mathbb{R})
    &&& characteristic\;class
    \\
    \uparrow && \uparrow  && \uparrow
    \\
    \Omega^\bullet(U \times \Delta^k)
    &\stackrel{A}{\leftarrow}&
    W(\mathfrak{g})
    &\stackrel{cs}{\leftarrow}&
    W(b^{n-1}\mathbb{R})
    &&&
    Chern-Simons\;form
    \\
    \uparrow && \uparrow && \uparrow
    \\
    \Omega^\bullet(U)
    &\stackrel{\langle -\rangle}{\leftarrow}&
    inv(\mathfrak{g})
    &\stackrel{\langle -\rangle}{\leftarrow}&
    inv(b^{n-1}\mathbb{R})
    &&&
    curvature\;characteristic\;form
  }
  
  \right\}
  \,.
\label{ChernWeilDiagram}\end{equation}
Here now the middle row is the evaluationn of the connection form inside the Chern-Simons element. This is the corresponding [[Chern-Simons form]]

\begin{displaymath}
\Omega^\bullet(U) \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{cs}{\leftarrow}
  W(b^{n-1}\mathbb{R})
  :
  CS(A)
\end{displaymath}
of the $\mathfrak{g}$-connection evaluated in the given Chern-Simons element. Its [[curvature]] is the [[curvature characteristic form]] $\langle F_A \rangle$ appearing in the bottom line of the diagram, which is obtained by evaluating the $\mathfrak{g}$-valued [[curvature]] in the given [[invariant polynomial]].

A more comprehensive account of this is at [[Chern-Weil homomorphism in Smooth∞Grpd]].

\hypertarget{chernsimons_action_functionals}{}\subsubsection*{{Chern-Simons action functionals}}\label{chernsimons_action_functionals}

By the above construction, every Chern-Simons element $cs \in W(\mathfrak{a})$ of degree $d$ on an [[∞-Lie algebroid]] $\mathfrak{a}$ induces an [[action functional]] on the space of [[∞-Lie algebroid valued forms]] on $\mathfrak{a}$ over a $d$-[[dimensional]] [[smooth manifold]] $\Sigma$

\begin{displaymath}
S_{cs} : \Omega(\Sigma, \mathfrak{a}) \to \mathbb{R}
\end{displaymath}
given by

\begin{displaymath}
(A,B,C, \cdots) \mapsto \int_\Sigma CS(A,B,C, \cdots)
  \,.
\end{displaymath}
This generalizes the action functional of ordinary [[Chern-Simons theory]] to general Chern-Simons elements. In the \hyperlink{Examples}{examples} below is a list of various [[quantum field theories]] that arise as generalized Chern-Simons theories this way.

For more details see [[schreiber:infinity-Chern-Simons theory]].

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

\hypertarget{StandardCS}{}\subsubsection*{{On semisimple Lie algebra-- Standard Chern-Simons action functional}}\label{StandardCS}

Let $\mathfrak{g}$ be a [[semisimple Lie algebra]]. For the following computations, choose a [[basis]] $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{r^a\}$ denotes the corresponding degree-shifted basis of $\mathfrak{g}^*[1]$.

Notice that in terms of this the differential of the CE-algebra is

\begin{displaymath}
d_{CE(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c
\end{displaymath}
and that of the Weil algebra

\begin{displaymath}
d_{W(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c
  + r^a
\end{displaymath}
and

\begin{displaymath}
d_{W(\mathfrak{g})} : r^a \mapsto  -C^a{}_{b c} t^b \wedge r^c
  \,.
\end{displaymath}
Let $P_{a b} r^a \wedge r^b \in W(\mathfrak{g})$ be the [[Killing form]] invariant polynomial. This being invariant

\begin{displaymath}
d_{W(\mathfrak{g})} P_{a b} r^a \wedge r^b 
  =
  2 P_{a b} C^{a}{}_{d e} t^d \wedge r^e \wedge r^b = 0
\end{displaymath}
is equivalent to the fact that the coefficients

\begin{displaymath}
C_{a b c} := P_{a a'}C^{a'}{}_{b c}
\end{displaymath}
are skew-symmetric in $a$ and $b$, and therefore skew in all three indices.

\begin{uprop}
A Chern-Simons element for the [[Killing form]] invariant polynomial $\langle -, - \rangle = P(-,-)$ is

\begin{displaymath}
\begin{aligned}
    cs &=
    P_{a b} t^a \wedge (d_{W(\mathfrak{g})} t^b) 
    + 
    \frac{1}{3} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c 
    \\
    & =
    P_{a b} t^a \wedge r^b
    -
    \frac{1}{6} 
    P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c 
  \end{aligned}
  \,.
\end{displaymath}
In particular the Killing form $\langle -,-\rangle$ is in transgression with the degree 3-cocycle

\begin{displaymath}
\mu = -\frac{1}{6}\langle -,[-,-]\rangle
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
We compute

\begin{displaymath}
\begin{aligned}
    d_{W(\mathfrak{g})}
    \left(
       P_{a b} t^a \wedge r^b
       +
       \frac{1}{2}P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c        
    \right)
    & = 
      P_{a b} r^a \wedge r^b
      \\
      & -\frac{1}{2} P_{a b} C^a{}_{d e} t^d \wedge t^e \wedge r^b
      \\
      & + P_{a b} C^b{}_{d e} t^a \wedge t^d \wedge r^e
      \\
      & - \frac{3}{6} 
       P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge r^c        
      \\
      & =
      P_{a b} r^a \wedge r^b
      \\
      & + \frac{1}{2}C_{a b c} t^a \wedge t^b \wedge r^c
      \\
      & - \frac{1}{2} 
       C_{a b c} t^a \wedge t^b \wedge r^c        
      \\
      & = P_{a b } r^a \wedge r^b
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
Under a [[Lie algebra-valued form]]

\begin{displaymath}
\Omega^\bullet(X) \stackrel{}{\leftarrow} W(\mathfrak{g}) : A
\end{displaymath}
this Chern-Simons element is sent to

\begin{displaymath}
cs(A) = P_{a b} A^a \wedge d A^b + \frac{1}{3} C_{a b c} A^a \wedge A^b \wedge A^c
  \,.
\end{displaymath}
If $\mathfrak{g}$ is a [[matrix Lie algebra]] then the Killing form is the [[trace]] and this is equivalently

\begin{displaymath}
cs(A) = tr(A \wedge d A) + \frac{2}{3} tr(A \wedge A \wedge A)
  \,.
\end{displaymath}
This is a familiar form of the standard [[Chern-Simons form]] in degree 3.

For $\Sigma$ a 3-[[dimensional]] [[smooth manifold]] the corresponding [[action functional]] $S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}$

\begin{displaymath}
S_{CS} : A \mapsto \int_\Sigma cs(A)
\end{displaymath}
is the standard action functional of [[Chern-Simons theory]].

\hypertarget{higher_csforms_on_semisimple_lie_algebras}{}\subsubsection*{{Higher CS-forms on semisimple Lie algebras}}\label{higher_csforms_on_semisimple_lie_algebras}

For $\mu \in CE(\mathfrak{g})$ any higher order cocycle, $CS_\mu(A)$ is the corresponding higher order Chern-Simons form.

\hypertarget{action_functional_for_chernsimons_supergravity}{}\paragraph*{{Action functional for Chern-Simons (super-)gravity}}\label{action_functional_for_chernsimons_supergravity}

Higher Chern-Simons elements on the [[Poincare Lie algebra]] $\mathfrak{g} = \mathfrak{iso}(d,1)$ or the [[super Poincare Lie algebra]] $\mathfrak{g} = \mathfrak{siso}(d,1)$ yield [[action functional]]s for [[gravity]] and [[supergravity]].

(\ldots{})

See (\hyperlink{Zanelli}{Zanelli}).

\hypertarget{fractional_secondary_pontryagin_classes}{}\paragraph*{{Fractional secondary Pontryagin classes}}\label{fractional_secondary_pontryagin_classes}

For instance for $\mu_7$ the 7-cocycle on a [[semisimple Lie algebra]], $CS_{\mu_7}(A)$ is the corresponding Chern-Simons 7-form, corresponding to the second [[Pontryagin class]].

Notice that this we may also think of as a 7-cocycle on the corresponding [[string Lie 2-algebra]]. As such it is the one that classifies the extension to the [[fivebrane Lie 6-algebra]]. The corresponding Chern-Simons 7-form appears as the local conneciton data in the [[Chern-Simons circle 7-bundle with connection]] that obstructions the lift from a [[differential string structure]] to a [[differential fivebrane structure]].

\hypertarget{BF}{}\subsubsection*{{On strict Lie 2-algebras -- BF-theory action functional}}\label{BF}

\begin{uprop}
Let $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1$ be a [[strict Lie 2-algebra]].

Then

\begin{itemize}%
\item every [[invariant polynomial]] $\langle -\rangle_{\mathfrak{g}_1} \in inv(\mathfrak{g}_1)$ on $\mathfrak{g}_1$ is a Chern-Simons element on $\mathfrak{g}$, restricting to the trivial [[∞-Lie algebra cocycle]];


\item for $\mathfrak{g}_1$ a [[semisimple Lie algebra]] and $\langle - \rangle_{\mathfrak{g}_1}$ the [[Killing form]], the corresponding Chern-Simons action functional on [[∞-Lie algebra valued forms]]

\begin{displaymath}
\Omega^\bullet(X) \stackrel{(A,B)}{\leftarrow}
  W(\mathfrak{g}_2 \to \mathfrak{g}_1)
  \stackrel{(\langle - \rangle_{\mathfrak{g}_1}, d_W \langle - \rangle_{\mathfrak{g}_1} )}{\leftarrow}
  W(b^{n-1} \mathbb{R})
\end{displaymath}


\end{itemize}
is the sum of the [[action functional]]s of [[topological Yang-Mills theory]] with [[BF-theory]] with [[cosmological constant]] (in the sense of [[gravity as a BF-theory]]):

\begin{displaymath}
CS_{\langle-\rangle_{\mathfrak{g}_1}}(A,B) 
  = 
  \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1}
  - 
  2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1}
  +
  2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1}   
  \,,
\end{displaymath}
where $F_A$ is the ordinary [[curvature]] 2-form of $A$.

\end{uprop}
This is from (\hyperlink{SSSI-BFtheory}{SSSI}).

\begin{proof}
For $\{t_a\}$ a [[basis]] of $\mathfrak{g}_1$ and $\{b_i\}$ a basis of $\mathfrak{g}_2$ we have

\begin{displaymath}
d_{W(\mathfrak{g})} : \sigma t^a \mapsto 
  d_{W(\mathfrak{g}_1)}
  + 
  \partial^a{}_i \sigma b^i
  \,.
\end{displaymath}
Therefore with $\langle -\rangle_{\mathfrak{g}_1} = P_{a_1 \cdots a_n} \sigma r^{a_1} \wedge \cdots \sigma t^{a_n}$ we have

\begin{displaymath}
d_{W(\mathfrak{g})} \langle - \rangle_{\mathfrak{g}_1}
  = 
  n P_{a_1 \cdots a_n}\partial^{a_1}{}_i  \sigma b^{i} \wedge \cdots \sigma t^{a_n}
  \,.
\end{displaymath}
The right hand is a polynomial in the shifted generators of $W(\mathfrak{g})$, and hence an [[invariant polynomial]] on $\mathfrak{g}$. Therefore $\langle - \rangle_{\mathfrak{g}_1}$ is a Chern-Simons element for it.

Now for $(A,B)$ an [[∞-Lie algebra-valued form]], we have that the 2-form curvature is

\begin{displaymath}
F_{(A,B)}^1 = F_A - \partial B
  \,.
\end{displaymath}
Therefore

\begin{displaymath}
\begin{aligned}
    CS_{\langle -\rangle_{\mathfrak{g}_1}}(A,B)
    & = \langle F_{(A,B)}^1\rangle_{\mathfrak{g}_1}
    \\
    & = 
    \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1}
    - 
    2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1}
    +
    2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1}   
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{Symplectic}{}\subsubsection*{{On symplectic $\infty$-Lie algebroids -- The AKSZ Lagrangian}}\label{Symplectic}

A [[symplectic Lie n-algebroid]] $(\mathfrak{a}, \omega)$ is a [[∞-Lie algebroid|Lie n-algebroid]] $\mathfrak{a}$ equipped with a binary non-degenerate [[invariant polynomial]] $\omega \in W(\mathfrak{a})$ of degree $n+2$.

The corresponding Chern-Simons elements of $\omega$ are the integrands for the [[action functional]]s of various [[TQFT]] [[sigma-model]]s.

With $\{-,-\} : CE(\mathfrak{a}) \otimes CE(\mathfrak{a}) \to CE(\mathfrak{a})$ the graded [[Poisson bracket]] induced by $\omega$ we have (see \hyperlink{Roytenberg}{Roytenberg}) that there exists a [[∞-Lie algebra cocycle]] $\mu \in CE(\mathfrak{a})$ such that

\begin{displaymath}
d_{CE(\mathfrak{a})} = \{\mu, -\}
  \,.
\end{displaymath}
So in particular $\mu$ being a cocycle means that

\begin{displaymath}
d_{CE(\mathfrak{a})} \mu = \{\mu, \mu\} = 0
  \,.
\end{displaymath}
\begin{uprop}
The cocycle $\mu$ is in transgression with the invariant polynomial $\frac{n}{2}\omega$ via the Chern-Simons element

\begin{displaymath}
\begin{aligned}
    cs &= \frac{1}{2 }\iota_{\epsilon} \omega - \mu
  \end{aligned}
  \,,
\end{displaymath}
where $\epsilon$ is the Euler vector field (\hyperlink{Roytenberg}{Roytenberg}).

\end{uprop}
Here $\mathbf{d}$ is the shift derivation in the [[Weil algebra]], in that $d_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d}$.

\begin{proof}
To safe typing signs, we write as if all functions were even graded. By standard reasoning the computation holds true then also for arbitrary grading.

Observe that

\begin{enumerate}%
\item on unshifted generators we have

\begin{displaymath}
(\mathbf{d}x^b) \omega_{a b} \{x^a  , -\} = \mathbf{d}
\end{displaymath}

\item we have graded commutators

\begin{itemize}%
\item $[\mathbf{d}, \iota_v] = N$ (the degree operator)

\end{itemize}
and

\begin{itemize}%
\item $[d_{CE(\mathfrak{a})}, \iota_v] = -d_{CE(\mathfrak{a})}\mathbf{d}^{-1}$.

\end{itemize}
(as one checks on generators).



\end{enumerate}
Therefore

\begin{displaymath}
\begin{aligned}
    d_{W(\mathfrak{a})} \frac{1}{2}\iota_{v} \omega
    &= 
    [d_{W(\mathfrak{a})}, \iota_{v}] \frac{1}{2}\omega
    \\
    & =
    (n - d_{CE(\mathfrak{a})} \mathbf{d}^{-1} ) \frac{1}{2}\omega
    \\
    & =
    \frac{n}{2}\omega - \omega_{a b} \{\mu, x^a\} \mathbf{d}x^b
    \\
    &= \frac{1}{2} \omega + \mathbf{d}\mu
  \end{aligned}
  \,,
\end{displaymath}
where in the first line we used that by definition of [[invariant polynomial]] $d_{W(\mathfrak{a})} \omega = 0$. Similarly, using that by definition $d_{CE(\mathfrak{a})} \mu = 0$ we have

\begin{displaymath}
d_{W(\mathfrak{a})} \mu = \mathbf{d}\mu
  \,.
\end{displaymath}
So in total we have

\begin{displaymath}
d_{W(\mathfrak{a})} (\frac{1}{2} \iota_\epsilon \omega - \mu)
  =
  \frac{1}{2}\omega
  \,.
\end{displaymath}
\end{proof}
\hypertarget{remark_3}{}\paragraph*{{Remark}}\label{remark_3}

In local coordinates where $\omega = \omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b$ we have

\begin{displaymath}
cs = n \omega_{a b} x^a \wedge \mathbf{d}x^b + \mu
  \,.
\end{displaymath}
The Chern-Simons action functional corresponding to this Chern-Simons element on $\mathfrak{a}$ is that considered in [[AKSZ theory]].

Below we spell out some low-dimensional cases explicitly.

\hypertarget{higher_phase_space_hamiltonian_and_lagrangian_mechanics}{}\paragraph*{{Higher phase space ---Hamiltonian and Lagrangian mechanics}}\label{higher_phase_space_hamiltonian_and_lagrangian_mechanics}

The symplectic Lie $n$-algebroid $(\mathfrak{P}, \omega)$ may be thought of as an [[n-symplectic manifold]] that models the [[phase space]] of a physical system.

This means for $(\mathfrak{g},\langle-\rangle) = (\mathfrak{P}, \omega)$ a symplectic Lie $n$-algebroid, the general diagram \eqref{TheDiagram} exhibiting the transgression between cocycles and invariant polynomials via Chern-Simons elements may be labeled in terms of [[Hamiltonian mechanics]], [[Lagrangian mechanics]] and [[symplectic geometry]] as follows

\begin{equation}
\itexarray{
    CE(\mathfrak{P})
    &\stackrel{H}{\leftarrow}&
    CE(b^{n}\mathbb{R})
    &&& Hamiltonian
    \\
    \uparrow && \uparrow
    \\
    W(\mathfrak{P})
    &\stackrel{L}{\leftarrow}&
    W(b^n \mathbb{R})
    &&&
    Lagrangian
    \\
    \uparrow && \uparrow
    \\
    inv(\mathfrak{P})
    &\stackrel{\omega}{\leftarrow}&
    inv(b^n \mathbb{R})
    &&&
    symplectic\;structure
  }
\label{PhysicsDiagram}\end{equation}
=--

See [[Hamiltonian]], [[Lagrangian]], [[symplectic structure]].

\hypertarget{on_a_symplectic_manifold__the_topological_particle}{}\paragraph*{{On a symplectic manifold -- The topological particle}}\label{on_a_symplectic_manifold__the_topological_particle}

For $X$ a [[smooth manifold]] we may regard its cotangent bundle $\mathfrak{a} = T^* X$ as a Lie 0-algebroid and the canonical 2-form $\omega \in W(\mathfrak{a}) = \Omega^\bullet(X)$ as a binary invariant polynomial in degree 2.

The Chern-Simons element is the canonical 1-form $\alpha$ which in local coordinates is $\alpha = p_i d q^i$.

The corresponding action functional on the line

\begin{displaymath}
\int_{\mathbb{R}} \gamma^* (p_i\, d q^i)
\end{displaymath}
is the familiar term for the action functional of the particle (missing the kinetic term, which makes it ``topological'').

\hypertarget{on_a_poisson_lie_algebroid__the_poisson_model}{}\paragraph*{{On a Poisson Lie algebroid -- The Poisson $\sigma$-model}}\label{on_a_poisson_lie_algebroid__the_poisson_model}

\begin{uprop}
Let $\mathfrak{a} = \mathfrak{P}(X,\pi)$ by a [[Poisson Lie algebroid]]. This comes with the canonical [[invariant polynomial]] $\omega = \mathbf{d} \partial_i \wedge \mathbf{d} x^i$.

The corresponding [[∞-Lie algebroid cocycle]] is

\begin{displaymath}
\mu_{\omega} = \pi
\end{displaymath}
and a Chern-Simons element for this is

\begin{displaymath}
cs_\omega = \pi^{i j} \partial_i  \wedge \partial_j 
  + \partial_i \wedge \mathbf{d}x^i
  \,.
\end{displaymath}
For $\Sigma$ a 2-[[dimensional]] [[smooth manifold]] the corresponding [[action functional]] on [[∞-Lie algebroid-valued forms]] $S : \Omega^\bullet(X, \mathfrak{P}(X,\pi)) \to \mathbb{R}$ is the actional functional of the [[Poisson sigma-model]]

\begin{displaymath}
S : (X, \eta) \mapsto \int_\Sigma (\eta \wedge d X + \pi(\eta \wedge \eta)
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
We compute in a local coordinte patch:

\begin{displaymath}
\begin{aligned}
    d_{W(\mathfrak{P}(X,\pi))} (\pi^{i j} \partial_i \wedge \partial_j
   + \partial_i \wedge \mathbf{d} x^i)
    = &
   \mathbf{d}x^k (\partial_k \pi^{i j}) \partial_i \wedge \partial_j
   \\
   & + 2 \pi^{i j} (\mathbf{d}\partial_i) \wedge \partial_j
   \\
   & - (\partial_i \pi^{j k}) \partial_j \wedge \partial_k \wedge 
       \mathbf{d}x^i
   \\
   & +
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
   \\
   & + (-)(-) 2\pi^{i j} \partial_i \wedge \mathbf{d}\partial_j
   \\
   = & 
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{on_higher_extensions_of_the_super_poincare_lie_algebra__supergravity}{}\subsubsection*{{On higher extensions of the super Poincare Lie algebra -- supergravity}}\label{on_higher_extensions_of_the_super_poincare_lie_algebra__supergravity}

See [[D'Auria-Fre formulation of supergravity]] for the moment.

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

\begin{itemize}%
\item [[∞-Lie algebroid cocycle]]


\item \textbf{Chern-Simons element}


\item [[invariant polynomial]]



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

A classical reference on transgression of differential forms from the fiber to the base of a [[fiber bundle]] is section 9 of.

\begin{itemize}%
\item [[Armand Borel]], \emph{Topology of Lie groups and characteristic classes} Bull. Amer. Math. Soc. Volume 61, Number 5 (1955), 397-432. (\href{http://projecteuclid.org/euclid.bams/1183520007}{EUCLID})

\end{itemize}
The general definition of Chern-Simons element on $\infty$-Lie algebras and $\infty$-Lie algebroids is  in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{$L_\infty$-connections} ()

\end{itemize}
The examples of the [[BF-theory]] invariant polynomials and Chern-Simons elements are in  and  and the BF-action functional itself is extracted below .

Dedicated discussion of $\infty$-Chern-Simons theory is at

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]] [[Urs Schreiber]], \emph{[[schreiber:∞-Chern-Simons functionals]]}

\end{itemize}
A comprehensive account is in

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

\end{itemize}
A survey of higher Chern-Simons elements and their action functionals as applied to [[gravity]] and [[supergravity]] is in

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

\end{itemize}
Symplectic Lie $n$-algebroids are discussed in

\begin{itemize}%
\item [[Dmitry Roytenberg]], \emph{Courant algebroids, derived brackets and even symplectic supermanifolds} PhD thesis (\href{http://arxiv.org/abs/math/9910078}{arXiv})

\emph{On the structure of graded symplectic supermanifolds and Courant algebroids} (\href{http://arxiv.org/abs/math/0203110}{arXiv})



\end{itemize}
A talk about the historical origins of the standard Chern-Simons forms see

\begin{itemize}%
\item [[Jim Simons]], \emph{Origin of Chern-Simons} talk at Simons Center for Geometry and Physics (2011) (\href{http://media.scgp.stonybrook.edu/video/video.php?f=20110728_1_qtp.mp4}{video})

\end{itemize}
[[!redirects Chern-Simons elements]]

[[!redirects generalized Chern-Simons theory]] [[!redirects generalized Chern-Simons theories]] [[!redirects ∞-Chern-Simons theory]]



\end{document}