3.3 Transitive multisymplectic group actions on spheres
4.1.2 Embedding of the observables in the Atiyah algebroid . 145
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.
Lemma 4.1.13 says that AP fits in a short exact sequence of Lie algebroids
0 R AP T M 0
where R denotes the trivial rank-1 bundle5 over M (a bundle of Abelian Lie algebras, with the constant section 1 mapping to E ∈ Γ(AP)) and the second map is the anchor. A principal connection θ on P provides a linear splitting of the above short exact sequence.
Lemma 4.1.23 (T M ⊕ R is isomorphic to AP). Let be π : P → M a S1 -principal bundle. Consider a connection 1-form θ. There is an isomorphism of vector bundles
σθ: T M ⊕ R AP
vc
vHθ+ c · E
∼
,
where the superscript Hθ denotes the horizontal lift on vectors and E is the fundamental vector field of the generator 1 in the Lie algebra of S1.
Remark 4.1.24 (Recovering the standard ω-twisted Lie algebroid). Considering the inverse of the isomorphism σθ introduced in lemma 4.1.23, one can pull back the Lie algebroid structure from AP to T M ⊕ R. As a result, we obtain the Lie algebroid (T M ⊕ R)ω, with anchor map given by the first projection onto T M and Lie bracket on sections given by:
X f
,Y
g
ω
:= σθ−1
σθ
X f
, σθY
g
=
= [X, Y ]
X(g) − Y (f) + ιXιYω
. To prove the last equality, observe that
[XHθ+ (π∗f)E, YHθ+ (π∗g)E] =
= [XHθ, YHθ] + [(π∗f)E, YHθ] + [XHθ,(π∗g)E] +
(([(π∗f(()E, (π((∗g()E] =(
= [XHθ, YHθ] +(((π∗f(()[E, Y((H(θ] +(((π∗g(()[XH((θ, E(] + (−LYHθ(π∗f) + LXHθ(π∗g)) E =
= [XHθ, YHθ] + π∗(X(g) − Y (f)) E
where the first cancellation occurs because E is vertical and π∗f and π∗g are constant along the fibres, and the second one follows from π∗([XHθ, E]) = θ([XHθ, E]) = 0. The claim is proved by noticing that
[XHθ, YHθ] = [X, Y ]Hθ + π∗(ιXιYω)E
5This is usually denoted asRM, we are employing a short-hand notation here.
since
θ([XHθ, YHθ] = LXHθιYHθθ − ιYHθιXHθd θ + ιYHθdιXHθθ =
= − ιYHθιXHθπ∗ω=
= − π∗(−ιYιXω) and
π∗[XHθ, YHθ] = [X, Y ] .
The upshot is that σθ is a Lie algebroid isomorphism (T M ⊕ R)ω ∼= AP between the standard ω-twisted Lie algebroid (see example 4.1.7) and the Atiyah algebroid of the S1-principal bundle.
Finally, noticing that Q(P, θ) ⊂ X(P )S1 = Γ(AP), we conclude that the sought embedding (see equation 4.1) can be obtained by composing6 Preqθ and σ−1θ . Proposition 4.1.25 (The Poisson algebra embeds into the sections of (T M ⊕ R)ω). Let be (M, ω) a prequantizable symplectic manifold.Consider the ω-twisted standard Lie algebroid (T M ⊕ R)ω. The Lie algebra morphism Ψ obtained by the composition of the maps introduced in lemmas 4.1.20, 4.1.19 and 4.1.23, i.e. the map obtained from the following diagram
C∞(M)ω Γ(T M ⊕ R)ω
Q(P, θ) Γ(AP)
Pr∼ eqθ
Ψ
∼ σ−1θ
,
is the Lie algebra embedding (4.1) appearing in the introduction.
Namely one has:
Ψ: C∞(M)ω Γ(T M ⊕ R)ω
f Xf
f
, (4.6)
where Xf denotes the Hamiltonian vector field pertaining to f . For the rest of this section, we will simply denote Ψ as
σθ−1◦Preqθ: C∞(M)ω→Γ(T M ⊕ R)ω.
Remark 4.1.26 (Independence from the choice of prequantization). We point out that the above expression (4.6) is a Lie algebra embedding even when ω does not satisfy the integrality condition. Furthermore, it does not depend on the connection θ implied by the prequantization procedure.
6We will tend, with a slight abuse of notation, to regard the map Preqθ introduced in equation (4.4) as a morphism C∞(M )ω→ X(P )S1.
4.1.3 Commutativity after twisting
In this subsection we show, by geometric arguments, the commutativity of the diagram (4.2) from the introduction.
Assume we have an action of a Lie group G on M, and denote by v : g → X(M) the corresponding infinitesimal action (a Lie algebra morphism). Assume the existence of an equivariant moment map
J: M → g∗.
This means that J satisfies ιvxω = −d(J∗(x)) (i.e. vx = XJ∗(x)) and that J∗: g → C∞(M)ω is a Lie algebra morphism7 (see reminder 2.4.5 in chapter 2).
Therefore, diagram (4.5) is extended to
C∞(M)ω Q(P, θ)
g X(M)
∼ Preqθ
X• π∗
v•
J∗ (4.7)
In particular, we obtain a Lie algebra morphism L0:= Preqθ◦J∗: g Q(P, θ)
x vxHθ + ((π∗J∗)(x)) · E , (4.8) lifting the infinitesimal action in the sense of the commutativity of the following diagram in the category of Lie algebras
Q(P, θ)
g X(M)
π∗
v•
L0 .
Twisting by an invariant one-form
Notice that the difference between any two connection 1-forms on the circle bundle π : P → M is basic, i.e. is the pullback of a 1-form on M.
Now we take α ∈ Ω1(M)G and use it to twist some of the above data, keeping
7This is equivalent to infinitesimal equivariance, i.e. LvyJ∗(x) = J∗([y, x]).
the G-action fixed: ω + dα is an invariant symplectic form on M (assuming it is non-degenerate), with moment map Jα determined by8
Jα∗: g C∞(M)ω+dα
x 7→ J∗(x) + ιvxα
. (4.9)
Furthermore, a prequantization of the symplectic manifold (M, ω + dα) is given by the same circle bundle P but with connection θ + π∗α.
We can repeat the procedure outlined above (see in particular equation (4.8)), obtaining a Lie algebra morphism Lα: g → Q(P, θ+π∗α) lifting the infinitesimal action. Since α is G-invariant, any lift to P of a generator vx preserves π∗α, hence we can view both L0 and Lαas maps
g→ Q(P, θ) ∩ Q(P, θ + π∗α)
which are Lie algebra morphisms lifting the infinitesimal action. There are “few”
such Lie algebra morphisms. (They are in bijection with moment maps for (M, ω), by diagram (4.7); if H1(g) = 0 then the moment map is unique [CdS01,
Theorem 26.5].) Hence the following is not a surprise.
Proposition 4.1.27. The Lie algebra morphisms L0 and Lα coincide.
Proof. Fix x ∈ g and write fx:= J∗(x). We have to show that L0(x) = Lα(x), i.e.
vxHθ+ (π∗fx) · E = vHxθ+π∗ α+ π∗(fx+ ιvxα) · E .
We do so decomposing T P as Hθ⊕RE. Since both the left-hand side and the right-hand side π-project to the same vector field (namely vx), we have to check that we obtain the same function applying θ to both vector fields. This is indeed the case, since applying θ to the vector field on the right we obtain
π∗(fx+ ιvxα) − (π∗α)(vxHθ+π∗ α) = π∗fx.
We can also repeat the construction of §4.1.2 using the connection θ + π∗α, yielding a Lie algebroid isomorphism
σθ+π∗α: (T M ⊕ R)ω+dα∼= AP.
8Indeed it can be checked easily that ιvx(ω+dα) = −d(J∗(x)+ιvxα) using the G-invariance of α (expressed as Lvxα = 0 for all x ∈ g).
The composition (σθ+π∗α)−1◦ σθ reads
τα: (T M ⊕ R)ω (T M ⊕ R)ω+dα
vc v
c+ ιvα
∼
(4.10)
and is often referred to as gauge transformation.
Commutativity
We end up with the following commutative diagram
C∞(M)ω Γ(T M ⊕ R)ω
g X(P )S1
C∞(M)ω+dα Γ(T M ⊕ R)ω+d α
Preqθ
τα
Jα∗
J∗ σ−1θ
σ−1θ+π∗ α Preqθ+π∗ α
where the left square commutes by proposition 4.1.27 and the right one by the very definition of τα (see equation (4.10)).
As we emphasized in remark 4.1.26, the composition σ−1θ ◦Preqθ: C∞(M)ω→ Γ(T M ⊕ R)ω does not depend on θ. Hence, after removing X(P )S1 from the above diagram, we obtain a commutative diagram that makes no reference to the prequantization bundle P :
C∞(M)ω Γ(T M ⊕ R)ω
g
C∞(M)ω+dα Γ(T M ⊕ R)ω+dα τα J∗
Jα∗
(4.11)
Remark 4.1.28. For a given α, in general, there is no linear map C∞(M)ω→ C∞(M)ω+dα making the left part of diagram (4.11) commute. Indeed such a map exists if, and only if, for all f ∈ C∞(M) we have
Xfω= Xf +ιω+dαXω
fα
where Xgν denotes the Hamiltonian vector field pertaining to the function g with respect to the symplectic form ν. The latter is equivalent to say that LXω
fα= 0. This explains why it is necessary to consider moment maps for a α-preserving action.
Remark 4.1.29. Diagram (4.11) commutes for any symplectic form ω, even for those that do not satisfy the integrality condition and therefore do not admit a prequantization bundle. This is immediate using the explicit expressions for the maps involved in eq. (4.6), (4.9) and (4.10). The discussion of this subsection – in particular proposition 4.1.27 – provides a geometric argument for the commutativity of diagram (4.11) in the integral case.