Contents

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.)

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

$QuantomorphismGroup \hookrightarrow Diff(P)$

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

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

in the cohesive (∞,1)-topos $\mathbf{H} =$ 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 $\nabla \in \mathbf{H}_{/\mathbf{B}U(1)_{conn}}$ in the slice (∞,1)-topos. Then the quantomorphism group is the automorphism group

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

in $\mathbf{H}$ (Sch).

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

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

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

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

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

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 $\mathbb{G}$-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 $X$)

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 groupoidhigher Courant groupoidgroupoid version of quantomorphism n-group
coefficient for cohomology:$\mathbf{B}\mathbb{G}$$\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$$\mathbf{B} \mathbb{G}_{conn}$
type of fiber ∞-bundle:principal ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-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 $\infty$-topos over the moduli stack of (higher) connections is in

and in section 4.4.17 of

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.

Revised on September 13, 2013 02:27:11 by Urs Schreiber (77.251.114.72)