# nLab 2-plectic geometry

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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

higher and integrated Kostant-Souriau extensions

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

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

| $\infty$ | 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-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-modelhigher symplectic geometry$(n+1)$d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension $(n+1)$discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
$n$symplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometry$d = n+1$ AKSZ sigma-model

## References

### General

Over smooth manifolds, the general setup is discussed in

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

### Application to the string $\sigma$-model

Applications to the 2-dimensional string sigma-model are discussed in

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

and in

Revised on November 7, 2013 02:36:35 by Urs Schreiber (77.251.114.72)