Hamiltonian form

For $\omega \in \Omega^{n+1}(X)$ an n-plectic geometry, and for $v \in \Gamma(X)$ a vector field, a **Hamiltonian form** for $v$ is, if it exists, a differential form $h \in \Omega^n(X)$ such that

$\iota_{v} \omega = \mathbf{d} h
\,.$

For $n = 1$ this reduces to the notion of a Hamiltonian function on a symplectic manifold.

If a Hamiltonian form for $v$ exists then $v$ is called a Hamiltonian vector field.

The Hamiltonian forms are the local classical observables/prequantum observables in higher prequantum field theory, often called *local currents*. They form the Poisson-bracket Lie n-algebra of local observables.

**higher and integrated Kostant-Souriau extensions**:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$

(extension are listed for sufficiently connected $X$)