synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
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 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.
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
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
(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.)
(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 function:
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
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. and def. 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.
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).
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.
(fields and their space of histories)
Given a spacetime $\Sigma$, then a type of fields on $\Sigma$ is a smooth fiber bundle
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
such that composed with the projection map it is the identity function, i.e. such that
The corresponding field space of histories is the smooth space of all these, to be denoted
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
such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.
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
is the space of sections restricted to the infinitesimal neighbourhood $N_\Sigma(S)$.
There is a canonical evaluation smooth function
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.
(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
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:
We write
for the subspace of differential forms on the jet bundle whose spacetime support is a compact subspace.
(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
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
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 $\Sigma_r$:
(relative transgression over manifolds with boundary)
$X$ be a smooth set;
$\Sigma_k$ be a compact smooth manifold of dimension $k$ with boundary $\partial \Sigma$
$n \geq k \in \mathbb{N}$;
$\omega \in \Omega^n_{X}$ 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. ) 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.
(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
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.
The transgression of this 0-form to the space of histories of fields
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
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
Its transgression is the function
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
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
By prop. the pullback of this form along the jet prolongation of fields is exact in the $\Sigma$-direction:
(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
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
while the pullback of $d \phi^a_{\mu_1\cdots \mu_k}$ vanishes at fixed $\Sigma$.
This means that
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.
We 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).
Last revised on August 1, 2018 at 11:55:44. See the history of this page for a list of all contributions to it.