nLab 2-plectic geometry

Idea

2-Plectic geometry is the higher generalization of symplectic geometry, the special case of n-plectic geometry (multisymplectic geometry) for $n = 2$. This is the input for higher prequantum geometry in degree 2.

As symplectic geometry naturally describes classical mechanics and, via geometric quantization, quantum mechanics, hence 1-dimensional quantum field theory, so 2-plectic geometry naturally describes 2-dimensional classical field theory and, via its higher geometric quantization, 2-dimensional QFT.

Examples

• A semisimple Lie group is canonically a 2-plectic manifold, with the canonical 3-form $\langle -, [-,-]\rangle$ on the Lie algebra (the canonical Lie algebra cocycle), extended to a left invariant differential form.

• A G₂-manifold is manifold of dimension 7 characterized by carrying a 2-plectic form.

• A (pre-)symplectic groupoid is a Lie groupoid equipped with a (pre-)2-plectic structure.

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 $X$)

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

• Nambu mechanics

References

General

Over smooth manifolds, the general setup is discussed in

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)

and considered in the general context of higher differential geometry/extended prequantum field theory in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber,

  Higher geometric prequantum theory,

  L-∞ algebras of local observables from higher prequantum bundles

Application to the string $\sigma$-model

Applications to the 2-dimensional string sigma-model (see at string Lie 2-algebra) are discussed in

• John Baez, Chris Rogers, Categorified Symplectic Geometry and the String Lie 2-Algebra, Homology Homotopy Appl. Volume 12, Number 1 (2010), 221-236. (arXiv:0901.4721,euclid).

• John Baez, Alexander Hoffnung, Chris Rogers, Categorified Symplectic Geometry and the Classical String (arXiv:0808.0246)

A survey of some (potential) applications of 2-plectic geometry in string theory and M2-brane models is in section 2 of

• Christian Saemann, Richard Szabo, Groupoid quantization of loop spaces (arXiv:1203.5921)

and in

• Christian Saemann, Richard Szabo, Quantization of 2-Plectic Manifolds (arXiv:1106.1890)