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Definition

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)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$
$n$geometrystructureunextended structureextension byquantum extension
$\infty$higher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$quantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
$n$n-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
$n$smooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected $X$)