geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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
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.
This perspective lends itself to a more abstract description: we may regard the prequantum circle bundle as being modulated by a morphism
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
in $\mathbf{H}$ (Sch).
From this it is clear what the quantomorphism ∞-group of an n-plectic ∞-groupoid should be: for
the morphism modulating a prequantum circle n-bundle, the corresponding quantomorphism $n$-group is again $Aut(\nabla)$, now formed in $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$
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.).
For $(X,\omega)$ a connected symplectic manifold there is a central extension of groups
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)
(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: | standard higher Atiyah groupoid | higher Courant groupoid | groupoid 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 ∞-bundle | principal ∞-connection without top-degree connection form | principal ∞-connection |
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
and in
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
The ILH group structure on the quantomorphism group is discussed in
The regular convenient Lie group structure is discussed in
A metric-structure on quantomorphisms groups is discussed in