3.3 Transitive multisymplectic group actions on spheres
4.1.1 Reminders on Atiyah algebroids and Prequantization
In this subsection, we succinctly review the language required to deliver a geometric interpretation of the -purely algebraic- problem of ascertaining the commutativity of diagram (4.2). We will need the notion of Atiyah algebroid and prequantization. The former is a certain Lie algebroid uniquely associated to any principal bundle and the latter is a construction involving S1-principal bundles on certain symplectic manifolds.
Lie algebroids
Informally, Lie algebroids are infinite dimensional Lie algebras controlled by a geometric data, namely, elements are sections of given vector bundle.
Definition 4.1.2 (Lie algebroid). We call Lie algebroid a triple (E, [·, ·]E, ρ) consisting of
• a vector bundle π : E M;
• a Lie algebra structure [·, ·]E on the space of section Γ(E);
• a vector bundle morphism ρ : E → T M (over the identity idM), called anchor;
such that:
(a) ρ induces a Lie algebra morphism at the level of sections2 ρ: (Γ(E), [·, ·]E) → (X(M), [·, ·]) ;
(b) [·, ·]E is compatible with the anchor in the sense of the Liebniz rule3: [X, f Y ]E= (ρ(X)f) Y + f [X, Y ]E ∀f ∈ C∞(M); X, Y ∈ Γ(E) . (ρ(X) in the above equation has to be interpreted as the unique derivation on the associative commutative algebra C∞(M), with respect to the point-wise product, associated to the smooth vector field ρ(X) ∈ X(M).) Notation 4.1.3. If the anchor is a surjective map ρ : E T M, the Lie algebroid is said to be transitive.
Example 4.1.4 (Lie Algebras). Any Lie algebra g can be seen as a Lie algebroid over a 0-dimensional manifold (i.e. a point) {∗}. In this sense, a Lie algebroid is a "many points version" (see "horizontal categorification" [na20k]) of a Lie algebra.
Example 4.1.5 (Tangent bundle). Given a manifold M, the corresponding tangent bundle is Lie algebroid with anchor given by the identity bundle map.
Example 4.1.6 (Standard Lie algebroid). Given any smooth manifold M, the vector bundle E = RM⊕ T M together with the standard projection ρ : RM ⊕ T M T M and the binary bracket
[·, ·]: Γ(T M ⊕ RM) ⊗ Γ(T M ⊕ RM) Γ(T M ⊕ RM)
X1
f1
⊗X2
f2
[X1, X2]
LX1f2− LX2f1
constitute a Lie algebroid called standard Lie algebroid.
Example 4.1.7 (ω-Twisted (standard) Lie algebroid). Consider a smooth manifold M. Let be ω ∈ Ω2(M) a closed 2-form , i.e. the manifold (M, ω) is a pre-1-plectic manifold. Then the vector bundle E = RM ⊕ T M together with the standard projection ρ : RM ⊕ T M T M and the binary bracket
[·, ·]ω: Γ(T M ⊕ RM) ⊗ Γ(T M ⊕ RM) Γ(T M ⊕ RM)
X1
f1
⊗X2
f2
[X1, X2]
LX1f2− LX2f1− ω(X1, X2)
2Note that this condition follows directly from condition (b).
3Algebraically, it tells that [X, ·] is a derivation on the C∞(M )-module of sections Γ(E).
constitute a Lie algebroid called ω-twisted (standard) Lie algebroid.
Principal connections and Atiyah algebroids
Given a finite dimensional Lie group G, consider a principal bundle G ,→ P M .We denote by ˆξ ∈ X(P ) the fundamental vector field of ξ ∈ g with respect to the action R : G P of the group on P from the right, namely
ˆξ p:= d
d tRexp{tξ}(p) t=0 ;
(c.f. remark 3.0.2). Le us recall the following two definitions:
Definition 4.1.8 (Connection of a Principal bundle). Given a G-principal bundle G ,→ P M , we call a connection on P any g-valued differential 1-form θ ∈ Ω1(P, g) satisfying the two following property
• θ reproduces the fundamental vector fields, i.e. θ(ˆξ) = ξ;
• θ is G-equivariant: Rg∗θ= Adg−1θ.
Definition 4.1.9(Curvature 2-form). Given a connection θ ∈ Ω1(P, g), we call curvature of θthe g-valued differential 2-form dAθ ∈Ω2(P, g) such that
dAθ(v1, v2) = d θ(v1, v2) + [θ(v1), θ(v2)] ∀vi∈ X(P ) . The connection θ is said flat if dAθ= 0.
Principal connections are special cases of Ehresmann connections:
Reminder 4.1.10 (Ehresmann connections). Let π : E → M be a smooth fibre bundle. Consider the tangent bundle τ : T E → E. One can introduce the unique vertical bundle V := ker(d π : T E → T M) where d π is the differential of the smooth map π (the tangent map T π). V is a subbundle of T E whose fibres Ve= Te(Eπ(e)) consist of vectors on T E which are tangent to the fibres of E.
An Ehresmann connection of E is any smooth subbundle H of E such that T E= H ⊕ V .
Consider now a principal bundle π : P → M with connection encoded by a 1-form θ as in definition 4.1.8. The subbundle Hθ:= ker(θ) defines a invariant Ehresmann connection on P corresponding by θ. Hence T P = ker(θ) ⊕ ker(d π).
Reminder 4.1.11 (Horizontal lifts). Let be π : P → M a vector bundle and consider an Ehresmann connection encoded by an horizontal subbundle H of P. Vector fields on P decompose uniquely in a vertical and a horizontal part.
Given a vector field X ∈ X(M), we call the horizontal lift of X the unique (horizontal) vector field XH ∈ X(P ) such that the following diagram commutes
in the category of smooth manifolds:
P H T P
M T M
XH
π
d π
X
.
Consider a G-principal bundle G ,→ P M, denoted simply as π : P → M.
The action R : G P is free and proper, therefore, there is a well-defined quotient. The same property holds for the lift of the action R to G T P . In other words, we have the following commutative diagram in the category of smooth manifold
T P T M
G
P M ,
d π
τP τM
π
where the vertical arrows τP and τM denote the canonical fibrations of the tangent bundles over their corresponding base manifolds. Observe that, while the lower horizontal line is a G-principal bundles, the upper one never inherits the structure of G-principal bundle since the Lie group does not act transitively on the fibers of d π.
This justify the following definition:
Definition 4.1.12 (Atiyah algebroid). Given a G-principal bundle π : P → M, we call Atiyah algebroid the Lie algebroid obtained by taking the quotient, with respect of G, of the tangent bundle τP : T P → P . Namely, it is given by the vector bundle
AP ∼= T P G
P G
∼= M
τP
,
whose sections correspond to G-invariant vector fields over P , i.e. Γ(AP) ∼= X(P )G, together with the restriction of the standard Lie bracket on X(P ) to G-invariant vector fields and with anchor given by d π : T P → T M.
The upshot is that Atiyah algebroids are certain Lie algebroids naturally associated to principal bundles.
The following lemma gives another characterization of the Atiyah algebroid that will be used in the following.
Lemma 4.1.13 (Atiyah exact sequence ([Ati57, Thm.1])). Given a principal bundle π: P → M, its corresponding Atiyah algebroid AP fits in a short exact sequence in the category of Lie algebroids:
0 P ×Gg AP d π T M 0
where P ×Gg is the adjoint bundle of P , i.e. the vector bundle (P × g)/ ∼ where(p, adgξ) ∼ (Rg(p), ξ) (see [KSM93, §17.6.]).
Prequantization
In this subsection we concisely review some basic notions related to geometric quantization, tailored to our needs (see [Kos70, Sou66] for the original articles or [Woo97, Bry93, Car18] for a more recent review), in a finite dimensional environment.
Let (M, ω) be a connected symplectic manifold.
Definition 4.1.14 (Prequantum bundle). We call a prequantum bundle of the symplectic manifold (M, ω) the pair (P, ω) consisting of a S1-principal bundle
S1,→ P M ,
together with a connection θ ∈ Ω1(P, g) ∼= Ω1(P ) such that π∗ω= d θ
where π : P → M denotes the fibre projection encoding P .
A prequantum bundle (P, θ) is also called a prequantization of (M, ω). When a given symplectic manifold admits a prequantum bundle it is said to be prequantizable.
Not all symplectic manifolds admit a prequantum bundle. The following celebrated theorem provides a cohomological condition to the existence of a prequantization.
Theorem 4.1.15 (Weyl-Kostant integrality condition (see [Kos70] or [Woo97, Thm. 8.3.1])). Consider a symplectic structure ω on the connected manifold M.
The symplectic manifold(M, ω) is "prequantizable" (in the sense of definition 4.1.14) if and only if 2π1 [ω] is an integral class, i.e. it lies in the image of the mapping
Hsing2 (M, Z) Hsing2 (M, R) ∼ HdR2 (M) .
Remark 4.1.16 (Connections on circle bundles). Specializing definition 4.1.8 to the case of circle bundles, like in definition 4.1.14, has the following implications:
• since g ∼= R, the connection is an honest 1-form θ ∈ Ω1(P );
• since g is generated by 1 ∈ R, one has that θ(v1) = 1 ∈ R where v1denotes the fundamental vector field of the generator 1;
• recalling that S1 ∼= U(1) and g∗ ∼= R, it results that Ad∗g = id for any g ∈ S1. Therefore R∗gθ= θ, i.e. θ is S1-invariant.
Notation 4.1.17. We denote by E ∈ X(P ) the fundamental vector field, pertaining to the action of the 1-dimensional group S1on P , corresponding to generator 1.
Once one fixes a "preferred" differential form, it is natural to select the class of smooth maps that preserve such a structure.
Definition 4.1.18 (Infinitesimal quantomorphisms). Consider a prequantum bundle (P, θ) of (M, ω). We call an infinitesimal quantomorphism any vector field on P preserving the connection θ. We denote by
Q(P, θ) := {Y ∈ X(P ) | LYθ= 0}
the Lie subalgebra of X(P ) consisting of infinitesimal quantomorphisms.
Lemma 4.1.19 ([Vau14, §2.2]). Consider a prequantum bundle (P, θ). The infinitesimal quantomorphisms are automatically S1-invariant, i.e.
Q(P, θ) ⊂ X(P )S1 .
Proof. Consider E ∈ X(P ), the generator of the action of S1 on P . Notice that E is determined by θ being the unique vector fields such that ιEθ = 1 and ιEd θ = 0. Hence if X ∈ X(P ) preserves θ, then it will preserve E too.
More precisely, let be X ∈ X(P ) such that LXθ= 0. One has that [X, E] is an horizontal vector field since
ι[X,E]θ= LXιEθ − ιELXθ= 0 ,
hence [X, E] is projectable and completely determined by its projection onto M. Similarly, one has that ι[X,E]d θ = 0. Noticing that
ι[X,E]d θ = ι[X,E]π∗ω= ιπ∗[X,E]ω ,
the non-degeneracy of ω implies [X, E] = LEX = 0 for any X ∈ Q(P, θ).
Assume that 2π1[ω] is an integral class and fix a prequantization circle bundle P → M. As above, we denote by E ∈ X(P ) the unique infinitesimal generator pertaining to the action of the 1-dimensional group S1 and by Hθ:= ker(θ) the invariant Ehresmann connection on P corresponding by θ. It is known that a prequantization provides a Lie algebra isomorphism between the observables Poisson algebras and the infinitesimal quantomorphisms.
Lemma 4.1.20(Kostant [Kos70] (see also [Vau14, Thm. 2.8])). Consider a prequantizable presymplectic manifold(M, ω), let be (P, θ) a prequantum bundle.
One has a Lie algebra isomorphism
Preqθ: C∞(M)ω Q(P, θ) f XfHθ+ (π∗f) · E
∼
, (4.4)
where XHθ denotes the horizontal lift of a vector field X on M using the Ehresmann connection Hθ (see reminder 4.1.10 and 4.1.11) and π∗f ∈ C∞(P ) is the pullback of f along π: P → M.
The Lie algebra isomorphism Preqθ is the first ingredient to realize the map (4.1) anticipated in the introduction of this chapter.
Remark 4.1.21. In the definition of Preqθ is implied the commutation of the following diagram in the category of Lie algebras:
C∞(M)ω Q(P, θ)
X(M)
Preqθ
∼
X•
π∗
. (4.5)
Notice that the vertical map has a one-dimensional kernel given by the constant functions on M, i.e. ker(X•) = R. Hence the same holds for the map π∗ in the diagram giving the pushforward of projectable vector fields via π, i.e
Preqθ(ker(X•)) = ker(π∗|Q(P,θ)) ,
where the right-hand side consists of vertical vector fields preserving θ, i.e.
elements of ker(π∗|Q(P,θ)) are constant multiples of E.
Remark 4.1.22 (About the Quantum name). We briefly explain why the term
"quantum" appears in this context.
Associating a prequantum bundle to a prequantizable symplectic manifold is the first step of a 3-step procedure called "geometric quantization (scheme)"
essentially due to Kostant, Kirillov and Souriau (KKS).
Roughly, a "quantization scheme" is procedure to associate to any symplectic
manifold (prequantizable in some sense), taken together with the corresponding Poisson algebra of observables (C∞(M), {·, ·}), a Hilbert C-vector space H, taken together with the algebra of self-adjoint operators on H.
If one understands states of a classical mechanical system as points on a (finite dimensional) manifold M, the corresponding "quantum" states will be vectors of an (infinite dimensional) complex vector spaces H with unitary norm. The keypoint is the linearity of H. This property provides a framework making possible to encompass the phenomenon of "superposition of states". In particular, given ϕ ∈ H, the vectors eiλϕrepresents the same "physical state" for any λ ∈ R.
Once this point is clear, the reason why we have named "prequantization" the act of associating a S1-bundle P over M should begin to emerge. Intuitively, a S1 bundle over M is the attachment of a 1-dimensional circle to any classical state p ∈ M. On the other hand, it is well-known that S1 is diffeomorphic to the unitary circle in the complex plane C, i.e. the Lie group U(1). Hence points in P can be seen as classical states taken together with a certain phase factor eiλ∈C.
Nevertheless, being P a generic smooth manifold, is not yet the sought linear space. According to the (KKS) procedure, the prequantum Hilbert space is represented by a certain subclass of complex-valued smooth function over P. Furthermore, being possible to regard vector fields on P as derivations C∞(P ) → C∞(P ), the images of the morphism given in equation (4.4) can be seen as prequantum versions of the classical observables of C∞(M), hence the name "prequantization map"4. We do not insist here on further details and we refer the interested reader to the fundamental manuals of geometric quantization [Bry93] and [Woo97]. We only stress that what we loosely described here are quantization schemes for ordinary, i.e. point-like, mechanical systems.
The mathematical foundation of quantization procedures for ∞-dimensional mechanical systems, i.e. field theories, is still largely incomplete.