Notes on classical gauge theories
Jord Boeijink
June 21, 2011
1
Disclaimer
These notes are based on the book by Bleecker [1]. There is nothing in these notes that is not already available
in Chapter 1-3 in this book. Initially, I wrote this text for my own convenience as a preparation for two tutorial
lectures on classical gauge theories at the CIMPA-school in Bangkok 2011. On request of some students I
decided to make these notes available on the school’s website.
These notes are far from complete: in the first place one will only find sketches of proof indicating the main
steps of the proof. To find a more detailed proof the reader will be referred to [1]. In the second place, I did not
have the time to read through the text thoroughly so there are probably quite some typos. Lastly, during the
tutorial I also mentioned some facts about Lie groups and their Lie algebras. I have omitted this part in these
notes since a nice overview can also be found in [1], Chapter 0.3. Actually, the entire Chapter 0 in [1] is a nice
introduction to differential geometry so I recommend reading this chapter if you are not familiar with notions
as tangent bundle, push-forward of tangent vectors and pull-backs of one-forms.
In Sections 2 and 3 I will introduce principal fibre bundles and connections thereon. Together these two
notions form the natural setting for gauge theories. Next, I will introduce the particle fields as sections of
associated bundles of the principal fibre bundle in Section 4. In the final section 5 I will tell how the theory is
related to physics. The dynamics, however, will not be discussed in these notes. Enjoy!
2
Principal fibre bundles
Classical gauge theories are mathematically described using principal fibre bundles. In these notes we will
explain how principal fibre bundles form a natural language for gauge theories. The relation with physics might
not seem obvious at the beginning, but at the end we will try to clarify this relation. In this section we will
give the definition of a principal fibre bundle.
Definition 2.1 (see Bleecker [1], Definition 1.1.1). A principal fibre bundle π : P → M with structure group
G consists of smooth manifolds P , M and a Lie group G together with a smooth surjective projection map
π : P → M , where the Lie group G has a free smooth right action on P and π −1 (π(p)) = {pg : g ∈ G}. If
x ∈ M , then π −1 (x) is called the fibre above x. Furthermore, we require that for each x ∈ M there exists an open
set U ∋ x and a diffeomorphism TU : π −1 (U ) → U × G of the form φU (p) = (π(p), sU (p)), where sU : P → G
has the property sU (pg) = sU (p)g for all g ∈ G, p ∈ π −1 (U ). The map TU is called a local trivialisation (or, in
physics, a choice of gauge).
Let TU : π −1 (U ) → U × G and TV : π −1 (V ) → V × G be two local trivialisations of a principal fibre bundle
π : P → M . The transition function from U to V is the map gU V : U ∩ V → G defined, for x ∈ π(p) ∈ U ∩ V ,
by gU V (x) = sV (p)sU (p)−1 . This is independent of the choice of p ∈ π −1 (x).
Proposition 2.2 ([1], Theorem 1.1.5). A principal bundle π : P → M with structure group G is a trivial bundle
if and only if it admits a global section M → P .
Proof. Let σ : U → P be a given local section. Then define the smooth map TU : π −1 (U ) → U × G by
TU (σ(x)g) = (π(p), g). We verify that indeed TU (σ(x)gh) = (π(p), gh) so that TU is a local trivialisation.
Conversely, let TU : π −1 (U ) → U × G be a given local trivialisation, then define a section σ : U → π −1 (U ) by
σ(x) = φ−1
u (x, e). Thus, we see that local sections correspond to local trivialisations. In particular, this implies
that a global section exists if and only if the bundle is trivial.
1
2
3 CONNECTIONS AND GAUGE POTENTIALS
Example 2.3 (Hopf fibration and square root).
1. Identify R3 with C × R by (x1 , x2 , x3 ) ↔ (z = x1 +
4
2
ix2 , x = x3 ) and R with C by identifying (x1 , x2 , x3 , x4 ) ↔ (z1 = x1 + ix2 , z2 = x3 + ix4 ). Then the unit
sphere S 2 in R3 is identified with {(z, x) | |z|2 +x2 = 1} and S 3 is identified with {(z1 , z2 ) | |z1 |2 +|z2 |2 = 1}.
The Hopf fibration is defined by
p(z1 , z2 ) = (2z1 z2∗ , |z1 |2 |z2 |2 ).
Then p maps S 3 onto S 2 as can easily be checked: 4|z1 |2 |z2 |2 + (|z1 |2 − |z2 |2 )2 = (|z1 |2 + |z2 |2 )2 = 1. It
can be shown that p maps elements of S 3 to the same points in S 2 if and only if these points are the
same up to a factor λ ∈ U (1). The bundle is not globally trivial, but for the Hopf fibration it is enough
to remove a single point m from S 2 , thus one can take U = S 2 − {m} as trivialising neighbourhoods, and
any point in S 2 has a neighborhood of this form. Hence, p : S 3 → S 2 is a principal U (1)-bunle.
2. The map z 7→ z 2 in S 1 induces a principal bundle π : S 1 → S 1 with structure group Z2 . It is locally trivial
since, locally, on the circle there always exists a smooth square root function. Since the total space is S 1
and not S 1 × Z2 this bundle is not trivial. Hence, there does not exist a continuous square root function
on S 1 .
3
Connections and gauge potentials
For fibre bundles in general, it is clear what is meant by moving along the fibre: i.e. a path γ : [0, 1] → P
stays in the fibre if and only if π(γ(t)) = const, or equivalenty π∗ ( dγ
dt (t)) = 0 for all t ∈ [0, 1]. Without further
structure it is not possible to say what it means to move in the horizonal direction, i.e. ”in the direction of the
base manifold”. The extra structure we need in order to speak of horizontal movement is a connection.
Definition 3.1 ([1], Definition 1.2.1). A connection assigns to each p ∈ P a subspace Hp ⊂ Tp P such that
• Tp P = Vp ⊕ Hp , where Vp = {X ∈ Tp P | π∗ (X) = 0}.1
• Rg∗ Hp = Hp
• For each p ∈ P , there exist a neighbourbood U and vector fields X1 , . . . Xn on U such that Hp is spanned
by X1 (p), . . . Xn (p) for all p ∈ U .
We will also use the following equivalent definition:
Definition 3.2 ([1], Definition 1.2.2). A connection is a g-valued 1-form ω on P satisyfing the following
conditions:
1. for the fundamental vector fields A∗ (p) :=
d
dt
(p exp(tA))|t=0 ,
ωp (A∗ ) = A,
for all p ∈ P ;
2. for any g ∈ G
Rg∗ ω = adg−1 ω,
where adg−1 ∈ Aut(g) is the adjoint
action of g on g. To be more precise, we require (Rg∗ ω)(p) = adg−1 ωp ,
i.e. ωpg (Rg∗ Xp ) = adg−1 ωp (Xp , where ad : G 7→ GL(g) denotes the adjoint action of G on g.
The equivalence of Definition 3.1 and 3.2 can be seen as follows: given a connection in the form of Definition
3.1, one defines a g-valued one-form ω by ω(A∗ ) = A and ωp (Xp ) = 0 for all Xp ∈ Hp . Conversely, given a
connection 1-form ω as in Definition 3.2 define Hp = {X ∈ Tp P | ωp (Xp ) = 0}. Since the action of G is free the
map A 7→ A∗p is injective,2 so one obtains a vector space isomorphism g ∼
= Vp . From part 1 of Definition 3.2 it
follows that Hp ⊕ Vp = Tp P .
1 Note
that Vp is dim G = dim g-dimensional.
d
d
d
is not so difficult to see: let A be such that A∗p = 0, then dt
p exp(tA) = ds
p exp((s+t)A)|s=0 = ds
p exp(sA) exp(tA)|s=0 =
d
∗
Rexp(tA)∗ ds p exp(sA)|s=0 = Rexp(tA)∗ Ap = 0. Since the action of G on P is free, it follows that exp(tA) = eG for all t. Hence,
A = 0.
2 This
3
Exercise 3.3. Show that [A∗ , B ∗ ]p = [A, B]∗p for all A, B ∈ g, where [·, ·] on vector fields is the usual Lie-bracket,
d
d
i.e. [X, Y ] = dt
φ−1
t∗ Yφt (p) t=0 , where dt φt = X in a neighbourhood of p.
Proposition 3.4. The vector fields on M can be identified with the G-invariant horizontal vector fields on P .
Proof. The maps π∗ establish an isomorphism between Hp and Tπ(p) M , so it indeeds identifies a subspace of
Tp P with the tangent space of M . This allows us to lift vector fields on M to unique horizontal vector fields
e = X,
e since Rg∗ Hp = Hpg . Conversely, every G-invariant
on P . This horizontal lift is G-invariant: i.e. Rg∗ X
e
e
e is G-invariant and π is
vector field X is a lift of a vector field X = π∗ X on M , which is well-defined since X
3
surjective.
e of X one has [A∗ , X]
e = 0 for all A ∈ g.
Lemma 3.5. For a horizontal lift X
e = d φ−1
eφ (p)
e
Proof. Use the formula [A∗ , X]
X
, where φt (p) = p exp(tA), and the G-invariance of X
t∗
t
dt
e = X)
e for all t).
(G-invariance implies φt∗ X
t=0
Remark 3.6. By choosing a local trivialisation (= a local section σ : U → P ), one can pull back ω to a 1-form
ωU = σ ∗ ω on U ⊂ M . In the particular case that G is a matrix Lie group, and two local sections σ : U → P
and σ : V → P are given such that U ∩ V 6= ∅, then the transformation rule between ωU and ωV is given by
(see Bleecker [1], Definition 1.2.3 and Theorem 1.2.5)
−1
−1
ωV = g U
V dgU V + gU V ωU gU V .
(1)
Conversely, a collection of local g-valued 1-forms {ωUi }, where {Ui } is a cover of M such that P is locally trivial
on the Ui , satisfying the transformation rule (1) glue together into a global g-valued 1-form ω on P satisfying
the conditions of Definition 3.2. The pull-backs ωU on M are known as gauge potentials in physics. The
transformation property (1) is what physicists may recognize as the transformation rule of a gauge potential in
gauge theories. A connection on P is therefore also known as a gauge potential. The choice of the local section
is called a choice of gauge.
Definition 3.7. A gauge theory on M with gauge group G consists of a principal G-bundle π : P → M endowed
with a connection ω.
4
Associated bundles and particles
Now we have defined the setting of gauge theories, one can introduce particle fields as sections of associated
vector bundles of the bundle P . These associated bundles are induced by representations of the group G. It
turns out that sections of associated bundles with fibres isomorphic to the vector space V can also be considered
as G-equivariant function on P with values in V . The connection ω on P carries over to a covariant derivative
∇ on the associated vector bundle E. However, using the identification of sections of E with V -valued Gequivariant functions it is possible to work on the principal bundle P only. I prefer to do the latter, but since
the language of associated bundles is closer to the formulation of gauge theories in physics, I will sometimes
show how definitions on P can be reinterpreted using associated bundles.
Definition 4.1. Let P be a principal G-bundle and F a smooth manifold on which G acts smoothly from the
left (no more conditions on the action of G are needed). Take the direct product P ×F and identify two elements
(p1 , f1 ), (p2 , f2 ) if and only if (p1 g, g −1 f1 ) = (p2 , f2 ) for some g ∈ G. This action of G is free and transitive
since the action of G on P is, therefore the quotient P ×G F is a manifold and even a fibre bundle with fibre
F under the projection map π(p, f ) = π(p). The bundle P ×G F is called an associated bundle of P . One can
show that local sections of P induce local trivialisations of P ×G F . If F = V is a finite-dimensional real or
complex vector space, then the fibres of P ×G F inherit the same structure if we define
[(p, v1 )] + [(p, v2 )] = [p, v1 + v2 ],
λ[(p, v)] = [p, λv],
(p ∈ P, v1 , v2 , v ∈ V, λ ∈ R, C).
One can introduce particle fields to a gauge theory by introducing an associated vector bundle of P . Sections
of this associated bundles are interpreted as the particle fields.
3 For any X ∈ Γ(T M ) the map p 7→ h X
p π(p) is a smooth vector field on P . Smoothness follows easily from the third condition
in Definition 3.1.
4
4
ASSOCIATED BUNDLES AND PARTICLES
Example 4.2. Let G = SU (2), and consider the fundamental representation ρ : SU (2) → GL2 (C). Then
Γ(P ×SU (2) C2 ) form a C ∞ (M )-module of particle fields that are C2 -valued locally.
If E is an associated vector bundle of P with fibre V , the C ∞ (M )-module of sections Γ∞ (E) can be identified
with V -valued G-equivariant functions on P .
Lemma 4.3. In general, if P is a G-principal fibre bundle and E := P ×G V is an associated vector bundle,
there is a natural isomorphism of C ∞ (M ) ∼
= C ∞ (P )G -modules between Γ(M, E) and C ∞ (P, V )G , where the last
space is the space of all G-equivariant V -valued functions on P . A function f : P → V is called G-equivariant
if for all g ∈ G and p ∈ P : f (pg) = g −1 · f (p).4
Proof. Let f be such an equivariant function. Then s ∈ Γ(M, E) is defined by s(x) := [p, f (p)], where π(p) = x.
Since f is G-equivariant this is well-defined: [pg, f (pg)] = [pg, g −1 · f (p)] = [p, f (p)] so that s(x) is independent
of the choice of p in the fibre of x. Conversely, given s ∈ Γ(M, E), a G-equivariant function f : P → V is defined
by s(x) = [p, f (p)].5
This identification is very useful: to study (sections of) associated bundles it is enought to consider equivariant functions on P . From now on, we will implictly make this identification. Using this identification it is
easy to carry over structures on P to similar structures on the associated bundles. One of these structures is
the connection. It induces a covariant derivative on associated vector bundles.
Definition 4.4 ([1], Definition 3.1.2). Let ρ : G → GL(V ) be a finite-dimensional representation of G. Given
k
a connection ω one can introduce the space Ω (P, V ) of differential forms on P such that φ(X1 , . . . Xk ) = 0 if
∗
one of the Xi is a vertical vector field, and Rg φ = g −1 φ.
The ’vertical condition’ implies that we are actually considering forms on the manifold M and the equivariance condition means that we are considering sections of the associated bundle Γ(P ×G V ).
1
Proposition 4.5. Two connections ω, ω ′ differ by an element in Ω (P, g).
Proof. It is easily checked that (ω − ω ′ )(X) = 0 on vertical fields X = A∗ by the first condition in Definition
3.2. The property Rg∗ (ω − ω ′ ) = ad g−1 (ω − ω ′ ) follows from the second condition in the same definition.
Proposition 4.5 has a familiar corollary.
Corollary 4.6. For a given ω the space Λ1 (P, g) is in one-to-one corresponde with the space of all connection
1-forms C on P through the assignment τ 7→ τ + ω.
A connection on a principal bundle P naturally induces a connection on any associated bundle E = P ×G V .
Definition 4.7 ([1], Definition 3.1.3). For a connection ω the covariant derivative on P ×G V is defined by
Dω φ = (dφ)H ,
•
•+1
This is indeed a map Ω (P, V ) → Ω
(P, V ), since
Rg∗ Dω φ = Rg∗ (dφ)H = (Rg∗ dφ)H = (dRg∗ φ)H = (d(adg−1 )φ)H = adg−1 (dφ)H .
e
e
If g acts on V , then one has an action of Ω(P,
g) on Ω(P,
V ) given by
X
.
(−1)σ φ(Xσ(1) , · · · Xσ(k) ) · τ (Xσ(k+1) , · · · Xσ(k+) ),
(φ ∧ τ )(X1 , · · · Xk+l ) :=
σ∈Sk+l
e
e
where φ ∈ Ω(P,
g) and τ ∈ Ω(P,
V ).
4 The isomorphism C ∞ (M ) ∼ C ∞ (P )G is just a special case of this lemma, where G acts trivially on C. Note that f ∈ C ∞ (M )
=
can be identified with a G-invariant function f˜ on P through f˜ = f ◦ π.
5 If one wants to check that the constructed f or s are smooth one needs the fact the action of G on P × V is proper and free.
This implies that E = P ×G V has a natural smooth structure with respect to the quotient topology (note that in the text we
already used the fact that E is smooth). Moreover, with this smooth structure the quotient map P × V → E is smooth and P × V
is a principal fibre bundle over E. Using the fact that locally the quotient map P × V → E has a smooth inverse (namely, a smooth
local section from E to P × V ) one can show that the defined s or f in the proof of the Lemma are smooth if (and only if) the
other is smooth.
5
This way the set Ω(P, g) = Ω(P ) ⊗ g is a differential
graded Lie-algebra, where g acts on itself in the adjoint
.
representation. We will denote [φ1 , φ2 ] := φ1 ∧ φ2 .
With a differential graded Lie-algebra we mean that the bracket [·, ·] satisfies
(1)
(2)
[φ1 , φ2 ] = −(−1)kl [φ2 , φ1 ],
(−1)mk [[φ1 , φ2 ], φ3 ] + (−1)lk [[φ2 , φ3 ], φ1 ] + (−1)lm [[φ3 , φ1 ], φ2 ] = 0,
(3)
d[φ1 , φ2 ] = [dφ1 , φ2 ] + (−1)k [φ1 , dφ2 ].
.
k
Theorem 4.8 ([1], Theorem 3.1.5). For τ ∈ Ω (P, V ) one has ∇ω τ = dτ + ω ∧ τ .
Proof. This can be proved point-wise, so if vp ∈ Tp P is horizontal, we can assume that vp = Xp , where X is
a G-invariant horizontal vector field. Similarly, if vp ∈ Tp P is vertical, we can assume that vp = A∗p for some
A ∈ g.
.
If all the fields X1 , . . . Xk are horizontal, then (ω ∧ τ )(X1 , . . . Xk ) = 0 and XiH = Xi so that both sides
coincide.
If at least two vector fields of the X1 , . . . Xk+1 are vertical at the point p and are extended to fundamental
.
vector fields, and the other vectors are extended to G-invariant horizontal fields, then ∇ω τ = 0 = ω ∧ τ , so we
must show that dτ = 0. Well,
dτ (X1 , . . . Xk+1 ) =
X
(−1)i+1 Xi τ (X1 , . . . , X̂i , . . . Xk+1 ) + (−1)i+j τ ([Xi , Xj ], X2 , . . . , X̂i , . . . , X̂j , . . . , Xk+1 ),
k+1
X
i=1
i<j
which is zero when at least two of the Xi are vertical (because [A∗ , B ∗ ] = [A, B]∗ ).
If precisely one of the Xi is vertical, say X1 , then [X1 , Xi ] = 0 for all i and one needs to show that
X1 (τ (X2 , . . . , Xk )) + ω(X1 ) · τ (X2 , . . . Xk ) = 0,
which follows from (gt = exp(tA))
X1 (τ (X2 , . . . , Xk ))
=
=
d
d −1
[τ (Rgt∗ X2 , . . . , Rgt∗ Xk+1 )] =
gt · τ (X2 , . . . , Xk )
dt
dt
−A · τ (X2 , . . . , Xk+1 ) = −ω(X1 ) · τ ((X2 , . . . , Xk+1 ).
The theorem now follows by linearity.
•
Corollary 4.9. If τ ∈ Ω (P, g), one has ∇ω τ = dτ + [ω, τ ].
.
•
2
Proposition 4.10. For τ ∈ Ω (P, V ) one has ∇2ω τ = Fω τ , where Fω = dω + 12 ω ∧ ω ∈ Ω (P, g) is the curvature
of ∇ω .
.
.
.
.
.
.
.
.
.
Proof. One checks that ∇ω (dτ +ω ∧ τ ) = d2 τ +dω ∧ τ −ω ∧ dτ +ω ∧ dτ +ω ∧ (ω ∧ τ ) = dω ∧ τ + 12 (ω ∧ ω) ∧ τ
.
(check that indeed a factor 21 appears because of the definitions of ∧). It remains to check that Fω (X1 , X2 ) =
(dω + 12 [ω, ω])(X1 , X2 ) vanishes when either X1 or X2 is vertical. This is again a point-wise calculation so we
assume that X1 = A∗ . Note that [ω, ω](X1 , X2 ) = 2[ω(X1 ), ω(X2 )] so that
Fω (X1 , X2 ) = dω(X1 , X2 ) + [ω(X1 ), ω(X2 )] = X1 (ω(X2 )) − X2 (ω(X1 )) − ω([X1 , X2 ]) + [ω(X1 ), ω(X2 )].
Since ω(X1 ) = A is constant, the second term vanishes. It remains to show that X1 (ω(X2 )) − ω([X1 , X2 ]) +
[ω(X1 ), ω(X2 )] = 0. If X2 is G-invariant horizontal, all terms are zero. If X2 = B ∗ for some B ∈ g, then
X1 (ω(X2 )) = 0 and ω([X1 , X2 ]) = [ω(X1 ), ω(X2 )] because [A∗ , B ∗ ] = [A, B]∗ .
Proposition 4.11 (Bianchi identity). ∇ω Fω = 0.
Proof. Note that ∇ω Fω = dFω + [Fω , ω] = d2 ω + 21 d([ω, ω]) + [ω, dω] + [ω, [ω, ω]] = 21 d([ω, ω]) + [dω, ω] = 0, since
d([ω, ω]) = [dω, ω] − [ω, dω] = −2[ω, dω]. We also used the Jacobi indentity to show that [ω, [ω, ω]] = 0
6
5
6 GAUGE GROUP AND GAUGE ALGEBRA
Relation with physics
Let P be a principal G-bundle and assume for simplicity that G is a matrix Lie group (like SU (N )). Let
ρ : G → GL(V ) be a finite-dimensional representation of G. Construct the associated bundle P ×G V . Then
a local section of P will induce a local trivialisation of P ×G V . Let s be a section of the associated bundle
P ×G V ). On some local trivialisation (U, σ) of P , induced by a local section σ : U → P , the section s can
be considered as a V -valued function. This is the point of view most physicists take. If we consider s as a
G-equivariant function s : P → V , then on U ⊂ M the section s is realised as a V -valued function as s ◦ σ. In
these local coordinates the covariant derivative s 7→ (ds)H takes the form
.
.
(ds)H ◦ σ = σ ∗ (ds + ω ∧ s) = dσ ∗ s + σ ∗ ω ∧ σ ∗ s = (d + ωU )σ ∗ s.
That is, on a local trivialisation a connection ω takes the form d + ωU , where ωU transforms according to the
rule (1). This is the usual form of a connection one encounters in physics.
1
The stament that two connections differ by Ω (P, g) translates, on the level of vector bundles, to the perhaps
more familiar statement that two connection ∇, ∇′ differ by an element Γ(M, T M ∗ ⊗ E). The curvature
Fω = dω + 12 ω ∧ ω lives in Ω2 (P, g) and therefore corresponds to an element F ∈ Γ(T M ∗ ⊗ T M ∗ ⊗ E), which
is also known as the curvature of the connection ∇ on E.
The local form on M of the curvature Fω = dω + 21 [ω, ω] is Fu = dωU + 12 [ωU , ωU ], the usual field strength
tensor in field theory.
Until now we have seen the following: G-gauge theories are naturally described by principal G-bundles with
a connection ω that plays the role of the gauge potential. If one wants to consider particles fields that transform
according to a representation ρ : G → V one can identify these particle fields with sections of the associated
bundles P ×G V , which in turn are identified with V -valued equivariant function on P . In physics one considers
local trivialisation of the bundles (i.e. a choice of gauge), which corresponds to choosing a local section σ of P .
Under this local trivialisation a section s takes the form s ◦ σ. A connection acts as d + ωU and under a gauge
−1
−1
∗
transformation ωU transforms as ωU = gU
V ωV gU V gU V dgU V and the curvature FU = σ Fω is the field strength
tensor. It satisfies the homogeneous field equation dFu = [Fu , ωu ].
6
Gauge group and gauge algebra
In this section we will define the gauge group and its Lie algebra. These notions were not discussed in the
lectures and here I will only briefly give the definitions.
The gauge group is defined as the group of G-equivariant diffeomorphisms preserving the fibres: GA(P ) =
{f ∈ Diff(P ) | f equivariant and f ◦ π = π}. This group is isomorphic to C ∞ (P, G)G , where G acts on itself
in the adjoint representation, and f ∈ GA(P ) is related to a τ ∈ C ∞ (M, G)G by f (p) = pτ (p). We have
isomorphisms GA(P ) ∼
= C ∞ (M, G)G ∼
= Γ(P ×Ad G). The bundle Ad P := P ×Ad G is called the adjoint bundle
of P and it is obtained as an associated bundle op P by the adjoint action of G on itself. The fibres have the
structure of a group since Ad g ∈ Aut G for all g ∈ G. The group structure of the fibres naturally induces a
group structure on the section Γ(M, Ad P ). From now on we will work with the group C ∞ (M, G)G .
k
The group C ∞ (M, G)G acts on Γ(P ×G V ) as (f φ)(p) = f (p)φ(p) or more generally, on Ω (P, V ) as (f ·
φ)(X1 , . . . Xk ) = f (p) · φ(X1 , . . . Xk ). The covariant derivative transforms as ∇ω 7→ f ∇ω f −1 =: ∇fω so that
f · ∇ω φ = ∇fω (f φ). This implies that the curvature transforms as F 7→ f Fω f −1 and ω as ω 7→ f df −1 + f ωf −1 .
Locally, f can be considered as a change of local trivialisation, σ 7→ f ◦ σ.
Note that for any gauge transformation f the 1-form f ∗ ω is again a connection. Two connections on P are
called equivalent if they are related by a gauge transformation as above.
Definition 6.1. The gauge algebra is the Lie algebra of infinitesimal gauge transformations Γ(M, ad P ) ∼
=
C ∞ (P, g)G , where G acts in the adjoint representation on g. Here ad P := P ×ad g.
The Lie-algebra structure on Γ(M, ad P ) is given by [H1 , H2 ](p) = [H1 (p), H2 (p)]. Moreover, there is a map
Exp : Γ(M, ad P ) → Γ(M, Ad P ) given by
Exp(H)(p) = exp(H(p)),
which is well-defined since
Exp(H)(pg) = expH(pg) = exp adg−1 H(p) = Adg−1 exp(H(p)) = Adg−1 (Exp(H)(p)).
7
REFERENCES
k
The gauge algebra acts on Ω (P, V ) as (H · φ)(X1 , . . . Xk ) = H(p) · φ(X1 , . . . Xk ). Equivalently, it can be
d
(Exp(tH) · φ)t=0 .
defined by H · φ = dt
Definition 6.2. The actional functional for matrix Lie groups is given by
Z
S(ω) = Tr Fω ∧ ∗Fω ,
M
where ∗ denotes the Hodge star operator.
If one looks for a local extremum, one finds the Yang-Mills equation:
∇ω (∗F ) = 0,
which is similar to the Bianchi identity ∇ω F = 0, which is always satisfied.
References
[1] Bleecker. Gauge theories and variational principles. Dover Publications (1981)