nLab Heisenberg n-group

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.

Properties

Relation to parameterized WZW models

• parameterized WZW model

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).

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory, 2013 (arXiv:1304.0236)