nLab jet bundle

Context

Differential geometry

differential geometry

synthetic differential geometry

Contents

Idea

For $p:P\to X$ a surjective submersion of smooth manifolds and $k\in ℕ$, the order $k$- jet bundle ${J}_{k}P\to X$ is the bundle whose fiber over a point $x\in X$ is the space of equivalence classes of germs of sections of $p$ where two germs are considered equivalent if their first $k$ derivatives coincide.

Definition

(…)

General abstract

We discuss a general abstract definition of jet bundles.

Let

• $H$ be a cohesive (∞,1)-topos

• equipped with infinitesimal cohesion

$H\stackrel{↪}{\stackrel{\stackrel{{\Pi }_{\mathrm{inf}}}{←}}{\stackrel{\stackrel{}{⟶}}{\underset{}{←}}}}{H}_{\mathrm{th}}$\mathbf{H} \stackrel{\hookrightarrow}{\stackrel{\overset{\Pi_{inf}}{\leftarrow}}{\stackrel{\overset{}{\longrightarrow}}{\underset{}{\leftarrow}}}} \mathbf{H}_{th}
• and equipped with an (∞,2)-sheaf

Mod $:\phantom{\rule{thickmathspace}{0ex}}{H}^{\mathrm{op}}\to$ Stab(∞,1)Cat

For $X\in H$, write ${\Pi }_{\mathrm{inf}}\left(X\right)$ for the corresponding de Rham space object.

Notice that we have the canonical morphism

$i:X\to {\Pi }_{\mathrm{inf}}\left(X\right)$i : X \to \mathbf{\Pi}_{inf}(X)

(“inclusion of constant paths into all infinitesimal paths”).

Definition

Write

$\mathrm{Jet}:H/X\stackrel{\stackrel{{i}^{*}}{←}}{\underset{\mathrm{Jet}:={i}_{*}}{\to }}{H}_{{\Pi }_{\mathrm{inf}}}\left(X\right)$Jet : \mathbf{H}/X \stackrel{\overset{i^*}{\leftarrow}}{\underset{Jet := i_*}{\to}} \mathbf{H}_{\mathbf{\Pi}_{inf}}(X)

for the corresponding base change geometric morphism.

Its direct image we call the jet bundle (∞,1)-functor .

In the context of D-schemes this is (BeilinsonDrinfeld, 2.3.2). See (Paugam, section 2.3) for a review. There this is expressed dually in terms of algebras in D-modules. We indicate how the translation works

Remark

In terms of differential homotopy type theory this means that forming “jet types” of dependent types over $X$ is the dependent product operation along the unit of the infinitesimal shape modality

$\mathrm{jet}\left(E\right)≔\prod _{X\to {\Pi }_{\mathrm{inf}}X}E\phantom{\rule{thinmathspace}{0ex}}.$jet(E) \coloneqq \underset{X \to \Pi_{inf}X}{\prod} E \,.
Definition

A quasicoherent (∞,1)-sheaf on $X$ is a morphism of (∞,2)-sheaves

$X\to \mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$X \to Mod \,.

We write

$\mathrm{QC}\left(X\right):=\mathrm{Hom}\left(X,\mathrm{Mod}\right)$QC(X) := Hom(X, Mod)

A D-module on $X$ is a morphism of (∞,2)-sheaves

${\Pi }_{\mathrm{inf}}\left(X\right)\to \mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{\Pi}_{inf}(X) \to Mod \,.

We write

$\mathrm{DQC}\left(X\right):=\mathrm{Hom}\left({\Pi }_{\mathrm{inf}}\left(X\right),\mathrm{Mod}\right)$DQC(X) := Hom(\mathbf{\Pi}_{inf}(X), Mod)

for the stable (∞,1)-category of D-modules.

The Jet algebra functor is the left adjoint to the forgetful functor from commutative algebras over $𝒟\left(X\right)$ to those over the structure sheaf $𝒪\left(X\right)$

$\left(\mathrm{Jet}⊣F\right):{\mathrm{Alg}}_{𝒟\left(X\right)}\stackrel{\stackrel{\mathrm{Jet}}{←}}{\underset{F}{\to }}{\mathrm{Alg}}_{𝒪\left(X\right)}\phantom{\rule{thinmathspace}{0ex}}.$(Jet \dashv F) : Alg_{\mathcal{D}(X)} \stackrel{\overset{Jet}{\leftarrow}}{\underset{F}{\to}} Alg_{\mathcal{O}(X)} \,.

Application

Typical Lagrangians in quantum field theory are defined on jet bundles. Their variational calculus is governed by Euler-Lagrange equations.

Examples of sequences of infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$←$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

The abstract characterization of jet bundles as the direct images of base change along the de Rham space projection is noticed on p. 6 of

The explicit description in terms of formal duals of commutative monoids in D-modules is in

An exposition of this is in section 2.3 of

Standard textbook references include

• G. Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory, Lectures for theoreticians, arXiv:0908.1886

• Shihoko Ishii, Jet schemes, arc spaces and the Nash problem, arXiv:math.AG/0704.3327

• D. J. Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.

A discussion of jet bundles with an eye towards discussion of the variational bicomplex on them is in chapter 1, section A of

• Ian Anderson, The variational bicomplex (pdf)

Discussion of jet-restriction of the Haefliger groupoid is in

• Arne Lorenz, Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method, Thesis (pdf)

Revised on November 5, 2013 02:17:23 by Urs Schreiber (145.116.129.122)