nLab Hamiltonian symplectomorphism infinity-group

Idea

The generalization of the traditional notion to symplectomorphism group of a symplectic manifold to n-plectic spaces in higher geometry

Definition

For $\mathbf{H}$ a cohesive (∞,1)-topos equipped with differential cohesion, let $\mathbb{G} \in Grp(\mathbf{H})$ by an ∞-group equipped with braided ∞-group structure. Write $\mathbf{B}\mathbb{G}_{conn}$ for the corresponding moduli ∞-stack of $\mathbb{G}$-principal ∞-connections (see there). For $\nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn}$ a $\mathbb{G}$-principal ∞-connection on some object $X \in \mathbf{H}$, which here we call a prequantum ∞-bundle. the corresponding quantomorphism ∞-group by definition comes with a canonical homomorphism

$$\mathbf{QuantMorph}(\nabla) \to \mathbf{Aut}(X)$$

to the automorphism ∞-group of $X$. The Hamiltonian symplectomorphism $\infty$-group $\mathbf{HamSym}(\nabla)$ is the 1-image of this map

$$\mathbf{QuantMorph}(\nabla) \to \mathbf{HamSymp}(\nabla) \hookrightarrow \mathbf{Aut}(X) \,.$$

Related concepts

higher and integrated Kostant-Souriau extensions:

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

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

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\infty$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)

References

• differential cohomology in a cohesive topos

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher prequantum geometry?