nLab Kostant-Souriau extension

Idea

Traditional notion

Given a presymplectic manifold $(X,\omega)$ there is the Poisson bracket Lie algebra $\mathfrak{Poisson}(X.\omega)$ of Hamiltonians and their Hamiltonian vector fields. This is a extension of Lie algebras of the Lie algebra of just Hamiltonian vecotor fields. Over a connected manifold it is an extension by $\mathbb{R}$. The Lie integration of this extension is the quantomorphism group-extension of the group of Hamiltonian symplectomorphisms.

These extensions are called Kostant-Souriau extensions after Bertram Kostant and Jean-Marie Souriau. They play a central role in geometric quantization.

Refinement to higher geometry

For $(X,\omega)$ a pre-n-plectic manifold there is an L-∞ algebra extension that generalizes the Kostant Souriau extension: the Poisson bracket L-∞ algebra (Rogers).

These in turn are special cases of L-∞ algebra extension/∞-group extensions which are variants of the higher Atiyah groupoid-extensions that exist in general in cohesive higher geometry (Hgpt), as indicated in the following table:

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

The traditional Kostant-Souriau extension appears for instance prop. 2.3.9 in

• Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993)

The refinement to n-plectic geometry is due to

• Chris Rogers, $L_\infty$ algebras from multisymplectic geometry ,

  Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).

A general characterization in higher geometry is in

• Higher geometric prequantum theory