Circle actions and geometric quantisation

0.3 MB·English

by  Romero Solha

Circle actions and geometric quantisation
Romero Solha∗†
3 January 8, 2013
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2

n
a Abstract
J

7 The aim of this paper is to present unifying proofs for results in

geometric quantisation by exploring the existence of symplectic circle
]
G actions. It provides an extension of Rawnsley’s results on the Kostant
S complex, and gives another proof for S´niatycki’s and Hamilton’s the-
.
h orems; as well, a partial result for the focus-focus contribution to
t geometric quantisation.
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Contents

1
v
0 1 Introduction 2
2
1.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 3
2
1
. 2 Geometric quantisation 3
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0 2.1 Prequantisation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3
2.2 Geometric quantisation `a la Kostant . . . . . . . . . . . . . . 5
1
:
v
3 Resolution approach 6
i
X
r 4 Circle actions and homotopy operators 9
a
5 The Bohr-Sommerfeld condition 14
∗Department of Applied Mathematics I, Universitat Polit`ecnica de Catalunya. Email
address: [email protected]
†Partially supported by EU-Brazil Startup 2009-2010 project; Contact And Symplec-
tic Topology, ESF programme; Geometry, Mechanics and Control Theory network; until
December 2012, the DGICYT/FEDER project MTM2009-07594: Estructuras Geomet-
ricas: Deformaciones, Singularidades y Geometria Integral; and Geometria Algebraica,
Simplectica, Aritmetica y Aplicaciones with reference: MTM2012-38122-C03-01,starting
in January 2013.
1
6 Applications I: local and semilocal computations 16
6.1 The cylinder: polarisation by circles . . . . . . . . . . . . . . . 16
6.2 The complex plane: polarisation by circles . . . . . . . . . . . 17
6.3 Symplectic vector spaces: linear polarisation . . . . . . . . . . 18
6.4 Direct product type with a regular component . . . . . . . . . 18
6.5 Direct product type with an elliptic component . . . . . . . . 20
6.6 Neighbourhood of a Liouville fibre . . . . . . . . . . . . . . . . 21
6.7 Neighbourhood of an elliptic fibre . . . . . . . . . . . . . . . . 22
6.8 Focus-focus contribution to geometric quantisation . . . . . . 23
7 Applications II: global computations 25
7.1 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.2 Locally toric manifolds . . . . . . . . . . . . . . . . . . . . . . 26
7.3 Almost toric manifolds . . . . . . . . . . . . . . . . . . . . . . 27
References 27
1 Introduction
Geometric quantisation tries to associate a Hilbert space to a symplectic
manifold via a complex line bundle. Although it is possible to describe the
canonical quantisation using this language, most of the difficulties arise when
one tries to mimic this procedure for symplectic manifolds which are not
naturally cotangent bundles. Those appear in the context of reduction and
are far from being artificial mathematical models.
The first difficulty is to isolate in a global way position and momentum,
in order to define wave functions as sections of a complex line bundle over
the symplectic manifold. The second issue, that will not be address here, is
how to define a Hilbert structure; however, all examples treated in this paper
have a natural one.
As suggested by Kostant, wave functions will be associated to elements of
higher cohomology groups and the quantum phase space will be built from
these groups. Although the honest-to-goodness quantum phase space will
not be constructed.
At least two approaches can be used to compute the cohomology groups:
Cˇech and de Rham. The results of Hamilton [3] and Hamilton and Miranda
[4] are based on a Cˇech approach. Here a de Rham approach, as in [8, 7], is
used: by finding a resolution for the sheaf, as suggested in [4].
This paper follows closely Rawnsley’s ideas [7] and explores the existence
of circle actions to provide an alternative proof for the theorems of S´niatycki
2
[8] and Hamilton [3]. The tools developed here highlight and unravel the role
played by symplectic circle actions in known results in geometric quantisa-
tion. Not only that, this approach casts some light on a conjecture about
the contributions coming from focus-focus type of singularities.
Throughout this paper and otherwise stated, all the objects considered
will be C∞; manifolds are real, Hausdorff, paracompact and connected;
C∞(V) denotes the set of complex valued functions over V; and the units
are such that ~ = 1.
1.1 Acknowledgements
There is no way to overemphasise the importance of Eva Miranda in this
work: she introduced the author of this paper to the subject, and also read
and commented on drafts of the paper.
2 Geometric quantisation
2.1 Prequantisation
Thissubsection dealswithsomeconceptsneededtodefinewavefunctions.
The first attempt was to see them as sections of a complex line bundle over
the symplectic manifold, the so-called prequantum line bundle. The other
notion described here, the polarisation, is a way to define a global distinction
between momentum and position.
Definition 2.1. A symplectic manifold (M,ω) such that [ω] is integral is
called prequantisable. A prequantum line bundle of (M,ω) is a hermitian
line bundle over M with connection, compatible with the hermitian structure,
(L,∇ω) that satisfies curv(∇ω) = −iω.
Example 2.1. Any exact symplectic manifoldsatisfies [ω] = 0, inparticular:
cotangent bundles with the canonical symplectic structure. In that case the
trivial line bundle is an example of a prequantum line bundle. ♦
The following theorem1 [5] provides a relation between the above defini-
tions:
Theorem 2.1 (Kostant). A symplectic manifold (M,ω) admits a prequan-
tum line bundle (L,∇ω) if and only if it is prequantisable.
1This result is also attributed to Andr´e Weil, Introduction `a l’´etude des vari´et´es
k¨ahl´eriennes (1958).
3
Lemma 2.1. The potential 1-forms of ∇ω for each unitary section defined
in a subset of M are cohomologous. Conversely, if there is an unitary section
over a subset of M, any 1-form which is cohomologous to the potential 1-form
of ∇ω is the potential 1-form of an unitary section. Moreover, any subset of
M where ω = dθ has an unitary section such that θ is its potential 1-form.
Proof: Let s and r be two unitary sections defined over N ⊂ M, and
Θ and ϑ the associated potential 1-forms. If s = eifr for some real valued
f ∈ C∞(N), then
−iΘ⊗s = ∇ωs = ∇ω(eifr) = [deif −ieifϑ]⊗r = −i[−df +ϑ]⊗eifr ⇒
ϑ−Θ = df . (1)
Conversely, by the same computation, if s has Θ = ϑ− df as potential
1-form, then r = e−ifs is a unitary section having ϑ as potential 1-form.
For ω = dθ over N ⊂ M, let A = {A } be a contractible open cover
j j∈I
of N such that each A is a local trivialisation of L with unitary section s
j j
(this can always be obtained, e.g. using a convenient cover made of balls
with respect to a riemannian metric). Each unitary section s has Θ as a
j j
potential 1-form of ∇ω and since
curv(∇ω) = −idθ| = −idΘ ⇒ (2)
Aj Aj j
there exists real valued functio(cid:12)ns f ∈ C∞(A ) such that θ| = Θ −df . By
(cid:12) j j Aj j j
the above argument, the unitary sections r = e−ifjs have θ| as potential
j j Aj
1-forms.
Any two sections r and r such that A ∩A 6= ∅ share the same potential
j k j k
1-form, and because of that, they differ by a nonzero constant function,
r = c r at A ∩A . Trivially, c can be extended to the same constant
j jk k j k jk
over A , and c r is a section defined over A such that its restriction to
k jk k k
A ∩A is exactly r , and it still has θ| as potential 1-form. Hence, they
j k j A
k
can be glued together, using the gluing condition of sheaves, to a unitary
section r defined over N and having θ as potential 1-form. (cid:4)
A real polarisation P is an integrable subbundle of TM whose leaves are
lagrangian submanifolds. But due to the example below, another definition
is considered.
Foran integrable system F : M2n → Rn ona symplectic manifold, the Li-
ouvilleintegrabilityconditionimpliesthatthedistributionofthehamiltonian
vector fields of the components of the moment map generates a lagrangian
foliation (possible) with singularities. This is an example of a generalised
real polarisation, i.e.: an integrable distribution on TM whose leaves are
lagrangian submanifolds, except for some singular leaves.
4
Definition 2.2. A real polarisation P is an integrable (in the Sussmann’s
[10] sense) distribution of TM whose leaves are generically lagrangian. The
complexification of P is denoted by P and will be called polarisation.
From now on (L,∇ω) will be a prequantum line bundle and P the com-
plexification of a real polarisation of (M,ω).
2.2 Geometric quantisation `a la Kostant
The original idea of geometric quantisation is to associate a Hilbert space
to a symplectic manifold via a prequantum line bundle and a polarisation.
Usually this is done using flat global sections of the line bundle. In case these
global sections do not exist, one can define geometric quantisation via higher
cohomology groups by considering cohomology with coefficients in the sheaf
of flat sections.
The existence of global flat sections is a nontrivial matter. Actually
Rawnsley [7] (and also proposition 4.3 in this paper) showed that the ex-
istence of a S1-action may be an obstruction for the existence of nonzero
global flat sections.
In order to use flat sections as analogue for wave functions one is forced
to work with delta functions with support over Bohr-Sommerfeld leaves, or
deal with sheaves and higher order cohomology groups. Both approaches can
be found in the literature2, but here only the sheaf approach is treated: as
suggested by Kostant.
Definition 2.3. Let J denotes the space of local sections s of a prequantum
line bundle L such that ∇ωs = 0 for all vector fields X of a polarisation
X
P. The space J has the structure of a sheaf and it is called the sheaf of flat
sections.
Considering the triplet prequantisable symplectic manifold (M,ω), pre-
quantum line bundle (L,∇ω), and polarisation P:
Definition 2.4. The quantisation of (M,ω,L,∇ω,P) is given by
Q(M) = Hˇk(M;J) , (3)
k≥0
M
where Hˇk(M;J) are Cˇech cohomology groups with values in the sheaf J. In
this case, one implicitly assumes the extra structures and calls M a quantis-
able manifold.
2Rawnsley cites works of Simms, S´niatycki and Keller in [7].
5
Remark 2.1. Even though Q(M) is just a vector space and a priori 3 has no
Hilbert structure, it will be called quantisation. The true quantisation of the
triplet (M,ω,L,∇ω,P) shall be the completion of the vector space Q(M),
after a Hilbert structure is given, together with a Lie algebra homomorphism
(possibly defined over a small set) between the Poisson algebra of C∞(M)
and operators on the Hilbert space. In spite of the problems that may exist
in order to define geometric quantisation using Q(M), the first step is to
compute this vector space.
Remark 2.2. Flat sections behave in a different fashion for the Ka¨hler case.
This paper does not deal with this case, however much can be found in the
literature (e.g.: [2, 3] and references therein).
3 Resolution approach
Following Rawnsley [7], given a prequantisable symplectic manifold with
polarisation, it is possible to construct a fine resolution for the sheaf of flat
sections. The propositions contained in this section are extensions of the
results in [7]; it is mainly an opportunity to fix notation, the replacement
of a subbundle of TM by an integral distribution offers no obstruction and,
therefore, proofs are omitted.
The restriction of the connection ∇ω to the polarisation induces a linear
operator
∇ : Γ(L) → Γ(P∗)⊗C∞(M) Γ(L) (4)
satisfying (by definition) the following property:
∇(fs) = d f ⊗s+f∇s , (5)
P
for f ∈ C∞(M) and s ∈ Γ(L), where d is the restriction of the exterior
P
derivative to the distribution directions.
Remark 3.1. Although P is not a subbundle of TMC when it is singular,
for the sake of simplicity, the same notation for vector bundles will be used
for it: Γ(P) = hX1,...,XniC∞(M), where X1,...,Xn ∈ Γ(TM) generates
the real distribution P, and Γ( kP∗) is the space of smooth alternating
C∞(M)-multilinear maps from the Cartesian product of k copies of Γ(P) to
V
C∞(M).
3As previously mentioned, for the concrete cases presented in this paper Q(M) does
admit Hilbert structures.
6
Definition 3.1. The space of line bundle valued polarised forms is S •(L) =
P
SPk(L), where SPk(L) = Γ( kP∗)⊗C∞(M) Γ(L).
k≥0
M V
Wherefore, ∇ : S0(L) → S1(L) and S •(L) has a module structure,
P P P
which enables an extension of ∇ to a derivation of degree 1 on the space of
line bundle valued polarised forms, as follows:
Definition 3.2. Denoting Γ( kP∗) by Ωk(M), Ω •(M) = Ωk(M) is
P P P
k≥0
the space of polarised forms. V M
If α ∈ Ωk(M) and β = β ⊗s ∈ Sl (L),
P P
α∧β = α∧(β ⊗s) = (α∧β)⊗s (6)
defines a left multiplication of the ring Ω •(M) on S •(L).
P P
Definition 3.3. The derivation on S •(L) is given by the degree +1 map
P
d∇ : S •(L) → S •(L),
P P
d∇β = d∇(β ⊗s) = d β ⊗s+(−1)lβ ∧∇s . (7)
P
Proposition 3.1. If α ∈ Ωk(M) and β ∈ Sl (L), then
P P
d∇(α∧β) = d α∧β +(−1)kα∧d∇β , (8)
P
and
d∇ ◦d∇β = curv(∇ω) ∧β . (9)
P
Since ω = i curv(∇ω) vanishes along P, d(cid:12)∇ ◦d∇ = 0:
(cid:12)
Corollary 3.1. d∇ is a coboundary operator.
Remark 3.2. TheonlypropertyofLbeingusedinthispaperistheexistence
of flat connections along P; any complex line bundle would do, not only a
prequantum one. In particular: the tensor product between a prequantum
line bundle and a bundle of half forms normal to P —the results here work
if metaplectic correction is included.
If Sk(L) denotes the associated sheaf of Sk(L), one can extend d∇ to
P P
a homomorphism of sheaves, d∇ : Sk(L) → Sk+1(L). S0(L) ∼= S, the
P P P
sheaf of sections of the line bundle L, and J is isomorphic to the kernel of
d∇ : S → S1(L). Because d∇ ◦d∇ = 0, one is able to build a sequence.
P
7
Definition 3.4. The Kostant complex is
0 −→ J ֒→ S −∇→ S1(L) −d→∇ ··· −d→∇ Sn(L) −d→∇ 0 . (10)
P P
The sheaves Sk(L) are fine; Γ(L) and Ωk(M) are free modules over the
P P
ring of functions of M, and by that, they admit partition of unity. Hence,
via a Poincar´e lemma, the abstract de Rham theorem [1] offers a proof for
the following:
Theorem 3.1. The Kostant complex is a fine resolution for J. Therefore,
each of its cohomology groups, Hk(S •(L)), are isomorphic to Hˇk(M;J).
P
Remark 3.3. There are particular situations in which a Poincar´e lemma is
available, and only in these cases theorem 3.1 holds. This is true4 when P is
a subbundle of TM, and it can be extended to a more general setting, as it is
announced in [9]. Using symplectic circle actions, this paper provides a proof
when P is given by a locally toric singular lagrangian fibration (subsection
6.7) or an almost toric fibration in dimension 4 (subsection 6.8).
As expected, the notions of interior product and Lie derivative are avail-
able for S •(L). The Lie derivative can be seen as a derivation along a flow,
P
but for that, a nontrivial notion of pullback is needed.
Definition 3.5. The contraction between line bundle valued polarised forms
and sections of P is given by a map i : Γ(P)×S •(L) → S •(L), which is
P P
a degree -1 map on S •(L), i.e.: for each X ∈ Γ(P) and β = β⊗s ∈ Sl (L)
P P
it holds that
i (∇s) = ∇ s , (11)
X X
i β = i (β ⊗s) = (ı β)⊗s . (12)
X X X
Proposition 3.2. If X ∈ Γ(P), α ∈ Ωk(M) and β ∈ Sl (L), then i ◦i = 0
P P X X
and
i (α∧β) = (ı α)∧β +(−1)kα∧i β . (13)
X X X
Using Cartan’s magic formula it is possible to define a Lie derivative on
S •(L):
P
Definition 3.6. The Lie derivative £∇ : Γ(P)×S •(L) → S •(L) is defined
P P
by:
£∇(β) = i ◦d∇β +d∇ ◦i β . (14)
X X X
Proposition 3.3. The Lie derivative £∇ commutes with the derivation d∇.
4Both S´niatycki and Rawnsley attribute this to Kostant, a proof is provided in [7].
8
For Ωk(M) the pullback still makes sense if one restricts to diffeomor-
P
phisms that preserve the polarisation P, but problems arise when one twists
it with Γ(L). A way to compare elements of L is by parallel transport, which
in general is path dependent. When it does not, the pullback on S •(L) is
P
well-defined (for diffeomorphisms that preserve the polarisation). When it
depends on the path, it is possible to make sense of pullbacks over paths.
Now, if X ∈ Γ(P) its flow φ already encodes both a curve and a diffeo-
t
morphism:
Definition 3.7. The pullback φ∗ of α = α⊗s ∈ Sk(L) is defined by
t P
φ∗α = (φ∗α)⊗Π−1(s◦φ ) ; (15)
t t φt t
where, by the bundle automorphism property of the parallel transport, Π
φt(p)
denotes the parallel transport between p and φ (p) through the integral curve
t
of the flow.
The following proposition justifies, somehow, the choices made for the
interior product and pullback.
Proposition 3.4. The Lie derivative £∇ can be characterised by
d
£∇(α) = φ∗α . (16)
X dt t t=0
(cid:12)
(cid:12)
Corollary 3.2. The Lie derivative £∇ com(cid:12)mutes with the pullback φ∗.
X t
Proposition 3.5. If X ∈ Γ(P) with flow φ , α ∈ Ωk(M) and β ∈ Sl (L),
t P P
then
φ∗(α∧β) = (φ∗α)∧φ∗(β) , (17)
t t t
and the pullback φ∗ commutes with d∇.
t
4 Circle actions and homotopy operators
This section explains the construction of an almost homotopy operator
for the Kostant complex when one has a symplectic S1-action, and how this
implies the vanishing of the stalks of points with nontrivial holonomy. Most
results of this section were previously provided in [7] with slightly less general
hypothesis; some proofs automatically hold (propositions 4.1, 4.2 and 4.3),
but one (lemma 4.2) had to be adapted.
Let X ∈ Γ(P) be a generator of a symplectic S1-action. If φ stands for
t
the flow of X at time t, it is possible to define an induced action on Sk(L)
P
9
by φ ∗. Denoting by per(γ) the period5 of the closed orbit γ (eventually
t
constant in the case of a fixed point) of X, it holds that:
φ ∗(α⊗s) = φ∗ α⊗Π−1 (s◦φ ) = α⊗[Q(γ)−1s] = Q(γ)−1α ,
per(γ) per(γ) φ per(γ)
per(γ)
Q(γ) is the holonomy of γ, and
per(γ) d
[Q(γ)−1 −1]α = φ ∗α−φ ∗α = (φ∗α) dt
per(γ) 0 dt t
Z0
per(γ) d per(γ) d
= φ∗ α dt = φ∗(φ∗α) dt
Z0 ds t+s (cid:12)s=0 Z0 ds s t (cid:12)s=0
per(γ) (cid:12) per(γ) (cid:12)
= £∇(φ∗α)(cid:12)dt = [i ◦d∇ +d∇ ◦(cid:12)i ](φ∗α) dt
X t X X t
Z0 Z0
per(γ) per(γ)
= i d∇(φ∗α) dt +d∇ ◦i φ∗α dt .
X t X t
Z0 ! Z0 !
Using that the pullback commutes with the derivative (proposition 3.5) one
gets from the last equation
per(γ) per(γ)
[Q(γ)−1 −1]α = i φ∗(d∇α) dt +d∇ ◦i φ∗α dt ,
X t X t
Z0 ! Z0 !
(18)
which resembles the equation satisfied by a homotopy operator.
per(γ)
Proposition 4.1. The expression J (α) = i φ∗α dt defines a
X X t
Z0 !
degree −1 derivation on S •(L).
P
Proof: Propositions 3.2 and 3.5 imply that J is a derivation, and the
X
degree comes from the fact that i has degree −1. (cid:4)
X
The equation 18 implies that J satisfies
X
[Q(γ)−1 −1]α = J (d∇α)+d∇J (α) , (19)
X X
for any α ∈ Sk(L) if k ≥ 1, whilst for k = 0 it becomes
P
[Q(γ)−1 −1]α = J (d∇α) , (20)
X
since S−1(L) is empty and J has degree −1.
P X
5Indeed, per(γ)∈C∞(M): for each p∈M the function per(γ) gives the period of the
orbit of X passing through p.
10