# nLab Heisenberg n-group

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

A Heisenberg $n$-group is a sub-∞-group of a quantomorphism n-group of a n-plectic space $(G, \omega)$ equipped with ∞-group structure, on those whose underlying n-plectomorphisms act by lect ∞-action of $G$ on itself. This is the higher refinement of the traditional notion of Heisenberg group.

## Definition

For $(G, \omega)$ an n-plectic geometry with higher geometric prequantization $(G, \nabla)$ and for $G$ equipped with ∞-group structure, the corresponding Heisenberg $\infty$-group is the sub-∞-group of the quantomorphism n-group on those elements whose corresponding n-plectomorphism is given by such a left action.

The corresponding Lie n-algebra is the Heisenberg Lie n-algebra.

## Examples

### String 2-group is Heisenberg 2-group of WZW gerbe

For $G$ a compact simply connected simple Lie group, there is the “WZW gerbe”, hence the circle 2-bundle with connection on $G$ whose curvature 3-form is the left invariant extension $\langle \theta \wedge [\theta \wedge \theta]\rangle$ of the canonical Lie algebra 3-cocycle to the group

$\mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2 \,.$
###### Proposition

The string 2-group is the smooth 2-groupo of automorphism of $\mathcal{L}_{WZW}$ which cover the left action of $G$ on itself (hence the “Heisenberg 2-group” of $\mathcal{L}_{WZW}$ regarded as a prequantum 2-bundle)

$\mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G) \,,$

This is due to (Fiorenza-Rogers-Schreiber 13, section 6.2.1).

higher and integrated Kostant-Souriau extensions:

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

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$
$n$geometrystructureunextended structureextension byquantum extension
$\infty$higher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$quantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
$n$n-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
$n$smooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected $X$)

## References

Last revised on October 17, 2013 at 22:35:46. See the history of this page for a list of all contributions to it.