Daniel Berwick Evans. Supersymmetries and supercharges. N is a spacetime (a supermanifold) (a Riemannian or Lorentzian manifold). F is a superspace of fields.

Lagrangian L ∈ Ω^{0,n}(F × N).

We have classical solutions M ⊂ F. Variational 1-forms. γ ∈ Ω_{loc}^{1,n−1}(F × N). Pick γ such that E(L) = δL + dγ ∈ Ω_{loc}^{1,n}(F × M) does not contain variations of derivatives of fields,

just variations of the fields themselves.

Remarks: (1) If L only depends on the 1-jet of fields, γ is uniquely determined. (2) By Taken’s theorem γ is generally only determined up to a d-closed (d-exact) form. (3) For the usual mechanics the restriction of γ to M is ∑_{i} p_{i}dq^{i}.

Examples: (1) Classical particles on a Riemannian manifold M. N = R, F = Map(R, M), L = (1/2)|ẋ|^{2}dt.

δL = ⟨δ_{∇}ẋ, ẋ⟩dt = ⟨δ_{∇}dx, ẋ⟩ = −⟨dδ_{∇}x, ẋ⟩ = −d⟨δx, ẋ⟩ + ⟨δx, d_{∇}ẋ⟩ = γ + E(L), E(L) = ⟨δx, ∇ẋẋ. Here ẋdt = dx.

Superparticle. N = R, F = {x: R → M, ψ ∈ Γ(x^{∗}πT M)}, L = (1/2)(|ẋ|^{2} + ⟨ψ, ∇ẋψ⟩)dt. δL = −d⟨ψ, δ_{∇}ψ⟩ + ⟨δ_{∇}ψ, d_{∇}ψ⟩ + ⟨δx, R(ψ, ψ)dx⟩ − d⟨δx, ẋ⟩ + ⟨δx, ∇ẋ, ẋ⟩.

Equations of motion: ∇ẋẋ = (1/2)R(ψ, ψ)ẋ, ∇ẋψ = 0. γ_{sp} = ⟨ẋ, δx⟩ + (1/2)⟨ψ, δ_{∇}ψ⟩. Symplectic structures on the space of classical solutions M. ω = δγ ∈ Ω_{loc}^{2,n−1}(F × N). ω = (d + δ)L on

M × N, L = L + γ. Hence (d + δ)ω = 0 on M × N.

Classical mechanics: dω = 0 on M, hence ω(t) = ω. In general, Ω(t) := ∫_{{t}×S} ω. By Stokes this does not depend on t.