nLab transgression of differential forms

Idea

Given a differential form $\omega$ of degree $n$ on some smooth space $X$ and given a 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 $\omega$ with respect to $\Sigma$

This construction has a variety of immediate generalizations, for instance $\Sigma$ may have boundary and 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 form on the jet bundle of a field bundle. Here the transgression of the Lagrangian itself (along jet prolongations of fields) is the corresponding action functional, the transgression of its 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. These examples are discussed below at Transgression of variational differential forms.

Definition

There are two definitions of transgression of differential forms: A traditional formulation is def. 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. below.

That these two definitions are indeed equivalent is the content of prop. below

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 smooth sets. For the following discussion we assume background on smooth sets as introduced in

• geometry of physics – smooth sets

• geometry of physics – differential forms.

(This entry itself here overlaps with geometry of physics – integration, where more background may be found.)

Recall form the discussion there that a smooth set $X$ is 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 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 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.)

Via parameterized integration of differential forms

Definition

(parameterized integration of differential forms)

Let

1. $X$ be a smooth set;

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

3. $\Sigma_k$ be a 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 $\Sigma$:

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

Definition

(transgression of differential forms to mapping spaces)

Let

1. $X$ be a smooth set;

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

3. $\Sigma_k$ be a 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 function:

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

where $[\Sigma,\omega]$ is the mapping space functor on morphisms and $\int_{\Sigma}$ is the parameterized integration of differential forms from def. .

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 $\Sigma$:

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

Via pullback along the evaluation map

Definition

(transgression of differential forms to mapping space via evaluation map)

Let

1. $X$ be a smooth set;

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

3. $\Sigma_k$ be a 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 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 $\Sigma$.

Proposition

The two definitions of transgression of differential forms to mapping spaces from def. and def. are equivalent.

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. 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 naturality square for adjuncts, and so the claim follows by its 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).

Transgression of variational differential forms

An important variant of transgression of differential forms is the transgression of 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.

Definition

(fields and their space of histories)

Given a spacetime $\Sigma$, then a type of fields on $\Sigma$ is a smooth fiber bundle

$$\array{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }$$

called the field bundle,

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

$$\Phi \colon \Sigma \longrightarrow E$$

such that composed with the projection map it is the identity function, i.e. such that

$$fb \circ \Phi = id \phantom{AAAAAAA} \array{ && E \\ & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}} \\ \Sigma & = & \Sigma } \,.$$

The corresponding field space of histories is the smooth space of all these, to be denoted

$$\Gamma_\Sigma(E) \in \mathbf{H} \,.$$

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

$$\array{ U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E \\ (u,x) &\mapsto& \Phi_u(x) }$$

such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.

$$\array{ && E \\ & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}} \\ U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma } \,.$$

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 space of histories of fields restricted to $S$, to be denoted

$$\Gamma_{S}(E) \coloneqq \Gamma_{N_\Sigma(S)}( E\vert_{N_\Sigma S} ) \in \mathbf{H}$$

is the space of sections restricted to the infinitesimal neighbourhood $N_\Sigma(S)$.

There is a canonical evaluation smooth function

(1)$$ev_S \;\colon\; N_\Sigma S \times \Gamma_{S}(E) \longrightarrow E$$

which takes a pair consisting of an element in $N_\Sigma S$ and a field configuration to the value of the field configuration at that point.

Definition

(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

$$\array{ J^\infty_\Sigma(E)\vert_S &\overset{\iota_S}{\hookrightarrow}& J^\infty_\Sigma(E) \\ \downarrow &(pb)& \downarrow \\ S &\hookrightarrow& \Sigma }$$

be the corresponding restriction of the jet bundle of $E$.

The 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:

$$supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma | \iota_{\{x\}^\ast A = 0} \} )$$

We write

$$\Omega^{r,s}_{\Sigma,cp}(E) \coloneqq \left\{ A \in \Omega^{r,s}_\Sigma(E) \;\vert\; supp_\Sigma(A) \, \text{is compact} \right\} \;\hookrightarrow\; \Omega^{r,s}_\Sigma(E)$$

for the subspace of differential forms on the jet bundle whose spacetime support is a compact subspace.

Definition

(transgression of variational differential forms to field space of histories)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. ), 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 transgression of variational differential forms to $\Sigma_r$ is the function

$$\tau_{\Sigma_r} \;\colon\; \Omega^{r,\bullet}_{\Sigma,cp}(E) \overset{ }{\longrightarrow} \Omega^\bullet\left( \Gamma_{\Sigma_r}(E) \right)$$

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

$$\Phi_{(-)} \;\colon\; U \times N_\Sigma \Sigma_r \longrightarrow E$$

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 $\Sigma_r$:

$$(\tau_{\Sigma}A)_{\Phi_{(-)}} \;\coloneqq\; \int_{\Sigma_r} (j^\infty_\Sigma(\Phi_{(-)}))^\ast \;\in\; \Omega^\bullet(U) \,.$$

Properties

Relative transgression over manifolds with boundary

Proposition

(relative transgression over manifolds with boundary)

1. $X$ be a smooth set;

2. $\Sigma_k$ be a compact smooth manifold of dimension $k$ with boundary $\partial \Sigma$

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

4. $\omega \in \Omega^n_{X}$ 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. ) 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

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.

Variational transgression picks out the vertical differential forms

Example

(some transgressions of variational differential forms)

We spell out the result transgression of variational differential forms (def. ) 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. ).

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

$$\Phi_{(-)} \;\colon\; U \times \Sigma \longrightarrow E \,.$$

Let $b \in C^\infty_{cp}(\Sigma)$ be any bump function on spacetime. Its product with the volume form (as in example ) is then a horizontal $p+1$-form on the jet bundle with compact spacetime support.

$$b dvol_\Sigma \in \Omega^{0,0}_{\Sigma,cp}(E)$$

The transgression of this 0-form to the space of histories of fields

$$\tau_\Sigma (b dvol_\Sigma) \in \Omega^0( \Gamma_\Sigma(E) )$$

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

$$\tau_\Sigma (b dvol_\Sigma)\vert_\Phi \colon u \mapsto \int_\Sigma b dvol_\Sigma$$

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

$$\phi^a \, b dvol_\Sigma \; \in \Omega^{p+1,0}( E ) \,.$$

Its transgression is the function

$$\tau_\Sigma( \phi^a \, b dvol_\Sigma )_\Phi \;=\; \left( u \mapsto \int_\Sigma \Phi^a_{(u)}(x) b(x) dvol_\Sigma(x) \right)$$

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

$$\tau_\Sigma( \phi^a_{,\mu} \, b dvol_\Sigma )_\Phi \;=\; \left( u \mapsto \int_\Sigma \frac{\partial \Phi^a_{u}}{\partial x^\mu} b(x) dvol_\Sigma(x) \right)$$

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

$$d \alpha \in \Omega^{p+1,s}_{\Sigma,cp}(E) \,.$$

By prop. the pullback of this form along the jet prolongation of fields is exact in the $\Sigma$-direction:

$$(j^\infty_\Sigma\Phi_{(-)})^\ast(d \alpha \wedge b dvol_\Sigma) = d_\Sigma (j^\infty_\Sigma\Phi_{(-)})^\ast\alpha \wedge b dvol_\Sigma \,,$$

(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

$$\delta \alpha \phi^a_{,\mu_1 \cdots \mu_k} \, b dvol_\Sigma \in \Omega^{p+1,1}_\Sigma(E)$$

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. , for 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

$$d_U \frac{ \partial^k \Phi_{(-)}}{\partial x^{\mu^1} \cdots \partial x^{\mu_k}}$$

while the pullback of $d \phi^a_{\mu_1\cdots \mu_k}$ vanishes at fixed $\Sigma$.

This means that

$$\tau_\Sigma( \delta \phi^a_{,\mu_1 \cdots \mu_k} ) = d \tau_{\Sigma}( \phi^a_{_\mu_1 \cdots \mu_k} )$$

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.

Examples

We discuss some examples and applications:

• Gauge coupling action functional of charged particle

• Transgression of Killing form to symplectic form of Chern-Simons theory

Gauge coupling action functional of charged particle

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 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

$$\nabla = A \;\colon\; X \to \mathbf{\Omega}^1 \,,$$

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 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

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 oriented 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 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).