# 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

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

Mod ${H}^{\mathrm{op}}\to$ 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

###### 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
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)

Revised on May 2, 2013 21:43:25 by Urs Schreiber (76.125.224.116)