nLab
quantomorphism group

Idea

Given a (pre)symplectic manifold $(X,\omega )$, its quantomorphism group is the Lie group that integrates the Lie bracket inside the Poisson algebra of $(X,\omega )$. This is a circle group-central extension of the group of Hamiltonian symplectomorphisms. It extends and generalizes the Heisenberg group of a symplectic vector space.

(Warning on terminology: A more evident name for the quantomorphism group might seem to be “Poisson group”. But this already means something different, see Poisson Lie group.)

Traditional construction

Over a symplectic manifold $(X,\omega )$ an explicit construction of the corresponding quantomorphism group is obtained by choosing $(P\to X,\nabla )$ a prequantum circle bundle, regarded with an Ehresmann connection 1-form $A$ on $P$, and then defining

$$\mathrm{QuantomorphismGroup}\hookrightarrow \mathrm{Diff}(P)$$QuantomorphismGroup \hookrightarrow Diff(P)

to be the subgroup of the diffeomorphism group $P\stackrel{\simeq }{\to }P$ on those diffeomorphisms that preserve $A$. In other words, the quantomorphism group is the group of equivalences of bundles with connection that need not cover the identity diffeomorphism on the base manifold $X$.

Notice that the tuple $(P,A)$ is a regular contact manifold (see the discussion there), and so the quantomorphism group is equivalently that of contactomorphisms $(P,A)\to (P,A)$ of weight 0.

In higher geometry

This perspective lends itself to a more abstract description: we may regard the prequantum circle bundle as being modulated by a morphism

$$\nabla :X\to BU(1{)}_{\mathrm{conn}}$$\nabla : X \to \mathbf{B} U(1)_{conn}

in the cohesive (∞,1)-topos $H=$ Smooth∞Grpd, with domain the given symplectic manifold and codomain the smooth moduli stack for circle bundles with connection. This in turn may be regarded as an object $\nabla \in H_{/BU(1{)}_{\mathrm{conn}}}$ in the slice (∞,1)-topos. Then the quantomorphism group is the automorphism group

$$\mathrm{QuantMorph}(X.\nabla )≔\prod _{BU(1{)}_{\mathrm{conn}}}\mathrm{Aut}(\nabla )$$\mathbf{QuantMorph}(X.\nabla) \coloneqq \underset{\mathbf{B}U(1)_{conn}}{\prod} \mathbf{Aut}(\nabla)

in $H$ (Sch).

From this it is clear what the quantomorphism ∞-group of an n-plectic ∞-groupoid should be: for

$$\nabla :X\to B^{n}U(1{)}_{\mathrm{conn}}$$\nabla : X \to \mathbf{B}^n U(1)_{conn}

the morphism modulating a prequantum circle n-bundle, the corresponding quantomorphism $n$-group is again $\mathrm{Aut}(\nabla )$, now formed in $H_{/B^{n}U(1{)}_{\mathrm{conn}}}$

Properties

Smooth structure

The quantomorphism group for a symplectic manifold may naturally be equipped with the structure of a group object in ILH manifolds (Omori, Ratiu-Schmid), as well as in convenient manifolds (Vizman, prop.).

Group extension

Proposition

For $(X,\omega )$ a connected symplectic manifold there is a central extension of groups

$$1\to U(1)\to \mathrm{QuantomorphismGroup}(X,\omega )\to \mathrm{HamiltonianSymplectomorphisms}(X,\omega )\to 1.$$1 \to U(1) \to QuantomorphismGroup(X,\omega) \to HamiltonianSymplectomorphisms(X,\omega) \to 1 \,.

This is due to (Kostant). It appears also (Brylinski, prop. 2.4.5).

higher and integrated Kostant-Souriau extensions

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

$$(\Omega 𝔾)\mathrm{FlatConn}(X)\to \mathrm{QuantMorph}(X,\nabla )\to \mathrm{HamSympl}(X,\nabla )$$(\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
$\mathrm{\infty }$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega 𝔾)$-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 $X$)

Related concepts

• Hamiltonian action

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:	Heisenberg ∞-group	$\hookrightarrow$	quantomorphism ∞-group	$\hookrightarrow$	∞-bisections of higher Courant groupoid	$\hookrightarrow$	∞-bisections of higher Atiyah groupoid
L-∞ algebras:	Heisenberg L-∞ algebra	$\hookrightarrow$	Poisson L-∞ algebra	$\hookrightarrow$	Courant L-∞ algebra	$\hookrightarrow$	twisted vector fields

higher Atiyah groupoid

higher Atiyah groupoid:	standard higher Atiyah groupoid	higher Courant groupoid	groupoid version of quantomorphism n-group
coefficient for cohomology:	$B𝔾$	$B(B{𝔾}_{\mathrm{conn}})$	$B{𝔾}_{\mathrm{conn}}$
type of fiber ∞-bundle:	principal ∞-bundle	principal ∞-connection without top-degree connection form	principal ∞-connection

References

General

Original accounts are

• Jean-Marie Souriau, Structure des systemes dynamiques Dunod, Paris (1970)

  Translated and reprinted as (see section V.18 for the quantomorphism group):

  Jean-Marie Souriau, Structure of dynamical systems - A symplectic view of physics, Brikhäuser (1997)

• Bertram Kostant, Quantization and unitary representations, in Lectures in modern analysis and applications III. Lecture Notes in Math. 170 (1970), Springer Verlag, 87—208

A textbook account is in section II.4 of

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

and in

• Rudolf Schmid, Infinite-dimensional Lie groups with applications to mathematical physics

The description in terms of automorphism in the slice $\mathrm{\infty }$-topos over the moduli stack of (higher) connections is in section 4.4.17 of

• differential cohomology in a cohesive topos

and at ∞-geometric prequantization.

Smooth manifold structure

The ILH group structure on the quantomorphism group is discussed in

• H. Omori, Infinite dimensional Lie transformation groups, Springer lecture notes in mathematics 427 (1974)

• T. Ratiu, R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Math. Z. 177 (1981)

The regular convenient Lie group structure is discussed in

• Cornelia Vizman, Some remarks on the quantomorphism group (pdf)

A metric-structure on quantomorphisms groups is discussed in

• Y. Eliashberg,; L. Polterovich, Partially ordered groups and geometry of contact transformations. Geom.Funct.Anal.10(2000),no.6, 1448-1476.