2 Principal Bundles and Gauge

Theory

2.1 Physical Motivation

In many areas of theoretical physics we encounter fields defined on a spacetime
M taking values in some other space F . For instance, presume F as a real
finite-dimensional vector space, then φ ∶ M → F is a vector field. More
generally, we can consider a family of spaces {Fx }x∈M varying over the
points on M , that is φ(x) ∈ Fx for each x ∈ M . A field φ is then understood
as a section from the spacetime manifold into the bundle of spaces over
M . This is exactly the idea encoded in the mathematical theory of fiber
bundles. Namely, fiber bundles provide a tool to describe the global structure
of physical fields.
We turn back for a moment to the case of a fixed target space F over M . The
corresponding bundle is called a trivial fiber bundle and a field is essentially
the same as the data of a global function φ ∶ M → F . Schematically, we have
the following picture
M ×F
pr1 φ

M
where pr1 ○ φ = idM . Locally, every fiber bundle can be given this product
structure, meaning that infinitesimally no global twist of the fields is visible.
All the fibers, that is the family of target spaces {Fx }x∈M , are equivalent
to a common space F . However, the crucial point is that the fibers have
automorphisms. This means that the way they are equivalent to the common
space F can vary over the spacetime manifold, encoding the global structure
of the physical fields.
One particular type of bundle is the principal G-bundle, whose fiber is a Lie
group G. Principal bundles are of enormous importance in many fields of
© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019
C. Keller, Chern-Simons Theory and Equivariant Factorization
Algebras, BestMasters, https://doi.org/10.1007/978-3-658-25338-7_2
18 2 Principal Bundles and Gauge Theory

theoretical physics, especially for describing gauge theories in a geometrical

setting. In this chapter we first give a brief overview on bundles in general,
presenting the most important definitions and basic concepts. We then
review the notion of principal bundles with an emphasis on their role in
gauge theory. In doing so, we equip the principal bundles with the geometric
data of a connection. We show that the local description of connections leads
to the notion of gauge fields as used in physics. Moreover, we introduce the
curvature of a connection, known in the physics literature as the gauge field
strength.
The main reference for this chapter is the book by C. Isham [Ish13], presenting
various topics in differential geometry and their application to modern
theoretical physics, as well as the book by H. Baum [Bau14], discussing the
mathematical concepts of gauge theory. In the following, proofs are often
omitted and we refer to [Ish13], [Bau14] and [KN96] for more details.

2.2 Fiber Bundles

Definition 2.1. Let E, M , and F be topological spaces and π ∶ E → M a
continuous surjection. A fiber bundle is a tuple (E, M, π) subjected to the
condition of local triviality, i.e. we require that for every x ∈ M there exists
an open neighborhood U ⊂ M and a homeomorphism ψ ∶ π −1 (U ) → U × F ,
such that the following diagram is commutative
ψ
π −1 (U ) U ×F

π pr1

Remark 2.1. The space E is referred to as the total space and the space
M as the base space of the bundle. The continuous surjection π ∶ E → M is
called the projection and F is called the fiber of the bundle.
Remark 2.2. A smooth fiber bundle is a fiber bundle where the spaces
E, M, F are smooth manifolds, the projection π is a smooth surjection and
the inverse images π −1 ({x}) are all diffeomorphic to the fiber F .
2.2 Fiber Bundles 19

Definition 2.2. Let (E, M, π) and (Ẽ, M̃ , π̃) be fiber bundles. A bundle
map is a pair (ϑ, χ), where ϑ ∶ E → Ẽ and χ ∶ M → M̃ , such that the
following diagram is commutative

ϑ
E Ẽ
π π̃
χ
M M̃

Remark 2.3. Definition 2.2 implies that for all x ∈ M the map ϑ restricts
to a map π −1 ({x}) → π̃ −1 ({χ(x)}), in other words ϑ is fiber-preserving.
Definition 2.3. A fiber bundle (E, M, π) with fiber F is called trivializable
if it is isomorphic to the product bundle (M × F, M, pr1 ), where pr1 is the
projection onto the first factor.

Roughly speaking, global sections assign to each point in the base space
an element of the fiber. However, global sections do not always exist, but
demanding the bundles to be locally trivial ensures the existence of local
sections. The following definition formalizes these notions.
Definition 2.4. A local section of a fiber bundle (E, M, π) is a map

sU ∶ U → E,

for an open subset U ⊂ M , such that the image of each point x ∈ U lies in
the fiber π −1 ({x}) over x. More precisely, we have

π ○ sU = idU .

Similarly, a global section is a map s ∶ M → E such that π ○ s = idM .

2.2.1 Vector Bundles

If the fiber of a bundle is equipped with the structure of a real vector space,
the resulting bundle is called a vector bundle. The space of sections of a
vector bundle carries a natural structure of a vector space, generalizing the
idea of the linear space of functions on a manifold.
Definition 2.5. A real vector bundle of dimension k is a fiber bundle
(E, M, π) in which each fiber has the structure of a k-dimensional real vector
20 2 Principal Bundles and Gauge Theory

space. Moreover, we require that for every x ∈ M there exists an open

neighborhood U ⊂ M and a local trivialization ψ ∶ π −1 (U ) → U × Rk such
that the restriction ψ∣π−1 ({y}) ∶ π −1 ({y}) → {y} × Rk is a linear isomorphism
for all y ∈ U .

Definition 2.6. Let (E, M, π) and (Ẽ, M̃ , π̃) be two vector bundles. A
vector bundle map is a bundle map (ϑ, χ), in which the restriction of ϑ ∶
E → Ẽ to each fiber is a linear map.
Remark 2.4. The space of sections of a vector bundle (E, M, π) carries
a natural structure of a real vector space. Indeed, let s, s̃ ∶ M → E be two
sections. Then, we can define

(s + s̃)(x) ∶= s(x) + s̃(x)

(rs)(x) ∶= rs(x)

for all x ∈ M and r ∈ R.

2.3 Principal Fiber Bundles

A principal fiber bundle is a bundle whose fibers are diffeomorphic to a Lie
group G. More precisely, the total space carries an action of the Lie group
G, turning each fiber into a right G-torsor. The essential definitions are the
following.
Definition 2.7. Let P and M be smooth manifolds, π ∶ P → M a smooth
surjection and G a Lie group. A principal G-bundle is a fiber bundle
(P, M, π, G), together with a right G-action

Γ∶P ×G→P
(p, g) ↦ Rg (p) = p.g

such that
– G preserves the fibers and acts freely and transitively on them;
– there exist a G-equivariant local trivialization of the bundle.

Remark 2.5. The manifold P is referred to as the total space and the
manifold M as the base space of the bundle. The smooth surjection π ∶ P →
M is called the projection and the Lie group G is called the structure group.
2.3 Principal Fiber Bundles 21

Remark 2.6. G-equivariant local triviality requires that for each x ∈ M

there exists an open neighborhood U ⊂ M and a diffeomorphism ψ ∶ π −1 (U ) →
U × G, such that the following diagram is commutative
ψ
π −1 (U ) U ×G

π pr1

Hence, we can write ψ(p) = (π(p), φ(p)) for some G-equivariant diffeomor-
phism
φ ∶ π −1 (U ) → G, satisfying φ(p.g) = φ(p)g.
On non-empty overlaps Uα ∩ Uβ , we have two trivializing maps ψα (p) =
(π(p), φα (p)) and ψβ (p) = (π(p), φβ (p)) for each p ∈ π −1 ({x}), where x ∈
Uα ∩ Uβ . From the G-equivariance of φ it follows that there has to be a map

φα (p)φβ (p)−1 ∶ G → G,

which is independent of the choice of p ∈ π −1 ({x}). We can identify this map

with an element of G, more precisely, we can define transition functions φαβ
on each non-empty overlap Uα ∩ Uβ

φαβ ∶ Uα ∩ Uβ → G, φαβ (x) ∶= φα (p)φβ (p)−1 ,

for any p ∈ π −1 ({x}). Notice that in physics literature the transition functions
are often referred to as local gauge transformations.
Definition 2.8. Let (P, M, π, G) and (P̃ , M̃ , π̃, G) be two principal G-
bundles. A principal bundle map is a bundle map (ϑ, χ), where ϑ is G-
equivariant in the sense that

ϑ(p.g) = ϑ(p).g,

for all p ∈ P and g ∈ G.

Remark 2.7. We denote with BunG (M ) the category whose objects are
principal G-bundles over M and whose morphisms are principal bundle
maps.
Lemma 2.1. Let (ϑ, idM ) be a principal bundle map between the principal
bundles (P, M, π) and (P̃ , M, π̃), covering the identity on M . Then ϑ is an
isomorphism.
22 2 Principal Bundles and Gauge Theory

Remark 2.8. It follows from lemma 2.1 that the category BunG (M ) is a
groupoid.

Recall that a bundle (P, M, π, G) is trivializable if it is isomorphic to the

product bundle (M × G, M, pr1 , G). Lemma 2.2 states that trivializability
of a principal bundle is equivalent to the existence of a global section in the
bundle. Hence, in general, principal bundles do not admit global sections
unless they are trivial.
Lemma 2.2. A principal G-bundle is trivializable if and only if it admits
a global section.
Remark 2.9. Since we require local triviality in our definition for principal
bundles it is guaranteed that local sections exist. In fact, there are local
sections sU ∶ U → π −1 (U ) canonically associated to the local trivialization
ψ ∶ π −1 (U ) → U × G, defined so that for every x ∈ U we have ψ(sU (x)) =
(x, e).
Remark 2.10. Simply connected Lie groups play a special role in theories
involving the use of principal bundles. Namely, for G a simply connected
Lie group and M a manifold of dimension d ≤ 3, any principal bundle
(P, M, π, G) admits a global section and hence is trivializable.

2.3.1 Basics of Lie Groups and Lie Algebras

In discussing principal bundles we have to deal with some basic technology
and results from the theory of Lie groups and their associated Lie algebras.
Therefore, this section gives a short outline on this topic and introduces the
main tools and notions that are needed later on.

Lie Groups

Recall that Lie groups are groups which are also differentiable manifolds
so that the group operations are smooth. Lie groups are of enormous
significance in describing continuous symmetries of mathematical objects
and are therefore encountered in many areas of modern theoretical physics
and mathematics.
Definition 2.9. A real Lie group is a group G that is a differentiable
manifold in such a way that the following maps are smooth
2.3 Principal Fiber Bundles 23

– (Group multiplication)

µ ∶ G × G → G , µ((g, g̃)) = gg̃;

– (Inverse element)
i ∶ G → G , i(g) = g −1 .
Definition 2.10. Let G be a Lie group. For every g ∈ G we define left
translation as the map

lg ∶ G → G, h ↦ gh,

and similarly, we define right translation as the map

rg ∶ G → G h ↦ hg,

for all h ∈ G.
Remark 2.11. Every g ∈ G defines a smooth map Adg ∶ G → G by Adg =
lg ○ rg−1 , called the adjoint map. That is

Adg (h) = ghg −1 ,

for all h ∈ G.
Example 2.1. The Lie group U (n), called the unitary group, is defined by

U (n) ∶= {A ∈ GL(n, C) ∣ AA† = 1},

where GL(n, C) is the complex general linear group in n-dimensions, i.e. the
Lie group of complex n × n matrices with non-zero determinant. U (n) is
a compact Lie group with real dimension n2 . For n = 1 we get the circle
U (1) ∶= {z ∈ C ∣ ∣z∣ = 1}. This is a 1-dimensional real Lie group with group
multiplication given by the multiplication law for complex numbers. It is a
compact, connected Lie group but it is not simply connected.

Lie Algebras

One can approach the study of Lie groups by means of their associated
Lie algebras which algebraically encode parts of the Lie group’s geometry.
They describe how Lie groups ’look like’ locally, however in general failing
to capture global topological features. We review the basic notions of Lie
24 2 Principal Bundles and Gauge Theory

algebras associated to Lie groups. Moreover, we show that one can use left
invariant vector fields on the Lie group G to identify the tangent space of G
at the identity element with its Lie algebra.

Definition 2.11. A vector field X on a Lie group G is left invariant if

lg ∗ Xh = Xlg (h) = Xgh ,

for all g, h ∈ G.
Remark 2.12. The set of all left invariant vector fields on a Lie group G is
denoted by L(G). Very often we also adapt the notation L(G) = g.
Remark 2.13. The set of left invariant vector fields on G is a real vector
space. Furthermore, given two left invariant vector fields X1 and X2 on G,
their commutator is again a left invariant vector field

lg ∗ [X1 , X2 ] = [lg ∗ X1 , lg ∗ X2 ]
= [X1 , X2 ].

Thus, X1 , X2 ∈ L(G) implies [X1 , X2 ] ∈ L(G) and we call the set L(G) the
Lie algebra of G.
Lemma 2.3. There is an isomorphism of the vector space L(G) of left
invariant vector fields on the Lie group G with the tangent space Te G at the
identity element e ∈ G
≃
Te G Ð
→ L(G), A ↦ LA ,

where we define the left-invariant vector field on G by

LA
g ∶= lg ∗ A,

for all g ∈ G and A ∈ Te G.

Remark 2.14. L(G) is a vector space with dimension dim(Te G) = dim(G).
Remark 2.15. We can identify the Lie algebra of G with Te G. The Lie
bracket on Te G is constructed from the commutator on the left invariant
vector fields L(G) using the isomorphism described in lemma 2.3. Explicitly,
the Lie bracket of two elements A, B ∈ Te G is defined as

[A, B] ∶= [LA , LB ]e .
2.3 Principal Fiber Bundles 25

A key result in the theory of Lie groups is that every left invariant vector field
on a Lie group is complete. We use this result to introduce the exponential
map from the Lie algebra L(G) to the Lie group G. It allows to reconstruct
the group structure locally.
Definition 2.12. The exponential map exp ∶ Te G → G is defined for A ∈ Te G
by
exp A ∶= exp tA∣ ,
t=1
where t → exp tA, for all t ∈ R, is the unique integral curve of the left invariant
vector field LA , passing at t = 0 through e ∈ G.
Remark 2.16. Recall from remark 2.11 that for each g ∈ G we have the
adjoint map Ad ∶ G → G, preserving the identity. Thus, we get a linear
representation of the group G on its Lie algebra Te G, known as the adjoint
representation adg = Adg ∗ ∶ Te G → Te G, for all g ∈ G. The representation is
defined on all elements X ∈ Te G by
d
adg (X) = g exp(tX)g −1 ∣ .
dt t=0

Transformation Groups

We now turn to Lie groups that are of special interest for our discussion,
namely groups that act on a space via automorphisms. We first recall the
definition of a right group action on a manifold and then introduce the
notion a fundamental vector fields on a manifold induced by the Lie algebra
L(G).
Definition 2.13. A right action of a Lie group G on a differentiable manifold
P is a homomorphism g ↦ Rg from G into the group of diffeomorphisms
Diff(P ) with the property that the map Γ ∶ P × G → P , defined by

Γ ∶ P × G → P, (p, g) ↦ Rg (p) = p.g,

is smooth.
Definition 2.14. Let G be a Lie group together with a right action Γ ∶
P × G → P on a manifold P . This action induces a map L(G) → X(P ),
assigning to every A ∈ L(G) the fundamental vector field X̃ A on P defined
by
d
X̃pA ∶= p. exp tA∣ .
dt t=0
26 2 Principal Bundles and Gauge Theory

Remark 2.17. Alternatively, we can define the fundamental vector field

induced by A ∈ Te G as
X̃pA ∶= Pp ∗ A,
where Pp (g) ∶= Γ(p, g) = p.g.

The following theorem guarantees that the map of definition 2.14 is a

morphism of Lie algebras. In other words, the vector fields on the manifold
P represent the Lie algebra of G homomorphically.
Theorem 2.1. Let P be a manifold on which a Lie group G has a right
action. Then the map A ↦ X̃ A , which associates to each A ∈ Te G the
fundamental vector field X̃ A ∈ X(P ), is a homomorphism of L(G) ≃ Te G
into the Lie algebra of all vector field on P , i.e.

[X̃ A , X̃ B ] = X̃ [A,B] ,

for all A, B ∈ Te G.

The Maurer-Cartan Form

Every Lie group carries a canonical 1-form, the Maurer-Cartan form, defined
globally on the Lie group. The Maurer-Cartan form defines a linear map of
the tangent space at each element of the Lie group into its Lie algebra and
thus entails infinitesimal information about the group structure.
Definition 2.15. The Maurer-Cartan form is the L(G)-valued 1-form θ on
G defined by
θg = lg−1 ∗ ∶ Tg G → Te G,
for all g ∈ G.
Remark 2.18. In other words, θ associates with any v ∈ Tg G the left
invariant vector field on G whose value at g ∈ G is precisely the given tangent
vector v.
Remark 2.19. The Maurer-Cartan form θ is left invariant, i.e.

lg∗ (θh ) = θg−1 h

for all g, h ∈ G. Moreover, it satisfies the Maurer-Cartan structure equation

1
dθ = − [θ ∧ θ],
2
2.4 Connections on Principal Bundles 27

where [− ∧ −] is the bilinear product obtained by composing the wedge

product of the two L(G)-valued 1-forms with the bracket of the Lie algebra
L(G).
More generally, for any smooth manifold M we can define a bilinear product
[− ∧ −] on the space Ω● (M, g) of Lie algebra-valued differential forms by

[α ⊗ x ∧ β ⊗ y] ∶= α ∧ β ⊗ [x, y],

for x, y ∈ g and α, β ∈ Ω● (M, R).

2.4 Connections on Principal Bundles

In general, a connection is a mathematical tool that enables to define the
concept of parallel translation on a bundle. Thus, a connection should
provide a natural way to compare and connect points in adjacent fibers.
Here, we give three equivalent characterizations, namely connections as
horizontal distributions, as Lie algebra-valued 1-forms on the total space
and as their local representatives.

2.4.1 Connections as Horizontal Distributions

Motivated by the idea of parallel translation, we note that a connection
should provide a consistent way of moving from fiber to fiber through the
bundle. This suggests that we should look for vector fields whose flow
lines point from one fiber to another. This idea will be made precise by
introducing connections as horizontal distributions. In the following, let
(P, M, π, G) be a principal G-bundle and let x ∈ M and p ∈ π −1 ({x}).

Definition 2.16. Let Tp P denote the tangent space at the point p ∈ P . The
vertical subspace Vp P of Tp P is defined by

Vp P ∶= {v ∈ Tp P ∣ π∗ v = 0}.

Remark 2.20. Recall that for each A ∈ g we can assign the fundamental
vector field X̃ A on P that represents the Lie algebra g homomorphically. A
28 2 Principal Bundles and Gauge Theory

vector X̃pA is tangent to the fiber and thus belongs to the vertical subspace
Vp P . Indeed, by definition 2.14 we have

d
π∗ X̃pA = π(p. exp tA)∣
dt t=0
d
= π(p)∣
dt t=0
= 0.

We are interested in constructing vectors that point away from the fibers
rather than along them. In other words, we are looking for vectors that live
in a space complementary to the vertical subspace Vp P . However, per se
there is no natural choice for such complement spaces. This is exactly what
a connection provides, a consistent way of assigning a horizontal subspace at
each point of the total space.
Definition 2.17. A connection on a principal bundle (P, M, π, G) is a
smooth assignment of a subspace Hp P of Tp P to each point p ∈ P such that
– T p P ≃ V p P ⊕ Hp P ;
– Rg ∗ Hp P = Hp.g P ,
for all p ∈ P and g ∈ G. In other words, a connection is a G-invariant
distribution H ⊂ T P complementary to the vertical distribution V .

2.4.2 Connections as Lie Algebra-valued 1-Forms

There is an alternative to characterize connections, namely by means of
certain 1-forms on P with values in the Lie algebra g.
Definition 2.18. A connection 1-form on a principal bundle (P, M, π, G)
is a 1-form ω ∈ Ω1 (P ; g) satisfying the following properties:
– ωp (X̃ A ) = A, for all p ∈ P and A ∈ g;
– Rg∗ ω = adg−1 ○ ω, for all g ∈ G.
Remark 2.21. We denote with BunG ω (M ) the category whose objects
are principal G-bundles over M equipped with a connection and whose
morphisms are principal bundle maps.
Remark 2.22. Let ω ∈ Ω1 (P ; g) be a 1-form as given in definition 2.18.
Then the distribution H = ker(ω) defines a connection on P .
2.4 Connections on Principal Bundles 29

2.4.3 Local Representatives of the Connection 1-Form

Recall that we have canonical local sections sα ∶ Uα → π −1 (Uα ) associated
to the local trivialization of the principal bundle. Thus, we can pullback
the connection 1-form ω to obtain a local representative of the connection
1-form on the base manifold M

Aα ∶= s∗α ω ∈ Ω1 (Uα ; g).

Local representatives are often referred to as gauge fields in the physics

literature. Since ω is defined globally, we have ωα = ωβ on π −1 (Uα ∩ Uβ ).
The following lemma shows how the corresponding local representatives are
related on Uα ∩ Uβ .
Lemma 2.4. Let ω ∈ Ω1 (P ; g) be a connection on a principal bundle
(P, M, π, G) and let sα ∶ Uα → π −1 (Uα ) and sβ ∶ Uβ → π −1 (Uβ ) be two
local sections on Uα , Uβ ⊂ M . Moreover, let φαβ ∶ Uα ∩ Uβ → G be the unique
transition function such that sβ (x) = sα (x).φαβ (x), for all x ∈ Uα ∩ Uβ .
Denote with Aα and Aβ the local representatives of ω with respect to sα and
sβ . Then for Uα ∩ Uβ ≠ 0 the two local representatives are related by

Aβ = adφαβ −1 ○ Aα + φ∗αβ θ,

where θ is the Maurer-Cartan form on G.

Proof. Let x ∈ Uα ∩ Uβ and X ∈ Tx M . First, use the relation sβ (x) =

sα (x).φαβ (x) to factorize the local section sβ as
sα ×φαβ Γ
Uα ∩ Uβ ÐÐÐÐ→ π −1 (Uαβ ) × G Ð
→ π −1 (Uαβ )
x ↦ (sα (x), φαβ (x)) ↦ sα (x).φαβ ,

where Γ denotes the right action of G on P . We get

(Aβ )x (X) = ((sα × φαβ )∗ Γ∗ ω)x (X)

= (Γ∗ ω)(sα (x),φαβ (x)) (sα∗ X, φαβ ∗ X).
30 2 Principal Bundles and Gauge Theory

We now make use of the following isomorphism, given here for a general
product manifold U × V

Tr U ⊕ Ts V ≃ T(r,s) (U × V )
(ū, v̄) ↦ is∗ ū + jr ∗ v̄,

with the injections is ∶ U → U × V , is (x) ∶= (x, s), and jr ∶ V → U × V ,

jr (y) ∶= (r, y), respectively, defined for each r ∈ U and s ∈ V . Define
functions ig ∶ P → P × G, ig (p) ∶= (p, g), and jp ∶ G → P × G, jp (g) ∶= (p, g),
for all p ∈ P and g ∈ G. We can re-express the local representative as

(Aβ )x (X) = ωsα (x).φαβ (x) ((Γ ○ iφαβ (x) )∗ sα∗ X + (Γ ○ jsα (x) )∗ φαβ ∗ X)
= (Rφ∗ αβ (x) ω)sα (x) (sα∗ X) + ωsα (x).φαβ (x) ((Γ ○ jsα (x) )∗ φαβ ∗ X)

where Γ ○ ig = Rg denotes the right action of the element g ∈ G on P . Now,

since ω is a connection 1-form we have

(Rφ∗ αβ (x) ω)sα (x) (sα∗ X) = adφαβ (x)−1 ○ ωsα (x) (sα∗ X)
= adφαβ (x)−1 ○ (s∗α ω)x (X)
= adφαβ (x)−1 ○ (Aα )x (X).

We know that φαβ ∗ X = LYφαβ (x) = lφαβ (x) ∗ Y is a left invariant vector field
for some Y ∈ Te G. More precisely, according to the definition of the Maurer-
Cartan form we have Y = θφαβ (x) (φαβ ∗ X). Furthermore, notice that Γ ○ jp ∶=
Pp ∶ G → P , g ↦ p.g, defines the right action of the group G for every p ∈ P
and we can use that the left invariant vector field LY on G and the induced
fundamental vector field X̃ Y on P are Pp -related for each p ∈ P , that is

Pp ∗ LYg = X̃p.g
Y
,

for all g ∈ G. Let us apply these observations to the case at hand

ωsα (x).φαβ (x) (Psα (x) ∗ φαβ ∗ X) = ωsα (x).φαβ (x) (Psα (x) ∗ LYφαβ (x) )
= ωsα (x).φαβ (x) (X̃sYα (x).φαβ (x) )
= θφαβ (x) (φαβ ∗ X)
= (φ∗αβ θ)x X
2.4 Connections on Principal Bundles 31

where we used that ω is a connection 1-form, i.e. we have ω(X̃ Y ) = Y .

Putting everything together we find

(Aβ )x (X) = adφαβ (x)−1 ○ (Aα )x (X) + (φ∗αβ θ)x X.

Remark 2.23. For G a matrix Lie group, the relation between local repre-
sentatives stated in lemma 2.4 assumes the following simpler form

Aβ = φ−1
αβ Aα φαβ + φαβ dφαβ .

Remark 2.24. Given an open cover {Uα }α∈A and a family of 1-forms Aα ∈
Ω1 (Uα ; g) satisfying the relation in lemma 2.4, one can construct a globally
defined connection 1-form ω ∈ Ω1 (P ; g). Thus, the local representatives
provide an alternative way to characterize connections on a principal bundle.

2.4.4 The Affine Space of Connections

Theorem 2.2. Every principal G-bundle admits a connection.

Since the theorem guarantees the existence of connections, we can raise the
question of how to describe the space of connections on a principal bundle.
Before giving the answer, we have to settle some terminology. First, we
define the notion of a fiber bundle associated to a given principal bundle.
Then we introduce the so-called basic forms on a principal bundle. Both
notions are used in lemma 2.6 to state the key identification between forms
on the total- and the base space that allows to characterize the space of
connections.

Associated Fiber Bundles

The associated fiber bundle construction is a method to build a wide variety

of fiber bundles that are related with a given principal bundle in a specific
way.
32 2 Principal Bundles and Gauge Theory

Definition 2.19. Let (P, M, π, G) be a principal G-bundle, F a space on

which G acts from the left via automorphisms and let ρ ∶ G → Aut(F ) be
the corresponding representation. Define the quotient

P ×(G,ρ) F ∶= (P × F )/ ∼,

where the equivalence relation ∼ is generated by the G-action

(p, f ) ∼ (p, f ).g = (p.g, ρ(g −1 ).f ).

There is a canonical projection

π̃ ∶ P ×(G,ρ) F → M ; π̃([p, f ]) ∶= π(p),

turning (P ×(G,ρ) F, M, π̃, F ) into a fiber bundle with fiber F , which is said to
be associated with the principal G-bundle (P, M, π, G) via the representation
ρ.

Remark 2.25. Let F be a space with a trivial G action, i.e. ρ(g −1 ).f = f for
all g ∈ G and f ∈ F . Then, the associated bundle is canonically isomorphic
to the trivial bundle (M × F, M, pr1 ). The isomorphism is given by [p, f ] ↦
(π(p), f ).

Example 2.2. Let F be a vector space and ρ a linear representation,

Aut(F ) ≃ GL(F ), then the associated bundle is a vector bundle. In particu-
lar, taking F to be the Lie algebra g together with the adjoint representation
ad ∶ G → GL(g) we obtain the adjoint bundle denoted by ad(P ) ∶= P ×G,ad g.
Example 2.3. Let F be a smooth manifold and let G act on F via diffeo-
morphisms, Aut(F ) = Diff(F ). In particular, taking F to be the Lie group
G together with the adjoint representation Ad ∶ G → Diff(G) we obtain the
adjoint principal bundle denoted by Ad(P ) ∶= P ×G,Ad G.

An important observation is that sections of an associated bundle can be

completely described in terms of functions on the corresponding principal
bundle.
Lemma 2.5. Let (P, M, π, G) be a principal G-bundle and E ∶= P ×(G,ρ) F
the fiber bundle associated to the principal bundle via ρ ∶ G → Aut(F ).
Sections Γ(M, E) of the associated bundle are in one-to-one correspondence
with maps
2.4 Connections on Principal Bundles 33

γ ∶ P → F,
satisfying the following equivariance property

γ(p.g) = ρ(g −1 ).γ(p).

Remark 2.26. We denote the space of equivariant smooth maps by

C ∞ (P, F )(G,ρ) .

Basic Forms

In the following, let (P, M, G, π) be a principal G-bundle, V a vector space

and ρ ∶ G → GL(V ) the corresponding linear representation of G and
E ∶= P ×(G,ρ) V the associated vector bundle. We want to generalize lemma
2.5 to vector bundle-valued forms.

Definition 2.20. A k-form ω ∈ Ωk (P ; V ) is called

– horizontal if ωp (X1 , . . . , Xk ) = 0 in case that one of the Xi ∈ Tp P is
vertical;
– equivariant if Rg∗ ω = ρ(g −1 ) ○ ω;
– basic if it is both equivariant as well as horizontal.

Remark 2.27. The space of basic k-forms on the total space P is denoted
by Ωkhor (P ; V )(G,ρ) .

The next lemma shows that there is a one-to-one correspondence between

basic k-forms on P and k-forms on M taking values in the associated bundle
E.
Lemma 2.6. The space Ωkhor (P ; V )(G,ρ) of basic k-forms on P is isomorphic
to the space Ωk (M ; E) of k-forms on M with values in E.

Proof. Given a basic k-form ξ ∈ Ωkhor (P ; V )(G,ρ) , the corresponding k-form

ηξ ∈ Ωk (M ; E) on the base manifold is defined for each x ∈ M by

ηξ x (X1 , . . . , Xk ) ∶= [p, ξp (Y1 , . . . , Yk )],

for any p ∈ π −1 ({x}) and Xj ∈ Tx M , Yj ∈ Tp P with π∗ Yj = Xj for j = 1, . . . , k.

The definition is independent of the choice of the vectors Yj ∈ Tp P , since
34 2 Principal Bundles and Gauge Theory

for any other vector Ỹj ∈ Tp P satisfying π∗ Ỹj = Uj , we have π∗ (Yj − Ỹj ) = 0.
It follows that Yj − Ỹj is vertical and as ξ is a horizontal k-form we have
ξp (Y1 , . . . , Yj − Ỹj , . . . , Yk ) = 0. Moreover, the definition is also independent
of the choice of p ∈ π −1 ({x}) since ξ is an equivariant k-form. Indeed, let
Ỹj ∈ Tp.g P such that π∗ Ỹj = Uj for all j = 1, . . . , k. Then, we have

[p.g, ξp.g (Ỹ1 , . . . , Ỹk ] = [p.g, ξp.g (Rg ∗ Y1 , . . . , Rg ∗ Yk )]

= [p.g, Rg∗ ξp (Y1 , . . . , Yk )]
= [p.g, ρ(g −1 )ξp (Y1 , . . . , Yk )]
= [p, ξp (Y1 , . . . , Yk )].

Conversely, let η ∈ Ωk (M ; E) be a k-form and define ξη ∈ Ωkhor (P ; V )(G,ρ) by

ξη p (Y1 , . . . , Yk ) ∶= ι−1
p ηx (π∗ Y1 , . . . , π∗ Yk ),

where ιp ∶ V → π̃ −1 ({x}) is the inclusion of the fiber and is defined by

ιp (f ) ∶= [p, f ] for any p ∈ π −1 ({x}). This so defined k-form ξη is basic.
Indeed, since for any v ∈ V we have ιp (v) = ιp.g (ρ(g −1 ).v) and it follows

(Rg∗ ξη )p (Y1 , . . . , Yk ) = ξη p.g (Rg ∗ Y1 , . . . , Rg ∗ Yk )

= ι−1
p.g ηx ((π ○ Rg )∗ Y1 , . . . , (π ○ Rg )∗ Yk )

= ι−1
p.g ηx (π∗ Y1 , . . . , π∗ Yk )

= ι−1
p.g ιp (ξη p (Y1 , . . . , Yk ))

= ρ(g −1 ).ξη p (Y1 , . . . , Yk )

where we used that π ○ Rg = π. This shows that ξη is equivariant. It is also

horizontal, as for any vertical vector Ỹj ∈ Tp P we have π∗ Ȳj = 0

The Space of Connections

We now use lemma 2.6 to describe the space of connections on a principal

bundle.
Corollary 2.1. Let (P, M, π, G) be a principal bundle. The space of
connections A(P ) on P is an affine space modeled over the vector space
Ω1 (M ; ad(P )), where ad(P ) = P ×G,ad g is the adjoint bundle.
2.5 Gauge Transformations 35

2.5 Gauge Transformations

Definition 2.21. Let (P, M, π, G) be a principal bundle. A gauge transfor-
mation is an automorphism of the principal bundle, i.e. a principal bundle
map ϑ making the following diagram commute
ϑ
P P
π π

Remark 2.28. The set of all gauge transformations forms a group under
composition of automorphisms. We denote the group of gauge transforma-
tions with G(P ).
Remark 2.29. The group of gauge transformations G(P ) can be identified
with the space of sections in the adjoint bundle Ad(P ) = P ×G,Ad G

G(P ) ≃ Γ(M ; Ad(P )).

Indeed, by lemma 2.5 we can identify Γ(M ; Ad(P )) with the space of equivari-
ant maps C ∞ (P ; G)(G,Ad) , that is maps γ satisfying γ(p.g) = Ad(g −1 )γ(p) =
g −1 γ(p)g. For each such γ we can define a map ϑγ ∶ P → P by

ϑγ (p) ∶= p.γ(p).

This is an element of G(P ). Indeed, it is G-equivariant

ϑγ (p.g) = p.g.γ(p.g)
= p.gg −1 γ(p)g
= ϑγ (p).g,

for all g ∈ G and satisfies π ○ ϑγ = π since π(ϑγ (p)) = π(p.γ(p)) = π(p).

Conversely, given ϑ ∈ G(P ), we define γϑ ∈ C ∞ (P ; G)(G,Ad) by γϑ (p) ∶=
φ(p)−1 φ(ϑ(p)), where φ ∶ π −1 (U ) → G is a local trivialization.
36 2 Principal Bundles and Gauge Theory

2.5.1 The Action of the Group of Gauge Transformations on the

Space of Connections
There is a natural action of the group of gauge transformations G(P ) on the
space of connections A(P ).
Lemma 2.7. Let H ⊂ T P be a connection on a principal bundle (P, M, π, G)
and let ϑ ∈ G(P ) be a gauge transformation. Then, we have that

H ϑ ∶= ϑ∗ H

is also a connection on P .
Lemma 2.8. Let ω ∈ Ω1 (P ; g) be a connection 1-form on a principal bundle
(P, M, π, G) and let ϑ ∈ G(P ) be a gauge transformation. Then, we have
that
– ϑ∗ ω is also a connection on P ;
– ϑ∗ ω = adγϑ−1 ○ ω + γϑ∗ θ,

where θ is the Maurer-Cartan form on G and γϑ ∈ C ∞ (P ; G)(G,Ad) is the

equivariant map associated to ϑ ∈ G(P ).
Remark 2.30. Given a connection 1-form ω and the gauge-transformed
∗
connection 1-form ω ϑ ∶= ϑ−1 ω, the corresponding local representatives
Aα = s∗α ω and Aϑα = s∗α ω ϑ are related via

Aα = adϑ̄−1
α
○ Aϑα + ϑ̄∗α θ,

where sα ∶ Uα → π −1 (Uα ) is a local section and ϑ̄α is a function ϑ̄α ∶ Uα → G

defined via
ϑ̄α (π(p)) ∶= φα (ϑ(p))φα (p)−1 ,
where φα ∶ π −1 (Uα ) → G is the local trivialization map canonically associated
to the local section. Also notice that if G is a matrix Lie group, we can write
ϑ
Aα = ϑ̄−1
α Aα ϑ̄α + ϑ̄α dϑ̄α .

2.6 The Curvature of a Connection

Throughout, let (P, M, π, G) be a principal bundle equipped with a connec-
tion 1-form ω and denote with ad(P ) = P ×G,ad G the adjoint bundle over M .
2.6 The Curvature of a Connection 37

Given a connection, we show how to define a covariant exterior derivative on

the space of differential forms on M with values in the adjoint bundle ad(P ).
The obstruction of the covariant exterior derivative to define a complex
is then measured by the curvature of the connection. We also show the
transformation properties of the connection under gauge transformations.

2.6.1 The Covariant Exterior Derivative

Our goal is to define the covariant exterior derivative

d¯ω ∶ Ωk (M ; ad(P )) → Ωk+1 (M ; ad(P )),

induced by the connection 1-form ω. By lemma 2.6 we can identify the

forms on M with values in the adjoint bundle with horizontal, equivariant
g-valued forms on P . Thus, we are searching for a map that assigns to a
basic g-valued form on P another basic form.
Definition 2.22. Let V be a vector space and denote with Ω● (P ; V ) the
space of differential forms on P with values in V . This space is equipped
with a linear map d ∶ Ωk (P ; V ) → Ωk+1 (P ; V ), ω ↦ dω, defined for k > 0 by
k
dω(X0 , . . . , Xk ) = ∑i=0 (−1)i Xi ω(X0 , . . . , X/ i , . . . , Xk )+
i+j
∑i<j (−1) ω([Xi , Xj ], X0 , . . . , X/ i , . . . , X/j , . . . , Xk ),

satisfying d ○ d = 0 and thus, turning (Ω● (P ; V ), d) into a complex.

Remark 2.31. Let ξ ∈ Ωkhor (P ; g)(G,ad) be a basic form. Then we have that
dξ is equivariant. Indeed, we have

Rg∗ (dξ) = d(Rg∗ ξ)

= d(adg−1 ○ ξ)
= adg−1 ○ dξ.

However, dξ need not be horizontal. Thus, the differential d is not the map
we are looking for.

The data of a connection allows us to define a projection on horizontal forms,

i.e. on forms that vanish on vertical vectors. This horizontal projection is
then used to define the exterior derivative dω ξ as the horizontal part of dξ.
38 2 Principal Bundles and Gauge Theory

Definition 2.23. Let H ⊂ T P be a connection in a principal bundle

(P, M, π, G). Define the horizontal projection

h ∶ TP → TP

as a collection of linear maps hp ∶ Tp P → Tp P , satisfying

⎧
⎪v,
⎪ if v ∈ Hp P,
hp (v) ∶= ⎨
⎪
⎪0, if v ∈ Vp P,
⎩
for all p ∈ P .
Remark 2.32. The horizontal projection induces a dual map h∗ ∶ T ∗ P →
T ∗ P , defined for every p ∈ P by

(h∗p (f ))(v) = f (hp (v))

for f ∈ Tp∗ P and v ∈ Tp P . More generally, we can extend h to a map

k k
hk ∶ ⋀(T P ) → ⋀(T P )

by setting hk (v1 ∧⋅ ⋅ ⋅∧vk ) = h(v1 )∧⋅ ⋅ ⋅∧h(vk ). For simplicity we write hk = h.

Then, for ω ∈ Ωk (P ) we have h∗ ω(v1 , . . . , vk ) = ω(h(v1 ), . . . , h(vk )).
Definition 2.24. Let (P, M, π, G) be a principal bundle equipped with a
connection 1-form ω. The covariant exterior derivative dω is defined as

dω ∶ Ωk (P ; g) → Ωk+1 (P ; g), ξ ↦ dξ ○ h.

Lemma 2.9. The exterior covariant derivative dω maps horizontal, equiv-

ariant g-valued k-forms into horizontal, equivariant g-valued (k + 1)-forms
on P
dω ∶ Ωkhor (P ; g)(G,ad) → Ωk+1
hor (P ; g)
(G,ad)
.

The following lemma gives an explicit formula for the covariant exterior
derivative.
Lemma 2.10. For every k-form ξ ∈ Ωkhor (P ; g)(G,ad) we have

dω ξ = dξ + [ω ∧ ξ].
2.6 The Curvature of a Connection 39

Remark 2.33. By lemma 2.6, the exterior covariant derivative dω induces

an exterior covariant derivative d¯ω on the space of differential forms on M
with values in the adjoint bundle, we thus have a commutative diagram
dω
Ωkhor (P ; g)(G,ad) Ωk+1
hor (P ; g)
(G,ad)

≃ ≃
d¯ω
Ωk (M ; ad(P )) Ωk+1 (M ; ad(P ))

2.6.2 The Curvature 2-Form

We have characterized the exterior derivative d¯ω on Ω● (M ; ad(P )), however,
this is not a differential. Indeed, let ξ ∈ Ωkhor (P ; g)(G,ad) represent a k-form
on M with values in the adjoint bundle. Then, we have

dω dω ξ = dω (dξ + [ω ∧ ξ]
= d2 ξ + d[ω ∧ ξ] + [ω ∧ dξ] + [ω ∧ [ω ∧ ξ]]
1
= [dω + [ω ∧ ω] ∧ ξ].
2
This finding motivates the following definition.

Definition 2.25. Let ω ∈ Ω1 (P ; g) be a connection 1-form on a principal

bundle (P, M, π, G). The curvature 2-from Ω is defined by

Ω ∶= dω ω ∈ Ω2 (P ; g).

Remark 2.34. Since the connection 1-form ω is equivariant, we have by

lemma 2.9 that Ω is a horizontal, equivariant 2-form. Hence, it can be
identified with a 2-form F ∈ Ω2 (M ; ad(P )) defined globally on the base
manifold M with values in the adjoint bundle.
Lemma 2.11. The curvature 2-form Ω satisfies the following structure
equation
1
Ω = dω + [ω ∧ ω].
2
Definition 2.26. A connection 1-form on a principal bundle is called flat if
its curvature is identically zero. A flat principal bundle is a principal bundle
equipped with a flat connection.
40 2 Principal Bundles and Gauge Theory

2.6.3 Local Representatives of the Curvature 2-Form and Gauge

Transformations
We can pullback the curvature 2-form Ω ∈ Ω2 (P ; g) along local sections
sα ∶ Uα → π −1 (Uα ) to obtain local representatives of the curvature 2-form

Fα ∶= s∗α Ω ∈ Ω2 (Uα ; g).

Notice that in the physics literature the local representatives are often
referred to as gauge field strength.
Lemma 2.12. Let Ω ∈ Ω2 (P ; g) be the curvature of a connection 1-form ω
on a principal bundle (P, M, π, G) and let sα ∶ Uα → π −1 (Uα ) and sβ ∶ Uβ →
π −1 (Uβ ) be two local sections on Uα , Uβ ⊂ M . Moreover, let φαβ ∶ Uα ∩ Uβ →
G be the unique transition function such that sβ (x) = sα (x).φαβ (x), for all
x ∈ Uα ∩ Uβ . Denote with Fα and Fβ the local representatives of Ω with
respect to sα and sβ . Then for Uα ∩ Uβ ≠ 0 the two local representatives are
related by
Fβ = adφαβ −1 ○ Fα .
Remark 2.35. Given a curvature 2-form Ω and a gauge transformation
ϑ ∈ G(P ), the gauge-transformed curvature 2-form is given by
∗
Ωϑ = ϑ−1 Ω.

Furthermore, one can show that the local representatives Fα = s∗α Ω and
Fαϑ = s∗α Ωϑ transform as
Fαϑ = adϑ̄ ○ Fα ,
where ϑ̄ ∶ Uα → G.