nLab
jet bundle

Idea

For $p:P\to X$ a surjective submersion of smooth manifolds and $k\in \mathbb{N}$, 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

Concrete

(…)

General abstract

We discuss a general abstract definition of jet bundles.

Let

• $H$ be a cohesive (∞,1)-topos

• equipped with infinitesimal cohesion

  $$H\stackrel{\hookrightarrow }{\stackrel{\stackrel{{\Pi }_{\inf }}{\leftarrow }}{\overrightarrow{\underset{}{\leftarrow }}}}{H}_{\mathrm{th}}$$\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

  of quasicoherent (∞,1)-sheavess.

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

Notice that we have the canonical morphism

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

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

Definition

Write

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

$$X\to \mathrm{Mod}.$$X \to Mod \,.

We write

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

for the stable (∞,1)-category of quasicoherent (∞,1)-sheaves.

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

$${\Pi }_{\inf }(X)\to \mathrm{Mod}.$$\mathbf{\Pi}_{inf}(X) \to Mod \,.

We write

$$\mathrm{DQC}(X):=\mathrm{Hom}({\Pi }_{\inf }(X),\mathrm{Mod})$$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 $𝒟(X)$ to those over the structure sheaf $𝒪(X)$

$$(\mathrm{Jet}⊣F):{\mathrm{Alg}}_{𝒟(X)}\stackrel{\stackrel{\mathrm{Jet}}{\leftarrow }}{\underset{F}{\to }}{\mathrm{Alg}}_{𝒪(X)}.$$(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.

Related concepts

• jet space

• metric jet

• h-principle

Examples of sequences of infinitesimal and local structures

first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
	$\leftarrow$ differentiation		integration $\to$
derivative		Taylor series		germ		smooth function
tangent vector		jet		germ of curve		curve
Lie algebra		formal group		local Lie group		Lie group
Poisson manifold		formal deformation quantization		local strict deformation quantization		strict 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

• Jacob Lurie, Notes on crystals and algebraic D-modules (pdf)

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

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras

An exposition of this is in section 2.3 of

• Frédéric Paugam, Homotopical Poisson Reduction of gauge theories (pdf)

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)