this page is going to contain one chapter of geometry of physics
previous chapters: differential forms, geometry of physics -- smooth homotopy types
By the discussion in Differential forms and Principal connections, differential forms and more generally connections may be regarded as infinitesimal measures of change, of displacement. The discussion in Differentiation showed how to extract from a finite but 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 and is then called the 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 (“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 connection) is an invariant under those gauge transformations of $\omega$ that are trivial on the boundary of $\Sigma$, and it is the 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) as well as in the famous characterization of flat connections, discussed above in Flat 1-connections:
for $X$ a connected manifold and for $G$ a Lie group, the operation of sending a 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
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 $G$-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, where gauge equivalence classes of differential forms are shown to be equivalently assignments of parallel transport to 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 $\nabla$ 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-dimensional manifold $S^1$. Accordingly, by the above bijection, forming the gauge equivalence class of $\gamma^* \nabla$ means to find a group homomorphism
modulo conjugation (modulo nothing if $G$ is 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
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 we discuss the classical examples of integration/parallel transport and their various generalizations in detail. Then in the Semantic Layer we show how indeed all these constructions are obtained forming equivalence classes in the (∞,1)-topos of smooth homotopy types, hence by truncation (followed, to obtain the correct cohesive structure, by concretification, def. \ref{ConcreteObjectsAndConcretification}).
For $n \in \mathbb{N}$ let
be the standard unit cube.
Let
be a differential n-form.
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
be the function that sends
Then
Let $\Sigma$ be a closed oriented smooth manifold of dimension $k$
For $n \in \mathbb{N}$, $n \geq k$, define the morphism of smooth spaces
by declaring that over a coordinate chart $U \in$ CartSp it is the ordinary integration of differential forms over smooth manifolds
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
$f(0) = 1$
$f$ satisfies the differential equation
where on the right we have the differential of the left action of the group on itself.
In this case one writes
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
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, then the equivalence classes of this relation form
and the quotient map coincides with the parallel transport
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
is the parallel transport
if $X$ is connected then forming the holonomy of flat $G$-connections
is an equivalence, $\pi_1(X)$ the fundamental group. If $X$ is not connected then
is an equivalence.
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 $\Sigma$:
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 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 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 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.
There are two definitions of transgression of differential forms: A traditional formulation is def. 4 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. 2 below.
That these two definitions are indeed equivalent is the content of prop. 1 below
(parameterized integration of differential forms)
Let
$X$ be a smooth set;
$n \geq k \in \mathbb{N}$;
$\Sigma_k$ be a compact smooth manifold of dimension $k$.
Then we write
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 $\Sigma$:
(transgression of differential forms to mapping spaces)
Let
$X$ be a smooth set;
$n \geq k \in \mathbb{N}$;
$\Sigma_k$ 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
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
to the differential form corresponding to the following composite smooth funtion:
where $[\Sigma,\omega]$ is the mapping space functor on morphisms and $\int_{\Sigma}$ is the parameterized integration of differential forms from def. 2.
More explicitly in terms of plots this means equivalently the following
A plot of the mapping space
is equivalently a smooth function of the form
The smooth function $[\Sigma,\omega]$ takes this smooth function to the plot
which is equivalently a differential form
Finally the smooth function $\int_\Sigma$ takes this to the result of integration of differential forms over $\Sigma$:
(transgression of differential forms to mapping space via evaluation map)
Let
$X$ be a smooth set;
$n \geq k \in \mathbb{N}$;
$\Sigma_k$ 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
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 $\Sigma$.
The two definitions of transgression of differential forms to mapping spaces from def. 3 and def. 4 are equivalent.
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. 4 we get
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 $\mathrm{ev}$:
Hence the integral is now
This is the operation of the top horizontal composite in the following naturality square for adjuncts, and so the claim follows by its commutativity:
(here we write $\mathbf{H}(-,-)$ for the hom functor of smooth sets).
(relative transgression over manifolds with boundary)
$X$ be a smooth set;
$n \geq k \in \mathbb{N}$;
$\Sigma_k$ be a compact smooth manifold of dimension $k$ with boundary $\partial \Sigma$
$\omega \in \Omega^n_{X}$ a closed differential form.
Write
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. 3) to $\Sigma$ and to $\partial \Sigma$, respectively, are related by
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.
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
Now we compute as follows:
where in the second but last step we used Stokes' theorem.
We next discuss some examples and applications:
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
the electromagnetic potential.
Let then $\Sigma = S^1$ be the circle. The transgression of the electromagnetic potential to the loop space of $X$
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.
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
Over a coordinate patch $U \in$ CartSp this sends a differential 1-form $A \in \Omega^1(U)$ to the differential 4-form
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
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
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
where $F_A^{\Sigma}$ is the curvature component with all legs along $\Sigma$ and where
is the variational derivative of $A$.
This means that in the 4-form
only the last term gives a 2-form contribution on $U$. Hence we find that the transgressed 2-form is
When restricted further to flat forms
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).
(…)
integration/higher holonomy is
and higher Chern-Simons action functionals induced from
are
here $\mathbf{L}$ is the Lagrangean.
(…)