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transgression of differential forms

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Integration theory

Contents

Idea

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

τ Σω| Φ () Σ(Φ ()) *ω. \tau_{\Sigma} \omega\vert_{\Phi_{(-)}} \coloneqq \int_\Sigma (\Phi_{(-)})^\ast \omega \,.

This differential form τ Σω\tau_\Sigma \omega on the mapping space is called the transgression of ω\omega with respect to Σ\Sigma

This construction has a variety of immediate generalizations, for instance Σ\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

(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 XX is defined by specifying, in a consistent way, what counts as a smooth functions UXU \to X from a Cartesian space UU (a “plot” of XX). Given two smooth sets XX and YY then a smooth function f:XYf \;\colon\; X \longrightarrow Y is a function that takes plots UϕXU \overset{\phi}{\to} X of XX to plots fϕ:UYf \circ \phi \colon U \to Y of YY.

A key example of a smooth set which is in general not a smooth manifold is the mapping space [X,Y][X,Y] between two smooth sets XX and YY, hence the set of all smooth functions XYX \to Y equipped with a smooth structure itself. Namely a plot ϕ ():U[X,Y]\phi_{(-)} \colon U \to [X,Y] is defined to be a smooth function ϕ ()():U×XY\phi_{(-)}(-) \colon U \times X \to Y out of the Cartesian product of UU with XX to YY, 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 nn \in \mathbb{N} the smooth set Ω n\mathbf{\Omega}^n which is the “smooth classifying space” for differential n-forms, defined by the rule that a smooth function ϕ:UΩ n\phi \colon U \to \mathbf{\Omega}^n is equivalently a smooth differential nn-form on UU (to be thought of as the pullback of a “universal nn-form” on Ω n\mathbf{\Omega}^n along ϕ\phi). It follows from this in particular that for XX any smooth manifold then smooth functions XΩ nX \to \mathbf{\Omega}^n are equivalent to smooth nn-forms on XX. Accordingly we may say that for XX any smooth set (which may be far from being a smooth manifold) then a differential nn-form on XX is equivalently a smooth function XΩ nX \to \mathbf{\Omega}^n. Under this identification the operation of pullback of differential forms along some smooth function f:YXf \colon Y \to X is just composition of smooth functions f *ω:YfXωΩ nf^\ast \omega \colon Y \overset{f}{\to} X \overset{\omega}{\to} \mathbf{\Omega}^n.

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

(There is a smooth space that has more right to be called “the” classifying space of differential nn-foms on Σ\Sigma, namely the concretification 1[Σ,Ω n]\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. XX be a smooth set;

  2. nkn \geq k \in \mathbb{N};

  3. Σ k\Sigma_k be a compact smooth manifold of dimension kk.

Then we write

Σ:[Σ k,Ω n]Ω nk \int_{\Sigma} \;\colon\; [\Sigma_k, \mathbf{\Omega}^n] \longrightarrow \mathbf{\Omega}^{n-k}

for the smooth function which takes a plot ω ():U[Σ,Ω k]\omega_{(-)} \colon U \to [\Sigma, \mathbf{\Omega}^k], hence equivalently a differential nn-form ω ()()\omega_{(-)}(-) on U×ΣU \times \Sigma to the result of integration of differential forms over Σ\Sigma:

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

(transgression of differential forms to mapping spaces)

Let

  1. XX be a smooth set;

  2. nkn \geq k \in \mathbb{N};

  3. Σ k\Sigma_k be a compact smooth manifold of dimension kk.

Then the operation of transgression of differential nn-forms on XX with respect to Σ\Sigma is the function

τ Σ Σ[Σ,]:Ω n(X)Ω nk([Σ,X]) \tau_\Sigma \coloneqq \int_\Sigma [\Sigma,-] \;\colon\; \Omega^n(X) \to \Omega^{n-k}([\Sigma,X])

from differential nn-forms on XX to differential nkn-k-forms on the mapping space [Σ,X][\Sigma,X] which takes the differential form corresponding to the smooth function

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

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

τ Σω Σ[Σ,ω]:[Σ,X][Σ,ω][Σ,Ω n] ΣΩ nk, \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 [Σ,ω][\Sigma,\omega] is the mapping space functor on morphisms and Σ\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

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

is equivalently a smooth function of the form

ϕ ()():U×ΣX. \phi_{(-)}(-) \;\colon\; U \times \Sigma \to X \,.

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

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

which is equivalently a differential form

(ϕ ()()) *ωΩ n(U×Σ). (\phi_{(-)}(-))^\ast \omega \in \Omega^n(U \times \Sigma) \,.

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

τ Σω| ϕ ()= Σ(ϕ ()()) *ωΩ nk(U). \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. XX be a smooth set;

  2. nkn \geq k \in \mathbb{N};

  3. Σ k\Sigma_k be a compact smooth manifold of dimension kk.

Then the operation of transgression of differential nn-forms on XX with respect to Σ\Sigma is the function

τ Σ Σev *:Ω n(X)ev *Ω n(Σ×[Σ,X]) ΣΩ nk([Σ,X]) \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 nn-forms on XX to differential nkn-k-forms on the mapping space [Σ,X][\Sigma,X] which is the composite of forming the pullback of differential forms along the evaluation map ev:[Σ,X]×ΣXev \colon [\Sigma, X] \times \Sigma \to X with integration of differential forms over Σ\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 γ:U[Σ,X]\gamma \colon U \to [\Sigma, X] the pullbacks of the two forms to UU coincide.

For def. we get

γ * Σev *A= Σ(γ,id Σ) *ev *AΩ n(U) \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 (()×Σ[Σ,])( (-)\times \Sigma \dashv [\Sigma,-])-adjunct γ˜:U×ΣΣ\tilde \gamma : U \times \Sigma \to \Sigma of γ\gamma, which is given by applying the left adjoint ()×Σ(-)\times \Sigma and then postcomposing with the adjunction counit ev\mathrm{ev}:

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

Hence the integral is now

= Σγ˜ *A. \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:

γ˜ H(U×Σ,X) H(U×Σ,A) H(U×Σ,Ω n+k) Σ(U) Ω n(U) γ H(U,[Σ,X]) H(U,[Σ,A]) H(U,[Σ,Ω n+k]) H(U, Σ) H(U,Ω n) \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 H(,)\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 Σ\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 Σ\Sigma, then a type of fields on Σ\Sigma is a smooth fiber bundle

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

called the field bundle,

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

Φ:ΣE \Phi \colon \Sigma \longrightarrow E

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

fbΦ=idAAAAAAA E Φ fb Σ = Σ. 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

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

This is a smooth set by declaring that a smooth family Φ ()\Phi_{(-)} of field configurations, parameterized over any Cartesian space UU is a smooth function

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

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

E Φ ()() fb U×Σ pr 2 Σ. \array{ && E \\ & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}} \\ U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma } \,.

More generally, let SΣS \hookrightarrow \Sigma be a submanifold of spacetime. We write N Σ(S)ΣN_\Sigma(S) \hookrightarrow \Sigma for its infinitesimal neighbourhood in Σ\Sigma.

If EfbΣE \overset{fb}{\to} \Sigma is a field bundle then the space of histories of fields restricted to SS, to be denoted

Γ S(E)Γ N Σ(S)(E| N ΣS)H \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 Σ(S)N_\Sigma(S).

There is a canonical evaluation smooth function

(1)ev S:N ΣS×Γ S(E)E ev_S \;\colon\; N_\Sigma S \times \Gamma_{S}(E) \longrightarrow E

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

Definition

(spacetime support)

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

For every subset SΣS \subset \Sigma let

J Σ (E)| S ι S J Σ (E) (pb) S Σ \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 EE.

The spacetime support supp Σ(A)supp_\Sigma(A) of a differential form AΩ (J Σ (E))A \in \Omega^\bullet(J^\infty_\Sigma(E)) on the jet bundle of EE is the topological closure of the maximal subset SΣS \subset \Sigma such that the restriction of AA to the jet bundle restrited to this subset vanishes:

supp Σ(A)Cl({xΣ|ι {x} *A=0}) supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma | \iota_{\{x\}^\ast A = 0} \} )

We write

Ω Σ,cp r,s(E){AΩ Σ r,s(E)|supp Σ(A)is compact}Ω Σ r,s(E) \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 EfbΣE \overset{fb}{\to} \Sigma be a field bundle over a spacetime Σ\Sigma (def. ), with induced jet bundle J Σ (E)J^\infty_\Sigma(E)

For Σ rΣ\Sigma_r \hookrightarrow \Sigma be a submanifold of spacetime of dimension rr \in \mathbb{N}, then transgression of variational differential forms to Σ r\Sigma_r is the function

τ Σ r:Ω Σ,cp r,(E)Ω (Γ Σ r(E)) \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 AΩ Σ,cp r,(E)A \in \Omega^{r,\bullet}_{\Sigma,cp}(E) to the differential form τ Σ rΩ (Γ Σ r(E))\tau_{\Sigma_r} \in \Omega^\bullet(\Gamma_{\Sigma_r}(E)) which to a smooth family on field configurations

Φ ():U×N ΣΣ rE \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 Σ (Φ ())j^\infty_\Sigma(\Phi_{(-)}) followed by the integration of differential forms over Σ r\Sigma_r:

(τ ΣA) Φ () Σ r(j Σ (Φ ())) *Ω (U). (\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. XX be a smooth set;

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

  3. nkn \geq k \in \mathbb{N};

  4. ωΩ X n\omega \in \Omega^n_{X} a closed differential form.

Write

()| Σ[ΣΣ,X]:[Σ,X][Σ,X] (-)\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 Σ\Sigma to smooth functions on the boundary Σ\partial \Sigma.

Then the operations of transgression of differential forms (def. ) to Σ\Sigma and to Σ\partial \Sigma, respectively, are related by

d(τ Σ(ω))=(1) k+1(()| Σ) *τ Σ(ω)AAAAAAAA[Σ,X] τ Σ(ω) Ω nk ()| Σ (1) k+1d [Σ,X] τ Σ(ω) Ω nk+1. 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 Σ\Sigma happens to have no boundary (is a closed manifold) then transgression over Σ\Sigma takes closed differential forms to closed differential forms.

Proof

Let ϕ ()():U×ΣX\phi_{(-)}(-) \colon U \times \Sigma \to X be a plot of the mapping space [Σ,X][\Sigma, X]. Notice that the de Rham differential on the Cartesian product U×ΣU \times \Sigma decomposes as

d=d U+d Σ. d = d_U + d_\Sigma \,.

Now we compute as follows:

dτ Σω| ϕ () =d U Σ(ϕ ()()) *ω =(1) k Σd U(ϕ ()()) *ω =(1) k Σ(ddΣ)(ϕ ()()) *ω =(1) k Σd(ϕ ()()) *ω(1) k Σd Σ(ϕ ()()) *ω =(1) k Σ(ϕ ()()) *dω=0(1) k Σd Σ(ϕ ()()) *ω =(1) k Σd Σ(ϕ ()()) *ω =(1) k Σ(ϕ ()()) *ω =(1) kτ Σω| ϕ () \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 Γ Σ(E)\Gamma_\Sigma(E) of fields (def. ).

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

Φ ():U×ΣE. \Phi_{(-)} \;\colon\; U \times \Sigma \longrightarrow E \,.

Let bC cp (Σ)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+1p+1-form on the jet bundle with compact spacetime support.

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

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

τ Σ(bdvol Σ)Ω 0(Γ Σ(E)) \tau_\Sigma (b dvol_\Sigma) \in \Omega^0( \Gamma_\Sigma(E) )

is the differential form on Γ Σ(E)\Gamma_\Sigma(E) which restricted to the given family of field configurations Φ ():uΦ u\Phi_{(-)} \colon u \mapsto \Phi_u yields the function

τ Σ(bdvol Σ)| Φ:u Σbdvol Σ \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 bb against the given volume form.

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

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

Its transgression is the function

τ Σ(ϕ abdvol Σ) Φ=(u ΣΦ (u) a(x)b(x)dvol Σ(x)) \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 Φ u\Phi_{u} in the family the value its aa-component integrated against bdvol Σb dvol_\Sigma.

Similarly the transgression of ϕ ,μ a\phi^a_{,\mu} is the function

τ Σ(ϕ ,μ abdvol Σ) Φ=(u ΣΦ u ax μb(x)dvol Σ(x)) \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 μ\muth derivative of its aath component against bdvol Σb dvol_\Sigma.

Next consider a horizontally exact variational form

dαΩ Σ,cp p+1,s(E). 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 Σ\Sigma-direction:

(j Σ Φ ()) *(dαbdvol Σ)=d Σ(j Σ Φ ()) *αbdvol Σ, (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 Σd = d_U + d_\Sigma for the de Rham differential on U×ΣU \times \Sigma). It follows that the integral over Σ\Sigma vanishes.

Now let

δαϕ ,μ 1μ k abdvol ΣΩ Σ p+1,1(E) \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 Σ (Φ ()):U×ΣJ Σ (E)j^\infty_\Sigma(\Phi_{(-)}) \colon U \times \Sigma \to J^\infty_\Sigma(E) has two contributions: one from the variation along Σ\Sigma, the other from variation along UU.

By prop. , for fixed uUu \in U the pullback along the jet prolongation vanishes.

On the other hand, for fixed sΣs \in \Sigma, the pullback of dϕ μ 1μ k a\mathbf{d} \phi^a_{\mu_1\cdots \mu_k} is

d U kΦ ()x μ 1x μ k d_U \frac{ \partial^k \Phi_{(-)}}{\partial x^{\mu^1} \cdots \partial x^{\mu_k}}

while the pullback of dϕ μ 1μ k ad \phi^a_{\mu_1\cdots \mu_k} vanishes at fixed Σ\Sigma.

This means that

τ Σ(δϕ ,μ 1μ k a)=dτ Σ(ϕ μ1μ k a) \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 UU) 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

Let XHX \in \mathbf{H} and consider a circle group-principal connection :XBU(1) conn\nabla \colon X \to \mathbf{B}U(1)_{conn} over XX. By the discussion in Dirac charge quantization and the electromagnetic field above this encodes an electromagnetic field on XX. 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

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

the electromagnetic potential.

Let then Σ=S 1\Sigma = S^1 be the circle. The transgression of the electromagnetic potential to the loop space of XX

S 1[S 1,A]:[S 1,X][S 1,A][S 1,Ω 1] S 1Ω 0 \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 AA is S em= S 1[S 1,A]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 𝔤\mathfrak{g} be a Lie algebra with binary invariant polynomial ,:𝔤𝔤\langle -,-\rangle \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}.

For instance 𝔤\mathfrak{g} could be a semisimple Lie algebra and ,\langle -,-\rangle its Killing form. In particular if 𝔤=𝔰𝔲(n)\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

F ()F ():Ω 1(,𝔤)F ()Ω 2(,𝔤)()()Ω 4(,𝔤𝔤),Ω 4 \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 UU \in CartSp this sends a differential 1-form AΩ 1(U)A \in \Omega^1(U) to the differential 4-form

F AF AΩ 4(U). \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

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

Now let Σ\Sigma be an oriented closed smooth manifold. The transgression of the above 4-form to the mapping space out of Σ\Sigma yields the 2-form

ω ΣF ()F ():Ω 1(Σ,𝔤)[Σ,BG conn][Σ,F ()F ()][Σ,Ω 4] ΣΩ 2 \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 Σ\Sigma.

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

F A=F A Σ+δA, F_A = F_A^{\Sigma} + \delta A \,,

where F A ΣF_A^{\Sigma} is the curvature component with all legs along Σ\Sigma and where

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

is the variational derivative of AA.

This means that in the 4-form

F AF A=F A ΣF A Σ+2F A ΣδA+δAδAΩ 4(Σ×U) \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 UU. Hence we find that the transgressed 2-form is

ω= ΣδAδA:Ω 1(Σ,𝔤)Ω 2. \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

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

which is the phase space of 𝔤\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 07:55:44. See the history of this page for a list of all contributions to it.