Clifford Geometry: A Seminar
Joseph C. Va´rilly and Jose´ M. Gracia-Bond´ıa
Universidad de Costa Rica, 2060 San Jose´, Costa Rica
Diciembre de 1995
Contents
1 Vector bundles and their classification 4
1.1 Generalities on manifolds 4
1.2 Principal bundles and vector bundles 5
1.3 Hermitian vector bundles 7
1.4 Operations on vector bundles 7
1.5 Equivalent bundles 8
1.6 Sections of vector bundles 9
1.7 Local sections and transition functions 10
1.8 Cˇech cocycles 12
1.9 The Cˇech cohomology of S2 13
1.10 Line bundle classification 14
1.11 The second cohomology group 15
1.12 Classification of Hermitian line bundles 16
1.13 Classification of vector bundles 17
2 Complex projective spaces 19
2.1 Complex manifolds 19
2.2 Local charts for complex projective spaces 20
2.3 The Ka¨hler form 21
2.4 The Fubini–Study metric 22
2.5 The Riemann sphere 24
3 The de Rham complex and Hodge duality 24
3.1 The de Rham complex 24
3.2 The Riemannian volume form 25
3.3 The Hodge star operator 26
3.4 The Hodge Laplacian 31
1
4 The Hodge Laplacian on the 2-sphere 33
4.1 The rotation group in three dimensions 33
4.2 The Hodge operators on the sphere 35
4.3 Eigenvectors for the Laplacian 36
4.4 Spectrum of the Hodge Laplacian 38
4.5 The Hodge–Dirac operator 40
5 Connections on vector bundles 42
5.1 Modules of vector-valued forms 42
5.2 Connections 45
5.3 Curvature of a connection 46
5.4 A curvature formula 47
5.5 From de Rham cohomology to Cech cohomology 48
5.6 Line bundles over CPm 51
5.7 Connections on the hyperplane bundle 53
5.8 Characteristic classes 54
5.9 Chern classes and the Chern character 57
5.10 Classification of line bundles over CPm 60
5.11 The Levi-Civita connection on the tangent bundle 61
6 Clifford algebras 62
6.1 Superspaces and superalgebras 62
6.2 Clifford algebras 65
6.3 Clifford actions 66
6.4 Complex Clifford algebras 68
6.5 The Fock space of spinors 70
6.6 The Pin and Spin groups 72
6.7 The spin representation 76
7 Global Clifford modules 77
7.1 Clifford algebra bundles 77
7.2 Existence of spin structures 80
7.3 Spinc structures 83
7.4 The spinor module 85
7.5 The spin connection 86
7.6 Local coordinate formulas 89
8 Dirac operators and Laplacians 91
8.1 Connections and differential forms 91
8.2 Divergence of a vector field 93
8.3 The Hodge–Dirac operator revisited 94
8.4 Dirac operators 95
8.5 Laplacians 98
2
8.6 The Lichnerowicz formula 101
9 The Dirac operator on the Riemann sphere 102
9.1 Coordinates on the Riemann sphere 102
9.2 Sections and gauge transformations 104
9.3 The spin connection over the sphere 106
9.4 The Dirac operator over the sphere 107
9.5 The spinor Laplacian 110
9.6 The SU(2) action on the spinor bundle 112
9.7 Equivariance of the Dirac operator 115
9.8 Angular momentum operators 117
9.9 Spinor harmonics 119
9.10 The spectrum of the Dirac operator 122
10 Construction of representations of SU(2) 124
10.1 Characters of the maximal torus 124
10.2 Twisting connections 126
10.3 Twisted Dirac operators 127
10.4 The group action on the twisted spinor bundles 128
10.5 The Borel–Weil theorem 131
A Calculus on manifolds 135
A.1 Differential manifolds 135
A.2 Tangent spaces 136
A.3 Vector fields 136
A.4 Lie groups 138
A.5 Fibre bundles 139
A.6 Tensors and differential forms 141
A.7 Calculus of differential forms 142
A.8 The de Rham complex 145
A.9 Volume forms and integrals 146
3
1 Vector bundles and their classification
The procedure commonly known as geometric quantization associates a Hilbert space to
certain symplectic manifolds, and thereby allows a bridge to be built from the algebra of
functions on the manifold to an operator algebra. It has been used, with considerable success,
to reconstruct unitary representations of Lie groups from their symplectic homogeneous
spaces; this is known as the Kirillov orbit method. An alternative method of constructing
these representations, at least for compact groups, arises from the study of Dirac operators
on these same homogeneous manifolds. The purpose of these notes is to clarify the roˆle
of the Dirac operator. Hopefully, this will enable us to build a bridge between geometric
quantization and noncommutative geometry, in which the key concept is a generalization of
the Dirac operator to “noncommutative manifolds”.
The main geometric objects with which we shall be concerned are vector bundles with
some extra algebraic structure, such as a metric or a symplectic structure (or both). Roughly
speaking, a vector bundle consists of the disjoint union E of a collection of isomorphic vector
spaces Ex, indexed by the points of a differential manifold M , together with some smoothness
conditions relating E and M ; an example is the tangent bundle of the manifold M . The
algebraic structure of the “fibres” Ex transfers to give an interesting algebraic structure to
the manifold E. The set of all possible vector bundles of a given species on a manifold M ,
up to a suitable equivalence relation, holds important information about the topology of
the manifold M (in fact, it holds all the topological information we need to know for the
purposes of noncommutative geometry) and we begin our study with this classification.
We have gathered in Appendix A a compendium of definitions and basic facts on differ-
ential manifolds, vector fields and differential forms, with which we assume the reader to be
acquainted. We refer to it for the notations used below.
1.1 Generalities on manifolds
Recall that a differentiable manifold of (real) dimension n is a (paracompact) topological
space M provided with an atlas, i.e., a collection of charts (Uj, φj), where the domains Uj
form a (locally finite) open covering U of M , each φj : Uj → Rm is a homeomorphism, and
the “transition maps” φi ◦ φ−1j are smooth functions on each φj(Ui ∩ Uj). If n = 2m, we
can regard R2m as Cm; we say that M is a complex manifold if the transition maps are
holomorphic (as multi-variable vector-valued complex functions); it suffices that they satisfy
the Cauchy–Riemann equations.
We shall be mainly interested in the case where M is compact; then its differentiable
structure is defined by a finite atlas.
A useful property of the chart domains Uj is that they be contractible, that is, that there
exist x0 ∈ Uj and a smooth function f : [0, 1] × Uj → Uj with f(0, x) = x0, f(1, x) = x;
informally, Uj may be “deformed” to a point {x0}. The intersection of two such domains
need not be contractible: just think of the sphere S2 covered by two large polar caps which
overlap at the equator. It is good to know, however, that by using another (equivalent) atlas,
we can avoid this difficulty. What we need is that the chart domains form a “good covering”
4
of M [12].
Definition 1.1. An open covering U := {Uj} of a topological space is a good covering1 if
every nonempty finite intersection Uj1 ∩ · · · ∩ Ujr is contractible. A differentiable manifold
M always has a good covering: take a Euclidean metric on M and cover M by an atlas for
which each Uj is “geodesically convex” (that is, any two of its points can be connected by
a minimal geodesic lying within Uj).
2 All finite intersections are also geodesically convex,
and hence are contractible (since one can deform to a point x0 by retreating along geodesics
emanating from that point). So we can replace the original atlas of M by a (possibly larger)
atlas whose domains form a good covering. From now on, we will assume that this has been
done.
Exercise 1.1. Show that the following recipe defines a good covering {U1, U2, U3, U4} of the
sphere S2. Take four points on S2 that are vertices of a regular tetrahedron (e.g., if a = 1/
√
3,
take (±a,±a,±a) with either one or three positive signs). Connect these points by six great-
circle arcs, which determine four spherical triangles Fj. Finally, let Uj := {x ∈ S2 : d(x, Fj) <
 } for some small  (say,  = 1/4).
1.2 Principal bundles and vector bundles
Recall that a fibre bundle E
pi−→M or simply E−→M (see Appendix A for notation) is a
triple (E,M, pi), where the manifolds E and M are its total space and base space respectively,
and pi : E → M is a surjective submersion, subject to two conditions: (a) that each fibre
Ex := pi
−1({x}) is diffeomorphic to a fixed manifold F (the “typical fibre”), and (b) for some
atlas {(Uj, φj)} of M there is a family of local trivializations of E, i.e., diffeomorphisms
ψj : pi
−1(Uj)→ Uj × F (1.1)
such that pi(ψ−1j (x, v)) = x for all x ∈ Uj, v ∈ F .
Definition 1.2. Let G be a Lie group. A principal G-bundle P
η−→M is a fibre bundle
whose fibres are diffeomorphic to G, together with a free right action of G on the total space
P , whose orbits are the fibres Px = η
−1(x). We use the notation χj : η−1(Uj)→ Uj ×G for
the local trivializations.
Exercise 1.2. Suppose that G is a Lie group and that H is a closed subgroup of G (so that
H is also a Lie group); then H acts freely on G by right multiplication g · h := gh. If
η : G→ G/H is the quotient map, check that G η−→G/H is a principal H-bundle.
Before examining where principal bundles come from, let us describe a general recipe for
manufacturing new fibre bundles using a given principal bundle. Indeed, this recipe is the
main reason why principal bundles are used at all.
1This is also known as a “contractible covering” [38], or a “Leray covering” [58].
2It is a basic proposition of Riemannian geometry that each point has a geodesically convex neighbour-
hood; see [32] or [36] for the proof.
5
Definition 1.3. Let P
η−→M be principal G-bundle, and let F be a manifold on which the
Lie group G acts on the left. The product manifold P × F carries a right action of G, given
by
(p, v) · g := (p · g, g−1 · v). (1.2)
Let E := P ×G F be the set of orbits of this action; we denote the orbit of (p, v) by [p, v].
Notice that [p · g, v] = [p, g · v] on account of (1.2).
Define pi : E →M : [p, v] 7→ η(p). For each p ∈ P , the map v 7→ [p, v] is a diffeomorphism
from F to Eη(p), and E
pi−→M is a fibre bundle with typical fibre F , which is said to be
associated to the given principal G-bundle.
Definition 1.4. A vector bundle E
pi−→M is a fibre bundle whose “typical fibre” is a (real
or complex) vector space V , such that each fibre Ex is a vector space of dimension dimV ,
and such that for each x ∈ Uj, the map v 7→ ψ−1j (x, v) is a linear isomorphism from V
onto Ex.
A real vector bundle with typical fibre V = R is called a real line bundle; a complex
vector bundle with typical fibre V = C is called a complex line bundle. When we say
simply “line bundle”, we shall usually mean a complex line bundle.
Examples of (real) vector bundles over M are the tangent bundle TM −→M , the cotan-
gent bundle T ∗M −→M , and its exterior powers ΛrT ∗M −→M : see Appendix A.
Given a representation ρ : G→ GL(V ) of a Lie group, and a principal G-bundle P η−→M ,
we can form the associated bundle by taking g · v ≡ ρ(g)v; thus
[p · g, v] = [p, ρ(g)v] for p ∈ P, v ∈ V, g ∈ G. (1.3)
The fibres Ex become vector spaces by taking [p, u]+ [p, v] := [p, u+v], and λ[p, v] := [p, λv];
and the linearity of each ρ(g) shows that E
pi−→M is a vector bundle with typical fibre V .
We can now reverse the recipe of Definition 1.3 in order to associate a principal bundle
to a given vector bundle:
Definition 1.5. Let E
pi−→M be a vector bundle3 with typical fibre V . Let Px, for x ∈M ,
denote the set of linear isomorphisms p : V → Ex (each such p is called a frame for Ex).
If g ∈ GL(V ), then p ◦ g is again a frame, so that GL(V ) acts on the right (freely and
transitively) on Px by composition p 7→ p ◦ g. The disjoint union P =
⊎
x∈M Px, with
η(p) := x for p ∈ Px, defines a principal GL(V )-bundle P η−→M , which is called the frame
bundle of E
pi−→M .
Exercise 1.3. Using the identity representation of the group GL(V ) on V , check that the
vector bundle associated to the frame bundle P
η−→M by the recipe (1.2) is the original
vector bundle E
pi−→M .
3We use a notation which covers the real and complex cases simultaneously; thus GL(V ) denotes either
GLR(V ) or GLC(V ), according as V is a real or complex vector space.
6
1.3 Hermitian vector bundles
If V carries some extra structure (e.g., an inner product or an orientation), we can restrict to
the subgroupG ≤ GL(V ) which preserves that structure. We need to impose a corresponding
structure on the fibres Ex and consider only those frames p : V → Ex which are structure-
preserving. Under p 7→ p ◦ g, these comprise a principal G-bundle associated to E pi−→M .
More generally, if we are given a Lie group G with a representation ρ : G → GL(V ), we
define a right action of G on frames by p · g := p ◦ ρ(g). If, and sometimes this is “a big
if”, we can select a subset of frames for each Ex for which this action is transitive and free
(due to some extra structure given on the vector bundle), these will form the fibres Qx of a
principal G-bundle Q
η′−→M . Since [p · g, v] = [p, ρ(g)v] for p ∈ Q, v ∈ V , the vector bundle
associated, via ρ, to the new principal bundle is still E
pi−→M .
For instance, we could ask that a real vector bundle E−→M be Euclidean, i.e., that
each fibre Ex carry a real inner product gx(·, ·), which depends smoothly on x. If V is a real
vector space with a positive-definite inner product q(·, ·), we consider “orthogonal frames” p
satisfying gx(p(u), p(v)) = q(u, v) for all u, v ∈ V . It is clear that such frames form a principal
O(n)-bundle associated to E
pi−→M . Such an isomorphism p is determined by choosing an
orthonormal basis in (Ex, gx) (as the image under p of a fixed orthonormal basis in (V, q)),
so that this bundle is often called the “orthonormal frame bundle”.
Alternatively, if E
pi−→M is a complex vector bundle, we could ask that it be Hermitian,
i.e., that each fibre Ex carry a (sesquilinear) inner product hx(·, ·), depending smoothly on x.
Then if V is a complex Hilbert space with inner product 〈· | ·〉, we consider “unitary frames”
p satisfying hx(p(u), p(v)) = 〈u | v〉 for u, v ∈ V . Such frames form a principal U(n)-bundle
associated to E
pi−→M .
1.4 Operations on vector bundles
Definition 1.6. Given any (real or complex) vector bundle E−→M , one can form a dual
vector bundle E∗−→M whose fibre E∗x is the dual vector space4 of Ex. Fix a basis
{v1, . . . , vn} for the typical fibre V . A frame p : V → Ex is defined by selecting a basis
{e1, . . . , en} for Ex and setting p(vj) := ej for each j; matching the dual bases {v′1, . . . , v′n},
{e′1, . . . , e′n} gives a frame p′ : V ∗ → E∗x for the dual bundle. Notice that (p ◦ g)′ = p′ ◦ g−t by
change-of-basis formulae, where g−t := (g−1)t is the contragredient matrix to g ∈ GL(V ).
Exercise 1.4. Verify that E∗−→M is the vector bundle associated to the frame bundle of
E−→M , via the representation ρ(g) := g−t of GL(V ) on V ∗.
Exercise 1.5. Check that the cotangent bundle T ∗M −→M is the dual bundle to the tangent
bundle TM −→M .
Definition 1.7. Given any two vector bundles E−→M , E ′−→M over the same base space,
we can form two new vector bundles over M : their Whitney sum E ⊕ E ′−→M and their
4The dual space of V is V ∗ := HomR(V,R) or V ∗ := HomC(V,C), the space of R-linear or C-linear forms
on V , according as V is a real or complex vector space.
7
tensor product E⊗E ′−→M , whose fibres at x ∈M are respectively the direct sum Ex⊕E ′x
and the algebraic tensor product Ex ⊗ E ′x.
The k-th exterior power of E−→M is the vector bundle ΛkE−→M whose fibre at x is
ΛkEx. (For k = 0, we take Λ
0E := M × R.)
These operations can be combined; for instance, the exterior algebra bundle Λ•E−→M
is the Whitney sum of all exterior powers, for k = 0, 1, . . . , r, where r = dimEx is the rank
of E−→M .
Definition 1.8. The complexification of a real vector bundle E−→M is the complex
vector bundle EC−→M with EC := E ⊗R C, i.e., (Ex)C := Ex ⊗R C = Ex ⊕ i Ex for
each x ∈M .
We write TCM and T
∗
CM for the total spaces of the complexified tangent and cotangent
bundles of M .
1.5 Equivalent bundles
Definition 1.9. A morphism of two fibre bundles E
pi−→M and E ′ pi′−→M ′ is a pair of smooth
maps (τ, σ), with τ : E → E ′ and σ : M → M ′, such that pi′ ◦ τ = σ ◦ pi, i.e., such that the
following diagram commutes:
E
τ−−−→ E ′
pi
y ypi′
M
σ−−−→ M ′
and, in particular, τ(Ex) ⊆ E ′σ(x) for each x ∈M .
A morphism of vector bundles is a bundle morphism for which τ : Ex → E ′σ(x) is linear.
When M ′ = M , we usually take σ = idM .
Definition 1.10. Let E
pi−→M be a vector bundle and let φ : N → M be a smooth map.
Write φ∗E := { (u, y) ∈ E ×N : pi(u) = φ(y) } and define p¯i : φ∗E → N and φ˜ : φ∗E → E by
p¯i(u, y) := y and φ˜(u, y) := u. Then p¯i−1(y) = Eφ(y), and so φ∗E
p¯i−→N is a vector bundle,
called the pullback bundle of E
pi−→M via φ. Moreover, (φ˜, φ) is a bundle morphism; in
other words, we have a commutative diagram of smooth maps:
φ∗E
φ˜−−−→ E
p¯i
y ypi
N
φ−−−→ M
Exercise 1.6. Show that the pullback bundle has the following universal property: if E ′ pi
′−→N
is a vector bundle over N and if ρ : E ′ → E is a map such that (ρ, φ) is a bundle morphism
from this bundle to E
pi−→M , then there is a unique bundle morphism (τ, idN), with τ : E ′ →
φ∗E, so that ρ = φ˜ ◦ τ . (In other words, any bundle morphism with base map φ factors
through the pullback bundle.)
8
Definition 1.11. Let E
pi−→M , E ′ pi′−→M be two vector bundles over the same base space.
A vector bundle equivalence between them is an invertible vector bundle morphism
(τ, id), which is given by a diffeomorphism τ : E → E ′ satisfying pi′ ◦ τ = pi and such that
τ : Ex → E ′x is a linear isomorphism for each x.
We shall denote by [E] the equivalence class of the vector bundle E
pi−→M ; for the set of
equivalence classes with typical fibre V , we write Vect(M ;V ).
A vector bundle E−→M is trivial if it is equivalent to the product bundle M×F pr1−→M .
Note, from Exercise 1.6, that the pullback bundle is unique up to equivalence, and so
defines a unique [φ∗E] ∈ Vect(N ;V ).
Exercise 1.7. Show that the tangent bundle TS1 of the unit circle S1 is trivial, by producing
a pair of local trivializations which together form a cylinder.
Exercise 1.8. Let E−→M be an arbitrary vector bundle over M ; write Er = M × V
with dimV = r, so that Er−→M denotes the trivial vector bundle of rank r. Show that
E ⊗Er−→M is equivalent to the Whitney sum E ⊕ · · · ⊕E−→M of r copies of E−→M .
Definition 1.12. Let P
η−→M , P ′ η′−→M be two principal G-bundles over the same base
space. An equivalence between them is an invertible bundle morphism (χ, id) which is G-
equivariant, that is, χ(p·g) = χ(p)·g for any p ∈ P . Note that χ : P → P ′ is a diffeomorphism.
We shall denote by [P ] the equivalence class of P
η−→M ; for the set of such equivalence
classes we write Prin(M ;G).
Proposition 1.1. Two vector bundles over M with the same typical fibre V are equivalent if
and only if their frame bundles are equivalent as principal GL(V )-bundles. The association
recipe thus yields a bijection [E]↔ [P ] between Vect(M ;V ) and Prin(M ;GL(V )).
Exercise 1.9. Prove this, using the defining relation (1.3) of an associated vector bundle.
1.6 Sections of vector bundles
Definition 1.13. A smooth section of a vector bundle E
pi−→M is a smooth map s : M →
E such that pi ◦ s = idM , i.e., s(x) ∈ Ex for each x ∈M . The totality of smooth sections will
be denoted by Γ(E), or by Γ(M,E) if it is necessary to specify the base space M . Notice that
Γ(E) is a module for the commutative algebra of functions C∞(M); the action of C∞(M) is
just scalar multiplication in each fibre:5
(fs)(x) := f(x)s(x),
for s ∈ Γ(E), f ∈ C∞(M).
5It would perhaps be more convenient, in view of an eventual translation to the language of noncom-
mutative geometry, to write the multiplication on the right : (sf)(x) = s(x)f(x); but as this conflicts with
traditional habits, and could be confusing in the case that s is a vector field, we will for the moment retain
the usual notation of multiplying by scalars f(x) on the left.
9
“Global” smooth sections s : M → E can sometimes be hard to find. For instance, a line
bundle admits a nonvanishing global section only if it is trivial. To see this, notice that any
v ∈ Ex is of the form λs(x), for a unique λ ∈ C, since s(x) 6= 0; so λs(x) 7→ (x, λ) is a vector
bundle equivalence between E and the trivial line bundle M × C.6
Lemma 1.2. If L−→M is a line bundle, then its tensor product with its dual line bundle,
namely L⊗ L∗−→M , is a trivial line bundle.
Proof. Indeed, Lx⊗L∗x ' End(Lx), so the tensor product bundle has an obvious nonvanishing
section s0, whose value at each x is the identity operator on the line Lx. Since each End(Lx)
is a one-dimensional vector space, any L ⊗ L∗−→M is a line bundle, with a nonvanishing
global section.
Corollary 1.3. The set of equivalence classes of line bundles over M has the structure of
an abelian group.
Proof. Let L−→M , L′−→M be any two line bundles over M , and define:
[L] [L′] := [L⊗ L′], [L]−1 := [L∗]. (1.4)
Let L0 := M × C so that [L0] is the trivial bundle class,7 and note that [L ⊗ L0] = [L]
by Exercise 1.8. Since [L ⊗ L∗] = [L0] by the previous Lemma, the inverse of [L] is [L∗].
Moreover, the flip map u⊗v 7→ v⊗u from Lx⊗L′x to L′x⊗Lx determines a bundle equivalence
between L⊗ L′ and L′ ⊗ L, so that the product (1.4) is commutative.
We shall soon identify this group with a cohomology group of M .
1.7 Local sections and transition functions
The question that now arises is how to deal with bundles that are not trivial, and how to
give an effective description of such bundles. Since there are no nonvanishing global sections,
we must make use of nonvanishing local sections sj ∈ Γ(Uj, E), where U := {Uj} is an open
covering of M by chart domains which admit local trivializations ψj as in (1.1). Thus
ψj(sj(x)) ≡ (x, fj(x)) (1.5)
where fj : Uj → V \{0} is a smooth nonvanishing function. (Indeed, to say that sj is smooth
is the same as saying that its local representative fj is a smooth function.)
A global section s ∈ Γ(M,E) is determined, via (1.5), by such a family of local represen-
tatives fj : Uj → V (which may now take zero values).
Let r = dimV be the rank of the vector bundle E
pi−→M . Suppose we can find a set
sj = (sj1, . . . , sjr) with each sjk ∈ Γ(Uj, E) so that {sj1(x), . . . , sjr(x)} is a basis for the
6Here we are falling into the sloppy habit of referring to a fibre bundle by naming its total space only. If
the base space M is fixed, this does no harm.
7We use complex line bundles only to be specific; the argument for real line bundles is identical.
10
fibre Ex at each x ∈ Uj. Then if tj ∈ Γ(Uj, E) is any smooth local section, we can find
smooth functions h1j , . . . , h
r
j ∈ C∞(Uj) such that tj =
∑r
k=1 h
k
j sjk on Uj. Thus Γ(Uj, E) is
determined by the set of local sections sj; and globally, the module Γ(M,E) is determined
by a family {(Uj, sj)} of such sets, one for each local chart of M . Such a family is sometimes
called a local system of sections [38] for the vector bundle E−→M .
Note that sj is a local frame over Uj and may be regarded as a local section of the frame
bundle P
η−→M . To be precise, choose and fix a basis {v1, . . . , vr} for V , and let pj ∈ Px,
for x ∈ Uj, be the linear isomorphism from V to Ex determined by pj(vk) := sjk(x) for
k = 1, . . . , r. Then pj ∈ Γ(Uj, P ). Conversely, any such local section pj determines the local
frame sj.
Suppose that (Ui, si) is another local frame and that the chart domains Ui and Uj overlap:
Ui ∩ Uj 6= ∅. Then for x ∈ Ui ∩ Uj, the isomorphisms pi, pj from V to Ex determined by
pi(vk) := sik(x), pj(vk) := sjk(x) are related by pj = pi ◦ gij(x) for some gij(x) ∈ GL(V ). In
the notation of associated bundles, we have
[pj, v] = [pi ◦ gij(x), v] = [pi, gij(x)v] (1.6)
for v ∈ V ; or equivalently,
ψi ◦ ψ−1j (x, v) = (x, gij(x)v), (1.7)
i.e., gij is the expression in local coordinates of the transition between the local trivializations
ψi, ψj of the vector bundle. Thus each gij : Ui ∩ Uj → GL(V ) is a smooth function.
Exercise 1.10. Show that a principal bundle P
η−→M is trivial if and only if it admits a
global smooth section q : M → P .
Definition 1.14. Let E−→M be a vector bundle, with typical fibre V , for which { (Uj, sj) :
j ∈ J } is a local system of pointwise linearly independent local sections. The family of
smooth functions gij : Ui∩Uj → GL(V ), defined whenever Ui∩Uj 6= ∅, such that si = gij ·sj
on Ui ∩ Uj, satisfies the consistency conditions
gii = id on Ui, gijgjk = gik on Ui ∩ Uj ∩ Uk. (1.8)
(The notation gij · sj denotes the natural action of GL(V ) on E, as expressed by (1.6)
or (1.7).) The set {gij} is called a family of transition functions for the vector bundle
E−→M .
Suppose now that the vector bundle E−→M carries extra structure, for instance a
Hermitian metric. Then we can assume that the basis sj(x) for Ex respects this structure
(continuing the Hermitian-metric example, we may take it to be an orthonormal basis, with
sjk(x) = pj(vk) for a fixed orthonormal basis of V ). Then the transition functions gij map
Ui ∩ Uj into the subgroup G of GL(V ) which preserves the appropriate structure (in our
example, the unitary group of V ). In summary, every gij(x) belongs to the structure group
G of the principal bundle P
η−→M to which the vector bundle is associated.
This suggests that we should study vector bundles by first classifying the corresponding
frame bundles, and then invoking Proposition 1.1 to pass the classification to vector bundles.
11
This procedure works because the frame bundle is entirely determined by the transition
functions. Specifically, we have the following “patching-together” construction.
Lemma 1.4. Let M be a manifold with an atlas of local charts { (Uj, φj) : j ∈ J }; let G be
a Lie group and suppose we are given a family of smooth functions gij : Ui∩Uj → G, defined
for Ui ∩ Uj 6= ∅, that satisfies (1.8). Then there exists a principal G-bundle P η−→M and a
family of local sections pj ∈ Γ(Uj, P ) such that pi(x) · gij(x) = pj(x) whenever x ∈ Ui ∩ Uj.
Proof. Let Q denote the disjoint union
⊎
j∈J Uj × G and let P be the quotient space of Q
formed by identifying (x, h)i ∈ Ui × G with (x, gij(x)h)j ∈ Uj × G whenever x ∈ Ui ∩ Uj.
The condition (1.8) simply says that (x, h)i ∼ (x, gij(x)h)j is an equivalence relation on Q,
so the quotient space is well defined. Write η[(x, h)i] := x; then it may be checked that P
inherits from Q the structure of a differential manifold, of dimension dimM + dimG, such
that η : P → M is a submersion. It remains to check that G acts freely and transitively on
the right on each fibre η−1(x); but it is obvious that the right action of G on Q given by
(x, h)i · g := (x, hg)i
preserves equivalence classes and drops to a right action on P whose orbits are the fibres
of η.
If we are also given a representation ρ : G→ GL(V ), we may then create a vector bundle
with transition functions {gij} by association, using (1.3). Conclusion: one can always patch
together a vector bundle with base M and structure group G from a set of transition functions
satisfying (1.8) and a representation of G.
The remaining question is how can one describe the equivalence of vector bundles in
terms of the transition functions, in order to obtain a manageable classification. The answer
is provided by the theory of Cˇech cohomology.
1.8 Cˇech cocycles
A family of transition functions forms what is called a “Cˇech 1-cocycle” with values in the
structure group G. Some difficulties arise in the cohomology theory of these objects for
noncommutative structure groups, so we shall assume for the present that this group is
abelian. This still covers many cases of interest, such as the groups C, C×, R, U(1), Z,
and Z2.
Definition 1.15. Let U = {Uj : j ∈ J } be an open covering of a manifold M and let
A be an abelian group; we will write the group operation additively. For r ∈ N, a Cˇech
r-cochain over U with coefficients in A is a family of elements cj0j1...jr ∈ A, indexed by the
collections of (r+ 1) sets {Uj0 , Uj1 , . . . , Ujr} ⊂ U for which Uj0 ∩Uj1 ∩ · · · ∩Ujr 6= ∅. The set
of all r-cochains is denoted Cr(U, A); it is an abelian group.
One often needs a broader definition, where A is replaced by a family of abelian groups
A := {Aj0j1...jr} satisfying certain compatibility relations.8 We will always take Aj0j1...jr to be
8To be precise, A should be a sheaf of abelian groups over M . The interested reader may consult [14] or
[58] for the full story.
12
the collection of smooth functions from Uj0 ∩Uj1 ∩· · ·∩Ujr into a fixed abelian group A. For
r ∈ N, a Cˇech r-cochain over U with coefficients in A is then a family of smooth functions
fj0j1...jr : Uj0 ∩Uj1 ∩ · · · ∩Ujr → A, defined whenever Uj0 ∩Uj1 ∩ · · · ∩Ujr 6= ∅. The set of all
such families is denoted Cr(U, A).
For instance, the chart maps φj of a manifold M form a 0-cochain in C
0(U,Rn) (or in
C0(U,Cm), if M is a complex manifold). The transition functions of a complex line bundle
gij form a 1-cochain in C
1(U,C×); the transition functions of a Hermitian line bundle form
a 1-cochain in C1(U, U(1)).
Definition 1.16. The Cˇech complex is the cochain complex9 (C•(U, A), δ) is defined as
follows. The coboundary operator δr : C
r(U, A)→ Cr+1(U, A) is given by
(δa)ij := ai − aj, (δb)ijk := bij − bik + bjk
for a ∈ C0(U, A), b ∈ C1(U, A), and (δc)j0...jr :=
∑r
k=0(−1)kcj0...jr−k−1jr−k+1...jr in general. It
is immediate that δ2 = 0 by cancellation of terms, so we have a complex
C0(U, A)
δ−→C1(U, A) δ−→C2(U, A) δ−→· · ·
which gives rise to cohomology groups in the standard way: Zr(U, A) := { c ∈ Cr(U, A) :
δc = 0 } are the Cˇech cocycles, Br(U, A) := { δb : b ∈ Cr−1(U, A) } are the Cˇech cobound-
aries, and the quotient group Hr(U, A) := Zr(U, A)/Br(U, A) is the rth Cˇech cohomology
group of the covering U with coefficients in A.
Open coverings of M form a directed set (under refinement); we can eliminate U by
taking a direct limit: the rth Cˇech cohomology group of the manifold M (with coefficients
in A) is defined as10
Hˇr(M,A) := lim−→
U
Hr(U, A).
An essential result from topology [14] is that this limit is already attained when U is a good
covering: Hˇr(M,A) = Hr(U, A) in this case.
1.9 The Cˇech cohomology of S2
Let us compute the Cˇech cohomology (with real coefficients) of the sphere S2. We know
(Exercise 1.1) that it has a good covering U = {U1, U2, U3, U4} by open neighbourhoods
of the four spherical triangles Fj obtained by projecting an inscribed regular tetrahedron
outward from the centre. Each Ui ∩ Uj is a neighbourhood of the edge Fi ∩ Fj, and each
Ui ∩ Uj ∩ Uk is a neighbourhood of the vertex Fi ∩ Fj ∩ Fk; all are contractible. Thus 0-
cochains are labelled by the faces of the tetrahedron, 1-cochains are labelled by its edges,
and 2-cochains are labelled by its vertices. Therefore
C0(U,R) ' R4, C1(U,R) ' R6, C2(U,R) ' R4,
9See Appendix A for generalities on cochain complexes and their cohomology groups.
10We use the notation Hˇ to distinguish Cˇech cohomology from singular or de Rham cohomology; although
we shall see that this is often unnecessary.
13
and since U1 ∩ U2 ∩ U3 ∩ U4 = ∅, we have Cr(U,R) = 0 for r ≥ 3. Thus the Cˇech complex
reduces to
C0(U,R) δ0−→C1(U,R) δ1−→C2(U,R) δ2−→ 0. (1.9)
Now a ∈ ker δ0 iff a1 = a2 = a3 = a4, so Hˇ0(S2,R) = Z0(U,R) ' R; and (by linearity of δ0)
B1(U,R) = im δ0 ' R3.
The elements b ∈ ker δ1 = Z1(U,R) satisfy bij − bik + bjk = 0 for {i, j, k} ⊂ {1, 2, 3, 4};
these 4 linear equations for the six bij form a system of rank 3; and hence Z
1(U,R) ' R3.
Thus the sequence (1.9) is exact at C1(U,R), and so Hˇ1(S2,R) = 0.
Finally, B2(U,R) = im δ1 ' R6/ ker δ1 ' R3. Since Z2(U,R) = C2(U,R) ' R4, we
conclude that Hˇ2(S2,R) ' R.
Exercise 1.11. Compute the de Rham cohomology (Appendix A) of S2. Show that closed
0-forms on S2 are constant functions, that the volume form Ω = sin θ dθ dφ is a closed 2-form
with nonzero integral, and that if β is another 2-form with
∫
S2 β =
∫
S2 Ω then β−Ω is exact.
If α = f(θ, φ) dθ + g(θ, φ) sin θ dφ is a closed 1-form, use the Poincare´ lemma to show that,
away from either the north or the south pole, α is of the form d(sin θ h(θ, φ)), and hence show
that α is exact. Conclude that Hˇr(S2,R) ' HrdR(S2) for every r. Is that just a coincidence?
1.10 Line bundle classification
Let L−→M be a complex line bundle with a local system {(Uj, sj)} of nonvanishing local
sections and with transition functions gij : Ui ∩ Uj → C× such that si = gijsj on Ui ∩ Uj.
(Since the fibres are one-dimensional, we get11 G = GL(1,C) = C× and sj 7→ gijsj is just
the module action of functions on sections; this simplifies the notation considerably.) We
may and shall assume that U = {Uj} is a good covering. The fundamental result we need is
the following [57].
Proposition 1.5. The family of transition functions g := {gij} is a Cˇech 1-cocycle for the
good covering U; its cohomology class [g] ∈ Hˇ1(M,C×) is independent of the local system of
sections sj, and depends only on the equivalence class [L] of the complex line bundle L−→M .
Moreover, the correspondence [L] 7→ [g] ∈ Hˇ1(M,C×) is an isomorphism of abelian groups.
Proof. From its definition, g is clearly a Cˇech 1-cochain in C1(U,C×). The consistency
condition (1.8) says that (δg)ijk := gijgjk/gik ≡ 1 on Ui ∩ Uj ∩ Uk, which means that g
is a Cˇech 1-cocycle. If a different local system {(Uj, tj)} is chosen, then ti = hisi, where
hi : Ui → C× is smooth, i.e., h ∈ C0(U,C×). Clearly ti = (hi/hj)gijtj, so that the transition
functions for the new local system form the 1-cocycle g + δh (in additive notation). Thus
the line bundle determines the class [g] in Hˇ1(M,C×).
The frame bundle P −→M of a line bundle is formed simply by deleting the zero section
from L, i.e., by taking Px = Lx \ {0} for each x ∈ M . A nonvanishing local section sj
of L−→M may thus also be regarded as a section of the frame bundle P −→M . From
Exercise 1.4, we see that the transition functions for the dual line bundle L∗−→M are
11The notation C× denotes the multiplicative group of nonzero complex numbers; we revert to multiplica-
tive notation when working with this group.
14
1/gij. On passing to additive notation, we conclude that the dual bundle determines the
class −[g] ∈ Hˇ1(M,C×).
Similarly, if g′ij are transition functions for another line bundle L
′−→M , then the tensor
product bundle L ⊗ L′−→M has transition functions gijg′ij, and so determines the class
[g] + [g′] ∈ Hˇ1(M,C×). Therefore [L] 7→ [g] is a homomorphism from the group of line
bundle classes to the group Hˇ1(M,C×).
If [g] = 0 in Hˇ1(M,C×), then g is a coboundary δf , i.e., gij = fi/fj where each fi : Ui →
C× is a smooth function. But then f−1i si = f
−1
j sj whenever Ui ∩ Uj 6= ∅, and so there is a
global nonvanishing section s ∈ Γ(L) given by s := f−1j sj on Uj; and therefore L−→M is a
trivial line bundle. Hence the kernel of homomorphism [L] 7→ [g] is zero. From Lemma 1.4, a
line bundle can always be patched together from a given Cˇech 1-cocycle in C1(U,C×), having
this cocycle as its set of transition functions; this shows that [L] 7→ [g] is surjective.
1.11 The second cohomology group
Consider the short exact sequence of abelian groups:
0−→Z ι−→C −→C×−→ 0, (1.10)
where ι is inclusion and (z) := e2piiz. One may form Cˇech cochains with values in any of these
groups. If c ∈ Zr(U,C×), then c = (b) with b ∈ Cr(U,C); since (δb) = δc = 0, we can find
a ∈ Cr+1(Z) with ι(a) = δb; since ι(δa) = δ(ιa) = 0, we have δa = 0 and so a ∈ Zr+1(U,Z).
One checks that [c] 7→ [a] is a well-defined homomorphism12 ∂r : Hˇr(M,C×)→ Hˇr+1(M,Z),
called the Bockstein homomorphism [23, 58]. We therefore get a long exact sequence in
cohomology:
· · · −→ Hˇ1(M,C)−→ Hˇ1(M,C×) ∂−→ Hˇ2(M,Z)−→ Hˇ2(M,C)−→· · ·
Notice that the discrete group Z need not be underlined: if U is a good covering, an element
of Cr(U,Z) is a family of Z-valued smooth functions with connected domains, i.e., a family
of constant functions; thus Cr(U,Z) = Cr(U,Z) for all r, and so Hˇr(M,Z) = Hˇr(M,Z).
Exercise 1.12. Verify that ∂ does not depend on the choices of a, b and c within their
cohomology classes, and that ker ∂r = imH
r and im ∂r = kerH
r+1ι.
Exercise 1.13. If U is a good covering of M and {ψj} is a partition of unity subordinate
to U, define, for c ∈ Zr+1(U,C), an element b ∈ Cr(U,C) by bj0...jr−1 :=
∑
j cj0...jr−1jψj (this
is a locally finite sum). Show that δb = c and conclude that Hˇk(M,C) = 0 for k > 0.
Proposition 1.6. The Bockstein homomorphism ∂ is an isomorphism of abelian groups
between Hˇ1(M,C×) and Hˇ2(M,Z).
Proof. This follows from the preceding exercise, but it is instructive to produce the isomor-
phism explicitly. Let g ∈ Z1(U,C×); since each Ui∩Uj is contractible, we can find a smooth
12This is just the standard construction of the “connecting homomorphism” in homological algebra.
15
function fij : Ui ∩ Uj → C such that  ◦ fij = gij; we could write fij = (2pii)−1 log gij, but
this is not quite correct since the complex logarithm is “multi-valued”. Define
aijk := fij − fik + fjk : Ui ∩ Uj ∩ Uk → C (1.11)
whenever Ui ∩ Uj ∩ Uk 6= ∅. Then exp(2piiaijk) = gijgjk/gik ≡ 1, and so aijk is Z-valued.
Thus a is an element of C2(U,Z). Now (1.11) says that a = δf in C2(U,C), and so
(δa)ijkl := aijk−aikl+aijl−ajkl = 0 on Ui∩Uj ∩Uk∩Ul; these are algebraic relations among
Z-valued functions, and so δa = 0 in C3(U,Z); hence a ∈ Z2(U,Z).
If we take g′ij := (hi/hj)gij with h ∈ C0(U,C×), we can find a smooth function ki : Ui → C
such that  ◦ ki = hi; then  ◦ (fij + ki − kj) = g′ij. In other words, we modify f to f + δk
in C1(U,C), and a = δf is unchanged. Therefore we obtain a well-defined homomorphism
∂ : [g] 7→ [a] from Hˇ1(M,C×) to Hˇ2(M,Z).
To see that this is an isomorphism, let {ψj} be a smooth partition of unity subordinate
to U. Suppose a ∈ Z2(U,Z) is any Cˇech 2-cocycle; define f ∈ C1(U,C) by fij :=
∑
r aijrψr,
and define g ∈ C1(U,C×) by gij := exp(2piifij). Then
fij − fik + fjk =
∑
r
(aijr − aikr + ajkr)ψr =
∑
r
aijkψr = aijk (1.12)
on using δa = 0; hence δf = a. This shows that ∂ is onto.
If ∂[g] = 0 in Hˇ2(U,Z), we can arrange, by suitably choosing a representative g in its
class, that fij − fik + fjk ≡ 0 on each Ui ∩ Uj ∩ Uk. Define k ∈ C0(U,C) by ki :=
∑
r firψr;
then ki − kj =
∑
r(fir − fjr)ψr =
∑
r fijψr = fij, and so f = δk. If hi := exp(2piiki), then
g = δh in C1(U,C×), and so [g] = 0 in Hˇ2(U,C×). Hence ∂ is one-to-one, which establishes
that ∂ : Hˇ1(M,C×)→ Hˇ2(M,Z) is an isomorphism.
The upshot is that complex line bundles over M are classified by the integral Cˇech
cohomology group Hˇ2(M,Z), at the very small price of going up to the second level in
cohomology. This is a first taste of “quantization”, that is, an unexpected discreteness
which appears in a seemingly continuous family of objects. The culprit here is the exact
sequence (1.10), due to the periodicity of the exponential function.
1.12 Classification of Hermitian line bundles
We are particularly interested in Hermitian line bundles L−→M , which have an inner
product in each fibre (and therefore the local sections sj can be chosen so that sj(x) ∈ Lx is
a unit vector). These have structure group U(1), and the previous arguments show that the
group of equivalence classes of Hermitian line bundles is isomorphic to the Cˇech cohomology
group Hˇ1(M,U(1)).
Exercise 1.14. Construct this isomorphism in detail.
We have a short exact sequence of abelian groups:
0−→Z ι−→R −→U(1)−→ 0,
16
where (t) := e2piit for t ∈ R. The corresponding long exact sequence is
· · · −→ Hˇ1(M,R)−→ Hˇ1(M,U(1)) ∂−→ Hˇ2(M,Z)−→ Hˇ2(M,R)−→· · ·
and again Hˇ1(M,R) = Hˇ2(M,R) = 0 by the partition-of-unity construction. Here also,
the Bockstein homomorphism is an isomorphism between Hˇ1(M,U(1)) and Hˇ2(M,Z). In
this case, the image of each gij is not the whole of the unit circle U(1), since Ui ∩ Uj is
contractible; by passing a half-line from the origin through an omitted point of U(1), we can
choose a branch of the logarithm for which fij := (2pii)
−1 log gij is well-defined. We then
have
aijk :=
1
2pii
(log gij − log gik + log gjk) ∈ Z (1.13)
since exp(2piiaijk) ≡ 1 (note that the three branches of the logarithm on the right hand
side of (1.13) need not be the same). Once again, we get a ∈ C2(U,Z) with δa = 0, so
a ∈ Z2(U,Z). This leads to the following result.
Proposition 1.7. The Bockstein homomorphism ∂ is an isomorphism of abelian groups
between Hˇ1(M,U(1)) and Hˇ2(M,Z).
Exercise 1.15. Write out the proof, using the arguments of Proposition 1.6.
The Cˇech cohomology groups obviously depend only on the topology of M , and are
computed by elementary topological arguments. However, it is desirable to replace them by
de Rham cohomology groups, so that one can work with differential forms. This we do in
Section 3.
1.13 Classification of vector bundles
In order to classify vector bundles of rank higher than 1 (up to equivalence), we face the
obstacle that the group structure on the line bundles does not extend to the higher-rank
case. In fact, it turns out that the most useful classification of vector bundles relies on a
weaker equivalence relation, called stable equivalence, which does not distinguish between a
vector bundle E−→M and the Whitney sum E⊕E0−→M whenever the second summand
E0−→M is a trivial bundle. This notion arises from the following fundamental property of
vector bundles.
Proposition 1.8. Let E−→M be a vector bundle over a compact manifold M . Then we
can find another vector bundle E ′−→M such that the Whitney sum E⊕E ′−→M is a trivial
vector bundle.
Proof. Let U = {U1, . . . , Um } be a finite open covering of M by chart domains, and let
{s1j , . . . , skj} be linearly independent sections in Γ(Uj, E) (where k is the rank of E). Let
{ψj}1≤j≤m be a smooth partition of unity subordinate to U. Let σrj ∈ Γ(E) be defined as
ψjs
r
j on Uj and as 0 on the complement of Uj; notice that the vectors σ
r
j (x) span the fibre
Ex, for any x ∈M .
17
Write n = km, and define f : M × Cn → E (in the case of complex fibres; the real-fibre
case is analogous) by f(x, t) :=
∑
j,r tjrσ
r
j (x); then f is a surjective bundle map, i.e., (f, idM)
is a bundle morphism. Write E ′x := { (x, t) : t ∈ Cn, f(x, t) = 0 }. Choose some hermitian
(or Riemannian) metric on M , and let Fx be the orthogonal complement of E
′
x in {x}×Cn;
one checks that the Fx form the fibres of a vector bundle F −→M , and that (f, idM) is a
bundle equivalence between this bundle and E−→M . Since F ⊕E ′ = M ×Cn, we thereby
obtain an invertible bundle map from M × Cn to E ⊕ E ′.
Definition 1.17. Let M be a compact manifold. We say that two vector bundles E−→M
and F −→M are stably equivalent if there exists a trivial bundle E0−→M such that
E ⊕ E0 and F ⊕ E0 are equivalent. We denote by [[E]] the stable equivalence class of E.
Exercise 1.16. Show that E−→M and F −→M are stably equivalent iff [E ⊕G] = [F ⊕G]
for some third vector bundle G−→M (which need not be trivial).
The equivalence classes of vector bundles over M form an abelian semigroup with identity,
under the operation [E] + [F ] := [E ⊕ F ]. One would like to embed this in an abelian group
in some canonical way. In fact there is a standard construction of such a group, by abstract
nonsense. For any abelian semigroup A with identity, let K(A) be the abelian group with
the following universal property: there is a unital semigroup homomorphism θ : A→ K(A)
such that, whenever G is a group and γ : A→ G is a unital semigroup homomorphism, there
is a unique group homomorphism κ : K(A)→ G for which γ = κ◦θ. Clearly, K(A) is unique
up to isomorphism, and it is called the Grothendieck group of A.
Exercise 1.17. Check that a group with the desired universal property is given by the
following construction. Define an equivalence relation on A × A by (a, b) ∼ (a′, b′) iff
a + b′ + c = a′ + b + c for some c ∈ A, and let K(A) be the set of equivalence classes
with the obvious sum operation, where the class of (b, a) is inverse to the class of (a, b); let
θ(a) be the class of (a, 0). Check also that each element of K(A) is of the form θ(a)− θ(b)
for some elements a, b ∈ A.
Exercise 1.18. What is K(N)? What is K(A) if A is the multiplicative semigroup Z \ {0}?
An abelian semigroup A is said to “allow cancellation” if a + c = b + c implies a = b,
for any c ∈ A. Regrettably, the semigroup Vect(M) of (ordinary) equivalence classes of
vector bundles over M does not usually allow cancellation. For example, the tangent bundle
TS2 → S2 is not trivial, whereas the normal bundle (of lines from the origin of R3 through
the sphere S2) is trivial, and their Whitney sum is the trivial bundle S2 × R3 → S2; thus
[TS2] + [S2 ×R] = [S2 ×R2] + [S2 ×R], so that cancellation is not allowed. In this case, the
homomorphism θ is not injective.
Definition 1.18. The K-theory of a compact manifold M is the Grothendieck group
K0(M) := K(Vect(M)). Two vector bundles E−→M and F −→M have the same image
in K0(M) iff they are stably equivalent; thus any element of K0(M) can be written as a
difference of two stable equivalence classes, [[E]]− [[F ]].
18
Exercise 1.19. Prove the assertion that θ([E]) = θ([F ]) iff [[E]] = [[F ]]. Show also that any
element of K0(M) can be written as [E] − [Ok] for some k, where Ok−→M denotes the
trivial bundle of rank k.
It turns out that K0(M) carries important topological information about the manifold M ;
in that context, vector bundles may be viewed as auxiliary tools to study topological spaces.
There is a companion group, called K1(M), which together with K0(M) forms a cohomology
theory distinct from the Cˇech and de Rham cohomologies. This theory is fully developed in
the monograph of Atiyah [3].
2 Complex projective spaces
We illustrate the general theory with a look at a particular class of manifolds, namely
the complex projective spaces. These are complex manifolds, that is, differentiable mani-
folds whose transition maps are holomorphic; thus we may use complex local coordinates
to describe them. They possess three important features: (a) a Hermitian metric; (b) a
distinguished nondegenerate closed 2-form, which is known as a “symplectic structure”; (c)
a “complex structure”, which at each point specifies a linear automorphism of the tangent
space whose square is −1. Moreover, any two of these three structures determine the third;
a complex manifold with such a triple structure is called a Ka¨hler manifold. The complex
projective spaces are perhaps the simplest examples of compact Ka¨hler manifolds.
2.1 Complex manifolds
Definition 2.1. A complex manifold of complex dimension m is a differentiable manifold
of real dimension n = 2m which has an atlas of local charts (Uj, φj), with φj : Uj → Cm,
such that the transition maps φi ◦ φ−1j are holomorphic. If (x1, . . . , xm, y1, . . . , ym) are local
(real) coordinates on Uj, write z
k := xk + iyk, z¯k := xk − iyk; then (z1, . . . , zm, z¯1, . . . , z¯m) is
an alternative system of local (real) coordinates on Uj.
Complex-valued 1-forms1 are locally generated by dz1, . . . , dzm, dz¯1, . . . , dz¯m, where we
write dzk := dxk + i dyk, dz¯k := dxk − i dyk. Complex-valued vector fields in X(M,C) are
locally generated by ∂/∂z1, . . . , ∂/∂zm, ∂/∂z¯1, . . . , ∂/∂z¯m, where ∂/∂zk := 1
2
(∂/∂xk−i∂/∂yk)
and ∂/∂z¯k := 1
2
(∂/∂xk+ i∂/∂yk); more precisely, X(Uj,C) is a C∞(Uj,C)-module with these
generators. (The notation is chosen so that dzj(∂/∂zk) = δjk and dz
j(∂/∂z¯k) = 0.)
In particular, we may write the differential of f ∈ C∞(M,C) as
df =
∂f
∂zk
dzk +
∂f
∂z¯k
dz¯k ≡ ∂f + ∂¯f, (2.1)
where, here and in the future, we use the Einstein summation convention of summing over
repeated upper and lower indices.2
1In this section, all vector fields and differential forms will be taken complex-valued unless stated other-
wise.
2To avoid possible misunderstandings, we emphasize that we do not sum over repeated upper or repeated
lower indices, unless a summation sign appears explicitly.
19
Exercise 2.1. Check that the decomposition df = ∂f + ∂¯f does not depend on the local
coordinate system, on account of the holomorphicity of the transition maps.
Exercise 2.2. A smooth map f : M → N is called holomorphic iff all its local expressions ψi◦
f ◦ φ−1j have holomorphic Cartesian components. Verify that f ∈ C∞(M,C) is holomorphic
iff ∂¯f = 0.
Definition 2.2. In view of (2.1), we have a splitting A1(M) = A1,0(M) ⊕ A0,1(M) of
C∞(M,C)-modules, and more generally,
Ar(M) =
⊕
p+q=r
Ap,q(M),
where each Ap,q(M) is locally spanned by r-forms of the type
f(z1, . . . , zm, z¯1, . . . , z¯m) dzi1 ∧ · · · ∧ dzip ∧ dz¯j1 ∧ · · · ∧ dz¯jq .
Thus the algebra of differential forms is bigraded : A•(M) =
⊕
p,q A
p,q(M). Again by (2.1),
the exterior derivative splits as d = ∂ + ∂¯, where ∂ : Ap,q(M) → Ap+1,q(M), ∂¯ : Ap,q(M) →
Ap,q+1(M). The identity d2 = 0 yields the three identities
∂2 = 0, ∂∂¯ = −∂¯∂, ∂¯2 = 0, (2.2)
on taking account of the grading degrees. (These identities say that the spaces Ap,q(M) form
the vertices of a “double complex”.)
Exercise 2.3. Verify that the conjugation ω 7→ ω¯ on A•(M,C) = A•(M,R)⊗RC is such that
∂ω = ∂¯ω¯ and that ω¯ ∈ Aq,p(M) whenever ω ∈ Ap,q(M).
2.2 Local charts for complex projective spaces
Definition 2.3. The m-dimensional complex projective space CPm is the set of complex lines
through the origin in Cm+1, i.e., the one-dimensional complex subspaces of Cm+1. For any
nonzero v ∈ Cm+1, the line 〈v〉 ≡ Cv lies in CPm, and η : Cm+1 \ {0} → CPm : v 7→ 〈v〉 is a
quotient map.3 If (z0, z1, . . . , zm) denotes coordinates in Cm+1 (with respect to the standard
orthonormal basis), we may regard each zj as a linear form on Cm+1; these cannot vanish
simultaneously on Cm+1 \ {0}, so we get the following chart domains for CPm:
Uj := { 〈v〉 ∈ CPm : zj(v) 6= 0 }, j = 0, 1, . . . ,m.
Since 〈v〉 /∈ Uj iff 〈v〉 ⊂ ker zj, we can identify the complement of Uj with the set of lines in
the hyperplane ker zj, which is homeomorphic to CPm−1. In particular, each Uj is open and
dense in CPm, since its complement is a lower-dimensional submanifold.
3This quotient map actually defines the topology of CPm.
20
For 〈v〉 ∈ Uj and k 6= j, we define
wkj (〈v〉) :=
zk(v)
zj(v)
. (2.3)
This is well-defined since the fraction is unchanged under v 7→ λv with λ ∈ C×). Now4
φ(〈v〉) := (w0j , . .
j
∨. . , wmj ) (2.4)
is a homeomorphism from Uj onto Cm, and (w0j , . .
j
∨. . , wmj ) is a system of local coordinates
for the chart (Uj, φj).
The wkj are Cartesian coordinates corresponding to the “homogeneous” coordinates z
k,
that is, (w0j , w
1
j , . . . , w
m
j ) = [z
0 : z1 : · · · : zm] in the standard notations.
On the overlap Ui ∩ Uj, we have
wki (〈v〉) = wkj (〈v〉)
zj(v)
zi(v)
=
wkj (〈v〉)
wij(〈v〉)
for k /∈ {i, j}, so the transition map φi ◦ φ−1j , when written as u 7→ v, is given by rational
functions: vj = 1/ui and vk = uk/ui for other k, and hence is holomorphic. Thus the atlas
{ (Uj, φj) : j = 0, 1, . . . ,m } makes CPm a complex manifold.
Since CPm \ Um ≈ CPm−1 and Um ≈ Cm ≈ R2m, we have (by induction) that CPm
is topologically a cell complex with one k-dimensional cell in every even dimension k =
0, 2, . . . , 2m and no odd-dimensional cells. This allows us to compute the singular homology
groups of the complex projective spaces by standard topological techniques [23]: we get
Hk(CPm,Z) = Z if k = 0, 2, . . . , 2m; Hk(CPm,Z) = 0 for all other k. Thus CPm is a
“torsion-free” space, i.e., all its integral homology groups are free; it is known that then the
singular cohomology groups with integer coefficients can be computed by duality:
Hk(CPm,Z) = Hom(Hk(CPm,Z),Z) =
{
Z if k = 0, 2, . . . , 2m,
0 otherwise.
(2.5)
2.3 The Ka¨hler form
Definition 2.4. For v = (z0, z1, . . . , zm) ∈ Cm+1, we write ‖v‖2 = ∑k |zk|2 = ∑k zkz¯k, and
define
Φ0(v) := i ∂∂¯ log ‖v‖2 = i ∂
(‖v‖−2∑kzk dz¯k)
=
i
‖v‖4
(
‖v‖2
∑
k
dzk ∧ dz¯k −
∑
r,s6=j
z¯rzs dzr ∧ dz¯s
)
,
4The notation
j∨ indicates that the index j is omitted from the sequence 0, 1, . . . ,m.
21
which is a 2-form in A1,1(Cm+1 \ {0}). It is clear that Φ0 is homogeneous of degree 0, i.e.,
Φ0(λv) = Φ0(v) for λ ∈ C×. This means that there is a 2-form Φ ∈ A1,1(CPm) such that
Φ0 = η
∗Φ.
To find an expression for Φ in local coordinates on Uj, we may identify Uj with a subset
of the hyperplane zj = 1 in Cm+1, by using (2.3) with denominator 1. With this convention,
the norm squared of a vector v in this hyperplane defines a positive function Qj on Uj (which
tends to infinity at the boundary); in fact,
‖v‖2 = Qj(〈v〉) := 1 +
∑
k 6=j
wkj w¯
k
j (2.6)
for 〈v〉 ∈ Uj with local coordinates (2.4). Thus
Φ = i ∂∂¯ logQj =
i
Q2j
(
Qj
∑
k 6=j
dwkj ∧ dw¯kj −
∑
r,s 6=j
w¯rjw
s
j dw
r
j ∧ dw¯sj
)
. (2.7)
The 2-form Φ is in fact real-valued, as is evident from its local expression (2.7), or
alternatively by noting that Φ = −i ∂¯∂ logQj = Φ on account of (2.2); this Φ ∈ A2(CPm,R)
is called the Ka¨hler form on CPm.
It is important to note that Φ = −∂∂¯Qj is a closed 2-form. This follows at once from
(2.2), since dΦ = (∂ + ∂¯)Φ = −∂2(∂¯Qj) + ∂¯2(∂Qj) = 0.
Exercise 2.4. A real-valued 2-form β ∈ A2(M) is called nondegenerate if each βx ∈ Λ2T ∗xM
is nondegenerate as an alternating R-bilinear form, i.e., βx(u, v) = 0 for all v ∈ TxM implies
u = 0 in TxM . Show that β is nondegenerate iff for any local expression β
∣∣
U
= brs dx
r ∧ dxs,
the matrix of local coefficients [brs] is nonsingular at each point of U . Verify that the Ka¨hler
form Φ is nondegenerate on CPm by showing that the matrix A with entries Qjδrs− w¯rjwsj is
positive definite (apply the Schwarz inequality to show that z¯tAz > 0 for all z ∈ Cm+1\{0}).
Exercise 2.5. Show that Φ∧m, the m-fold exterior power of Φ, is a volume form on CPm,
i.e., a 2m-form which is nonzero at each point of CPm when regarded as a section of the line
bundle Λ2mT ∗CPm → CPm.
2.4 The Fubini–Study metric
Definition 2.5. Let M be a differential manifold of even (real) dimension n = 2m. An
almost complex structure on M is an operator J : X(M) → X(M) which is C∞(M,R)-
linear,5 and which satisfies J2 = − id. Thus (JX)x = Jx(Xx) where Jx ∈ EndR(TxM) with
(Jx)
2 = − id for each x ∈ M .6 Almost complex structures need not exist; if one does exist,
we say that (M, J) is an almost complex manifold.
5In other words, J can be identified with a tensor on M , of bidegree (1, 1), given by (X,α) 7→ α(J(X)).
One also says that J is a “tensorial operator”.
6An almost complex structure may therefore also be defined as a bundle automorphism (J, id) of the
tangent bundle TM −→M for which J2 = − id.
22
It turns out that M need not be a complex manifold in order to possess an almost complex
structure; a known example is the sphere S6 which has an almost complex structure related
to the Cayley numbers. See [15, 58] for some discussion of this. Here we will stick to complex
manifolds.
A complex structure can always be defined locally if dimM is even; it suffices to take
local coordinates (x1, . . . , xm, y1, . . . , ym) on a chart domain U and let
J
(
∂
∂xk
)
=
∂
∂yk
, J
(
∂
∂yk
)
= − ∂
∂xk
. (2.8)
Now J may be amplified to a C∞(U,C)-linear operator on X(U,C) in the natural way; from
(2.8) we derive
J
(
∂
∂zk
)
= i
∂
∂zk
, J
(
∂
∂z¯k
)
= −i ∂
∂z¯k
.
On a complex manifold, these local definitions patch together to give a global definition
of J. For instance, on CPm we have
J
(
∂
∂wkj
)
= i
∂
∂wkj
, J
(
∂
∂w¯kj
)
= −i ∂
∂w¯kj
, (2.9)
which is independent of the chart (Uj, φj).
Exercise 2.6. Verify this independence on an overlap Ui ∩ Uj of charts of CPm.
Lemma 2.1. Let M be a manifold with an almost complex structure J, and let Φ ∈ A2(M)
be a real-valued nondegenerate 2-form on M which is invariant under J, i.e., Φ(JX, JY ) =
Φ(X, Y ) for X, Y ∈ X(M). Then the recipe g(X, Y ) := Φ(X, JY ) defines a symmetric tensor
on M which is also J-invariant.
Proof. Clearly g is a tensor of bidegree (2, 0); symmetry follows from invariance, since
g(Y,X) = Φ(Y, JX) = −Φ(JX, Y ) = Φ(JX, J2Y ) = Φ(X, JY ) = g(X, Y ).
When M = CPm, Φ is the Ka¨hler form (2.7), and J is given by (2.9), we obtain on Uj:
g =
2
Q2j
(
Qj
∑
k 6=j
dwkj · dw¯kj −
∑
r,s 6=j
w¯rjw
s
j dw
r
j · dw¯sj
)
. (2.10)
Exercise 2.7. Check that g is positive definite, and thus defines a Riemannian metric on CPm.
(Recall that the matrix with entries Qjδ
rs − w¯rjwsj is positive definite, by Exercise 2.4.)
Definition 2.6. The metric (2.10) is the Fubini–Study metric on CPm.
23
2.5 The Riemann sphere
When m = 1, the complex projective space CP1 is identified with the Riemann sphere
C∞ = C unionmulti {∞}, by identifying [z0 : z1] with z = z1/z0 ∈ C if z0 6= 0, and [1 : 0] with ∞.
Notice that z ≡ w10 is the local complex coordinate in U0 = C∞\{∞}. By writing z = x+iy,
we can also identify C∞ with the two-sphere S2, regarding the latter as a submanifold of R3,
via the stereographic projection
f(z) :=
(
2x
1 + x2 + y2
,
2y
1 + x2 + y2
,
−1 + x2 + y2
1 + x2 + y2
)
(2.11)
and f([1 : 0]) := (0, 0, 1).
The Ka¨hler form on C∞ is easily found. We have Q0(〈v〉) = 1+ |z|2 for v = (z0, z1) ∈ C2,
so (2.7) reduces to
Φ =
i dz ∧ dz¯
(1 + |z|2)2 = −
2 dx ∧ dy
(1 + x2 + y2)2
. (2.12)
Exercise 2.8. Write (u1, u2, u3) to denote the right hand side of (2.11). Check that Φ = f ∗Ω,
where Ω ∈ A2(R3) is given by Ω = (du1 ∧ du2)/2u3. The usual spherical coordinates on S2
are defined by the map h : R2 → S2 where h(θ, φ) := (sin θ cosφ, sin θ sinφ, cos θ). Check
that h∗Ω = −1
2
sin θ dθ ∧ dφ.
The Riemannian metric (2.10) reduces to g = 2(1 + |z|2)−2 dz · dz¯ or equivalently g =
2(1 + x2 + y2)−2 (dx2 + dy2), where dx2 denotes the symmetric product dx · dx.
Exercise 2.9. Show that g = f ∗(1
2
G) where G = (du1)2 + (du2)2 + (du3)2 is the standard
Riemannian metric on R3, and that h∗G = dθ2 + sin2 θ dφ2.
We may regard (θ, φ) as local coordinates on the Riemann sphere. Thus we write simply:
Φ = −1
2
(sin θ dθ ∧ dφ), g = 1
2
(dθ2 + sin2 θ dφ2). (2.13)
It may prove useful to have available some relations between the local coordinate systems
(z, z¯) and (θ, φ). We list a few identities that follow from the previous formulae:
z =
sin θ
1− cos θe
iφ, dz =
eiφ
1− cos θ (−dθ + i sin θ dφ), Q0 =
2
1− cos θ . (2.14)
Exercise 2.10. Verify the formulae (2.14) and derive (2.13) directly.
Exercise 2.11. Show that the complex structure J on S2 satisfying g(X, Y ) = Φ(X, JY ) is
given by J(∂/∂θ) = − csc θ ∂/∂φ and J(csc θ ∂/∂φ) = ∂/∂θ in spherical coordinates.
3 The de Rham complex and Hodge duality
3.1 The de Rham complex
Definition 3.1. If M is an n-dimensional differential manifold, its de Rham complex is
the cochain complex
A0(M)
d−→A1(M)→ · · · → Ak(M) d−→Ak+1(M)→ · · · → An(M)
24
where Ak(M) = Ak(M,R) denotes the real-valued differential forms on M , and d is the
exterior derivation. We recall (see Appendix A) that its k-cocycles are the closed differential
k-forms ZkdR(M) := {ω ∈ Ak(M) : dω = 0 }, and its k-coboundaries are the exact differential
k-forms BkdR(M) := { dβ : β ∈ Ak−1(M) }. The k-th de Rham cohomology group HkdR(M) :=
Hk(A•(M), d) is a real vector space.
The 0-cocycles in Z0dR(M) are locally constant smooth functions, so H
0
dR(M) = Rm if M
has exactly m connected components. In particular, H0dR(M) 6= 0. If U is a contractible
manifold, then by the Poincare´ lemma,1 HkdR(U) = 0 for each k = 1, 2, . . . , n.
If M is an orientable compact manifold (without boundary), with volume form ν ∈
An(M), then dν = 0 and ν is not exact since
∫
M
ν 6= 0. (If µ ∈ An(M) is exact, with
µ = dβ, say, then
∫
M
µ =
∫
M
dβ = 0 by Stokes’ theorem, since ∂M = ∅; thus, a volume form
cannot be exact.) Therefore HndR(M) 6= 0. Moreover, since the integral vanishes on exact
(n−1)-forms, again by Stokes’ theorem, we see that [ω] 7→ ∫
M
ω is a well-defined linear form
on HndR(M).
3.2 The Riemannian volume form
Definition 3.2. Let g be a Riemannian metric on the compact oriented manifold M . Then
g : X(M)×X(M)→ C∞(M) is a symmetric C∞(M)-bilinear form such that each gx : TxM×
TxM → R is positive definite. The pair (M, g) is called a Riemannian manifold.
If U ⊆M is a chart domain with local coordinates x1, . . . , xn, let gij := g(∂/∂xi, ∂/∂xj) ∈
C∞(U); then [gij] is a symmetric matrix, whose determinant we write as det g (by a slight
abuse of notation, since this determinant is coordinate-dependent), and we have g = gij dx
i ·
dxj on U .
For each x ∈M , there is a vector space isomorphism gˆx : TxM → T ∗xM given by gˆx(u) :=
[v 7→ gx(u, v)]; these determine a diffeomorphism gˆ : TM → T ∗M such that (gˆ, idM) is
an equivalence between the tangent and cotangent bundles. They also determine tensorial
operators X 7→ X[ : X(M)→ A1(M), given by X[(Y ) := g(X, Y ), and its inverse α 7→ α] :
A1(M)→ X(M), given by α(Y ) =: g(α], Y ). These “musical isomorphisms” [8] allow us to
identify vector fields and 1-forms and to use them interchangeably, as the occasion demands.
The metric g defines thus defines bilinear pairings, not only of vector fields but also of 1-
forms and indeed of differential forms of any degree; we will denote all these pairings by (· | ·)
whenever a fixed g is given. Thus (X | Y ) := g(X, Y ) for X, Y ∈ X(M); (α | β) := g(α], β])
for α, β ∈ A1(M); and the pairing on Ak(M) (for k > 1) is determined by
(α1 ∧ · · · ∧ αk | β1 ∧ · · · ∧ βk) := det[(αi | βj)]
for α1, . . . , αk, β1, . . . , βk ∈ A1(M).
Definition 3.3. Let (M, g) be an oriented Riemannian manifold. Choose a local orthonormal
frame X1, . . . , Xn for X(U), i.e., vector fields such that g(Xr, Xs) = δrs, which is oriented,
1A complex (C•, d) is called “acyclic” if Hk(C•, d) = 0 for k > 0. Thus the Poincare´ lemma says that
the de Rham complex of a contractible manifold is acyclic.
25
that is, ν(X1, . . . , Xn) > 0 where ν is a volume form on M which defines the orientation.
Let θˆk := X[k ∈ A1(U), so that θˆ1, . . . , θˆn is a local (oriented) orthonormal frame for A1(U).
Then
Ω := θˆ1 ∧ θˆ2 ∧ · · · ∧ θˆn ∈ An(M) (3.1)
is a volume form on M , since (Ω |Ω) = 1 and in particular Ωx 6= 0 in ΛnT ∗xM for all x ∈M .
This Ω is called the Riemannian volume form on (M, g).
To see that Ω is in fact independent of the choice of oriented orthonormal frame, we argue
as follows. If Y1, . . . , Yn is another oriented orthonormal frame for X(V ), where U ∩ V 6= ∅,
then Yi := a
j
iXj where a = [a
j
i ] is a smooth function on U ∩ V with values in SO(n), the
group of orthogonal n × n matrices of determinant 1. (That is to say, a is a local section
of the principal SO(n)-bundle of oriented orthonormal frames on M .) Now ϑˆk := Y [k gives
the corresponding oriented orthonormal basis for A1(V ), and thus ϑˆk = akl θˆl, from which
it follows that ϑˆ1 ∧ · · · ∧ ϑˆn = (det a) θˆ1 ∧ · · · ∧ θˆn = Ω on U ∩ V . Therefore, Ω is defined
globally by (3.1).
One can express Ω in terms of local coordinates x1, . . . , xn on U for which the ∂/∂xj
need not be orthonormal, i.e., gij 6= δij in general. The local coordinates should, however,
be compatible with the orientation, which means that det g > 0. If y1, . . . , yn are other
local coordinates on V , and if g˜ij := g(∂/∂y
i, ∂/∂yj) ∈ C∞(V ), then det g˜ = J2 det g, where
J := det[∂yr/∂xj] is the Jacobian of the transition function (which is positive). Hence√
det g˜ dy1 ∧ · · · ∧ dyn =
√
det g dx1 ∧ · · · ∧ dxn (3.2)
on U ∩ V , and thus the coordinate-independent expression √det g dx1 ∧ · · · ∧ dxn defines a
volume on M . In particular, one may choose y1, . . . , yn so that g˜ij = δij, so det g˜ = 1 and
dy1, . . . , dyn is an orthonormal frame: so the volume form (3.2) coincides with Ω of (3.1). In
other words,
Ω =
√
det g dx1 ∧ · · · ∧ dxn (3.3)
in any oriented local coordinate system.
3.3 The Hodge star operator
Definition 3.4. Let (M, g) be a compact oriented Riemannian manifold of dimension n,
and write m := bn/2c (so n = 2m or else n = 2m − 1). Define the Hodge star operator
? : A•(M,C)→ A•(M,C) as follows. Choose a local orthonormal frame X1, . . . , Xn for X(U)
on some chart domain U ⊂ M , and let θˆ1, . . . , θˆn be the corresponding local orthonormal
frame for A1(U), determined by θˆi(Xj) := δ
i
j. Define ? on A
•(U,C) by
? := im((θˆ1)− ι(X1)) . . . ((θˆn)− ι(Xn)), (3.4)
where ι(X) denotes contraction with the vector field X and (α) : ω 7→ α ∧ ω denotes
exterior product with the 1-form α. It is readily checked that the right hand side of (3.4) is
independent of these local orthonormal frames, and therefore defines an operator on all of
A•(M,C).
26
To simplify products of noncommuting quantities as in (3.4), we use the following nota-
tion: if aj1 , aj2 , . . . , ajk are elements of some algebra, where the indices jr are in increasing
order, write
→∏
1≤r≤k
ajr := aj1aj2 . . . ajk . (3.5)
Thus ? = im
∏→
1≤r≤k (θˆ
r)− ι(Xr). It is also convenient to write J := {j1, . . . , jk} and denote
the right hand side of (3.5) by
∏→
j∈J aj, where it is understood that the indices j ∈ J are
arranged in increasing order.
Exercise 3.1. Let V be an n-dimensional oriented real vector space with a positive definite
symmetric bilinear form q. For u ∈ V , α ∈ V ∗, define operators ι(u) : ΛkV ∗ → Λk−1V ∗
and (α) : ΛkV ∗ → Λk+1V ∗ by [ι(u)η](v1, . . . , vk−1) := η(u, v1, . . . , vk−1) and (α)η := α ∧ η.
Let {e1, . . . , en}, {e′1, . . . , e′n} be oriented orthonormal bases for (V, q) (so the change-of-basis
matrix [q(ei, e
′
j)] has determinant +1) and let {ζ1, . . . , ζn}, {ζ ′1, . . . , ζ ′n} be the respective
dual bases for V ∗. Show that
∏→
1≤r≤n (ζ
′r) − ι(e′r) =
∏→
1≤r≤n (ζ
r) − ι(er) as operators on
Λ•V ∗.
Exercise 3.2. Use the previous exercise to show that the right hand side of (3.4) is indepen-
dent of the given local orthonormal frames {Xr} and {θˆk}.
Notice that when m is even, i.e., when n is of the form 4k or 4k + 3 for some integer k,
then it is not necessary to use complex-valued forms since ? takes A•(M,R) to A•(M,R).
However, for n of the form 4k+ 1 or 4k+ 2, we have im = ±i and complex forms are needed.
More traditional treatments of Hodge duality [1, 28, 33, 57] use different sign conventions,
so that ? operates on A•(M,R) in all cases, but the important involutivity property (see
Lemma 3.3 below) is then lost.
Lemma 3.1. If J := {j1, . . . , jk} ⊆ {1, . . . , n} and if J ′ := {1, . . . , n} \ J = {i1, . . . , in−k}
is its complement, with indices written in increasing order in both cases, let η(J, J ′) = ±1 de-
note the sign of the shuffle permutation which reorders (1, 2, . . . , n) as (j1, . . . , jk, i1, . . . , in−k).
Write, for brevity, θˆJ := θˆj1∧· · ·∧θˆjk , where θˆ1, . . . , θˆn is a local orthonormal basis of A1(M).
Then the Hodge star operator is given explicitly on this basis by the recipe:
?θˆJ = im(−1)nk+k(k−1)/2η(J, J ′) θˆJ ′ . (3.6)
27
Proof. This identity results from the following calculation:
?θˆJ = im
→∏
1≤j≤n
((θˆj)− ι(Xj)) θˆj1 ∧ · · · ∧ θˆjk
= im(−1)k(k−1)/2
→∏
1≤j≤n
((θˆj)− ι(Xj)) θˆjk ∧ · · · ∧ θˆj1
= im(−1)k(k−1)/2η(J ′, J)
→∏
i∈J ′
(θˆi)
→∏
j∈J
(−ι(Xj)) θˆjk ∧ · · · ∧ θˆj1
= im(−1)k(k−1)/2(−1)kη(J ′, J)
→∏
i∈J ′
(θˆi) 1
= im(−1)k(k−1)/2(−1)k(−1)k(n−k)η(J, J ′) θˆJ ′ .
Here, one first reverses the order of the exterior product of 1-forms θˆj1 ∧ · · · ∧ θˆjk , by a
permutation of sign (−1)k(k−1)/2. Then the operators (θˆj)− ι(Xj) act successively on forms
of type θˆj ∧ βj, yielding −βj at each stage, until only the constant (−1)k remains; next, the
operators (θˆi)− ι(Xi), with i ∈ J ′, act successively on forms of type γj, yielding θˆj ∧ γj at
each stage, until θˆJ
′
is created. Lastly, the identity η(J, J ′)η(J ′, J) = (−1)k(n−k) is used: this
is just the observation that the permutation which interchanges the blocks of indices J and
J ′ —while preserving the order within each block— is a product of k(n− k) transpositions.
Now (3.6) follows on noticing that (−1)k(−1)k(n−k) = (−1)nk(−1)k(k+1) = (−1)nk.
Corollary 3.2. The Hodge star operator maps Ak(M,C) into An−k(M,C).
The next calculation shows that the presence of the factor im is what ensures that ? is
an involution, i.e., an operator whose square is the identity.
Lemma 3.3. ?? = id.
Proof. From (3.6) it is clear that ??θˆJ = CJ θˆ
J for some constant CJ . One computes:
CJ = i
2m(−1)nk+k(k−1)/2(−1)n(n−k)+(n−k)(n−k−1)/2η(J, J ′)η(J ′, J)
= (−1)m(−1)n2(−1)(k(k−1)+(n−k)(n−k−1))/2(−1)k(n−k)
= (−1)m(−1)n2(−1)(n2−n(2k+1)+2k2)/2(−1)nk−k2
= (−1)m(−1)(3n2−n)/2 = (−1)m(−1)(n2+n)/2 = (−1)m(−1)m(2m±1) = +1.
This establishes that ?? = id, which also implies that ?mapsAk(M,C) onto An−k(M,C).
Exercise 3.3. If X ∈ X(M), show that ?(X[)? = (−1)nι(X) as operators from Ak(M) to
Ak−1(M). (Reduce to the special case X = Xj.)
28
Consider the case M = R3; though not compact, it is an oriented Riemannian manifold
with the usual euclidean metric; here θˆk = dxk, and ? is determined by
?1 = −Ω = −dx1 ∧ dx2 ∧ dx3,
?dx1 = dx2 ∧ dx3, ?dx2 = −dx1 ∧ dx3, ?dx3 = dx1 ∧ dx2,
on account of (3.6).
When M = C with local coordinates x, y, take z = x + iy. For the usual metric, we
have θˆ1 = dx, θˆ2 = dy, and so ?dx = i dy, ?dy = −i dx. Using the coordinates z, z¯, we then
have ?dz = dz, ?dz¯ = −dz¯. Thus, ?df = ?(∂f + ∂¯f) = ∂f − ∂¯f , so that f ∈ C∞(C,C) is
holomorphic iff ∂¯f = 0 (by the Cauchy–Riemann equations) iff ?df = df . More generally,
any α ∈ A•(M,C) may be written as α = α+ + α− where α+ is selfdual, i.e., ?α+ = α+,
and α− is antiselfdual, i.e., ?α− = −α−. When n = 2m and α ∈ Am(M), we have α+, α− ∈
Am(M) also. When n = 2 and M is a compact complex manifold (i.e., a Riemann surface),
f ∈ C∞(M,C) is holomorphic iff df is selfdual.
An important example is M = S2 with the Riemannian metric g = dθ2 + sin2 θ dφ2. In
spherical coordinates, a local orthonormal basis is given by
θˆ1 = dθ, θˆ2 = sin θ dφ.
The star operator on S2 is determined by
?1 = iΩ = i sin θ dθ ∧ dφ, ?dθ = i sin θ dφ,
and thus ?Ω = −i and ?(sin θ dφ) = −i dθ.
For a general system of coordinates, the star operator can be described as follows. On
a chart domain U , a k-form ω ∈ Ak(U) can be written as ω = ωj1...jk dxj1 ∧ · · · ∧ dxjk .
New coefficients with “raised indices” are defined by ωr1...rk = gr1j1 . . . grkjkωj1...jk where [g
rs]
denotes the matrix inverse to the matrix [gij]. Let t1...tn := 0 if t1, . . . , tn contains a repeated
index, and otherwise let t1...tn := ±1 be the sign of the permutation (1, . . . , n) 7→ (t1, . . . , tn).
Then ?ω is given by
?(ωj1...jk dx
j1 ∧ · · · ∧ dxjk)
= im(−1)nk+k(k−1)/2
√
det g
(n− k)!i1...in−kr1...rnω
r1...rk dxi1 ∧ · · · ∧ dxin−k . (3.7)
Exercise 3.4. Check the validity of (3.7) by showing that its right hand side is invariant
under a general change of (oriented) frame, and that it reduces to (3.6) for an orthonormal
frame.
Exercise 3.5. Work out the action of ? for the Fubini–Study metric on CPm.
The Hodge star operator relates the exterior product of forms to the Riemannian volume
form, as follows.
29
Lemma 3.4. For α, β ∈ Ak(M,C),
α ∧ ?β = im(−1)nk+k(k−1)/2(α | β) Ω. (3.8)
Proof. Since both sides of (3.8) are C∞(M)-bilinear in (α, β), it suffices to verify equality
for a local basis; thus we may take α = θˆI , β = θˆJ where I and J are subsets of {1, . . . , n}.
Now (θˆI | θˆJ) = 1 or 0 according as I = J or not. If I 6= J then θˆI ∧ ?θˆJ = 0 from (3.6),
since I∩J ′ 6= ∅. Also, θˆJ ∧?θˆJ = im(−1)nk+k(k−1)/2η(J, J ′)θˆJ ∧ θˆJ ′ = im(−1)nk+k(k−1)/2Ω.
Definition 3.5. The complex vector space Ak(M,C) becomes a prehilbert space under the
positive definite Hermitian form
〈〈α | β〉〉 :=
∫
M
(α¯ | β) Ω,
where the integral over M is normalized by the requirement that
∫
M
Ω = 1. Its completion
with respect to this “integrated inner product”2 is a Hilbert space which may be denoted
L2,k(M). Notice that 〈〈α | β〉〉 = 〈〈β¯ | α¯〉〉, i.e., the conjugation of forms is antiunitary.
The inner product may be extended to all of A•(M,C) by declaring that k-forms and l-
forms be orthogonal for k 6= l; the completion of A•(M,C) is the Hilbert-space direct sum
L2,•(M) :=
⊕n
k=0 L
2,k(M).
From (3.8), it follows that
〈〈α | β〉〉 = i−m(−1)nk+k(k−1)/2
∫
M
α¯ ∧ ?β.
Lemma 3.5. The Hodge star operator is isometric, i.e., if α, β ∈ Ak(M,C) then
〈〈?α | ?β〉〉 = 〈〈α | β〉〉.
Proof. From (3.6) it follows that ?α¯ = (−1)m?α for any α ∈ A•(M,C). Applying (3.8)
twice,
〈〈?α | ?β〉〉 = i−m(−1)n(n−k)+(n−k)(n−k−1)/2
∫
M
?α ∧ β
= im(−1)n2+n(n−1)/2+k(k+1)/2
∫
M
?α¯ ∧ β
= im(−1)n(n+1)/2+k(k+1)/2(−1)k(n−k)
∫
M
β ∧ ?α¯
= (−1)m(−1)n(n+1)/2〈〈β¯ | α¯〉〉 = 〈〈α | β〉〉
since 1
2
n(n+ 1) = m(2m± 1) ≡ m mod 2.
Since ?? = id on L2,k(M), this isometry also satisfies 〈〈?α|γ〉〉 = 〈〈α|?γ〉〉 for α ∈ Ak(M,C),
γ ∈ An−k(M,C); thus the Hodge star operator extends to a selfadjoint unitary operator on
L2,k(M)⊕ L2,n−k(M).
2The double brackets are intended to distinguished the integrated inner product from the C∞(M)-valued
form (· | ·).
30
3.4 The Hodge Laplacian
Definition 3.6. The codifferential δ : Ak(M,C)→ Ak−1(M,C) is the adjoint of the exte-
rior derivative with respect to the integrated inner product. In other words, if α ∈ Ak(M,C),
β ∈ Ak−1(M,C), δα is determined by the relation
〈〈δα | β〉〉 := 〈〈α | dβ〉〉. (3.9)
The Riesz theorem “guarantees” that there is a unique element δα ∈ L2k−1(M) satisfy-
ing (3.9), though it is not immediately clear that δα ∈ Ak−1(M,C). That this is indeed
the case, and that δ takes Ak(M,R) to Ak−1(M,R), is a consequence of the following basic
identity.
Lemma 3.6. δ = (−1)n+1?d?.
Proof. If α ∈ Ak(M,C) and β ∈ Ak−1(M,C), then dβ¯ ∧ ?α+ (−1)k−1β¯ ∧ d(?α) = d(β¯ ∧ ?α)
is an exact n-form, whose integral is zero by Stokes’ theorem. Thus
〈〈β | δα〉〉 = 〈〈dβ | α〉〉 = i−m(−1)nk+k(k−1)/2
∫
M
dβ¯ ∧ ?α
= i−m(−1)nk+k(k−1)/2
∫
M
(−1)kβ¯ ∧ d(?α)
= (−1)nk+k(k−1)/2(−1)k(−1)n(k−1)+(k−1)(k−2)/2〈〈β | ?d?α〉〉
= (−1)n+1〈〈β | ?d?α〉〉,
and so δα = (−1)n+1?d?α since β is arbitrary.
Clearly δ2 = 0, and in particular (A•(M), δ) is a chain complex. We say that ω ∈ Ak(M)
is coclosed if δω = 0, or coexact if ω = δζ for some ζ ∈ Ak+1(M).
Definition 3.7. The Hodge Laplacian is the operator ∆ on A•(M) defined as
∆ := (d+ δ)2 = dδ + δd.
Notice that ∆ takes Ak(M) to Ak(M), since d raises and δ lowers degrees by one. [The
restriction of ∆ to 0-forms is called the Laplace–Beltrami operator on C∞(M).]
A k-form γ is called harmonic if ∆γ = 0. We denote the vector space of harmonic
k-forms by Harmk(M).
Lemma 3.7. A k-form γ is harmonic iff dγ = δγ = 0.
Proof. If γ is both closed and coclosed, then clearly ∆γ = d(δγ) + δ(dγ) = 0. On the other
hand, for any ω ∈ A•(M),
〈〈ω |∆ω〉〉 = 〈〈ω | dδω〉〉+ 〈〈ω | δdω〉〉 = 〈〈δω | δω〉〉+ 〈〈dω | dω〉〉 ≥ 0,
so that ∆γ = 0 implies dγ = δγ = 0.
31
The Laplacian ∆ commutes with the operators ?, d and δ. Indeed, ?dδ = (−1)n+1?d?d? =
δd? and ?δd = (−1)n+1d?d = dδ?, so ?∆ = ?dδ + ?δd = δd?+ dδ? = ∆?. Also, d∆ = dδd =
∆d and δ∆ = δdδ = ∆δ since d2 = δ2 = 0.
On the Hilbert space L2,•(M), ∆ = dδ+ δd = dd∗ + d∗d is a formally selfadjoint positive
operator with domain A•(M,C); however, it is generally an unbounded operator. It can be
made bounded by defining a larger Hilbert space norm on A•(M,C) and completing to obtain
a smaller Hilbert space H2n,•(M), called a Sobolev space. We shall not go into this matter
here; we refer to [28] for the details. It turns out that ∆ is then bounded as an operator from
H2n,•(M) to L2,•(M), which is, in fact, an elliptic differential operator.3 Elliptic operators
have two main properties. Firstly, they are Fredholm operators. This implies both that
ker ∆ is finite-dimensional, and that ∆ has closed range in L2,•(M). Secondly, a generalized
solution of an elliptic differential operator is in fact smooth.4 This means that ∆u = 0 for
u ∈ H2n,•(M) only when u in fact lies in A•(M,C). In other words, ker ∆ = Harm•(M,C).
The Fredholm property of ∆ now implies that Harm•(M,C) is a finite-dimensional vector
space.
The theory of the Hodge Laplacian culminates in the fundamental theorem of Hodge,
which states that any k-form on a compact oriented Riemannian manifold can be uniquely
decomposed as a sum of a closed k-form, a coclosed k-form, and a harmonic k-form.
Theorem 3.8. (Hodge). Let (M, g) be a compact oriented Riemannian manifold. Then
for each k = 0, 1, . . . , n there is an orthogonal direct sum
Ak(M) = dAk−1(M)⊕ δAk+1(M)⊕ Harmk(M). (3.10)
That the summands are orthogonal follows from the identities 〈〈dα |δβ〉〉 = 〈〈d2α |β〉〉 = 0,
〈〈dα | γ〉〉 = 〈〈α | δγ〉〉 = 0, 〈〈δβ | γ〉〉 = 〈〈β | dγ〉〉 = 0, for α ∈ Ak−1(M), β ∈ Ak+1(M),
γ ∈ Harmk(M). If P denotes the orthogonal projector on the Hilbert space L2,k(M) whose
range is Harmk(M), then if ω ∈ Ak(M) the form ω − Pω lies in the range of the selfadjoint
operator ∆; it turns out that this range is closed, even if one restricts ∆ to its original domain
Ak(M). Hence, ω−Pω = ∆η for some η ∈ Ak(M); and therefore ω = d(δη)+δ(dη)+Pω.
The full proof of Hodge’s theorem depends on verifying that ∆ is elliptic and identifying
carefully the range of ∆. For the original treatment of Hodge, one can examine his book [33];
for more modern proofs, in the somewhat more general setting of an “elliptic complex”, one
may consult [23, 28]. The inverse operator ∆−1, defined on the orthogonal complement of
Harmk(M), is an integral operator, namely the “Green operator” for the partial differential
equation ∆α = 0.
Corollary 3.9. Each de Rham class [ω] ∈ HdR(M) contains exactly one harmonic form;
thus γ 7→ [γ] is an isomorphism of Harmk(M,R) onto HkdR(M), and therefore every de Rham
cohomology space of M is finite dimensional.
3This means that the “principal symbol” of ∆, which is a certain matrix-valued function on T ∗M , becomes
invertible after deleting a neighbourhood of the zero section of the cotangent bundle. In fact, the principal
symbol σ∆ of the Laplacian is given by σ∆(ξx) = −(ξx|ξx) := −gx(ξ]x, ξ]x) for ξx ∈ TxM .
4A differential operator whose generalized solutions are automatically smooth is called “hypoelliptic”.
Any elliptic operator is hypoelliptic.
32
Proof. If ω = dα + δβ + γ in Ak(M,R), with γ harmonic, then ω is closed iff dδβ = 0 iff
(β | dδβ) = (δβ | δβ) = 0 iff δβ = 0. For ω = dα + γ ∈ ZkdR(M), we clearly have [ω] = [γ].
Moreover, if ω′ ∈ ZkdR(M) with [ω′] = [ω], then ω′ = ω + dζ for some ζ ∈ Ak−1(M), so
ω′ = d(α + ζ) + γ; hence ω′ and ω have the same harmonic component γ.
Finally, notice that ?d = (−1)n−1δ?, so that whenever ω = dα+ δβ + γ in Ak(M,C), we
also have ?ω = d((−1)n−1?β) + δ((−1)n−1?α) + ?γ in An−k(M,C). The uniqueness of the
Hodge decomposition says that the star operator is a linear isomorphism of Harmk(M,C)
onto Harmn−k(M,C); if desired, one can match real harmonic forms by γ 7→ i−m?γ. Passing
to cohomology, this yields a well-defined R-linear isomorphism
[γ] 7→ [i−m?γ] : HkdR(M) ∼−→Hn−kdR (M).
This isomorphism is called Poincare´ duality.5
4 The Hodge Laplacian on the 2-sphere
In this section, we investigate in detail the Hodge Laplacian on the 2-sphere S2, both as an
illustration of the Hodge theory in general, and in order to introduce a first example of a Dirac
operator. The sphere is, of course, an oriented Riemannian manifold, but it is also round,
i.e., it is a homogeneous space under the group of rotations of R3. The rotation invariance
of the Laplacian ∆ facilitates a complete description of all its eigenvalues and eigenvectors,
which in turn leads to a corresponding spectral description of the Dirac operator D/ = d+ δ.
4.1 The rotation group in three dimensions
Definition 4.1. The rotation group SO(3) consists of all 3× 3 real matrices A satisfying
AtA = 13 and detA = 1. Any rotation belongs to a one-parameter subgroup { exp tN :
t ∈ R }, where N belongs to the Lie algebra so(3) of the rotation group, i.e., the 3× 3 real
matrices satisfying N t +N = 0, TrN = 0. Write
N =
 0 −n3 n2n3 0 −n1
−n2 n1 0
 , (4.1)
and suppose (n1)2 + (n2)2 + (n3)2 = 1. Then if n = (n1, n2, n3) ∈ R3, the matrix identities
N2 = nnt − 13, N3 = −N hold, and it follows that exp tN = I + N sin t + N2(1 − cos t) ∈
SO(3). Now the rotation action of SO(3) on R3 (or on S2) is given explicitly by
(exp tN)x = (I +N sin t+N2(1− cos t))x
= x+ (n× x) sin t+ (n(n · x)− x)(1− cos t)
= x cos t+ (n× x) sin t+ n(n · x)(1− cos t). (4.2)
5For noncompact Riemannian manifolds, one can define “compactly supported de Rham cohomology”
H•dR,c(M) by starting from the cochain complex of differential forms with compact support. Then Poincare´
duality is a family of isomorphisms between HkdR(M) and H
n−k
dR,c(M). In the compact case, both cohomologies
coincide.
33
Let L1, L2, L3 be the generators of the rotation group, i.e., the elements of so(3) given by
replacing n in (4.1) by the standard orthonormal basis vectors; so N = n1L1 + n
2L2 + n
3L3.
It is immediate that [L1, L2] = L3, [L2, L3] = L1 and [L3, L1] = L2, or more compactly,
[Li, Lj] = ij
kLk.
It is very convenient to introduce the matrices L± := L1 ± i L2 (which belong to the
complexified Lie algebra so(3,C)). Then
L21 + L
2
2 + L
2
3 = L−L+ + L
2
3 − iL3 (4.3)
as 3× 3 complex matrices.1
Definition 4.2. The action of SO(3) on S2 induces an action of C∞(S2) by (R · f)(x) :=
f(R−1x); more generally, SO(3) acts on A•(S2) by R · ω := (R−1)∗ω. For the corresponding
action of the Lie algebra so(3) on C∞(S2), the homomorphism property of the group action
induces a Leibniz rule, so that N ∈ so(3) acts on C∞(S2) as a vector field N˜ , called the
fundamental vector field of N , which is given explicitly by
N˜f(x) :=
d
dt
∣∣∣∣
t=0
f(exp(−tN)x).
Since the rotation action on SO(3) is a restriction of an action on R3, the same formula
yields fundamental vector fields in X(R3). In particular, since exp(−tL1)x = (x1, x2 −
tx3, x3 + tx2) + O(t2) and similarly for exp(−tL2)x and exp(−tL3)x, on account of (4.2),
the corresponding fundamental vector fields on R3 are
L˜1 = x
3 ∂
∂x2
− x2 ∂
∂x3
, L˜2 = x
1 ∂
∂x3
− x3 ∂
∂x1
, L˜3 = x
2 ∂
∂x1
− x1 ∂
∂x2
. (4.4)
On transforming (4.4) to spherical coordinates (r, θ, φ), one finds that the L˜j are inde-
pendent of r and of ∂/∂r (as expected). Indeed,
L˜± = e±iφ
(
∓i ∂
∂θ
+ cot θ
∂
∂φ
)
, L˜3 = − ∂
∂φ
, (4.5)
which may be regarded as vector fields either in X(R3 \ {0}) or in X(S2).
Exercise 4.1. Verify the formulae (4.5) for the fundamental vector fields.
The sphere S2 is a homogeneous space for the rotation group, and may be identified with
the coset space SO(3)/SO(2), where SO(2) is the subgroup of east-west rotations which
fix the north pole. One may therefore consider a system of local coordinates for S2 which
1Any Lie algebra g gives rise to an associative algebra, called its universal enveloping algebra U(g), whose
elements are polynomial combinations of elements of g, reduced by the commutation relations among such
elements. The Poincare´–Birkhoff–Witt theorem [35] proves that the natural map from g to U(g) is injective,
so that g may be regarded as a subspace of U(g). Now (4.3) may be regarded as an identity in U(so(3,C)).
34
privileges the cartesian coordinate x3. We adopt the following local coordinates for the
remainder of this chapter:
ζ := x1 + ix2 = eiφ sin θ, x3 = cos θ.
We also write ζ¯ := x1 − ix2 = e−iφ sin θ, which supplies a third local coordinate for R3. Let
us abbreviate ∂3 := ∂/∂x
3, ∂ζ := ∂/∂ζ, ∂¯ζ := ∂/∂ζ¯. In these coordinates, the fundamental
vector fields are given by
L˜+ = −2ix3∂¯ζ + iζ∂3, L˜− = 2ix3∂ζ − iζ¯∂3, L˜3 = iζ¯∂¯ζ − iζ∂ζ . (4.6)
A polynomial in the variables L˜j gives a differential operator on S2. For instance, the
operator corresponding to (4.3) satisfies
(L˜−L˜+ + L˜23 − iL˜3)ζ = (L˜3 − i)L˜3ζ = (L˜3 − i)(−iζ) = −2ζ, (4.7)
since ∂¯ζ , and therefore also L˜+, vanishes on holomorphic functions of ζ.
4.2 The Hodge operators on the sphere
Lemma 4.1. The Hodge star operator on A•(S2) is determined by the relations
?(dζ ∧ dx3) = ζ, ?(dζ) = x3 dζ − ζ dx3,
in the (ζ, x3) coordinates.
Proof. First observe that
dζ ∧ dx3 = (ieiφ sin θ dφ+ eiφ cos θ dθ) ∧ (− sin θ dθ)
= ieiφ sin2 θ dθ ∧ dφ = iζΩ,
so that ?1 = iΩ = ζ−1 dζ ∧ dx3, and ?(ζ) = dζ ∧ dx3; reciprocally, ?(dζ ∧ dx3) = ζ. Hodge
duals of 1-forms are given by
? dζ = eiφ ?(i sin θ dφ) + eiφ cos θ ?(dθ) = eiφ dθ + eiφ cos θ(i sin θ dφ),
which simplifies to ?(dζ) = x3 dζ − ζ dx3, and reciprocally, ?(x3 dζ − ζ dx3) = dζ.
The codifferential δ = −?d? is now easily found. For instance, δ(dζ) = ?d(ζ dx3−x3 dζ) =
?(2dζ ∧ dx3) = 2ζ, and since δ(ζ) = 0, it follows that
∆(ζ) = (dδ + δd)ζ = δ(dζ) = 2ζ. (4.8)
35
Definition 4.3. Since rotations act on differential forms through R · ω := (R−1)∗ω, their
generators act as Lie derivatives :
Ljω :=
d
dt
∣∣∣∣
t=0
(exp(−tLj)∗ω) = LeLjω
for j = 1, 2, 3. Lie derivatives commute with exterior derivation: LXd = dLX as operators
on A•(S2), for any X ∈ X(S2); and, in particular, Ljd = dLj for j = 1, 2, 3.
An operator T on A•(S2) is rotation-invariant if T (R · ω) = R · (Tω) for any R ∈
SO(3). Since the three one-parameter subgroups { exp(tLj) : t ∈ R } generate SO(3), this
holds iff TLj = LjT for j = 1, 2, 3. For instance, the Hodge star operator ? = i((θˆ
1) −
ι(θˆ1]))((θˆ2)−ι(θˆ2])) is unchanged if the local oriented orthonormal frame {θˆ1, θˆ2} is replaced
by {R · θˆ1, R · θˆ2}; this says that ? is invariant under rotations, and therefore ?Lj = Lj? for
j = 1, 2, 3.
Since δ = −?d?, ∆ = dδ + δd, it follows that δLj = Ljδ and ∆Lj = Lj∆ for j = 1, 2, 3.
In particular, the Hodge Laplacian is rotation-invariant.
Exercise 4.2. Use the Cartan identity LX = ιXd+ dιX to show that LjΩ = 0 for j = 1, 2, 3.
What can one conclude from this?
The Laplace–Beltrami operator ∆0 (the restriction of ∆ to C
∞(S2)) is thus a second-order
differential operator which commutes with rotations. It therefore represents a quadratic
element in the centre of the algebra U(so(3)). Now it is known [34, 35] that this centre
is the polynomial algebra R[C] generated by the “Casimir element” C = L21 + L22 + L23.
Thus ∆0 = a(L˜
2
1 + L˜
2
2 + L˜
2
3) for some constant a; ∆ and each Lj commutes with d, and so
∆ = a(L21 + L
2
2 + L
2
3) = a(L−L+ + L
2
3 − iL3). From (4.7) and (4.8) one sees that a = −1,
and therefore
∆ = −L−L+ − L23 + iL3. (4.9)
Exercise 4.3. Show directly, using only the commutation relations [Li, Lj] = ij
kLk, that any
quadratic polynomial in L1, L2, L3 commuting with each Lj must be a multiple of L
2
1+L
2
2+L
2
3.
4.3 Eigenvectors for the Laplacian
Since ∆ is invariant under rotations, any subspace of differential forms which is stable under
the SO(3) action is mapped by ∆ into another such subspace. The search for eigenvec-
tors under ∆ should therefore start with the irreducible subspaces of the representation
R 7→ (R−1)∗ of the compact group SO(3) on A•(S2). This is the point of view adopted by
Folland [26], who obtained a complete spectral decomposition of the Hodge Laplacian on a
sphere of any dimension. Here we show how this works for the 2-sphere.
It is useful to recall some facts about representations of compact Lie groups, which may
be found in many places, e.g. [13, 37, 53]. Any irreducible unitary representation of a com-
pact group G acts on a finite dimensional Hilbert space, and the Hilbert space of any unitary
representation may be written as a (possibly infinite) direct sum of irreducible subrepresen-
tations. Moreover, by the Peter–Weyl theorem, all such irreducible representations already
36
occur in the decomposition of the regular representation of G on L2(G), and their carrier
spaces are in fact subspaces of C∞(G).
In the case G = SO(3), the identification SO(3)/SO(2) ≈ S2 goes as follows: elements of
SO(3) are parametrized by three local coordinates (φ, θ, ψ), called “Euler angles”, and SO(2)
is regarded as the one-parameter subgroup { expψL3 : ψ ∈ R }; the remaining angles (θ, φ) as
the spherical coordinates on S2. Thus functions on S2 are identified with functions on SO(3)
which are constant on SO(2)-cosets, i.e., functions which do not depend on the variable ψ; in
this way the representation of SO(3) on C∞(S2) —or on its completion L2(S2)— becomes a
subrepresentation of the regular representation on L2(SO(3)). Its irreducible subspaces are
spanned by the spherical harmonics Ylm(θ, φ), where l = 0, 1, 2, . . . and m = −l, . . . , l− 1, l;
indeed Hl := span{Ylm : m = −l, . . . , l } is an irreducible subspace of dimension 2l + 1,
and the Ylm form an orthonormal basis for L
2(S2). It turns out, again by the Peter–Weyl
theorem, that SO(3) has (up to equivalence) exactly one irreducible unitary representation
of each odd dimension, so the orthonormality of the spherical harmonics shows that in the
decomposition of the rotation action on C∞(S2), each such representation occurs once only,
and that there are no other irreducible subrepresentations.2
There is a general principle for finding irreducible representations of a compact Lie
group G, called the “theorem of the highest weight” [13, 35, 37]. One finds a maximal
torus3 in G, that is, a subgroup T ≤ G which is a torus, i.e., isomorphic to Tk for some k,
with k maximal; if G = SO(3), then k = 1 and T := SO(2) ' T will do. In a representation
space for G, one looks for a joint eigenvector for the torus T ; when G = SO(3), this is
just an eigenvector v for L3, whose eigenvalue is called a “weight” of T . Within gC one
finds k “raising elements” which annihilate v (since the weight is “highest”) and k “lowering
elements” which, applied successively to v, generate a basis for an irreducible representation
space V ; the commutation relations in gC ensure that applying other generators does not
enlarge the space V . When G = SO(3), there is one raising element, namely L+, and one
lowering element, namely L−.
The upshot of the general theory is this. Within each space of k-forms on the sphere
(k = 0, 1, 2), we must find forms α satisfying L+α = 0 and L3α = cα for some eigenvalue c,
and such that the vector space span{Lr−α : r ∈ N } is finite dimensional. The identity (4.9)
guarantees that α is also an eigenvector for the Laplacian ∆.
Definition 4.4. From (4.6), the identity L+f = 0 is satisfied whenever f ∈ C∞(S2) is
holomorphic in ζ and independent of x3. Since (L3f)(ζ) = −iζ f ′(ζ), this f is an eigenvector
for L3 iff it is of the form f(ζ) = ζ
l for some l; since ζ l = eilφ sinl θ, the smoothness of f
forces the condition l ∈ N. Define φ0l ∈ A0(S2) by φ0l(ζ, x3) := ζ l, for l = 0, 1, 2, . . . .
Exercise 4.4. Show that Lk−φ0l is a linear combination of terms (x
3)k−2rζ¯rζ l−k+r and check
that Lk−φ0l 6= 0 for k = 0, 1, . . . , 2l but L2l+1− φ0l = 0. Conclude that the functions {R · φ0l :
2For other compact groups, the completeness of the decomposition of the natural representation on 0-
forms on a homogeneous space may be obtained from a counting argument based on Frobenius reciprocity.
For the case of SO(n) and the sphere Sn−1, we refer to Folland [26].
3Any compact connected Lie group is the union of all its maximal tori, and any two maximal tori are
conjugate, by a theorem of Weyl: see [13] for a proof.
37
R ∈ SO(3) } span an irreducible representation space for SO(3), of dimension 2l + 1, for
each l ∈ N.
Exercise 4.5. Prove that the functions {Lr−φ0l : l ∈ N, r = 0, . . . , 2l } span a dense subset
of L2(S2) by showing that any homogeneous polynomial in the variables ζ, ζ¯, x3 is a linear
combination of these.
Exercise 4.6. Check that 〈〈φ0l | φ0m〉〉 = 0 for l 6= m, and that 〈〈α | L−β〉〉 = 〈〈L+α | β〉〉 for
α, β ∈ A0(S2), using (4.6). Show that [L+, L−] = −2iL3 and [L3, L−] = iL− and conclude
that L+L
r
−φ0l = alrL
r−1
− φ0l for some constant alr. Deduce that the functions L
r
−φ0l, when
suitably normalized, yield an orthonormal basis for L2(S2).
We have thus identified a complete set of irreducible subrepresentations of the rotation
action on A0(S2). Note that L3(ζ l) = −ilζ l, so the L3-eigenvalue is −il. It is immediate
from (4.9) that ∆(ζ l) = (l2 + l) ζ l. We take stock that
L+φ0l = 0, L3φ0l = −il φ0l, ∆φ0l = l(l + 1)φ0l. (4.10)
It is now clear that the Laplace–Beltrami operator ∆0 is a formally selfadjoint, positive
operator on L2(S2), with spectrum sp(∆0) = { l(l + 1) : l ∈ N }. Since ∆0 commutes with
L−, each Lr−φ0l (r = 0, . . . , 2l) is an eigenvector for the eigenvalue l(l + 1), which therefore
has multiplicity 2l+ 1. The set sp(∆0), with these multiplicities, is sometimes referred to as
the spectrum of the Riemannian manifold S2 [8].
Note, in particular, that ker ∆0 = Harm
0(S2) is the one-dimensional space spanned by
φ00, i.e., the space of constant functions. Thus, the only harmonic functions on S2 are the
constants.
4.4 Spectrum of the Hodge Laplacian
Definition 4.5. Since ∆ commutes with the exterior derivative and the Hodge star operator,
we can manufacture more eigenvectors by applying these operators to the φ0l. Since dφ00 = 0
because φ00 is constant, dφ0l is an eigenvector only for l ≥ 1. Define
ψ1l := l
−1 dφ0l = ζ l−1 dζ,
φ1l := −?ψ1l = −ζ l−1?dζ = ζ l dx3 − ζ l−1x3 dζ,
for l = 1, 2, 3, . . . . Since L+, L3 and ∆ each commutes with d and ?, we obtain from (4.10):
L+ψ1l = 0, L3ψ1l = −il ψ1l, ∆ψ1l = l(l + 1)ψ1l,
L+φ1l = 0, L3φ1l = −il φ1l, ∆φ1l = l(l + 1)φ1l.
Thus the 1-forms ψ1l and φ1l are highest-weight vectors for irreducible representations of
SO(3), of dimension 2l + 1; in fact, the representation spaces are spanned respectively by
{Lr−ψ1l : r = 0, . . . , 2l } and {Lr−φ1l : r = 0, . . . , 2l }.
Exercise 4.7. Show that 〈〈ψ1l | ψ1m〉〉 = 〈〈φ1l | φ1m〉〉 = 0 for l 6= m, and that 〈〈ψ1l | φ1m〉〉 = 0
for all l,m ≥ 1.
38
Exercise 4.8. Check that 〈〈α | L−β〉〉 = 〈〈L+α | β〉〉 for α, β ∈ A1(S2), and conclude that the
1-forms {Lr−ψ1l,Lr−φ1l : l ≥ 1, r = 0, . . . , 2l } are orthogonal.
The 1-forms ψ1l are exact, so applying d to them yields only zero. However, the 1-forms
φ1l are coexact, since φ1l = −?dφ0l = δ(?φ0l), so we may define
ψ2l := (l + 1)
−1 dφ1l = ζ l−1 dx3 ∧ dζ = iζ lΩ
for l = 1, 2, 3, . . . . Since the expression iζ lΩ also makes sense for l = 0, we also define
ψ20 := iΩ. Then
L+ψ2l = 0, L3ψ2l = −il ψ2l, ∆ψ2l = l(l + 1)ψ2l
for l = 0, 1, 2, . . . . Furthermore, since ?1 = iΩ, we have ?ψ2l = φ0l for l ∈ N.
We may summarize this information with two commutative diagrams:
φ0l
?←−−− ψ2l
l−1d
y x(l+1)−1d
ψ1l
−?−−−→ φ1l
φ0l
?−−−→ ψ2l
(l+1)−1δ
x yl−1δ
ψ1l
−?←−−− φ1l
(4.11)
where l ≥ 1. The first diagram takes stock of the foregoing definitions, and the vertical
arrows in the second diagram are formed by composing three arrows from the first, using
the identity δ = −?d?. From this it is evident that the ψkl are exact forms and the φkl are
coexact forms. A complete circuit of four arrows in either diagram corresponds to applying
the operator (l(l + 1))−1(−?d?d− d?d?) = (l(l + 1))−1∆, which acts as the identity at each
vertex.
We now have a complete set of eigenforms for ∆. Indeed, we have shown already that the
Lr−φ0l are a complete set of eigenforms of degree 0. Since ? is a bijection between 0-forms and
2-forms, the Lr−ψ2l, for l ∈ N, r = 0, . . . , 2l densely span A2(S2). For the 1-forms, we may
use the Hodge decomposition (3.10); the diagrams (4.11) show that d : δA1(S2) → dA0(S2)
and δ : dA1(S2) → δA2(S2) are bijections, that the ψ1l densely span dA0(S2), and that the
φ1l densely span δA
2(S2). It remains only to observe that there are no nonzero harmonic
1-forms. This can be seen by noting that by writing any 1-form as α = f(ζ) dζ+g(x3) dx3 +
h(ζ, x3)(x3dζ − ζ dx3), since f(ζ) dζ + g(x3) dx3 is exact, and h(x3dζ − ζ dx3) = ?(h dζ) is
coexact. Thus, {Lr−ψ1l,Lr−φ1l : l ≥ 1, r = 0, . . . , 2l } forms an orthogonal basis for A1(S2).
The spectrum of the Hodge Laplacian is therefore sp(∆) = { l(l + 1) : l ∈ N }, with
multiplicities 4l(l + 1) for the eigenvalue l(l + 1) when l ≥ 1, and multiplicity 2 for the
zero eigenvalue. Indeed, ker ∆ = span{φ00, ψ20} = Harm•(S2). If K denotes the operator on
L2,•(S2) which is zero on Harm•(S2) and inverts ∆ on its orthogonal complement, then K is a
bounded selfadjoint operator and, moreover, is compact, since its spectrum sp(K) = { (l(l+
1))−1 : l ∈ N } accumulates only at 0 and consists of eigenvalues with finite multiplicities.
Also, ∆K = K∆ = I − P , where P is the orthogonal projector of rank 2 with range
Harm•(S2).
39
Corollary 4.2. The de Rham cohomology spaces of S2 are given by
H•dR(S2) ' Harm•(S2) ' R⊕ 0⊕ R, (4.12)
on decomposing ker ∆ by degrees of forms.
Notice that this coincides with the Cˇech cohomology Hˇ•(S2,R), computed in subsec-
tion 1.9.
Definition 4.6. An important topological invariant of a manifold M is its Euler character-
istic4
χ(M) :=
dimM∑
k=0
(−1)k dimHkdR(M).
For the 2-sphere, (4.12) yields χ(S2) = 1− 0 + 1 = 2.
4.5 The Hodge–Dirac operator
Definition 4.7. The Hodge–Dirac operator on the 2-sphere S2 is the operator D/ := d+δ,
whose square is the Hodge Laplacian: D/ 2 = ∆.
Let Aeven(S2) := A0(S2) ⊕ A2(S2) be the algebra of differential forms of even degree on
S2, and write Aodd(S2) := A1(S2) to denote the odd-degree forms. Then D/ is an odd operator
in the sense that it interchanges forms of even and odd parities: D/ (Aeven(S2)) ⊆ Aodd(S2)
and D/ (Aodd(S2)) ⊆ Aeven(S2).
From (4.11), the action of D/ is given explicitly by
D/φ0l = l ψ1l, D/ψ1l = (l + 1)φ0l,
D/ψ2l = l φ1l, D/ φ1l = (l + 1)ψ2l, (4.13)
for l = 0, 1, 2, . . . on the left, and l = 1, 2, 3, . . . on the right. It follows that
D/
(√
l + 1φ0l ±
√
l ψ1l
)
= ±
√
l(l + 1)
(√
l + 1φ0l ±
√
l ψ1l
)
,
D/
(√
l + 1ψ2l ±
√
l φ1l
)
= ±
√
l(l + 1)
(√
l + 1ψ2l ±
√
l φ1l
)
.
Since these eigenforms for D/ densely span L2,•(S2), one concludes that D/ is a formally
selfadjoint5 unbounded linear operator on L2,•(S2), with spectrum sp(D/ ) = {±√l(l + 1) :
l ∈ N }.
Exercise 4.9. What are the multiplicities of the eigenvalues of D/ ?
4The Euler characteristic may be computed as the integral of a certain differential form over M , as we
shall see later.
5The term “formally selfadjoint” means that 〈〈D/ω | η〉〉 = 〈〈ω |D/η〉〉 for ω, η ∈ A•(S2); operator theorists
would say that D/ is “symmetric”. With some more work, one can check that D/ is essentially selfadjoint,
which means that it has an extension to a larger domain in the Hilbert space L2,•(S2) which is a closed,
unbounded selfadjoint operator. See [39] for a proof of this.
40
It also follows from (4.13) that kerD/ = span{φ00, ψ20}, and that D/ has dense range, so
that cokerD/ = 0; moreover, the restriction of D/ to L2,•(S2) 	 Harm•(S2) has a compact
inverse. Thus D/ is a Fredholm operator.6 Its index is given by
indD/ := dim(kerD/ )− dim(cokerD/ ) = 2− 0 = 2.
Corollary 4.3. indD/ = χ(S2).
This is a first example of an index theorem, wherein a certain integer obtained by an
integral over the manifold (namely, the Euler characteristic) turns out to be equal to the
Fredholm index of an operator (namely, the Hodge–Dirac operator) which is bound up with
the geometric structure of the manifold. For a wider discussion of index theorems in modern
geometry and topology, we refer to [9, 28, 39, 42].
There is an equivalent method of computing the index from the grading of A•(S2) into
forms of even and odd degree. Write L2,•(S2) = L2,even(S2) ⊕ L2,odd(S2) where L2,even(S2)
and L2,odd(S2) are the respective completions of Aeven(S2) and Aodd(S2); this is then a graded
Hilbert space. The odd operator may be written as
D/ =
(
0 D/ 1
D/ 0 0
)
,
where D/ 0 : L
2,even(S2)→ L2,odd(S2) and D/ 1 : L2,odd(S2)→ L2,even(S2). The selfadjointness of
D/ says that D/ 1 = D/
†
0, and this may be verified directly from (4.13) also. Now cokerD/ 1 '
(kerD/ 0)
⊥ and cokerD/ 0 ' (kerD/ 1)⊥, so the index of D/ equals
indD/ = dim(kerD/ 0)− dim(kerD/ 1). (4.14)
From (4.13), the right hand side of (4.14) equals 2− 0 = 2, as expected.
Exercise 4.10. Show that l 〈〈ψ1l |ψ1l〉〉 = (l+1) 〈〈φ0l |φ0l〉〉 and that l 〈〈φ1l |φ1l〉〉 = (l+1) 〈〈ψ2l |
ψ2l〉〉. Deduce from (4.13) that D/ 0 and D/ 1 are adjoints of each other.
The Hodge–Dirac operator is one of many operators which are collectively known as Dirac
operators. Some common properties are: (i) they are unbounded selfadjoint Fredholm oper-
ators; (ii) the Hilbert spaces on which they act are graded into “even” and “odd” subspaces,
which the Dirac operators interchange; (iii) their squares are “generalized Laplacians” (of
which we shall have more to say later) acting on Riemannian manifolds.
Definition 4.8. As a second example, let us redefine the grading of differential forms by
splitting A•(S2) into the (±1)-eigenspaces for the Hodge star operator: thus A+(S2) :=
{ω : ?ω = ω } is the space of selfdual forms, and A−(S2) := { η : ?η = −η } is the set of
antiselfdual forms. Now δ? = −?d implies that D/? = −?D/ , so the same operator D/ = d+ δ
interchanges selfdual and antiselfdual forms: D/ (A±(S2)) ⊆ A∓(S2).
6Conventionally, Fredholm operators are taken to be bounded, whereas D/ and ∆ are not. One could
remedy this by redefining the norm on the domain space, but then the domain and range would lie in
different Hilbert spaces. Instead, we use the alternative definition [22] of a Fredholm operator as a closed,
possibly unbounded, operator between Hilbert spaces which has dense domain, finite-dimensional kernel and
finite-codimensional (therefore closed) range. In this sense, the operator closures of D/ and ∆ are unbounded
Fredholm operators on L2,•(S2).
41
The diagrams (4.11) show that the forms ψ2l + φ0l (l ≥ 0) and φ1l − ψ1l (l ≥ 1) are
selfdual, whereas ψ2l − φ0l (l ≥ 0) and φ1l + ψ1l (l ≥ 1) are antiselfdual. Since D/ commutes
with L−, applying powers to L− to these forms generates a complete set [i.e., a set which
spans a dense subspace of L2,•(S2)]. Thus we may rewrite (4.13) as
D/ (ψ2l + φ0l) = l (φ1l + ψ1l), D/ (φ1l − ψ1l) = (l + 1) (ψ2l − φ0l),
D/ (φ1l + ψ1l) = (l + 1) (ψ2l + φ0l), D/ (ψ2l − φ0l) = l (φ1l − ψ1l),
where l ≥ 1 in all cases; and D/ (ψ20 ± φ00) = 0.
Observe that, in the new grading of L2,•(S2), the odd operator D/ can be written as
D/ =
(
0 D/ −
D/ + 0
)
, (4.15)
where D/ ± : L2,±(S2)→ L2,∓(S2). Now we define its index as
indD/ := dim(kerD/ +)− dim(kerD/ −), (4.16)
which equals 1− 1 = 0 since the kerD/ ± are one-dimensional spaces, spanned by (ψ20±φ00).
The corresponding topological quantity [9] arises from the bilinear form s([α], [β]) :=∫
S2 α ∧ β on H1dR(S2). The integral depends only on the cohomology classes [α] and [β], and
s is antisymmetric; we assign to s a “signature” of zero.7 The index theorem for this case is
the equality of this zero signature with the zero index for D/ as defined by (4.15).
5 Connections on vector bundles
Line bundles are classified by integral Cˇech 2-cocycles, according to the theory developed
in Section 1. In this chapter, we show how to produce, for a given Hermitian line bundle
E−→M , a Cˇech 2-cocycle which is associated to its class. This 2-cocycle comes from the
curvature form of a connection on the line bundle. A connection, or covariant derivative,
is a general structure which supplements that of a vector bundle with a notion of “parallel
displacement” among neighbouring fibres. There are several possible ways to introduce
connections; we shall adopt here an algebraic approach, in the spirit of Cartan calculus of
vector fields and forms.
5.1 Modules of vector-valued forms
Lemma 5.1. Suppose that E−→M and E ′−→M are two vector bundles over the (compact)
manifold M . If τ : E → E ′ is a bundle map, i.e., a smooth map such that (τ, idM) is a vector
bundle morphism, there is an C∞(M)-linear map τ∗ : Γ(E) → Γ(E ′) given by τ∗s := τ ◦ s;
and the correspondence τ 7→ τ∗ satisfies (idE)∗ := idΓ(E) and (τ ◦ σ)∗ = τ∗ ◦ σ∗.
7A less trivial example arises on considering a Riemannian manifold M of dimension n = 4k; the corre-
sponding formula yields a symmetric bilinear form q([α], [β]) :=
∫
M
α ∧ β on H2kdR(M), and the topological
invariant is the signature of this bilinear form.
42
Proof. The C∞(M)-linearity of τ means that τ∗(fs) = fτ∗s for s ∈ Γ(E), f ∈ C∞(M), which
follows from the linearity of τ : Ex → E ′x for each x ∈ M , since τ∗(fs)(x) := τ(f(x)s(x)) =
f(x)τ(s(x)) = (f τ∗s)(x). The remaining assertions are obvious.
Let A := C∞(M) be the algebra of smooth functions on M . One way to restate the
preceding Lemma is to say that E 7→ Γ(E), τ 7→ τ∗ is a covariant functor Γ: Vect(M) →
Mod(A) from the category of vector bundles over M to the category of A-modules.
Exercise 5.1. Verify that Γ(E∗) = Γ(E)∗, where E∗ → M is the dual vector bundle to
E−→M , and the notation E∗, for an A-module E, denotes the module of A-linear maps
from E to A.
Definition 5.1. If A is any (real or complex) algebra, a bimodule E over A is a vector
space with bilinear operations A × E → E and E × A → E, usually simply as (a, s) 7→ as
and (s, a) 7→ sa, satisfying 1 s = s 1 = s, a(a′s) = (aa′)s, and (sa′)a = s(a′a), for s ∈ E,
a, a′ ∈ A. The tensor product E ⊗A E′ of two such bimodules is the bimodule1 whose
elements are finite sums
∑
j sj ⊗ s′j with sj ∈ E and s′j ∈ E′, subject only to the relations
(sa)⊗s′−s⊗ (as′) = 0, for s ∈ E, s′ ∈ E′, a ∈ A. When A is commutative, the identification
sa = as makes each A-module into an A-bimodule; the foregoing recipe defines the tensor
product of A-modules in that case.
Proposition 5.2. Let A := C∞(M) be the algebra of smooth functions on a compact ma-
nifold M , and let E−→M , E ′−→M be vector bundles over M . Then there is a canonical
isomorphism of A-modules:
Γ(E)⊗A Γ(E ′) ' Γ(E ⊗ E ′).
Proof. If s ∈ Γ(E), s′ ∈ Γ(E ′), let s ⊗ s′ denote the section x 7→ s(x) ⊗ s′(x) of the tensor
product bundle E⊗F −→M . We provisionally denote by s⊗As′ the element of Γ(E)⊗AΓ(E ′)
given by the definition of the tensor product ofA-modules. Let θ : Γ(E)⊗AΓ(E ′)→ Γ(E⊗E ′)
be the A-linear map determined by θ(s⊗As′) := s⊗s′; the claim is that θ is an isomorphism.
If U is a chart domain in M over which the bundles E−→M and E ′−→M are trivial,
then E ⊗ E ′−→M is also trivial over U . Indeed, we have seen in subsection 1.7 that any
local section t ∈ Γ(U,E) is of the form t = ∑rk=1 hksk, where {s1, . . . , sr} is a local system
of sections for E over U , and h1, . . . , hr ∈ C∞(U); in other words, the sections {s1, . . . , sr}
generate Γ(U,E) freely over C∞(U). If {s′1, . . . , s′l} is a local system of sections for E ′
over U , they generate Γ(U,E ′) freely as a C∞(U)-module, and it is clear that { sj ⊗ s′k : j =
1, . . . , r; k = 1, . . . , l } generate Γ(U,E⊗E ′) as a free C∞(U)-module, since { sj(x)⊗ s′k(x) :
j = 1, . . . , r; k = 1, . . . , l } is a basis for Ex ⊗ E ′x for each x ∈ U . In summary, θ is an
isomorphism whenever the bundles E−→M and E ′−→M (and consequently E⊗E ′−→M)
are trivial.
In the general case, there are vector bundles F −→M and F ′−→M such that E ⊕
F −→M and E ′ ⊕ F ′−→M are trivial,2 by Proposition 1.8. Let ι : E → E ⊕ F and
1The bimodule operations on E⊗AE′ are, of course, defined by a(s⊗s′) := (as)⊗s′ and (s⊗s′)a := s⊗(s′a).
2The use of Proposition 1.8 (existence of the supplementary bundle) is the only point in this proof where
the compactness of M is used. It should be said that, with some work to establish the existence of a finite
trivialising open covering of M , the compactness assumption can be dropped: we refer to [17] for a proof.
43
σ : E ⊕ F → E be the extension and restriction maps: ι(u) := (u, 0), σ(u, v) := u, and let
ι′ : E ′ → E ′⊕F ′, σ′ : E ′⊕F ′ → E ′ be similarly defined. Then σ ◦ ι = idE, so σ∗ ◦ ι∗ = idΓ(E)
and also σ′∗ ◦ ι′∗ = idΓ(E′); thus, ι∗ and ι′∗ are injective, whereas σ∗ and σ′∗ are surjective.
Now Ex⊗E ′x is a direct summand of the vector space (Ex⊕Fx)⊗(E ′x⊕F ′x) for each x ∈M ;
this yields bundle maps ι′′ : E⊗E ′ → (E⊕F )⊗ (E ′⊕F ′), σ′′ : (E⊕F )⊗ (E ′⊕F ′)→ E⊗E ′
satisfying σ′′∗ ◦ ι′′∗ = idΓ(E⊗E′). Finally, let
ι∗ ⊗ ι′∗ : Γ(E)⊗A Γ(E ′)→ Γ(E ⊕ F )⊗A Γ(E ′ ⊕ F ′)
be defined by (ι∗ ⊗ ι′∗)(s ⊗ s′) := ι∗s ⊗ ι′∗s′, and let σ∗ ⊗ σ′∗ : Γ(E ⊕ F ) ⊗A Γ(E ′ ⊕ F ′) →
Γ(E)⊗A Γ(E ′) be defined similarly. We thus have two commutative diagrams:
Γ(E)⊗A Γ(E ′) θ−−−→ Γ(E ⊗ E ′)
ι∗⊗ι′∗
y yι′′∗
Γ(E ⊕ F )⊗A Γ(E ′ ⊕ F ′) Θ−−−→ Γ((E ⊕ F )⊗ (E ′ ⊕ F ′))
(5.1)
where Θ is the isomorphism of free A-modules already obtained, and
Γ(E)⊗A Γ(E ′) θ−−−→ Γ(E ⊗ E ′)
σ∗⊗σ′∗
x xσ′′∗
Γ(E ⊕ F )⊗A Γ(E ′ ⊕ F ′) Θ−−−→ Γ((E ⊕ F )⊗ (E ′ ⊕ F ′))
(5.2)
From (5.1), θ is injective, since ι∗⊗ ι′∗ and ι′′∗ are injective and Θ is bijective; and (5.2) shows
analogously that θ is surjective. Thus θ is an A-linear isomorphism in the general case.
Corollary 5.3. Each A-linear map from Γ(E) to Γ(E ′) is of the form τ∗ for some bundle
map τ : E → E ′.
Proof. The maps τ : E → E ′ form the total space of the vector bundle Hom(E,E ′)−→M ,
whose fibres are Hom(Ex, E
′
x) ' E∗x ⊗ E ′x. Thus τ can be identified with a section of the
vector bundle E∗ ⊗ E ′−→M . On the other hand, an A-linear map from Γ(E) to Γ(E ′)
belongs to HomA(Γ(E),Γ(E
′)) ' Γ(E)∗ ⊗A Γ(E ′) ' Γ(E∗) ⊗A Γ(E ′); it is easily seen that
τ∗ ∈ HomA(Γ(E),Γ(E ′)) corresponds to θ−1(τ) ∈ Γ(E∗)⊗AΓ(E ′) under these identifications.
Since θ is bijective, these account for all A-linear maps from Γ(E) to Γ(E ′).
Definition 5.2. Consider the vector bundle E ′ = ΛrT ∗M , whose smooth sections are the
r-forms over M : Ar(M) = Γ(ΛrT ∗M). Define
Ar(M,E) := Γ(E)⊗A Ar(M) ' Γ(E ⊗ ΛrT ∗M), (5.3)
using Proposition 5.2. The elements of the A-module Ar(M,E) are finite sums of the form∑
k sk ⊗ ωk, where sk ∈ Γ(E), ωk ∈ Ar(M); we shall refer to them as “E-valued r-forms
over M”.
On a complex manifold, we may also define Ap,q(M,E) := Γ(E)⊗A Ap,q(M).
44
5.2 Connections
Definition 5.3. A connection on a complex vector bundle E−→M is a C-linear3 map
∇ : Γ(E)→ A1(M,E) such that
∇(sf) = (∇s)f + s⊗ df for all s ∈ Γ(E), f ∈ C∞(M). (5.4)
This Leibniz rule4 shows that the definition is local, i.e., that ∇s is determined by its
restrictions to chart domains Uj. To see that, let {fj} be a smooth partition of unity
subordinate to {Uj}, and notice that
∑
j dfj = d
(∑
j fj
)
= d(1) = 0, and thus ∇s =∑
j∇(sfj) by (5.4).
This locality is immediately useful in showing that connections exist on any vector bundle.
First, if E−→M is a trivial bundle, E ≈M×Cr, so that Γ(E) ' Ar, the exterior derivative
d : Ar → A1(M)r : (h1, . . . , hr) 7→ (dh1, . . . , dhr) is a connection, since d(hkf) = dhk f+hk df
for k = 1, . . . , r, i.e., d(h f) = (dh)f+h⊗df for h ∈ Ar. In the general case, over each chart
domain Uj of M one can choose a local system of sections sj = (sj1, . . . , sjr) for E−→M ;
the expansion t =
∑
k sjk h
k gives an isomorphism ψ∗j : Γ(Uj, E) → C∞(Uj)r : t 7→ h. (This
is just the pullback to sections of the local trivialization ψj of (1.1)). Since A
1(Uj, E) '
Γ(Uj, E)⊗C∞(Uj)A1(Uj), we get an isomorphism ψ∗j ⊗ id : A1(Uj, E)→ A1(Uj)r on tensoring
with A1(Uj). Now define ∇(j) := (ψ∗j ⊗ id)−1 ◦d◦ψ∗j , so that ∇(j) : Γ(Uj, E)→ A1(Uj, E) is a
connection on pi−1(Uj)−→Uj. Finally, take any smooth partition of unity {fj} subordinate
to the cover {Uj}, and define ∇s :=
∑
j∇(j)(sfj); one checks that ∇ is a connection on the
vector bundle E−→M .
Exercise 5.2. Carry out this check.
The Leibniz rule (5.4) means that ∇ is not itself A-linear. However, if ∇0 and ∇1 are
two connections on E−→M , it is immediate from (5.4) that (∇1−∇0)(sf) = (∇1s−∇0s)f
for s ∈ Γ(E) and f ∈ A. By Corollary 5.3, ∇1 −∇0 = α∗ for some α ∈ Γ(EndE ⊗ T ∗M) =
A1(M,EndE). Therefore,
∇1s = ∇0s+ α ◦ s for s ∈ Γ(E); (5.5)
conversely, given ∇0, this equation defines a connection ∇1 when α ∈ A1(M,EndE). This
says that the set of all connections on E−→M is an affine space, based on the vector space
A1(M,EndE).
Definition 5.4. If X ∈ X(M) is a vector field, the contraction ιX : A1(M)→ A is A-linear,
since ιX(fβ) = f β(X) = f ιXβ for f ∈ A, β ∈ A1(M). Thus it extends to an A-linear map
from A1(M,E) = Γ(E)⊗A A1(M) to Γ(E)⊗A A = Γ(E) by ιX(s⊗ β) := s(ιXβ) = s β(X).
If ∇ : Γ(E)→ A1(M,E) is a connection, we write ∇X := ιX ◦ ∇ : Γ(E)→ Γ(E). This is
a C-linear map satisfying the Leibniz rule:
∇X(sf) = (∇Xs)f + s(Xf). (5.6)
3For real vector bundles, we require only that ∇ be an R-linear map.
4It is convenient to regard Γ(E) and A1(M,E) as right A-modules; if one wishes to regard them as left
A-modules by identifying fs = sf , the Leibniz rule can equivalently be written as ∇(fs) = f(∇s) + df ⊗ s.
45
Definition 5.5. If E−→M is a Hermitian vector bundle, the metric h on E defines a A-
sesquilinear map Γ(E)×Γ(E)→ A by (s|t) : x 7→ hx(s(x), t(x)), for s, t ∈ Γ(E). This extends
to a sesquilinear form Γ(E) × Ak(M,E) → Ak(M) by tensoring, i.e., (s | t ⊗ ω) := (s | t)ω
when ω ∈ Ak(M).
A connection ∇ on E−→M is compatible with the metric if
(∇s | t) + (s | ∇t) = d(s | t) for all s, t ∈ Γ(E). (5.7)
Notice that this is yet another variant of the Leibniz rule. If X ∈ X(M), a compatible
connection satisfies (∇Xs | t) + (s | ∇Xt) = X(s | t).
Exercise 5.3. In general, for any connection ∇ on a vector bundle E−→M , one may define
a dual connection ∇∗ on the dual bundle E∗−→M by stipulating that the following Leibniz
rule should hold: 〈∇∗ξ, s〉+〈ξ,∇s〉 = d〈ξ, s〉, where 〈·, ·〉 denotes the evaluation map Γ(E∗)×
Γ(E) → A. Verify that, for given ∇ and ξ ∈ Γ(E∗), this Leibniz rule determines a well-
defined element ∇∗ξ of A1(M,E∗), and that the operator ∇∗ thus obtained is a connection.
Exercise 5.4. Suppose that E−→M and E ′−→M are equivalent vector bundles and that
τ : E → E ′ is an invertible bundle map. Extend τ∗ to an operator from Ak(M,E) to
Ak(M,E ′) by tensoring: τ∗(s⊗ω) := τ∗s⊗ω for s ∈ Γ(E), ω ∈ Ak(M). If ∇ is a connection
on E−→M , show that ∇′ := τ∗ ◦ ∇ ◦ τ−1∗ is a connection on E ′−→M .
5.3 Curvature of a connection
Definition 5.6. If ∇ : Γ(E)→ A1(M,E) is a connection on a vector bundle E−→M , there
is a canonical way to extend∇ to a linear map∇ : Ak(M,E)→ Ak+1(M,E). SinceAk(M,E)
is a vector space generated by elements of the form s ⊗ ω, for s ∈ Γ(E), ω ∈ Ak(M), it
suffices to define
∇(s⊗ ω) := (∇s) ∧ ω + s⊗ dω for s ∈ Γ(E), ω ∈ Ak(M). (5.8)
If η ∈ A•(M), it follows that
∇(s⊗ (ω ∧ η)) = (∇s) ∧ ω ∧ η + s⊗ dω ∧ η + (−1)k s⊗ ω ∧ dη,
from which one obtains the “graded Leibniz rule”:
∇(ζ ∧ η) = (∇ζ) ∧ η + (−1)k ζ ∧ dη if ζ ∈ Ak(M,E), η ∈ A•(M). (5.9)
Exercise 5.5. Verify that the extended map ∇ : Ak(M,E)→ Ak+1(M,E) is well-defined, i.e.,
if
∑
j sj ⊗ ωj =
∑
r tr ⊗ ηr, with sj, tr ∈ Γ(E) and ωj, ηr ∈ Ak(M), then
∑
j∇(sj) ⊗ ωj =∑
r∇(tr)⊗ ηr.
The iterated map ∇2 : Γ(E)→ A2(M,E) satisfies
∇2(sf) = ∇((∇s)f)+∇(s⊗ df)
= (∇2s)f −∇s ∧ df +∇s ∧ df + s⊗ d(df) = (∇2s)f,
for s ∈ Γ(E), f ∈ A. Thus ∇2 is an A-linear map from Γ(E) to A2(M,E) = Γ(E⊗Λ2T ∗M);
by Corollary 5.3, ∇2s = ωs with ω ∈ Γ(EndE ⊗ Λ2T ∗M) = A2(M,EndE). This “matrix-
valued 2-form” ω is called the curvature of the connection ∇.
46
Exercise 5.6. If τ : E → E ′ is an invertible bundle map between equivalent vector bundles
over M , and if ∇, ∇′ are connections on E and E ′ related by ∇′ = τ∗ ◦ ∇ ◦ τ−1∗ , show that
their curvatures are likewise related by ω′ = τ∗ ◦ ω ◦ τ−1∗ .
On the trivial bundle E = M ×Cr, let d be the connection (h1, . . . , hr) 7→ (dh1, . . . , dhr)
given by the exterior derivative. From (5.5), we can write ∇s = ds + αs where α ∈
A1(M,EndE) = A1(M)r×r is a matrix of 1-forms on M . Explicitly, if s ∈ Ar has k-th
component hk, then αs has k-th component αkl h
l, so α = [αkl ] ∈ A1(M)r×r. In this case, the
curvature ω is a matrix of 2-forms:
ωs = ∇2s = d(ds+ αs) + α ∧ (ds+ αs)
= ((dα)s− α ∧ ds) + (α ∧ ds+ (α ∧ α)s) = (dα + α ∧ α) s, (5.10)
so that ω = dα + α ∧ α in A2(M)r×r. In components, ωkl = dαkl + αkm ∧ αml .
5.4 A curvature formula
Lemma 5.4. The curvature ω of a connection ∇ on a line bundle E−→M satisfies
ω(X, Y )s = ∇X∇Y s−∇Y∇Xs−∇[X,Y ]s (5.11)
for all X, Y ∈ X(M) and s ∈ Γ(E).
Proof. Since ω ∈ A2(M,EndE) = Γ(EndE)⊗AA2(M), the evaluation on the pair of vector
fields X, Y yields ω(X, Y ) ∈ Γ(EndE), so that ω(X, Y )s ∈ Γ(E). Let the right hand
side of (5.11) be denoted provisionally by F (X, Y )s. We claim that s 7→ F (X, Y )s is A-
linear. To see that, denote by f˜ the operator on Γ(E) of right multiplication by f ∈ A;
then the Leibniz rule (5.6) can be rewritten as [∇X , f˜ ] = X˜f . The desired A-linearity of
F (X, Y ) = [∇X ,∇Y ]−∇[X,Y ] follows from
[F (X, Y ), f˜ ] = [[∇X ,∇Y ], f˜ ]− [∇[X,Y ], f˜ ]
= [∇X , [∇Y , f˜ ]] + [[∇X , f˜ ],∇Y ]− [∇[X,Y ], f˜ ]
= [∇X , Y˜ f ] + [X˜f,∇Y ]− ([X, Y ]f )˜
= (X(Y f)− Y (Xf)− [X, Y ]f )˜ = 0,
on using the Jacobi identity. Thus, by Corollary 5.3, F (X, Y ) lies in Γ(EndE), for each
X, Y ∈ X(M). Furthermore, the formula ιhX = h˜ιX for h ∈ A, which entails ∇hX = h˜∇X ,
shows that F is A-bilinear in (X, Y ), and so F ∈ A2(M,EndE).
We must now show that ω and F coincide. It is enough to check this locally, since if
{fj} is a partition of unity on M , the identities ω(X, Y )s =
∑
j ω(X, Y )sfj, F (X, Y )s =∑
j F (X, Y )sfj show that it suffices to prove ω(X, Y )s = F (X, Y )s when X, Y and s vanish
outside a chart domain Uj. Thus we may suppose that E−→M is a trivial bundle and
indeed that E = M × Cr, whereupon ω = dα + α ∧ α if α ∈ A2(M)r×r is the matrix of
1-forms satisfying ∇ = d+ α.
47
Let us denote by h the element (hi, . . . , hr) of A
r = Γ(E); then ∇Xh = ιX(dh+ αh) has
k-th component Xhk + αkl (X)h
l. Thus F (X, Y )h has k-th component
X(Y hk + αkl (Y )h
l) + αkl (X)Y h
l + αkm(X)α
m
l (Y )h
l − Y (Xhk + αkl (X)hl)
− αkl (Y )Xhl − αkm(Y )αml (X)hl − [X, Y ]hk − αkl ([X, Y ])hl
= X(αkl (Y ))h
l − Y (αkl (X)hl)− αkl ([X, Y ])hl + (αkm(X)αml (Y )− αkm(Y )αml (X))hl
= dαkl (X, Y )h
l + (αkm ∧ αml )(X, Y )hl,
that is, F (X, Y ) is a matrix of 2-forms with components (dαkl + α
k
m ∧ αml )(X, Y ). Hence
F = dα + α ∧ α = ω in A2(M,EndE).
Exercise 5.7. Verify that F (gX, hY )s = F (X, Y )s gh for g, h ∈ A.
The connection ∇ is determined by local 1-forms αj ∈ A1(Uj,EndE), and by a local
system sj in Γ(Uj, E) with respect to which the identification h↔ sjkhk is made. It should
be carefully noted that the local 1-forms αj do not patch together to give a global 1-form
α ∈ A1(M,EndE) unless E−→M is a trivial bundle, since there is no global d for which
∇ = d+α unless the vector bundle is trivial.5 However, the local 2-forms ωj = dαj+αj∧αj ∈
A2(Uj,EndE) do patch together, since they are restrictions to the Uj of the global 2-form ω.
When L−→M is a line bundle, there is a simplification: by Lemma 1.2, the line bundle
EndL−→M is trivial, so the curvature form ω belongs to A2(M), since A2(M,EndL) '
A2(M) ⊗A Γ(EndL) ' A2(M) ⊗A A = A2(M) via canonical isomorphisms. Furthermore,
there are local 1-forms αj ∈ A1(M) such that ω = dαj on each chart domain Uj, so that the
curvature ω is locally exact and hence is a closed 2-form on M .
The curvature form ω depends on the connection ∇, but its de Rham class [ω] ∈ H2dR(M)
does not. To see that, recall from (5.5) that if ∇0 and ∇1 are two connections on L, with
respective curvatures ω0 and ω1, then ∇1s − ∇0s = α s for some α ∈ A1(M), and hence
ω1−ω0 = dα, an exact 2-form. Thus the class [ω] depends only on the line bundle L−→M .
Exercise 5.8. Suppose L−→M and L′−→M are equivalent line bundles and that τ : L →
L′ is an invertible bundle map. Show that τ∗ ∈ Γ(Hom(L,L′)) intertwines the canonical
isomorphisms Γ(EndL) ' A and Γ(EndL′) ' A, and deduce that the connections ∇ and
τ∗ ◦ ∇ ◦ τ−1∗ have the same curvature. Conclude that the class [ω] depends only on the
equivalence class [L] of the line bundle.
5.5 From de Rham cohomology to Cech cohomology
We have now associated to any line bundle L−→M , by very different procedures, two
second-degree cohomology classes, namely the Cˇech class obtained directly from its transition
5The local 1-forms αj may be globalized by pulling back to the frame bundle P
η−→M ; recall that the
local system sj may be regarded as a local section of the frame bundle. It is possible to construct a global
1-form α˜ ∈ A1(P,EndV ) which incorporates each η∗αj , and different connections given rise to different α˜:
this is called the connection 1-form for ∇, for which one may consult [9, 39, 52] or any standard text on
differential geometry.
48
functions and the de Rham class of the curvature of an arbitrary connection. We are led to
suspect that these two cohomologies are related in some underlying fashion. This is indeed
the case: one way of proving the de Rham theorem, which says that the cohomology of
real-valued differential forms on the manifold M depends only on the topology of M (i.e.,
not on its differential structure) is to show that the de Rham cohomology is isomorphic to
Cˇech cohomology with constant real coefficients.6 For the full proof, we refer to [12] or [17,
Appendix E]. We need only the second-degree case of this isomorphism; however, its proof
illustrates the general method.
Proposition 5.5. Let M be a compact manifold. Then H2dR(M) ' Hˇ2(M,R) by a canonical
isomorphism.
Proof. Select a finite good covering7 U = {Uj} for M . We must show that H2dR(M) '
H2(U,R). We first define maps between Z2dR(M) and Z2(U,R), which have some ambiguities
that can be removed by passing to cohomology, thereby yielding well-defined R-linear maps
between H2dR(M) and H
2(U,R).
Start with a closed 2-form ω in A2(M); denote by ω := {ωj} the set of restrictions of ω to
each set Uj of the good covering; this is an element of C
0(U,A2), the set of Cˇech 0-cochains
with 2-form coefficients. Since dω = 0 and each Uj is contractible, the Poincare´ lemma shows
that there is αj ∈ A1(Uj) with dαj = ωj for each j; write α := {αj} ∈ C0(U,A1). Now on
each nonvoid overlap Ui ∩ Uj we have d(αi − αj) = ωi − ωj = 0, so that αi − αj = dfij with
f := {fij} ∈ C1(U,A0). On each nonvoid Ui ∩ Uj ∩ Uk we have d(fij − fik + fjk) = 0 by
cancellation, so that aijk := fij−fik+fjk is a constant real-valued function (since Ui∩Uj∩Uk
is connected); since aijk − aijl + aikl − ajkl = 0 on Ui ∩Uj ∩Uk ∩Ul by cancellation, we have
δa = 0, i.e., a ∈ Z2(U,R).
There are some ambiguities in the choices of αj and fij, so this process does not give a
well-defined map ω 7→ a. Firstly, we could replace each αj by αj + dgj, where g := {gj} ∈
C0(U,A0); then fij becomes fij + gi − gj, which leaves aijk unchanged. Secondly, we could
replace each fij by fij + cij, where the cij are constant functions, i.e., c := {cij} ∈ C1(U,R);
this changes a to a + δc, and [a] ∈ H2(U,R) is left unchanged. Therefore ω 7→ [a] is
well-defined. Thirdly, we could replace ω by ω+ dβ, adding an exact form; then αj becomes
αj + βj, and fij = (αi + βi) − (αj + βj) is unchanged because βi and βj agree on Ui ∩ Uj.
Therefore, [a] depends only on the de Rham class of ω, so [ω] 7→ [a] is a well-defined R-linear
map from H2dR(M) to H
2(U,R).
To go the other way, start from a ∈ Z2(U,R), and take a smooth partition of unity {ψj}
subordinate to the covering U. As in the proof of Proposition 1.6, define f ∈ C1(U,A0)
by fij :=
∑
r aijrψr; then fij − fik + fjk = aijk just as in (1.12), since δa = 0; moreover,
dfij − dfik + dfjk = 0 since the aijk are constant. Define α ∈ C0(U,A1) by αj :=
∑
k ψk dfjk;
now αi − αj =
∑
k ψk dfij = dfij on Ui ∩Uj, and so dαi − dαj = 0 there, which says that the
6Another way is to establish an isomorphism between de Rham cohomology and singular cohomology.
This is thoroughly dealt with in [23].
7This is the only point at which compactness is invoked; and the compactness assumption may be removed
whenever the existence of a finite good covering can be established independently.
49
local 2-forms ωj := dαj patch together to give a global 2-form ω on M . Since dωj = 0, ω is
closed, i.e., ω ∈ Z2dR(M).
The correspondence a 7→ ω depends on the partition of unity {ψj}, but it is clear that
a 7→ f 7→ α 7→ ω retraces the earlier path ω 7→ α 7→ f 7→ a; thus we have shown that the
linear map [ω] 7→ [a] is surjective. Moreover, if [a] = 0, then a = δc with c ∈ C1(U,R), so
that fij :=
∑
r(cij−cir+cjr)ψr = cij−hi+hj where hj :=
∑
r cjrψr and thus dfij = dhj−dhi;
but then αj =
∑
k ψk(dhk − dhj) = βj − dhj, where βj is the restriction to Uj of the 1-form
β :=
∑
k ψk dhk ∈ A1(M); thus ω = dβ is exact, i.e., [ω] = 0. Hence [ω] 7→ [a] is injective,
and is therefore an isomorphism (or real vector spaces) between H2dR(M) and H
2(U,R).
The foregoing proof can be cast in a more algebraic framework, as follows. The abelian
groups (actually, vector spaces) Cpq := Cp(U,Aq) of q-form-valued Cˇech p-cochains are
related by two coboundary operators, δ : Cpq → Cp+1,q and d : Cpq → Cp,q+1, satisfying
δd− dδ = 0; if we introduce ∂ := (−1)pd on Cpq, we get δ∂+ ∂δ = 0, so we obtain a “double
complex”. By introducing D := δ + ∂ : Cpq → Cp+1,q ⊕ Cp,q+1, and Em := ⊕p+q=mCpq,
we obtain a new cochain complex (E•, D), sometimes called the Cˇech–de Rham complex.
(For instance, a 1-cochain for this complex is of the form α ⊕ f where α and f are as in
the proof of Proposition 5.5, and D(α ⊕ f) = dα ⊕ (δα − df) ⊕ δf , which evaluates to
ω ⊕ 0 ⊕ a. A general theorem [12, 17] asserts that the k-th cohomology groups for the
complexes (A•(M), d), (E•, D), and (C•(U,M), δ) are isomorphic, for any k ∈ N.
Definition 5.7. The standard inclusion ι : Z→ R induces an injection of Cˇech cohomology
groups ι∗ : Hˇ2(M,Z)→ Hˇ2(M,R). We regard Hˇ2(M,Z) as a subset of Hˇ2(M,R) by identi-
fying it with its image under ι∗. We say that a de Rham cohomology class [ω] ∈ H2dR(M) is
integral if the corresponding Cˇech class [a] lies in Hˇ2(M,Z).
Theorem 5.6. Let L−→M be a Hermitian line bundle over a compact manifold, and let ∇
be a connection on it, compatible with the metric, with curvature ω. Then [(2pii)−1ω] is an
integral de Rham class, corresponding to the class [L] of the line bundle in Hˇ2(M,Z).
Proof. Let U = {Uj} be a covering of M by chart domains, and let {sj} be a local system
of nonvanishing sections for the line bundle L−→M . Then sj ∈ Γ(Uj, L) for each j. By the
locality property of ∇, we have ∇sj ∈ A1(Uj, L) = Γ(Uj, L)⊗C∞(Uj)A1(Uj), so ∇sj = sj⊗αj
for some αj ∈ A1(Uj) (because sj generates Γ(Uj, L) as a C∞(Uj)-module). We may take sj
to be normalized with respect to the metric, i.e., (sj | sj) = 1 on Uj. Since ∇ is compatible,
we find that αj is a purely imaginary 1-form, because α¯j + αj = (∇sj | sj) + (sj | ∇sj) = 0.
By (5.8), ∇2sj = (sj ⊗ αj) ∧ αj + s ∧ dαj = s ∧ dαj, so that ω = dαj on Uj. Let us
write βj := (2pii)
−1αj; then the βj are real-valued 1-forms, satisfying dβj = (2pii)−1ω. Thus
[(2pii)−1ω] ∈ H2dR(M) is a real de Rham class.
The local sections sj are related by si = gijsj = sjgij on Ui ∩ Uj, where the gij are the
U(1)-valued transition functions of the line bundle (see Definition 1.14). Then
sj ⊗ gijαi = si ⊗ αi = ∇si = ∇(sjgij) = (sj ⊗ αj)gij + sj ⊗ dgij
50
on Ui ∩ Uj; since sj does not vanish there, we get
αi = g
−1
ij αjgij + g
−1
ij dgij = αj + g
−1
ij dgij on Ui ∩ Uj.
This gives
βi − βj = (2pii)−1g−1ij dgij = (2pii)−1 d(log gij)
for a suitable branch of the logarithm, and fij := (2pii)
−1 log gij is a real-valued function on
Ui ∩ Uj satisfying βi − βj = dfij. Now aijk := fij − fik + fjk gives the Cˇech 2-cocycle a ∈
C2(U,R) such that [a] corresponds to [(2pii)−1ω] under the isomorphism of Proposition 5.5.
On the other hand, the proof of Proposition 1.6 constructs this very same Cˇech class [a] as
the element of Hˇ2(M,Z) which corresponds to [L] under the Bockstein isomorphism between
Hˇ1(M,U(1)) and Hˇ2(M,Z). Hence, [(2pii)−1ω] is integral and corresponds to [L].
In particular, we see that a closed 2-form on M is the curvature of some compatible
connection only if it equals 2pii times an integral 2-form.8
5.6 Line bundles over CPm
Definition 5.8. Any element of CPm is a line through the origin in Cm+1, and two such
lines intersect only at the origin. The tautological line bundle L−→CPm is given by
taking the disjoint union of these lines; the fibre at any element of CPm is the very same
line. More explicitly, we can take L to be
L := { (`, v) ∈ CPm × Cm+1 : v ∈ ` }, (5.12)
and define pi(`, v) := `.
Here L−→CPm is manifestly a subbundle of a trivial vector bundle. Indeed, let `⊥ be
the subspace of Cm+1 orthogonal to ` (with respect to the usual inner product on Cm+1),
and if E := { (`, u) ∈ CPm × Cm+1 : u ∈ `⊥ }, then the Whitney sum L⊕ E = CPm × Cm+1
is trivial.
Exercise 5.9. If η : Cm+1\{0} → CPm is the quotient map, Cm+1\{0} η−→CPm is a principal
C×-bundle. Show that the tautological line bundle is associated to this principal bundle via
the representation ρ of C× on C given by the multiplication ρ(λ)µ := λµ.
Definition 5.9. The dual of the tautological line bundle on CPm is the hyperplane bundle
H −→CPm, where H` = `∗ (the one-dimensional space of linear functionals on `).
8This observation is the launching point for the theory of geometric quantization, which seeks to represent
certain functions on a symplectic manifold (M,Ω), i.e., a manifold equipped with a closed nondegenerate
2-form, by operators on a Hilbert space. The elements of this Hilbert space come from sections of a certain
Hermitian line bundle over M , equipped with a compatible connection whose curvature is (2pii~)−1Ω (where
~ is a positive constant, identified with Planck’s constant). Theorem 5.6 shows that such a line bundle exists
iff [~−1Ω] is integral; this is a discreteness condition on the original symplectic form Ω, hence the use of the
word “quantization”.
51
The tensor product bundles L⊗k−→CPm and H⊗l−→CPm, for k, l ∈ N, give more
examples of line bundles over CPm. (We take L⊗0 = H⊗0 := CPm × C by convention.) We
aim to show that these are distinct, and that any line bundle over CPm is equivalent to one
on this list. Since [L]−1 = [H] in the group of line bundle classes, and since H2dR(CP
m) = R,
we have Hˇ2(CPm,Z) ' Z, so it suffices to verify that [H] corresponds to a generator of the
infinite cyclic group Hˇ2(CPm,Z).
Definition 5.10. A complex vector bundle E−→M on a complex manifold M is a holo-
morphic vector bundle if its transition functions gij : Ui∩Uj → GL(m,C) are holomorphic.
The space of holomorphic sections of this line bundle is denoted by O(M,E) or simply by
O(E).
Exercise 5.10. Compute the transition functions for the tautological and hyperplane bundles
over CPm and hence show that these are holomorphic line bundles.
Consider first the trivial bundle CPm×C−→CPm; a holomorphic section of this bundle
is of the form s(x) = (x, f(x)), where f : CPm → C is a holomorphic function. If (U, φ) is
a chart with φ(U) = Cm, then f ◦ φ−1 : Cm → C is an entire holomorphic function on Cm,
which is bounded since CPm is compact and f is continuous; thus, by Liouville’s theorem,
f is constant. More generally, any holomorphic function on a compact complex manifold is
constant.
It turns out that the tautological line bundle has no global holomorphic sections, other
than the zero section. The hyperplane bundle H −→CPm, by contrast, has nontrivial global
holomorphic sections. For any f ∈ (Cm+1)∗, define sf ∈ Γ(CPm, H) by sf (`) := f
∣∣
`
. Con-
versely, any holomorphic section of H −→CPm is of the form s(`) := g∣∣
`
, where g : Cm+1 → C
is a holomorphic function whose restriction to each one-dimensional subspace is linear, i.e.,
g is homogeneous of degree one. Since only first-degree terms can then occur in the Taylor
series of g, the function g is itself linear, i.e., g ∈ (Cm+1)∗ and s = sg. Thus, f 7→ sf is a
linear bijection between (Cm+1)∗ and O(H).
In particular, O(H) is finite-dimensional, with dimO(H) = m + 1. A basis is given
by σ0, . . . , σm, where σj = szj and z
j ∈ (Cm+1)∗ be the j-th coordinate function, for j =
0, 1, . . . ,m.
The standard inner product 〈〈· | ·〉〉 on Cm+1 gives metrics on the line bundles L and H.
We may write (u | v)` := 〈〈u | v〉〉 for u, v ∈ L`, i.e., u, v ∈ ` ⊂ Cm+1. Now if φ, ψ ∈ `∗ are
given by φ(v) = 〈〈u | v〉〉, ψ(v) = 〈〈w | v〉〉 for v ∈ `, then
(φ | ψ)`∗ := 〈〈u | v〉〉〈〈w | v〉〉〈〈v | v〉〉 =
〈〈w | v〉〉〈〈v | u〉〉
〈〈v | v〉〉 ,
which is independent of any nonzero v ∈ `. When ` ∈ Uj, we have the equality
〈〈v | v〉〉 = z0z¯0 + · · ·+ zmz¯m = zj z¯j
(
1 +
∑
k 6=j
wkj w¯
k
j
)
= |zj(v)|2Qj(`),
52
and in particular
(σj | σj) = Q−1j =
(
1 +
∑
k 6=j
wkj w¯
k
j
)−1
on Uj. (5.13)
By continuity, (σj | σj) = 0 and thus σj vanishes on the complement of Uj.
5.7 Connections on the hyperplane bundle
Definition 5.11. When M is a complex manifold and ∇ is a connection on a holomorphic
vector bundle E−→M , we say that ∇ is compatible with the holomorphic structure if ∇s ∈
A1,0(E) whenever s ∈ O(E). If the vector bundle is also Hermitian, we say that ∇ is
a canonical connection if it is compatible with both the metric and the holomorphic
structure.
Proposition 5.7. On the hyperplane bundle H −→CPm, there is a unique canonical con-
nection.
Proof. Let {Uj, σj} be the local system of holomorphic sections of the previous Section.
Since σj is nonvanishing on Uj (e.g., on account of (5.13)), the argument of the proof of
Theorem 5.6 shows that any connection ∇ on the hyperplane bundle is given on each Uj by
∇σj = σj ⊗ αj, for some αj ∈ A1(Uj).
Thus ∇ is compatible with the holomorphic structure iff each αj lies in A1,0(Uj).
We may write αj = ajk dw
k
j with each ajk ∈ C∞(Uj), so that ∇ is also compatible with
the metric iff
Q−1j
(
ajk dw
k
j + a¯jk dw¯
k
j
)
= (σj | ∇σj) + (∇σj | σj) = d(σj | σj)
= d(Q−1j ) = −Q−2j d
(∑
k 6=j
wkj w¯
k
j
)
iff ajk = −Q−1j w¯kj for all k 6= j. Thus ∇ satisfies
∇σj = −σj ⊗Q−1j
∑
k 6=j
w¯kj dw
k
j ,
so this calculation establishes existence and uniqueness of the canonical connection.
Proposition 5.8. The curvature of the canonical connection on the hyperplane bundle is
−iΦ, where Φ is the Ka¨hler form on CPm.
Proof. The curvature ω satisfies ω = dαj on Uj, so
ω = dαj = Q
−1
j
∑
k 6=j
dwkj ∧ dw¯kj +Q−2j dQj ∧
∑
k 6=j
w¯kj dw
k
j
= Q−2j
(
Qj
∑
k 6=j
dwkj ∧ dw¯kj −
∑
r,s6=j
w¯rjw
s
j dw
r
j ∧ dw¯sj
)
= −iΦ,
by comparison with (2.7).
53
The element of H2dR(CP
m) which corresponds to the equivalence class of the hyperplane
bundle is therefore [−(2pi)−1Φ].
5.8 Characteristic classes
We have seen that the curvature of a Hermitian line bundles over M is a closed 2-form ω
on M and [(2pii)−1ω] is an integral cohomology class. For vector bundles of higher rank,
it is possible to obtain integral cohomology classes of even degree from the matrix-valued
curvature ω by taking suitable traces of its exterior powers. These classes are topological
invariants of the manifold M .
Definition 5.12. Let q˜ : (Cr×r)k → C be a symmetric k-linear map on the vector space
of complex r × r matrices. One says that q˜ is invariant (under the adjoint representation
Ad(g)A := gAg−1 of GL(r,C)) if
q˜(gA1g
−1, . . . , gAkg−1) = q˜(A1, . . . , Ak) (5.14)
for all g ∈ GL(r,C), A ∈ Cr×r.
The function q : Cr×r → C defined by q(A) := q˜(A,A, . . . , A) is called a homogeneous
polynomial of degree k; when q˜ is invariant, we call it an invariant polynomial on Cr×r.
With the “polarization formula” q˜(A1, . . . , Ak) := (k!)
−1∑
|J |=s(−1)k−sq(Aj1 + · · ·+Ajs), the
map q˜ may be recovered from q.
More generally, an invariant polynomial on Cr×r is a finite sum of homogeneous invariant
polynomials of various degrees. (A constant function is an invariant polynomial of degree
zero.)
For instance, in the expansion det(t−A) = ∑rk=0(−1)kqk(A)tr−k, each qk is an invariant
homogeneous polynomial; here qr(A) = detA, q1(A) = trA, and q2(A) =
∑
i<j λi(A)λj(A),
where the λj(A) are the eigenvalues of A. Notice that 2q2(A) = (trA)
2 − tr(A2).
Lemma 5.9. If q : Cr×r → C is an invariant polynomial on Cr×r, then for A1, . . . , Ak, B ∈
Cr×r we have
∑k
j=1 q˜(A1, . . . , [B,Aj], . . . , Ak) = 0.
Proof. Set g = etB in (5.14), and differentiate with respect to t at t = 0; the result follows
on noting that (d/dt)
∣∣
t=0
etBAe−tB = BA− AB = [B,A].
If Er = M×Cr, so that Er−→M is the trivial bundle of rank r, then q yields a polynomial
map from Γ(EndEr) = A
r×r = C∞(M,Cr×r) to A = C∞(M) by writing q(A)(x) := q(A(x));
and q˜ similarly defines an A-valued symmetric k-linear map on Ar×r. The invariance prop-
erty is q(gAg−1) := q(A), where g ∈ C∞(M,GL(r,C)). Since τ∗(v) := gv for v ∈ Ar = Γ(Er)
defines a bundle automorphism (τ, idM) of Er−→M , the invariance condition may be reex-
pressed as q ◦ Ad τ = q for every invertible bundle map τ : Er → Er.
Exercise 5.11. Verify that for any vector bundle E−→M of rank r, the recipe q(A)(x) :=
q(A(x)) yields a map q : Γ(EndE)→ A (and by polarization, a k-linear map q˜ : Γ(EndE)k →
A), such that q(τ∗ ◦ A ◦ τ−1∗ ) = q(A) for any invertible bundle map τ : E → E.
54
These may be extended to maps q, q˜ on A•(M,EndE) with values in A•(M) by defining
q˜(A1 ⊗ η1, . . . , Ak ⊗ ηk) := q˜(A1, . . . , Ak) η1 ∧ · · · ∧ ηk. (5.15)
The invariance property of Lemma 5.9 may be expressed in this context as
k∑
j=1
(−1)s0(s1+···+sj−1)q˜(ω1, . . . , [[β, ωj]], . . . , ωk) = 0 (5.16)
whenever β ∈ As0(M,EndE) and ωj ∈ Asj(M,EndE) for j = 1, . . . , k. The bracket on
A•(M,EndE) is defined as the linear extension of the recipe:
[[A⊗ η,B ⊗ ζ]] := [A,B] η ∧ ζ (5.17)
for A,B ∈ Γ(EndE), η, ζ ∈ A•(M).
Exercise 5.12. Check the invariance formula (5.16).
Exercise 5.13. If α ∈ Ak(M,EndEr), β ∈ Al(M,EndEr) are matrix-valued forms, deduce
from (5.17) that
[[α, β]] = α ∧ β − (−1)klβ ∧ α
in Ak+l(M,EndEr).
Exercise 5.14. By taking the exterior derivative of the right hand side of (5.15), show that
d(q˜(ω1, . . . , ωk)) =
k∑
j=1
(−1)s1+···+sj−1 q˜(ω1, . . . , dωj, . . . , ωk) (5.18)
when each ωj ∈ Asj(M,EndEr) is a matrix-valued form.
It is convenient to introduce the notation q′(ω; θ) :=
∑k
j=1 q˜(ω, . . . , θ, . . . , ω), where θ
appears in the j-th place and ω elsewhere; if ω has even degree, it follows from (5.16) that
q′(ω; [[β, ω]]) = 0 for any β ∈ A•(M,EndE). Also, if ω ∈ Aeven(M,EndEr) is a matrix-valued
form of even degree, then (5.18) gives d(q(ω)) = q′(ω; dω) ∈ Aodd(M,EndEr).
Proposition 5.10. Let ∇ be a connection on a vector bundle E−→M , with curvature
ω ∈ A2(M,EndE). If q is an invariant polynomial on Cr×r, then q(ω) is closed in Aeven(M).
Proof. To show that d(q(ω)) = 0 on M , it is enough to show that d(q(ω)) = 0 on each chart
domain U . Thus we can suppose that E−→M is a trivial bundle, that ∇ = d + α with
α ∈ A1(M,EndE) and that ω = dα + α ∧ α.
Now Ak(M,EndE) ' (Ak(M))r×r by the triviality of the bundle, and the exterior deriva-
tive d : A•(M,EndE)→ A•(M,EndE) is an antiderivation, so
dω = d(α ∧ α) = dα ∧ α− α ∧ dα = ω ∧ α− α ∧ ω = [[ω, α]], (5.19)
which is known as the Bianchi identity for the curvature. Now
d(q(ω)) = q′(ω; dω) = −q′(ω; [[α, ω]]) = 0
by the invariance of q.
55
Exercise 5.15. A connection ∇ on E−→M , with curvature ω, yields a connection on
the vector bundle EndE−→M , also denoted ∇, by adopting the Leibniz rule ∇(as) =
(∇a)s + a(∇s) as a definition, i.e., by setting (∇a)s := ∇(as) − a(∇s) for a ∈ Γ(EndE),
s ∈ Γ(E). Check that a 7→ ∇a is indeed a connection. This extends to a linear map
∇ : Ak(M,EndE)→ Ak+1(M,EndE) by (5.8). Use (5.8) and (5.9) to show that∇2β = ω∧β
for any β ∈ A1(M,E), and deduce that (∇ω)s = ∇(∇2s) −∇2(∇s) = 0 for any s ∈ Γ(E).
Finally, show that the equation ∇ω = 0 reduces to the Bianchi identity (5.19) locally,9 when
ω is of the form ω = dα + α ∧ α.
Proposition 5.11. Let ∇ be a connection on a vector bundle E−→M , with curvature ω,
and let q be an invariant polynomial. Then the cohomology class [q(ω)] is independent of ∇.
Proof. Let ∇0, ∇1 be two connections on E−→M . Then, by (5.5), ∇1 = ∇0 + β with
β ∈ A1(M,EndE). Set ∇t := (1− t)∇0 + t∇1 = ∇0 + tβ for 0 ≤ t ≤ 1; this is a connection
on E−→M , whose curvature we denote by ωt. The relation [q(ω0)] = [q(ω1)] follows from
the transgression formula:
q(ω1)− q(ω0) = d
(∫ 1
0
q′(ωt; β) dt
)
.
To verify this relation, it suffices to show that (d/dt)q(ωt) = d(q
′(ωt; β)) for 0 < t < 1.
By k-linearity of q˜, we have
d
dt
q(ωt) =
d
dt
q˜(ωt, . . . , ωt) =
k∑
j=1
q˜(ωt, . . . , (d/dt)ωt, . . . , ωt) = q
′(ωt; (d/dt)ωt).
To see that q′(ωt; (d/dt)ωt) = d(q′(ωt; β)), we may compare these two 2k-forms over any
chart domain U ⊂ M ; or equivalently, we may assume that the bundle E−→M is trivial,
that ∇t = d+ αt and ωt = dαt + αt ∧ αt. In that case, αt = α0 + tβ, and
d
dt
ωt =
d
dt
(
dα0 + t dβ + (α0 + tβ) ∧ (α0 + tβ)
)
= dβ + α0 ∧ β + β ∧ α0 + 2t(β ∧ β) = dβ + αt ∧ β + β ∧ αt
= dβ + [[αt, β]].
9The identity ∇ω = 0 for any connection is therefore called the Bianchi identity.
56
It remains to compute
d(q′(ωt; β)) =
k∑
j=1
d(q˜(ωt, . . . , β, . . . , ωt))
=
∑
i<j
q˜(ωt, . . . , dωt, . . . , β, . . . , ωt) + q
′(ωt; dβ)−
∑
i>j
q˜(ωt, . . . , β, . . . , dωt, . . . , ωt)
= −
∑
i<j
q˜(ωt, . . . , [[αt, ωt]], . . . , β, . . . , ωt) + q
′(ωt; dβ)
+
∑
i>j
q˜(ωt, . . . , β, . . . , [[αt, ωt]], . . . , ωt)
=
∑
j=1k
q˜(ωt, . . . , [[αt, β]], . . . , ωt) + q
′(ωt; dβ) = q′(ωt; dβ + [[αt, β]])
= q′(ωt; (d/dt)ωt),
(with dωt in the i-th position and β in the j-th position in the double summations). Here we
have used the Bianchi identity dωt + [[αt, ωt]] = 0 and the invariance property (5.16) applied
to the form q˜(ωt, . . . , dωt, . . . , β, . . . , ωt) and also to q˜(ωt, . . . , β, . . . , dωt, . . . , ωt).
If E ′−→M is another vector bundle equivalent to E−→M , and it τ : E → E ′ is an
invertible bundle map, then by Exercises 5.4 and 5.6, the recipes ∇′ := τ∗ ◦ ∇ ◦ τ−1∗ and
ω′ := τ∗◦ω◦τ−1∗ match connections and curvatures on both vector bundles; by the invariance
property (5.14) of the polynomial q, we have q(ω′) = q(ω) in A2k(M). Thus the cohomology
class [q(ω)] depends only on the equivalence class of the vector bundle E−→M , and we may
denote it by q([E]), or more simply by q(E) ∈ H2kdR(M)⊗R C.
If E−→M is a Hermitian vector bundle, we consider only bundle maps τ : E → E ′ which
preserve the fibre metrics. Thus we may use polynomials q which are only invariant under
the unitary group U(r) rather than GL(r,C); in (5.16) we may only use forms β with either
iβ or β itself real-valued. If a connection ∇ on E−→M is compatible with the metric, then
−iω is real-valued, and the cohomology class [q(−iω)] belongs to H2kdR(M).
5.9 Chern classes and the Chern character
Definition 5.13. Let E−→M be a Hermitian vector bundle, together with a compatible
connection ∇ whose curvature is ω ∈ A2(M,EndE). Define a U(r)-invariant polynomial on
Cr×r by
c(A) := det
(
1r − 1
2pii
A
)
,
which may be written as a sum of homogeneous polynomials
c(A) = 1 + c1(A) + c2(A) + · · ·+ cr(A) (5.20)
where c1(A) = (i/2pi) trA, cr(A) = (i/2pi)
r detA; and if {iλ1, . . . , iλr} are the eigenvalues
of A, then ck(A) = (−1/2pi)k
∑
λj1λj2 . . . λjk ; the invariant homogeneous polynomials ck are
real-valued on the Lie algebra u(r) = iRr×r of the unitary group U(r).
57
The class ck(E) ∈ H2kdR(M) is called the k-th Chern class, and c(E) ∈ HevendR (M) is
called the total Chern class of the vector bundle.
Exercise 5.16. If E∗−→M is the dual vector bundle to E−→M , and if ∇∗ is the dual
connection to ∇ (see Exercise 5.3), show that ∇∗ has curvature −ωt ∈ A2(M,End(E∗)), and
conclude that ck(E
∗) = (−1)kck(E).
Exercise 5.17. If φ : N →M is a smooth map, and if ∇ is a connection on a Hermitian vector
bundle E−→M with curvature ω, find a connection ∇′ on the pullback bundle φ∗E−→N
whose curvature is φ∗ω ∈ A2(N,End(φ∗E)). Conclude that ck(φ∗E) = φ∗ck(E) ∈ H2kdR(N).
For Hermitian line bundles (r = 1), the total Chern class is just c(L) := 1 + c1(L). From
Theorem 5.6, we know that c1(L) is an integral cohomology class, so c(L) is also integral.
This integrality property holds for Chern classes of any Hermitian vector bundle. Indeed,
if E−→M and E ′−→M are two Hermitian vector bundles with compatible connections ∇
and ∇′ and respective curvatures ω and ω′, then ∇⊕∇′ is a connection on E ⊕ E ′−→M ,
with curvature ω ⊕ ω′ ∈ A2(M,End(E ⊕ E ′)). Clearly
c(ω ⊕ ω′) = det
(
1r − ω
2pii
)
∧ det
(
1r′ − ω
′
2pii
)
= c(ω) ∧ c(ω′), (5.21)
on account of (5.15). Now the wedge product of closed forms induces a product of cohomology
classes, since (η+ dζ)∧ (η′+ dζ ′) = η ∧ η′+ d(η ∧ ζ ′+ ζ ∧ (η′+ dζ)) if dη = 0 and dη′ = 0, so
[η ∧ η′] is not affected by adding an exact form to either η or η′; in other words, the recipe
[η] [η′] := [η ∧ η′] is a well-defined product10 making H•dR(M) into a ring. The even-degree
classes form a commutative subring HevendR (M). On passing to cohomology, (5.20) becomes
c(E ⊕ E ′) = c(E) c(E ′) in HevendR (M). (5.22)
The integral 2-forms generate an integral subring Heven(M,Z). By (5.21), c(E) is integral,
i.e., lies in Heven(M,Z), whenever E−→M is a Whitney sum of Hermitian line bundles.
An important splitting principle (see [12] for a proof) asserts that some pullback bundle
φ∗E−→N can be split (into a sum of line bundles) in such a way that c(E) 7→ φ∗(c(E)) =
c(φ∗E) is injective, and therefore c(E) is integral since c(φ∗E) is.
Definition 5.14. The Chern character ch(E) of the Hermitian vector bundle E−→M of
rank r is given by the invariant power series
ch(A) := tr
(
exp((2pi)−1i A)
)
= 1 + ch1(A) + ch2(A) + · · · , (5.23)
or equivalently by the invariant polynomial obtained by discarding the terms chk with 2k >
dimM , since ω∧k ∈ A2k(M,EndE) and hence chk(ω) = 0 for 2k > dimM . This polynomial
is found explicitly by writing the eigenvalues of A as {2piiµ1, . . . , 2piiµr}, expanding the
function e−µ1 + · · ·+ e−µr in a Taylor series, and discarding high-degree terms.
10This product is usually called the cup product in de Rham cohomology.
58
Exercise 5.18. Check that ch1(E) = c1(E), ch2(E) =
1
2
(c1(E)
2 − 2c2(E)), and ch3(E) =
1
6
(c1(E)
3 − 3c1(E)c2(E) + c3(E)).
In general, the chk(E) are polynomial combinations of the Chern classes cj(E) with ratio-
nal coefficients, on account of the 1/k! terms in the Taylor series expansion; thus they might
not be integral classes, but rational classes, i.e., elements of Heven(M,Q) = Heven(M,Z)⊗ZQ.
Proposition 5.12. The Chern character satisfies the homomorphism properties:
ch(E ⊕ E ′) = ch(E) + ch(E ′),
ch(E ⊗ E ′) = ch(E) ch(E ′).
Proof. The invariant power series ch(A) of (5.22) satisfies ch(A⊕A′) = ch(A) + ch(A′) and
ch(A⊗ 1 + 1⊗ A′) = ch(A) ch(A′); where ⊕ and ⊗ denote the usual direct sum and tensor
product of matrices. These identities are simple to check for diagonal matrices, therefore
hold for diagonalizable matrices by invariance, and hence hold generally, since the set of
diagonalizable r × r matrices is dense in Cr×r.
Given connections ∇, ∇′ with curvatures ω, ω′ on the respective vector bundles E−→M
and E ′−→M , the curvatures of ∇⊕∇′ and ∇⊗∇′ are ω ⊕ ω′ ∈ A2(M,End(E ⊕E ′)) and
ω ⊗ 1 + 1⊗ ω′ ∈ A2(M,End(E ⊗ E ′)). On replacing A, A′ by ω, ω′ in the foregoing matrix
identities, bearing in mind (5.15), and on passing to cohomology, one obtains the desired
formulae (5.23) for ch(E ⊕ E ′) and ch(E ⊗ E ′).
Exercise 5.19. Explain how the tensor product connection ∇⊗∇′ is defined, and check the
given formula for its curvature.
A trivial bundle Er = M × Cr−→M has a “flat” connection (i.e., a connection with
zero curvature) namely d, and thus ch(Er) = 1. Thus, if two vector bundles E−→M and
F −→M are stably equivalent, then ch(E)−ch(F ) is an integral multiple of 1 in Heven(M,Q).
If one defines the “reduced K-theory” K˜0(M) of M as the quotient of K0(M) by Z (on
identifying r ∈ N with [[Er]] ∈ K0(M)), the tensor product of vector bundles makes K˜0(M) a
commutative ring, and thus the Chern character defines a ring homomorphism ch: K˜0(M)→
Heven(M,Q).
For Hermitian line bundles L−→M and L′−→M , Proposition 5.12 yields the identity
c1(L⊗ L′) = c1(L) + c1(L′), (5.24)
by extracting the component in H2dR(M) from the formula ch(L⊗ L′) = ch(L) ch(L), using
ch1(L) = c1(L). Thus c1 determines a homomorphism from the group of line bundle classes
to the additive group H2(M,Z) of integral de Rham classes. Since c1(L) = [(i/2pi)ω] when ω
is the curvature of a compatible connection on L−→M , and c1(L∗) = −c1(L), we conclude
that [L] 7→ c1(L∗) is the isomorphism described in Theorem 5.6.
To show that [L] 6= [L′], it is enough to show that c1(L) 6= c1(L′). Moreover, if dimM =
2m, then c1(L)
m = [(i/2pi)nω∧m] is an integral 2m-form, and so
∫
M
(i/2pi)nω∧m ∈ Z, since
the identification of H2mdR (M) with R is precisely the map [ν] 7→
∫
M
ν.
59
5.10 Classification of line bundles over CPm
Proposition 5.13. The group of classes of line bundles over CPm is an infinite cyclic group,
generated by the class [H] of the hyperplane bundle.
Proof. Since we already know that Hˇ2(CPm,Z) ≡ Z, we need only check that [H] is a
generator. Equivalently, we must check that c1(H) = [(2pi)
−1Φ] is a generator for H2(M,Z).
For this, it is enough to show that
∫
CPm(2pi)
−mΦ∧m = 1.
Since the complement of the chart domain U0 is a lower-dimensional submanifold (diffeo-
morphic to CPm−1), we need only show that the integral over U0 equals 1. We may therefore
use the formula (2.7) (with j = 0) for the Ka¨hler form Φ. Then∫
U0
Φ∧m =
∫
Cm
(
iQ−10
m∑
k=1
dwk0 ∧ dw¯k0 − iQ−20
m∑
k,l=1
w¯k0w
l
0 dw
k
0 ∧ dw¯l0
)∧m
. (5.25)
At the point (r, 0, . . . , 0) ∈ Cm, Q0 simplifies to 1 + r2, and the integrand on the right hand
side of (5.24) becomes(
i(1 + r2)−2 dw10 ∧ dw¯10 + i(1 + r2)−1
m∑
k=2
dwk0 ∧ dw¯k0
)∧m
=
imm!
(1 + r2)m+1
m∧
k=1
dwk0 ∧ dw¯k0 =
2mm!
(1 + r2)m+1
m∧
k=1
dxk ∧ dyk
=
2mm!
(1 + r2)m+1
λ, (5.26)
where λ is Lebesgue measure on Cm, and wk0 = xk + iyk give Cartesian coordinates on Cm.
Interpreting r as a polar coordinate on Cm, one can write λ = r2m−1 dr d2m−1θ, with θ ∈
S2m−1 being the angular part. The right hand side of (5.25) is invariant under the unitary
group U(m), as is the Ka¨hler form, so it represents the integrand at all points, not just at
(r, 0, . . . , 0).
The volume of the sphere S2m−1 is Ω2m = 2 pim/(m− 1)! (see [1, 8, 28], for instance), so
the desired integral is∫
CPm
(2pi)−mΦ∧m =
m!
pim
∫
Cm
r2m−1
(1 + r2)m+1
dr d2m−1θ
=
m!
pim
Ω2m
∫ ∞
0
r2m−1
(1 + r2)m+1
dr =
∫ ∞
0
2mr2m−1
(1 + r2)m+1
dr
=
∫ ∞
0
mtm−1
(1 + t)m+1
dt =
∫ 1
0
mum−1 du = 1,
on substituting t = r2, u = t/(1 + t).
Corollary 5.14. The hyperplane bundle is not trivial, since c1(H) 6= 0.
60
This completes the classification of Hermitian line bundles over CPm, since any line bundle
L′−→CPm satisfies c1(L′) = k c1(H) for some k ∈ Z; thus L′ ∼ H⊗k if k > 0, L′ ∼ L⊗(−k)
if k < 0, and L′ is trivial iff k = 0. Furthermore, k is precisely the integral over CPm of
(−2pii)−mω∧mL′ where ωL′ is the curvature of any compatible connection on L′−→CPm.
5.11 The Levi-Civita connection on the tangent bundle
Definition 5.15. LetM be a Riemannian manifold, and let∇ be a connection on the tangent
bundle TM −→M . The fundamental 1-form θ is the unique element of A1(M,TM) sat-
isfying ιXθ = X for all X ∈ X(M) = Γ(TM). The torsion of ∇ is T := ∇θ ∈ A2(M,TM).
Exercise 5.20. Show that the contraction map ιX : A
1(M,E) → Γ(E) of Definition 5.4 ex-
tends to an A-linear map ιX : A
k(M,E)→ Ak−1(E) such that ιX(ζ∧η) = (ιXζ)∧η+(−1)kζ∧
ιXη for ζ ∈ Ak(M,E), η ∈ A•(M), provided we define ιXs := 0 for s ∈ Γ(E) = A0(M,E).
We can then define T (X, Y ) := ιY (ιXT ) ∈ X(M).
Lemma 5.15. If X, Y ∈ X(M), then T (X, Y ) = ∇XY −∇YX − [X, Y ].
Proof. It is not hard to show that∇Xζ = ∇(ιXζ)+ιX(∇ζ) and∇X(ιY ζ) = ιY (∇Xζ)+ι[X,Y ]ζ
for ζ ∈ A1(M,TM). For the particular case ζ = θ, these identities give
T (X, Y ) = ιY (ιX∇θ) = ιY (∇Xθ −∇(ιXθ)) = ιY (∇Xθ −∇X)
= ∇X(ιY θ)− ι[X,Y ]θ −∇YX = ∇XY − [X, Y ]−∇YX, (5.27)
as claimed.
Exercise 5.21. Verify the aforementioned formulae for ∇Xζ and ∇X(ιY ζ) by applying (5.8)
and (5.9) (and their analogues for ιX) in the case ζ = Z ⊗ α with Z ∈ X(M), α ∈ A1(M).
Proposition 5.16. If M is a Riemannian manifold, there is a unique connection ∇ on the
tangent bundle TM −→M , which is compatible with the Euclidean metric on TM and is
torsion-free.
Proof. Compatibility with the metric demands that (∇X | Y ) + (X | ∇Y ) = d(X | Y ), as in
(5.7), where (X | Y ) = g(X, Y ) denotes the bilinear pairing on X(M) = Γ(TM) determined
by the Euclidean metric g. By Lemma 5.15, ∇ is torsion-free iff ∇XY −∇YX = [X, Y ] for
X, Y ∈ X(M).
Thus Z(X |Y ) = ιZ(∇X |Y )+ ιZ(X |∇Y ) = (∇ZX |Y )+(X |∇ZY ). A short calculation
then shows that
2(∇ZX | Y ) = X(Y | Z)− Y (Z |X) + Z(X | Y )
+ (X | [Y, Z]) + (Y | [Z,X])− (Z | [X, Y ]), (5.28)
which establishes the uniqueness of ∇. On the other hand, it is easy to show that the right
hand side is A-linear in Y and Z, hence is of the form ιZ(DX | Y ), where D : X(M) →
A1(M,TM). By replacing X by hX (with h ∈ A) on the right hand side of (5.27) and then
simplifying, one verifies the Leibniz rule for D, which proves the existence of the desired
connection.
61
Definition 5.16. The unique metric-compatible torsion-free connection on the tangent bun-
dle of a Riemannian manifold M is called its Levi-Civita connection. Its curvature, in
A2(M,TM), is usually denoted by R, and is also called the Riemannian curvature ten-
sor11 of M .
Exercise 5.22. Verify the following property of the Riemannian curvature tensor:
R(X, Y )Z +R(Y, Z)X +R(Z,X)Y = 0 for X, Y, Z ∈ X(M), (5.29)
using only Lemma 5.4, Lemma 5.15, and the Jacobi identity.
Exercise 5.23. Verify the following symmetry properties of the Riemannian curvature tensor:
(W |R(X, Y )Z) = −(R(X, Y )W | Z) = (X |R(W,Z)Y ),
for W,X, Y, Z ∈ X(M).
6 Clifford algebras
A Clifford algebra is an associative algebra which is generated by starting with a real vector
space and defining a product of vectors in such a way that the square of any vector is a scalar.
This can be done consistently if the generating vector space is Euclidean, i.e., if it carries a
symmetric bilinear form. For the zero bilinear form, the corresponding algebra is just the
exterior algebra on the given vector space; otherwise, it has the same underlying vector space
as the exterior algebra, but with a modified product operation. The Clifford algebra has
interesting matrix representations, whose representation spaces are called “Clifford modules”.
All of these spaces are graded into an “even” part and an “odd” part, so we begin with a
general discussion of vector spaces and algebras which are Z2-graded.
6.1 Superspaces and superalgebras
Definition 6.1. A superspace is just a vector space with a given Z2-grading: V = V +⊕V −;
here V + is called the even subspace and V − is called the odd subspace.1
A superalgebra is an algebra whose underlying vector space is a superspace: A =
A+ ⊕ A−, where the product respects the grading,2 i.e., A+ · A+ ⊆ A+, A− · A− ⊆ A+,
A+ · A− ⊆ A−, and A− · A+ ⊆ A−.
11Since A2(M,TM) = X(M) ⊗A A2(M), the map (X,Y, α) 7→ α(R(X,Y )) is a tensor of bidegree (2, 1)
on M .
1The unfortunate prefix “super”, which is simply a synonym for “Z2-graded”, was introduced by Feliks
Berezin [7] about 30 years ago, and has since become fashionable. Berezin wished to extend the calculus of
Gaussian integrals by regarding an exterior algebra as a “space of functions of anticommuting variables”.
This crazy idea works astonishingly well.
2If we regard the exponents + and − as the elements of the additive group Z2, these four inclusions may
be collected as the formula Ai · Aj ⊆ Ai+j . In this way we can define a G-graded algebra for any abelian
group G, although only the cases G = Z2 and G = Z are commonly used.
62
Definition 6.2. The exterior algebra Λ•V of a vector space V is a Z-graded algebra,
whose subspace of degree k is Λk(V ) for k = 0, 1, . . . , dimV (and for k < 0 or k > dimV ,
one sets Λk(V ) := {0}), since Λk(V ) ∧ Λl(V ) ⊆ Λk+l(V ) for k, l ∈ Z.
But Λ•V is also a superalgebra, since we may define
Λ+(V ) :=
⊕
k even
Λk(V ), Λ−(V ) :=
⊕
k odd
Λk(V ).
Indeed, any Z-graded algebra becomes a superalgebra, by collecting the subspaces of even
degree and of odd degree in this manner.
Definition 6.3. If V = V + ⊕ V − is a superspace, then EndV is a superalgebra, with
End+ V := End(V +)⊕ End(V −),
End− V := Hom(V +, V −)⊕ Hom(V −, V +),
Exercise 6.1. Guess the definition of a supermodule. Any superspace V is a supermodule
for the superalgebra EndV .
Definition 6.4. We say an element a of a superalgebra A is homogeneous if either a ∈ A+
or a ∈ A−; its parity #a is defined as #a := 0 if a ∈ A+, #a := 1 if a ∈ A−. Analogously,
in a Z-graded algebra, we define the degree of a homogeneous element as #a := k if a ∈ Ak.
There is an important, if somewhat informal, sign rule in superalgebra, which says that
in any calculation in which the order of multiplication of homogeneous elements is reversed
(i.e., ab is changed to ba), a sign factor of (−1)#a#b must be inserted. Thus, for example, we
say that a superalgebra is “supercommutative” if ab = (−1)#a#bba for homogeneous elements
a, b ∈ A; in other words, even elements commute with both even and odd elements, but two
odd elements anticommute. Notice that the exterior algebra Λ•V is supercommutative.
Definition 6.5. The superbracket in a superalgebra is the bilinear operation A×A→ A
defined, for a, b homogeneous, by
[[a, b]] := ab− (−1)#a#bba.
A superalgebra is supercommutative iff all supercommutators [[a, b]] vanish, i.e., if the
superbracket is trivial. The superbracket satisfies the properties:
[[a, b]] + (−1)#a#b[[b, a]] = 0,
[[a, [[b, c]]]] = [[[[a, b]], c]] + (−1)#a#b[[b, [[a, c]]]]. (6.1)
A vector space with a bilinear operation (of any kind) which satisfies (6.1) is called a Lie
superalgebra.
Several bracket notations are in general use. Usually one writes [a, b] = ab − ba, and
anticommutators ab+ ba are denoted {a, b} or sometimes [a, b]+; thus [[a, b]] = [a, b] if either
a or b is even, and [[a, b]] = {a, b} if both a and b are odd. Warning : many superalgebraists
use [a, b] to denote a supercommutator, even when a and b are odd.
63
Definition 6.6. A supertrace on a superalgebra A is a linear form τ which vanishes on
supercommutators, i.e., τ([[a, b]]) = 0 for all a, b ∈ A.
When A = EndE for E a superspace, we may write a ∈ A+ and b ∈ A− as
a =
(
a+ 0
0 a−
)
, b =
(
0 b−
b+ 0
)
, (6.2)
and we define a supertrace on EndE by
Str(a+ b) := Tr(a+)− Tr(a−). (6.3)
Exercise 6.2. Write the supercommutator [[a, b]] for a, b homogeneous elements of EndE
in the matrix notation (6.2) —there are four cases— and thus verify that (6.3) defines a
supertrace.
The tensor product U ⊗V of two superspaces U and V is a superspace, with the grading:
(U ⊗ V )+ := (U+ ⊗ V +)⊕ (U− ⊗ V −),
(U ⊗ V )− := (U+ ⊗ V −)⊕ (U− ⊗ V +). (6.4)
The tensor product of two superalgebras A and B, with the standard product recipe (a ⊗
b)(a′ ⊗ b′) := aa′ ⊗ bb′ is not, however, a superalgebra, since this product does not respect
the grading (6.4). However, one defines a graded tensor product, denoted A ⊗ B, whose
underlying superspace is A⊗B, by using the multiplication rule:
(a⊗ b)(a′ ⊗ b′) := (−1)#a′#baa′ ⊗ bb′ (6.5)
(for a, a′, b, b′ homogeneous) in view of the passage of a′ to the left of b.
Exercise 6.3. Verify that the product (6.5) is compatible with the grading (6.4) on the
superspace A⊗B.
Exercise 6.4. If U and V are (ungraded) vector spaces, show that Λ•(U ⊕ V ) ' Λ•U ⊗ Λ•V
as superalgebras.
Exercise 6.5. If U and V are superspaces, define an action of EndU ⊗ EndV on the super-
space U ⊗ V in such a way that EndU ⊗ EndV ' End(U ⊗ V ) as superalgebras.
Exercise 6.6. Let A be a supercommutative superalgebra, and let V be a superspace. Es-
tablish the following identity for homogeneous elements of A ⊗ EndV :
[[a⊗ T, b⊗ S]] = (−1)#T #bab⊗ [[T, S]].
Show that the linear map StrA : A ⊗ EndV → A given by StrA(a ⊗ T ) := a Str(T ) is an
A-valued supertrace, in the sense that it vanishes on supercommutators.
Definition 6.7. A superbundle over a smooth manifold M is a vector bundle E−→M
which is a Whitney sum of two vector bundles: E = E+ ⊕ E−. The fibres Ex = E+x ⊕ E−x
are superspaces.
64
The space of sections Γ(E) is a Z2-graded C∞(M)-module: Γ(E) = Γ(E+) ⊕ Γ(E−).
The E-valued forms on M may also be graded by degree, and so, in accordance with (6.4),
A•(M,E) carries the total Z2-grading:
A+(M,E) := Aeven(M,E+)⊕Aodd(M,E−),
A−(M,E) := Aeven(M,E−)⊕Aodd(M,E+).
6.2 Clifford algebras
In this section we treat briefly the algebraic theory of Clifford algebras over Euclidean vector
spaces. The presentation follows the treatment in [9], and the appendix to [54]. For a more
detailed exposition of the algebraic theory, see [27] or [39]. All these rely on the fundamental
paper of Atiyah, Bott and Shapiro [4].
Definition 6.8. Let V be a real vector space, with dimR V = n, and q a positive definite
symmetric bilinear form on V ; we shall call the pair (V, q) a Euclidean vector space.3 The
Clifford algebra C`(V ) = C`(V, q) is an associative algebra generated by the elements
of V subject (only) to the relation v2 = −q(v, v) 1. More precisely, one defines C`(V, q) :=
T(V )/I(q), where T(V ) is the tensor algebra over V and I(q) is the ideal generated by
{ v⊗ v+ q(v, v) 1 : v ∈ E }. The canonical mapping of V into C`(V, q) is injective, so V may
be regarded as a subspace of C`(V, q); we then have the fundamental relation
uv + vu = −2 q(u, v) for u, v ∈ V. (6.6)
Exercise 6.7. Check that (6.6) is a consequence of v2 = −q(v, v).
Proposition 6.1. The algebra C`(V, q) satisfies the following universal property: if A is a
real algebra with identity and f : V → A is a linear mapping such that f(v)2 = −q(v, v) 1,
then there is a unique algebra homomorphism f˜ : C`(V, q)→ A such that f = f˜ ∣∣
V
.
Exercise 6.8. Give the proof of Proposition 6.1.
A Clifford algebra is Z2-graded, as follows. By taking f(v) := −v in the previous propo-
sition, with A = C`(V, q), we see that there is a unique automorphism α of C`(V, q) ex-
tending f , such that α2 = id. (From the definition, each element of C`(V, q) is a linear
combination of products v1v2 . . . vk with v1, . . . , vk ∈ V ; clearly we have α(v1v2 . . . vk) =
(−1)k v1v2 . . . vk.) We write C`±(V, q) to denote the (±1)-eigenspaces of α. Thus C`(V, q) is
a superalgebra. Notice that V ⊂ C`−(V, q).
Now take A to be the opposite algebra of C`(V, q) (the same vector space with the
product reversed), and f(v) := v; we get an involutive antiautomorphism of C`(V, q) given
by (v1 . . . vk)
! := vk . . . v1. We define another antiautomorphism a 7→ a¯, called conjugation,
by a¯ := α(a)! = α(a!).
Many properties of the Clifford algebra come from the following simple proposition, due
to Chevalley [16].
3A Clifford algebra can be defined if q is a symmetric bilinear form of any signature. However, we shall
use only forms which are either positive definite or (occasionally) negative definite.
65
Proposition 6.2. Let (V, q) and (W, r) be two Euclidean vector spaces, and let q ⊕ r be the
symmetric bilinear form on V ⊕W given by (q⊕ r)(v1 +w1, v2 +w2) := q(v1, v2) + r(w1, w2).
Then there is an isomorphism of superalgebras C`(V ⊕W, q ⊕ r) ' C`(V, q) ⊗ C`(W, r).
Proof. Define f : V ⊕ W → C`(V, q) ⊗ C`(W, r) by f(v + w) := v ⊗ 1 + 1 ⊗ w. Using
(6.5), we see that f(v + w)2 = −(q ⊕ r)(v + w, v + w) 1 ⊗ 1; therefore f extends to an
algebra homomorphism f˜ : C`(V ⊕W, q ⊕ r) → C`(V, q) ⊗ C`(W, r). Since the elements
v ⊗ 1 + 1 ⊗ w generate C`(V, q) ⊗ C`(W, r) as an algebra, f˜ is surjective; to see that it is
injective, it suffices to compute f˜ on a basis for C`(V ⊕W, q ⊕ r) generated by bases for V
and W .
If e1, . . . , en is an orthonormal basis of (V, q), write eJ := ej1 . . . ejk for J = {j1, . . . , jk} ⊆
{1, 2, . . . , n} with 1 ≤ j1 < · · · < jk ≤ n; and let e∅ := 1. Then { eJ : J ⊆ {1, 2, . . . , n} } is a
basis of C`(V, q). In particular, dim C`(V, q) = 2n and dim C`+(V, q) = dim C`−(V, q) = 2n−1.
Exercise 6.9. Use the previous Proposition to verify this basis, by induction on n.
Suppose V = Rn and that qn is the standard Euclidean inner product on Rn; in that
case, we abbreviate C`(n) := C`(Rn, qn). Since C`(1) = span{1, e1} with e21 = −1, we
have C`(1) ' C (as real algebras); with C`+(1) = R, C`−(1) = iR. Next, C`(2) =
span{1, e1, e2, e1e2} ' H, the algebra of quaternions. In both these cases, a 7→ a¯ denotes the
usual conjugation.
Since C`(R,−q1) = span{1, eˆ1} with eˆ21 = +1, we have C`(R,−q1) ' R⊕ R, so C`(V, q)
depends on the signature of the form q. Also, C`(R2,−q2) ' R2×2, by taking eˆ1 =
(
1 0
0 −1
)
,
eˆ2 =
(
0 1
1 0
)
. A complete list of the algebras C`(Rn,±qn) is given in [4], and reproduced
in [27, 39]. A basic fact is what is called Bott periodicity : C`(n + 8) ' C`(n) ⊗ C`(8)
(ungraded tensor product). Since C`(8) ' R16×16 is a real matrix algebra, this means that
C`(n+ 8) ' C`(n)16×16, so one only need determine C`(n) for n = 1, 2, . . . , 8.
Exercise 6.10. Show that C`(3) ' H ⊕ H. The remaining cases are C`(4) ' H2×2, C`(5) '
C4×4, C`(6) ' R8×8, C`(7) ' R8×8 ⊕ R8×8.
6.3 Clifford actions
Definition 6.9. A Clifford module for the Clifford algebra C`(V, q) is a superspace F =
F+ ⊕ F− together with an even homomorphism (called a Clifford action) c : C`(V, q) →
EndF . In other words, c(a)F± ⊆ F± if a ∈ C`+(V, q), and c(b)F± ⊆ F∓ if b ∈ C`−(V, q).
Suppose that F has a (Euclidean or hermitian) inner product (·|·), and denote the adjoint
of A ∈ EndF by A†, that is, (x |A†y) := (Ax |y)) for x, y ∈ F . We say that F is a selfadjoint
Clifford module if c(a)† = c(a¯) for all a ∈ A. Notice that is enough to see that c(v)† = −c(v)
for all v ∈ V , i.e., that each c(v) is skewadjoint.
Write EndC`(V,q) F := {R ∈ EndF : [[c(a), R]] = 0, for a ∈ C`(V, q) } to denote the
subalgebra of EndF consisting of operators which supercommute with the Clifford action.
Exercise 6.11. Check that [[RS, c(v)]] = R[[S, c(v)]]+(−1)#S[[R, c(v)]]S for R, S homogeneous
elements of EndF , and conclude that EndC`(V,q) F is indeed an algebra.
66
Definition 6.10. Let (V, q) be a finite-dimensional real vector space with a nondegenerate
bilinear form.4 The dual space V ∗ of R-linear forms on V is a real vector space of the same
dimension; we may define the so-called musical isomorphisms [8], namely [ : V → V ∗ and
] : V ∗ → V , by
u[(v) := q(u, v), q(λ], v) := λ(v).
They are mutually inverse: (u[)] = u, (λ])[ = λ.
If W is a complex vector space with a hermitian inner product 〈〈· | ·〉〉 (or a Hilbert
space,5 not necessarily finite-dimensional), we define [ : W → W ∗ and ] : W ∗ → W similarly
by u[(v) := 〈〈u | v〉〉, 〈〈λ] | v〉〉 := λ(v). In the complex case, the musical isomorphisms are
antilinear.
Definition 6.11. Let (V, q) be an Euclidean vector space, and let Λ•V be its exterior algebra.
For v ∈ V , define the exterior multiplication (v) and the contraction ι(v[) in End(Λ•V )
by
(v)(v1 ∧ · · · ∧ vk) := v ∧ v1 ∧ · · · ∧ vk,
ι(v[)(v1 ∧ · · · ∧ vk) :=
k∑
j−1
(−1)j−1q(v, vj)v1 ∧ · ·
j
∨· · ∧ vk. (6.7)
(The notation
j∨ indicates that the term with index j is missing from the exterior product.)
For k = 0, we define (v)1 := v, ι(v[)1 := 0. Note that ι(v[) is a graded derivation:6
ι(v[)(α ∧ β) = ι(v[)α ∧ β + (−1)#αα ∧ ι(v[)β for α, β ∈ Λ•V.
On Λ•V we use the real inner product
(u1 ∧ · · · ∧ um, v1 ∧ · · · ∧ vn) := δmn det
[
q(uk, vl)
]
It is easily seen that (v)† = ι(v[) for v ∈ V . Define
c(v)α := (v)α− ι(v[)α, for α ∈ Λ•V. (6.8)
Exercise 6.12. Check that (v)† = ι(v[) and that
(u)ι(v[) + ι(v[)(u) = q(v, u).
Conclude that c(v)2 = −q(v, v) 1 for v ∈ V .
By Proposition 6.1, c extends to a Clifford action of C`(V, q) on Λ•V . Since c(v)† = −c(v),
this action is selfadjoint.
4This definition applies to both symmetric and antisymmetric bilinear forms.
5In the infinite-dimensional case, the bijectivity of the musical isomorphisms is a restatement of the Riesz
theorem.
6It could have been defined as the unique graded derivation that takes u to q(v, u).
67
Definition 6.12. The R-linear map σ : C`(V, q)→ Λ•V given by
σ(a) := c(a) 1 (6.9)
for a ∈ C`(V, q), is a vector-space isomorphism (but not an algebra isomorphism) between
the Clifford algebra and the exterior algebra, called [9] the symbol map. (It is easy to see
that σ is surjective; since C`(V, q) and Λ•V have the same dimension 2n, it is also injective.)
Note that σ(1) = 1 and σ(v) = v for v ∈ V .
The inverse isomorphism Q : Λ•V → C`(V, q) may be called [9] a quantization map,
since it maps a supercommutative algebra to an algebra which is no longer supercommuta-
tive.7
Exercise 6.13. Show that σ(uv) = u∧v−q(u, v) 1, so that Q(u∧v) = uv+q(u, v) 1. Establish
the general formula:
Q(v1 ∧ · · · ∧ vk) = 1
k!
∑
τ∈Sk
(−1)τ vτ(1) . . . vτ(k), (6.10)
where (−1)τ denotes the sign of the permutation τ . In particular, Q(ej1∧· · ·∧ejk) = ej1 . . . ejk
if {e1, . . . , en} is an orthonormal basis for V and j1 < · · · < jk.
Exercise 6.14. The map Q does not preserve the Z-grading of Λ•V , but does preserve its
filtration. A filtration of an algebra A is a system of subspaces Aj with Aj ⊆ Aj+1 and
AiAj ⊆ Ai+j. For A = C`(V, q), take Aj to be the subspace generated by products at most
j vectors in V . There is an associated graded algebra, namely grA :=
⊕
j Aj/Aj−1 and a
canonical “symbol map” σ : A → grA. Check that for A = C`(V, q), the associated graded
algebra is none other than Λ•V with the symbol map given by (6.9).
6.4 Complex Clifford algebras
Definition 6.13. The complexification C`(V, q) := C`(V, q) ⊗R C can be regarded as the
Clifford algebra over C of the complexified vector space VC with the same quadratic form
q amplified to VC: any vector in VC can be written uniquely as w = u + iv with u, v ∈ V ,
and one defines q(w,w′) := q(u, u′)− q(v, v′) + iq(u, v′)− iq(v, u′); notice that the amplified
bilinear form q is not positive definite on VC. On VC all nondegenerate symmetric bilinear
forms are equivalent, so we may abbreviate C`(V, q) to C`(V ); in particular, C`(V, q) '
C`(n) := C`(Cn, qn) if dimR V = n.
Exercise 6.15. Prove that C`(1) ' C⊕ C and that C`(2) ' C2×2.
7Quantum field theory deals with Fermi fields, which are systems of operators {φ(v) : v ∈ V } satisfying
an “anticommutation relation” akin to (6.6). It helps to imagine an analogous situation in which the
anticommutators vanish, i.e., the algebra generated by the Fermi fields is replaced by a supercommutative
algebra; restoration of the scalar terms q(u, v) then corresponds to quantizing the latter system.
68
Exercise 6.16. Prove that C`(n+2) ' C`(n)⊗C`(2) (ungraded tensor product)8 by showing
that the C-linear map f : Cn+2 → C`(n) ⊗ C`(2) given by f(e1) := 1 ⊗ e1, f(e2) := 1 ⊗ e2,
f(ej) := i ej−2⊗ e1e2 for j ≥ 3, satisfies {f(ej), f(ek)} = −2δjk(1⊗ 1), and hence extends to
the desired isomorphism.
From this we find that C`(2m) ' CN×N with N = 2m; and C`(2m+1) ' CN×N ⊕CN×N .
In particular, C`(V, q) is a simple matrix algebra iff dimR V is even. From now on, we
consider only the case that V has even dimension n = 2m.
Definition 6.14. Any finite-dimensional module for the matrix algebra A = CN×N is of the
form F = S1 ⊕ · · · ⊕ Sr where each Sk has dimension N and A acts irreducibly on each Sk.
Another way to express this is to say that F = W ⊗ S, where dimC S = N and A acts
trivially on W , that is, a · (w ⊗ s) = w ⊗ (a · s). Therefore, if dimV is even, any Clifford
module for C`(V ) is of the form F = W ⊗ S, with c(a)(w ⊗ s) := w ⊗ c(a)s, where S is an
irreducible Clifford module, unique up to equivalence, called the spinor module (or “spinor
space”) for C`(V ). As algebras, C`(V ) and EndC(S) are isomorphic.
Exercise 6.17. Define HomC`(V )(S, F ) to be the vector space of C-linear maps T : S → F
which intertwine the Clifford actions of C`(V ) on S and on F , i.e., c(a)(Ts) := T (c(a)s) for
all s ∈ S, a ∈ C`(V ). Check that the map T⊗s 7→ Ts gives an isomorphism HomC`(V )(S, F )⊗
S ' F , and deduce that HomC`(V )(S, F ) ' W and EndC`(V ) F ' EndCW .
Definition 6.15. Let (V, q) be an oriented Euclidean vector space of even dimension n = 2m
(over R), and let {e1, . . . , en} be an orthonormal basis for (V, q) which is compatible with
the given orientation.9 We define the chirality element of C`(V ) as
γ := im e1e2 . . . en. (6.11)
If {e′1, . . . , e′n} is another oriented orthonormal basis, then e′j =
∑n
k=1 gjkek for g ∈ SO(n) an
orthogonal matrix of determinant +1, and so im e′1e
′
2 . . . e
′
n = det(g)γ = γ: thus the recipe
(6.11) is independent of the chosen basis.
The chirality element γ satisfies γ2 = 1 (that is why the factor im belongs in the defini-
tion), and it anticommutes with the copy of V within C`(V ). Indeed, ejγ = −γej since ej
anticommutes with every ek except ej itself; by linearity, vγ = −γv for all v ∈ V . Therefore
γvγ = −v for v ∈ V , so by Proposition 6.1 γaγ = α(a) for all a ∈ C`(V ). The Z2-grading
on C`(V ) induced by γ (by declaring an element even or odd according as it commutes or
anticommutes with γ) coincides, fortunately, with the original grading given by α.
Notice that if n = 4k is a multiple of 4, then γ belongs to the real Clifford algebra
C`(V, q), and it is not necessary to use the complexification C`(V ).
8This is the “Bott periodicity” identity for complex Clifford algebras.
9Recall that an orientation of V is a choice of a positive direction in the real line ΛnV ; a basis {v1, . . . , vn}
for V is compatible with the orientation iff v1 ∧ · · · ∧ vn is positive.
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6.5 The Fock space of spinors
The availability of γ means that any (ordinary) complex module F under the Clifford algebra
can be given a Z2-grading by taking F± to be the (±1)-eigenspaces of c(γ) ∈ EndC F . In
particular, the spinor module is a superspace S = S+ ⊕ S−, with C`(V ) ' EndC S as
superalgebras. We now give an explicit construction of the spinor module.
Definition 6.16. A Euclidean vector space (V, q) of even real dimension n = 2m can be
made into a complex Hilbert space by choosing an orthogonal complex structure J on (V, q);
this is an R-linear operator on V satisfying:
J2 = −1 and q(Ju, Jv) = q(u, v) for u, v ∈ V. (6.12)
Now we regard V as a complex vector space via the rule (α+ iβ)v := αv+βJv for α, β ∈ R.
The hermitian form
〈〈u | v〉〉 := q(u, v) + iq(Ju, v)
is positive definite since q is. We shall denote the complex Hilbert space thus obtained by
(V, q, J). Notice that i〈〈u | v〉〉 = 〈〈u | Jv〉〉 = −〈〈Ju | v〉〉 for u, v ∈ V .
Exercise 6.18. For (V, q) = (R4, q4), find all J ∈ EndR V = R4×4 satisfying (6.12). Show that
the set of such J has two connected components, each homeomorphic to a 2-sphere S2.
Exercise 6.19. If g ∈ EndR(V ), check that g is a C-linear map on the Hilbert space (V, q, J)
iff gJ = Jg. Conclude that { g ∈ GLR(V ) : gJ = Jg } = GLC(V, q, J). If O(V, q) denotes the
orthogonal group of those g ∈ GLR(V ) for which q(gu, gv) = q(u, v) for all u, v ∈ V , show
that UJ(V ) := O(V, q) ∩ GLC(V, q, J) is the unitary group of the Hilbert space (V, q, J),
i.e., the group of complex automorphisms satisfying 〈〈gu | gv〉〉 = 〈〈u | v〉〉 for u, v ∈ V .
Exercise 6.20. Let J(V, q) denote the set of all orthogonal complex structures on a Euclidean
vector space (V, q). Show that J(V, q) is empty if dimR V is odd, and nonempty if dimR V
is even (produce an example). Show that the orthogonal group O(V, q) acts transitively on
J(V, q) by J 7→ gJg−1, and that the isotropy subgroup of any J is the unitary group UJ(V ).
Exercise 6.21. Check that {u1, Ju1, . . . , um, Jum} is an orthonormal basis for (V, q) when
{u1, . . . , um} is an orthonormal basis for (V, q, J), and deduce that O(V, q) ' O(R2m, q2m) ≡
O(2m) and that UJ(V ) ' Ui(Cm) ≡ U(m). Conclude that J(V, q) is diffeomorphic to the
quotient manifold O(2m)/U(m) (which, as it happens, has two connected components).10
Definition 6.17. Choose and fix an orthogonal complex structure J on (V, q). Denote by
FJ(V ) or simply by F(V ) the complex exterior algebra Λ
•(V, q, J) over the Hilbert space
(V, q, J). Then F(V ) is a complex vector space of dimension 2m, called the Fock space11
10The identity component SO(V, q) of the group O(V, q), which consists of those g with determinant +1,
satisfies SO(V, q) ' SO(2m), so one of the two components of J(V, q) is the set of “orientation-preserving”
J , which is diffeomorphic to SO(2m)/U(m).
11This is sometimes called a fermion Fock space to distinguish it from the Hilbert space formed from
a symmetric algebra over V (suitably completed), which is known as a “boson Fock space”; these spaces
describe many-particle states of fermions and bosons in quantum field theory.
70
over (V, q, J). The Fock space is itself a complex Hilbert space, with the inner product
determined by
〈〈u1 ∧ · · · ∧ um | v1 ∧ · · · ∧ vn〉〉 := δmn det
[〈〈uk | vl〉〉].
We can choose (by induction on dimV ) an orthonormal basis {e1, . . . , en} for V so that
Je2r−1 = e2r for r = 1, . . . ,m.
It is useful to identify (V, q, J) with an m-dimensional complex subspace W of the com-
plexification VC = V ⊗R C; take W := { v − iJv ∈ VC : v ∈ V }. Notice that W is isotropic
for the amplified bilinear form q on VC, i.e., q(u − iJu, v − iJv) = 0 for u, v ∈ V , on ac-
count of (6.12); and moreover, dimCW = m =
1
2
dimC VC, so W is a maximally isotropic
subspace.12 The conjugate subspace W := { v + iJv ∈ VC : v ∈ V } satisfies W ∩W = 0,
W ⊕W = VC, and indeed W is the orthogonal complement of W under the inner product
on VC given by
〈〈w | z〉〉 := 2 q(w¯, z).
The operator P+ :=
1
2
(I − iJ) is then a unitary isomorphism between (V, q, J) and W , so we
may identify these Hilbert spaces. The Riesz theorem allows us to identify the dual space
V ∗ with W , and if w = P+v, we identify v[ with w¯ := P−v := 12(I + iJ)v. An orthonormal
basis for W is {z1, . . . , zm}, where zk := P+e2k−1; notice that P+e2k = P+(Je2k−1) = izk.
We may also identify F(V ) with the complex exterior algebra Λ•W . Now define c(v)
on Λ•W , for v ∈ V , by
c(v)α := (P+v)α− ι(P−v)α, (6.13)
where ι(z¯)(w1 ∧ · · · ∧ wk) =
∑k
j=1(−1)j−1〈〈z | wj〉〉w1 ∧ · ·
j
∨· · ∧ wk, in accordance with (6.7),
and extend to VC by c(v1 + iv2) := c(v1) + ic(v2).
Exercise 6.22. Show that {(w), ι(z¯)} = 〈〈z | w〉〉 for w, z ∈ W .
Exercise 6.23. Show that c(w) = (w) and c(z¯) = −ι(z¯) for w, z ∈ W . Conclude that c
extends to a selfadjoint Clifford action on Λ•(W ).
Proposition 6.3. The operator c(γ) on the Fock space is the grading operator on the super-
space Λ•W .
Proof. The assertion is that the (±1)-eigenspaces of c(γ) are the even and odd subspaces
Λ±W ; in other words, c(γ)α = (−1)kα for all α ∈ ΛkW .
First of all, notice that zkz¯k − z¯kzk = i e2k−1e2k, so that γ =
∏→
1≤k≤m(zkz¯k − z¯kzk).
Moreover, c(zkz¯k− z¯kzk) = [c(zk), c(z¯k)] = [ι(z¯k), (zk)]. Now a direct calculation shows that,
for α = zj1 ∧ · · · ∧ zjk , we have [ι(z¯r), (zr)]α = ±α, with negative sign iff r ∈ {j1, . . . , jk}.
Since Λ•W is generated (as a complex vector space) by such α, we obtain c(γ)α = (−1)kα.
12If Z ≤ VC has dimension k, the subspace of vectors w with q(w, z) = 0 for all z ∈ Z has dimension
2m − k, since q is nondegenerate; thus an isotropic subspace can have dimension at most m. A maximally
isotropic subspace is also called a polarization for q.
71
6.6 The Pin and Spin groups
Definition 6.18. Let (V, q) be a Euclidean vector space with q positive definite. The
invertible elements of C`(V, q) form a group which includes all nonzero scalars and all nonzero
v ∈ V (since v−1 = −v/q(v, v)); and the “twisted conjugation” φ(a)b := α(a)ba−1 provides a
linear action of this group on C`(V, q). The invertible elements which preserve the subspace
V under this action form the Clifford group Γ(V, q) := { a : φ(a)(V ) ⊂ V }. The nonzero
scalars t ∈ R× lie in Γ(V, q), and so do the nonzero vectors u ∈ V \ {0}, since if v ∈ V ,
φ(u)v = −uvu−1 = (vu− 2q(u, v))u−1 = v − 2q(u, v)
q(u, u)
u, (6.14)
so u ∈ Γ(V, q). Geometrically, (6.14) is the reflection in the hyperplane orthogonal to u,
since φ(u)u = −u and φ(u)v = v iff q(u, v) = 0.
Lemma 6.4. The kernel of φ : Γ(V, q)→ GLR(V ) is the subgroup R× of nonzero scalars.
Proof. If a ∈ kerφ, let a = a+ + a− with a± ∈ C`±(V, q); taking even and odd parts of
the equation α(a)v = va gives a±v = ±va± for v ∈ V . If {e1, . . . , en} is an orthonormal
basis for V , and if V1 denotes the hyperplane in V orthogonal to e1, then a
+ = b+ + e1b
−
with b± ∈ C`±(V1, q). Now the equation a+e1 = e1a+ yields b+e1 + b− = e1b+ − b−, so
b− = 0; hence a+ ∈ C`(V1, q). A similar argument shows that a− ∈ C`(V1, q). An obvious
induction argument now shows that a has only a scalar component when expanded in the
basis { eJ : J ⊆ {1, . . . , n} } of C`(V, q), and so a ∈ R×.
If v ∈ V , then vv¯ = vα(v) = −v2 = q(v, v) 1, so the quadratic form q(·, ·) extends to
the map a 7→ aa¯ of C`(V, q) into itself. If a = ∑J aJeJ is the expansion of a with respect
to the basis { eJ : J ⊆ {1, 2, . . . , n} } of C`(V, q), then aa¯ and a¯a have the same scalar
component
∑
J a
2
J . If a ∈ Γ(V, q) and v ∈ V , then φ(a)v = u implies u = α(a)−1va, and so
u = u! = a!va¯−1 = φ(a¯)v; in consequence, φ(aa¯)v = φ(a)φ(a¯)v = v, so that aa¯ ∈ kerφ =
R×. Thus a 7→ aa¯ = a¯a yields a group homomorphism of Γ(V, q) into R×. Its kernel is
Pin(V, q) := { a ∈ Γ(V, q) : aa¯ = 1 }.
If a ∈ Γ(V, q) and v ∈ V , then
q(φ(a)v, φ(a)v) = α(a)va−1a¯−1v¯α(a¯) = α(aa¯)(a¯a)−1vv¯ = −v2 = q(v, v),
since aa¯ = a¯a is a scalar, and so φ(a) lies in the orthogonal group O(V, q). Restriction to
Pin(V, q) yields a homomorphism φ : Pin(V, q)→ O(V, q) which is surjective since the orthog-
onal group is generated by the reflections13 (6.14), and v ∈ Pin(V, q) whenever q(v, v) = 1.
Definition 6.19. The Spin group of the Euclidean vector space (V, q) is defined as the
even part of Pin(V, q), i.e.,
Spin(V, q) := C`+(V, q) ∩ Pin(V, q) := { a ∈ Γ(V, q) ∩ C`+(V, q) : aa¯ = 1 }.
13This is the Cartan–Dieudonne´ theorem, which can be proved by writing an orthogonal matrix in normal
form as a direct of k plane rotation matrices and possibly a −1 diagonal entry, and by recalling that a plane
rotation is a product of two reflections in lines.
72
The image φ(Spin(V, q)) is the subgroup SO(V, q) of O(V, q) generated by products of an
even number of reflections. This is the rotation group, also called the special orthogonal
group, since g ∈ O(V, q) lies in SO(V, q) iff det g = 1.
The homomorphisms φ : Pin(V, q) → O(V, q) and φ : Spin(V, q) → SO(V, q) have the
same kernel {±1} ' Z2, by Lemma 6.4. In particular, we have a short exact sequence of Lie
groups:
1−→Z2−→ Spin(V, q) φ−→SO(V, q)−→ 1,
so that Spin(V, q) is a double covering of the rotation group.
Exercise 6.24. Show that Pin(V, q) is the set of products {v1 . . . vk} of unit vectors (i.e.,
vectors vj ∈ V such that q(vj, vj) = 1 for each j), by showing that these products form a
normal subgroup which φ maps onto O(V, q). Deduce that Spin(V, q) is the set of products
{v1 . . . v2r} of an even number of unit vectors.
Exercise 6.25. If u, v ∈ V are unit vectors in V , with u 6= ±v, and if a(t) := 1 + q(u, v) sin 2t
for 0 ≤ t ≤ pi
2
, define w(t) := a(t)−1/2(cos t u + sin t v), z(t) := a(t)−1/2(cos t v + sin t u), and
b := (1 − q(u, v)2)−1/2(uv + q(u, v) 1). Show that t 7→ −w(t)w(−t) is a continuous path in
Spin(V, q) from 1 to −1 through b, and that t 7→ w(t)z(−t) is a continuous loop in Spin(V, q)
from uv to b and back. Deduce that Spin(V, q) is a connected group.14
On passing to the complexification, we may regard Spin(V, q) as a subgroup of the in-
vertible elements of the complex Clifford algebra C`(V ); the circle group U(1), regarded as
complex scalars is another such subgroup. Now if λ, µ ∈ U(1) and a, b ∈ Spin(V, q), then
λa = µb in C`(V ) iff (µ, b) = (λ, a) or else (µ, b) = (−λ,−a); thus Spin(V, q) and U(1)
generate the subgroup
Spinc(V ) ' (Spin(V, q)× U(1))/Z2,
where the quotient map is defined by the relation (λ, a) ∼ (−λ,−a). Define φc(λa)v :=
λ2φ(a)v for λa ∈ Spinc(V ); then φc maps Spinc(V ) onto SO(V, q)× U(1) with kernel {±1},
so there is another short exact sequence of Lie groups:
1−→Z2−→ Spinc(V ) φ
c−→SO(V, q)× U(1)−→ 1.
The Lie algebra of the spin group Spin(V, q) is readily identified as a subspace of the
Clifford algebra C`(V, q). Indeed, let A+2 be the subspace of C`
+(V, q) with basis {1}∪{ eiej :
i < j } with {e1, . . . , en} an (arbitrary) orthonormal basis of V (compare with Exercise 6.14),
and let C2(V, q) be the image of Λ2V under the quantization map Q.
Lemma 6.5. C2(V, q) = { b ∈ A+2 : b¯ = −b }.
Proof. Since Q(u ∧ v) = uv + q(u, v) 1, it follows that C2(V, q) ⊆ A+2 . Counting dimensions,
dimC2(V, q) = dim Λ2V = 1
2
n(n− 1) = dimA+2 − 1, so C2(V, q) is a hyperplane in A+2 . Now
14Since the fundamental group of SO(V, q) is Z2, this shows that the covering map φ is nontrivial and
hence that Spin(V, q) is the universal covering group of SO(V, q).
73
eiej = ejei = −eiej for i < j, so the map b 7→ b¯ + b is a scalar-valued R-linear form on A+2 ,
whose kernel is a hyperplane; and since
Q(u ∧ v) = vu+ q(u, v) 1 = −uv − q(u, v) 1 = −Q(u ∧ v)
for u, v ∈ V , it coincides with C2(V, q).
If b ∈ C2(V, q) and w ∈ V , then [[b, w]] = [b, w] lies in V also; to see that, it suffices to
take b = uv + q(u, v) 1, whereupon
[b, w] = uvw − wuv = uvw + uwv − uwv − wuv
= 2q(u,w)v − 2q(v, w)u ∈ V. (6.15)
The only elements of A+2 which commute with V are scalars, so [b, w] vanishes for all w ∈ V
iff b = 0 in C2(V, q). Hence τ(b)v := [b, v] defines an injective R-linear map τ : C2(V, q) →
EndV .
Proposition 6.6. The map τ is a Lie algebra isomorphism from C2(V, q) onto the Lie
algebra of antisymmetric operators so(V, q) := {A ∈ EndV : q(Au, v) ≡ −q(u,Av) }.
Proof. First notice that C2(V, q) is a Lie algebra under the bracket [b, d] := bd − db of
C`+(V, q). Indeed, on taking b = uv + q(u, v) 1, d = wz + q(w, z) 1, one finds that
[b, d] = [uv, wz] = [uv, w]z + w[uv, z] ∈ A+2
using (6.15), and [b, d] = [d¯, b¯] = [−d,−b] = −[b, d], so [b, d] ∈ C2(V, q). Also from (6.15),
q([b, w], z) = 2q(u,w)q(v, z)− 2q(v, w)q(u, z) = −q(w, [b, z]),
which shows that τ(b) ∈ so(V, q). Since dim so(V, q) = 1
2
n(n − 1) = dimC2(V, q) and τ is
injective, its image is all of so(V, q). Finally, the Jacobi identity yields
τ([b, d])v = [[b, d], v] = [[b, v], d] + [b, [d, v]] = −τ(d)τ(b)v + τ(b)τ(d)v,
so τ([b, d]) = [τ(b), τ(d)] where the latter bracket is the commutator bracket in EndV (which
preserves so(V, q)); thus, τ is a Lie algebra isomorphism.
Lemma 6.7. If A ∈ so(V, q) and {e1, . . . , en} is an orthonormal basis for V , then
τ−1(A) =
1
2
∑
j<k
q(Aej, ek) ejek =
1
4
n∑
j,k=1
q(Aej, ek) ejek. (6.16)
Proof. The series in (6.16) are equal since A is antisymmetric, and they lie in C2(V, q) by
Lemma 6.5; if b denotes either series, we show that q(er, τ(b)es) = q(er, Aes) for all r, s.
Indeed,
τ(b)es =
1
2
∑
j<k
q(Aej, ek) [ejek, es] =
1
2
∑
j<k
q(Aej, ek) (2δjsek − 2δksej)
=
∑
k>s
q(Aes, ek) ek −
∑
j<s
q(Aej, es) ej =
n∑
k=1
q(Aes, ek) ek,
by antisymmetry of A, so q(er, τ(b)es) = q(Aes, er) = q(er, Aes) as required.
74
The Lie algebra C2(V, q) gives rise to a Lie group, namely, the group of invertible elements
of C`+(V, q) generated by the elements exp b := 1 +
∑
k≥1 b
k/k!, for b ∈ C2(V, q), i.e., by the
image of C2(V, q) under the exponential map in the Clifford algebra. Notice that if a = exp b
and w ∈ V , then
awa−1 = (exp b)w(exp(−b)) =
∑
k,l≥0
1
k! l!
bkw(−b)l
=
∑
r≥0
1
r!
r∑
k=0
(
r
k
)
bkw(−b)r−k =
∑
r≥0
1
r!
[b, [b, . . . , [b, w] . . . ]]
=
∑
r≥0
1
r!
τ(b)rw, (6.17)
so that awa−1 ∈ V , and hence a ∈ Γ(V, q). From the relation b¯ = −b it follows that
aa¯ = (exp b)(exp(−b)) = 1, and so a ∈ Spin(V, q).
Exercise 6.26. If b = uv + q(u, v) 1 with u, v ∈ V , show that
q(τ(b)w, τ(b)w) ≤ 16 q(u, u)q(v, v)q(w,w).
If Σ denotes the series on the right hand side of (6.17), verify its convergence by obtaining
the estimate q(Σ,Σ) ≤ e8 s(b)q(w,w) where s(b)2 = q(u, u)q(v, v).
Proposition 6.8. The exponential map exp: C2(V, q)→ Spin(V, q) is surjective.
Proof. The equation (6.17) reads
φ(exp b) = exp(τ(b)), (6.18)
where the second exponential map is the matrix exponential in EndV . Now any rotation
in SO(V, q) is of the form expA for some antisymmetric operator A ∈ so(V, q), as is seen
by expressing the matrix of the rotation in canonical form. Therefore, {φ(exp b) : b ∈
C2(V, q) } = SO(V, q).
Also, if u, v are orthogonal unit vectors in V and if θ ∈ R, then θuv ∈ C2(V, q), and
exp(θuv) = cos θ + sin θ uv. In particular, −1 = exp(piuv) lies in exp(C2(V, q)). Now if
b = τ−1(A) with A ∈ so(V, q), choose an orthonormal basis for V such that q(Aej, ek) = 0
unless k = j ± 1. From (6.16), b = ∑mr=1 bre2r−1e2r with br = 12q(Ae2r−1, e2r). Thus b
commutes with pie1e2, and so − exp b = exp(pie1e2 + b). It follows that exp(C2(V, q)) is a
subset of Spin(V, q) doubly covering SO(V, q), and hence is all of Spin(V, q).
Thus we may regard C2(V, q) as the Lie algebra of Spin(V, q). It should be noted that
Spin(V, q) and Pin(V, q) are compact Lie groups, since they are closed subsets of the “unit
sphere” in C`(V, q) determined by the condition
∑
J a
2
J = 1. Therefore, Proposition 6.8
exemplifies the well-known fact that the exponential mapping from the Lie algebra to a
compact connected Lie group is surjective [13].
75
6.7 The spin representation
Proposition 6.9. Let S := Λ•CW be the spinor module for C`(V ). Define c : Pin(V, q) →
EndC S by restriction of the Clifford action c : C`(V )→ EndC S given by (6.13). Then c is
an irreducible unitary representation of the Lie group Pin(V, q).
Proof. Since c : C`(V )→ EndC S is multiplicative, its restriction to Pin(V, q) is a represen-
tation of this group on S. Since the Clifford action is selfadjoint (see Exercise 6.24), we have
c(a)† = c(a¯) = c(a−1) = c(a)−1 for a ∈ Pin(V, q), because aa¯ = 1; thus c(a) is a unitary
operator on S.
The irreducibility of c follows from Schur’s Lemma: we show that any T ∈ EndC S
commuting with { c(a) : a ∈ Pin(V, q) } is a scalar operator. Indeed, T must commute
with c(w) = (w) for any w ∈ W , since V \ {0} ⊂ Pin(V, q) and W ⊂ VC; and T must
also commute with c(z¯) = −ι(z¯) for z ∈ W . If Ω is a unit vector in the one-dimensional
space Λ0CW ' C, then ι(z¯)TΩ = T (ι(z¯)Ω) = 0, so TΩ = tΩ ∈ Λ0CW for some t ∈ C. Now
T (w1∧· · ·∧wr) = T ((w1) . . . (wr)Ω) = (w1) . . . (wr)(tΩ) = t w1∧· · ·∧wr, so T = t idS.
Definition 6.20. The irreducible representation given by the previous Proposition is called
the pin representation of the group Pin(V, q). Its restriction to the subgroup Spin(V, q) is
called the spin representation.
Proposition 6.10. The spin representation is not irreducible, but it is the direct sum of two
inequivalent irreducible representations on the subspaces S+ := Λ+CW and S
− := Λ−CW .
Proof. Recall Proposition 6.3: c(γ) is the grading operator on S = S+ ⊕ S−. Now the chi-
rality element γ anticommutes with every v ∈ V , and so γa = aγ for all a ∈ Spin(V, q) by
Exercise 6.24. Thus c(γ) commutes with { c(a) : a ∈ Spin(V, q) }, so by Schur’s lemma the
spin representation cannot be irreducible; in fact, it is the direct sum of two subrepresenta-
tions on the ±1-eigenspaces of c(γ), which are S+ and S−.
If T ∈ EndC S± commutes with c(a) for all a ∈ Spin(V, q), then T must commute with
c(w1)c(w2) = (w1)(w2) and with c(z¯1)c(z¯2) = ι(z¯1)ι(z¯2) for w1, w2, z1, z2 ∈ W , so T is
a scalar operator, by adapting the argument of the proof of Proposition 6.9. Thus both
subrepresentations are irreducible.
Notice that dimS+ = dimS− = 2m−1, so these “half-spin representations” have the same
dimension. However, they are not equivalent, since if R ∈ Hom(S+, S−) is an intertwining
operator, then Rc(e2j−1e2j) = c(e2j−1e2j)R for j = 1, . . . ,m and hence Rc(γ) = c(γ)R; but
then Rα = Rc(γ)α = c(γ)Rα = −Rα for all α ∈ S+, so R = 0; inequivalence follows from
Schur’s lemma.
Exercise 6.27. Show that γ ∈ Spinc(V ).
The whole theory of Clifford algebras and spin representations may be extended, with a
little care, to infinite-dimensional vector spaces with a symmetric bilinear form. The Clifford
algebra must be complete, so its complexification is defined as the unique C∗-algebra with
the desired universal property. The orthogonal group must be restricted to the subgroup of
operators g ∈ EndR V whose antilinear part, with respect to a fixed complex structure, is
76
Hilbert–Schmidt; this is necessary to guarantee convergence of series similar to (6.16). For
an exposition of the infinite-dimensional spin representation which extrapolates from the
present treatment, we refer to [2]. An alternative method, wherein the spin representation
is defined by permuting certain “Gaussian” elements in the Fock space, is developed in [31],
based on ideas of [44].
7 Global Clifford modules
Given a compact Riemannian manifold (M, g), we can form a Clifford bundle C`(M)−→M
by gluing together the complex Clifford algebras C`(TxM, gx). The space of smooth sections
Γ(C`M) is an algebra, and its completion to the space of continuous sections Γcont(C`M)
is canonically a C∗-algebra. Naturally, we want to consider vector bundles of representation
spaces for the algebras C`(TxM, gx), whose section spaces are modules over the algebra
Γ(C`M). However, although the Clifford algebra bundle can be built over any Riemannian
manifold, topological obstructions can prevent putting together the corresponding Clifford
modules. We shall restrict our attention to oriented, even-dimensional Riemannian manifolds
and we shall say that (M, g) is a spin manifold if a Clifford module corresponding to the
irreducible spinor representation exists. Such manifolds may have none, one, or several
inequivalent spin structures; our first task is to sort out the possibilities.
7.1 Clifford algebra bundles
Definition 7.1. Let M be a compact manifold, and let E−→M be a real vector bundle of
rank n with a Euclidean metric g. For any x ∈M , (Ex, gx) is a Euclidean vector space, and
we can form the Clifford algebra C`(Ex, gx). These form the fibres of a real vector bundle
C`E−→M of rank 2n, which extends the vector bundle E−→M in the sense that there
is an injective bundle map j : E−→C`E; since C`(Ex, gx) ' Λ•Ex as vector spaces, just
take C`E to be the total space of the exterior algebra bundle Λ•E−→M , and let j be
the map which identifies each vx ∈ Ex with its canonical image in Λ1Ex. The declaration
j(vx)
2 := −gx(vx, vx) determines, by Proposition 6.1, a bilinear product in each fibre which
depends smoothly on x ∈M , and also determines isomorphisms (C`E)x ' C`(Ex, gx).
The complexification of C`E−→M is C`E−→M , the complex Clifford algebra bundle;
notice that C`E = C`EC.
The module of smooth sections Γ(C`E) is thus an algebra under the pointwise Clifford
product (κξ)(x) := κ(x)ξ(x). If the rank of E−→M is even, the centre of this algebra is
just A = C∞(M), by identifying smooth functions on M with scalar-valued sections of the
Clifford algebra bundle.1
1The continuous sections Γcont(C`E) form a C∗-algebra whose centre is the algebra C(M) of continuous
complex-valued functions on M . This C∗-algebra is noncommutative, but in a fairly trivial way: it is “Morita-
equivalent” to the commutative C∗-algebra C(M). This means [46] that Γcont(C`E)⊗K ' C(M)⊗K, where
K is the elementary C∗-algebra of compact operators on a separable infinite-dimensional Hilbert space.
In particular, since K is a simple C∗-algebra, there is a bijective correspondence between the irreducible
representations of these Morita-equivalent C∗-algebras.
77
Exercise 7.1. Let Q→M be the principal O(n)-bundle of orthonormal frames for E−→M .
Check that any g ∈ O(n) extends to an automorphism ρ(g) of the Clifford algebra C`(n),
and that ρ(g)ρ(g′) = ρ(gg′) for g, g′ ∈ O(n). Then show that C`E−→M is the vector
bundle associated to Q→M via the representation ρ of O(n).
Definition 7.2. The Clifford bundle over a Riemannian manifold (M, g) is the bundle
of Clifford algebras C`(M)−→M obtained by taking the tangent bundle TM −→M as the
generating vector bundle, with the Riemannian metric g defining the Euclidean structure
on TM . It would be more consistent to write C` TM rather than C`(M), but this is not
usually done, for the following reason. The cotangent bundle T ∗M −→M is equivalent to
the tangent bundle via the bundle map gˆ : TM → T ∗M induced by the metric (see the
discussion after Definition 3.2), and this in turn defines a Euclidean structure g−1 on the
cotangent bundle by
g−1(α, β) := g(α], β]) for α, β ∈ A1(M), (7.1)
where the vector field α] is determined by g(α], Y ) := α(Y ). This extends to a bundle
equivalence gˆ : C` TM → C` T ∗M . Thus we may and shall regard sections of C`(M)−→M
as Clifford products of vector fields or of forms, according to momentary convenience. We
shall, as before, write the scalar products in (7.1) as (α | β) and (α] | β]) respectively.
Exercise 7.2. Verify that the map gˆ : TM → T ∗M extends to an equivalence of the Clifford
algebra bundles C` TM and C` T ∗M by applying Proposition 6.1 on the fibres.
Exercise 7.3. Let gij := g(∂/∂x
i, ∂/∂xj) be the matrix entries of the metric g in local coordi-
nates, and let [grs] be the inverse matrix to [gij]; show that (dx
r)] = grj∂/∂xj and conclude
that g−1(dxr, dxs) = grs, thereby justifying the notation g−1 for the metric in T ∗M .
In view of Proposition 5.2, the sections of the Clifford algebra bundle may be constructed
by an alternative method. Starting from a Euclidean vector bundle E−→M , one can define
the tensor bundle T(E)−→M as the (infinite) Whitney sum of the tensor product bundles
E⊗k−→M , where E⊗0 = M × R, the trivial rank-one bundle. The ideals I(gx) of T(Ex)
glue together to give a bundle of ideals I(g)−→M , and C`E → M may be defined as the
quotient bundle T(E)/I(g)−→M . The module of sections Γ(I(g)) is an A-submodule of
Γ(T(E)), and the quotient A-module may be identified with Γ(C`E). Note, in particular,
that Γ(I(g)) is generated as an A-module by the elements s⊗ s+ g(s, s), for s ∈ Γ(E).
Proposition 7.1. Any connection ∇ on a Euclidean vector bundle E−→M which is com-
patible with the metric on E extends to a connection, also denoted by ∇, on C`E−→M ,
such that
∇X(κξ) = (∇Xκ)ξ + κ(∇Xξ) (7.2)
for X ∈ X(M), κ, ξ ∈ Γ(C`E); that is, each ∇X is a derivation on the algebra Γ(C`E).
Proof. Compatibility of ∇ with the metric is expressed by (5.7), which is equivalent to the
condition that (∇Xs | t) + (s |∇Xt) = X(s | t) for s, t ∈ Γ(E), X ∈ X(M). The operators ∇X
78
on Γ(E) extend to Γ(T(E)) by setting ∇X(s1⊗ · · · ⊗ sk) :=
∑k
j=1 s1⊗ · · · ⊗∇Xsj ⊗ · · · ⊗ sk;
it is clear that X 7→ ∇X(s1 ⊗ · · · ⊗ sk) is a connection on the bundle E⊗k−→M for each k.
Moreover, each∇X : Γ(T(E))→ Γ(T(E)) is anA-linear derivation, i.e., it satisfies the Leibniz
rule
∇X(σ ⊗ τ) = (∇Xσ)⊗ τ + σ ⊗ (∇Xτ). (7.3)
On applying the extended ∇ to a generator of the module of ideals Γ(I(g)), we get
∇X [s⊗ s+ (s | s)] = (s+∇Xs)⊗ (s+∇Xs) + (s+∇Xs | s+∇Xs)
− s⊗ s− (s | s)−∇Xs⊗∇Xs− (∇Xs | ∇Xs).
Therefore, ∇X preserves Γ(I(g)), and so drops to the quotient algebra Γ(C`E). The deriva-
tion property (7.3) survives on passing to the quotient, and (7.2) follows.
One may eliminate the vector field X in the usual way, by noting that both sides of (7.2)
are A-linear in X, and by extending the Clifford product in Γ(C`E) to an A-bilinear form
from Γ(C`E)×A1(M,C`E) to A1(M,C`E). Thus (7.2) may be rewritten as
∇(κξ) = (∇κ)ξ + κ(∇ξ) for κ, ξ ∈ Γ(C`E), (7.4)
which is an equation in A1(M,C`E).
Definition 7.3. The Levi-Civita connection on the Clifford bundle over M is the deriva-
tion ∇ : Γ(C`(M)) → A1(M,C`(M)) which extends the Levi-Civita connection on the tan-
gent bundle (or equivalently, its dual connection on the cotangent bundle). Whenever con-
venient, we shall denote any of these three connections by ∇LC .
Definition 7.4. A Clifford module over a Riemannian manifold M is a module of sec-
tions Γ(F ) of a superbundle F −→M , together with a Clifford action of the sections of the
complexified Clifford bundle Γ(C`(M)). Such an action is defined as an A-bilinear map
Γ(C`(M)) × Γ(F ) → Γ(F ), written (κ, s) 7→ c(κ)s, such that c(1)s = s and c(κ)c(ξ)s =
c(κξ)s for all s ∈ Γ(F ) and κ, ξ ∈ Γ(C`(M)), or else as a homomorphism c : Γ(C`(M)) →
Γ(EndF ), which is even in the sense that c : Γ(C`±(M))→ Γ(End± F ).
The evenness of the action is equivalent to the condition that for any α ∈ A1(M), the
operator c(α) interchanges Γ(F+) and Γ(F−).
If F −→M is a Hermitian vector bundle, we say that the Clifford action is selfadjoint
if c(κ)† = c(κ¯) for κ ∈ C`(M), or equivalently, if c(α)† = −c(α) for any (real-valued)
α ∈ A1(M). More precisely, this condition reads
(c(α)s | t) + (s | c(α)t) = 0,
for s, t ∈ Γ(F ), using the scalar product obtained from the hermitian structure on F .
Obvious examples of Clifford modules are Γ(C`(M)) itself, with the left multiplication
cl(κ)ξ := κξ; and Γ(C`(M)) itself, with conjugated right multiplication cr(κ)ξ := ξκ¯. An-
other example is the module of complex-valued exterior forms A•(M,C), with the Clifford
action obtained by applying the recipe (6.8) to each fibre:
c(α)ω := α ∧ ω − ι(α])ω.
79
However, these examples are “too large”, since the Clifford actions on the fibres are reducible,
and it is to be expected that the global module should also be decomposable into a direct
sum of several submodules. Therefore we seek irreducible Clifford modules, where each fibre
is linearly isomorphic to the Fock space S = Λ•W of subsection 6.5 (provided n = dimM
is even). In that case, after choosing an oriented orthonormal basis for T ∗xM , the spin
representation of the group Spin(n) := Spin(Rn, qn) yields a homomorphism c : Spin(n) →
GLC(Fx); all such homomorphisms form a principal Spin(n)-bundle associated to the vector
bundle F −→M via the spin representation. There is, however, a topological obstruction
to the existence of such a principal bundle. Before we can proceed, we must identify this
obstruction, in order to know whether irreducible Clifford modules are available at all.
7.2 Existence of spin structures
Definition 7.5. A real vector bundle E−→M is orientable if its transition functions
gij : Ui ∩ Uj → GL(r,R) can be chosen to satisfy det gij > 0 for all i, j. A Euclidean
vector bundle has transition functions with values in the orthogonal group O(r); thus it is
orientable iff one can choose the gij to satisfy det gij = +1. (This may be expressed by saying
that the structure group of E can be reduced from O(r) to SO(r).)
To see whether a Euclidean vector bundle is orientable or not, take a good covering U =
{Uj} for M and choose transition functions gij for U. Then det gij : Ui ∩ Uj → {±1} = Z2,
and since gijgjk = gik on each nonempty Ui ∩ Uj ∩ Uk, we conclude that det g is a Cˇech
1-cocycle2 in C1(U,Z2). Any set of transition functions is of the form g′ij = figijf−1j with
fj : Uj → O(r), as follows from the proof of Proposition 1.5. (If a local system of sections
sj for E transforms as si = gijsj, and an equivalent system of local sections is given by
s′j := fjsj, the new transition functions satisfy g
′
ijfj = figij.) Hence detf ∈ C0(U,Z2) and
det g′ = det g + δ(detf). Thus the class w1(E) := [det g] ∈ Hˇ1(M,Z2) depends only on the
equivalence class [E] of the vector bundle. It is usually called the first Stiefel–Whitney
class of E [23, 39]. It is also customary to write w1(M) := w1(TM) when (M, g) is a
Riemannian manifold. We may summarize the situation as follows.
Lemma 7.2. A Euclidean vector bundle E−→M is orientable iff w1(E) = 0 in Hˇ1(M,Z2).
A Riemannian manifold (M, g) is orientable iff w1(M) = 0.
Proof. The condition w1(E) = 0 means that transition functions g
′
ij can be chosen to satisfy
det g′ij = +1 identically.
Suppose now that E−→M is an oriented Euclidean vector bundle (i.e., that w1(E) = 0),
with transition functions gij : Ui∩Uj → SO(r) for a good covering U of M . Since each Ui∩Uj
is contractible, one can lift these maps to the double covering Spin(r) of SO(r), i.e., we can
find smooth functions hij : Ui ∩ Uj → Spin(r) such that φ(hij) = gij for all i, j. The relation
φ(hijhjkh
−1
ik ) = gijgjkg
−1
ik = 1 in SO(r) shows that
aijk := hijhjkh
−1
ik = ±1
2We identify {±1} with Z2, in order to use additive notation when combining Cˇech cocycles.
80
in Spin(r). Thus a is a Cˇech 2-cochain in C2(U,Z2); indeed, a is a Cˇech 2-cocycle, as a
short calculation shows.
Exercise 7.4. Compute δa, to check that it is trivial in C3(U,Z2).
We can make random changes of signs of some of the hij, i.e., we can let h
′
ij := bijhij
where bij : Ui ∩Uj → {±1} are arbitrary (but constant) sign functions. Then b ∈ C1(U,Z2),
and a′ijk := h
′
ijh
′
jk(h
′
ik)
−1 yields a′ ∈ C2(U,Z2) with a′ = a + δb. Therefore, the class
w2(E) := [a] ∈ Hˇ2(U,Z2) depends only on the equivalence class [E] of the vector bundle;
it is called the second Stiefel–Whitney class of E. One writes w2(M) := w2(TM) when
(M, g) is an oriented Riemannian manifold.
Definition 7.6. A spin structure on an oriented Riemannian manifold (M, g) of dimen-
sion n is a principal Spin(n)-bundle P
η−→M together with a bundle map τ : P → Q where
Q
θ−→M is the principal SO(n)-bundle of oriented orthonormal frames for the tangent bun-
dle TM −→M , such that τ(p · g) = τ(p) ◦ φ(g) for p ∈ P , g ∈ Spin(n). We say that M is a
spin manifold if it admits at least one spin structure.
By Lemma 1.4, such a principal Spin(n)-bundle may be assembled from transition func-
tions hij : Ui∩Uj → Spin(n) satisfying φ(hij) = gij, where the gij are the transition functions
of the frame bundle, provided only that hijhjk = hik on each nonempty Ui ∩ Uj ∩ Uk. Thus
M admits a spin structure iff w2(M) = 0 in Hˇ
2(M,Z2).
Since τ : P → Q reproduces the double covering φ : Spin(n)→ SO(n) on each fibre, the
map τ is two-to-one.
Inequivalent spin structures may arise from different choices of hij covering the same
gij. The only freedom here comes from the sign changes bij = ±1 mentioned above, where
h′ij = bijhij determine another spin structure on M . Since hijhjk = hik and similarly for
the h′ij, it follows that bijbjk = bik also; thus b is a Cˇech 1-cocycle. Now if b = δc with
c ∈ C0(U,Z2), i.e., if bij = ci/cj with all cj = ±1, then h′ij = (ci/cj)hij as functions from
Ui ∩Uj to the group Spin(n), so the corresponding principal Spin(n)-bundles are equivalent.
Conversely, a principal bundle equivalence yields relations h′ij = (ai/aj)hij for some smooth
functions aj : Uj → Spin(n) satisfying ai = ±aj on overlaps; thus, signs cj can be chosen so
that ciai = cjaj on overlaps, and consequently h
′
ij = (ci/cj)hij. In summary, the two spin
structures are equivalent iff b = δc for some c, which gives the following result.
Lemma 7.3. If M is a Riemannian manifold with w1(M) = 0 and w2(M) = 0, the inequiv-
alent spin structures on M are classified by the Cˇech cohomology group Hˇ1(M,Z2).
The computation of the Stiefel–Whitney classes for particular manifolds involves either
a good deal of combinatorial calculation (see [28], for instance) or general theorems from
topology [23, 39, 41]. One such general theorem is particularly useful: if E−→M is a complex
vector bundle and ER−→M is the underlying real vector bundle, then w1(ER) = 0 and
w2(ER) is the image of the first Chern class c1(E) under the canonical homomorphism from
Hˇ2(M,Z) to Hˇ2(M,Z2) obtained from the standard homomorphism j : Z → Z2 (namely,
81
reduction modulo 2).3 That w1(ER) = 0 should be no surprise, because a complex vector
bundle is always orientable. To see that, notice firstly that the unitary group U(r) can
be regarded as a subgroup of SO(2r), since U(r) permutes orthonormal bases in Cr, and
any such basis {e1, . . . , er} yields an orthonormal basis {e1, f1, . . . , er, fr} for R2r, by setting
fr := ier; and since U(r) is connected, it is therefore contained
4 in the identity component
SO(2r) of the orthogonal group of R2r. Now, with respect to any Hermitian metric on E, a
local orthonormal basis of sections for Γ(U,E) gives a local orthonormal basis of sections for
Γ(U,ER), and the U(r)-valued transition functions of E may thus be regarded as SO(2r)-
valued transition functions of ER.
Since the first Chern class c1(H) of the hyperplane bundle H → CPm is a generator of
Hˇ2(CPm,Z) ' Z, so that c1(H)↔ 1 under this group isomorphism, its modulo-2 reduction is
not zero: therefore, w2(HR) 6= 0 in Hˇ2(CPm,Z2). Now it can be shown [28] that the tangent
bundle TCPm has the following property: if Er−→CPm denotes the trivial real bundle of
rank r, then TCPm ⊕ E2 and HR ⊕ · · · ⊕HR (with (m + 1) summands) are equivalent real
bundles over CPm. These facts suffice to decide which of the CPm are spin manifolds.
Proposition 7.4. The complex projective space CPm is a spin manifold iff m is odd; and
for odd m, the spin structure on CPm is unique.
Proof. The isomorphism C`(V, q) ⊗ C`(W, r) ' C`(V ⊕W, q⊕r) of Proposition 6.2, restricted
to the spin subgroups, shows that Spin(V, q) and Spin(W, r) may be regarded as commuting
subgroups of Spin(V ⊕W, q⊕r). Thus we may embed the direct product Spin(k)×Spin(l) as a
subgroup of Spin(k+l); and φmaps this to the usual embedding of SO(k)×SO(l) in SO(k+l).
Thus, whenever E−→M and F −→M are oriented Euclidean vector bundles, the transition
functions gij⊕g′ij of E⊕F take values in SO(k)×SO(l), and lift to hij⊕h′ij in Spin(k)×Spin(l).
The second Stiefel–Whitney class therefore satisfies5 the relation w2(E⊕F ) = w2(E)+w2(F ).
When M = CPm, we thereby obtain
w2(CPm) + 0 = w2(TCPm ⊕ E2) = w2(HR ⊕ · · · ⊕HR) = (m+ 1)w2(HR). (7.5)
Any nontrivial element of the group Hˇ2(CPm,Z2) is of order two,6 and w2(HR) 6= 0, so the
right hand side of (7.5) vanishes iff m is odd.
To get uniqueness, we must show that Hˇ1(CPm,Z2) = 0. This follows from the Bockstein
homomorphism construction of subsection 1.11, applied to the exact sequence of abelian
groups
0−→Z 2−→Z j−→Z2−→ 0,
3See Appendix B of [39] for a proof.
4If one chooses an orthonormal basis in Cr for which g ∈ U(r) is diagonal, i.e., g(ej) = eiαjej , then
g(ej) = cosαj ej + sinαj fj , g(fj) = − sinαj ej + cosαj fj , so the image of g in SO(2r) is a direct sum of
2× 2 rotation blocks.
5When E and F are not orientable, this additivity breaks down; there is a “product formula” of Whitney
[41] which yields the relation w2(E ⊕ F ) = w2(E) + w1(E)w1(F ) + w2(F ).
6Indeed, Hˇ2(CPm,Z2) is a vector space over the field Z2.
82
where ‘2’ denotes multiplication by two, and j : Z→ Z2 is reduction modulo 2. The Bockstein
construction yields a long exact sequence in Cˇech cohomology
· · · → Hˇ1(M,Z2) ∂−→ Hˇ2(M,Z) 2∗−→ Hˇ2(M,Z) j∗−→ Hˇ2(M,Z2) ∂−→ Hˇ3(M,Z)→ · · ·
for any compact manifold M . Let us chase this diagram backwards from Hˇ3(M,Z) in the
case M = CPm. We know that Hˇ3(CPm,Z) = 0 from (2.5), so j∗ is surjective.7 We know also
that Hˇ2(CPm,Z) = Z (these are the Chern classes found in subsection 5.10), so the image of
2∗, which equals the kernel of j∗, is the subgroup 2Z of “even” Chern classes, and 2∗ is just
multiplication by 2, which is injective. This forces ∂ : Hˇ1(CPm,Z2)→ Hˇ2(CPm,Z) to be the
zero homomorphism, and thus j∗ : Hˇ1(CPm,Z)→ Hˇ1(CPm,Z2) is onto; but Hˇ1(CPm,Z) = 0,
again by (2.5), so we conclude that Hˇ1(CPm,Z2) = 0.
7.3 Spinc structures
What can be done about manifolds like CP2, which are complex manifolds but do not admit a
spin structure? Since our immediate aim is to describe an irreducible module for the complex
Clifford bundle over such manifolds, we could relax our requirements slightly by replacing
the required structure group of the tangent bundle by Spinc(n) rather than Spin(n). (Recall
that Spinc(n) is the product of the group Spin(n) and the unitary scalars U(1) within the
complex Clifford algebra C`(V ).) We say that a (compact, oriented, Riemannian) manifold
M admits a spinc structure if there is a principal Spinc(n)-bundle P c
η−→M and a bundle
map τ : P c → Qc, satisfying τ(p · g) = τ(p) ◦ φc(g) for p ∈ P and g ∈ Spinc(n); here Qc is
a principal SO(n) × U(1)-bundle of the form Q × R−→M , where8 Q is the SO(n)-bundle
of oriented orthonormal frames of the tangent bundle, as before, and R−→M is a principal
U(1)-bundle which we may choose as we please.
Now a principal U(1)-bundle is just the frame bundle of a complex line bundle L→ M ,
so (up to equivalence) these are classified by Hˇ2(M,Z). The modulo-2 reduction j∗[L]
is an element of Hˇ2(M,Z2), where the second Stiefel-Whitney class also lives. Suppose
that w2(M) = j∗[L∗] in H2(M,Z2), i.e., that w2(M) + j∗[L] = 0; then one can find local
sections h˜ij : Ui ∩ Uj → Spinc(n) such that φc(h˜ij) = λijgij are the transition functions
of TM ⊕ L−→M , which patch together properly to give the required principal bundle
P c−→M .
Exercise 7.5. Assume that w2(M) + j∗[L] = 0 and construct P c−→M as indicated.
If a spinc structure exists, M is called a spinc manifold. The foregoing argument says
that this is the case iff w2(M) ∈ j∗(Hˇ2(M,Z)). Moreover, inequivalent spinc structures are
parametrized by the classes of complex line bundles, i.e., by Hˇ2(M,Z).
7Actually, the formula (2.5) refers to singular or de Rham cohomology; but we can establish an iso-
morphism between de Rham and Cˇech cohomology in degree 3 by a simple modification of the proof of
Proposition 5.5. For the isomorphism between the de Rham–Cˇech cohomologies in any degree, see [12, 17].
8The fibrewise product of principal bundles corresponds, by association, to the Whitney sum of vector
bundles; thus, if Q and R are the frame bundles of vector bundles E and F over M , with respective structure
groups G and H, then Q×R−→M may be defined as the frame bundle of E⊕F −→M , which is a principal
G×H-bundle.
83
Lemma 7.5. The complex projective space CPm is a spinc manifold for any positive inte-
ger m.
Proof. In the proof of Proposition 7.4, we verified the surjectivity of the homomorphism
j∗ : Hˇ2(CPm,Z)→ Hˇ2(CPm,Z2), without reference to the parity of m.
In fact, any compact complex manifold carries a spinc structure. This follows from the
relation w2(M) = j∗(c1(TholM)), where TholM −→M is the holomorphic tangent bundle,
i.e., the complex vector bundle whose fibres Tx,holM are spanned by the tangent vectors
(∂/∂z1)x, . . . , (∂/∂z
m)x. To get a more concrete construction, we first regard U(m) as a
subgroup of SO(2m) and consider the homomorphisms τ : U(m)→ SO(2m)×U(1) given by
τ(g) := (g, det g), and σ : U(m) → Spinc(2m) defined as follows. For each g ∈ U(m) there
is an orthonormal basis {e1, . . . , em} of Cm which diagonalizes g, i.e., g(ej) = eiαjej; write
fj := iej and aj := e
iαj/2(cos 1
2
αj + (sin
1
2
αj)ejfj) ∈ C`+(2m); then σ(g) := a1a2 . . . am ∈
Spinc(2m).
Exercise 7.6. Check that all the aj commute,
9 that σ is a well-defined homomorphism, and
that φc ◦ σ = τ .
Exercise 7.7. The linear map on R2m determined by Jej := fj, Jfj := −ej is a complex
structure; and U(m) = { g ∈ SO(2m) : gJ = Jg }. Write k := e1f1 + · · ·+ emfm ∈ C`+(2m)
and check that exp(pik) = (−1)m and φ(exp(pi
4
k)) = J . Define the metaunitary group MU(m)
as { a ∈ Spin(2m) : ak = ka } and verify that φ(MU(m)) = U(m). Show that the centre of
the group MU(m) equals { exp(θk) : −pi < θ ≤ pi } ' U(1) if m is odd, whereas the centre
is {± exp(θk) : 0 ≤ θ < pi } ' U(1)× Z2 if m is even.
Now we can exhibit a spinc structure on a complex manifold M . Take any Hermitian
metric on M and let Q′−→M be the unitary frame bundle of the holomorphic tangent
bundle TholM −→M . Associate to it, firstly, a principal Spinc(2m)-bundle P −→M via
the homomorphism σ, as in (1.3); and secondly, a principal U(1)-bundle R−→M via the
homomorphism det : U(m)→ U(1). The latter is just the frame bundle of the complex line
bundle K∗−→M where K −→M is the so-called canonical line bundle on M , defined by
Γ(K) := Am,0(M). Since φc ◦ σ = τ , the principal bundle P −→M yields the required spinc
structure.
Of course, a spin manifold is automatically a spinc manifold. It suffices to use the trivial
U(1) bundle R = M × U(1) to build a spinc structure from a given spin structure.
Exercise 7.8. Complete the following construction of the spin structure for CPm, with m odd,
due to Dabrowski and Trautman [21]. Show that CPm is diffeomorphic to the homogeneous
space U(m+ 1)/(U(1)×U(m)) and that the unitary frame bundle is given by Q′ := U(m+
9For a fixed orthonormal basis {e1, . . . , em} of Cm, the elements a1, . . . , am and the scalars eiθ1 generate
a subgroup T of Spinc(2m) which is isomorphic to an (m + 1)-dimensional torus Tm+1; moreover, since
φc(a1a2 . . . am) is a block diagonal matrix over R2m, the centralizer of T in Spinc(2m) is T itself, so it is a
maximal torus. If b ∈ Spinc(2m), then by pulling back φc(b) to U(m) via τ and diagonalizing the resulting
unitary matrix, one sees that b lies in some conjugate subgroup aTa−1. This exemplifies the well-known
theorem of Weyl [13] that the maximal tori in a compact connected Lie group are conjugate and cover the
whole group.
84
1)/U(1). Let MU(m) be the metaunitary group of Exercise 7.7 and let P ′ := MU(m +
1)/U(1), where the subgroup U(1) of MU(m + 1) is given as { exp(θk) : 0 ≤ θ < pi }, the
identity component of the centre. Check that the inclusion MU(m) ⊂MU(m+1) drops to a
free right action of MU(m) on P ′, and that the double covering φ : MU(m+ 1)→ U(m+ 1)
drops to a double covering τ ′ : P ′ → Q′, which intertwines the respective actions of MU(m)
and U(m). Finally, show how the inclusions MU(m) ↪→ Spin(2m) and U(m) ↪→ SO(2m)
associate to these new principal bundles P and Q together with a double covering τ : P → Q
which yields the desired spin structure.
Catalogues of spin manifolds and spinc manifolds are given in several places, e.g., [29, 39].
As well as the classes of manifolds just discussed, it is worth mentioning that any sphere
Sn is spin; any simply connected Lie group is spin; any orientable 2-dimensional manifold
is spin (e.g., any Riemann surface); any 3-dimensional manifold is spin; and any orientable
4-dimensional manifold is spinc. As counterexamples, we note that CP2 is an orientable
4-dimensional manifold which is not spin; and the homogeneous space SU(3)/SO(3) is an
orientable 5-dimensional manifold which is not spinc.
7.4 The spinor module
Definition 7.7. Let M be a spin manifold of even dimension n = 2m, and let P −→M be
the principal Spin(n)-bundle defining its spin structure.10 Let S = Λ•CW be an irreducible
Clifford module for C`(n), i.e., a Fock space of complex dimension 2m, and let c : Spin(n)→
EndC S denote the spin representation. Let S(M)−→M be the complex vector bundle
associated to the spin structure P via the spin representation c. It is called the spinor
bundle over M . Its module of sections Γ(S(M)) is an irreducible Clifford module, and will
be called the spinor module for the algebra Γ(C`(M)).
Proposition 7.6. If M is an even-dimensional spin manifold, any Clifford module Γ(F ) is
of the form Γ(W ⊗ S(M)), where Γ(W ) is a trivial Clifford module, i.e., the Clifford action
is (κ,w) 7→ w for κ ∈ Γ(C`(M)), w ∈ Γ(W ).
Proof. Let Γ(F ) be given. If A = C∞(M), any A-linear map from Γ(S(M)) to Γ(F ) is of
the form τ∗ where τ : S(M)→ F is a bundle map, i.e., τ∗ is an element of Γ(Hom(S(M), F )).
Thus any such map which intertwines the two Clifford actions belongs to Γ(W ), where W :=
HomC`(M)(S(M), F ) is the vector bundle over M with fibres Wx = HomC`(n)(S(M)x, Fx).
From Exercise 6.17 it follows that wx ⊗ σx 7→ wx(σx) gives an vector space isomorphism
Wx ⊗ S(M)x ' Fx. (If wx(σx) = 0 for any nonzero σx, then 0 = c(a)[wx(σx)] = wx(c(a)σx)
for all a, so wx = 0 by irreducibility of S(M)x.) The intertwining property wx(c(a)σx) =
c(a)[wx(σx)] shows that Wx⊗S(M)x becomes a C`(n)-module via the recipe c(a)[wx⊗σx] :=
wx ⊗ c(a)σx, and the aforementioned isomorphism intertwines this action with the given
action on Fx. Globally, we obtain an invertible bundle map from W ⊗S(M) to F and hence
an A-linear isomorphism from Γ(W ⊗ S(M)) to Γ(F ) which intertwines the action
c(κ)[w ⊗ σ] := w ⊗ c(κ)σ (7.6)
10If M carries more than one spin structure, we choose and fix a particular one.
85
with the given Clifford action on Γ(F ).
Exercise 7.9. Prove that Γ(EndC`(M) F ) ' Γ(EndCW ); in other words, match A-linear op-
erators commuting with the Clifford action on Γ(F ) to A-linear operators on Γ(W ).
Definition 7.8. The passage from the irreducible spinor module Γ(S(M)) to a Clifford
module of the form Γ(W ⊗S(M)) whose Clifford action is given by (7.6) is called a twisting
by the vector bundle W −→M (which in principle may be any complex vector bundle). We
refer to Γ(W ⊗ S(M)) as a twisted Clifford module.
Notice, in particular, that any Clifford module Γ(F ) where F −→M has minimal rank
2m is obtained by twisting the spinor module with a complex line bundle. Clearly, then,
the twisting may affect the global topology of F , but for algebraic properties of the Clifford
action it is enough to study the spinor module.
7.5 The spin connection
The next task is to show that the spinor module admits a distinguished connection, to be
called the “spin connection”, which satisfies a “Leibniz rule” with respect to the Clifford
action. This is not a trivial matter, as the spinor module is not “tensorial”, that is, its
existence has topological obstructions not faced by modules of tensors, vector fields, or
differential forms. Therefore the algebraic techniques used in Section 5 to construct new
connections from old ones are not enough to produce a suitable connection on the spinor
bundle.
What is needed is to use the spin representation. This we have defined, in Section 6,
as a group representation of Spin(V, q) as unitary operators on S which does not drop to a
representation of the special orthogonal group SO(V, q). It may, however, be regarded as a
projective (or “double-valued”) representation of SO(V, q). The corresponding infinitesimal
representations of Lie algebras are not troubled by this topological problem,11 and there is
a linear isomorphism τ : C2(V, q)→ so(V, q) between the Lie algebras.
Definition 7.9. Let µ˙ : so(V, q)→ End+C(S) be the linear map given by
µ˙(A) := c(τ−1(A)), (7.7)
Note that µ˙(A) is skewhermitian since the Clifford action is selfadjoint, so c(b)† = c(b¯) =
−c(b) for b ∈ C2(V, q) by Lemma 6.5. We call µ˙ the derived spin representation.
Lemma 7.7. The derived spin representation satisfies the identity
[µ˙(A), c(v)] = c(Av) (7.8)
for all A ∈ so(V, q), v ∈ V .
11This is because V is finite-dimensional. In the infinite-dimensional case, the covering map φ has a
1-dimensional kernel (a circle), which gives rise to a “spin anomaly” in the infinitesimal representation of
SO(V, q). For details, consult [31].
86
Proof. It is equivalent to show that [τ−1(A), v] = Av as elements of the Clifford algebra
C`(V, q). However, this follows immediately from the formula of Lemma 6.7 for τ−1(A), and
from (6.15):
[τ−1(A), v] =
1
4
n∑
j,k=1
q(Aej, ek) [ejek, v]
=
1
2
n∑
j,k=1
q(Aej, ek)
(
q(ej, v)ek − q(ek, v)ej
)
=
1
2
n∑
j=1
q(ej, v)Aej +
1
2
n∑
k=1
q(ek, v)Aek = Av,
on using the antisymmetry of A.
It would be helpful to have a version of (7.8) which is valid when vectors v ∈ V are
replaced by arbitrary elements b ∈ C`(V, q). To accomplish this, one needs to extend the
operator A on V to a linear operator on C`(V, q). In fact, A extends as a derivation of the
Clifford algebra: define Aˆ(1) := 0, Aˆ(v) := Av for v ∈ V , and
Aˆ(v1 . . . vk) :=
k∑
j=1
v1 . . . vj−1(Avj)vj+1 . . . vk (7.9)
for v1, . . . , vk ∈ V .
Exercise 7.10. Check that the definition (7.9) is consistent by first verifying that Aˆ(uv) +
Aˆ(vu) = 0 for u, v ∈ V . Then show that (7.8) extends to the identity [µ˙(A), c(b)] = c(Aˆb)
for all A ∈ so(V, q), b ∈ C`(V, q).
Now we can translate these algebraic identities to relations among sections of bundles over
a spin manifold M , by applying them at each fibre. Thus, if A ∈ Γ(EndE) where E−→M
is a Euclidean vector bundle, and if (As | t) = −(s | At) for s, t ∈ Γ(E), then A(x) ∈ so(Ex)
for x ∈ M , so we may write A ∈ Γ(so(E)). In particular, if E = TM is the tangent
bundle, the notation A ∈ Γ(so(TM)) means that A is an operator on Γ(TM) = X(M)
satisfying g(AX, Y ) = −g(X,AY ) for X, Y ∈ X(M). We write µ˙(A) to denote the section
x 7→ µ˙(A(x)) of the spinor bundle S(M)−→M . This defines a map µ˙ : Γ(so(TM)) →
Γ(S(M)). Furthermore, by tensoring these A-modules of sections with Ak(M), we obtain
maps µ˙ : Ak(M, so(TM))→ Ak(M,S(M)).
If ∇ is a connection on the tangent bundle compatible with the metric, such as the Levi-
Civita connection for instance, then, locally at least, ∇ = d + α where α ∈ A1(U, so(TM))
in view of the metric compatibility condition (5.7). Thus µ˙(α) makes sense as an element of
A1(U, S(M)), where U is the domain of α.
Theorem 7.8. Let M be an even-dimensional spin manifold and let ∇ be the Levi-Civita
connection on the complex Clifford bundle C`(M)−→M . Then there is a connection ∇S on
87
the spinor bundle S(M)−→M satisfying the following Leibniz rule:
∇S(c(κ)σ) = c(∇κ)σ + c(κ)(∇Sσ), (7.10)
for all κ ∈ Γ(C`(M)), σ ∈ Γ(S(M)).
Proof. Let U = {Uj} be a covering of M by chart domains and let {fj} be a smooth
partition of unity subordinate to U, with fj(x) > 0 iff x ∈ Uj. Then fj∇ is a connection
on Uj compatible with the metric fjg. Suppose for the moment that there exist connections
∇Sj on each restriction of the spinor bundle S(M)−→M to Uj, satisfying ∇Sj (c(κ)σ) =
c(fj∇κ)σ+c(κ)(∇Sj σ), whenever σ vanishes outside Uj; then the recipe ∇Sτ :=
∑
j∇Sj (fjτ),
for any τ ∈ Γ(S(M)), defines a connection satisfying (7.10).
Therefore, we may replace M by any chart domain Uj; or we may equivalently suppose
that the vector bundles C`(M)−→M and S(M)−→M are trivial, and that the Levi-Civita
connection may be written as ∇ = d + α with α ∈ A1(M, so(TM)). The formula ∇ =
d+α, defined initially on vector fields, is also applicable to Clifford products of vector fields,
provided ακ is interpreted as the derivation action of α on κ ∈ Γ(C`(M)), in view of the
Leibniz rule (7.4).
Now introduce a connection ∇S on S(M) by defining
∇S := d+ µ˙(α). (7.11)
From (7.8), extended to Clifford algebra, we get [µ˙(α), c(κ)] = c(ακ) on Γ(S(M)), and then
∇S(c(κ)σ) = d(c(κ)σ) + µ˙(α)(c(κ)σ)
= c(dκ)σ + c(κ) dσ + [µ˙(α), c(κ)]σ + c(κ)µ˙(α)σ
= c(dκ)σ + c(ακ)σ + c(κ)
(
dσ + µ˙(α)σ
)
= c(∇κ)σ + c(κ)(∇Sσ),
verifying (7.10).
We have constructed one solution ∇S to (7.10). If ∇˜S is another connection on S(M) sat-
isfying the same module-derivation property, then ∇˜S−∇S is given by β ∈ A1(M,EndS(M))
such that β(c(κ)σ) ≡ c(κ)(βσ). That is to say, β lies in A1(M,EndC`(M) S(M)) ' A1(M) in
view of Exercise 7.9 and the irreducibility of S(M). Therefore, ∇S is unique up to addition
of a scalar action by a 1-form on M .
We may apply the same reasoning to any Clifford module, with the following result.
Proposition 7.9. Let Γ(F ) = Γ(W ⊗ S(M)) be a Clifford module over a spin manifold M ,
and let ∇˜ be a connection on F which is a module derivation, i.e., ∇˜(c(κ)τ) ≡ c(∇κ)τ +
c(κ)(∇˜τ). Then there is a unique connection ∇W on W such that ∇W ⊗ 1 + 1⊗∇S = ∇˜.
Proof. Since τ ∈ Γ(F ) is a finite sum of the form ∑j wj ⊗ σj, we may rewrite the module
derivation property of ∇˜ as ∇˜(w ⊗ c(κ)σ) = w ⊗ c(∇κ)σ + c(κ)(∇˜(w ⊗ σ)). Let ∇W0 be an
arbitrary connection on W . Then we also get
[∇W0 ⊗ 1 + 1⊗∇S, 1⊗ c(κ)](w ⊗ σ) = w ⊗ c(∇κ)σ,
88
so β := ∇˜− (∇W0 ⊗1 + 1⊗∇S) lies in A1(M,EndF ) and commutes with the Clifford action,
that is, β ∈ A1(M,EndC`(M) F ). By Exercise 7.9 it is of the form β = α ⊗ 1 for a unique
α ∈ A1(M,EndW ). Writing ∇W := ∇W0 +α now gives ∇˜ = ∇W ⊗1+1⊗∇S, as desired.
Lemma 7.10. The curvature of the spin connection is µ˙(R), where R denotes the Rieman-
nian curvature, R ∈ A2(M, so(TM)).
Proof. It suffices to check this on a chart domain U , where we may write ∇S = d+µ˙(α). The
curvature form of the spin connection is given by (5.10) as ωS := µ˙(dα+α∧α) = µ˙(R).
7.6 Local coordinate formulas
Definition 7.10. On a chart domain U of M with local coordinates {x1, . . . , xn}, we shall
write the basic vector fields as ∂j ≡ ∂/∂xj. If M is Riemannian, the Christoffel symbols
Γkij of the Levi-Civita connection on the chart domain U are the functions in C
∞(U) defined
by
∇∂i∂j = Γkij∂k, (7.12)
or equivalently, ∇∂j = Γkij dxi ⊗ ∂k. In particular, Γkij = dxk
(∇∂i∂j). Notice also that Γk•j
give the matrix components αkj of α ∈ A1(U,EndTM).
Exercise 7.11. Check that the Levi-Civita connection on the cotangent bundle is determined
locally by ∇ dxk = −Γkij dxi ⊗ dxj, or equivalently, ∇∂i dxk = −Γkij dxj.
Exercise 7.12. From the definition (5.27) of the Levi-Civita connection, show that
Γkij =
1
2
gkl(∂igjl + ∂jgil − ∂lgij), (7.13)
using the relations [∂i, ∂j] = 0.
Exercise 7.13. Show that the Christoffel symbols on the sphere S2 are given in spherical
coordinates by
Γφθφ = Γ
φ
φθ = cot θ, Γ
θ
φφ = − sin θ cos θ, Γkij = 0 otherwise, (7.14)
by applying (7.13) to the metric g := dθ2 + sin2 θ dφ2.
Exercise 7.14. The components of the Riemann curvature tensor may be written [52] as
Rijkl := dx
i(R(∂k, ∂l)∂j). Verify the “Ricci identities”:
Rijkl = ∂kΓ
i
lj − ∂lΓikj + ΓmljΓikm − ΓmkjΓilm,
by using the curvature formula (5.11).
It is somewhat more common to use the components Rijkl = gimR
m
jkl = (∂i |R(∂k, ∂l∂j)).
One sees from (5.11) that, for k, l fixed, the matrix with (i, j)-entry Rijkl is antisymmetric.
Exercise 7.15. Verify that the curvature of the spin connection is given locally by
ωS(∂k, ∂l) = −14Rijkl c(dxi) c(dxj), (7.15)
on account of (7.7).
89
We now express the spin connection itself in local coordinates. Fix a chart domain
U ⊂ M , and write g = gij dxi · dxj there. Since the matrix G = [gij] is positive definite, we
can find a matrix H = [hαj ] of functions in C
∞(U) such that H tH = G. Indeed, if G1/2 is
the positive definite square root of G, we may take H = AG1/2 where A is an orthogonal
matrix at each point of U with A : U → SO(n) smooth.12 Choose and fix such an H, and let
H−1 = [h˜rβ] be its inverse matrix. Recall that g
−1 = grs ∂r · ∂s is the metric on the cotangent
bundle T ∗M −→M ; thus we have
hαi δαβh
β
j = gij, h˜
i
αδ
αβh˜jβ = g
ij.
Orthonormal bases for A1(U) and X(U) are then given by
θα := hαj dx
j, Eβ := h˜
r
β∂r.
(We use the convention of reserving Latin indices for coordinate bases and Greek indices for
orthonormal bases.) By construction, we get
g(Eα, Eβ) = δαβ, g
−1(θα, θβ) = δαβ,
and also (θα)] = Eα, (Eβ)
[ = θβ on U .13
Locally, a smooth section of the spinor bundle looks like a smooth map from an open
subset U of M (which can actually be taken as the complement of the closure of an arbitrarily
small open set in M [19]) to the Hilbert space S of the spinor representation. Let { γα ≡
γα : α = 1, . . . , n } be a fixed set of unitary skewadjoint operators on S with the property
γαγβ + γβγα = −2 δαβ.
[For instance, take γα := c(eα) where {e1, . . . , en} is an orthonormal basis for (V, q).] We set
c(dxr) := h˜rβγ
β. (7.16)
From (7.7) it is immediate that
c(dxr)c(dxs) + c(dxs)c(dxr) = −2grs = −2(dxr | dxs), (7.17)
so that (7.16) in fact defines a local Clifford action of A1(U) on Γ(U, S). Conversely, a given
Clifford action on the spinor bundle, which necessarily satisfies (7.17), together with a given
fixed set of γα, defines via (7.16) matrices [h˜rβ] and [h
α
j ] satisfying (7.7).
14
It is convenient to introduce “mixed” or “orthogonal” Christoffel symbols Γ˜βiα, Γ̂
β
α in
C∞(U) by
∇∂iEα =: Γ˜βiαEβ; ∇EEα =: Γ̂βαEβ.
12We require detA = 1 so as to preserve the orientation when passing to orthonormal bases in (7.7).
13These last relations give rise to the common phrasing: “we need not worry about raising and lowering
indices, so long as we deal with orthonormal bases”.
14The possibility of varying the Clifford action by premultiplying H by arbitrary SO(n)-valued functions,
provided only that these be compatible with changes of local charts, reflects the noncanonical nature of the
spinor bundle.
90
The Γ˜βiα and Γ̂
β
α are antisymmetric in the indices α, β; for instance,
Γ˜βiα + Γ˜
α
iβ = g(∇∂iEα, Eβ) + g(Eα,∇∂iEβ) = ∂i(δαβ) = 0,
on account of the compatibility with the metric: α ∈ A1(U, so(TM)).
The components of the spin connection are given by the EndS-valued functions
ωi :=
1
4
Γ˜βiα γ
αγβ, ν :=
1
4
Γ̂βα γ
αγβ =
1
4
h˜iΓ˜
β
iα γ
αγβ. (7.18)
Exercise 7.16. Show that hαj Γ˜
β
iα = −∂i(hβj ) + Γkijhβk . Deduce the formula
ωi =
1
4
(
Γkijgkl − ∂i(hαj )δαβhβl
)
c(dxj) c(dxl),
that expresses the spin connection in a coordinate basis.
Exercise 7.17. Let X = aj∂j be the local expression of a vector field on M . Show that the
contraction of X with the spin connection is given locally as an operator on C∞(U, S) by:
∇SXσ = aj(∂jσ + ωj)σ,
and, in particular, that ∇SEσ = Eσ + ν(x)σ.
Exercise 7.18. Check that ∇S is a hermitian connection on the spinor module.
8 Dirac operators and Laplacians
A Dirac operator is an operator of odd parity on a Clifford module over a spin (or spinc) ma-
nifold, which is a first-order differential operator whose corresponding Leibniz rule involves
Clifford multiplication by differentials of functions. Its square is therefore an even-parity
second-order differential operator, which gives a far-reaching generalization of the Laplace-
Beltrami operator on a Riemannian manifold: we may call this square a “generalized Lapla-
cian”, using the terminology of [9]. Moreover, any Dirac operator over a compact manifold
is elliptic, which implies that its inverse (off its finite-dimensional kernel) is a compact op-
erator, so it has a discrete spectrum of eigenvalues which give precise information about the
geometry of the manifold. Even more importantly, as Connes [18] has pointed out, the Dirac
operator determines the Riemannian metric, and therefore serves as a gateway for reformu-
lating the entire corpus of Riemannian geometry in analytic terms, allowing its extension to
discrete spaces, fractals, spaces of tilings and group orbits, and many other contexts whose
geometric structure seemed only a few short years ago to be hopelessly beyond reach.
8.1 Connections and differential forms
Definition 8.1. Let ∇ be a connection on the cotangent bundle T ∗M −→M of a compact
manifold M ; the Leibniz rule ∇(ξ ∧ ω) = (∇ξ)∧ ω+ ξ ∧ (∇ω) extends it to a connection on
91
the exterior product bundle Λ•T ∗M −→M , so that ∇ maps A•(M) to A1(M)⊗A A•(M).1
The exterior product defines an A-linear map ˆ : A1(M)⊗AA•(M)→ A•(M) by ˆ(ω⊗η) :=
ω ∧ η. The composition ˆ ◦ ∇ is an R-linear endomorphism of A•(M). In local coordinates
∇ω = dxj ⊗∇∂jω, and so ˆ(∇ω) = (dxj)∇∂jω where (dxj) denotes exterior multiplication
by dxj.
We introduce also a contraction map ιˆ : X(M)⊗A A•(M)→ A•(M) by ιˆ(X ⊗ η) := ιXη.
If (M, g) is a Riemannian manifold, we identify X(M) with A1(M) via the metric g, and
write ιˆ(α⊗η) := ια]η. The composed map ιˆ◦∇ is another R-linear endomorphism of A•(M).
Lemma 8.1. Let ∇ be a connection on the cotangent bundle T ∗M −→M ; and let T ∈
A2(M,TM) be the torsion of the dual connection ∇∗ on the tangent bundle. Then for any
ω ∈ A•(M),
ˆ(∇ω) = dω − ιTω, (8.1)
where ιT : A
k(M)→ Ak+1(M) denotes contraction with the torsion.
Proof. Write ∇ω = βk ⊗ ηk ∈ A1(M) ⊗ A•(M); then ∇Xω = βk(X) ηk by contraction.
Therefore, if X, Y ∈ X(M) and ω ∈ A1(M),
ˆ(∇ω)(X, Y ) = (βk ∧ ηk)(X, Y ) = βk(X)ηk(Y )− βk(Y )ηk(X)
= (∇Xω)(Y )− (∇Y ω)(X)
= X(ω(Y ))− Y (ω(X))− ω(∇∗XY −∇∗YX)
= dω(X, Y )− ω(∇∗XY −∇∗YX − [X, Y ])
= dω(X, Y )− ω(T (X, Y )). (8.2)
This establishes (8.1) for 1-forms. Moreover, ιY ιX(ˆ(∇ω)) = ιY ιX(dω)− ιT (X,Y )ω, for a form
ω ∈ Ak(M) of any degree, by an easy extension of the above argument.
Exercise 8.1. Generalize (8.2) to higher-degree forms.
Corollary 8.2. If ∇ is a connection on the cotangent bundle T ∗M −→M whose dual con-
nection is torsion-free, then ˆ ◦ ∇ equals the exterior derivation, d.
In particular, ˆ◦∇LC = d for the Levi-Civita connection ∇LC (on the cotangent bundle).
There is a kind of dual formula relating the Levi-Civita connection with the codifferen-
tial δ. Before revealing that, it is useful to consider the divergence of a vector field on a
Riemannian manifold.
1We find it more convenient in this Section to write Ar(M,E) = Ar(M)⊗A Γ(E), reversing the order of
the tensor product given in (5.3). The Leibniz rule (5.4) is adapted accordingly, and the contraction operator
ιX : A1(M,E)→ Γ(E) is now given by ιX(β ⊗ s) := β(X) s.
92
8.2 Divergence of a vector field
Definition 8.2. Let X be a vector field on an oriented manifold M with volume form ν.
The divergence of X relative to ν is the function divν X ∈ C∞(M) determined by
(divν X)ν := LXν.
If f ∈ C∞(M) is never zero, fν is another volume form with (divfν X) fν = LX(fν) =
(Xf)ν + fLXν = (Xf + f divν X)ν, so that
divfν X = divν X +
Xf
f
. (8.3)
If (M, g) is a Riemannian manifold with Riemannian volume form Ω, we write divX ≡
divΩX.
Since LXΩ = d(ιXΩ) by Cartan’s identity, the identity
∫
M
(divX)Ω = 0 (the divergence
theorem) is an immediate consequence of Stokes’ theorem.
Over a chart domain U , we may consider the local volume form µ = dx1 ∧ · · · ∧ dxn. If
X = Xj ∂j ∈ X(U), then by computing LX(dx1 ∧ · · · ∧ dxn) we obtain the standard formula
divµX = ∂jX
j. Since Ω =
√
det g µ on U by (3.3), we derive from (8.3) a local formula for
the Riemannian divergence:
divX = ∂jX
j +Xj ∂j(ln
√
det g) =
1√
det g
∂j(X
j
√
det g).
Lemma 8.3. The Riemannian divergence of a vector field is obtained from the Levi-Civita
connection on the tangent bundle via the local formula
divX = dxj(∇LC∂j X). (8.4)
Proof. The connection ∇LC is determined over a chart domain U by the Christoffel symbols
Γkij of (7.12), satisfying Γ
k
ij = dx
k
(∇LC∂i ∂j). The Leibniz rule now yields, for X = Xj ∂j ∈
X(U),
dxj(∇LC∂j X) = ∂jXj + ΓjjkXk.
The verification of (8.4) thus reduces to checking the formula
Γjjk = ∂k(ln
√
det g).
This relation is well known [36], but it is instructive to see how it goes. Recall that the
Christoffel symbols are given by (7.13):
Γkij =
1
2
gkl(∂igjl + ∂jgil − ∂lgij).
93
If [Gij] denotes the adjugate matrix to [gij] (that is, the matrix of cofactors), then by Cramer’s
rule gij = Gij/ det g. Now gjl(∂jgkl − ∂lgkj) = 0 since the matrix [gij] is symmetric, and so
Γjjk =
1
2
gjl ∂kgjl =
1
2 det g
Gjl ∂kgjl =
1
2 det g
∂ det g
∂gjl
∂kgjl
=
1
2 det g
∂k(det g) =
1
2
∂k(ln det g),
on using the Cramer expansion det g = gjlG
jl.
If we choose local orthonormal bases {θα} for A1(U) and {Eβ} for X(U) with Eα = (θα)],
we may write, as in (7.7), θα = hαj dx
j and Eβ = h˜
k
β∂k. Then θ
α(∇LCEαX) = hαj dxj(h˜kα∇LC∂k X),
so we obtain the alternative formula to (8.4), using local orthonormal bases:
divX = θα(∇LCEαX). (8.5)
Yet another divergence formula, for a particular set of vector fields, is the following.
Lemma 8.4. Let M be an oriented Riemannian manifold. Given ζ ∈ Ak(M) and η ∈
Ak−1(M), define Z ∈ X(M) by ω(Z) := (ζ | ω ∧ η) for ω ∈ A1(M). The divergence of this
vector field is given locally by
divZ = Eα(ζ | θα ∧ η),
where {θ1, . . . , θn} and {E1, . . . , En} are local orthonormal bases of 1-forms and vector fields
respectively, with Eα = (θ
α)].
Proof. Since ω 7→ (ζ |ω∧η) is A-linear on A1(M), it indeed defines a vector field Z ∈ X(M).
Since Ω = θ1 ∧ · · · ∧ θn locally, we find that ιZΩ =
∑n
α=1(−1)α−1(ζ | θα ∧ η) θ1 ∧ · ·
α∨· · ∧ θn,
and it follows that LZΩ = d(ιZΩ) = Eα(ζ | θα ∧ η) Ω.
8.3 The Hodge–Dirac operator revisited
Lemma 8.5. If ∇LC denotes the Levi-Civita connection on the cotangent bundle of an
oriented Riemannian manifold M , and δ : A•(M) → A•(M) is the codifferential, then ιˆ ◦
∇LC = −δ.
Proof. Choose ζ ∈ Ak(M) and η ∈ Ak−1(M) with k ≥ 1, and define Z ∈ X(M) by ω(Z) :=
(ζ | ω ∧ η). Choose local orthonormal bases {θ1, . . . , θn} and {E1, . . . , En} of 1-forms and
vector fields, with Eα ≡ Eα = (θα)]. Since (ξ | β ∧ ω) = (ιβ]ξ | ω) for any ξ, ω ∈ A•(M) and
β ∈ A1(M), we obtain
(ζ | dη) = (ζ | ˆ(∇LCη)) = (ζ | (θα)∇LCEα η)
= (ιEαζ | ∇LCEα η) = Eα(ιEαζ | η)− (∇LCEα ιEαζ | η)
= Eα(ζ | θα ∧ η)− (ιEα∇LCEα ζ | η)
= divZ − (ιˆ(∇LCζ) | η),
94
where we have used Corollary 8.2, the metric compatibility of ∇LC , and the commutation
relation [∇X , ιY ] = ι[X,Y ] (see Exercise 5.21). On multiplying both sides by Ω and integrating,
the term divZ is killed by Stokes’ theorem, and we obtain 〈〈δζ |η〉〉 := 〈〈ζ |dη〉〉 = −〈〈ιˆ(∇LCζ)|
η〉〉; since η is arbitrary, this gives ιˆ(∇LCζ) = −δζ.
Corollary 8.6. If cˆ : Γ(C`(M)) ⊗ A•(M) → A•(M) denotes the Clifford action on the
de Rham algebra of an oriented Riemannian manifold, and if ∇LC denotes the Levi-Civita
connection, then
cˆ ◦ ∇LC = d+ δ. (8.6)
Proof. If ∇ω = βk ⊗ ηk, then cˆ(∇ω) = c(βk)ηk = (βk)ηk − ι(βk)]ηk, so that cˆ = ˆ− ιˆ. Thus
(8.6) follows immediately from Corollary 8.2 and Lemma 8.5.
In subsection 4.5, we referred to d+δ as the “Hodge–Dirac operator”, which we obtained,
in a somewhat ad-hoc fashion, as a square root of the Hodge Laplacian. The formula (8.6)
reveals its true nature as the composition of a Clifford action and a connection which is
compatible with this action (recall Proposition 7.1). This opens the way for the general
definition of Dirac operators.
8.4 Dirac operators
Definition 8.3. Let Γ(F ) be a selfadjoint Clifford module over a compact oriented Rieman-
nian manifold M , that is, let F −→M be a Hermitian vector bundle and let there be given
a selfadjoint Clifford action cˆ : Γ(C`(M) ⊗ F ) → Γ(F ), where we write cˆ(κ ⊗ ψ) := c(κ)ψ.
Let ∇ be a Hermitian connection on F −→M that satisfies the Leibniz rule ∇(c(κ)ψ) =
c(∇LCκ)ψ+ c(κ)(∇ψ), for κ ∈ Γ(C`(E)), ψ ∈ Γ(F ). The Dirac operator associated to the
connection ∇ and the Clifford action cˆ is
D/ := cˆ ◦ ∇.
Via Γ(F )
∇−→A1C(M)⊗AΓ(F ) cˆ−→Γ(F ), this D/ is a C-linear endomorphism of Γ(F ). In local
coordinates, D/ψ = c(dxj)∇∂jψ.
There are many Dirac operators to be found. First of all, if M is a spin manifold, we may
consider the irreducible Clifford module Γ(S), i.e., the spinor module, with its spin connection
∇S. Then D/ S := cˆ ◦ ∇S is the original Dirac operator.2 Any other Clifford module Γ(S)
on M is obtained by twisting, that is, S = W ⊗ F where W −→M is a Hermitian vector
bundle carrying the trivial Clifford action, and the compatible connection is determined,
via Proposition 7.9, by an arbitrary Hermitian connection on W . The Dirac operator d+ δ
on the de Rham algebra arises in this way, since Λ•T ∗M ' S ⊗ S ′ where S ′−→M is the
superbundle obtained from the spinor bundle by taking the opposite grading: (S ′)± := S∓.
2D/ S is usually called the Dirac operator on the spinor module.
95
Exercise 8.2. If c : C`(V )→ EndC Λ•CW is the Clifford action on Fock space, and if v ∈ V is
a unit vector, show that c′(a) := −c(vav) is another selfadjoint Clifford action, whose restric-
tion to Spin(V, q) switches the two irreducible subrepresentations of the spin representation.
Conclude that there is an irreducible Clifford module S ′−→M on any spin manifold, whose
“even” subspace Γ(S ′+) is the “odd” subspace Γ(S−) of the spinor module, and viceversa.3
Exercise 8.3. Define a Clifford action on the conjugate Fock space c¯ : C`(V ) → EndC Λ•CW
by setting c¯(v)α¯ := (P−v)α¯ − ι(P+v)α¯ for v ∈ VC, α¯ ∈ W [compare with (6.13)]. Check
that c¯(w) = −ι(w) for w ∈ W and c¯(z¯) := (z¯) for z ∈ W . If γ denotes, as usual, the
chirality element of C`(V ), show that c¯(γ)α¯ = (−1)k+1α¯ whenever α¯ ∈ ΛkCW [compare with
Proposition 6.3]. If S = Λ•CW is the spinor module for C`(V ), the dual Hilbert space S∗ is
identified with Λ•CW via
〈z¯1 ∧ · · · ∧ z¯r, w1 ∧ · · · ∧ ws〉 := δrs det
[
2q(z¯k, wl)
]
,
that is, the conjugation C : w1∧· · ·∧ws 7→ w¯1∧· · ·∧w¯s is the (antilinear) Riesz isomorphism
of S with S∗. Check that C intertwines the action c¯ on S∗ with the action c on S, and
conclude that S∗ is an irreducible Clifford module for C`(V ) whose Z2-grading is given by
(S∗)+ = C(S−) and (S∗)− = C(S+).4
Even if M is not spin, there may be many Clifford modules with compatible connections;
the de Rham algebra with the Levi-Civita connection is again the prime example. In any
case, since the obstruction to the existence of a spin structure is global, we can always
write Γ(U, F ) = Γ(U,W ⊗ S) over a chart domain (where W may depend on U) and from
any one compatible connection we can manufacture others by altering ∇W over U only.
Alternatively, we may vary the Clifford action by redefining (7.16) (which amounts to an
action of the SO(n) frame bundle), as explained in subsection 7.6. Moreover, we have the
further freedom of making a smooth change in the metric g and thereby changing the Clifford
action c on Γ(F ) (and also the connection, if necessary, to preserve compatibility). Thus any
one Dirac operator gives rise to a large family of “smoothly perturbed” Dirac operators on
the same Clifford module.
Any Dirac operator is a first-order differential operator. To avoid any possible confusion
of terminology, we make a formal definition.
Definition 8.4. Let E−→M be a vector bundle. Any A ∈ Γ(EndE) defines a C-linear
operator on Γ(E) by left multiplication, that is, (As)x := Ax(sx) for s ∈ Γ(E), x ∈M . Any
connection5 ∇ on E−→M , when contracted by vector fields X ∈ X(M), provides other
C-linear operators ∇X on Γ(E). The differential operators on E−→M are defined as
the elements of the subalgebra D(M,E) generated by Γ(EndE) and by {∇X : X ∈ X(M) }.
3In more sophisticated terms, the spinor bundles S and S′ over a spin manifold are associated to the spin
structure P −→M via the representations c and c′ of Spin(2m); these vector bundles are inequivalent since
the representations c and c′ are inequivalent.
4We are indebted to William Ugalde for clarification on this point.
5In view of (5.5), two such connections differ by the action of an element of Γ(EndE), so only a single
connection is needed here.
96
A finite sum of operators of the form A∇X1 . . .∇Xr with r ≤ k (and r = k for at least
one summand) is called a differential operator of order k. By a partition-of-unity argument,
a differential operator is of order k if it can be written as such a sum in each chart domain
separately. Thus a Dirac operator D/ = c(dxj)∇∂j is a differential operator of first order.
As an operator on the superspace Γ(F ), the Dirac operator D/ is odd, that is, D/ (Γ(F±)) ⊆
Γ(F∓), since cˆ : A1C(M)⊗A Γ(F±)→ Γ(F∓). Thus we may write
D/ =:
(
0 D/ −
D/ + 0
)
, (8.7)
where D/ ± : Γ(F±)→ Γ(F∓). Its square is an even operator: D/ 2± : Γ(F±)→ Γ(F±); and
D/ 2 =
(
D/ −D/ + 0
0 D/ +D/ −
)
.
Last but not least, a Dirac operator is essentially selfadjoint. This means that D/ : Γ(F )→
Γ(F ) extends uniquely to a selfadjoint operator on the Hilbert space L2(F ) obtained by
completing the space of smooth sections Γ(F ) with respect to the inner product 〈〈φ | ψ〉〉 :=∫
M
(φ | ψ) Ω. Regrettably, D/ is always an unbounded operator, so this selfadjoint extension
is still only densely defined. Rather than worry about identifying the precise domain of the
extended D/ , we will stick to the original domain Γ(F ); a full proof of essential selfadjoint-
ness (see [39], for instance) shows that nothing is thereby lost, as the closure of D/ on this
domain is the full selfadjoint extension.6 We shall therefore show merely that D/ is formally
selfadjoint, that is,
〈〈D/φ | ψ〉〉 = 〈〈φ |D/ψ〉〉 for φ, ψ ∈ Γ(F ). (8.8)
Proposition 8.7. The Dirac operator D/ is formally selfadjoint.
Proof. The argument is similar to that of Lemma 8.5. By invoking partitions of unity, we
can reduce the problem to verifying (8.8) when φ, ψ are smooth sections of F −→M which
vanish outside some chart domain; thus we write D/ = c(θα)∇Eα , where {θ1, . . . , θn} and
{E1, . . . , En} are local orthonormal bases of 1-forms and vector fields, with Eα ≡ Eα = (θα)].
Now
(φ |D/ψ) = (φ | c(θα)∇Eαψ) = −(c(θα)φ | ∇Eαψ)
= −Eα(c(θα)φ | ψ) + (∇Eαc(θα)φ | ψ)
= −Eα(c(θα)φ | ψ) + (c(∇LCEα θα)φ | ψ) + (c(θα)∇Eαφ | ψ)
= −Eα(c(θα)φ | ψ) + (c(∇LCEα θα)φ | ψ) + (D/φ | ψ), (8.9)
6In other words, Γ(F ) is a “core” for D/ [45]. The compactness of M is not indispensable here: essential
selfadjointness can be proven under the weaker assumption that the Riemannian manifold M is complete
[39, 53, 61].
97
where we have used the skewadjointness of c(θα), the hermiticity of the connection ∇, and
the compatibility of ∇ with the Levi-Civita connection on the cotangent bundle. We claim
that we can find a vector field Z ∈ X(M), depending on φ and ψ, such that divZ =
−Eα(c(θα)φ|ψ)+(c(∇LCEα θα)φ|ψ). Then integration over M of both sides of (8.9) (multiplied
by the Riemannian volume form Ω) yields the desired relation (8.8).
The vector field Z is defined simply by
ω(Z) := (φ | c(ω)ψ) = −(c(ω)φ | ψ),
since the right hand side is A-linear in ω ∈ A1(M). Now
−Eα(c(θα)φ | ψ) + (c(∇LCEα θα)φ | ψ) = Eα(θα(Z))− (∇LCEα θα)(Z) = θα(∇LCEαZ)
from the definition of a dual connection. The divergence formula (8.5) says that the right
hand side equals divZ, as claimed.
8.5 Laplacians
Definition 8.5. Let ∇E : Γ(E) → A1(M,E) = Γ(T ∗M ⊗ E) be a connection on a vector
bundle E−→M over a Riemannian manifold (M, g), and let ∇˜E := ∇LC ⊗∇E be its tensor
product with the Levi-Civita connection on the cotangent bundle of M . Then ∇˜E maps
Γ(T ∗M ⊗ E) to A1(M,T ∗M ⊗ E) = Γ(T ∗M ⊗ T ∗M ⊗ E), so it can be composed with ∇E.
Contraction with the metric g−1 on T ∗M gives an A-linear map Trg : Γ(T ∗M⊗T ∗M⊗E)→
Γ(E). The composition of these three maps yields the following operator on Γ(E):
∆E := −Trg ◦∇˜E ◦ ∇E,
called the Laplacian associated to the connection ∇E. The minus sign is a convention7
which assures that ∆E is a positive operator whenever ∇E is a Hermitian connection.
To express ∆E in a more tractable form, we shall compute the result of contracting
the operator ∇˜E ◦ ∇E with two vector fields X, Y ∈ X(M). If s ∈ Γ(E), we can write
∇Es = βk ⊗ sk, and so
ιY ιX(∇˜E∇Es) = ιY ∇˜EX(βk ⊗ sk) = ιY (βk ⊗∇EXsk +∇Xβk ⊗ sk)
= βk(Y )∇EXsk + (∇Xβk)(Y ) sk
= ∇EX(βk(Y )sk)−X(βk(Y )) + (∇Xβk)(Y ) sk
= ∇EX(βk(Y )sk)− βk(∇XY ) sk = ∇EX(∇EY s)− ι(∇XY )∇Es
where we have written the Levi-Civita connection simply as ∇. In other words,
ιY ιX(∇˜E ◦ ∇E) = ∇EX∇EY −∇E∇XY .
7Actually, the usual convention is to use the opposite sign; however, this results in operators, such as∑
j ∂
2
j on Rn, which are negative definite.
98
Since g−1 = gij∂i ·∂j on a chart domain, we get immediately from (7.12) the local expression
for the Laplacian:
∆E = −gij(∇E∂i∇E∂j − Γkij∇E∂k). (8.10)
With a local orthonormal basis of vector fields {e1, . . . , en}, this takes a slightly simpler
form:8
∆E = −
n∑
α=1
(∇Eeα∇Eeα −∇E∇eαeα). (8.11)
It is immediate from (8.10) or (8.11) that the Laplacian ∆E is a second-order differential
operator.
Exercise 8.4. Use the formulas of Exercise 7.11 to give an alternative derivation of (8.10).
Exercise 8.5. Show that the Laplacian ∆0 associated to the standard connection d on the
trivial line bundle S2 × C−→ S2 is
∆0 = −
(
∂2
∂θ2
+ cot θ
∂
∂θ
+
∂2
∂φ2
)
,
i.e., the Laplace–Beltrami operator on A0(S2).
Proposition 8.8. Let ∇E be a Hermitian connection on a vector bundle E over a Rie-
mannian manifold (M, g). Then its Laplacian is a formally selfadjoint and positive operator
on Γ(E).
Proof. The strategy of the proof should by now be familiar: we choose sections s, t ∈ Γ(E)
and with them construct a vector field Z ∈ X(M) such that (s|∆Et) = divZ+Tr(∇Es|∇Et).
(The last term is locally expressed as
∑n
α=1(∇Eeαs |∇Eeαt), where {e1, . . . , en} are orthonormal
vector fields.) Multiplying these functions by Ω and integrating over M , we get the relation
〈〈s |∆Et〉〉 = 〈〈∇Es | ∇Et〉〉, (8.12)
where these brackets denote integrated inner products on Γ(E) and A1(M,E) respectively.9
Positivity follows from setting s = t, and formal selfadjointness follows by repeating the
argument with the roˆles of s and t interchanged.
The vector field Z is defined by
g(Z,X) := −(s | ∇EXt) for X ∈ X(M).
Its divergence may be computed with the formula (8.5):
divZ = θα(∇eαZ) = g(∇eαZ, eα) = −g(Z,∇eαeα) + eα g(Z, eα)
= (s | ∇E∇eαeαt)− eα(s | ∇Eeαt) = (s |∆Et)− (∇Eeαs | ∇Eeαt),
(with summation over α), so divZ = (s |∆Et)− Tr(∇Es | ∇Et) as claimed.
8We rename the vector fields Eα to eα temporarily to avoid a notational clash with the exponent E
denoting a vector bundle.
9On account of (8.12), the Laplacian is often denoted ∇∗∇, as in [39], for instance.
99
We already know a second-order differential operator on the bundle Λ•T ∗M −→M ,
namely the “Hodge Laplacian”, which we now write ∆Hodge. It is natural to ask whether this
operator equals the Laplacian ∆LC associated to the Levi-Civita connection on the exterior
bundle. It turns out that they are not the same: indeed, they differ by a differential operator
of order zero, that is essentially the Ricci tensor of Riemannian geometry [8, 36, 39].
Definition 8.6. The Ricci tensor on a Riemannian manifold is the symmetric tensor Ric
of bidegree (2, 0) obtained from the Riemann curvature tensor R by defining Ric(X, Y ) as
the trace of the A-bilinear form (W,Z) 7→ (W | R(Z, Y )X) on X(M). Locally, Ric(X, Y ) =
dxk(R(∂k, Y )X). In terms of the components of the Riemann curvature tensor, we get
Ric(∂i, ∂j) = R
k
ikj.
By contracting with the metric, we obtain the curvature scalar10 K := Trg ◦Ric of the
Riemannian manifold (M, g); notice that K ∈ C∞(M). Locally, K = gijRkikj.
Exercise 8.6. Show that the Riemannian curvature tensor on the sphere S2 satisfies Rφθφθ = 1
and Rθφθφ = sin
2 θ, and deduce that Ric = g: the Ricci tensor coincides with the metric
on S2. Conclude that K = 2 on S2: the sphere is a surface of constant (Gaussian) curvature.
We can identify Ric with a tensor of bidegree (0, 2) on M , also called Ric, via the metric g;
or better, with the element of Γ(EndT ∗M) defined by Ric(ω)(X) := Ric(ω], X). The relation
between the Hodge Laplacian and the connection Laplacian, as operators on A•(M), is then
given by the so-called Weitzenbo¨ck formula:
∆Hodge = ∆
LC + Ric . (8.13)
We shall not prove this here, though we have the means to do so; consult [9] or [39] for the
details.
Two things are notable about the formula (8.13). First of all, ∆Hodge = (d + δ)
2 is the
square of the Dirac operator d+ δ on the Clifford module A•(M), so the formula says that,
for the de Rham complex at least, the square of the Dirac operator is almost, but not quite,
a Laplacian; more precisely, it differs from the Laplacian by a differential operator of lower
order that, by the way, depends only on the curvature of the connection which defines both
the Dirac operator and the Laplacian. This is one of a family of formulae due variously to
Bochner, Weitzenbo¨ck and Lichnerowicz, which say that D/ 2 − ∆ is a curvature-dependent
multiplication operator.
The second noteworthy feature is that ∆Hodge and ∆
LC are positive (formally) selfadjoint
operators. What happens if the Ricci operator is positive too? For one thing, both ∆LC and
Ric must then vanish on the kernel of the Hodge Laplacian, namely, on the harmonic forms.
Thus, for example, on S2 the Ricci operator kills 0-forms and 2-forms but acts as the identity
on 1-forms since the Ricci tensor coincides with the metric; the upshot of (8.13) is then that
any harmonic 1-form on S2 must be zero (as we already noted in Section 4), and therefore
H1dR(S2) = 0. This result is an example of a vanishing theorem, whereby certain cohomology
groups reduce to zero —a topological result— on account of positivity properties of certain
analytic operators.
10The curvature scalar is also called the Gaussian curvature when dimM = 2.
100
8.6 The Lichnerowicz formula
Proposition 8.9. Let M be a compact spin manifold, let D/ S = cˆ ◦ ∇S denote the Dirac
operator on the irreducible spinor module Γ(S), and let ∆S be the Laplacian associated to the
spin connection ∇S. Then
(D/ S)2 = ∆S +
K
4
, (8.14)
where K is the curvature scalar of M .
Proof. It suffices to prove this on any chart domain; that is, we must show that, in a local
coordinate basis, (
c(dxj)∇S∂j
)2
= −gij(∇S∂i∇S∂j − Γkij∇S∂k)+ 14K.
The left hand side is
c(dxi)∇S∂ic(dxj)∇S∂j = c(dxi) c(dxj)∇S∂i∇S∂j + c(dxi) c(∇∂idxk)∇S∂k
= c(dxi) c(dxj)
(∇S∂i∇S∂j − Γkij∇S∂k) (8.15)
= 1
2
[[c(dxi), c(dxj)]]
(∇S∂i∇S∂j − Γkij∇S∂k)+ 12c(dxi) c(dxj)[∇S∂i ,∇S∂j ],
where we have used the symmetry Γkij = Γ
k
ji due to the zero torsion ∇∂i∂j = ∇∂j∂i of the
Levi-Civita connection. Now [[c(dxi), c(dxj)]] = −2gij —recall (7.17)— and [∇S∂k ,∇S∂l ] =
ωS(∂k, ∂l) = −14Rijkl c(dxi) c(dxj) by (7.15), so by taking (8.10) into account we arrive at
(D/ S)2 = ∆S − 1
8
Rijkl c(dx
k) c(dxl) c(dxi) c(dxj).
It remains only to check that the second term on the right reduces to 1
4
K. We may rewrite it
as 1
8
Rjikl c(dx
k) c(dxl) c(dxi) c(dxj) since Rijkl is antisymmetric in the indices i, j by (5.29).
Since the antisymmetrization of Rjikl in the indices i, k, l vanishes by (5.28), it follows from
Exercise 8.7 below that this term reduces to
1
8
Rjikl
(−gkl c(dxi) c(dxj) + gil c(dxk) c(dxj)− gik c(dxl) c(dxj)),
and since Rjiklg
kl = 0 by antisymmetry of Rjikl in k, l, again by (5.29), this in turn reduces
to
1
4
Rijklg
ik c(dxl) c(dxj) = 1
4
Rkjki c(dx
i) c(dxj) = 1
4
gijRkikj =
1
4
K,
due to the symmetry Ricij := R
k
ikj of the Ricci tensor.
Exercise 8.7. If u, v, w are three vectors in a Euclidean vector space (V, q), denote their
antisymmetrized product by a := 1
6
(uvw+vwu+wuv−uwv−wvu−vuw) ∈ C`(V, q). Show
that uvw = a− q(v, w)u+ q(u,w)v − q(u, v)w.
The identity (8.14) is due to Lichnerowicz [40]. The proof generalizes in a straight-
forward manner [9, 27, 39] to the case of a twisted Clifford module. The only differences
consist in replacing the Laplacian ∆S by ∆F , where F = W ⊗ S, and the curvature term
1
2
c(dxi) c(dxj)ωS(∂i, ∂j) in (8.15) by
1
2
c(dxi) c(dxj)ωF (∂i, ∂j). Now by Proposition 7.9, the
101
curvature ωF can be decomposed as ωF = ωW +ωS, where ωW is the curvature of some com-
patible connection on W . This yields an extra term of the form 1
2
c(dxi) c(dxj)ωW (∂i, ∂j),
which, in view of (6.10), we may write as Q(ωW ), where Q : A•(M,W )→ C`(M) is the ex-
tension of the quantization map of Definition 6.12 to the (trivial) Clifford module A•(M,W ).
The result is a generalization of (8.14) known as the Bochner–Weitzenbo¨ck formula:
(D/ F )2 = ∆F + 1
4
K +Q(ωW ).
Exercise 8.8. When F = Λ•T ∗M ' S ⊗ S, verify (8.13) by showing that Ric−1
4
K = Q(ω0)
for a suitable ω0 ∈ A2(M,S).
9 The Dirac operator on the Riemann sphere
This section is devoted to a detailed exploration of a single but fundamental example: the
Dirac operator on the irreducible spinor module over the sphere S2. While the sphere is
undoubtedly the simplest possible even-dimensional compact spin manifold, its Dirac oper-
ator exemplifies the full complexity of the general case while remaining directly accessible
by elementary computations. We give here an account of its action on spinors, show its
equivariance under the Lie group SU(2) of symmetries of the spinor module, compute its
spectrum and exhibit a full set of eigenspinors.
Surprisingly, such an account is not to be found in the current literature on spinors, so this
exposition breaks some new ground. The ingredients have been available for a long time, and
there is no reason why this story could not have been told thirty years ago. Indeed, before
the geometrical theory of Dirac operators was developed at all, the eigenspinors for the Dirac
operator on the sphere were considered (in 1938) by Schro¨dinger [47], who put his finger on
the basic module property (7.10) of the spin connection (albeit not in so many words). A
generation later, Newman and Penrose [43] introduced a family of functions on the sphere
that they called “spinor harmonics”, which generalize the ordinary spherical harmonics and
constitute the eigenspinors, as we shall see.
In order to keep the development of the Dirac operator on S2 fairly self-contained, we
begin by reviewing some elementary notions. Everything can be done with elementary
calculus, provided one takes great care to get the signs and the constants right from the
beginning.
9.1 Coordinates on the Riemann sphere
The most direct way to reveal the Dirac operator on the sphere is to regard it as the Riemann
sphere CP1, and use complex coordinates. Recall that CP1 may be described by homogeneous
coordinates [z0 : z1] with z0, z1 not both zero, and is covered by two chart domains U0 :=
{ [z0 : z1] : z0 6= 0 } and U1 := { [z0 : z1] : z1 6= 0 }. We shall use the local complex coordinates
z := z1/z0 on U0, ζ := z
0/z1 on U1.
The coordinate change is just z = ζ−1, ζ = z−1 on U0 ∩ U1.
102
We identify CP1 with the sphere S2 of unit vectors (u1, u2, u3) ∈ R3 by the stereographic
projections :
z =
u1 + iu2
1− u3 on S
2 \ {N}, ζ = u
1 − iu2
1 + u3
on S2 \ {S}
where N = (0, 0, 1) and S = (0, 0,−1) are the north and south poles. This identifies U0 with
S2 \ {N} and U1 with S2 \ {S}. It is important to notice that both stereographic projections
are orientation reversing.
We may also use the standard spherical coordinates (θ, φ), satisfying u1 = sin θ cosφ,
u2 = sin θ sinφ, u3 = cos θ. The stereographic projections are then expressed as
z = eiφ cot θ
2
= eiφ
sin θ
1− cos θ = e
iφ1 + cos θ
sin θ
,
ζ = e−iφ tan θ
2
= e−iφ
sin θ
1 + cos θ
= e−iφ
1− cos θ
sin θ
. (9.1)
The positive functions (2.6) on U0 and U1 are given by
Q0(z) := 1 + zz¯ =
2
1− cos θ , Q1(ζ) := 1 + ζζ¯ =
2
1 + cos θ
. (9.2)
Observe that Q0(z)/Q1(z
−1) = zz¯.
The local 1-forms dz, dz¯ on U0 and dζ, dζ¯ on U1 are given —recall (2.14)— by
dz =
eiφ
1− cos θ (−dθ + i sin θ dφ), dζ =
e−iφ
1 + cos θ
(dθ − i sin θ dφ),
and their complex conjugates; notice that dζ = −e−2iφQ1/Q0 dz on U0 ∩U1. The basic local
vector fields are given by
∂
∂z
= e−iφ
1− cos θ
2
(
− ∂
∂θ
− i
sin θ
∂
∂φ
)
,
∂
∂ζ
= eiφ
1 + cos θ
2
(
∂
∂θ
+
i
sin θ
∂
∂φ
)
, (9.3)
and their complex conjugates; now ∂/∂ζ = −e+2iφQ0/Q1 ∂/∂z on U0 ∩ U1.
The Riemannian metric g on S2 is the usual one:
g = dθ2 + sin2 θ dφ2 = 4dz · dz¯(1 + zz¯)2 = 4dζ · dζ¯(1 + ζζ¯)2 (9.4)
(although (2.13) differs from this by a factor of two). The Riemannian volume form is
Ω = sin θ dθ ∧ dφ = −2i Q−20 dz ∧ dz¯ = −2i Q−21 dζ ∧ dζ¯, where the minus sign [compare
with (2.12)] indicates the reversal of orientation in passing from (θ, φ) coordinates to (z, z¯)
or (ζ, ζ¯) coordinates.
103
9.2 Sections and gauge transformations
A section of a complex line bundle L−→ S2 is given by a pair of local sections over U0, U1
respectively. Once we have chosen basic sections s0 ∈ Γ(U0, L) and s1 ∈ Γ(U1, L) which are
nonvanishing, any global section s ∈ Γ(L) is determined by a pair of smooth functions f0,
f1 on C such that
s(x) = f0(z) s0(x) for x ∈ U0, s(x) = f1(ζ) s1(x) for x ∈ U1,
where z, ζ are the respective coordinates of the point x ∈ S2. The basic local sections
are related by the transition function of the line bundle s0 = g01s1 (since the sphere is
covered by only two charts, one transition function suffices), which implies a corresponding
relation between f0(z) and f1(ζ), called a gauge transformation. In fine: a global section is
determined by a related pair of functions, and the line bundle is identified by the particular
gauge transformation relating them.
We start with a brief mention of the holomorphic line bundles. The tautological line
bundle L−→CP1 is defined, as in (5.12), by the fibres Lx := { (λz0, λz1) ∈ C2 : λ ∈ C } for
x = [z0 : z1]; its basic sections are defined as s˜0(x) := (1, z) for x ∈ U0, s˜1(x) := (ζ, 1) for
x ∈ U1. This yields s˜1(x) = z−1s˜0(x) for x ∈ U0 ∩ U1; the transition functions are therefore
g˜10(z) = z
−1, g˜01(ζ) = ζ−1, which of course are holomorphic on U0 ∩ U1.
The hermitian metric on L is obtained from the natural inclusion of the fibres in C2; this
means that (s˜0 | s˜0) = ‖(1, z)‖2 = 1 + zz¯ = Q0(z) and (s˜1 | s˜1) = ‖(ζ, 1)‖2 = 1 + ζζ¯ = Q1(ζ).
Its dual, the hyperplane bundle H −→CP1 has local sections σ˜0, σ˜1 with (σ˜0 | σ˜0) = Q−10 on
U0, (σ˜1 | σ˜1) = Q−11 on U1 —recall (5.13)— and so they extend smoothly to global sections
on S2 just by setting σ˜0(N) := 0, σ˜1(S) := 0. The (holomorphic) transition functions are
g˜01(ζ) = ζ and g˜10(z) = z. A global holomorphic section σ˜ ∈ O(CP1, H) is given by a pair of
holomorphic functions f0, f1 with σ˜(x) = f0(z) σ˜0(x) and σ˜(x) = f1(ζ) σ˜1(x) for all x. That
means that f0, f1 are entire functions whose possible singularities at infinity can only be
poles, and f0(z) = zf1(z
−1) for z ∈ C×; this relation identifies a Taylor series and a Laurent
series, and can only hold if f0, f1 are of the form f0(z) = a + bz, f1(ζ) = b + aζ for some
a, b ∈ C: we have once again established that O(CP1, H) ' C2.
From now on, we shall normalize all basic sections and thus use only U(1)-valued tran-
sition functions. Thus we replace the basic sections of L by
s0 := Q0(z)
−1/2s˜0, s1 := Q1(ζ)−1/2s˜1,
and likewise σ0 := Q0(z)
1/2σ˜0, σ1 := Q0(ζ)
1/2σ˜1. This gives
s1(x) = z
−1
√
Q0(z)
Q1(z−1)
s0(x) =
√
zz¯
z
s0(x) = (z¯/z)
1/2s0(x),
so the transition function is g10(z) = (z¯/z)
1/2 = e−iφ. A section s : CP1 → L is then given
by a pair of functions (f0, f1) such that f0(z)s0(x) ≡ f1(ζ)s1(x) on U0 ∩ U1, i.e., such that
f0(z) ≡ (z¯/z)1/2f1(z−1), f1(ζ) ≡ (ζ¯/ζ)1/2f0(ζ−1), (9.5)
104
where the second equation is of course redundant. The formula (9.5) exhibits the U(1) gauge
transformation of the tautological line bundle.
It is clear from (9.5) that f0 and f1 cannot both be holomorphic, and in general neither
is holomorphic. Therefore, it would be more correct to write f0(z, z¯) and f1(ζ, ζ¯) to signal
the dependence of these smooth functions on both real coordinates. We shall do so whenever
the need arises.
For the hyperplane bundle H −→CP1, the very same argument shows that a global
section is given by a pair of functions (h0, h1) such that h0(z)σ0(x) ≡ h1(ζ)σ1(x) on U0 ∩U1,
and which are therefore related by the gauge transformation
h0(z) ≡ (z/z¯)1/2h1(z−1), h1(ζ) ≡ (ζ/ζ¯)1/2h0(ζ−1). (9.6)
Exercise 9.1. Show that for any Hermitian line bundle E−→CP1, the sections in Γ(E) are
described by pairs of functions (f0, f1) satisfying the relation f0(z) ≡ (z/z¯)k/2f1(z−1), where
k ∈ Z and k[H] is the Chern class of E.
We may decompose the complexified tangent bundle as TCS2 = T 1,0S2 ⊕ T 0,1S2, where
the sections of the holomorphic tangent bundle T 1,0S2 are locally of the form f0(z, z¯) ∂/∂z
or f1(ζ, ζ¯) ∂/∂ζ. [The sections of the “antiholomorphic tangent bundle” T
0,1S2 are of the
form h0(z, z¯) ∂/∂z¯ or h1(ζ, ζ¯) ∂/∂ζ¯.] Then T
1,0S2−→ S2 is a Hermitian line bundle, under
the metric determined by
〈∂/∂z | ∂/∂z〉 := g(∂/∂z¯, ∂/∂z) = 4
(1 + zz¯)2
.
As normalized local sections we take
Ez :=
1
2
Q0(z)
∂
∂z
over U0, −Eζ := −12Q1(ζ)
∂
∂ζ
over U1.
Since z = ζ−1 gives ∂/∂z = −ζ2 ∂/∂ζ, we find that Ez = −ζ2Q0(z)/Q1(ζ)Eζ = (z¯/z)(−Eζ).
Thus a section of the holomorphic tangent bundle is given by a pair of functions f0, f1
satisfying f0(z)Ez ≡ f1(ζ)(−Eζ) on U0 ∩ U1, or equivalently
f0(z) ≡ (z/z¯) f1(z−1).
This establishes that [T 1,0S2] = 2[H] in Hˇ(S2,Z), i.e., the line bundles T 1,0S2 and H ⊗ H
are equivalent.
Exercise 9.2. Conclude from Ez = (z/z¯)(−Eζ) that the complex line bundle T 0,1S2 is equiv-
alent to L⊗ L.
Exercise 9.3. Write T ∗CS2 = Λ1,0T ∗S2⊕Λ0,1T ∗S2, where A1,0(S2) = Γ(Λ1,0T ∗S2) has elements
f0(z, z¯) dz = f1(ζ, ζ¯) dζ and A
0,1(S2) = Γ(Λ0,1T ∗S2) consists of all h0(z, z¯) dz¯ = h1(ζ, ζ¯) dζ¯.
Use g−1 to define Hermitian metrics on both these line bundles, write down suitable normal-
ized local sections, and verify that Λ1,0T ∗S2 ∼ L⊗ L and Λ0,1T ∗S2 ∼ H ⊗H.
105
Definition 9.1. Let S−→ S2 denote the irreducible spinor bundle. Since T ∗CS2 ∼ S⊗S∗, we
expect that S ∼ L⊕H, and that S+ ∼ L, S− ∼ H. Due to the noncanonical nature of the
spinor bundle, we bypass the construction of basic local sections and define a spinor ψ over
S2 directly as a pair of functions on each chart, denoted ψ±N(z, z¯) and ψ
±
S (ζ, ζ¯) respectively,
satisfying the gauge transformation rules:
ψ+N(z, z¯) ≡ (z¯/z)1/2ψ+S (z−1, z¯−1), ψ−N(z, z¯) ≡ (z/z¯)1/2ψ−S (z−1, z¯−1),
ψ+S (ζ, ζ¯) ≡ (ζ¯/ζ)1/2ψ+N(ζ−1, ζ¯−1), ψ−S (ζ, ζ¯) ≡ (ζ/ζ¯)1/2ψ−N(ζ−1, ζ¯−1). (9.7)
It is immediate from (9.5) and (9.6) that (ψ+N , ψ
+
S ) determines a section of L and (ψ
−
N , ψ
−
S )
determines a section of H.1
9.3 The spin connection over the sphere
Lemma 9.1. The Levi-Civita connection on the sphere is determined by the local formulae
∇∂i∂j = −
2
1 + x21 + x
2
2
(xi∂j + xj∂i − δijxk ∂k), (9.8)
with x1 ≡ x1, x2 ≡ x2, where x1 + ix2 = z on U0 and x1 + ix2 = ζ on U1.
Proof. The metric (9.4) is given by g = 4(1 + x21 + x
2
2)
−2(dx21 + dx
2
2) on both charts, so
gij = 4(1 + x
2
1 + x
2
2)
−2δij and gij = 14(1 + x
2
1 + x
2
2)
2δij. Also, ∂kgij = −16xk(1 + x21 + x22)−3δij.
From (7.13) we get at once
Γkij = −
2
1 + x21 + x
2
2
(xiδ
k
j + xjδ
k
i − xkδij),
from which (9.8) follows.
Local orthonormal bases of vector fields (on both charts) are given by E1 :=
1
2
(1 + x21 +
x22)∂/∂x
1, E2 :=
1
2
(1 + x21 + x
2
2)∂/∂x
2. With these bases, we get
∇∂iEα = xi∂α + 12(1 + x21 + x22)∇∂i∂α =
2
1 + x21 + x
2
2
(δiαx
β Eβ − xαEi),
or equivalently,
Γ˜βiα =
2
1 + x21 + x
2
2
(δiαx
β − δβi xα).
The spin connection components are given by (7.18) as ωi =
1
4
Γ˜βiα γ
αγβ, which yields
ω1 =
1
2(1 + x21 + x
2
2)
(xβγ1γβ − xαγαγ1) = x2
1 + x21 + x
2
2
γ1γ2,
ω2 =
1
2(1 + x21 + x
2
2)
(xβγ2γβ − xαγαγ2) = − x1
1 + x21 + x
2
2
γ1γ2, (9.9)
1We label the local spinor coefficients N and S (north and south) in order to reduce the clutter of
numerical indices.
106
To return to complex notation, we notice that (ω1 + iω2)(z) = − iz1+zz¯γ1γ2 over U0 while
(ω1 + iω2)(ζ) = − iζ1+ζζ¯γ1γ2 over U1. These are related by a gauge transformation: (ω1 +
iω2)(z) = (z/z¯) (ω1 + iω2)(ζ).
9.4 The Dirac operator over the sphere
Definition 9.2. We fix a local Clifford action on the spinor module by choosing a particular
matrix function H˜ := [h˜rβ(z)] such that H˜
tH˜ = G−1 ≡ [gij(z)] = 1
4
(1 + zz¯)2 I over U0. Let us
(arbitrarily) choose the positive square root H˜ := G−1/2 = 1
2
(1+zz¯) I. The Dirac operator
on the chart U0 is then defined as
D/N := c(dx
j)∇S∂j = h˜jβ(z)γβ(∂j + ωj(z))
= 1
2
(1 + zz¯)(γβ∂β + γ
βωβ(z))
= 1
2
(1 + zz¯)(γ1 ∂/∂x1 + γ2 ∂/∂x2)− 1
2
(x1γ
1 + x2γ
2).
Here we have used the identities γ1γ1γ2 = −γ2 and γ2γ1γ2 = +γ1.
It turns out that the Dirac operator over the chart U1, which we shall write as D/ S, is
now completely determined: the Dirac operator on the spinor module is determined by its
restriction to a single chart, no matter how small!2 Indeed, this is a characteristic feature of
the spinor module [19]. However, in order to preserve the illusion of a symmetrical treatment
of both charts, we shall anticipate the corresponding formulas for D/ S. We therefore choose
the matrix [h˜rβ(ζ)] to be the negative square root of G
−1 ≡ [gij(ζ)] = 1
4
(1 + ζζ¯)2 I, that is,
h˜rβ(ζ) := −12(1 + ζζ¯)δrβ. This leads to
D/ S = −12(1 + ζζ¯)(γ1 ∂/∂x1 + γ2 ∂/∂x2) + 12(x1γ1 + x2γ2). (9.10)
Explicitly, ∂/∂x1 = ∂/∂z+∂/∂z¯ and ∂/∂x2 = i(∂/∂z−∂/∂z¯) on U0, and similar formulas
hold on U1 with z replaced by ζ. This allows us to express the Dirac operators properly in
complex coordinates:
D/N =
1
2
(1 + zz¯)
(
(γ1 + iγ2)
∂
∂z
+ (γ1 − iγ2) ∂
∂z¯
)
− 1
4
(
z(γ1 − iγ2) + z¯(γ1 + iγ2)),
D/ S = −12(1 + ζζ¯)
(
(γ1 + iγ2)
∂
∂ζ
+ (γ1 − iγ2) ∂
∂ζ¯
)
+ 1
4
(
ζ(γ1 − iγ2) + ζ¯(γ1 + iγ2)). (9.11)
These formulae are still a bit cumbersome; to obtain a simpler picture, we need to select a
particular representation for the operators γ1 and γ2.
2On more general compact spin manifolds, the restriction of the Dirac operator to a single chart determines
its restriction to neighbouring charts through its interaction with the gauge transformations; this in turn
determine their neighbours, and so on.
107
Definition 9.3. The Fock space of R2 is the two-dimensional complex superspace Λ0C ⊕
Λ1C ' C⊕ C. We may represent γ1 and γ2 as anticommuting odd operators of square −1;
a suitable choice is
γ1 :=
(
0 1
−1 0
)
, γ2 :=
(
0 −i
−i 0
)
. (9.12)
The grading operator is iγ1γ2 =
(
1 0
0 −1
)
. Since γ1+iγ2 =
(
0 2
0 0
)
and γ1−iγ2 =
(
0 0
−2 0
)
,
the expressions (9.11) simplify to
D/N =
(
0 ðz
−ð¯z 0
)
, D/ S =
(
0 −ðζ
ð¯ζ 0
)
, (9.13)
where ðz is the first-order differential operator3
ðz := (1 + zz¯)
∂
∂z
− 1
2
z¯. (9.14)
The operator ðz was introduced by Newman and Penrose [43] and further studied by
Goldberg et al. [30], with particular attention to its eigenfunctions. Since
ðzψ = (1 + zz¯)
∂ψ
∂z
− 1
2
∂
∂z
(1 + zz¯)ψ = (1 + zz¯)3/2
∂
∂z
(
(1 + zz¯)−1/2ψ
)
, (9.15)
we get the identity ðz = Q3/20 (∂/∂z)Q
−1/2
0 in the algebra of differential operators on U0.
The formal selfadjointness of D/ is not perhaps apparent from (9.13), since the term −1
2
z¯
in (9.14) seems to have the adjoint −1
2
z rather than +1
2
z. However, the inner product of
spinors 〈〈φ |ψ〉〉 involves integration over the sphere, i.e., integration over C with respect to to
the area form 2i(1 + zz¯)−2 dz ∧ dz¯. Thus, if φ, ψ are two spinors, then (9.15) gives
(1 + zz¯)−2φ+Nðzψ
−
N = (1 + zz¯)
−1/2φ+N
∂
∂z
(
(1 + zz¯)−1/2ψ−N
)
,
and it follows that 〈〈φ+ | ðzψ−〉〉 = −〈〈ð¯zφ+ | ψ−〉〉 on integrating by parts.
Proposition 9.2. The restriction of the Dirac operator to U1 is determined by its restriction
to U0.
Proof. We must show that the form (9.13) of the operator D/ S is completely determined by
that of D/N . This is possible because these operators act respectively on functions ψ
±
S and
ψ±N which are linked by the gauge transformations (9.7), and because the Dirac operator is
odd, so the gauge transformations for both spinor parities must be invoked.
3The letter ð, from the Icelandic alphabet, is pronounced “edth”.
108
Given a spinor ψ with components ψ+ and ψ−, let φ = D/ψ. Then (ðzψ−N)(z, z¯) =
φ+N(z, z¯) = (z¯/z)
1/2φ+S (z
−1, z¯−1). On the other hand, with ζ = z−1 we get
∂ψ−N
∂z
(z, z¯) =
∂
∂z
(
(z/z¯)1/2ψ−S (z
−1, z¯−1)
)
= 1
2
(zz¯)−1/2ψ−S (z
−1, z¯−1)− (z/z¯)1/2z−2∂ψ
−
S
∂ζ
(z−1, z¯−1)
= (ζζ¯)1/2
(
−ζ ∂ψ
−
S
∂ζ
(ζ, ζ¯) + 1
2
ψ−S (ζ, ζ¯)
)
,
and so the operator ðz transforms as follows:
(ðzψ−N)(z, z¯) = (1 + zz¯)
∂ψ−N
∂z
(z, z¯)− 1
2
z¯ψ−N(z, z¯)
= (ζζ¯)−1/2(1 + ζζ¯)
(
−ζ ∂ψ
−
S
∂ζ
(ζ, ζ¯) + 1
2
ψ−S (ζ, ζ¯)
)
− (ζ¯/ζ)
1/2
2ζ¯
ψ−S (ζ, ζ¯)
= (ζ/ζ¯)1/2
(
−(1 + ζζ¯)∂ψ
−
S
∂ζ
(ζ, ζ¯) + 1
2
(ζ−1 + ζ¯)ψ−S (ζ, ζ¯)− 12ζ−1ψ−S (ζ, ζ¯)
)
= −(ζ/ζ¯)1/2ðζψ−S (ζ, ζ¯) = −(z¯/z)1/2ðζψ−S (z−1, z¯−1). (9.16)
We conclude that φ+S = −ðζψ−S . If we apply complex conjugation to (9.16) and replace ψ−N
by ψ+N and ψ
−
S by ψ
+
S , we find that
φ−N(z, z¯) = −(ð¯zψ+N)(z, z¯) = (z/z¯)1/2ð¯ζψ+S (z−1, z¯−1),
and it follows that φ−S = ð¯ζψ
+
S .
We have recovered the expression of (9.13) for D/ S from the corresponding for D/N and
from (9.7), without using the recipe (9.10) for D/ S. This means that the “choice” h˜
r
β(ζ) :=
−1
2
(1 + ζζ¯)δrβ which led to (9.10) is actually forced by the action of D/ on spinors.
Exercise 9.4. Use (9.1) and (9.3) to obtain expressions for ðz and ðζ in spherical coordinates
(θ, φ). Check that eiφðz = −e−iφðζ − csc θ on U0 ∩ U1.
Proposition 9.2 shows that the only freedom in the choice of the Dirac operator, once
the metric and the spin connection are given, lies in selecting the matrix function H˜ =
[h˜rβ(z)] which gives the local Clifford action on U0. The condition H˜
tH˜ = G−1 fixes the
matrix H˜ up to premultiplication by an arbitrary SO(2) matrix function on U0. We may
alternatively think of this as the freedom to select any local orthonormal bases of tangent
vectors compatible with the orientation of S2, since this amounts to picking a section of the
SO(2)-frame bundle. To express this in our complex-coordinate notation, we must bear in
mind that the stereographic projections (θ, φ) 7→ (z, z¯) and (θ, φ) 7→ (ζ, ζ¯) are orientation
reversing ; such a local orthonormal basis is therefore given by(
E1
E2
)
:=
(− cosα sinα
sinα cosα
)(
∂/∂θ
csc θ ∂/∂φ
)
, (9.17)
where α(θ, φ) is a smooth real-valued function which we call the spin gauge [25].
109
Exercise 9.5. Check that Ez =
1
2
(E1 − iE2) if and only if α ≡ −φ, and −Eζ = 12(E1 − iE2)
if and only if α ≡ φ.
Exercise 9.6. Compute the mixed Christoffel symbols Γ˜βiα with the spin gauge (9.17) (use
Exercise 7.16) and verify that Γ˜2θ1 = ∂α/∂θ, Γ˜
2
φ1 = ∂α/∂φ− cos θ.
Exercise 9.7. Use the results of the Exercises 9.5 and 9.6 to obtain the following local expres-
sions in spherical coordinates for the spin connection ∇S on the chart domains U0 and U1:
∇Sθ =
∂
∂θ
, ∇Sφ =
∂
∂φ
∓ 1
2
(1± cos θ)γ1γ2 (9.18)
where the upper signs are for U0 and the lower signs are for U1.
Exercise 9.8. Compute the Dirac operator in spherical coordinates with the spin gauge (9.17),
using the results of Exercise 7.16: check that D/ is given by
D/ − = eiα
(
− ∂
∂θ
− i
sin θ
∂
∂φ
+
1
2 sin θ
(∂α
∂φ
− cos θ
)
− i
2
∂α
∂θ
)
,
D/ + = e
−iα
(
∂
∂θ
− i
sin θ
∂
∂φ
− 1
2 sin θ
(∂α
∂φ
− cos θ
)
− i
2
∂α
∂θ
)
,
with the conventions of (8.7) and (9.12).4
9.5 The spinor Laplacian
Lemma 9.3. The spinor Laplacian ∆S on the sphere S2 is given locally by
∆S = −
(
∂2
∂θ2
+ cot θ
∂
∂θ
+
∂2
∂φ2
)
+
(
1± cos θ
2 sin θ
)2
± 1± cos θ
sin2 θ
γ1γ2
∂
∂φ
, (9.19)
where the upper signs are for U0 and the lower signs are for U1.
Proof. It suffices to check that the general local formula (8.10) for the Laplacian specializes,
in view of (7.14) and (9.18), to
∆S = − ∂
2
∂θ2
− 1
sin2 θ
(
∂
∂φ
∓ 1
2
(1± cos θ)γ1γ2
)2
− 1
sin2 θ
(
sin θ cos θ
∂
∂θ
)
,
from which (9.19) is immediate.
It is convenient to rewrite the spinor Laplacian in complex coordinates. From (9.3) we
derive
∂
∂φ
= i
(
z
∂
∂z
− z¯ ∂
∂z¯
)
= −i
(
ζ
∂
∂ζ
− ζ¯ ∂
∂ζ¯
)
on U0 ∩ U1.
4The formulae (9.18) have been obtained by Dray [25] as the expressions of the Newman–Penrose operators
−ð and ð¯ on quantities of “spin-weights” − 12 and + 12 respectively.
110
Using (9.2) and (9.3) it is readily checked that
(1 + zz¯)2
∂2
∂z ∂z¯
= (1 + ζζ¯)2
∂2
∂ζ ∂ζ¯
=
∂2
∂θ2
+ cot θ
∂
∂θ
+
∂2
∂φ2
.
Since (1 + cos θ/ sin θ)2 = zz¯ and (1− cos θ/ sin θ)2 = ζζ¯ by (9.1), we arrive at
∆S = −(1 + zz¯)2 ∂
2
∂z ∂z¯
+ 1
4
zz¯ + 1
2
(1 + zz¯) iγ1γ2
(
z
∂
∂z
− z¯ ∂
∂z¯
)
over U0,
= −(1 + ζζ¯)2 ∂
2
∂ζ ∂ζ¯
+ 1
4
ζζ¯ + 1
2
(1 + ζζ¯) iγ1γ2
(
ζ
∂
∂ζ
− ζ¯ ∂
∂ζ¯
)
over U1. (9.20)
Lemma 9.4. The Dirac operator and the spinor Laplacian on S2 are related by
D/ 2 = ∆S + 1
2
.
Proof. This follows, of course, from the Lichnerowicz formula (8.14), since the sphere has
constant scalar curvature K ≡ 2; but it is instructive to make a direct verification. From
(9.14) we get
−ðzð¯z =
(
(1 + zz¯)
∂
∂z
− 1
2
z¯
)(−(1 + zz¯) ∂
∂z¯
+ 1
2
z
)
= −(1 + zz¯)2 ∂
2
∂z ∂z¯
+ 1
2
(1 + zz¯)
(
z
∂
∂z
+ z¯
∂
∂z¯
)
−1
4
zz¯ + (1 + zz¯)
(
1
2
− z¯ ∂
∂z¯
)
= −(1 + zz¯)2 ∂
2
∂z ∂z¯
+ 1
2
+ 1
4
zz¯ + 1
2
(1 + zz¯)
(
z
∂
∂z
− z¯ ∂
∂z¯
)
, (9.21)
whereas
−ð¯zðz =
(−(1 + zz¯) ∂
∂z¯
+ 1
2
z
)(
(1 + zz¯)
∂
∂z
− 1
2
z¯
)
= −(1 + zz¯)2 ∂
2
∂z ∂z¯
+ 1
2
+ 1
4
zz¯ − 1
2
(1 + zz¯)
(
z
∂
∂z
− z¯ ∂
∂z¯
)
(9.22)
by a similar calculation. The different signs of the last terms on the right in (9.21) and (9.22)
signify the presence of the grading operator iγ1γ2 in the representation (9.12). Thus
D/ 2N =
(−ðzð¯z 0
0 −ð¯zðz
)
= −(1 + zz¯)2 ∂
2
∂z ∂z¯
+ 1
2
+ 1
4
zz¯ + 1
2
(1 + zz¯) iγ1γ2
(
z
∂
∂z
− z¯ ∂
∂z¯
)
. (9.23)
There is an identical formula for D/ S, on replacing z by ζ. Now a glance at (9.20) gives the
Lichnerowicz formula D/ 2 = ∆S + 1
2
.
111
9.6 The SU(2) action on the spinor bundle
Definition 9.4. The Lie group SU(2) of unitary matrices of determinant 1,
g =
(
α β
−β¯ α¯
)
with αα¯ + ββ¯ = 1,
acts transitively on the sphere S2 by rotations. The identification g ↔ (α, β) ∈ C2 shows
that SU(2) is topologically the sphere S3, so it is compact. If [z0 : z1] are homogeneous
coordinates of a point in CP1, the action is given by g · [z0 : z1] = [αz0 + βz1 : −β¯z0 + α¯z1].
On the charts U0 and U1, the action of SU(2) is given by Mo¨bius transformations :
g · z = αz + β−β¯z + α¯ , g
′ · ζ = α¯ζ − β¯
βζ + α
, (9.24)
which clearly satisfies (g · z)−1 = g′ · ζ. Notice that g 7→ g′ is an (inner) automorphism of
SU(2), since
g′ :=
(
α¯ −β¯
β α
)
=
(
0 i
i 0
)(
α β
−β¯ α¯
)(
0 −i
−i 0
)
.
Exercise 9.9. Any isometry (rotation or reflection) of the sphere S2 takes circles to circles
and takes antipodal pairs of points to antipodal pairs of points. Conversely, any smooth
bijective transformation of S2 with these two geometrical properties is either a rotation or
a reflection. Prove this converse, using the fact that a circle-preserving transformation of
S2 corresponds, under stereographic projection, to a circle-preserving transformation of C∞,
which is either a Mo¨bius transformation z 7→ (αz + β)/(γz + δ) or a conjugate Mo¨bius
transformation z 7→ (αz¯ + β)/(γz¯ + δ); and that a Mo¨bius transformation is determined by
the images of three points in C∞.
Exercise 9.10. The antipode of z = eiφ cot 1
2
θ is ei(pi+φ) cot 1
2
(pi−θ) = −1/z¯. Use this fact and
the preceding exercise to prove that any rotation of S2 is given by a Mo¨bius transformation
of the form z 7→ (αz + β)/(−β¯z + α¯).
Definition 9.5. The elements of SU(2) are conveniently described in terms of the Pauli
matrices :
σ1 :=
(
0 1
1 0
)
, σ2 :=
(
0 −i
i 0
)
, σ3 :=
(
1 0
0 −1
)
.
If ~n = (n1, n2, n3) denote Cartesian coordinates in R3 of a point ~n ∈ S2, we write ~n · ~σ :=
n1σ1 + n2σ2 + n3σ3; then any g ∈ SU(2) may be written in the form
g = exp
(
1
2
iψ ~n · ~σ) = cos 1
2
ψ + i sin 1
2
ψ ~n · ~σ (9.25)
with ~n ∈ S2 and −pi < ψ ≤ pi. This is called the angle-axis parametrization of SU(2). We
may identify the sphere S2 with the submanifold { g : Tr g = 0 } of SU(2), where ~n ∈ S2
corresponds to i~n · ~σ ∈ SU(2); the rotation action ρ of SU(2) on the sphere is given by
conjugation: i~n · ~σ 7→ g(i~n · ~σ)g−1 =: iρ(g)~n · ~σ. The isotropy subgroup of the point ~n
consists of elements of the form (9.25) with ψ arbitrary, which form a subgroup isomorphic
to U(1). Thus S2 ≈ SU(2)/U(1) as a homogeneous space.
112
Exercise 9.11. Show that ρ(g)~n = ~n if and only if g is of the form (9.25) for some ψ ∈ R.
Exercise 9.12. Show that there is a group isomorphism between Spin(3) and SU(2), such
that if {e1, e2, e3} is an orthonormal basis of R3, then the elements e2e3, e3e1, e1e2 ∈ Spin(3)
correspond respectively to iσ1, iσ2, iσ3 ∈ SU(2).
The quotient mapping η : SU(2)→ S2 may be described more economically by regarding
S2 as CP1 and proceeding as follows.
Definition 9.6. The Hopf fibration η : SU(2)→ CP1 is the map given by
η
(
α β
−β¯ α¯
)
:=
β
α¯
. (9.26)
It is immediate that (
α β
−β¯ α¯
)(
eiψ/2 0
0 e−iψ/2
)
η7−→ β
α¯
,
so SU(2)
η−→CP1 is a principal U(1)-bundle, where the free right action of U(1) on the fibres
η−1(β/α¯) is given simply by multiplication on the right by the diagonal elements of SU(2).
Exercise 9.13. The Hopf fibration decomposes the sphere S3 ≈ SU(2) into a disjoint union of
circles (the fibres), any two of which are linked. Regard S3 as R3unionmulti{∞} via the stereographic
projection (α, β) 7→ (w, t) where
w :=
α
1−=β ∈ C, t :=
<β
1−=β ∈ R.
Check that χ(w, t) := (2t+ i(ww¯ + t2 − 1))/2w¯ ∈ C∞ is the expression in (w, t)-coordinates
of the Hopf fibration. Deduce that the equation χ(w, t) = z represents the circle obtained
by cutting the sphere ww¯+ t2− i(z¯w− zw¯) = 1 with the equatorial plane z¯w+ zw¯− 2t = 0;
in particular, χ(w, t) = ∞ is the t-axis (including ∞) and χ(w, t) = 0 is the unit circle in
the ww¯-plane. Show that any other circle χ(w, t) = z cuts the ww¯-plane obliquely at two
points, one inside and one outside the unit circle; conclude that the circles χ(w, t) = 0 and
χ(w, t) = z are linked.
The group SU(2) acts on itself by left translations λ(g) : h 7→ gh; since these commute
with right translations by U(1), they define a left action on the quotient manifold CP1,
which is just the aforementioned rotation action. Indeed, if h = (1 + zz¯)−1
(
1 z
−z¯ 1
)
, then
η(h) = z, and from
1
1 + zz¯
(
α β
−β¯ α¯
)(
1 z
−z¯ 1
)
=
1
1 + zz¯
(
α− βz¯ αz + β
−β¯ − α¯z¯ −β¯z + α¯
)
it follows that η(gh) = g · η(h) ≡ ρ(g)η(h) for g, h ∈ SU(2). Schematically, we get a
commutative diagram
SU(2)
λ(g)−−−→ SU(2)
η
y xη
S2 ρ(g)−−−→ S2
113
which says that each pair of maps (λ(g), ρ(g)) is a morphism of the principal U(1)-bundle
SU(2)
η−→CP1.
We may say that the group SU(2) acts “equivariantly” on the principal bundle. The
corresponding type of group action on vector bundles is defined as follows.
Definition 9.7. A homogeneous vector bundle with symmetry group G (a Lie group)5 is
a vector bundle E−→M together with a pair of (left) actions τ : G×E → E and ρ : G×M →
M such that each (τ(g), ρ(g)) is a vector bundle morphism on E−→M . We shall call such
a pair a bundle action of G.
A Hermitian homogeneous vector bundle is a Hermitian vector bundle with a bundle
action for which each τ(g)x ∈ End(Ex) is unitary. If E = E+⊕E− is a superbundle, we say
the bundle action of G is even if τ(g)x ∈ End+(Ex) for each x ∈M ; in other words, if both
subbundles E±−→M are G-homogeneous under the bundle action (τ, ρ).
We seek to define an even action of SU(2) on the spinor bundle S−→ S2. This can be
pictured as a commutative diagram
S±
τ(g)−−−→ S±
pi±
y xpi±
S2 ρ(g)−−−→ S2
which, in terms of the spinor components ψ± ∈ Γ(S±), means that
τ(g)ψ±N(z, z¯) = A
±
N(g, z)ψ
±
N(g
−1 · z, (g−1 · z)−),
τ(g)ψ±S (ζ, ζ¯) = A
±
S (g
′, ζ)ψ±S (g
′−1 · ζ, (g′−1 · ζ)−), (9.27)
for g ∈ SU(2). By unitarity, the multipliers A± must be U(1)-valued functions; and they
must satisfy the following consistency conditions in order that (9.27) define a group action:
A±N(gh, z) = A
±
N(g, z)A
±
N(h, g
−1 · z), (9.28)
A±S (g
′h′, ζ) = A±S (g
′, ζ)A±S (h
′, g′−1 · ζ). (9.29)
Exercise 9.14. Show that A(g, z) := (βz¯+ α¯)k/(β¯z+α)k is a formal solution to the equation
A(gh, z) = A(g, z)A(h, g−1 · z), for g, h ∈ SU(2), z ∈ C∞; and that this solution is well-
defined provided 2k is an integer.
Bearing in mind that
g−1 · z = α¯z − β
β¯z + α
, g′−1 · ζ = αζ + β¯−βζ + α¯ ,
the gauge transformation rule (9.7) for the spinors τ(g)ψ yields
A+S (g
′, ζ)ψ+S
(
αζ + β¯
−βζ + α¯ ,
α¯ζ¯ + β
−β¯ζ¯ + α
)
= (ζ¯/ζ)1/2A+N(g, ζ
−1)ψ+N
(
α¯− βζ
β¯ + αζ
,
α− β¯ζ¯
β + α¯ζ¯
)
. (9.30)
5This is often called a G-vector bundle, for short.
114
The gauge transformation
ψ+S
(
αζ + β¯
−βζ + α¯ ,
α¯ζ¯ + β
−β¯ζ¯ + α
)
=
(
α¯ζ¯ + β
−β¯ζ¯ + α
/
αζ + β¯
−βζ + α¯
)1/2
ψ+N
(
α¯− βζ
β¯ + αζ
,
α− β¯ζ¯
β + α¯ζ¯
)
shows that the coefficients in (9.30) must satisfy the relation
A+S (g
′, ζ)
A+N(g, ζ
−1)
=
(
ζ¯(αζ + β¯)(−β¯ζ¯ + α)
ζ(−βζ + α¯)(α¯ζ¯ + β)
)1/2
=
(
(α + β¯ζ−1)(−β¯ζ¯ + α)
(−βζ + α¯)(α¯ + βζ¯−1)
)1/2
,
or equivalently
A+S (g
′, ζ)
A+N(g, z)
=
(−β¯ζ¯ + α
−βζ + α¯
)1/2 / (
βz¯ + α¯
β¯z + α
)1/2
. (9.31)
The following solution of (9.29) is therefore consistent with the spinor gauge transforma-
tions:
A+N(g, z) :=
(
βz¯ + α¯
β¯z + α
)1/2
, A+S (g
′, ζ) :=
(−β¯ζ¯ + α
−βζ + α¯
)1/2
. (9.32)
Substituting these in (9.27) yields a bundle action of SU(2) on S+−→ S2.
The same procedure leads to a bundle action of SU(2) on S−−→ S2. One need only
replace the term (ζ¯/ζ)1/2 in (9.30) by (ζ/ζ¯)1/2 when invoking the gauge transformation rule
(9.7); this leads to the choice of A−N(g, z) and A
−
S (g
′, ζ) as the complex conjugates of (9.32):
A−N(g, z) :=
(
βz¯ + α¯
β¯z + α
)−1/2
, A−S (g
′, ζ) :=
(−β¯ζ¯ + α
−βζ + α¯
)−1/2
. (9.33)
We summarize the foregoing in a definition.
Definition 9.8. The Lie group SU(2) acts on the spinor bundle via (τ, ρ), where ρ is the
rotation action (9.24) on the Riemann sphere, and τ is given by (9.27), where the multipliers
A±N and A
±
S are defined by (9.32) and (9.33).
9.7 Equivariance of the Dirac operator
Lemma 9.5. Let T ∈ End+(Γ(U0, S)) be a transformation of the form
(Tψ+N)(z, z¯) := a(z, z¯)
1/2ψ+N
(
b(z), b(z)
)
,
(Tψ−N)(z, z¯) := a(z, z¯)
−1/2ψ−N
(
b(z), b(z)
)
. (9.34)
where a is a smooth function on C× and b is a rational function on C∞. Then Tðz = ðzT
as operators from Γ(U0, S
−) to Γ(U0, S+) if and only if a, b satisfy the pair of differential
equations
(1 + zz¯)
db
dz
= a(z, z¯)(1 + b(z)b(z)),
(1 + zz¯)
∂a
∂z
= a(z, z¯)2b(z)− z¯a(z, z¯). (9.35)
115
Proof. It is enough to notice that
T (ðzψ−N)(z, z¯) = a
1/2(1 + bb¯)
∂ψ−N
∂z
(b, b¯)− 1
2
a1/2b¯ψ−N(b, b¯),
whereas
ðz(Tψ−N)(z, z¯) = (1 + zz¯)
{
−1
2
a−3/2
∂a
∂z
ψ−N(b, b¯) + a
−1/2 db
dz
∂ψ−N
∂z
(b, b¯)
}
− 1
2
z¯a−1/2ψ−N(b, b¯),
using both halves of (9.34); and then to equate coefficients of ψ−N and ∂ψ
−
N/∂z.
Proposition 9.6. The Dirac operator on the sphere is equivariant under the action of SU(2)
on the spinor bundle, i.e., τ(g)D/ = D/ τ(g) on Γ(S), for all g ∈ SU(2).
Proof. To show that τ(g)D/ = D/ τ(g) for all g, it is enough to check this for g belonging to
a collection of one-parameter subgroups which generate SU(2). Since αα¯ + ββ¯ = 1, we can
write α = exp( i
2
φ+ i
2
ψ) cos 1
2
θ, β = exp( i
2
φ− i
2
ψ) sin 1
2
θ; we thereby see that any g ∈ SU(2)
is of the form k(φ)h(θ)k(ψ), where6
k(φ) =
(
eiφ/2 0
0 e−iφ/2
)
, h(θ) =
(
cos 1
2
θ sin 1
2
θ
− sin 1
2
θ cos 1
2
θ
)
. (9.36)
Now τ(k(t)) is of the form (9.34) with a(z, z¯) = e−it, b(z) = e−itz, so the equations (9.35)
reduce to the identities
(1 + zz¯)e−it = e−it(1 + zz¯), 0 = e−2iteitz¯ − z¯e−it.
On the other hand, τ(h(t)) is of the form (9.34) with
a(z, z¯) =
z¯ sin 1
2
t+ cos 1
2
t
z sin 1
2
t+ cos 1
2
t
, b(z) =
z cos 1
2
t− sin 1
2
t
z sin 1
2
t+ cos 1
2
t
. (9.37)
for which
∂a
∂z
= − sin 1
2
t
z¯ sin 1
2
t+ cos 1
2
t
(z sin 1
2
t+ cos 1
2
t)2
,
db
dz
=
1
(z sin 1
2
t+ cos 1
2
t)2
. (9.38)
From this it is easy to check that (9.37) satisfies the equations (9.35) for all t ∈ R.
Thus τ(g)ðzψ−N = ðzτ(g)ψ
−
N for all g ∈ SU(2). By applying complex conjugation, we
obtain ð¯zτ(g−1) = τ(g−1)ð¯z on functions ψ+N , and so τ(g)D/N = D/Nτ(g) for all g. Replacing z
by ζ and g by g′ and adjusting a few signs, the same calculations show that τ(g)D/ S = D/ Sτ(g)
for all g, as expected.
Exercise 9.15. Verify directly that (9.37) satisfies the equations (9.35).
6The parameters (φ, θ, ψ) in this product are the so-called Euler angles for the group SU(2).
116
9.8 Angular momentum operators
Definition 9.9. The homomorphisms t 7→ g(~n; t) := exp(1
2
it ~n · ~σ) = cos 1
2
t + i sin 1
2
t ~n · ~σ,
for each ~n ∈ S2, yield all one-parameter subgroups of SU(2). The infinitesimal generators of
these subgroups are − i
2
~n · ~σ ∈ su(2), where su(2) is the Lie algebra of antihermitian 2 × 2
matrices. The corresponding generator J~n of the spinor action of this subgroup is
−i(J~nψ±N)(z, z¯) :=
d
dt
∣∣∣∣
t=0
τ(g(~n; t))ψ±N(z, z¯) =
d
dt
∣∣∣∣
t=0
at(z, z¯)
±1/2ψ±N
(
bt(z), bt(z)
)
, (9.39)
where the coefficient (−i) is inserted for convenience, so that J~n is formally selfadjoint (rather
than skewadjoint).
For the three cardinal directions, where ~n · ~σ = n1σ1 + n2σ2 + n3σ3, we write the gen-
erators simply as J1, J2, J3 respectively; these are commonly called the angular momentum
generators.7 As before, we write J± := J1 ± iJ2.
With the notations a˙0(z, z¯) :=
d
dt
∣∣
t=0
at(z, z¯) and b˙0(z) :=
d
dt
∣∣
t=0
bt(z), the definition (9.39)
simplifies to
J~n = −i
(
b˙0(z)
∂
∂z
+ b˙0(z)
∂
∂z¯
)
+ 1
2
a˙0(z, z¯) γ
1γ2 (9.40)
on recalling that iγ1γ2 = ±1 on Γ(S±).
Thus, for the one-parameter subgroup { k(−t) : t ∈ R }, where at(z, z¯) = eit and bt(z) =
eitz, we get a˙0(z, z¯) = i, b˙0(z) = iz, and so
J3 = z
∂
∂z
− z¯ ∂
∂z¯
+ 1
2
iγ1γ2 = −i ∂
∂φ
+ 1
2
iγ1γ2. (9.41)
For the one-parameter subgroup {h(−t) : t ∈ R }, at(z, z¯) and bt(z) are given by (9.37) with
t replaced by −t. In this case a˙0(z, z¯) = 12(z − z¯) and b˙0(z) = 12(z2 + 1), leading to
J2 =
i
2
(z2 + 1)
∂
∂z
+ i
2
(z¯2 + 1)
∂
∂z¯
− 1
4
(z − z¯) γ1γ2.
The one-parameter subgroup { cos 1
2
t+ i sin 1
2
t σ1 : t ∈ R } is generated by i2σ1, and at(z, z¯),
bt(z) are now given by
at(z, z¯) =
cos 1
2
t+ iz¯ sin 1
2
t
cos 1
2
t− iz sin 1
2
t
, bt(z) =
z cos 1
2
t− i sin 1
2
t
cos 1
2
t− iz sin 1
2
t
,
for which a˙0(z, z¯) =
i
2
(z¯ + z) and b˙0(z) =
i
2
(z2 − 1). Therefore
J1 = −12(z2 − 1)
∂
∂z
+ 1
2
(z¯2 − 1) ∂
∂z¯
− i
4
(z + z¯) γ1γ2.
7The suitability of interpreting these generators as angular momentum operators for a magnetic monopole
is discussed at length in [10].
117
The operators J± are therefore given by:
J+ = J1 + iJ2 = −z2 ∂
∂z
− ∂
∂z¯
− 1
2
z iγ1γ2,
J− = J1 − iJ2 = ∂
∂z
+ z¯2
∂
∂z¯
− 1
2
z¯ iγ1γ2. (9.42)
It follows from this and from (9.41) that [J+, J−] = 2J3. One also obtains that J21 + J
2
2 =
J−J+ + 12 [J+, J−] = J−J+ + J3.
Exercise 9.16. Show that the operators J± are given in spherical coordinates by
J± = e±iφ
(
± ∂
∂θ
+ i cot θ
∂
∂φ
− 1 + cos θ
2 sin θ
iγ1γ2
)
.
These operators arise in the theory of the magnetic monopole [10, 25] when the monopole
parameter8 µ = eg/~c takes the value µ = 1
2
.
Exercise 9.17. Show that over U1 the angular momentum generators satisfy the analogue
of (9.40), with z replaced by ζ. Verify that
J3 = −ζ ∂
∂ζ
+ ζ¯
∂
∂ζ¯
− 1
2
iγ1γ2,
J+ =
∂
∂ζ
+ ζ¯2
∂
∂ζ¯
− 1
2
ζ¯ iγ1γ2,
J− = −ζ2 ∂
∂ζ
− ∂
∂ζ¯
− 1
2
ζ iγ1γ2, (9.43)
in the coordinates (ζ, ζ¯) over U1.
There is one more operator worthy of mention, namely that corresponding to the “Casimir
element” X21 +X
2
2 +X
2
3 , where {X1, X2, X3} is an orthonormal basis for the Lie algebra su(2).
[The Casimir element belongs to the enveloping algebra U(su(2)).] The image of this element
under a representation of the Lie algebra is called a “Casimir operator”.
Definition 9.10. The Casimir operator for the spinor bundle action of SU(2) is the
operator defined by C := J21 + J
2
2 + J
2
3 = J−J+ + J3(J3 + 1) on Γ(S).
9
Proposition 9.7. The Casimir operator satisfies the relations
C = ∆S + 1
4
= D/ 2 − 1
4
. (9.44)
8The constants are c, the speed of light; ~, Planck’s constant; e, the electric charge of a particle whose
total angular momentum is J ; and g, the magnetic charge of the monopole. The condition that 2µ be an
integer is the quantization condition of Dirac [24], and arises from the necessary description of monopoles
by complex line bundles over the sphere.
9Strictly speaking the Casimir operator should be −C, on account of the factor (−i) in (9.39); but we
change the sign to obtain a positive operator. On Γ(S), it is formally selfadjoint.
118
Proof. Using (9.41) and (9.42), we compute that
C = J−J+ + J3(J3 + 1)
= −(1 + zz¯)2 ∂
2
∂z ∂z¯
+ 1
4
(1 + zz¯) + 1
2
(1 + zz¯) iγ1γ2
(
z
∂
∂z
− z¯ ∂
∂z¯
)
over U0, with an analogous formula over U1. A glance at (9.20) and (9.23) is enough to verify
(9.44).
This result shows that even in the simplest example of a spinor bundle over a compact
manifold, the three operators commonly referred to as “the Laplacian” are distinct, and
must be carefully distinguished. Though they only differ by constants, this has the important
consequence that their spectra are not the same. A similar shifting of the Laplacian occurs in
harmonic analysis on compact Lie groups [59], where the Casimir satisfies C = ∆+ 1
12
dimG.
For SU(2), a 3-dimensional group, this leads us to expect C = ∆ + 1
4
, which is nicely
confirmed by Proposition 9.7.
9.9 Spinor harmonics
We come, finally, to the matter of diagonalizing the Dirac operator by finding an explicit
basis of eigenspinors for D/ . In view of the SU(2) symmetry, we may suspect, by analogy with
the diagonalization of the Hodge–Dirac operator in Section 4, that the members of this basis
should be closely related to the spherical harmonics Ylm(θ, φ) on S2. We may also anticipate
that at some point we shall need to use some heavy artillery from the representation theory of
SU(2). However, we begin in a fairly pedestrian manner, with some polynomial calculations
over the chart U0.
Lemma 9.8. The identity
ðz
(
(1 + zz¯)−lzr(−z¯)s)= (1 + zz¯)−l((l + 1
2
− r)zr(−z¯)s+1 + rzr−1(−z¯)s)
holds for all r, s ∈ N and l ∈ R.
Exercise 9.18. Prove Lemma 9.8, and show also that the conjugate identity holds: −ð¯z
(
(1 +
zz¯)−lzr(−z¯)s)= (1 + zz¯)−l((l + 1
2
− s)zr+1(−z¯)s + szr(−z¯)s−1).
These calculations show one method of finding eigenspinors: take for ψ−N a linear combi-
nation of several terms of the form (1+zz¯)−lzr(−z¯)s, with a common value for the difference
of exponents (r−s), and choose the coefficients cleverly enough that the result of applying ðz
closely resembles a multiple of the original function. However, since we wish these functions
to be components of spinors, we must first consider the effect of the gauge transformation
rules.
Lemma 9.9. Let φ : C→ C be a smooth function of the form
φ(z, z¯) := (1 + zz¯)−l
∑
r,s∈N
a(r, s)zr(−z¯)s.
119
Then φ represents a section in Γ(U0, S
±) if and only if l + 1
2
is a positive integer, and
a(r, s) = 0 for r > l ∓ 1
2
or s > l ± 1
2
. Moreover, the coefficients must satisfy the symmetry
relations a(r, s) = (−1)l± 12a(l ∓ 1
2
− r, l ± 1
2
− s).
Proof. Suppose that φ represents a section in Γ(U0, S
+). Then by (9.7) we obtain
φ(z, z¯) = (z¯/z)1/2φ(z−1, z¯−1) = z−1/2z¯1/2(zz¯)l(1 + zz¯)−l
∑
r,s∈N
a(r, s)z−r(−z¯)−s
= (1 + zz¯)−l(−1)l+ 12
∑
r,s∈N
a(r, s)zl−
1
2
−r(−z¯)l+ 12−s,
where the exponents in the sum on the right hand side must also be nonnegative integers.
Thus l − 1
2
∈ N, and the nonnegativity of the exponents on the right guarantees that
r ∈ {0, 1, . . . , l − 1
2
} while s ∈ {0, 1, . . . , l + 1
2
}. The argument for sections in Γ(U0, S−)
is similar.
The structure of the symmetry relations among the coefficients, and the allowed ranges
of the exponents, suggests the introduction of the following spinors.
Definition 9.11. For each l ∈ N+ 1
2
= {1
2
, 3
2
, 5
2
, . . . }, and for each m ∈ {−l,−l + 1, . . . , l −
1, l},10 let Y ′lm ∈ Γ(S) be the spinors whose components over U0 are 2−1/2Y ±lm(z, z¯), where
Y +lm(z, z¯) := Clm(1 + zz¯)
−l ∑
r−s=m− 1
2
(
l − 1
2
r
)(
l + 1
2
s
)
zr(−z¯)s,
Y −lm(z, z¯) := Clm(1 + zz¯)
−l ∑
r−s=m+ 1
2
(
l + 1
2
r
)(
l − 1
2
s
)
zr(−z¯)s, (9.45)
where the constants Clm are defined as
11
Clm := (−1)l−m
√
2l + 1
4pi
√
(l +m)! (l −m)!
(l + 1
2
)! (l − 1
2
)!
. (9.46)
Also, let Y ′′lm ∈ Γ(S) be the spinor whose components are 2−1/2Y +lm(z, z¯) and −2−1/2Y −lm(z, z¯).
The simplest examples of (9.45) are
Y ′1
2
, 1
2
:=
1√
4pi
(
1/
√
1 + zz¯
z/
√
1 + zz¯
)
, Y ′1
2
,− 1
2
:=
1√
4pi
(
z¯/
√
1 + zz¯
−1/√1 + zz¯
)
.
10The range of allowed values of m is obtained by listing the possibilities for (r−s) in (9.45) and adjusting
by 12 . Notice that each m is a half-integer.
11The precise form of the constants Clm is obtained by looking ahead to the normalization 〈〈Y ′lm |Y ′lm〉〉 = 1;
for the present, we need only that the constants for Y +lm and Y
−
lm be the same.
120
The functions Y +lm and Y
−
lm were introduced by Newman and Penrose [43], using the
notations − 1
2
Ylm and 1
2
Ylm respectively. In fact, these appear as a subfamily of functions
sYlm with s ∈ {−l,−l+1, . . . , l−1, l} which, for l,m, s integers, they called “spin-s spherical
harmonics”; for s = 0 they reduce to the everyday spherical harmonics Ylm on S2. Newman
and Penrose also noted that their formulas make sense when l,m, s are all half-integers, and
christened such functions “spinor harmonics”; they were investigated further by Goldberg et
al [30]. Later, Dray [25] showed that these same functions occur as the spinor components
in the theory of the magnetic monopole [10].
Lemma 9.10. ðzY −lm = (l +
1
2
)Y +lm and −ð¯zY +lm = (l + 12)Y −lm.
Proof. Using Lemma 9.8, we find that ðzY −lm(z, z¯) equals
Clm(1 + zz¯)
−l ∑
r−s=m+ 1
2
(
l + 1
2
r
)(
l − 1
2
s
)
{(l + 1
2
− r)zr(−z¯)s+1 + rzr−1(−z¯)s}
= Clm(1 + zz¯)
−l ∑
j−k=m− 1
2
{
(l + 1
2
− j)
(
l + 1
2
j
)(
l − 1
2
k − 1
)
+ (j + 1)
(
l + 1
2
j + 1
)(
l − 1
2
k
)}
zj(−z¯)k. (9.47)
The term in braces can be simplified, using the binomial identities
k
(
r
k
)
= r
(
r − 1
k − 1
)
, (r − k)
(
r
k
)
= r
(
r − 1
k
)
,
to the form
(l + 1
2
)
(
l − 1
2
j
)(
l − 1
2
k − 1
)
+ (l + 1
2
)
(
l − 1
2
j
)(
l − 1
2
k
)
= (l + 1
2
)
(
l − 1
2
j
)(
l + 1
2
k
)
,
so the right hand side of (9.47) equals (l + 1
2
)Y +lm.
Corollary 9.11. The spinors Y ′lm and Y
′′
lm are eigenspinors for the Dirac operator, with
nonzero integer eigenvalues ±(l + 1
2
):
D/Y ′lm = (l +
1
2
)Y ′lm, D/ Y
′′
lm = −(l + 12)Y ′′lm,
and each eigenvalue ±(l + 1
2
) has multiplicity (2l + 1).
Proof. Just observe that(
0 ðz
−ð¯z 0
)(
Y +lm
Y −lm
)
=
(
(l + 1
2
)Y +lm
(l + 1
2
)Y −lm
)
,
(
0 ðz
−ð¯z 0
)(
Y +lm
−Y −lm
)
=
(−(l + 1
2
)Y +lm
(l + 1
2
)Y −lm
)
.
The multiplicity is just the number of possibilities for the index m, i.e., the (2l+ 1) elements
of {−l,−l + 1, . . . , l − 1, l}.
Exercise 9.19. For a fixed l ∈ N+ 1
2
, express J3Y
′
lm, J+Y
′
lm and J−Y
′
lm as linear combinations
of the spinors Y ′ln with n ∈ {−l,−l + 1, . . . , l − 1, l}.
121
9.10 The spectrum of the Dirac operator
Definition 9.12. A representative function for the group SU(2) is a function in L2(SU(2))
which may appear as a matrix element in some finite-dimensional unitary representation
of the group. It is a linear combination of the functions Djmn, indexed by j ∈ 12N ={0, 1
2
, 1, 3
2
, 2, . . . } and m,n ∈ {−j,−j + 1, . . . , j − 1, j}, which are defined in terms of the
Euler-angle presentation g = k(α)h(β)k(γ) ∈ SU(2) by
Djmn(α, β, γ) :=
√
(j + n)! (j − n)!
(j +m)! (j −m)! e
i(nα+mγ)(sin 1
2
β)2j
×
∑
r
(−1)j+m−r
(
j +m
r
)(
j −m
r −m− n
)
(cot 1
2
β)2r−m−n. (9.48)
The Hilbert space L2(SU(2)) is described by defining the Haar measure on SU(2) in terms
of the Euler angles as dg = (16pi2)−1 sin β dα dβ dγ, and the Parseval–Plancherel formula [37]
shows that the functions Djmn form an orthogonal basis for this Hilbert space:∫
SU(2)
|h(g)|2 dg =
∞∑
2j=0
(2j + 1)
j∑
m,n=−j
∣∣(Djmn | h)∣∣2.
On comparing the definitions (9.45), (9.46) of the functions Y ±lm(z, z¯) with (9.48), we see
from z = eiφ cot 1
2
θ that
Y +lm(z, z¯) =
√
2l + 1
4pi
Dl− 1
2
,m
(φ, θ, φ), Y −lm(z, z¯) =
√
2l + 1
4pi
Dl1
2
,m
(φ, θ, φ).
Exercise 9.20. Show that, with ζ = e−iφ tan 1
2
θ, the formulae
Y +lm(ζ, ζ¯) =
√
2l + 1
4pi
Dl− 1
2
,m
(φ, θ,−φ), Y −lm(ζ, ζ¯) =
√
2l + 1
4pi
Dl1
2
,m
(φ, θ,−φ)
express Y ±lm in terms of the SU(2) representative functions over U1.
We now (at last!) fix the normalization of the inner product of spinors by
‖ψ‖2 = 〈〈ψ | ψ〉〉 := 1
4pi
∫
S2
(ψ+ψ+ + ψ−ψ−) sin θ dθ ∧ dφ. (9.49)
Proposition 9.12. The spinors {Y ′lm, Y ′′lm : l ∈ N+ 12 ,m ∈ {−l, . . . , l} } form an orthonormal
basis for the Hilbert space L2(S).
Proof. We associate to each spinor ψ ∈ Γ(S) a pair of functions h± on SU(2) by
h±(φ, θ, ψ) :=
√
4pie±i(φ−ψ)/2 ψ±N(z, z¯) =
√
4pie∓i(φ+ψ)/2 ψ±S (ζ, ζ¯).
122
By integration over the ψ variable, we see that h+ is orthogonal to Djmn unless m = −12 , and
h− is orthogonal to Djmn unless m = +
1
2
. The Parseval–Plancherel formula then shows that
‖ψ‖2 = 1
4pi
∫
SU(2)
(|h+(g)|2 + |h−(g)|2) dg
=
∑
l,m
(2l + 1)
4pi
(∣∣(Dl− 1
2
,m
| h+)∣∣2 + ∣∣(Dl1
2
,m
| h−)∣∣2)
=
∑
l,m
∣∣∣∣ 14pi
∫
S2
Y +lmψ
+ Ω
∣∣∣∣2 + ∣∣∣∣ 14pi
∫
S2
Y −lmψ
−Ω
∣∣∣∣2
=
∑
l,m
∣∣〈〈Y ′lm | ψ〉〉∣∣2 + ∣∣〈〈Y ′′lm | ψ〉〉∣∣2. (9.50)
This is a Parseval identity for the orthonormal family {Y ′lm, Y ′′lm}, and so establishes com-
pleteness of this family in L2(S).
Corollary 9.13. The spectra of the Dirac operator, its square, the Casimir operator and the
spinor Laplacian are given by
sp(D/ ) = {±(l + 1
2
) : l ∈ N+ 1
2
} = Z \ {0},
sp(D/ 2) = { (l + 1
2
)2 : l ∈ N+ 1
2
},
sp(C) = { l(l + 1) : l ∈ N+ 1
2
},
sp(∆S) = { l2 + l − 1
4
: l ∈ N+ 1
2
}.
The respective multiplicities are: 2l + 1 for the eigenvalue ±(l + 1
2
) of D/ , and 2(2l + 1) for
each listed eigenvalue of D/ 2, C and ∆S.
Proof. The eigenvalues of D/ are those given by Corollary 9.11; the completeness relation
(9.50) shows that there are no others. The eigenvalues of C and ∆S follow from (9.43).
Notice that the Casimir eigenvalues have the form l(l + 1) (compare the spectrum of the
Hodge Laplacian), but with l half-integral in the present case.
We recall from (4.16) the definition of the index of (any) Dirac operator:
indD/ := dim(kerD/ +)− dim(kerD/ −),
We end with an important result.
Corollary 9.14. For the spinor module over S2, the index of the Dirac operator is zero.
The Atiyah–Singer index theorem [9, 28, 39, 42] asserts the existence of a characteristic
class whose integral coincides with the index of the Dirac operator. In fact, the characteristic
form is given by Aˆ(R) := det−1/2(j(R)), where R is the Riemannian curvature and j(x) :=
(sinh 1
2
x)/1
2
x. Since the power series j(x) is even, however, only forms of degree 4k appear in
123
the expansion of Aˆ(R) by degrees; in particular, the second-degree component is zero. Thus
for S2 or indeed any compact two-dimensional manifold M , the integral (2pii)−1
∫
M
Aˆ(R)
(called the “Aˆ-genus of M”) vanishes, and on that basis the vanishing of the index of the
Dirac operator is to be expected.
To get a Dirac operator with a nontrivial kernel, one only has to twist the irreducible
spinor module with a complex line bundle, such as H or L. In that context, the eigenspinors
(9.45) yield explicit solutions of the Seiberg–Witten equations [60].
10 Construction of representations of SU(2)
The theory of the Dirac operator on the Riemann sphere, developed in the previous chapter,
has a direct application to the construction of the irreducible unitary representations of the
group SU(2). There are three main routes to that goal. In view of the Peter–Weyl theorem,
an irreducible representation of a compact Lie group is finite-dimensional, unitarizable (that
is, the representation space may be provided with an inner product that is invariant under
the group action) and lies within the left regular representation of the group. The first route
to the irreducible unitary representations is an algebraic construction called the “theorem of
the highest weight” [13, 35, 37] which yields an abstract description of all such representa-
tions up to unitary equivalence, but does not exhibit them concretely. Secondly, the famous
Borel–Weil theorem [11, 50] constructs the representation spaces as spaces of holomorphic
sections of certain line bundles over coadjoint orbits of the compact Lie group in question;
this construction depends on the fact that these orbits are complex manifolds, and therefore
carry a spinc structure. It turns out that the maximal-dimensional coadjoint orbits in fact
carry a spin structure, compatible with the group action, and this opens a third path to the
construction of representations, whereby the representation spaces are the kernel spaces of
certain Dirac operators. (Indeed, this last recipe is related to the Borel–Weil–Bott construc-
tion by a “twisting” with a certain line bundle, so the second and third routes are thereby
equivalent.) Here we shall not attempt to give the construction for all compact groups, but
rather shall build up the particular example of SU(2), that contains all the features of the
general theory.
10.1 Characters of the maximal torus
Any compact connected Lie group G contains a maximal torus T , and any two such tori
are conjugate, by a basic theorem of Weyl [13]. For G = SU(2), we may take T to be the
subgroup of diagonal unitary matrices k(φ) of (9.36), so that T ' U(1). Its characters are
χm(k(φ)) = χm
(
eiφ/2 0
0 e−iφ/2
)
= eimφ/2 (10.1)
where m is necessarily an integer in order that χm be a well-defined homomorphism from
T to U(1). The Hopf map η of (9.26) allows us to regard S2 as the homogeneous space
SU(2)/T , whereby SU(2)−→ S2 becomes a principal U(1)-bundle.
124
Let x ∈ S2; if it is not the north or south pole, let ζ = z−1 be its local coordi-
nates. Two particular local sections of this principal bundle are γ0 ∈ Γ(U0, SU(2)) and
γ1 ∈ Γ(U1, SU(2)), given by
γ0(x) :=
1√
1 + zz¯
(
z¯ 1
−1 z
)
, γ1(x) :=
1√
1 + ζζ¯
(
1 ζ
−ζ¯ 1
)
.
It is immediate that η(γ0(x)) = z
−1 = ζ = η(γ1(x)) for x ∈ U0∩U1, which verifies that these
are sections and implies that γ0(x) = γ1(x)h(ζ) with h(ζ) ∈ T ; a quick calculation yields
h(ζ) =
(√
ζ/ζ¯ 0
0
√
ζ¯/ζ
)
,
so that χm(h(ζ)) = (ζ/ζ¯)
m/2.
Every complex line bundle over S2 may be constructed as an associated bundle to this
principal bundle via a suitable character of the maximal torus T . This can be seen directly
by exhibiting these associated bundles, since we have already classified all line bundles over
S2 in Proposition 5.13. However, it should be said that one can take a more principled point
of view. On each such line bundle, we shall eventually construct an equivariant bundle action
of SU(2); the equivalence classes of such actions generate a commutative semiring (under the
Whitney sum and tensor product of equivariant vector bundles), and by formal subtraction
(the Grothendieck construction) they generate an abelian group, denoted KSU(2)(S2) [48].
The characters of the maximal torus T form a ring R(T ), isomorphic to Z, and the matching
of equivariant line bundles and characters yields a canonical isomorphism KSU(2)(S2) ' R(T ).
The line bundle Lm−→ S2 associated to SU(2)−→ S2 via the character χm of T is given
by (1.2): the space Lm has elements [g, v] with g ∈ SU(2), v ∈ C, and [gh, v] = [g, χm(h)v]
whenever h ∈ T . The local sections s0 ∈ Γ(U0, Lm) and s1 ∈ Γ(U1, Lm) defined by sj(x) :=
[γj(x), 1] suffice to determine the class of L
m. Indeed,
s0(x) = [γ1(x)h(ζ), 1] = [γ1(x), χm(h(ζ))] = χm(h(ζ))s1(x) = (ζ/ζ¯)
m/2s1(x),
so the transition function of Lm is g01(ζ) = (ζ/ζ¯)
m/2 = e−imφ/2. As before, we represent a
general section s : S2 → Lm by a pair of functions (f0, f1) satisfying f0(z)s0(x) = f1(ζ)s1(x)
on U0 ∩ U1. We therefore get the gauge transformation rule
f1(ζ) ≡ (ζ¯/ζ)m/2f0(ζ−1).
From Exercise 9.1 (compare also (9.5) and (9.6), which are particular cases), we obtain that
the Chern class of Lm is m[L] = −m[H], so that Lm is indeed equivalent to the m-th tensor
power of the tautological line bundle, as the notation had anticipated.1
1The isomorphism KSU(2)(S2) ' R(T ) is thus given by m[L] 7→ χm.
125
10.2 Twisting connections
In subsection 9.3 we have seen that the spin connection ∇S is defined over U1 (say) by
∇S∂i = ∂i + ωi, where ζ = x1 + ix2 and ω1, ω2 are given by (9.9). Since ∂/∂ζ = 12(∂1 − i∂2),
∂/∂ζ¯ = 1
2
(∂1 + i∂2), we may rewrite (9.9) as
2
∇S∂/∂ζ =
∂
∂ζ
+
iζ¯
2(1 + ζζ¯)
γ1γ2, ∇S∂/∂ζ¯ =
∂
∂ζ¯
− iζ
2(1 + ζζ¯)
γ1γ2.
If we use the presentation (9.12) of γ1 and γ2, in which the grading operator iγ1γ2 is a
diagonal matrix, these become
∇S±∂/∂ζ =
∂
∂ζ
± ζ¯
2(1 + ζζ¯)
, ∇S±∂/∂ζ¯ =
∂
∂ζ¯
∓ ζ
2(1 + ζζ¯)
. (10.2)
Now S+ = L, the tautological bundle over S2. If f1, . . . , fm, with m ∈ N, are local
sections in Γ(U1, L), then their product is a local section in Γ(U1, L
m). Denote by ∇(m) the
tensor product of m copies of ∇S+ ; this is a connection on the vector bundle Lm−→ S2, for
which
∇(m)X (f1 . . . fm) =
m∑
j=1
f1 . . . fj−1∇S±X (fj) fj+1 . . . fm,
from which it follows that
∇(m)∂/∂ζ =
∂
∂ζ
+
mζ¯
2(1 + ζζ¯)
, ∇(m)
∂/∂ζ¯
=
∂
∂ζ¯
− mζ
2(1 + ζζ¯)
(10.3)
on Γ(U1, L
m).
Exercise 10.1. If m is a negative integer, let ∇(m) be the tensor product of (−m) copies of
∇S− , i.e., the connection on Lm = H−m dual to the connection ∇(−m) on L−m; show that
(10.3) holds also in this case.
Exercise 10.2. The curvature ωm of the connection ∇(m) is determined by the relation
ωm(∂/∂ζ, ∂/∂ζ¯) = [∇(m)∂/∂ζ ,∇(m)∂/∂ζ¯ ]. Check that ωm = −m(1 + ζζ¯)−2 dζ ∧ dζ¯.
Definition 10.1. Let S−→ S2 denote the irreducible spinor bundle; let Lm−→ S2 be the
complex line bundle with first Chern class m[L]; we call S⊗Lm−→ S2 a twisted spinor bundle.
Clearly S ⊗ Lm ∼ Lm+1 ⊕ Lm−1. The twisted spin connection ∇˜ := ∇S ⊗ 1 + 1 ⊗∇(m)
is determined, in view of (10.2) and (10.3), by
∇˜±∂/∂ζ =
∂
∂ζ
+
(m± 1)ζ¯
2(1 + ζζ¯)
, ∇˜±
∂/∂ζ¯
=
∂
∂ζ¯
− (m± 1)ζ
2(1 + ζζ¯)
. (10.4)
2In this Section, we shall write explicit formulas over the chart domain U1 with local coordinates (ζ, ζ¯);
we leave the corresponding formulas for U0 to the reader.
126
10.3 Twisted Dirac operators
Definition 10.2. The Dirac operator D/m on the twisted spinor bundle S ⊗ Lm−→ S2 is
given, in accordance with Definition 8.3, by D/m := c(dx
1)∇˜∂1 + c(dx2)∇˜∂2 = c(dζ)∇˜∂/∂ζ +
c(dζ¯)∇˜∂/∂ζ¯ . From (9.10), we get
c(dζ) = −1
2
(1 + ζζ¯)(γ1 + iγ2) =
(
0 −(1 + ζζ¯)
0 0
)
,
c(dζ¯) = −1
2
(1 + ζζ¯)(γ1 − iγ2) =
(
0 0
1 + ζζ¯ 0
)
,
so that D/m = c(dζ)∇˜−∂/∂ζ + c(dζ¯)∇˜+∂/∂ζ¯ ; explicitly,
D/m =
(
0 −(1 + ζζ¯)∂/∂ζ − 1
2
(m− 1)ζ¯
(1 + ζζ¯)∂/∂ζ¯ − 1
2
(m+ 1)ζ 0
)
=
(
0 −ðζ − 12mζ¯
ð¯ζ − 12mζ 0
)
=:
(
0 D/ −m
D/ +m 0
)
. (10.5)
From (9.15) we get ðζψ + 12mζ¯ψ = Q1(ζ)∂ψ/∂ζ +
1
2
(m − 1)(∂Q1/∂ζ)ψ, where we have
written Q1(ζ) = 1 + ζζ¯. Also, ð¯ζψ − 12mζψ = Q1(ζ)∂ψ/∂ζ¯ − 12(m+ 1)(∂Q1/∂ζ¯)ψ. Thus,
D/ −m = −Q1(ζ)−(m−3)/2
∂
∂ζ
Q1(ζ)
(m−1)/2,
D/ +m = Q1(ζ)
(m+3)/2 ∂
∂ζ¯
Q1(ζ)
−(m+1)/2. (10.6)
The kernel of D/ −m is a subspace of Γ(L
m−1) and the kernel of D/ +m is a subspace of
Γ(Lm+1). Suppose that ψ ∈ Γ(Lm+1), and let ψS(ζ, ζ¯) and ψN(z, z¯) be its component
functions over U1 and U0 respectively. These are related by the gauge transformation rule
ψN(ζ
−1, ζ¯−1) ≡ (ζ/ζ¯)(m+1)/2ψS(ζ, ζ¯). From (10.6), ψ ∈ kerD/ +m iff
a(ζ) := (1 + ζζ¯)−(m+1)/2ψS(ζ, ζ¯) (10.7)
is an entire holomorphic function on C. Moreover,
ψN(ζ
−1, ζ¯−1) = ((1 + ζζ¯)ζ/ζ¯)(m+1)/2a(ζ)
must be regular at ζ = ∞. Since Q1(ζ)(m+1)/2 = O(|ζ|)m+1, for nonnegative m this is only
possible if a(ζ) ≡ 0. If m is negative, the entire function a(ζ) is O(|ζ|)|m|−1 as |ζ| → ∞, so
that a(ζ) is a polynomial of degree at most |m| − 1. To sum up:
dim kerD/ +m =
{
0, if m ≥ 0,
|m|, if m < 0.
127
The kernel of D/ −m is found similarly. Suppose that φ ∈ Γ(Lm−1), with component func-
tions φS(ζ, ζ¯) and φN(z, z¯) related by φN(ζ
−1, ζ¯−1) ≡ (ζ/ζ¯)(m−1)/2φS(ζ, ζ¯). Then φ ∈ kerD/ −m
iff
b(ζ¯) := (1 + ζζ¯)(m−1)/2φS(ζ, ζ¯) (10.8)
is antiholomorphic. The regularity at∞ of φN(ζ−1, ζ¯−1) = ((1+ζζ¯)ζ¯/ζ)−(m−1)/2b(ζ¯), together
with Q1(ζ)
−(m−1)/2 = O(|ζ|)1−m, shows that b(ζ¯) ≡ 0 for m negative or zero, while for m
positive b(ζ¯) is a polynomial in ζ¯ of degree at most m− 1. In fine:
dim kerD/ −m =
{
m, if m > 0,
0, if m ≤ 0.
We have in particular shown the following result.
Proposition 10.1. The index of the Dirac operator D/m on the twisted spinor bundle S ⊗
Lm−→ S2 is given by
indD/m = dim kerD/
+
m − dim kerD/ −m = −m,
that is, by the integer that labels the first Chern class of the twisting line bundle Lm.
10.4 The group action on the twisted spinor bundles
Proposition 10.2. A bundle action of SU(2) on S ⊗ Lm−→ S2 is given by the formula
(9.27), where ψ± ∈ Γ(Lm±1), and where the multipliers A±N , A±S are determined by
A±N(g, z) :=
(
βz¯ + α¯
β¯z + α
)(m±1)/2
, A±S (g
′, ζ) :=
(−β¯ζ¯ + α
−βζ + α¯
)(m±1)/2
.
Proof. The analysis of the SU(2)-action on the untwisted spinor bundle, given in Section 9.6,
may be repeated verbatim, except that the gauge transformation factor (ζ¯/ζ)1/2 in (9.30)
must be replaced by (ζ¯/ζ)(m±1)/2. This leads to the necessary and sufficient condition (9.31),
with the exponent 1/2 replaced by (m± 1)/2, and the cocycles (10.2) satisfy that condition.
Proposition 10.3. The Dirac operator D/m is equivariant under the action of SU(2) on the
twisted spinor bundle S ⊗ Lm−→ S2 determined by (10.2).
Proof. Let T ∈ End+(Γ(S⊗Lm)) be an operator that commutes with D/m and is of the form
(Tψ±S )(ζ, ζ¯) = c±(ζ, ζ¯)ψ
±
S
(
b(ζ), b(ζ)
)
. Then, as in Lemma 9.5, the identity TD/ −m = D/
−
mT
—in Hom(Γ(Lm−1),Γ(Lm+1))— yields the relations
Q1(ζ)c−(ζ, ζ¯)b′(ζ) = Q1(b(ζ))c+(ζ, ζ¯),
Q1(ζ)
∂c−
∂ζ
= 1
2
(m− 1)(b(ζ)c+(ζ, ζ¯)− ζ¯c−(ζ, ζ¯)). (10.9)
128
Similarly, from TD/ +m = D/
+
mT in Hom(Γ(L
m+1),Γ(Lm−1)) we get
Q1(ζ)c+(ζ, ζ¯)b′(ζ) = Q1(b(ζ))c−(ζ, ζ¯),
Q1(ζ)
∂c+
∂ζ¯
= −1
2
(m+ 1)
(
b(ζ)c−(ζ, ζ¯)− ζc+(ζ, ζ¯)
)
. (10.10)
We check that these conditions are satisfied when T = T (g) is the operator corresponding
to the diagonal element g = k(φ) of SU(2), determined by (10.2). Here b(ζ) = g′−1 ·
ζ = eiφζ. Clearly Q1(b(ζ)) ≡ Q1(ζ), so that (10.9) and (10.10) reduce to the conditions
c+(ζ, ζ¯) = e
iφc−(ζ, ζ¯), ∂c−/∂ζ = 0, and ∂c+/∂ζ¯ = 0, which together imply that c− and c+
are independent of ζ and ζ¯. Since the prescription c±(ζ, ζ¯) := ei(m±1)φ/2, dictated by (10.2),
obeys c+ = e
iφc−, we see that T (g) commutes with D/m.
In the same way, it is easy to show that the operator T (h(θ)), associated to h(θ) of (9.36)
by b(ζ) = h(θ)′−1 · ζ and by (10.2), commutes with D/m. Since elements of the form k(φ)
and h(θ) generate SU(2), and since g 7→ T (g) is a homomorphism on account of the cocycle
relations (9.29), we conclude that T (g)D/m = D/mT (g) for all g ∈ SU(2).
Exercise 10.3. Complete the proof of the previous Proposition by verifying that the coefficient
functions of T (h(θ)) satisfy (10.9) and (10.10).
Corollary 10.4. The bundle action of SU(2) on S ⊗ Lm−→ S2 determined by (9.27) and
(10.2) restricts to a finite-dimensional unitary representation ρm of SU(2) on the kernel of
the Dirac operator D/m.
To exhibit the representation ρm, it is useful to notice that with g
′−1 ·ζ = (αζ+β¯)/(−βζ+
α¯), one has Q1(g
′−1 · ζ) = Q1(ζ)| − βζ + α¯|−2 on account of αα¯+ ββ¯ = 1. If m > 0, then for
ψ ∈ kerD/ −m ⊂ Γ(Lm−1), the bundle action (9.27) specializes to
ρm
(
α β
−β¯ α¯
)
ψS(ζ, ζ¯) =
(−β¯ζ¯ + α
−βζ + α¯
)(m−1)/2
ψS
(
αζ + β¯
−βζ + α¯ ,
α¯ζ¯ + β
−β¯ζ¯ + α
)
. (10.11)
Using (10.8), we can write ψS(ζ, ζ¯) = Q1(ζ)
−(m−1)/2b(ζ¯) with b a polynomial in ζ¯ of degree
less than m, and the previous formula simplifies to
ρm
(
α β
−β¯ α¯
)[
Q1(ζ)
−(m−1)/2b(ζ¯)
]
= Q1(ζ)
−(m−1)/2(−β¯ζ¯ + α)m−1 b
(
α¯ζ¯ + β
−β¯ζ¯ + α
)
.
The right hand side is clearly also Q
−(m−1)/2
1 times a polynomial in ζ¯ of degree less than m.
If we define ψk(ζ, ζ¯) := Q1(ζ)
−(m−1)/2ζ¯k, for k = 0, 1, . . . ,m − 1, these form an orthogonal
basis for kerD/ −m. For these basis vectors,
ρm
(
α β
−β¯ α¯
)
ψk(ζ, ζ¯) = Q1(ζ)
−(m−1)/2(α¯ζ¯ + β)k(−β¯ζ¯ + α)m−k−1, (10.12)
from which the matrix elements of the representation may be computed at once.
129
When m < 0, we obtain similar formulas for ρm. Indeed, for ψ ∈ kerD/ +m ⊂ Γ(Lm+1),
we obtain (10.11) with the exponent (m − 1)/2 replaced by (m + 1)/2. Since, by (10.7),
ψS(ζ, ζ¯) = Q1(ζ)
(m+1)/2a(ζ) with a a polynomial in ζ of degree less than m, we get
ρm
(
α β
−β¯ α¯
)[
Q1(ζ)
(m+1)/2a(ζ)
]
= Q1(ζ)
(m+1)/2(−βζ + α¯)|m|−1 a
(
αζ + β¯
−βζ + α¯
)
, (10.13)
and on the orthogonal basis ψk(ζ, ζ¯) := Q1(ζ)
−(|m|−1)/2ζk (k = 0, 1, . . . ,m − 1) for kerD/ +m,
we find that
ρm
(
α β
−β¯ α¯
)
ψk(ζ, ζ¯) = Q1(ζ)
−(|m|−1)/2(αζ + β¯)k(−βζ + α¯)|m|−k−1. (10.14)
From the explicit formulae (10.12) and (10.14), it is evident that ρ−m is the conjugate
representation to ρm.
Proposition 10.5. The representation ρm is irreducible.
Proof. If m = 0, there is nothing to prove. The case m < 0 mirrors the case m > 0,
on interchanging D/ +m with D/
−
m; thus we may take m > 0. Then kerD/m = kerD/
−
m is an
m-dimensional Hilbert space with the orthogonal basis {ψk : k = 0, 1, . . . ,m − 1 }. It is
immediate from (10.12) that this basis consists of joint eigenvectors for { ρm(g) : g ∈ T };
indeed, with g = k(φ) we get
ρm(k(φ))ψk = e
i(m−1)φ/2e−ikφψk, (10.15)
so the eigenvalues are generally distinct, and therefore any operator S on kerD/ −m that com-
mutes with each ρm(g) must have a diagonal matrix in this basis. On the other hand,
the one-parameter subgroup of operators t 7→ ρm(h(t)) mingles the basis elements ψk: for
h(t), one has α = cos 1
2
t, β = sin 1
2
t, and from (10.12) we get (d/dt)
∣∣
t=0
ρm(h(t))ψk =
1
2
kψk−1− 12(m−k−1)ψk+1 if 1 ≤ k ≤ m−2. A similar calculation with the subgroup h′(t) for
which α = cos 1
2
t and β = i sin 1
2
t gives (d/dt)
∣∣
t=0
ρm(h
′(t))ψk = 12ikψk−1 +
1
2
i(m−k−1)ψk+1.
Thus if S commutes with every ρm(g), it commutes with the “ladder operator” ψk 7→ kψk−1,
and hence S is scalar. Irreducibility of ρm now follows from Schur’s lemma.
In order to identify the unitary irreducible representation ρm of SU(2), it suffices to com-
pute its character. Since the character is a class function, it is determined by its restriction
to a maximal torus, so for m > 0 we obtain it immediately from (10.15):
χm(k(φ)) =
m−1∑
k=0
ei(m−1)φ/2e−ikφ =
eimφ/2
eiφ/2
1− e−imφ
1− e−iφ =
eimφ/2 − e−imφ/2
eiφ/2 − e−iφ/2 .
When m < 0, we similarly get from (10.14)
ρm(k(φ))ψk = e
i(m+1)φ/2eikφψk,
130
and so its character is:
χm(k(φ)) =
|m|−1∑
k=0
ei(m+1)φ/2eikφ =
eimφ/2
e−iφ/2
1− eimφ
1− eiφ =
eimφ/2 − e−imφ/2
eiφ/2 − e−iφ/2 .
Therefore the representations ρ−m and ρm are equivalent.
This equivalence is to be expected on general grounds. The Weyl group of SU(2), namely
the normalizer subgroup of the maximal torus T modulo T itself, is just the two-element
group W ' Z2, since the only nontrivial isomorphism σ of the diagonal subgroup T is the
interchange of diagonal elements, that can be implemented by conjugating with
(
0 1
−1 0
)
.
By (10.1), χm ◦ σ = χ−m; so two characters of T lead to equivalent representations of SU(2)
iff they lie in the same orbit under the action of the Weyl group. This exemplifies the general
relation [13, 49]:
R(G) ' R(T )W .
This construction of the unitary irreducible representations of SU(2) directly from the
equivariant Dirac operator exemplifies the “universal quantization map” of Vergne [55]:
Q : KSU(2)(S2)→ R(SU(2)),
which is already an instance of the index theorem of Atiyah, Segal and Singer [5, 6].
10.5 The Borel–Weil theorem
The construction of the irreducible unitary representations of SU(2) on the kernel spaces
of Dirac operators, developed in the preceding sections, produces two families of equivalent
representations, according as one twists the standard bundle with a tensor power of the
tautological bundle (Lm, with m > 0), or with a tensor power of the hyperplane bundle
(Hn ∼ L−n, with n > 0). Now only the latter line bundles admit nonzero holomorphic
sections, while only the former admit nonzero antiholomorphic sections. The celebrated
construction of Borel and Weil does not deal with spinor bundles, but rather uses the fact
that the coadjoint orbits of compact Lie groups are complex manifolds and constructs the
desired representations on the spaces of holomorphic sections of holomorphic line bundles
over those orbits. By a simple application of Liouville’s theorem —see Section 5.6 for the
details in the case M = CP(m)— the compactness of the orbit implies that the space of
holomorphic sections is finite-dimensional. Indeed, in many cases, that of the trivial line
bundle for instance, the space of holomorphic sections reduces to zero. A variant of the
Borel–Weil construction produces representations on space of antiholomorphic sections of
the respective dual line bundles.
There is a simple relationship between the spinor-based and the Borel–Weil constructions:
one passes from one to the other by twisting or untwisting with a fixed line bundle that can
be obtained directly from the spin structure of the maximal coadjoint orbit G/T . Indeed,
this relationship is parallel to the replacement of the spin structure by a spinc structure
131
on a spin manifold: for that, one needs to identify a particular principal U(1)-bundle that
combines with the spin structure to yield a spinc structure. (See the discussion in Section 7.3,
and also Appendix D of [39].)
If M is a complex manifold of complex dimension m, let K := Λm,0T ∗M ; then K −→M
is the so-called canonical line bundle, and Γ(K) = Am,0(M). The first Chern class of the dual
bundle K∗, turns out [41] to be the element of Hˇ2(M,Z) whose modulo-2 reduction is the
Stiefel–Whitney class w2(M); thus w2(M) + j∗(c1(K)) = 0 in Hˇ2(M,Z2), in the notation of
Section 7.3. If M is spin, so that w2(M) = 0, then j∗(c1(K)) = 0 also, so that c1(K) is even;
in other words, there is a complex line bundle K1/2−→M such that K1/2 ⊗K1/2 ∼ K. In
principle, the “square root of the canonical bundle” K1/2 may depend on the spin structure
chosen for M .
For the case M = S2, we know that K ∼ L⊗ L = L2 by Exercise 9.3. Thus we can take
K1/2 := L. Its dual line bundle is K−1/2 := H. Therefore we find that
S ⊗K−1/2 = (L⊕H)⊗H ∼ E ⊕H2 ∼ Λ0,•T ∗S2
where E = S2×C denotes the trivial line bundle, and Λ0,•T ∗S2−→ S2 is the complex vector
bundle whose smooth sections form the module A0,•(S2) = A0,0(S2) ⊕ A0,1(S2). Over U1,
such a section may be written as f(ζ, ζ¯) + h(ζ, ζ¯) dζ¯.
Thus A0,•(S2) = Γ(S⊗L−1) is the domain of the twisted Dirac operator D/ −1. Of course,
(10.5) for m = −1 simplifies to
D/ −1 =
(
0 −Q1(ζ)∂/∂ζ + ζ¯
Q1(ζ)∂/∂ζ¯ 0
)
.
The operatorD/ +−1 : A
0,0(S2)→ A0,1(S2) has as kernel the holomorphic functions inA0,0(S2) =
C∞(S2), which are just the constant functions, by Liouville’s theorem. The factor Q1(ζ) =
1 + ζζ¯ is a normalization factor, that takes account of the metric on A0,1(S2).
To see that, consider the twisted spinor bundle S ⊗ Lm for any negative m, say m =
−(2j+1) with j a nonnegative half-integer.3 Clearly S⊗Lm = Λ0,•T ∗S2⊗H2j = H2j⊕H2j+2,
so that S+⊗Lm = H2j admits holomorphic sections. If s1 denotes a section of H normalized
by (s1 |s1) = 1, then s2j1 is a normalized section of H2j. We may select s1 as s1 := Q1(ζ)1/2σ1,
where σ1 is the holomorphic section of H −→CP(1) of Section 5.6: the normalization follows
from (5.13), whereby we have (σ1 | σ1) = Q1(ζ)−1.
It is convenient4 to choose the metric 4g−1 = Q1(ζ)2(∂/∂ζ) · (∂/∂ζ¯) on T ∗CM , so that the
normalized section of A0,1(S2) is just s21 = Q1(ζ)−1 dζ¯. It follows that σ21 = Q1(ζ)−2 dζ¯.
Any section of A0,1(S2) is of the form h(ζ, ζ¯) dζ¯ = Q1(ζ)h(ζ, ζ¯) s21, and a section of
S⊗L−1 = L0⊕L−2 can be written as φ = f(ζ, ζ¯) +h(ζ, ζ¯) dζ¯. Then the Dirac operator D/ −1
can be identified as follows. Firstly,
D/ +−1f =
(
Q1(ζ)
∂f
∂ζ¯
)
s21 =
∂f
∂ζ¯
dζ¯ = ∂¯f
3By now it should be clear that it is preferable to label representations of SU(2) by nonnegative half-
integers j rather than the integers m.
4Without this normalization, the operator ∂¯ in the following formulae should be replaced by
√
2 ∂¯; this
convention is adopted in [28], for instance.
132
where ∂¯ is the Dolbeault operator of Section 2.1.
The twisted Dirac operator D/m, with m = −(2j + 1), is
D/m =
(
0 −Q1(ζ)∂/∂ζ + (j + 1)ζ¯
Q1(ζ)∂/∂ζ¯ + jζ 0
)
,
by rewriting (10.5). An element of Γ(H2j) can be written (over U1) as f(ζ, ζ¯)σ
2j
1 . Then
D/ +m(f(ζ, ζ¯)σ
2j
1 ) = D/
+
m(Q1(ζ)
−jf(ζ, ζ¯) s2j1 )
=
(
Q1(ζ)
∂
∂ζ¯
+ jζ
)[
Q1(ζ)
−jf(ζ, ζ¯)
]
s2j+21 = Q1(ζ)
−j+1∂f
∂ζ
s2j+21
= Q1(ζ)
2∂f
∂ζ
σ2j+21 =
∂f
∂ζ
dζ¯ ⊗ σ2j1 = ∂¯f ⊗ σ2j1 . (10.16)
The Dolbeault operator extends to A0,•(S2, H2j) := A0,•(S2)⊗Γ(H2j) by setting ∂¯(ω⊗s) :=
(∂¯ω) ⊗ s for s ∈ Γ(H2j). With this understanding,5 we have the relation D/ +−2j−1 = ∂¯ as
operators from A0,0(S2, H2j) to A0,1(S2, H2j).
In order to identify D/ −m, we must compute the adjoint of the Dolbeault operator. We
consider first the case m = −1.
Definition 10.3. The adjoint of ∂¯ : A0,0(S2) → A0,1(S2) is the operator ∂¯∗ : A0,1(S2) →
A0,0(S2) given by
〈〈∂¯∗(h dζ¯) | f〉〉 := 〈〈h dζ¯ | ∂¯f〉〉
where 〈〈· | ·〉〉 denotes the integrated inner product in either space of sections, defined as in
(9.49) by integrating the appropriate pairing of sections over S2 with respect to the volume
form (4pi)−1 sin θ dθ ∧ dφ = (2pii)−1Q1(ζ)−2 dζ ∧ dζ¯. We find that
〈〈∂¯∗(h dζ¯) | f〉〉 = 〈〈h dζ¯ | ∂¯f〉〉 = 1
2pii
∫
C
4g−1
(
h¯ dζ,
∂f
∂ζ¯
dζ¯
)
Q1(ζ)
−2 dζ ∧ dζ¯
=
1
2pii
∫
C
h¯
∂f
∂ζ¯
dζ ∧ dζ¯ = − 1
2pii
∫
C
∂h
∂ζ
f dζ ∧ dζ¯
= −〈〈Q1(ζ)2 ∂h/∂ζ¯ | f〉〉,
on integrating by parts, so that ∂¯∗(h dζ¯) = −(1 + ζζ¯)2 ∂h/∂ζ¯.
From this it follows easily that
D/ −−1(h dζ¯) = D/
−
−1(Q1h s
2
1) = (−Q1∂/∂ζ + ζ¯)(Q1h) = −Q21 ∂h/∂ζ¯ = ∂¯∗(h dζ¯). (10.17)
Thus D/ −−1 = ∂¯
∗ on A0,1(S2).
Exercise 10.4. Show D/ −−2j−1 = ∂¯
∗ as operators from A0,1(S2, H2j) to A0,0(S2, H2j).
5We could have written ∂¯(2j) to denote the extended operator, but we suppress the index to reduce
notational clutter.
133
We summarize these calculations as follows.
Proposition 10.6. The Dirac operator on the twisted spinor bundle S ⊗ L−2j−1 equals the
sum of the Dolbeault operator and its adjoint on the twisted bundle Λ0,•T ∗S2 ⊗H2j.
Proof. The formulae (10.16) and (10.17), plus the previous exercise, establish that D/ +−2j−1 =
∂¯ and D/ −−2j−1 = ∂¯
∗, thus
D/ −2j−1 = ∂¯ + ∂¯∗
on the module A0,•(S2, H2j).
The kernel of the Dirac operator D/ −2j−1 thus coincides with ker ∂¯ ⊕ ker ∂¯∗. The sec-
ond summand is zero since kerD/ −−2j−1 = 0. Thus kerD/
+
−2j−1 = ker ∂¯ consists of sections
ψS(ζ, ζ¯) s
2j
1 = f(ζ, ζ¯)σ
2j
1 ∈ A0,0(S2, H2j) for which ∂f/∂ζ¯ = 0; these are precisely the holo-
morphic sections f(ζ)σ2j1 ∈ O(H2j). Since
ψS(ζ, ζ¯) s
2j
1 = f(ζ)σ
2j
1 = f(ζ)Q1(ζ)
−j s2j1
it follows from (10.7) that f(ζ) is a polynomial of degree at most 2j. That is to say, we have
shown that dimO(H2j) = 2j+ 1. By suppressing the factors Q1(ζ)
−j in (10.13) and (10.14),
we arrive at the following representation pij of SU(2) on O(H
2j), that is by construction
equivalent to ρ−2j−1:
pij
(
α β
−β¯ α¯
)[
f(ζ)
]
= (−βζ + α¯)2j f
(
αζ + β¯
−βζ + α¯
)
,
and on the orthogonal basis { ξk := ζk σ2j1 : k = 0, 1, . . . , 2j } for O(H2j), we find that
pij
(
α β
−β¯ α¯
)
ξk = (αζ + β¯)
k(−βζ + α¯)2j−k σ2j1 .
This completes the passage from the Dirac-operator-kernel construction of the irreducible
representations of SU(2) to their realization on finite-dimensional spaces of holomorphic
sections of line bundles over S2. The content of the foregoing construction is the Borel–
Weil theorem for the compact group SU(2), that we may now state as follows.
Theorem 10.7. Every irreducible unitary representation of SU(2) can be realized on a space
of holomorphic sections of a holomorphic Hermitian line bundle over S2. This line bundle
is determined, up to equivalence, by a character χ of a maximal torus of SU(2) modulo the
action of the Weyl group Z2. Moreover, if E−→ S2 is a holomorphic Hermitian line bundle
whose Chern class is m[H] with m > 0, then O(E) carries the representation corresponding
to the character χm of the maximal torus.
This result is of course well known; what we have done is to show how it arises from
the equivariant index of the Dirac operator on the flag manifold S2. We remark that our
construction is equivalent to the standard induced representation recipe for each pij, though
this is usually achieved by studying the complexification of the compact Lie group in question
[56].
134
A Calculus on manifolds
In this Appendix, we briefly review the concepts and notations of calculus on manifolds, with
emphasis on the algebraic formulae which arise in differential geometry. Proofs are left to
the reader. General references are the books of Abraham, Marsden and Ratiu [1], Crampin
and Pirani [20], Singer and Thorpe [51], and Spivak [52].
A.1 Differential manifolds
Definition A.1. A differential manifold of finite dimension n is a paracompact Hausdorff
topological space M together with a family (or “atlas”) of local charts { (Uj, φj) : j ∈ J }
such that U := {Uj : j ∈ J } is a locally finite open covering of M , φj : Uj → Rn is a
homeomorphism onto an open subset of Rn, and the transition functions
φi ◦ φ−1j : φj(Ui ∩ Uj)→ φi(Ui ∩ Uj)
are smooth. A second atlas { (Vk, ψk) : k ∈ K } is declared equivalent to the first if every
φi ◦ ψ−1k is smooth (the “differentiable structure” of M is actually an equivalence class of
atlases). If M is compact, a finite atlas may be chosen.
If n = 2m is even, we can regard the chart maps φj as having images in Cm. We say
that M is a complex manifold if the transition functions are holomorphic maps between open
subsets of Cm.
Definition A.2. If M , N are two differential manifolds, a continuous map f : M → N is
smooth if for any pair of local charts (U, φ) for M and (V, ψ) for N , the composite map
ψ ◦f ◦φ−1 : φ(U ∩f−1(V ))→ ψ(V ) is smooth. When N = R, the set of all smooth functions
on M is a commutative algebra over R, which we denote by C∞(M,R); when N = C, the
smooth complex-valued functions on M forms a commutative C-algebra, C∞(M,C). We
often write simply C∞(M), if it is clear from the context whether real-valued or complex-
valued functions are to be used.
A diffeomorphism between M and N is a bijective smooth function f : M → N whose
inverse f−1 : N → M is also smooth. If such an f exists, we say that M and N are
diffeomorphic.
If (Uj, φj) is a local chart for M , we define x
1, . . . , xn ∈ C∞(Uj,R) by xk := prk ◦φj,
where prk : Rn → R is the k-th coordinate projector. We say (x1, . . . , xn) is a system of local
coordinates for M on the chart domain Uj.
The following two lemmas show that smooth functions are abundant.
Lemma A.1. If M is a differential manifold, and if V , W are two open subsets of M with
V ⊂ W , there exists f ∈ C∞(M,R) such that supp f ⊂ W , f ≡ 1 on V , and 0 ≤ f ≤ 1
on W \ V .
Lemma A.2. If M is a differential manifold, with atlas { (Uj, φj) : j ∈ J }, there exists
a smooth partition of unity subordinate to the locally finite covering U, that is, a family
{ fj : j ∈ J } ⊂ C∞(M,R) with 0 ≤ fj ≤ 1 and supp fj ⊂ Uj for each j, such that∑
j∈J fj(x) = 1 for all x ∈M (the sum is finite for each x).
135
If M , N are two manifolds with respective atlases {(Uj, φj)}, {(Vk, ψk)}, the product
manifold is the cartesian product M × N with atlas {(Uj × Vk, φj × ψk)}; its dimension is
dim(M ×N) = dimM + dimN .
A.2 Tangent spaces
Definition A.3. Let M be a differential manifold, x ∈ M . Let C∞(M,x) be the set of all
smooth functions f : Vf → R whose domain is an open neighbourhood of x in M ; this is a
commutative algebra that includes C∞(M). Indeed, C∞(M) =
⋂
x∈M C
∞(M,x).
A tangent vector at x is an R-linear map v : C∞(M,x) → R which satisfies the “local
Leibniz rule”:
v(fg) = v(f) g(x) + f(x) v(g), for all f, g ∈ C∞(M,x).
These form a real vector space TxM .
If (Uj, φj) is a local chart for M , with local coordinates (x
1, . . . , xn), then the directional
derivatives ∂
∂xj
∣∣
x
: f 7→ Dj(f ◦φ−1)(φ(x)) form a basis for TxM ; in particular, dimTxM = n.
If γ : I → M is a smooth curve, whose domain is an interval I ⊆ R, with γ(t0) = x, its
velocity vector at x is γ˙(t0) ∈ TxM defined by γ˙(t0)(f) := (f ◦ γ)′(t0).
Definition A.4. If f : M → N is smooth, and x ∈ M , the tangent mapping Txf : TxM →
Tf(x)N is the R-linear map
Txf(v) : h 7→ v(h ◦ f),
for v ∈ TxM , h ∈ C∞(N, f(x)). If (y1, . . . , yr) is a system of local coordinates near f(x) ∈ N ,
the matrix of Txf is has entries ∂f
k/∂xj|x := ∂∂xj
∣∣
x
(yk ◦ f).
Definition A.5. A smooth mapping f : M → N is an immersion if for all x ∈ M , the
tangent map Txf : TxM → Tf(x)N is injective. If each Txf is surjective, f is a submersion.
If f is both an immersion and a submersion, then dimM = dimN , and the Jacobian
matrix of Txf (in local coordinates) is invertible, for each x; by the inverse function theorem,
f is a diffeomorphism between a neighbourhood of x and a neighbourhood of f(x), for each x;
we say f is a local diffeomorphism. However, f need not be injective or surjective on all
of M , so it need not be a global diffeomorphism.
A.3 Vector fields
Definition A.6. Let M be a differential manifold. A vector field on M is a R-linear
operator X : C∞(M)→ C∞(M), which is a derivation of this algebra, that is, it satisfies the
Leibniz rule:
X(fg) = (Xf) g + f (Xg), for all f, g ∈ C∞(M).
These derivations form a vector space denoted by X(M). This is in fact a module for the
algebra C∞(M), if we define1 fX ∈ X(M) by fX(h) := f(Xh).
1There is a notational difficulty here, since expressions like fXh, while not ambiguous, are confusing. For
this reason, the derivation action of the vector field X on the function h is sometimes written X · h rather
than Xh; then the module structure can be defined by (fX) · h := f(X · h), and so forth.
136
If Y ∈ X(M) and x ∈M , the recipe Yxf := (Y f)(x) defines a tangent vector Yx ∈ TxM .
If (U, φ) is a local chart of M , any vector field X ∈ X(M) determines a vector field
X|U ∈ X(U) by restriction. If (x1, . . . , xn) is the local coordinate system for this chart, we
obtain n linearly independent local vector fields ∂
∂xj
∈ X(U) by writing
∂
∂xj
(f) := Dj(f ◦ φ−1) ◦ φ.
These form a basis for the C∞(U)-module X(U), that is, every X ∈ X(U) is of the form
X =
∑n
j=1 a
j ∂
∂xj
, with a1, . . . , an ∈ C∞(U).
Definition A.7. The Lie bracket of two vector fields X, Y ∈ X(M) is defined as
[X, Y ] : f 7→ X(Y f)− Y (Xf).
It is easy to check that this is a derivation of C∞(M); it is clearly skewsymmetric, and it
satisfies the Jacobi identity :
[X, [Y, Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0. (A.1)
Thus X(M) is an (infinite-dimensional) Lie algebra.
Definition A.8. If τ : M → N is a diffeomorphism, and X ∈ X(M), we define the pushout
τ∗X ∈ X(N) by:
τ∗X(h) := X(h ◦ τ) ◦ τ−1 for all h ∈ C∞(N).
For each x ∈M , we find that (τ∗X)τ(x) = Txτ(Xx). Note that this last formula makes sense
if τ : M → N is smooth and surjective, but not necessary invertible.
Lemma A.3. The pushout τ∗ : X(M) → X(N) is a Lie algebra homomorphism, i.e., τ∗ is
linear and τ∗[X, Y ] = [τ∗X, τ∗Y ] for X, Y ∈ X(M). If σ : N → R is a diffeomorphism, then
(σ ◦ τ)∗ = σ∗ ◦ τ∗.
Definition A.9. An integral curve of a vector fieldX ∈ X(M) is a smooth curve γ : I →M
such that γ˙(t) = Xγ(t) for all t ∈ I.
One can always find a unique integral curve γx for X satisfying γ(0) = x in some maximal
interval Ix 3 0, by the existence and uniqueness theorem for first-order ordinary differential
equations. We say the vector field X is complete if Ix = R for all x ∈ M ; if M is compact,
every vector field is complete. Write φt(x) := γx(t); then φt : M → M is a diffeomorphism
for all t ∈ R, and φt ◦ φs = φt+s for all t, s.
The one-parameter group of diffeomorphisms {φt} is called the flow generated by the
vector field X. The vector field may be recovered from the flow by noticing that
Xf = lim
t→0
f ◦ φt − f
t
=
d
dt
∣∣∣∣
t=0
(f ◦ φt).
137
A.4 Lie groups
Definition A.10. A Lie group is a differential manifold G which is also a group, for which
the multiplication (g, h) 7→ gh : G×G→ G and the inversion g 7→ g−1 : G→ G are smooth
maps.
The left translations λg : h 7→ gh and the right translations ρg : h 7→ hg are diffeomor-
phisms from G onto G. We usual write e to denote the identity element of G.
A finite-dimensional (real or complex) vector space V is an additive Lie group. The group
GLR(V ) of invertible linear operators on V is a Lie group (since it is a dense open subset
of the vector space EndR(V )); if V is a complex vector space, GLC(V ) is a Lie group. We
write GL(n,R) := GLR(Rn) and GL(m,C) := GLC(Cm).
Definition A.11. A vector field X ∈ X(G) on a Lie group G is left-invariant if (λg)∗X = X
for all g ∈ G. In that case, X is determined by its value at the identity, Xe ∈ TeG.
A Lie algebra is a real vector space with a skewsymmetric bilinear operation [·, ·] sat-
isfying the Jacobi identity (A.1). Since the Lie bracket [X, Y ] of two left-invariant vector
fields X, Y is also left-invariant, TeG becomes a (finite-dimensional) Lie algebra by defining
[Xe, Ye] := [X, Y ]e. We usually write g := TeG to denote this Lie algebra.
Definition A.12. If G is a Lie group with Lie algebra g, and if X ∈ g, let γX be the
integral curve of the corresponding left-invariant vector field such that γX(0) = e. Then
γX(s+ t) = γX(s) γX(t) for all s, t ∈ R, so that t 7→ γX(t) is a one-parameter subgroup of G;
also γtX(1) = γX(t) for t ∈ R.
We write expX := γX(1). This defines the exponential map exp: g → G, which
satisfies exp tX = γX(t), and thus t 7→ exp tX is a homomorphism.
Unless G is abelian, exp is not a homomorphism of the additive group g into G. However,
there is the important Campbell–Baker–Hausdorff formula:
exp tX exp tY = exp
(
t(X + Y ) + 1
2
t2[X, Y ] +O(t3)
)
,
and its corollary:
exp tX exp tY exp(−tX) = exp(tY + t2[X, Y ] +O(t3)). (A.2)
If G is a closed subgroup of GLR(V ), the Lie algebra can be identified with the subspace
of operators {X ∈ EndR(V ) : exp tX ∈ G for all t ∈ R }, and the Lie bracket becomes
[X, Y ] = XY − Y X ∈ EndR(V ). If W is a complex vector space, the Lie algebra of a
subgroup of GLC(W ) is likewise identified to a subspace of EndC(W ).
Definition A.13. A smooth left action of a Lie group G on a differential manifold M
is a smooth map Φ: G × M → M : (g, x) 7→ Φ(g, x) ≡ g · x, such that e · x = x and
g · (h · x) = (gh) · x, for g, h ∈ G, x ∈M .
Thus Φg : x 7→ g · x is a diffeomorphism of M for each g ∈ G.
If x ∈ M , the isotropy subgroup for x is Gx := {h ∈ G : h · x = x }, and the orbit of x
is G · x := { g · x ∈ M : g ∈ G } ⊆ M . The natural bijection g · x 7→ gGx between G · x
and the left-coset space G/Gx is a diffeomorphism when G/Gx is given a natural differential
structure for which the quotient map η : G→ G/Gx is a submersion.
138
We say that the action of G on M is free if no group element except e leaves any point
fixed; in that case, all isotropy groups are trivial and all orbits are diffeomorphic to G.
We say that that the action of G on M is transitive if there is only one orbit. If so, and
if H is the isotropy subgroup of some point of M , then M is diffeomorphic to G/H (and the
action of G on M corresponds to permutation of the left-cosets g′H 7→ gg′H). A manifold
with a transitive G-action is called a homogeneous space for the group G.
Definition A.14. A smooth right action of a Lie group G on a differential manifold M
is similarly defined, as a smooth map M × G → M : (x, g) 7→ x · g such that x · e = x and
(x · g) · h = x · (gh), for g, h ∈ G, x ∈M .
If H is a closed subgroup of G, then H acts (on the right) on G by right translations
g 7→ gh; the orbits of this action are the cosets gH.
Definition A.15. The adjoint action of a Lie group G on its Lie algebra g is the map
(g,X) 7→ Ad(g)X given by
Ad(g)X :=
d
dt
∣∣∣∣
t=0
g(exp tX)g−1.
It can be deduced from (A.2) that Ad(g)X ∈ g. The map g 7→ Ad(g) is a homomorphism
Ad: G→ GL(g).
Definition A.16. A representation of a Lie group G on a vector space V is a homomor-
phism ρ : G→ GL(V ) for which (g, v) 7→ ρ(g)v is a smooth map from G× V to V .
The derived representation of its Lie algebra g is the linear map ρ˙ : g→ End(V ) given by
ρ˙(X)v :=
d
dt
∣∣∣∣
t=0
ρ(exp tX)v.
In particular, the derived representation ad of the adjoint representation Ad is given, on
account of (A.2), as ad(X)Y = [X, Y ].
A.5 Fibre bundles
Definition A.17. A fibre bundle is a triple (E,M, pi), more usually written as E
pi−→M ,
where E and M are differential manifolds (called, respectively, the total space and the base
space), and pi : E → B is a surjective submersion,2 such that each fibre Ex := pi−1({x}) is
diffeomorphic to a fixed manifold F , called the “typical fibre”, and which is locally trivial in
the following sense. If U = {Uj : j ∈ J } is a covering of the base space by chart domains,
there are diffeomorphisms
ψj : pi
−1(Uj)→ Uj × F (A.3)
(called “local trivializations”) such that pi(ψ−1j (x, v)) = x for all x ∈ Uj, v ∈ F .
2We shall often omit explicit mention of the submersion pi and simply write E−→M to denote the fibre
bundle.
139
Definition A.18. If E
pi−→M and E ′ pi′−→M ′ are two fibre bundles, a bundle morphism
is a pair of smooth maps (τ, σ), with τ : E → E ′ and σ : M →M ′, such that pi′ ◦ τ = σ ◦ pi.
(We also say that τ is a lifting of the map σ between the base spaces.)
A bundle equivalence is a bundle morphism (τ, σ) such that both τ and σ are diffeo-
morphisms.
A fibre bundle E−→M is trivial if it is equivalent to the product bundle M × F pr1−→M
via a bundle morphism (τ, idM). Thus (A.3) says that any fibre bundle is locally a trivial
bundle.
Definition A.19. A vector bundle is a fibre bundle E
pi−→M whose typical fibre is a (real
or complex) vector space V , and whose fibres Ex are vector spaces of the same dimension,
such that the maps V → Ex : v 7→ ψ−1j (x, v) are linear isomorphisms.
The dimension dimV is called the rank of the vector bundle.
The tangent bundle TM −→M of a manifold M has total space TM := { (x, v) :
v ∈ TxM }, with pi(x, v) := x. The local trivialization ψj : pi−1(Uj) → Uj × Rn is given by
ψj(x, v) := (x; v
1, . . . , vn), with v =
∑
k v
k ∂
∂xk
∣∣
x
. The atlas { (pi−1(Uj), (φj×id)◦ψj) : j ∈ J }
makes TM a 2n-dimensional manifold. The fibre at x ∈M is the tangent space TxM .
The cotangent bundle T ∗M −→M is formed similarly; its fibres are the dual spaces
T ∗xM := (TxM)
∗. We define {dx1|x, . . . , dxn|x} as the dual basis in T ∗xM to the basis
{ ∂
∂x1
∣∣
x
, . . . , ∂
∂xn
∣∣
x
} of TxM , and if ξ =
∑
k ξk dx
k|x, then ψj(x, ξ) := (x; ξ1, . . . , ξn).
Definition A.20. A smooth section of a fibre bundle E
pi−→M is a smooth map s : M → E
such that pi ◦ s = idM , i.e., s(x) ∈ Ex for each x ∈ M . We denote the totality of smooth
sections by Γ(M,E), or simply by Γ(E) if the base space M is understood; it is a C∞(M)-
module, where the action of C∞(M) is just scalar multiplication on each fibre:
(fs)(x) := f(x) s(x),
for s ∈ Γ(E), f ∈ C∞(M).
If U ⊂ M is open, a smooth map s : U → pi−1(U) satisfying pi(s(x)) = x for x ∈ U is
called a local section of E
pi−→M ; all such maps form a vector space Γ(U,E), which is a
module over C∞(U).
From the definition, it is easy to see that a smooth section of the tangent bundle
TM −→M can be written in local coordinates on a chart domain Uj as X =
∑n
k=1 a
k ∂
∂xk
,
with each ak ∈ C∞(U). Thus X is nothing other than a vector field on M ; we have
Γ(TM) = X(M) as C∞(M)-modules.
A section of a trivial fibre bundle M × F pr1−→M is of the form s(x) = (x, f(x)) where
f : M → F is a smooth map. Thus sections of more general bundles can be thought of as
“functions” which take values in different sets at each point of their domains.
140
A.6 Tensors and differential forms
Definition A.21. A differential 1-form on a differential manifold M is a map α : X(M)→
C∞(M) that is C∞(M)-linear, that is:
α(X + Y ) = α(X) + α(Y ), α(fX) = f α(X),
for X, Y ∈ X(M) and f ∈ C∞(M). These form a real vector space A1(M), which becomes
a C∞(M)-module on defining fα : X 7→ f α(X).
If x ∈ M , then α(Y )(x) = fα(Y )(x) if f is any smooth function with f(x) = 1, whose
support is an (arbitrary small) neighbourhood of x. Thus α(Y )(x) depends only on Yx,
and so Yx 7→ α(Y )(x) is an element αx of the dual space T ∗xM of TxM ; by definition,
α(Y )(x) = αx(Yx).
Hence α can be identified with the section x 7→ αx of the cotangent bundle T ∗M −→M ;
and Γ(T ∗M) = A1(M) as C∞(M)-modules. In local coordinates over U , we can write
α =
∑n
k=1 fk dx
k with each fk ∈ C∞(U).
Definition A.22. A tensor of bidegree (p, q) on a manifold M is a multilinear map
T : X(M)p ×A1(M)q → C∞(M)
such that T (X1, . . . , Xp, α
1, . . . , αq) is C∞(M)-linear in each Xj and each αk. The tensor
is called covariant if q = 0, or contravariant if p = 0. Any such tensor T defines a smooth
section of a vector bundle over M whose fibre at x ∈M is (T ∗xM)⊗p ⊗ (TxM)⊗q.
Definition A.23. A Riemannian metric on M is a tensor g of bidegree (2, 0) that is
symmetric, i.e., g(X, Y ) = g(Y,X) for all X, Y ∈ X(M), and positive definite: g(X,X) > 0
for nonzero X ∈ X(M). Locally, we may take g = gij dxi · dxj, where [gij] is a positive-
definite symmetric matrix of elements of C∞(U) and dxi · dxj := 1
2
(dxi ⊗ dxj + dxj ⊗ dxi);
a Riemannian metric may be defined globally on M by taking g =
∑
j fjg
(j), where g(j) is a
metric on the chart domain Uj, and the functions fj form a smooth partition of unity on M .
The pair (M, g), consisting of a differential manifold with a Riemannian metric g, is
called a Riemannian manifold.
A Hermitian metric on M is a tensor h of bidegree (2, 0) with values in C∞(M,C), such
that h(X, Y ) = h(Y,X) for X, Y ∈ X(M) and h is positive definite. One often writes (X |Y )
instead of h(X, Y ). The same partition-of-unity argument shows that any manifold can be
given a Hermitian metric.
Definition A.24. A differential k-form on M is a covariant tensor ω : X(M)k → C∞(M)
that is alternating, which means that ω(Xσ(1), . . . , Xσ(k)) = (−1)σω(X1, . . . , Xk) for σ ∈ Sk.
The totality of k-forms on M is denoted Ak(M), and is a C∞(M)-module.
Let ΛkT ∗M −→M be the vector bundle whose fibre at x is ΛkT ∗xM , the k-th exterior
power of T ∗xM ; then A
k(M) = Γ(ΛkT ∗M).
The direct sum A•(M) :=
⊕n
k=0A
k(M) = Γ(Λ•T ∗M) is a Z-graded C∞(M)-module.
The zero-degree term isA0(M) := C∞(M). Under the exterior product, A•(M) is an algebra;
141
in fact, it is Z2-graded by the parity of the degree k, and is a supercommutative superalgebra,
since this property holds in each fibre of Λ•T ∗M −→M . Thus ω ∧ η = (−1)]ω ]ηη ∧ ω where
]ω = k for ω ∈ Ak(M).
Thus if ω ∈ Ak(M), η ∈ Al(M), and X1, . . . , Xk+l ∈ X(M), then
(ω ∧ η)(X1, . . . , Xk+l) = 1
k! l!
∑
σ∈Sk+l
(−1)σω(Xσ(1), . . . , Xσ(k)) η(Xσ(k+1), . . . , Xσ(k+l)),
and in particular, (α ∧ β)(X, Y ) = α(X)β(Y ) − α(Y )β(X) for α, β ∈ A1(M). Moreover, if
α1, . . . , αk ∈ A1(M) then α1 ∧ · · · ∧ αk ∈ Ak(M), with
(α1 ∧ · · · ∧ αk)(X1, . . . , Xk) = det[αi(Xj)].
In local coordinates, an element of Ak(U) is of the form ω =
∑
|J |=k fJ dx
j1 ∧ · · · ∧ dxjk ,
where J = {j1, . . . , jk} ⊆ {1, . . . , n}.
A.7 Calculus of differential forms
Definition A.25. If τ : M → N is a diffeomorphism, and ω ∈ Ak(N), we define the pull-
back τ ∗ω ∈ Ak(M) by:
τ ∗ω(X1, . . . , Xk) := ω(τ∗X1, . . . , τ∗Xk) ◦ τ.
In particular, τ ∗β(X) := β(τ∗X) ◦ τ for β ∈ A1(N). Thus (τ ∗β)x(Xx) := βτ(x)(Txτ(Xx)), so
that the linear map βτ(x) 7→ (τ ∗β)x : T ∗τ(x)N → T ∗xM is the transpose of the tangent map
Txτ : TxM → Tτ(x)N .
If τ : M → N is any smooth map, not necessarily a diffeomorphism, the pullback τ ∗ω of
ω ∈ Ak(N) is likewise defined by transposition:
(τ ∗ω)x((X1)x, . . . , (Xk)x) := ωτ(x)(Txτ((X1)x), . . . , Txτ((Xk)x)).
For k = 0, we get simply: τ ∗f := f ◦ τ .
Lemma A.4. The pullback τ ∗ : A•(N) → A•(M) is a degree-preserving homomorphism of
exterior algebras, that is, τ ∗ is linear and τ ∗(ω ∧ η) = τ ∗ω ∧ τ ∗η for ω, η ∈ A•(N). If
σ : N → R is a smooth map, then (σ ◦ τ)∗ = τ ∗ ◦ σ∗.
Definition A.26. The contraction of a k-form ω ∈ Ak(M) with a vector field X ∈ X(M)
is the (k − 1)-form ι(X)ω ≡ ιXω defined as
ιXω(X1, . . . , Xk−1) := ω(X,X1, . . . , Xk−1).
For f ∈ A0(M), we set ιXf := 0. If α ∈ A1(M), then ιXα = α(X) ∈ C∞(M).
142
Lemma A.5. Contraction with X ∈ X(M) is an odd derivation of the graded algebra A•(M),
that is,
ιX(ω ∧ η) = ιXω ∧ η + (−1)]ωω ∧ ιXη.
In particular,
ιX(α
1 ∧ · · · ∧ αk) =
k∑
j=1
(−1)j−1αj(X) (α1 ∧ · ·
j
∨· · ∧ αk)
for α1, . . . , αk ∈ A1(M).
Proposition A.6. There is a unique operator d : A•(M)→ A•(M), called exterior deriva-
tion, such that:
1. d is an odd derivation of degree +1, that is, d(Ak(M)) ⊂ Ak+1(M), and
d(ω ∧ η) = dω ∧ η + (−1)]ωω ∧ dη;
2. df(X) = Xf for f ∈ A0(M) = C∞(M);
3. d2 ≡ d ◦ d = 0;
4. d is natural with respect to restrictions, that is, if U ⊂M is open and ω ∈ A•(U), then
d(ω|U) = (dω)
∣∣
U
.
In local coordinates, d
(∑
J fJ dx
j1 ∧ · · · ∧ dxjk) = ∑J dfJ ∧ dxj1 ∧ · · · ∧ dxjk , where, for
f ∈ C∞(U), df = ∑nj=1(∂f/∂xj) dxj.
Lemma A.7. If ω ∈ Ak(M), its exterior derivative dω ∈ Ak+1(M) is given by
dω(X1, . . . , Xk+1) =
k+1∑
j=1
(−1)j−1Xj(ω(X1, . .
j
∨. . , Xk+1))
+
∑
i<j
(−1)i+jω([Xi, Xj], X1, . .
i∨. .
j
∨. . , Xk+1)).
In particular, if α ∈ A1(M), then dα(X, Y ) = X(α(Y ))− Y (α(X))− α([X, Y ]).
Lemma A.8. The exterior derivative commutes with pullbacks: if τ : M → N is a smooth
map, and ω ∈ Ak(N), then d(τ ∗ω) = τ ∗(dω) in Ak+1(M).
Definition A.27. If X ∈ X(M) is a complete vector field, whose flow is {φt}, the Lie
derivative of a differential form ω ∈ A•(M) is the form LXω defined by
LXω :=
d
dt
∣∣∣∣
t=0
φ∗tω = lim
t→0
φ∗tω − ω
t
. (A.4)
Notice that LXf = Xf for f ∈ A0(M).
143
If R is a contravariant tensor, we interpret φ∗tR as the pushout φ−t∗R by the inverse
diffeomorphism φ−t = φ−1t ; with this convention, (A.4) makes sense when ω is any tensor,
and LXω is a tensor of the same bidegree. In particular, if Y is a vector field,
LXY :=
d
dt
∣∣∣∣
t=0
φ−t∗Y =
d
dt
∣∣∣∣
t=0
Y (• ◦ φ−t) ◦ φt = [X, Y ].
Lemma A.9. The Lie derivative LX : A
•(M)→ A•(M) is an R-linear map satisfying:
1. LX is an even derivation of degree 0, that is, LX(A
k(M)) ⊂ Ak(M), and
LX(ω ∧ η) = LXω ∧ η + ω ∧ LXη; (A.5)
2. in particular, LX(fω) = (Xf)ω + f LXω;
3. LX(dω) = d(LXω); for all ω, η ∈ A•(M), f ∈ C∞(M).
Lemma A.10. The Lie derivative LX : A
•(M)→ A•(M) is given by Cartan’s formula:
LXω = ιX(dω) + d(ιXω),
i.e., LX = ιX ◦ d+ d ◦ ιX , for X ∈ X(M).
Corollary A.11. For any ω ∈ Ak(M), LXω is given by the formula:
LXω(X1, . . . , Xk) = X(ω(X1, . . . , Xk))−
k∑
j=1
ω(X1, . . . , [X,Xj], . . . , Xk)).
In particular, LXα(Y ) = X(α(Y ))− α([X, Y ]) for α ∈ A1(M), X, Y ∈ X(M).
Notice that this last formula can be rearranged as: LX(α(Y )) = LXα(Y ) + α(LXY ). In
other words, the Leibniz rule (A.5) for LX is valid not only for exterior products of forms,
but also for pairings of forms and vector fields. In passing, we note also that the Jacobi
identity (A.1) for vector fields can be written as LX([Y, Z]) = [LXY, Z] + [Y,LXZ].
Lemma A.12. The map X 7→ LX from X(M) to End(A•(M)) is a Lie algebra homomor-
phism; that is,
L[X,Y ]ω = LX(LY ω)− LY (LXω) (A.6)
for all X, Y ∈ X(M), ω ∈ A•(M).
Note that if we substitute a vector field Z for ω in (A.6), we again get the Jacobi identity.
144
A.8 The de Rham complex
Definition A.28. A cochain complex (C•, d) is a sequence of abelian groups and homo-
morphisms
C0
d0−→C1 → · · · → Cn dn−→Cn+1 dn+1−→Cn+2 → · · ·
such that dn+1 ◦ dn = 0 for all n. (The complex may terminate: it may happen that for
some N , Cn = 0 for n > N .) We usually suppress the index of d and write simply that
d2 = 0 at all stages. The elements of Cn are called “n-cochains”: c ∈ Cn is an n-cocycle if
dc = 0; it is an n-coboundary if c = db for some b ∈ Cn−1. The condition d2 = 0 says that
the totality of n-coboundaries Bn(C•) is a subgroup of the group Zn(C•) of n-cocycles. The
quotient group
Hn(C•) := Zn(C•)/Bn(C•)
is called the n-th cohomology group of the complex.
A morphism between two complexes (C•, d) and (K•, d′) is a set of homomorphisms
fn : C
n → Kn which intertwines the d-maps, i.e., fn+1 ◦ dn = d′n ◦ fn for all n. Thus
fn(Z
n(C•)) ⊆ Zn(K•) and fn(Bn(C•)) ⊆ Bn(K•), so that f induces a homomorphism
Hnf : Hn(C•)→ Hn(K•).
The components Cn of a complex may have more structure than that of an abelian group:
they could be vector spaces, modules over a commutative ring, etc.3 The cohomology groups
Hn(C•, d) inherit a similar structure.
Definition A.29. The de Rham complex of an n-dimensional differential manifold M is
the terminating complex
A0(M)
d−→A1(M)→ · · · → Ak(M) d−→Ak+1(M)→ · · · d−→An(M)
of C∞(M)-modules; here d is the exterior derivation. We say a k-form ω is closed if dω = 0,
and that ω is exact if ω = dη for some (k − 1)-form η; thus ZkdR(M) := Zk(A•(M), d)
comprises the closed k-forms and BkdR(M) := B
k(A•(M), d) comprises the exact k-forms.
The k-th de Rham cohomology group
HkdR(M) := H
k(A•(M), d)
is a real vector space. We shall denote by [ω] ∈ HkdR(M) the class of ω ∈ ZkdR(M).
By the de Rham theorems —see, for instance, [23]— if M is compact then HkdR(M) '
Hk(M,R), where the latter is the k-th singular cohomology group, which is a finite-dimen-
sional real vector space depending only on the topology of M .
The most important single fact about de Rham cohomology is the following proposition,
often called the Poincare´ lemma.
Proposition A.13. Suppose that U is a contractible manifold, i.e., for some x0 ∈ U , there
is a smooth map f : [0, 1]×U → U such that f(0, x) = x0 and f(1, x) = x, for x ∈ U . Then
H0dR(U) = R and HkdR(U) = 0 for k > 0.
3In fact, a complex can be formed from objects Cn and morphisms dn in any abelian category.
145
Proof. A contractible manifold has an atlas with a single chart (U, φ), so we can suppose
that U ⊆ Rn, that U is star-shaped about x0, and that f(t, x) = (1 − t)x0 + tx. If ω ∈
Ak(U), let η := ι∂/∂t(f
∗ω) ∈ Ak−1([0, 1] × U). Now define hk : Ak(M) → Ak−1(M) by
hk(ω) :=
∫ 1
0
η dt. Then one can check that the maps hk form a “cochain homotopy”, i.e.,
that hk+1 ◦ d + d ◦ hk = id for each k > 0, and h1 ◦ d = id−x0. The triviality of the
cohomology groups follows at once, since dω = 0 implies ω = d(hkω) for k > 0, and df = 0
implies f ≡ f(x0) for f ∈ A0(M).
A.9 Volume forms and integrals
Definition A.30. A volume form on an n-dimensional manifold M is a real n-form ν ∈
An(M) which is nonvanishing, i.e., νx 6= 0 in ΛnT ∗xM for all x ∈ M . A volume form need
not exist; we say that M is orientable if one exists.
We say that two volume forms µ, ν on M are equivalent if µ = fν for some f ∈ C∞(M)
with f(x) > 0 for all x ∈ M . (This is clearly an equivalence relation.) An equivalence class
for this relation is called an orientation on M ; a pair (M, ν) consisting of a manifold and a
volume form ν in a given equivalence class is called an oriented manifold.
In a local coordinate system on a chart domain U , we have ν = h dx1 ∧ · · · ∧ dxn with
h ∈ C∞(U,R) nonvanishing. Under a change of local coordinates in the overlap of two chart
domains, h is multiplied by a Jacobian factor, det[∂yi/∂xj].
Lemma A.14. A differential manifold M is orientable if and only if M has an atlas
{(Uj, φj)} all of whose transition functions φi ◦ φ−1j have positive Jacobians.
On an oriented manifold (M, ν), we therefore may and shall always choose an atlas such
that in every local coordinate system we have ν = h dx1 ∧ · · · ∧ dxn with h > 0. (We say
that the corresponding charts are “positively oriented”.)
Proposition A.15. Let (M, ν) be an oriented manifold. Then there is a unique linear form∫
M
: An(M)→ R, called the integral over M , such that if η ∈ An(M) vanishes outside the
domain of a positively oriented chart (U, φ) with local coordinate system (x1, . . . , xn), and if
η = f dx1 ∧ · · · ∧ dxn, then∫
M
η =
∫
φ(U)
f(x1, . . . , xn) dx1 . . . dxn,
where the right hand side is a Lebesgue integral on Rn.
Proof. Uniqueness of
∫
M
follows from the change-of-variables formula for multiple Lebesgue
integrals; existence follows by writing η =
∑
j fjηj where each ηj is supported in a chart
domain and {fj} is a partition of unity.
Lemma A.16. If (M, ν) and (N, ρ) are two oriented n-dimensional manifolds and if τ : M →
N is an orientation-preserving diffeomorphism (i.e., τ ∗ρ is equivalent to ν), then
∫
M
τ ∗η =∫
N
η for all η ∈ An(N).
146
An n-form on M is closed, since An+1(M) = 0. If η = dζ is an exact n-form, with
compact support in the domain of an oriented chart (U, φ), then∫
M
η =
∫
U
dζ =
∫
φ(U)
ψ∗(dζ) =
∫
φ(U)
d(ψ∗ζ),
where ψ = φ−1 : φ(U)→ U . Since ψ∗ζ = ∑j gj dx1∧· ·j∨· ·∧dxn for some gj ∈ C∞(U) having
compact supports in U , we conclude that d(ψ∗ζ) =
(∑
j ∂gj/∂x
j
)
dx1 ∧ · · · ∧ dxn; therefore,∫
U
dζ = 0 by the fundamental theorem of calculus. By a partition-of-unity argument, we
obtain
∫
M
dζ = 0 for any ζ ∈ An−1(M), so the integral vanishes on exact n-forms. (This is
the “boundaryless” case of Stokes’ theorem.) In consequence, [η] 7→ ∫
M
η is a well-defined
linear form on HndR(M).
147
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