geometry of physics -- integration

> this page is going to contain one chapter of _[[geometry of physics]]_

> previous chapters: _[[geometry of physics -- differential forms|differential forms]]_, _[[geometry of physics -- smooth homotopy types]]_

***

#Contents#
* table of contents
{:toc}

## Integration

By the discussion in _[Differential forms](#DifferentialForms)_ and _[Principal connections](#PrincipalConnections)_, [[differential forms]] and more generally [[connection on a bundle|connections]] may be regarded as [[infinitesimal space|infinitesimal]] [[measures]] of _change_, of _displacement_. The discussion in _[Differentiation](#Differentiation)_ showed how to extract from a _finite_ but [[cohesion|cohesive]] (e.g. smoothly continuous) displacement all its infinitesimal measures of displacements by [[differentiation]].

Here we discuss the reverse operation: _[[integration]]_ is a construction from a [[differential form]] of the corresponding finite cohesive displacement. More generally this applies to any [[connection on a bundle|connection]] and is then called the _[[nLab:parallel transport]]_ of the connection, a term again referring to the idea that a finite displacement proceeds pointwise in parallel to a given infinitesimal displacement.

Under good conditions this construction can proceed literally by "adding up all the infinitesimal contributions" and therefore [[integration]] is traditionally thought of as a generalization of forming [[sums]]. Therefore one has the notation "$\int_{\Sigma} \omega$"
for the [[integral]] of a [[differential form]] $\omega$ over a space $\Sigma$, as a variant of the notation "$\sum_{S} f$" for the [[sum]] of values of a [[function]] on a [[set]] $S$. For the case of integrals of connections the corresponding parallel transport expression is often denoted by an exponentiated integral sign "$\mathcal{P} \exp(\int_\Sigma \omega)$", referring to the fact that the passage from infinitesimal to finite quantities involves also the passage from [[Lie algebra]] data to [[Lie group]] data ("[[Lie integration|exponentiated Lie algebra data]]").

However, both from the point of view of [[gauge theory]] [[physics]] as well as from the [[general abstract]] perspective of [[cohesive homotopy type theory]] another characterization of integration is more fundamental: the [[integral]] $\int_\Sigma \omega$ of a [[differential form]] $\omega$ (or more generally of a [[circle n-bundle with connection|connection]]) is an [[invariant]] under those [[gauge transformations]] of $\omega$ that are trivial on the [[boundary]] of $\Sigma$, and it is the _[[universal construction|universal]]_ such invariant, hence is uniquely characterized by this property.

In traditional accounts this fact is referred to via the _[[Stokes theorem]]_ and its generalizations (such as the [[nonabelian Stokes theorem]]), which says that the [[integral]]/[[parallel transport]] is indeed _[[invariant]]_ under gauge transformations of differential forms/connections. That this invariance actually characterizes the integral and the parallel transport is rarely highlighted in traditional texts, but it is implicit for instance in the old "path method" of [[Lie integration]] (discussed below in _[Lie integration](#LieIntegration)_) as well as in the famous characterization of [[flat connections]], discussed above in _[Flat 1-connections](#Flat1Connections)_:

for $X$ a [[connected topological space|connected]] [[manifold]] and for $G$ a [[Lie group]], the operation of sending a [[flat connection|flat]] $G$-[[principal connection]] $\nabla$ to its [[parallel transport]] $\gamma \mapsto hol_{\gamma}(\nabla)$ around [[loops]] $\gamma\colon S^1 \to X$, hence to the [[integral]] of the connection around all possible loops (its [[holonomy]]),  for any fixed basepoint

$$
  hol \coloneqq \mathcal{P} \exp(\int_{(-)} (-))
  \;\colon\;
  H^1_{conn, flat}(X,G)
   \stackrel{\simeq}{\to}
  Hom_{Grp}(\pi_1(X) , G)/G
$$

exhibits a [[bijection]] between gauge  [[equivalence classes]] of connections and [[group]] homomorphisms from the [[fundamental group]] $\pi_1(X)$ of $X$ to the [[gauge group]] $G$ (modulo [[adjoint action|adjoint]] $G$-action from gauge transformations at the base point, hence at the integration boundary). This is traditionally regarded as a property of the definition of the [[parallel transport]] $\mathcal{P} \exp(\int_{(-)}(-))$ by [[integration]]. But being a [[bijection]], we may read this fact the other way round: it says that forming equivalence classes of flat $G$-connections is a way of computing their [[integral]]/[[parallel transport]].

We saw a generalization of this fact to non-closed forms and non-flat connections already in the discussion at _[Differential 1-forms as smooth incremental path measures](##1FormsAsSmoothFunctors)_, where gauge equivalence classes of differential forms are shown to be equivalently assignments of [[parallel transport]] to [[path groupoid|smooth paths]].

This is also implied by the above discussion: for $\nabla \in H^1_{conn}(X,G) $ any non-flat connection and $\gamma \colon S^1 \to X$ a trajectory in $X$, we may form the [[pullback of differential forms|pullback]] of $\nabla$ to $S^1$. There it becomes a necessarily flat connection $\gamma^* \nabla \in H^1_{conn, flat}(S^1,G)$, since the [[curvature]] [[differential 2-form]] necessarily vanishes on the 1-[[dimension|dimensional]] [[manifold]] $S^1$. Accordingly, by the above bijection, forming the gauge [[equivalence class]] of $\gamma^* \nabla$ means to find a group homomorphism

$$
  \mathbb{Z} \simeq \pi_1(S^1) \to G
$$

modulo conjugation (modulo nothing if $G$ is [[abelian group|abelian]], such as $G = U(1)$) and since $\mathbb{Z}$ is the [[free group]] on a single generator this is the same as finding an element

$$
  hol_\gamma(\nabla)
  =
  \mathcal{P} \exp(\int_\gamma \gamma^*\nabla)
  \in
  G
  \,.
$$

This total operation of first pulling back the connection and then forming its [[integration]] (by taking gauge equivalence classes) is called the _[[transgression]]_ of the original 1-form connection on $X$ to a 0-form connection on the [[loop space]] $[S^1,X]$.

Below in the [Model Layer](#IntegrationModelLayer) we discuss the classical examples of [[integration]]/[[parallel transport]] and their various generalizations in detail. Then in the [Semantic Layer](#IntegrationSemanticLayer) we show how indeed all these constructions are obtained forming [[equivalence classes]] in the [[(∞,1)-topos]] of [[smooth ∞-groupoids|smooth homotopy types]], hence by _[[truncated object in an (∞,1)-category|truncation]]_ (followed, to obtain the correct cohesive structure, by _concretification_, def. \ref{ConcreteObjectsAndConcretification}).

### Model Layer
 {#IntegrationModelLayer}

#### Integration

##### Integration over a coordinate patch

For $n \in \mathbb{N}$ let

$$
  C^n
  \coloneqq
  \{
    \vec x \in \mathbb{R}^n
    |
    \forall_i (0 \leq x_i \leq 1)
  \}
  \hookrightarrow
  \mathbb{R}^n
$$

be the standard unit [[cube]].

Let

$$
  \omega \in \Omega^n(\mathbb{R}^n)
$$

be a [[differential n-form]].

$$
  \omega = f \mathbf{d} x^1 \wedge \mathbf{d} x^2 \wedge \cdots \wedge \mathbf{d} x^n
  \,.
$$

Let $Partitions(C^k)$ be the [[poset]] whose elements are partitions of the unit $n$cube $C^n$ into $N^n$ subcubes, for $N \in \mathbb{N}$, ordered by inclusion.

Let

$$
  \sum_{(-)} \omega
  \colon
  Partitions(C^k)
  \to
  \mathbb{R}
$$

be the function that sends

$$
  \frac{1}{N^n}
  \sum_{x^1 = 0}^N
  \sum_{x^2 = 0}^N
  \cdots
  \sum_{k^n = 0}^N
  f( x^1, \cdots, x^n )
  \,.
$$

Then

$$
  \int_{C^k} \omega \colon \lim_{N} \sum_N \omega
  \,.
$$

##### Integration of differential forms over a manifold
 {#IntegrationOfDifferentialFormsOverASmoothManifold}

Let $\Sigma$ be a [[closed manifold|closed]] [[orientation|oriented]] [[smooth manifold]] of [[dimension]] $k$

+-- {: .num_defn #IntegrationOfDifferentialFormsInTermsOfSmoothModuli}
###### Definition

For $n \in \mathbb{N}$, $n \geq k$, define the
morphism of [[smooth spaces]]

$$
  \int_{\Sigma} \colon [\Sigma, \Omega^n] \to \Omega^{n-k}
$$

by declaring that over a [[coordinate chart]] $U \in$ [[CartSp]]
it is the ordinary [[integration of differential forms]] over smooth manifolds

$$
  \int_{\Sigma, U} : \Omega^n(\Sigma\times U) \to \Omega^{n-k}(U)
  \,.
$$

=--

##### Integration in ordinary differential cohomology

* [[fiber integration in ordinary differential cohomology]]

#### Holonomy

##### Parallel transport

given $A \in \Omega^1(\Delta^1, \mathfrak{g})$

we say $f \in C^\infty(\Delta^1, G)$ is the [[parallel transport]] of $A$ if

1. $f(0) = 1$

1. $f$ satisfies the [[differential equation]]

   $$
     \mathbf{d}f = A f
   $$

where on the right we have the differential of the left action of the group on itself.

In this case one writes

$$
  \mathcal{P} \exp\left(\int_{\Delta^1} A \right)
  \coloneqq
  f(1)
$$

and calls it the _[[path ordered integral]]_ of $A$. Here the enire left hand side is primitive notation.

In the case that $G = U(1)$ this reproduces the ordinary [[integral]]

$$
  \left(G = \mathbb{R}\right)
  \Rightarrow
  \mathcal{P} \exp(\int_{\Delta^1} A)
  =
  \exp(i \int_{\Delta^1} A)
  \in
  U(1)
$$

There is another way to express this [[parallel transport]],
related to [[Lie integration]]:

Define an [[equivalence relation]] on $\Omega^1(\Delta^1, \mathfrak{g})$ as follows: two 1-forms $A,A'$ are taken to be equivalent if there is a flat 1-form $\hat A \in \Omega^1_{flat}(D^2, \mathfrak{g})$ on the 2-[[disk]] such that its restriction to the upper semicircle is $A$ and the restriction to the lower semicircle is $\tilde A$.

If $G$ is [[simply connected topological space|simply connected]], then the [[equivalence classes]] of this relation form

$$
  \Omega^1(\Delta^1,\mathfrak{g})_{/\sim} \simeq
  G
$$

and the [[quotient]] map coincides with the [[parallel transport]]

$$
  \mathcal{P} \exp\left(\int_{\Delta^1} \left(-\right)\right)
  \colon
  \Omega^1(\Delta^1, \mathfrak{g})
  \to
  \Omega^1(\Delta^1, \mathfrak{g})_{/\sim}
  \simeq
  G
$$

Finally yet another perspective is this: consider the [[equivalence relation]] on $\Omega^1(\Delta^1, \mathfrak{g})$ where two 1-forms are regarded as equivalent if there is a [[gauge transformation]] $\lambda \in C^\infty(\Delta^1, G)$ with $\lambda(0) = e$ and $\lambda(1) = e$, then again

$$
  \mathcal{P} \exp\left(\int_{\Delta^1} \left(-\right)\right)
  \colon
  \Omega^1(\Delta^1, \mathfrak{g})
  \to
  \Omega^1(\Delta^1, \mathfrak{g})_{/\sim}
  \simeq
  G
$$

is the parallel transport

##### Holonomy of a flat principal connection

if $X$ is connected then forming the [[holonomy]] of flat $G$-connections

$$
  hol
  \colon
  G Bund_{\nabla, flat}(X)
  \stackrel{\simeq}{\to}
  Hom_{Grp}(\pi_1(X), G)
$$

is an [[equivalence]], $\pi_1(X)$ the [[fundamental group]]. If $X$ is not connected then

$$
  hol
  \colon
  G Bund_{\nabla, flat}(X)
  \stackrel{\simeq}{\to}
  Hom_{Grpd}(\Pi_1(X), \mathbf{B}G)
$$

is an equivalence.

#### Transgression
 {#Transgression}

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 $\Sigma$:

$$
  \tau_{\Sigma} \omega\vert_{\Phi}
    \coloneqq
  \int_\Sigma \Phi_{(-)}^\ast \omega
  \,.
$$

This differential form $\tau_\Sigma \omega$ on the mapping space is called the _[[transgression of differential forms|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 the [[mapping space]] may be generalized to a [[space of sections]] of a given [[fiber bundle]]. Also it is immediate to generalize the construction from [[smooth sets]] to [[super formal smooth sets]]. 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 is the corresponding _[[action functional]]_, the transgression of its [[Euler-Lagrange operator|Euler-Lagrange variational derivative]] is the 1-form whose vanishing is the _[[equations of motion]]_ and the transgression of the induced [[pre-symplectic current]] is the _[[pre-symplectic form]] on the [[covariant phase space]]_ of the field theory.

##### Transgression of differential forms
 {#TransgressionOfDifferentialForms}

###### Defintion

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

+-- {: .num_defn #ParameterizedIntegrationOfDifferentialForms}
###### Definition
**(parameterized [[integration of differential forms]])

Let

1. $X$ be a [[smooth set]];

1. $n \geq k \in \mathbb{N}$;

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

Then we write

$$
  \int_{\Sigma}
    \;\colon\;
  [\Sigma_k, \mathbf{\Omega}^n]
    \longrightarrow
  \mathbf{\Omega}^{n-k}
$$

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 $\Sigma$:

$$
  \int_{\Sigma} \omega_{(-)}(-) \coloneqq \int_\Sigma \omega_{(-)}
  \,.
$$

=--

+-- {: .num_defn #TransgressionOfDifferentialFormsToMappingSpaces}
###### Definition
**([[transgression of differential forms]] to [[mapping spaces]])

Let

1. $X$ be a [[smooth set]];

1. $n \geq k \in \mathbb{N}$;

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

Then the operation of _transgression of differential $n$-forms_ on $X$ with respect to $\Sigma$ is the [[function]]

$$
  \tau_\Sigma
   \coloneqq
  \int_\Sigma [\Sigma,-]
   \;\colon\;
  \Omega^n(X) \to \Omega^{n-k}([\Sigma,X])
$$

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

$$
  (X \stackrel{\omega}{\to} \Omega^n) \in \Omega^n(X)
$$

to the differential form corresponding to the following composite smooth funtion:

$$
  \tau_\Sigma \omega
   \coloneqq
  \int_{\Sigma} [\Sigma,\omega]
   \;\colon\;
  [\Sigma, X]
    \stackrel{[\Sigma, \omega]}{\to}
  [\Sigma, \Omega^n]
    \stackrel{\int_{\Sigma}}{\to}
  \Omega^{n-k}
  \,,
$$

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]]

$$
  \phi_{(-)} \;\colon\; U \to [\Sigma, X]
$$

is equivalently a [[smooth function]] of the form

$$
  \phi_{(-)}(-) \;\colon\; U \times \Sigma \to X
  \,.
$$

The smooth function $[\Sigma,\omega]$ takes this smooth function to the plot

$$
  U \times \Sigma \to X
   \overset{\phi_{(-)}(-)}{\longrightarrow}
  X
   \overset{\omega}{\longrightarrow}
  \mathbf{\Omega}^{n}
$$

which is equivalently a differential form

$$
  (\phi_{(-)}(-))^\ast \omega \in \Omega^n(U \times \Sigma)
  \,.
$$

Finally the smooth function $\int_\Sigma$ takes this to the result of [[integration of differential forms]] over $\Sigma$:

$$
  \tau_{\Sigma}\omega\vert_{\phi_{(-)}}
  \;=\;
  \int_\Sigma (\phi_{(-)}(-))^\ast \omega
  \;\in\;
  \Omega^{n-k}(U)
  \,.
$$

=--

+-- {: .num_defn #TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap}
###### Definition
**([[transgression of differential forms]] to mapping space via evaluation map)**

Let

1. $X$ be a [[smooth set]];

1. $n \geq k \in \mathbb{N}$;

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

Then the operation of _transgression of differential $n$-forms_ on $X$ with respect to $\Sigma$ is the [[function]]

$$
  \tau_\Sigma
   \coloneqq
  \int_\Sigma ev^\ast
   \;\colon\;
  \Omega^n(X)
    \overset{ev^\ast}{\longrightarrow}
  \Omega^n(\Sigma \times [\Sigma, X])
    \overset{\int_\Sigma}{\longrightarrow}
  \Omega^{n-k}([\Sigma,X])
$$

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 $\Sigma$.

=--

+-- {: .num_prop #EquivalenceOfTransgressionOfDifferentialFormsToMappingSpaces}
###### Proposition

The two definitions of transgression of differential forms to mapping spaces from def. \ref{TransgressionOfDifferentialFormsToMappingSpaces} and def. \ref{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap} are equivalent.

=--

+-- {: .proof}
###### 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

$$
  \gamma^\ast \int_\Sigma \mathrm{ev}^\ast A
  =
  \int_\Sigma (\gamma,\mathrm{id}_\Sigma)^\ast \mathrm{ev}^\ast A
  \;
  \in \Omega^n(U)
$$

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 $\mathrm{ev}$:

$$
  \array{
    U \times \Sigma
    &
     \overset{(\gamma, \mathrm{id}_\Sigma)}{\longrightarrow}
    &
    [\Sigma,X] \times \Sigma
    &
      \overset{\mathrm{ev}}{\longrightarrow}
    &
    X
  }
  \,.
$$

Hence the integral is now

$$
  \cdots = \int_{\Sigma} \tilde \gamma^\ast A
  \,.
$$

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]]:

$$
  \array{
    \tilde \gamma \in
    &
    \mathbf{H}(U \times\Sigma, X)
    &
      \overset{\mathbf{H}(U \times \Sigma,A)}{\longrightarrow}
    &
    \mathbf{H}(U \times \Sigma, \mathbf{\Omega}^{n+k})
    &
      \overset{\int_\Sigma(U)}{\longrightarrow}
    &
    \Omega^n(U)
    \\
    &
    {}^{\mathllap{\simeq}}\downarrow
      &&
    {}^{\mathllap{\simeq}}\downarrow
      &&
    {}^{\mathllap{\simeq}}\downarrow
    \\
    \gamma \in
    &
    \mathbf{H}(U,[\Sigma,X])
    &
      \overset{\mathbf{H}(U,[\Sigma,A])}{\longrightarrow}
    &
    \mathbf{H}(U,[\Sigma,\mathbf{\Omega}^{n+k}])
    &
      \overset{\mathbf{H}(U,\int_\Sigma)}{\longrightarrow}
    &
    \mathbf{H}(U,\mathbf{\Omega}^n)
  }
$$

(here we write $\mathbf{H}(-,-)$ for the [[hom functor]] of [[smooth sets]]).

=--

###### Properties

+-- {: .num_example }
###### Example
**(relative transgression over [[manifolds with boundary]])**

1. $X$ be a [[smooth set]];

1. $n \geq k \in \mathbb{N}$;

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

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

Write

$$
  (-\vert_{\partial \Sigma})
    \coloneqq
  [\partial \Sigma \hookrightarrow \Sigma, X]
  \;\colon\;
  [\Sigma, X]
   \longrightarrow
  [\partial \Sigma, X]
$$

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
$$
  d \left(
    \tau_{\Sigma}(\omega)
  \right)
  =
  (-1)^{k+1}
  (-\vert_{\partial \Sigma})^\ast \tau_{\partial \Sigma}(\omega)
    \phantom{AAAAAAAA}
  \array{
    [\Sigma, X]
      &\overset{ \tau_{\Sigma}(\omega) }{\longrightarrow}&
    \mathbf{\Omega}^{n-k}
    \\
    {}^{\mathllap{(-\vert_{\partial \Sigma}) }}\downarrow
      &&
    \downarrow^{\mathrlap{ (-1)^{k+1} d}}
    \\
    [\partial \Sigma, X]
      &\underset{ \tau_{\partial\Sigma}(\omega) }{\longrightarrow}&
    \mathbf{\Omega}^{n-k+1}
  }
  \,.
$$

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.

=--

+-- {: .proof}
###### 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

$$
  d = d_U + d_\Sigma
  \,.
$$

Now we compute as follows:

$$
  \begin{aligned}
    d \tau_{\Sigma}\omega\vert_{\phi_(-)}
    & =
    d_U \int_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma d_U (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma (d - d \Sigma) (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma d (\phi_{(-)}(-))^\ast \omega
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma (\phi_{(-)}(-))^\ast \underset{= 0}{\underbrace{d \omega}}
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -(-1)^k \int_{\partial \Sigma} (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -(-1)^k \tau_{\partial \Sigma} \omega \vert_{\phi_{(-)}}
  \end{aligned}
$$

where in the second but last step we used [[Stokes' theorem]].

=--

We next discuss some examples and applications:

* [Gauge coupling action functional of charged particle](#GaugeCouplingActionFunctionalOfChargedParticle)

* [Transgression of Killing form to symplectic form of Chern-Simons theory](#TransgressionOfKillingFormToSymplecticFormOfChernSimons)

###### Gauge coupling action functional of charged particle
 {#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 _[Dirac charge quantization and the electromagnetic field](#DiracChargeQuantizationAndElectromagneticField)_ 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

$$
  \nabla = A \colon X \to \Omega^1
  \,,
$$

the [[electromagnetic potential]].

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

$$
  \int_{S^1} [S^1, A]
   \;\colon\;
  [S^1, X]
    \stackrel{[S^1, A]}{\to}
  [S^1 , \Omega^1]
    \stackrel{\int_{S^1}}{\to}
  \Omega^0
  \simeq
  \mathbb{R}
$$

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 _[[Lorentz force]]_ law describing the [[force]] exerted by the [[background gauge field]] on the electron.

###### Transgression of Killing form to symplectic form of Chern-Simons theory
 {#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]]

$$
  \langle F_{(-)} \wedge F_{(-)} \rangle
  \colon
  \Omega^1(-,\mathfrak{g})
   \stackrel{F_{(-)}}{\to}
  \Omega^2(-, \mathfrak{g})
   \stackrel{(-)\wedge (-)}{\to}
  \Omega^4(-, \mathfrak{g}\otimes \mathfrak{g})
   \stackrel{\langle-,-\rangle}{\to}
  \Omega^4
$$

Over a [[coordinate patch]] $U \in $ [[CartSp]] this sends a differential 1-form $A \in \Omega^1(U)$ to the differential 4-form

$$
  \langle F_A \wedge F_A \rangle \in \Omega^4(U)
  \,.
$$

The fact that $\langle -, - \rangle$ is indeed an _[[invariant polynomial]]_ means that this indeed extends to a 4-form on the smooth [[groupoid of Lie algebra valued forms]]

$$
  \langle F_{(-)} \wedge F_{(-)}\rangle
  \colon
  \mathbf{B}G_{conn}
  \to
  \Omega^4
  \,.
$$

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

$$
  \omega
    \coloneqq
  \int_{\Sigma} \langle F_{(-)}\wedge F_{(-)}\rangle
   \colon
  \mathbf{\Omega}^1(\Sigma,\mathfrak{g})
   \hookrightarrow
  [\Sigma, \mathbf{B}G_{conn}]
    \stackrel{[\Sigma, \langle F_{(-)}\wedge F_{(-)}\rangle]}{\to}
  [\Sigma, \Omega^4]
    \stackrel{\int_{\Sigma}}{\to}
  \Omega^2
$$

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

$$
  F_A = F_A^{\Sigma} + \delta A
  \,,
$$

where $F_A^{\Sigma} $ is the curvature component with all legs along $\Sigma$ and where

$$
  \delta A
  \coloneqq
  - \sum_{i = 1}^n \frac{\partial}{\partial x^i} A \wedge \mathbf{d}x^i
$$

is the [[variational calculus|variational]] derivative of $A$.

This means that in the 4-form

$$
  \langle F_A \wedge F_A\rangle
  =
  \langle F_A^\Sigma \wedge F_A^\Sigma \rangle
  +
  2 \langle F_A^\Sigma \wedge \delta A\rangle
  +
  \langle \delta A \wedge \delta A\rangle
  \in
  \Omega^4(\Sigma \times U)
$$

only the last term gives a 2-form contribution on $U$. Hence we find that the transgressed 2-form is

$$
  \omega = \int_\Sigma \langle \delta A \wedge \delta A\rangle
   \colon
  \mathbf{\Omega}^1(\Sigma, \mathfrak{g})
   \to
  \Omega^2
  \,.
$$

When restricted further to flat forms

$$
  \mathbf{\Omega^1}_{flat}(\Sigma,\mathfrak{g})
  \hookrightarrow
  \mathbf{\Omega^1}(\Sigma,\mathfrak{g})
$$

which is the [[phase space]] of $\mathfrak{g}$-[[Chern-Simons theory]], then this is the corresponding [[symplectic form]] (by the discussion at _[Chern-Simons theory -- covariant phase space](Chern-Simons+theory#CovariantPhaseSpace)_).

##### Transgression of circle $n$-bundles with connection

* [[fiber integration in ordinary differential cohomology]]

##### Action functionals from transgression

(...)

#### Lie integration
 {#LieIntegration}

* [[Lie integration]]

### Semantic Layer
 {#IntegrationSemanticLayer}

#### Integration and higher holonomy

[[integration]]/[[higher holonomy]] is

$$
  \exp(2 \pi i \int_{\Sigma}(-))
  \colon
  [\Sigma_n, \mathbf{B}^n U(1)_{conn}]
  \to
  Conc \tau_0 [\Sigma, \mathbf{B}^n U(1)]
  \simeq
  U(1)
$$

#### Transgression

[[transgression]] is

$$
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
  \stackrel{\exp(2 \pi i \int_{\Sigma_k}) (-) }{\to}
  \mathbf{B}^{n-k} U(1)_{conn}
$$

#### Action functionals from Lagrangeans
 {#SemLayerActionFunctionalsFromLagrangeans}

and [[schreiber:infinity-Chern-Simons theory|higher Chern-Simons]] [[action functionals]] induced from

$$
  \mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}
$$

are

$$
  \exp\left(i S\left(-\right)\right)
  \coloneqq
  \exp(2 \pi \in \int_{\Sigma_k} [\Sigma_k, \mathbf{L}] )
  \colon
  [\Sigma_k, \mathbf{B}G]
  \stackrel{[\Sigma_k, \mathbf{L}]}{\to}
  [\Sigma_k, \mathbf{B}^n U(1)]
  \stackrel{\exp(2 \pi i\left(-\right))}{\to}
  \mathbf{B}^{n-k} U(1)_{conn}
$$

here $\mathbf{L}$ is the [[Lagrangean]].

### Syntactic Layer

(...)