For $(X,\omega)$ a symplectic manifold, a vector field $v \in \Gamma(T X)$ is called a Hamiltonian vector field if its contraction with the differential 2-form $\omega$ is exact: if there exists $\alpha \in C^\infty(X)$ such that
In this case $\alpha$ is called a Hamiltonian for $v$.
For $(X,\omega)$ an n-plectic manifold, a vector field $v \in \Gamma(T X)$ is called a Hamiltonian vector field if is contraction with the $(n+1)$-form $\omega$ is exact: there is $\alpha \in \Omega^{n-1}(X)$ such that
In this case $\alpha$ is called a Hamiltonian (n-1)-form for $v$.
We discuss now the notion of Hamiltonian vector fields in the full generality internal to a cohesive (∞,1)-topos $\mathbf{H}$. We write out the discussion for the case $\mathbf{H} =$ Smooth∞Grpd for convenience, but any other choice of cohesive $(\infty,1)$-topos works as well.
Consider the circle n-group $\mathbf{B}^{n-1}U(1)$ and the corresponding coefficient object $\mathbf{B}^n U(1)_{conn} \in \mathbf{H}$ for $U(1)$-differential cohomology in degree $(n+1)$, the smooth moduli stack of circle n-bundles with connection.
For any $X \in \mathbf{H}$, a morphism $\omega \colon X \to \Omega^{n+1}_{cl}$ is a pre-n-plectic structure on $X$. For instance $(X,\omega)$ might be a symplectic ∞-groupoid.
A higher geometric prequantization of $(X,\omega)$ is a lift $\nabla$ in
The quantomorphism n-group of this prequantization is
where
$\mathbf{Aut}(\nabla)$ is the automorphism ∞-group of $\nabla$ formed in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$
$\prod_{\mathbf{B}^n U(1)_{conn}} \colon \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \to \mathbf{H}$ is the dependent product (∞,1)-functor.
There is a canonical homomorphism of ∞-groups
to the automorphism ∞-group of $X$ (the diffeomorphism group of $X$), given as the restriction to invertible endomorphisms of the canonical morphism
which is discussed at internal hom – Examples – In slice categories.
The Hamiltonian symplectomorphism n-group $\mathbf{HamSymp}(X,\omega)$ of $(X,\omega)$ is the ∞-image of this morphism $p$, hence the factorization
of $p$ by an effective epimorphism followed by a monomorphism.
The corresponding ∞-Lie algebra
we call the $\infty$-Lie algebra of Hamiltonian vector fields on $(X,\omega)$.
More explicitly:
A Hamiltonian diffeomorphism $\phi$ on
on $(X, \omega)$ is an element $\phi \colon X \stackrel{\simeq}{\to} X$ in the automorphism ∞-group $\phi \in \mathbf{Aut}(X)$ such that it fits into a diagram of the form
in $\mathbf{H}$.
For $n = 1$ and $(X, \omega)$ an ordinary prequantizable symplectic manifold regarded as a smooth $\infty$-groupoid, this definition reproduces the ordinary definition of Hamiltonian vector fields above.
In particular it is independent of the choice of prequantum line bundle.
To compute the Lie algebra of this, we need to consider smooth 1-parameter families of Hamiltonian diffeomorphisms and differentiate them.
Assume first that the prequantum line bundle is trivial as a bundle, with the connection 1-form of $\nabla$ given by a globally defined $A \in \Omega^1(X)$ with $d A = \omega$. Then the existence of the diagram in def. 4 is equivalent to the condition
where $\alpha(t) \in C^\infty(X)$. Differentiating this at 0 yields the Lie derivative
where $v$ is the vector field of which $t \mapsto \phi(t)$ is the flow and where $\alpha' := \frac{d}{dt} \alpha$.
By Cartan calculus this is equivalently
and using that $A$ is the connection on a prequantum circle bundle for $\omega$
This says that for $v$ to be Hamiltonian, its contraction with $\omega$ must be exact. This is precisely the definition of Hamiltonian vector fields. The corresponding Hamiltonian function here is
In the general case that the prequantum bundle is not trivial, we can present it by a Cech cocycle on the Cech nerve $C(P_* X \to X)$ of the based path space surjective submersion (regarding $P_* X$ as a diffeological space and choosing one base point per connected component, or else assuming without restriction that $X$ is connected).
Any diffeomorphism $\phi = \exp(v) : X \to X$ lifts to a diffeomorphism $P_*\phi : P_* X \to P_* X$ by setting $P_* \phi(\gamma) : (t \in [0,1]) \mapsto \exp(t v)(\gamma(t))$. This way the Hamiltonian diffeomorphism is presented in the model structure on simplicial presheaves by a diagram
Now the same argument as above applies for $P_* X$.
An action of a Lie algebra by (flows of) Hamiltonian vector fields that can be lifted to a Hamiltonian action is equivalently given by a moment map. See there for details.
Every Hamiltonian vector field is in particular a symplectic vector field. Where a symplectic vector field only preserves the symplectic form, a Hamiltonian vector field also preserves the connection on its prequantum line bundle.
For $(X, \omega)$ a finite dimensional symplectic manifold, there is an exact sequence
This appears as (Brylinski, 2.3.3).
Let $(X, \omega)$ be a connected symplectic manifold. Then there is a central extension of Lie algebras
This is a special case of what is called the Kostant-Souriau central extension. See around (Brylinski, prop. 2.3.9).
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)
(extension are listed for sufficiently connected $X$)
The auto-symplectomorphisms on a symplectic manifold form a group, of which the symplectic vector fields generate the connected component. The Hamiltonian vector fields among the symplectic ones generate the group of Hamiltonian symplectomorphisms.
(…)
The Hamiltonian vector field of a given function may also be called its symplectic gradient.
The generalization to multisymplectic geometry/n-plectic geometry: Hamiltonian n-vector fields
A textbook reference is section II.3 in
For more references on the ordinary notion of Hamiltonian vector fields see the references at symplectic geometry and geometric quantization.
The notion of Hamiltonian vector field in n-plectic geometry is discussed in
The notion of Hamiltonian vector field for $n$-plectic cohesive $\infty$-groupoids is discussed in section 4.8.1 of
See also at higher geometric quantization.