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\begin{document}

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\section*{transgression of differential forms}

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

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration 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{preliminaries_on_smooth_sets}{Preliminaries on smooth sets}\dotfill \pageref*{preliminaries_on_smooth_sets} \linebreak
\noindent\hyperlink{via_parameterized_integration_of_differential_forms}{Via parameterized integration of differential forms}\dotfill \pageref*{via_parameterized_integration_of_differential_forms} \linebreak
\noindent\hyperlink{via_pullback_along_the_evaluation_map}{Via pullback along the evaluation map}\dotfill \pageref*{via_pullback_along_the_evaluation_map} \linebreak
\noindent\hyperlink{TransgressionOfVariationalDifferentialForms}{Transgression of variational differential forms}\dotfill \pageref*{TransgressionOfVariationalDifferentialForms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelativeTransgressionOverManifoldsWithBoundary}{Relative transgression over manifolds with boundary}\dotfill \pageref*{RelativeTransgressionOverManifoldsWithBoundary} \linebreak
\noindent\hyperlink{variational_transgression_picks_out_the_vertical_differential_forms}{Variational transgression picks out the vertical differential forms}\dotfill \pageref*{variational_transgression_picks_out_the_vertical_differential_forms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{GaugeCouplingActionFunctionalOfChargedParticle}{Gauge coupling action functional of charged particle}\dotfill \pageref*{GaugeCouplingActionFunctionalOfChargedParticle} \linebreak
\noindent\hyperlink{TransgressionOfKillingFormToSymplecticFormOfChernSimons}{Transgression of Killing form to symplectic form of Chern-Simons theory}\dotfill \pageref*{TransgressionOfKillingFormToSymplecticFormOfChernSimons} \linebreak


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

Given a [[differential form]] $\omega$ of degree $n$ on some [[smooth space]] $X$ and given a [[closed manifold|closed]] [[smooth manifold]] $\Sigma$ of [[dimension]] $k \leq n$, then there is canonically induced a differential form $\tau_\Sigma \omega$ of degree $n-k$ on the [[mapping space]] $[\Sigma,X]$: its restriction to any smooth family $\Phi_{(-)}$ of smooth functions $\Phi_u \colon \Sigma \to X$ is the result of first forming the [[pullback of differential forms]] of $\omega$ along $\Phi_{(-)}$ and then forming the [[integration of differential forms]] of the result over $\Sigma$:

\begin{displaymath}
\tau_{\Sigma} \omega\vert_{\Phi_{(-)}}
    \coloneqq
  \int_\Sigma (\Phi_{(-)})^\ast \omega
  \,.
\end{displaymath}
This differential form $\tau_\Sigma \omega$ on the mapping space is called the \emph{[[transgression]]} of $\omega$ with respect to $\Sigma$

This construction has a variety of immediate generalizations, for instance $\Sigma$ may have [[manifold with boundary|boundary]] and [[manifold with corners|corners]], and it may be a [[supermanifold]] and/or a [[formal manifold]]; and the [[mapping space]] may be generalized to a [[space of sections]] of a given [[fiber bundle]]. Finally, the construction also generalizes to coefficients richer than differential forms, such as [[cocycles]] in [[differential cohomology]], but this is no longer the topic of the present entry.

Important examples of transgression of differential forms appear in [[Lagrangian field theory]] (in the sense of [[physics]]) defined by a [[Lagrangian density|Lagrangian form]] on the [[jet bundle]] of a [[field bundle]]. Here the transgression of the Lagrangian itself (along [[jet prolongations]] of [[field (physics)|fields]]) is the corresponding \emph{[[action functional]]}, the transgression of its [[Euler-Lagrange operator|Euler-Lagrange variational derivative]] is the 1-form whose vanishing is the \emph{[[equations of motion]]} and the transgression of the induced [[pre-symplectic current]] is the \emph{[[pre-symplectic form]] on the [[covariant phase space]]} of the field theory. These examples are discussed below at \emph{\hyperlink{TransgressionOfVariationalDifferentialForms}{Transgression of variational differential forms}}.

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

There are two definitions of transgression of differential forms: A traditional formulation is def. \ref{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap} below, which transgresses by [[pullback of differential forms]] along the [[evaluation map]], followed by [[integration of differential forms]].

Another definition is useful, which makes more use of the existence of smooth classifying spaces for differential forms in [[smooth sets]], this we consider as def. \ref{ParameterizedIntegrationOfDifferentialForms} below.

That these two definitions are indeed equivalent is the content of prop. \ref{EquivalenceOfTransgressionOfDifferentialFormsToMappingSpaces} below

\hypertarget{preliminaries_on_smooth_sets}{}\subsubsection*{{Preliminaries on smooth sets}}\label{preliminaries_on_smooth_sets}

Since the concept of transgression of differential forms involves [[mapping spaces]] between, in particular, [[smooth manifolds]], it is most conveniently formulated in terms of the concept of [[generalized smooth spaces]] called \emph{[[smooth sets]]}. For the following discussion we assume background on [[smooth sets]] as introduced in

\begin{itemize}%
\item \emph{[[geometry of physics -- smooth sets]]}


\item \emph{[[geometry of physics -- differential forms]]}.



\end{itemize}
(This entry itself here overlaps with \emph{[[geometry of physics -- integration]]}, where more background may be found.)

Recall form the discussion there that a [[smooth set]] $X$ is \emph{defined} by specifying, in a consistent way, what counts as a smooth functions $U \to X$ from a [[Cartesian space]] $U$ (a ``plot'' of $X$). Given two smooth sets $X$ and $Y$ then a [[smooth function]] $f \;\colon\; X \longrightarrow Y$ is a function that takes plots $U \overset{\phi}{\to} X$ of $X$ to plots $f \circ \phi \colon U \to Y$ of $Y$.

A key example of a smooth set which is in general not a [[smooth manifold]] is the [[mapping space]] $[X,Y]$ between two smooth sets $X$ and $Y$, hence the set of all smooth functions $X \to Y$ equipped with a smooth structure itself. Namely a plot $\phi_{(-)} \colon U \to [X,Y]$ is defined to be a smooth function $\phi_{(-)}(-) \colon U \times X \to Y$ out of the [[Cartesian product]] of $U$ with $X$ to $Y$, hence a ``U-parameterized smooth family of smooth functions''.

An example of a smooth set which is far from being a smooth manifold is for $n \in \mathbb{N}$ the smooth set $\mathbf{\Omega}^n$ which is the ``smooth [[classifying space]]'' for [[differential n-forms]], defined by the rule that a smooth function $\phi \colon U \to \mathbf{\Omega}^n$ is equivalently a smooth differential $n$-form on $U$ (to be thought of as the [[pullback of differential forms|pullback]] of a ``universal $n$-form'' on $\mathbf{\Omega}^n$ along $\phi$). It follows from this in particular that for $X$ any [[smooth manifold]] then smooth functions $X \to \mathbf{\Omega}^n$ are equivalent to smooth $n$-forms on $X$. Accordingly we may say that for $X$ any [[smooth set]] (which may be far from being a smooth manifold) then a differential $n$-form on $X$ is equivalently a smooth function $X \to \mathbf{\Omega}^n$. Under this identification the operation of [[pullback of differential forms]] along some smooth function $f \colon Y \to X$ is just [[composition]] of smooth functions $f^\ast \omega \colon Y \overset{f}{\to} X \overset{\omega}{\to} \mathbf{\Omega}^n$.

These examples may be combined: the [[mapping space]] $[\Sigma, \mathbf{\Omega}^n]$ is a kind of smooth classifying space for differential forms \emph{on} $\Sigma$: a smooth function $\omega_{(-)} \colon U \to [\Sigma,\mathbf{\Omega}^n]$ into this space is, by the above, a [[differential n-form]] on the [[Cartesian product]] $U \times \Sigma$.

(There is a smooth space that has more right to be called ``the'' classifying space of differential $n$-foms on $\Sigma$, namely the [[concretification]] $\sharp_1 [\Sigma, \mathbf{\Omega}^n]$, but for the discussion of trangression actually the unconcretified space is the right one to use.)

\hypertarget{via_parameterized_integration_of_differential_forms}{}\subsubsection*{{Via parameterized integration of differential forms}}\label{via_parameterized_integration_of_differential_forms}

\begin{defn}
\label{ParameterizedIntegrationOfDifferentialForms}\hypertarget{ParameterizedIntegrationOfDifferentialForms}{}
\textbf{(parameterized [[integration of differential forms]])}

Let

\begin{enumerate}%
\item $X$ be a [[smooth set]];


\item $n \geq k \in \mathbb{N}$;


\item $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$.



\end{enumerate}
Then we write

\begin{displaymath}
\int_{\Sigma}
    \;\colon\;
  [\Sigma_k, \mathbf{\Omega}^n]
    \longrightarrow
  \mathbf{\Omega}^{n-k}
\end{displaymath}
for the [[smooth function]] which takes a plot $\omega_{(-)} \colon U \to [\Sigma, \mathbf{\Omega}^k]$, hence equivalently a differential $n$-form $\omega_{(-)}(-)$ on $U \times \Sigma$ to the result of [[integration of differential forms]] over $\Sigma$:

\begin{displaymath}
\int_{\Sigma} \omega_{(-)}(-) \coloneqq \int_\Sigma \omega_{(-)}
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{TransgressionOfDifferentialFormsToMappingSpaces}\hypertarget{TransgressionOfDifferentialFormsToMappingSpaces}{}
\textbf{(transgression of differential forms to [[mapping spaces]])}

Let

\begin{enumerate}%
\item $X$ be a [[smooth set]];


\item $n \geq k \in \mathbb{N}$;


\item $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$.



\end{enumerate}
Then the operation of \emph{transgression of differential $n$-forms} on $X$ with respect to $\Sigma$ is the [[function]]

\begin{displaymath}
\tau_\Sigma
   \coloneqq
  \int_\Sigma [\Sigma,-]
   \;\colon\;
  \Omega^n(X) \to \Omega^{n-k}([\Sigma,X])
\end{displaymath}
from differential $n$-forms on $X$ to differential $n-k$-forms on the [[mapping space]] $[\Sigma,X]$ which takes the differential form corresponding to the smooth function

\begin{displaymath}
(X \stackrel{\omega}{\to} \Omega^n) \in \Omega^n(X)
\end{displaymath}
to the differential form corresponding to the following composite smooth function:

\begin{displaymath}
\tau_\Sigma \omega
   \coloneqq
  \int_{\Sigma} [\Sigma,\omega]
   \;\colon\;
  [\Sigma, X] 
    \stackrel{[\Sigma, \omega]}{\to}  
  [\Sigma, \Omega^n]
    \stackrel{\int_{\Sigma}}{\to}
  \Omega^{n-k}
  \,,
\end{displaymath}
where $[\Sigma,\omega]$ is the [[mapping space]] [[functor]] on [[morphisms]] and $\int_{\Sigma}$ is the parameterized integration of differential forms from def. \ref{ParameterizedIntegrationOfDifferentialForms}.

More explicitly in terms of plots this means equivalently the following

A plot of the [[mapping space]]

\begin{displaymath}
\phi_{(-)} \;\colon\; U \to [\Sigma, X]
\end{displaymath}
is equivalently a [[smooth function]] of the form

\begin{displaymath}
\phi_{(-)}(-) \;\colon\; U \times \Sigma \to X
  \,.
\end{displaymath}
The smooth function $[\Sigma,\omega]$ takes this smooth function to the plot

\begin{displaymath}
U \times \Sigma \to X
   \overset{\phi_{(-)}(-)}{\longrightarrow}
  X 
   \overset{\omega}{\longrightarrow}
  \mathbf{\Omega}^{n}
\end{displaymath}
which is equivalently a differential form

\begin{displaymath}
(\phi_{(-)}(-))^\ast \omega \in \Omega^n(U \times \Sigma)
  \,.
\end{displaymath}
Finally the smooth function $\int_\Sigma$ takes this to the result of [[integration of differential forms]] over $\Sigma$:

\begin{displaymath}
\tau_{\Sigma}\omega\vert_{\phi_{(-)}}
  \;=\;
  \int_\Sigma (\phi_{(-)}(-))^\ast \omega
  \;\in\;
  \Omega^{n-k}(U)
  \,.
\end{displaymath}
\end{defn}
\hypertarget{via_pullback_along_the_evaluation_map}{}\subsubsection*{{Via pullback along the evaluation map}}\label{via_pullback_along_the_evaluation_map}

\begin{defn}
\label{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap}\hypertarget{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap}{}
\textbf{(transgression of differential forms to mapping space via evaluation map)}

Let

\begin{enumerate}%
\item $X$ be a [[smooth set]];


\item $n \geq k \in \mathbb{N}$;


\item $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$.



\end{enumerate}
Then the operation of \emph{transgression of differential $n$-forms} on $X$ with respect to $\Sigma$ is the [[function]]

\begin{displaymath}
\tau_\Sigma
   \coloneqq
  \int_\Sigma ev^\ast
   \;\colon\;
  \Omega^n(X)
    \overset{ev^\ast}{\longrightarrow} 
  \Omega^n(\Sigma \times [\Sigma, X])
    \overset{\int_\Sigma}{\longrightarrow}
  \Omega^{n-k}([\Sigma,X])
\end{displaymath}
from differential $n$-forms on $X$ to differential $n-k$-forms on the [[mapping space]] $[\Sigma,X]$ which is the [[composition|composite]] of forming the [[pullback of differential forms]] along the [[evaluation map]] $ev \colon  [\Sigma, X] \times \Sigma \to X$ with [[integration of differential forms]] over $\Sigma$.

\end{defn}
\begin{prop}
\label{EquivalenceOfTransgressionOfDifferentialFormsToMappingSpaces}\hypertarget{EquivalenceOfTransgressionOfDifferentialFormsToMappingSpaces}{}
The two definitions of transgression of differential forms to mapping spaces from def. \ref{TransgressionOfDifferentialFormsToMappingSpaces} and def. \ref{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap} are equivalent.

\end{prop}
\begin{proof}
We need to check that for all plots $\gamma \colon U \to [\Sigma, X]$ the pullbacks of the two forms to $U$ coincide.

For def. \ref{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap} we get

\begin{displaymath}
\gamma^\ast \int_\Sigma \mathrm{ev}^\ast A
  =
  \int_\Sigma (\gamma,\mathrm{id}_\Sigma)^\ast \mathrm{ev}^\ast A
  \;
  \in \Omega^n(U)
\end{displaymath}
Here we recognize in the integrand the pullback along the $( (-)\times \Sigma \dashv [\Sigma,-])$-[[adjunct]] $\tilde \gamma : U \times \Sigma \to \Sigma$ of $\gamma$, which is given by applying the [[left adjoint]] $(-)\times \Sigma$ and then postcomposing with the adjunction counit $\mathrm{ev}$:

\begin{displaymath}
\itexarray{
    U \times \Sigma
    &
     \overset{(\gamma, \mathrm{id}_\Sigma)}{\longrightarrow}
    &
    [\Sigma,X] \times \Sigma
    &
      \overset{\mathrm{ev}}{\longrightarrow}
    &
    X
  }
  \,.
\end{displaymath}
Hence the integral is now

\begin{displaymath}
\cdots = \int_{\Sigma} \tilde \gamma^\ast A
  \,.
\end{displaymath}
This is the operation of the top horizontal composite in the following [[natural transformation|naturality square]] for [[adjuncts]], and so the claim follows by its [[commuting diagram|commutativity]]:

\begin{displaymath}
\itexarray{
    \tilde \gamma \in 
    & 
    \mathbf{H}(U \times\Sigma, X)
    &
      \overset{\mathbf{H}(U \times \Sigma,A)}{\longrightarrow}
    &    
    \mathbf{H}(U \times \Sigma, \mathbf{\Omega}^{n+k})
    &
      \overset{\int_\Sigma(U)}{\longrightarrow}
    &
    \Omega^n(U)
    \\
    &
    {}^{\mathllap{\simeq}}\downarrow
      &&
    {}^{\mathllap{\simeq}}\downarrow
      &&
    {}^{\mathllap{\simeq}}\downarrow
    \\
    \gamma \in 
    & 
    \mathbf{H}(U,[\Sigma,X])
    &
      \overset{\mathbf{H}(U,[\Sigma,A])}{\longrightarrow}
    &
    \mathbf{H}(U,[\Sigma,\mathbf{\Omega}^{n+k}])
    &
      \overset{\mathbf{H}(U,\int_\Sigma)}{\longrightarrow}
    & 
    \mathbf{H}(U,\mathbf{\Omega}^n)
  }
\end{displaymath}
(here we write $\mathbf{H}(-,-)$ for the [[hom functor]] of [[smooth sets]]).

\end{proof}
\hypertarget{TransgressionOfVariationalDifferentialForms}{}\subsubsection*{{Transgression of variational differential forms}}\label{TransgressionOfVariationalDifferentialForms}

An important variant of transgression of differential forms is the transgression of [[variational bicomplex|variational differential forms]] along [[jet prolongation]].

In the following let $\Sigma$ be a fixed [[smooth manifold]]. We will refer to this as ``[[spacetime]]'', but for the present purpose it may be an smooth manifold without further structure.

\begin{defn}
\label{Fields}\hypertarget{Fields}{}
\textbf{([[field (physics)|fields]] and their [[space of histories]])}

Given a [[spacetime]] $\Sigma$, then a \emph{[[type]] of fields} on $\Sigma$ is a [[smooth set|smooth]] [[fiber bundle]]

\begin{displaymath}
\itexarray{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }
\end{displaymath}
called the \emph{[[field bundle]]},

Given a type of fields on $\Sigma$ this way, then a \emph{[[field (physics)|field]] [[trajectory]]} (or \emph{field history}) of that type on $\Sigma$ is a smooth [[section]] of this [[bundle]], namely a [[smooth function]] of the form

\begin{displaymath}
\Phi \colon \Sigma \longrightarrow E
\end{displaymath}
such that composed with the [[projection]] map it is the [[identity function]], i.e. such that

\begin{displaymath}
fb \circ \Phi = id
  \phantom{AAAAAAA}
  \itexarray{
    && E
    \\
    & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    \Sigma & = & \Sigma
  }
  \,.
\end{displaymath}
The corresponding \emph{field [[space of histories]]} is the [[smooth set|smooth]] [[space of sections|space of all these]], to be denoted

\begin{displaymath}
\Gamma_\Sigma(E) \in \mathbf{H}
  \,.
\end{displaymath}
This is a [[smooth set]] by declaring that a smooth family $\Phi_{(-)}$ of field configurations, parameterized over any [[Cartesian space]] $U$ is a smooth function

\begin{displaymath}
\itexarray{
    U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
    \\
    (u,x) &\mapsto& \Phi_u(x)
  }
\end{displaymath}
such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.

\begin{displaymath}
\itexarray{
    && E
    \\
    & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
  }
  \,.
\end{displaymath}
More generally, let $S \hookrightarrow \Sigma$ be a [[submanifold]] of spacetime. We write $N_\Sigma(S) \hookrightarrow \Sigma$ for its [[infinitesimal neighbourhood]] in $\Sigma$.

If $E \overset{fb}{\to} \Sigma$ is a [[field bundle]] then the \emph{[[space of histories]] of fields restricted to $S$}, to be denoted

\begin{displaymath}
\Gamma_{S}(E) \coloneqq \Gamma_{N_\Sigma(S)}( E\vert_{N_\Sigma S} ) \in \mathbf{H}
\end{displaymath}
is the [[space of sections]] restricted to the [[infinitesimal neighbourhood]] $N_\Sigma(S)$.

There is a canonical [[evaluation]] [[smooth function]]

\begin{equation}
ev_S
   \;\colon\;
  N_\Sigma S \times \Gamma_{S}(E)
    \longrightarrow
  E
\label{FieldEvaluation}\end{equation}
which takes a [[pair]] consisting of an [[generalized element|element]] in $N_\Sigma S$ and a field configuration to the value of the field configuration at that point.

\end{defn}
\begin{defn}
\label{SpacetimeSupport}\hypertarget{SpacetimeSupport}{}
\textbf{(spacetime support)}

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$, with induced [[jet bundle]] $J^\infty_\Sigma(E)$

For every [[subset]] $S \subset \Sigma$ let

\begin{displaymath}
\itexarray{
    J^\infty_\Sigma(E)\vert_S
      &\overset{\iota_S}{\hookrightarrow}&
    J^\infty_\Sigma(E)
    \\
    \downarrow &(pb)& \downarrow
    \\
    S &\hookrightarrow& \Sigma
  }
\end{displaymath}
be the corresponding restriction of the [[jet bundle]] of $E$.

The \emph{spacetime support} $supp_\Sigma(A)$ of a [[differential form]] $A \in \Omega^\bullet(J^\infty_\Sigma(E))$ on the [[jet bundle]] of $E$ is the [[topological closure]] of the maximal subset $S \subset \Sigma$ such that the restriction of $A$ to the jet bundle restrited to this subset vanishes:

\begin{displaymath}
supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma |  \iota_{\{x\}^\ast A = 0} \} )
\end{displaymath}
We write

\begin{displaymath}
\Omega^{r,s}_{\Sigma,cp}(E)
  \coloneqq
  \left\{
    A \in \Omega^{r,s}_\Sigma(E)
    \;\vert\;
    supp_\Sigma(A) \, \text{is compact}
  \right\}
    \;\hookrightarrow\;
  \Omega^{r,s}_\Sigma(E)
\end{displaymath}
for the subspace of differential forms on the jet bundle whose spacetime support is a [[compact subspace]].

\end{defn}
\begin{defn}
\label{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}\hypertarget{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}{}
\textbf{(transgression of variational differential forms to [[field (physics)|field]] [[space of histories]])}

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$ (def. \ref{Fields}), with induced [[jet bundle]] $J^\infty_\Sigma(E)$

For $\Sigma_r \hookrightarrow \Sigma$ be a submanifold of [[spacetime]] of dimension $r \in \mathbb{N}$, then \emph{transgression of variational differential forms to $\Sigma_r$} is the function

\begin{displaymath}
\tau_{\Sigma_r}
   \;\colon\;
  \Omega^{r,\bullet}_{\Sigma,cp}(E)
    \overset{  }{\longrightarrow}
  \Omega^\bullet\left( 
    \Gamma_{\Sigma_r}(E) 
   \right)
\end{displaymath}
which sends a differential form $A \in \Omega^{r,\bullet}_{\Sigma,cp}(E)$ to the differential form $\tau_{\Sigma_r} \in \Omega^\bullet(\Gamma_{\Sigma_r}(E))$ which to a smooth family on field configurations

\begin{displaymath}
\Phi_{(-)} \;\colon\; U \times N_\Sigma \Sigma_r \longrightarrow E
\end{displaymath}
assigns the differential form given by first forming the [[pullback of differential forms]] along the family of [[jet prolongation]] $j^\infty_\Sigma(\Phi_{(-)})$ followed by the [[integration of differential forms]] over $\Sigma_r$:

\begin{displaymath}
(\tau_{\Sigma}A)_{\Phi_{(-)}}
    \;\coloneqq\;
  \int_{\Sigma_r}  (j^\infty_\Sigma(\Phi_{(-)}))^\ast 
  \;\in\;
  \Omega^\bullet(U)
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelativeTransgressionOverManifoldsWithBoundary}{}\subsubsection*{{Relative transgression over manifolds with boundary}}\label{RelativeTransgressionOverManifoldsWithBoundary}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(relative transgression over [[manifolds with boundary]])}

\begin{enumerate}%
\item $X$ be a [[smooth set]];


\item $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$ with [[manifold with boundary|boundary]] $\partial \Sigma$


\item $n \geq k \in \mathbb{N}$;


\item $\omega \in \Omega^n_{X}$ a [[closed differential form]].



\end{enumerate}
Write

\begin{displaymath}
(-)\vert_{\partial \Sigma}
    \;\coloneqq\;
  [\partial \Sigma \hookrightarrow \Sigma, X]
  \;\colon\;
  [\Sigma, X]
   \longrightarrow
  [\partial \Sigma, X]
\end{displaymath}
for the smooth function that restricts smooth functions on $\Sigma$ to smooth functions on the [[boundary]] $\partial \Sigma$.

Then the operations of transgression of differential forms (def. \ref{TransgressionOfDifferentialFormsToMappingSpaces}) to $\Sigma$ and to $\partial \Sigma$, respectively, are related by

\begin{displaymath}
d \left(
    \tau_{\Sigma}(\omega)
  \right)
  = 
  (-1)^{k+1}
  ((-)\vert_{\partial \Sigma})^\ast \tau_{\partial \Sigma}(\omega)
    \phantom{AAAAAAAA}
  \itexarray{
    [\Sigma, X] 
      &\overset{ \tau_{\Sigma}(\omega) }{\longrightarrow}&
    \mathbf{\Omega}^{n-k}
    \\
    {}^{\mathllap{(-)\vert_{\partial \Sigma} }}\downarrow
      &&
    \downarrow^{\mathrlap{ (-1)^{k+1} d}}
    \\
    [\partial \Sigma, X]
      &\underset{ \tau_{\partial\Sigma}(\omega) }{\longrightarrow}&
    \mathbf{\Omega}^{n-k+1}
  }
  \,.
\end{displaymath}
In particular this means that if the compact manifold $\Sigma$ happens to have no boundary (is a [[closed manifold]]) then transgression over $\Sigma$ takes closed differential forms to closed differential forms.

\end{prop}
\begin{proof}
Let $\phi_{(-)}(-) \colon U \times \Sigma \to X$ be a plot of the mapping space $[\Sigma, X]$. Notice that the [[de Rham differential]] on the [[Cartesian product]] $U \times \Sigma$ decomposes as

\begin{displaymath}
d = d_U + d_\Sigma
  \,.
\end{displaymath}
Now we compute as follows:

\begin{displaymath}
\begin{aligned}
    d \tau_{\Sigma}\omega\vert_{\phi_(-)}
    & =
    d_U \int_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma d_U (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma (d - d \Sigma) (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma d (\phi_{(-)}(-))^\ast \omega
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma (\phi_{(-)}(-))^\ast \underset{= 0}{\underbrace{d \omega}}
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -(-1)^k \int_{\partial \Sigma} (\phi_{(-)}(-))^\ast \omega   
    \\
    & = 
    -(-1)^k \tau_{\partial \Sigma} \omega \vert_{\phi_{(-)}}
  \end{aligned}
\end{displaymath}
where in the second but last step we used [[Stokes' theorem]].

\end{proof}
\hypertarget{variational_transgression_picks_out_the_vertical_differential_forms}{}\subsubsection*{{Variational transgression picks out the vertical differential forms}}\label{variational_transgression_picks_out_the_vertical_differential_forms}

\begin{example}
\label{}\hypertarget{}{}
\textbf{(some transgressions of variational differential forms)}

We spell out the result transgression of variational differential forms (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of some variational differential forms on the [[jet bundle]] of a trivial vector field bundle to the [[space of histories]] $\Gamma_\Sigma(E)$ of fields (def. \ref{Fields}).

We describe the resulting transgressed differential forms restricted to any smooth family of field configurations

\begin{displaymath}
\Phi_{(-)} \;\colon\; U \times \Sigma \longrightarrow E
  \,.
\end{displaymath}
Let $b \in C^\infty_{cp}(\Sigma)$ be any [[bump function]] on spacetime. Its product with the [[volume form]] (as in example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}) is then a horizontal $p+1$-form on the jet bundle with compact spacetime support.

\begin{displaymath}
b dvol_\Sigma \in \Omega^{0,0}_{\Sigma,cp}(E)
\end{displaymath}
The transgression of this 0-form to the [[space of histories]] of fields

\begin{displaymath}
\tau_\Sigma (b dvol_\Sigma) \in \Omega^0( \Gamma_\Sigma(E) )
\end{displaymath}
is the differential form on $\Gamma_\Sigma(E)$ which restricted to the given family of field configurations $\Phi_{(-)} \colon u \mapsto \Phi_u$ yields the function

\begin{displaymath}
\tau_\Sigma (b dvol_\Sigma)\vert_\Phi \colon u \mapsto \int_\Sigma b dvol_\Sigma
\end{displaymath}
which is simply the constant function with value the integral of $b$ against the given volume form.

The constancy of this function is due to the fact that $b dvol_\Sigma$ does not depend on the field variables. So consider next the horizontal $(p+1)$-form

\begin{displaymath}
\phi^a \, b dvol_\Sigma \; \in \Omega^{p+1,0}( E )
  \,.
\end{displaymath}
Its transgression is the function

\begin{displaymath}
\tau_\Sigma( \phi^a \, b dvol_\Sigma  )_\Phi
   \;=\;
   \left(
     u
       \mapsto
     \int_\Sigma \Phi^a_{(u)}(x)  b(x) dvol_\Sigma(x)
   \right)
\end{displaymath}
which assigns to a given field configuration $\Phi_{u}$ in the family the value its $a$-component integrated against $b dvol_\Sigma$.

Similarly the transgression of $\phi^a_{,\mu}$ is the function

\begin{displaymath}
\tau_\Sigma( \phi^a_{,\mu} \, b dvol_\Sigma  )_\Phi
   \;=\;
   \left(
     u
       \mapsto
     \int_\Sigma \frac{\partial \Phi^a_{u}}{\partial x^\mu}  b(x) dvol_\Sigma(x)
   \right)
\end{displaymath}
which assigns to a field configuration the integral of the value of the $\mu$th derivative of its $a$th component against $b dvol_\Sigma$.

Next consider a horizontally exact variational form

\begin{displaymath}
d \alpha \in \Omega^{p+1,s}_{\Sigma,cp}(E)
  \,.
\end{displaymath}
By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative} the pullback of this form along the jet prolongation of fields is exact in the $\Sigma$-direction:

\begin{displaymath}
(j^\infty_\Sigma\Phi_{(-)})^\ast(d \alpha  \wedge b dvol_\Sigma) = d_\Sigma (j^\infty_\Sigma\Phi_{(-)})^\ast\alpha \wedge b dvol_\Sigma
  \,,
\end{displaymath}
(where we write $d = d_U + d_\Sigma$ for the de Rham differential on $U \times \Sigma$). It follows that the integral over $\Sigma$ vanishes.

Now let

\begin{displaymath}
\delta \alpha \phi^a_{,\mu_1 \cdots \mu_k} \, b dvol_\Sigma
  \in
  \Omega^{p+1,1}_\Sigma(E)
\end{displaymath}
be a variational (vertical) differential 1-form. Its [[pullback of differential forms]] along $j^\infty_\Sigma(\Phi_{(-)}) \colon U \times \Sigma \to J^\infty_\Sigma(E)$ has two contributions: one from the variation along $\Sigma$, the other from variation along $U$.

By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative}, for \emph{fixed} $u \in U$ the pullback along the jet prolongation vanishes.

On the other hand, for fixed $s \in \Sigma$, the pullback of $\mathbf{d} \phi^a_{\mu_1\cdots \mu_k}$ is

\begin{displaymath}
d_U \frac{ \partial^k \Phi_{(-)}}{\partial x^{\mu^1} \cdots \partial x^{\mu_k}}
\end{displaymath}
while the pullback of $d \phi^a_{\mu_1\cdots \mu_k}$ vanishes at fixed $\Sigma$.

This means that

\begin{displaymath}
\tau_\Sigma( \delta \phi^a_{,\mu_1 \cdots \mu_k} )
  =
  d \tau_{\Sigma}( \phi^a_{_\mu_1 \cdots \mu_k} )
\end{displaymath}
is the de Rham differential (on $U$) of the corresponding function discussed before.

In conclusion:

Under transgression the variational (vertical) derivative on the jet bundle turns into the ordinary de Rham derivative on the [[space of histories]] of fields.

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

We discuss some examples and applications:

\begin{itemize}%
\item \hyperlink{GaugeCouplingActionFunctionalOfChargedParticle}{Gauge coupling action functional of charged particle}


\item \hyperlink{TransgressionOfKillingFormToSymplecticFormOfChernSimons}{Transgression of Killing form to symplectic form of Chern-Simons theory}



\end{itemize}
\hypertarget{GaugeCouplingActionFunctionalOfChargedParticle}{}\subsubsection*{{Gauge coupling action functional of charged particle}}\label{GaugeCouplingActionFunctionalOfChargedParticle}

Let $X \in \mathbf{H}$ and consider a [[circle group]]-[[principal connection]] $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ over $X$. By the discussion in \emph{\hyperlink{DiracChargeQuantizationAndElectromagneticField}{Dirac charge quantization and the electromagnetic field}} above this encodes an [[electromagnetic field]] on $X$. Assume for simplicity here that the underlying [[circle principal bundle]] is trivialized, so that then the connection is equivalently given by a differential 1-form

\begin{displaymath}
\nabla = A \;\colon\; X \to \mathbf{\Omega}^1
  \,,
\end{displaymath}
the \emph{[[electromagnetic potential]]}.

Let then $\Sigma = S^1$ be the [[circle]]. The [[transgression]] of the electromagnetic potential to the [[loop space]] of $X$

\begin{displaymath}
\int_{S^1} [S^1, A]
   \;\colon\;
  [S^1, X]
    \stackrel{[S^1, A]}{\to}
  [S^1 , \Omega^1]
    \stackrel{\int_{S^1}}{\to}
  \Omega^0
  \simeq
  \mathbb{R}
\end{displaymath}
is the [[action functional]] for an [[electron]] or other electrically charged [[particle]] in the [[background gauge field]] $A$ is $S_{em} = \int_{S^1} [S^1, A]$.

The [[variational calculus|variation]] of this contribution in addition to that of the [[kinetic action]] of the electron gives the \emph{[[Lorentz force]]} law describing the [[force]] exerted by the [[background gauge field]] on the electron.

\hypertarget{TransgressionOfKillingFormToSymplecticFormOfChernSimons}{}\subsubsection*{{Transgression of Killing form to symplectic form of Chern-Simons theory}}\label{TransgressionOfKillingFormToSymplecticFormOfChernSimons}

Let $\mathfrak{g}$ be a [[Lie algebra]] with binary [[invariant polynomial]] $\langle -,-\rangle \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$.

For instance $\mathfrak{g}$ could be a [[semisimple Lie algebra]] and $\langle -,-\rangle$ its [[Killing form]]. In particular if $\mathfrak{g} = \mathfrak{su}(n)$ is a [[matrix Lie algebra]] such as the [[special unitary Lie algebra]], then the Killing form is given by the [[trace]] of the product of two matrices.

This pairing $\langle -,-\rangle$ defines a differential 4-form on the [[smooth space]] of [[Lie algebra valued 1-forms]]

\begin{displaymath}
\langle F_{(-)} \wedge F_{(-)} \rangle
  \colon
  \Omega^1(-,\mathfrak{g})
   \stackrel{F_{(-)}}{\to}
  \Omega^2(-, \mathfrak{g})
   \stackrel{(-)\wedge (-)}{\to}
  \Omega^4(-, \mathfrak{g}\otimes \mathfrak{g})
   \stackrel{\langle-,-\rangle}{\to}
  \Omega^4
\end{displaymath}
Over a [[coordinate patch]] $U \in$ [[CartSp]] this sends a differential 1-form $A \in \Omega^1(U)$ to the differential 4-form

\begin{displaymath}
\langle F_A \wedge F_A \rangle \in \Omega^4(U)
  \,.
\end{displaymath}
The fact that $\langle -, - \rangle$ is indeed an \emph{[[invariant polynomial]]} means that this indeed extends to a 4-form on the smooth [[groupoid of Lie algebra valued forms]]

\begin{displaymath}
\langle F_{(-)} \wedge F_{(-)}\rangle
  \colon
  \mathbf{B}G_{conn}
  \to 
  \Omega^4
  \,.
\end{displaymath}
Now let $\Sigma$ be an [[orientation|oriented]] [[closed manifold|closed]] [[smooth manifold]]. The [[transgression]] of the above 4-form to the [[mapping space]] out of $\Sigma$ yields the 2-form

\begin{displaymath}
\omega
    \coloneqq
  \int_{\Sigma} \langle F_{(-)}\wedge F_{(-)}\rangle
   \colon
  \mathbf{\Omega}^1(\Sigma,\mathfrak{g})
   \hookrightarrow
  [\Sigma, \mathbf{B}G_{conn}]
    \stackrel{[\Sigma, \langle F_{(-)}\wedge F_{(-)}\rangle]}{\to}
  [\Sigma, \Omega^4]
    \stackrel{\int_{\Sigma}}{\to}
  \Omega^2
\end{displaymath}
to the [[moduli stack]] of [[Lie algebra valued 1-forms]] on $\Sigma$.

Over a [[coordinate chart]] $U = \mathbb{R}^n \in$ [[CartSp]] an element $A \in \mathbf{\Omega}^1(\Sigma,\mathfrak{g})(\mathbb{R}^n)$ is a $\mathfrak{g}$-valued 1-form $A$ on $\Sigma \times U$ with no leg along $U$. Its [[curvature]] 2-form therefore decomposes as

\begin{displaymath}
F_A = F_A^{\Sigma} + \delta A
  \,,
\end{displaymath}
where $F_A^{\Sigma}$ is the curvature component with all legs along $\Sigma$ and where

\begin{displaymath}
\delta A
  \coloneqq 
  - \sum_{i = 1}^n \frac{\partial}{\partial x^i} A \wedge \mathbf{d}x^i
\end{displaymath}
is the [[variational calculus|variational]] derivative of $A$.

This means that in the 4-form

\begin{displaymath}
\langle F_A \wedge F_A\rangle
  = 
  \langle F_A^\Sigma \wedge F_A^\Sigma \rangle
  + 
  2 \langle F_A^\Sigma \wedge \delta A\rangle 
  + 
  \langle \delta A \wedge \delta A\rangle 
  \in
  \Omega^4(\Sigma \times U)
\end{displaymath}
only the last term gives a 2-form contribution on $U$. Hence we find that the transgressed 2-form is

\begin{displaymath}
\omega = \int_\Sigma \langle \delta A \wedge \delta A\rangle
   \colon
  \mathbf{\Omega}^1(\Sigma, \mathfrak{g})
   \to
  \Omega^2
  \,.
\end{displaymath}
When restricted further to flat forms

\begin{displaymath}
\mathbf{\Omega^1}_{flat}(\Sigma,\mathfrak{g})
  \hookrightarrow
  \mathbf{\Omega^1}(\Sigma,\mathfrak{g})
\end{displaymath}
which is the [[phase space]] of $\mathfrak{g}$-[[Chern-Simons theory]], then this is the corresponding [[symplectic form]] (by the discussion at \emph{\href{Chern-Simons+theory#CovariantPhaseSpace}{Chern-Simons theory -- covariant phase space}}).

[[!redirects transgressions of differential forms]]

[[!redirects transgression of differential n-forms]] [[!redirects transgressions of differential n-forms]]

[[!redirects transgression of variational differential forms]] [[!redirects transgressions of variational differential forms]]



\end{document}