Symplectic Geometry and Geometric Quantization Sophie de Buyl r , St´ ephane Detournay ♥ and Yannick Voglaire ♣ r Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures–sur–Yvette, France ♥ Universita’ degli Studi di Milano and INFN, sezione di Milano, 16 Via G. Celoria, 20133 Milano, Italy ♣ Universit´ e Catholique de Louvain, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium [email protected] — [email protected] — [email protected]Abstract: We review in a pedagogical manner the geometrical formulation of clas- sical mechanics in the framework of symplectic geometry and the geometric quanti- zation that associate to a classical system a quantum one. These notes are based on Lectures given at the 3rd Modave Summer School in Mathematical Physics by Sophie de Buyl and St´ ephane Detournay.
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Symplectic Geometry and Geometric Quantization
Sophie de Buylr, Stephane Detournay♥ and Yannick Voglaire♣
r Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures–sur–Yvette, France
♥ Universita’ degli Studi di Milano and INFN, sezione di Milano, 16 Via G. Celoria, 20133 Milano, Italy
♣ Universite Catholique de Louvain, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium
Abstract: We review in a pedagogical manner the geometrical formulation of clas-
sical mechanics in the framework of symplectic geometry and the geometric quanti-
zation that associate to a classical system a quantum one. These notes are based on
Lectures given at the 3rd Modave Summer School in Mathematical Physics by Sophie
de Buyl and Stephane Detournay.
Contents
2
Chapter 1
Foreword
The aim of these lectures will be to concisely present a mathematical approach to the question of quantization
of a physical system, called geometric quantization. To begin with, we will introduce how mathematicians
formalize the notion of a classical mechanical system. Then we will show how they define and perform its
quantization in this framework. The question of quantization consists in assigning a quantum system to a
classical one. This problem is still very timely, since there is in general no unique way of doing so. The
different approaches then try to be as general and natural as possible. The general idea is that the quantization
procedure should preserve the initial structure of the classical system as much as possible. Namely, if a
classical system possesses a symmetry -represented by a hamiltonian action (to be defined later) of this group
on the symplectic manifold modelling the classical phase space-, one would like the associated quantum
system to form a unitary representation of this group. If the action is transitive, this representation should
be irreducible. The latter condition reflects the constraint that the quantization of an elementary classical
system, when possible, should yield an elementary quantum system, these systems being defined as those
which cannot be decomposed in smaller parts without breaking the symmetry. As we will see, to reach these
objectives we will be led to introduce and use a lot of mathematical tools, such as symplectic manifolds,
hamiltonian actions, moment maps, line bundles, Cech cohomology, Chern classes, Kahler manifolds and
polarizations. The prerequisites are a basic knowledge of differential geometry (manifolds, exterior calculus
on manifolds, (co-)tangent bundles, Lie derivative, pull-back, push-forward essentially), that can be found
e.g. in [?] and of the philosophy of quantum mechanics. These lectures are mainly based on [?] and [?].
3
Chapter 2
Geometric formulation of Classical
Mechanics
2.1 Basic notions and examples
The fundamental object used to represent a classical phase space is a symplectic manifold. It consists in a
pair (M, ω), where M is a differentiable manifold and ω a closed 2-form on M (dω = 0) such that ωx is non
degenerate ∀x ∈ M, i.e. if x ∈ M and ωx(Y,Z) = 0∀Z ∈ TxM, then Y = 0.
The 2-form ω is called a symplectic structure on M. Being non degenerate, it establishes a linear isomor-
phism ∀x ∈ M between TxM and T?x M:
TxM → T?x M : X → iXω := ω(X, . ) .
Examples
1. Euclidean space: M = R2n with coordinates (q1, ..., qn, p1, ..., pn), ω = dpi ∧ dqi.
2. Cotangent bundle : M = T?N, where N is a manifold. M is the phase space of a system whose config-
uration space is N. M can be endowed with coordinates (qa, pb), where at each point x = (q1, ..., qn) in
N the components of a form α ∈ T?x N are (p1, ..., pn) (i.e. α = pidqi).
The symplectic form is given locally by ω = dqi ∧ dpi. It is closed but also exact (i.e. there exists a
one-form θ such that ω = −dθ). Indeed, let π : T?N → N be the canonical projection, with π(ξx) = x
if ξx ∈ T?x N. The Liouville or canonical one-form θ on M is defined as
θξx (X) := 〈ξx, (π∗)αx X〉 (2.1)
where (i) ξx ∈ T?N (we here make a common abuse of notation, using the symbol ξx either for an
element of T?N – where it is to be understood as (x, ξx) – or for an element of T?x N), (ii) X ∈ Tξx T
?N =
Tξx M, (iii) π? : T (T?N) → T N is the differential of π and (iv) 〈., .〉 is the duality between TxN and
T?x N. The 2-form ω = −dθ defines the canonical symplectic structure of the cotangent bundle. Let’s
4
work this out in local coordinates. Let U ⊂ N be an open subset with local coordinates (q1, · · · , qn).
We get local coordinates (q1(x), · · · , qn(x), p1, · · · , pn) on π−1(U) ⊂ T?N for a point ξx in T?N, with
ξx = pidqix. (2.2)
In local coordinates, X ∈ Tξx can be written
X = ai∂qi + b j∂p j . (2.3)
Hence, using
(π?)αx
∂
∂qi =∂
∂qi , (π?)αx
∂
∂pi= 0, (2.4)
one has with (??), (??) and (??)
θξx (X) = 〈pidqix, a
j ∂
∂q j〉 = pi ai = pi dqi(X). (2.5)
In local coordinates, for αx ∈ π−1(U), the Liouville one-form thus reads
θ = pidqi (2.6)
and hence, defining ω = −dθ one has
ω|π−1(U) = dqi ∧ dpi. (2.7)
The 2-form ω is in this case globally defined, its expression in local coordinates being given by (??).
3. Coadjoint orbits. The latter play an important role namely in Kirillov’s orbit method, to which we will
allude in Sect.??. Let G be a connected Lie group with Lie algebra G. Let G? be its dual, i.e. the space
of real linear forms on G. The group acts on G? by the so-called coadjoint action:
Ad? = G × G? → G? (2.8)
with Ad?g f = f Adg−1 , g ∈ G, f ∈ G?, i.e.
〈Ad?g f , X〉 = 〈 f , Adg−1 X〉, (2.9)
with Adg−1 X = g−1Xg, X ∈ G. If f ∈ G?, let θ f be the coadjoint orbit of f in G?, defined as
θ f = Ad?G f := Ad?g f | g ∈ G. (2.10)
If x ∈ θ f ⊂ G?, the tangent space Txθ f is spanned by the vectors Xx for any X ∈ G, where
Xx := x adX =: x [X, . ]. (2.11)
Indeed, we have
ddt |0〈Ad?e−tX x, 〉 =
ddt |0〈x, Ade−tX 〉
= 〈x, [X,Y]〉. (2.12)
5
One may define for all X∗x ,Y∗x ∈ Txθ f , and ∀x ∈ θ f ⊂ G
?
ωx(X∗x ,Y∗x ) = 〈x, [X,Y]〉 ,∀X,Y ∈ G. (2.13)
One may show (but it’s somewhat lengthy, see e.g. [?, ?, ?]) thatω indeed defines a symplectic structure
on θ f , i.e. that it is well-defined, nondegenerate and closed.
To close with this section, let us mention the important Darboux theorem: if (M, ω) is a symplectic manifold,
then ∀x ∈ M there exists an open neighborhood U of x in M and local coordinates (qi, p j), called canonical
coordinates on U so that
ω|U = dqi ∧ dpi. (2.14)
It thus states that any symplectic manifold locally looks like a cotangent bundle. However, in general, the
one-form θ need not be globally defined. In particular, on a compact manifold a global symplectic potential
θ such that ω = −dθ does not exist by virtue of Stokes’ theorem.
2.2 Observables and Poisson algebra
The state of a system in classical mechanics is specified by a point in phase space. An observable is then
simply a real-valued function on the manifold. If (M, ω) is a symplectic manifold, one can associate to each
such function f ∈ C∞(M) a vector field X f such that1
i(X f )ω = d f , (2.15)
where i is the contraction operator, which means that ωx(X f ,Y) = d f (Y) = Y( f ) for all Y ∈ TxM. This
relation defines X f , since ω is nondegenerate ∀x ∈ M. It is called the hamiltonian vector field generated by
f . Conversely, a vector field X on M is said hamiltonian if there exists a function fX such that i(X)ω = d fX .
The set of hamiltonian vector fields is denoted by Ham(M, ω).
If f , g ∈ C∞(M), one defines the Poisson bracket f , g by
f , g = ω(X f , Xg). (2.16)
Using (??), we may rewrite this as
f , g = iX fω(Xg) = d f (Xg) = Xg( f ) = −X f (g). (2.17)
In canonical coordinates, in which the symplectic form is given by (??), the hamiltonian vector fields and
Poisson bracket read as2
Xg =∂g∂pi
∂
∂qi −∂g∂qi
∂
∂pi(2.18)
f , g = −∂ f∂pi
∂g∂qi
+∂ f∂qi
∂g∂pi (2.19)
The Poisson bracket enjoys the following properties, ∀ f , g, h ∈ C∞(M):1Some references use the convention i(X f )ω = −d f , which changes some signs from place to place, see below.2by expressing ω(Xg, X) = dg(X) in local canonical coordinates
6
(i) skewsymmetry: f , g = −g, f
(ii) Jacobi identity: f , g, h + g, h, f + h, f , g = 0
(iii) derivation: f g, h = f g, h + f , hg
Properties (i) and (ii) give to the set of smooth functions on M the structure of an infinite-dimensional Lie
algebra, while the additional property (iii) endow it with a Poisson algebra structure. The additional property
(iv) X f ,g = −[X f , Xg],
where [., .] is the Lie bracket of vector fields, makes the application f → X f an anti-homomorphism of Lie
structure3. Property (i) follows from the fact that ω is a 2-form, i.e. an antisymmetric tensor, (ii) is a conse-
quence of dω = 0, using the definition of the exterior derivative4 (iii) can be shown by direct computation
and (iv) follows from the identities i([X,Y])ω = LX(i(Y)ω) − i(Y)LXω and LX = i(X) d + d i(X), where
L denotes the Lie derivative. For detailed proofs, see [?, ?, ?].
2.3 A flight over Hamiltonian and Lagrangian mechanics
In the Hamiltonian formulation of mechanics, a system with configuration space N is characterized by its
phase space, given by a symplectic manifold (M = T?N, ω) defining the kinematics (i.e. the state of a system
at a given time), and a Hamiltonian function H ∈ C∞(M) governing its dynamics (i.e. how it evolves). A
triple (M, ω,H) is called a Hamiltonian system. Hamilton’s equation take the following concise form in this
context:
i(XH)ω = dH. (2.20)
This equation determines a hamiltonian vector field XH whose integral curves (i.e. the curves whose tangent
vector at each point x is (XH)x) are the classical trajectories c(t) for given initial data c(0). By definition, one
has
XH|c(s) =ddt |t=s
c(t) =: c(s). (2.21)
In canonical coordinates, a curve or trajectory is written c(t) = (qi(t), p j(t)), with the obvious abuse of notation
qi(t) = qi(c(t)), and c(t) = qi ∂∂qi + pi
∂∂pi
. On the other hand, (??) gives
XH =∂H∂pi
∂
∂qi −∂H∂qi
∂
∂pi, (2.22)
so that (??) gives the familiar Hamilton’s equations
qi =∂H∂pi
, pi = −∂H∂qi . (2.23)
Of course, (??) supplemented with the fact that classical trajectories are integral curves of XH are not God-
given, and derive from variational principles. The physical/classical path of a general mechanical system
3Remark that if we set the signs such that i(X f )ω = −d f , see (??), then one gets X f ,g = [X f , Xg], i.e. a homomorphism of Lie
This follows directly from the identity LX = i(X) d + d i(X) (see also [?] for a more precise proof).
Remember that we defined the set of Hamiltonian vector fields, denoted by Ham(M, ω), as
Ham(M, ω) = vector fields X | ∃ f s.t. i(X)ω = d f . (2.44)
Obviously, Ham(M, ω) ⊂ aut(M, ω), since d2 = 0. Therefore, to each function f ∈ C∞(M) is associated a
vector field X f generating a one-parameter group of symplectomorphisms.
2.5 Lie groups of symplectomorphisms
The presence of symmetries in a classical system translates in the context of symplectic geometry in the
presence of an action of a Lie group G on the phase space M. Let M be a smooth manifold and G a Lie group.
The group G is called a Lie transformation group of M if to each g ∈ G is associated a diffeomorphism σg
of M such that
1. σg1σg2 (x) = σg1g2 (x) ∀x ∈ M,∀g1, g2 ∈ M
2. the map σ : G × M→M : (g, x)→σg(x) is C∞.
The group homomorphism G→Diff(M) : g→σg is called the action of G on M. It is called effective if e ∈ G
is the only element of G which leaves each x ∈ M fixed. It is said transitive if ∀x ∈ M, σG(x) = σg(x)|g ∈ G
is equal to M. If G = Lie(G), each X ∈ G defines a one-parameter group of diffeomorphisms ψXt :
ψXt (x) = σetX (x). (2.45)
The fundamental vector field5 X associated to X ∈ G is defined as
Xx =ddt |0
σe−tX (x). (2.46)
One can prove (see [?], pp. 4,34 and [?, ?]) that the map G→ vector fields: X→ X is a Lie algebra homomor-
phism:
[X, Y] = [X,Y]. (2.47)5Usually, the fundamental vector field associated to X is denote X∗, to avoid confusion with the dual or pull–back, we prefer here the
notation X.
10
An action σ : G→Diff(M) is called a symplectic action if σg is a symplectomorphism for all g ∈ G, i.e.
σ∗gω = ω ∀g ∈ G. (2.48)
From now on, we’ll restrict ourselves to symplectic actions. Following what we have seen in the previous
section, the vector field X ∈ aut(M, ω), ∀X ∈ G, since it generates a symplectomorphism. Thus d(i(X)ω) = 0.
If moreover, ∀X ∈ G a function λX such that
i(X)ω = dλX , (2.49)
exists, i.e. the fundamental vector fields X are hamiltonian ∀X ∈ G, then the action is said almost hamiltonian.
In that case one has
X = −λX , . . (2.50)
as follows from (??) and (??). Remark that with the notation (??), one has the equality X = XλX (where the
‘big’ X of XλX does not refer the element X ∈ G). If moreover the correspondence X→ λX is a Lie algebra
homomorphism (i.e. λkY+Z = kλY + λZ , ∀k ∈ R, λ[Y,Z] = λY , λZ), then the action is said hamiltonian. Once
one is given an almost hamiltonian action of G on M, one can always render it hamiltonian by extending G
to G = G × R and making an appropriate choice for λX (see [?], p39 and [?, ?]).
Example: coadjoint orbits
Let us now take M = θ f , see Sect.??. Being the orbit in g∗ of f under the group G, M obviously
admits an action of G:
σ : G × M→M : (g, x)→σ(g).x , (2.51)
whose fundamental vector fields are given by (??). One can show that G actually acts by sym-
plectomorphisms, that is
σ(g)∗ω = ω. (2.52)
Next, consider the action of a one-parameter subgroup of G, σ(exp(tZ)), with Z ∈ g. This gives a
one-parameter group of symplectomorphisms of the orbit M. It can be checked that its associated
fundamental vector field Z is Hamiltonian and
i(Z)ω = −dλZ , where λZ(x) = 〈x,Z〉. (2.53)
Furthermore, the correspondence g→ C∞(M) can be shown to be a Lie algebra homomorphism.
Therefore, the action of G on M given by (??) is Hamiltonian.
The importance of coadjoint orbits rests on the fact that the only symplectic spaces admitting a
Hamiltonian action of G are the covering spaces of the coadjoint orbits of G [?, ?, ?] .
11
2.6 Moment maps, hamiltonian systems with symmetry and integrals
of motion
Assume that a Lie group G has a hamiltonian action σ on (M, ω), and set, ∀Y ∈ G, Y = XλY , for a certain
The importance of the moment map appears namely when considering a hamiltonian system with G–symmetry,
consisting in a hamiltonian system (M, ω,H) and a hamiltonian action of G on M leaving the Hamiltonian
invariant:
σ∗gH = H ∀g ∈ G. (2.55)
Indeed, in this case, the invariance condition of H gives infinitesimally
LY H = 0
= Y(H)
= −λY ,H. (2.56)
Thus the functions λY , called the moment functions are integrals of motion forming through their Poisson
bracket a Lie algebra isomorphic to that of G.
Example: Let us consider a simple example to illustrate this [?]. Take M = R2n = x = (p, q) = (pi, q j).
Consider the hamiltonian action of the one-parameter group G = gt on M, with
gt : p j→ p j , q j→q j + t. (2.57)
The corresponding Hamiltonian field is
ddt |0
f (gt(x)) =ddt |0
f (p1, · · · , pn, q1 + t, · · · , qn + t) =
n∑j=1
∂ f∂q j
|x:= X f = −λX , f . (2.58)
With (??), one finds that
λX =
n∑j=1
p j. (2.59)
The invariance condition of H under G is H(pi, q j) = H(pi, q j + t), or infinitesimallyn∑
j=1
∂∂q j H(pk, qi) = 0. This
simply states that H can only depend upon the q j’s through the differences qi − q j. When this the case, one
can check that
H, λX = 0 (2.60)
which expresses the well-known fact that the total momentum is conserved in a system invariant under trans-
lations.
12
Chapter 3
Geometric Quantization
The issue of assigning a quantum system to a classical one, i.e. quantization, is still very timely since there
is no unique way of proceeding. The various approaches then try to be as general and natural as possible. It
is also asked that the quantization procedure preserve as much as possible the initial structure of the classical
system. For instance if a classical system possesses a symmetry group – through an hamiltonian action of this
group on the symplectic manifold modelling the classical phase space – one would like that the associated
quantum system belong to an unitary representation of this group. Moreover, if the action of the group is
transitive, this representation should be irreducible.
3.1 General Philosophy
We have seen that the mathematical geometrical framework to study a classical system is symplectic geom-
etry. The state of a system is specified by a point on a symplectic manifold (M, ω) which is called the phase
space, the observables are smooth functions on this manifold and their algebra is endowed with an additional
structure given by the Poisson bracket , .
Our purpose is to associate to such a classical system a quantum system. This means that we want to establish
a correspondence between the classical quantities and the quantum ones. The classical states — p ∈ M —
should become rays of an Hilbert space H (i.e. equivalence classes of elements of H , with v ∼ w if v = λw,
where λ is a scalar) and the classical observables — f on M — must correspond to self–adjoint operators
O f on this Hilbert space. The quantum observables are also endowed with a Poisson structure, namely the
commutator [ , ], see Table ??. These remarkable similarities led Dirac to formulate its quantification rules,
asking for the linear correspondence f → O f between classical and quantum observables in such a way that:
O1 = Id , [O f ,Og] = i~O f ,g . (3.1)
For instance, if we apply this to the canonical coordinates on R2n, we could get
qa → Oqa = qa , pa → Opa = −i~∂
∂qa , (3.2)
13
Table 3.1: Classical System vs. Quantum System
Classical System Quantum System
Phase Space symplectic manifold (M, ω) Hilbert spaceH
States point of M ray ofH
Algebra of Observables f ∈ C∞(M) Self–Adjoint Linear Operator onH
Structure on the Algebra of Observables Poisson bracket Commutator
where the operators act on the functions in L2(Rn) (depending only on the qa). Then to extend this correspon-
dence to operators, for example quadratic on the pa, it is necessary to make ordering choices. This method
therefore clearly depends on the coordinate choices which is a bad feature.
As mentioned above, an important issue in the quantization procedure is also to preserve the structure of the
classical phase space as much as possible. In particular, if a group acts on the classical system (M, ω) through
an Hamiltonian action, a ‘good’ quantization will translate this into the fact that the quantum states belong
to irreducible representations of this group. In the example of quantizing the canonical coordinates of R2n
mentioned above, the subspace of functions that depend only on the qa’s (and not on the pa’s) is clearly an
invariant subset under the action of the Oqa ’s and the Opa ’s given by equations (??). A solution to obtain
an irreducible representation of R2n (i.e. the group of invariance of the classical system1) is to restrict the
space of functions to quantize to be the subspace of C∞(R2n) that depend only on the qa’s, i.e. C∞(Rn). More
generally, if the quantization procedure furnish a reducible representation of group, a solution is to restrict
the quantization of observables to a Poisson subalgebraA of C∞(M). It is generally not an easy task to select
the appropriate subalgebraA.
The program of geometric quantization is a procedure of quantization based on the ideas of Dirac that is
applicable to any symplectic manifold, i.e. any phase space, which is independent of coordinate choices and
which ‘keeps’ tracks of the symmetry of the classical system with an Hamiltonian action G [in the sense that
the quantum states form irreducible representation of G]. A first step in the geometric quantization procedure,
namely the prequantization, consists in forgetting the irreducibility condition. It is an elegant construction
involving line bundles and connections on these line bundles and but unfortunately, when one ask for the irre-
ducibility condition to hold, this complicate the story. The will be done by the introduction of a polarization
of the space of functions on M (the aim is to select a subspace of functions which are quantize) as explained
in the sequel.
Geometric Quantization procedure : Let (M, ω) be a classical system and A a sub-algebra of observables.
A quantum system (H ,O) is said to be associated with this classical system if
1not really hamiltonian, this will be explained at the end of section ??
14
Q1. H is a complex separable Hilbert space. Its elements ψ are the quantum wave functions and the rays
λψ |λ ∈ C are the quantum states.
Q2. O is an application that maps a classical observable f ∈ A into an self-adjoint operator O f on H such
that
O1. Oλ f +g = λO f + Og ,
O2. O1 = IdH ,
O3. [O f ,Og] = i~O f ,g ,
O4. if there is a transitive and strongly hamiltonian action of a group G on M, thenH forms an irreducible
representation space of this group.
3.2 Mathematical Preliminaries
This section is devoted to introduce notions needed for the geometric quantization procedure. As explained
after, a line bundle is the correct object to define the states of a system, the observables are then connections
on this line bundle. Finally, as the quantum states should belong to an Hilbert space, the notion of hermiticity
on the fiber is introduced.
3.2.1 Line Bundle
A (complex) line bundle is a vector bundle whose fibers are one–dimensional (complex) vector spaces. Ex-
plicitly, it consists in a triple (L, π,M) such that
F1. L is a differentiable manifold and π : L→ M is a smooth surjective application,
F2. ∀x ∈ M, the fiber Ex = p−1(x) possesses a structure of one dimensional complex vector space,
F3. There exist an open covering Ua|a ∈ A of M and smooth functions sa : Ua → L such that
(a) ∀a ∈ A, π sa = IdUa ,
(b) ∀a ∈ A, the application
ψa : Ua × C→ π−1(Ua) : (m, z)→ z sa(m) , (3.3)
is a diffeomorphism.
The functions sa satisfying the condition (a) are called local sections of L and a collection (Ua, sa)| a ∈ A a
local system for L. The space of global sections, s : M → L : m → s(m), is denoted Γ(L). A line bundle is
depicted in Figure ??.
15
M
π
L
π−1(Ua) ψa
Ua × C
sa
Ua
Figure 3.1: Line Bundle
Remark 1: the condition F3 (b) implies that the sections are nowhere vanishing, otherwise ψa could not be a
diffeomorphism.
Remark 2: The notation sa(m) denotes an element of L but it is also sometimes used to designate its corre-
sponding element in the fiber at the point m, i.e. we have sa(m) = (m, sa(m)) and the meaning is clear accord-
ing to the context. For instance in equation (??), z sa(m) (where sa(m) ∈ L) means z.sa(m) = z.(m, sa(m)) =
(m, z.sa(m)) ∈ L. This abuse of notation will often be used in the sequel.
Transition functions
On the intersection Uab := Ua ∩ Ub of two open sets Ua and Ub, a transition function can be defined by
Ψab = ψ−1a ψb : Uab × C→ Uab × C : (m, z)→ (m,
sb(m)sa(m)
z) .
We can therefore define the functions gba : Uab → C? = C/0 : m → gba(m) =
sb(m)sa(m) such that Ψab(m, z) =
(m, zcba). These functions satisfy
gaa = 1 on Ua,
gabgba = 1 on Uab if not empty,
gabgbcgca = 1 on Uabc := Ua ∪ Ub ∪ Uc if not empty . (3.4)
The three conditions (??) are called cocycle conditions. For later purpose, these conditions can further be
expressed by setting
gab := e2πihab , hab : Uab → C smooth functions, (3.5)
16
as
haa = 0 on Ua,
hab = −hba on Uab if not empty,
hab + hbc + hca =: Cabc ∈ Z on Uabc = Ua ∪ Ub ∪ Uc if not empty . (3.6)
Cabc is constant on Uabc (since it is smooth on a connected set and can only take discrete values) and is totally
antisymmetric in its arguments.
Remark: An equivalent definition of a line bundle can be given in terms of the transition functions. The
condition F3 is replaced by asking that there exists an open covering Ua|a ∈ A of M such that on each
intersection Uab, there exists a transition function Ψab : Uab ×C→ Uab ×C : (m, z)→ (m, gba(m)z) such that
the conditions (??) are fulfilled.
Trivialisation
The collection Ua, ψa defined by a local system Ua, sa is called a local trivialisation of L since over each
open set Ua, the diffeomorphism ψa gives to the fiber Ex (for all x ∈ Ua) a “trivial” structure, i.e. it establishes
a isomorphism between π−1(Ua) and the direct product Ua × C. A line bundle is said to be trivial if L is
globally diffeomorphic to M × C.
Cech Cohomology
The Cech cohomology that we will now introduce, provides a useful tool to classify line bundles . Let G be
an abelian group and Ua |a ∈ A a contractible2 open covering of a smooth manifold M. A p–cohain on M
with values in G is a rule c which assigns to each collection Ua0 , ...,Uap of (p + 1) open sets in the covering
with non empty intersection (Ua0 ∪ ... ∪ Uap , φ) an element ca0...ap in G so that it is totally antisymmetric in
its arguments.3 The co–boundary δc of a p–cochain c is the (p + 1)–cochain defined by
δcα0..αp+1 :=p+1∑i=0
(−1)i cα0...αi...αp+1 , (3.7)
where αi means that the indice αi is omitted. One may check that δ2 = 0.
Example: Let us illustrate the notion of cochain and co–boundary on a very simple example. We
consider a manifold which is the union of three open sets U0, U1, U2 and the abelian group of
natural number N,+. A 1–cochain is a rule that assigns to U0,U1, U1,U2, U0,U2, U1,U0,
U2,U1 and U2,U0 some natural numbers c01, c12, c02,−c01,−c12 and −c02. Let us take a
concrete example c01 = 3, c12 = −1, c02 = 4. The co–boundary of c is a 2–cochain whose values
2Contractible means that any close curve in Ua can be smoothly deformed to a point.3Note that the order in which the open sets are given matters for the rule c.
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on the sets U0,U1,U2, U1,U0,U2, U2,U1,U0, U0,U2,U1, U2,U0,U1 and U1,U2,U0,
is given by the equation (??). For instance, one has δc012 = c12 − c02 + c01 = −2.
The pth–Cech cohomology space HpCech(M,G) is defined by
HpCech(M,G) :=
p − cochains c | δc = 0p − cochains c | c = δb
(3.8)
Theorem : The pth–cohomology space HpCech(M,G) is independent of the choice of the contractible covering
of M. Moreover, if G = R or C, HpCech(M,G) ∼ Hp
De Rham(M,G) (canonical isomorphism).
Let us now return to the transition functions of a line bundle and their cocycle conditions. Equation (??)
expresses that C is a 2–cochain. By explicit computation one may check that
Cbcd −Cacd + Cabd −Cabc = 0 if Uabcd , 0 . (3.9)
Thus C is a 2–cocycle for the Cech cohomology of M with values in Z (δC = 0). For line bundle (L, π,M),
let c(L) denote the class of C in H2Cech(M,Z):
c(L) := 2 cocycles c′ | c′ = C + δb, b is a 1 − cochain , (3.10)
it is called the Chern class of the line bundle.
Theorem : The line bundle (L, π,M) is trivial iff c(L) = 0. Moreover, two line bundles are equivalent iff they
have same Chern class and there is a bijection between the set of equivalence classes of line bundles on M
and H2(M,Z) (the latter classify the former).
3.2.2 Connection on a Line Bundle
A connection is an application ∇ that associates to each vector field X ∈ V(M)C an endomorphism ∇X :
Γ(L)→ Γ(L) such that
C1. ∇X+Y s = ∇X s + ∇Y s,
C2. ∇ f X s = f∇X s,
C3. ∇X f s = X( f )s + f∇X s,
for all s ∈ Γ(L), X,Y ∈ V(M)C and f ∈ C∞(M)C. The linear operator ∇X is the covariant derivative along X
for the connection ∇. Note that the condition C2 implies that ∇X s(m) depends on X only at the point m while
condition C3 indicates that ∇X s(m) depends on locally on the section s. We can focus on a open set Ua. Since
the connection is linear in X, we have for each sa,
∇X sa = 〈θa, X〉sa
for some 1–form θa ∈ Ω1C(Ua). Therefore, for an arbitrary section (which can be written locally as s = fasa
on Ua where fa ∈ C∞(Ua)), we have4
∇X s = (X( fa) − 2πi〈αa, X〉 fa)sa . (3.11)4multiplication of a section by a function means multiplication in the fiber, i.e. f s(m) = (m, f s(m)) locally
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The set of 1–forms αa on the Ua is called 1–form connection. This 1–form connection (Ua, θa) is not globally
defined on M.
On a non–empty intersection Uab, we have sa = gbasb and therefore
∇X sa = (X(cba) − 2πi〈αb, X〉gba)sb
= −2πi〈αa, X〉gbasb
which implies that
αb − αa =1
2πidgba
gba. (3.12)
The equation (??) implies that dαa |Uab = dαb |Uab . The 2–form Ω defined by
Ω|Ua = −dαa ∀a , (3.13)
is therefore globally well defined.
3.2.3 Curvature on a Line Bundle
The curvature of a connection is generally defined as the operator Curv∇(X,Y) : Γ(L)→ Γ(L) such that
Curv∇(X,Y) = ∇X∇Y − ∇Y∇X − ∇[X,Y] .
This operator is skewsymmetric and C∞(M)C–linear in X and Y and therefore should be given by a 2–form