Peter Teichner. Hamiltonian approach to classical field theory. Data for a classical Lagrangian field theory: (1) Smooth (oriented) manifold M (spacetime); (2) A

smooth fiber bundle E → M (fields on M are Γ(M, E)); (3) Lagrangian density λ ∈ Ω^{n,0}(JE).

Note: Get an action S: Φ → R: M × Φ → JE is the jet map. We have Ω^{n,0}(JE) ⊃ Ω^{n}(JE) → Ω^{n}(M × Φ) → C^{∞}(Φ) ∋ S.

Goal for today: Filtration on the space of jets, Euler-Lagrange equations, derive a Hamiltonian field theory picture.

Example: (Riemannian sigma-model.) M and N are Riemannian, E = M × N → M, Φ = C^{∞}(M, N), λ(ϕ) = ∥T ϕ(x)∥^{2}dvol_{M}.

Definition: J^{∞}E = {(x, ϕ) | x ∈ M ∧ ϕ is a local section of E near x}/ ∼. Here (x, ϕ_{1}) ∼ (x, ϕ_{2}) iff ϕ_{1} has k-contact with ϕ_{2} at x for all k.

Lemma: In coordinates this means that derivatives of ϕ_{i} coincide at point x. So we have a sequence of bundles M ← J^{0}E = E ← J^{1}E = T E ← J^{2}E ← · · ·. JE = lim_{k} J^{k}E. Define C^{∞}(JE) := colim C^{∞}(J^{k}E) and Ω^{∗}(JE) := colim Ω^{∗}(J^{k}E). For any bundle J → M we have sub-dga of Ω^{∗}(J) constructed as follows. Ω_{H}^{∗}(J) = C^{∞}(J) ⊗_{C^{∞}M}

π^{∗}Ω^{∗}M. This leads to the Serre spectral sequence for de Rham cohomology. Key geometric fact about JE: it is the differential ideal of “contact form”. Jet bundle has a canonical flat connection: Ω^{1}(JE) = Ω_{H}(JE) ⊕ C^{1}(JE). Here C^{∗}(JE) = {ω ∈ Ω^{∗}JE | j^{∗}(ω)(ϕ_{u}) = 0 for all local sections ϕ_{u}}. Note that this works only for infinite jets.