6Non-abelian gauge theory
III Advanced Quantum Field Theory
6.1 Bundles, connections and curvature
Vector bundles
To do non-abelian gauge theory (or even abelian gauge theory, really), we need
to know some differential geometry. In this section, vector spaces can be either
real or complex — our theory works fine with either.
So far, we had a universe
M
, and our fields took values in some vector space
V
. Then a field is just a (smooth/continuous/whatever) function
f
:
M → V
.
However, this requires the field to take values in the same vector space at each
point. It turns out it is a more natural thing to assign a vector space
V
x
for each
x ∈ M, and then a field φ would be a function
φ : M →
a
x∈M
V
x
≡ E,
where
`
means, as usual, the disjoint union of the vector spaces. Of course, the
point of doing so is that we shall require
φ(x) ∈ V
x
for each x. (∗)
This will open a can of worms, but is needed to do gauge theory properly.
We are not saying that each
x ∈ M
will be assigned a completely random
vector space, e.g. one would get
R
3
and another would get
R
144169
. In fact,
they will be isomorphic to some fixed space
V
. So what do we actually achieve?
While it is true that
V
x
∼
=
V
for all
x
, there is no canonical choice of such an
isomorphism. We will later see that picking such an isomorphism correspond to
picking a gauge of the field.
Now if we just have a bunch of vector spaces
V
x
for each
x ∈ M
, then we
lose the ability to talk about whether a field is differentiable or not, let alone
taking the derivative of a field. So we want to “glue” them together in some way.
We can write
E = {(x, v) : x ∈ M, v ∈ V
x
},
and we require
E
to be a manifold as well. We call these vector bundles. There
are some technical conditions we want to impose on
E
so that it is actually
possible to work on it.
There is a canonical projection map
π
:
E → M
that sends (
x, v
) to
x
. Then
we have
V
x
=
π
−1
(
{x}
). A natural requirement is that we require this map
π
to
be smooth. Otherwise, we might find someone making
E
into a manifold in a
really stupid way.
We now introduce a convenient terminology.
Definition
(Section)
.
Let
p
:
E → M
be a map between manifolds. A section
is a smooth map s : M → E such that p ◦ s = id
M
.
Now we can rewrite the condition (
∗
) as saying
φ
is a section of the map
π
:
E → M
. In general, our fields will usually be sections of some vector bundle.
Example.
Let
M
be any manifold, and
V
be any vector space. Then
M ×V
is
a manifold in a natural way, with a natural projection
π
1
:
M × V → M
. This
is called the trivial bundle.
Now trivial bundles are very easy to work with, since a section of the trivial
bundle is in a very natural correspondence with maps
M → V
. So we know
exactly how to work with them.
The final condition we want to impose on our vector bundles is not that
they are trivial, or else we have achieved absolutely nothing. What we want to
require is that near every point, the vector bundle looks trivial.
Definition
(Vector bundle)
.
Let
M
be a manifold, and
V
a vector space. A
vector bundle over
M
with typical fiber
V
is a manifold
E
with a map
π
:
E → M
such that for all
x ∈ M
, the fiber
E
x
∼
=
π
−1
(
{x}
) is a vector space that is
isomorphic to V .
Moreover, we require that for each
x ∈ M
, there exists an open neighbourhood
U
of
x
, and a diffeomorphism Φ :
U × V → π
−1
(
U
) such that
π
(Φ(
y, v
)) =
y
for
all y, and Φ(y, ·) : {y} × V → E
y
is a linear isomorphism of vector spaces.
Such a Φ is called a local trivialization of
E
, and
U
is called a trivializing
neighbourhood.
By definition, each point
x
is contained in some trivializing neighbourhood.
Thus, we can find a trivializing cover
{U
α
}
with a trivialization on each
U
α
such
that
S
U
α
= M.
There are some philosophical remarks we can make here. On
R
n
, every
bundle is isomorphic to a trivial bundle. If we only care about Euclidean (or
Minkowski) space, then it seems like we are not actually achieving much. But
morally, requiring that a bundle is just locally trivial, and not globally trivial
in some sense tells us everything we do is “local”. Indeed, we are mere mortals
who can at best observe the observable universe. We cannot “see” anything
outside of the observable universe, and in particular, it is impossible to know
whether bundles remain trivial if we go out of the observable universe. Even
if we observe that everything we find resembles a trivial bundle, we are only
morally allowed to conclude that we have a locally trivial bundle, and are not
allowed to conclude anything about the global geometry of the universe.
Another reason for thinking about bundles instead of just trivial ones is that
while every bundle over
R
n
is globally trivial, the choice of trivialization is not
canonical, and there is a lot of choice to be made. Usually, when we have a
vector space
V
and want to identify it with
R
n
, then we just pick a basis for
it. But now if we have a vector bundle, then we have to pick a basis at each
point in space. This is a lot of arbitrary choices to be made, and it is often more
natural to study the universe without making such choices. Working on a vector
bundle in general also prevents us from saying things that depends on the way
we trivialize our bundle, and so we “force” ourselves to say sensible things only.
Recall that for a trivial bundle, a section of the bundle is just a map
M → V
.
Thus, for a general vector bundle, if we have a local trivialization Φ on
U
, then
under the identification given by Φ, a section defined on
U
can be alternatively
be written as a map
φ
:
U → V
, which we may write, in coordinates, as
φ
a
(
x
),
for
a
= 1
, ··· , dim V
. Note that this is only valid in the neighbourhood
U
, and
also depends on the Φ we pick.
Example. Let M be any manifold. Then the tangent bundle
T M =
a
x∈M
T
x
M → M
is a vector bundle. Similarly, the cotangent bundle
T
∗
M =
a
x∈M
T
∗
x
M → M
is a vector bundle.
Recall that given vector spaces
V
and
W
, we can form new vector spaces by
taking the direct sum
V ⊕W
and the tensor product
V ⊗W
. There is a natural
way to extend this to vector bundles, so that if
E, F → M
are vector bundles,
then
E ⊕ F
and
E ⊗ F
are vector bundles with fibers (
E ⊕ F
)
x
=
E
x
⊕ F
x
and
(
E ⊗ F
)
x
=
E
x
⊗ F
x
. It is an exercise for the reader to actually construct these
bundles. We can also similarly extend then notion of exterior product
V
p
V
to
vector bundles.
In particular, applying these to
T M
and
T
∗
M
gives us tensor product bundles
of the form (T M)
⊗n
⊗ (T
∗
M)
⊗m
, whose sections are tensor fields.
In more familiar notation, (in a local trivialization) we write sections of
tangent bundles as
X
µ
(
x
), and sections of the cotangent bundle as
Y
µ
(
x
).
Sections of the tensor product are written as X
µ
1
,...,µ
n
ν
1
,...,ν
m
.
Example.
There is exactly one non-trivial vector bundle we can visualize.
Consider the circle S
1
:
Let’s consider line bundles on
S
1
, i.e. vector bundles with fiber
∼
=
R
. There is of
course the trivial bundle, and it looks like this:
However, we can also introduce a “twist” into this bundle, and obtain the M¨obius
band :
This is an example of a non-trivial line bundle. How do we know it is non-trivial?
The trivial bundle obviously has a nowhere-vanishing section. However, if we
stare at the M¨obius band hard enough, we see that any section of the M¨obius
band must vanish somewhere. Thus, this cannot be the trivial line bundle.
In fact, it is a general theorem that a line bundle has a nowhere-vanishing
section if and only if it is trivial.
We introduce a bit more notation which we will use later.
Notation.
Let
E → M
be a vector bundle. Then we write Ω
0
M
(
E
) for the
vector space of sections of
E → M
. Locally, we can write an element of this as
X
a
, for a = 1, ··· , dim E
x
.
More generally, we write Ω
p
M
(
E
) for sections of
E ⊗
V
p
T
∗
M → M
, where
V
p
T
∗
M
is the bundle of
p
-forms on
M
. Elements can locally be written as
X
a
µ
1
...µ
n
.
If V is a vector space, then Ω
p
M
(V ) is a shorthand for Ω
p
M
(V × M).
Let’s return to the definition of a vector bundle. Suppose we had two trivial-
izing neighbourhoods
U
α
and
U
β
, and that they have a non-trivial intersection
U
α
∩ U
β
. We can then compare the two trivializations on U
α
and U
β
:
(U
α
∩ U
β
) × V π
−1
(U
α
× U
β
) (U
α
∩ U
β
) × V.
Φ
α
Φ
β
Composing the maps gives us a map
t
αβ
: Φ
−1
α
◦ Φ
β
: (U
α
∩ U
β
) × V → (U
α
∩ U
β
) × V
that restricts to a linear isomorphism on each fiber. Thus, this is equivalently a
map U
α
∩ U
β
→ GL(V ). This is called the transition function.
These transition functions satisfy some compatibility conditions. Whenever
x ∈ U
α
∩ U
β
∩ U
γ
, we have
t
αβ
(x) · t
βγ
(x) = t
αγ
(x).
Note that on the left, what we have is the (group) multiplication of
GL
(
V
).
These also satisfy the boring condition
t
αα
=
id
. These are collectively known
as the cocycle conditions.
Exercise.
Convince yourself that it is possible to reconstruct the whole vector
bundle just from the knowledge of the transition functions. Moreover, given any
cover
{U
α
}
of
M
and functions
t
αβ
:
U
α
∩ U
β
→ GL
(
V
) satisfying the cocycle
conditions, we can construct a vector bundle with these transition functions.
This exercise is crucial. It is left as an exercise, instead of being spelt out
explicitly, because it is much easier to imagine what is going on geometrically
in your head than writing it down in words. The idea is that the transition
functions tell us how we can glue different local trivializations together to get a
vector bundle.
Now we want to do better than this. For example, suppose we have
V
=
R
n
, which comes with the Euclidean inner product. Then we want the local
trivializations to respect this inner product, i.e. they are given by orthogonal
maps, rather than just linear isomorphisms. It turns out this is equivalent to
requiring that the transition functions
t
αβ
actually land in O(
n
) instead of just
GL(n, R). More generally, we can have the following definition:
Definition
(
G
-bundle)
.
Let
V
be a vector space, and
G ≤ GL
(
V
) be a Lie
subgroup. Then a
G
-bundle over
M
is a vector bundle over
M
with fiber
V
,
equipped with a trivializing cover such that the transition functions take value
in G.
Note that it is possible to turn a vector bundle into a
G
-bundle into many
different ways. So the trivializing cover is part of the data needed to specify the
G-bundle.
We can further generalize this a bit. Instead of picking a subgroup
G ≤
GL
(
V
), we pick an arbitrary Lie group
G
with a representation on
V
. The
difference is that now certain elements of G are allowed to act trivially on V .
Definition
(
G
-bundle)
.
Let
V
be a representation,
G
a Lie group, and
ρ
:
G →
GL(V ) a representation. Then a G-bundle consists of the following data:
(i) A vector bundle E → M.
(ii) A trivializing cover {U
α
} with transition functions t
αβ
.
(iii)
A collection of maps
ϕ
αβ
:
U
α
∩ U
β
→ G
satisfying the cocycle conditions
such that t
αβ
= ρ ◦ ϕ
αβ
.
Note that to specify a
G
-bundle, we require specifying an element
ϕ
αβ
(
x
)
∈ G
for each
x ∈ M
, instead of just the induced action
ρ
(
ϕ
αβ
(
x
))
∈ GL
(
V
). This
is crucial for our story. We are requiring more information than just how the
elements in
V
transform. Of course, this makes no difference if the representation
ρ
is faithful (i.e. injective), but makes a huge difference when
ρ
is the trivial
representation.
We previously noted that it is possible to recover the whole vector bundle just
from the transition functions. Consequently, the information in (i) and (ii) are
actually redundant, as we can recover
t
αβ
from
ϕ
αβ
by composing with
ρ
. Thus,
a
G
-bundle is equivalently a cover
{U
α
}
of
M
, and maps
ϕ
αβ
:
U
α
∩ U
β
→ G
satisfying the cocycle condition.
Note that this equivalent formulation does not mention ρ or V at all!
Example.
Every
n
-dimensional vector bundle is naturally a
GL
(
n
) bundle —
we take ρ to be the identity map, andϕ
αβ
= t
αβ
.
Principal G-bundles
We are halfway through our voyage into differential geometry. I promise this
really has something to do with physics.
We just saw that a
G
-bundle can be purely specified by the transition
functions, without mentioning the representation or fibers at all. In some sense,
these transition functions encode the “pure twisting” part of the bundle. Given
this “pure twisting” information, and any object
V
with a representation of
G
on
V
, we can construct a bundle with fiber
V
, twisted according to this prescription.
This is what we are going to do with gauge theory. The gauge group is the
group
G
, and the gauge business is encoded in these “twisting” information.
Traditionally, a field is a function
M → V
for some vector space
V
. To do gauge
coupling, we pick a representation of
G
on
V
. Then the twisting information
allows us to construct a vector bundle over
M
with fiber
V
. Then gauge-
coupled fields now correspond to sections of this bundle. Picking different local
trivializations of the vector bundle corresponds to picking different gauges, and
the transition functions are the gauge transformations!
But really, we prefer to work with some geometric object, instead of some
collection of transition functions. We want to find the most “natural” object
for
G
to act on. It turns out the most natural object with a
G
-action is not a
vector space. It is just G itself!
Definition
(Principal
G
-bundle)
.
Let
G
be a Lie group, and
M
a manifold.
A principal
G
-bundle is a map
π
:
P → M
such that
π
−1
(
{x}
)
∼
=
G
for each
x ∈ M
. Moreover,
π
:
P → M
is locally trivial, i.e. it locally looks like
U × G
,
and transition functions are given by left-multiplication by an element of G.
More precisely, we are given an open cover
{U
α
}
of
M
and diffeomorphisms
Φ
α
: U
α
× G → π
−1
(U
α
)
satisfying π(Φ
α
(x, g)) = x, such that the transition functions
Φ
−1
α
◦ Φ
β
: (U
α
∩ U
β
) × G → (U
α
∩ U
β
) × G
is of the form
(x, g) 7→ (x, t
αβ
(x) · g)
for some t
αβ
: U
α
∩ U
β
→ G.
Theorem.
Given a principal
G
-bundle
π
:
P → M
and a representation
ρ
:
G → GL
(
V
), there is a canonical way of producing a
G
-bundle
E → M
with
fiber V . This is called the associated bundle.
Conversely, given a
G
-bundle
E → M
with fiber
V
, there is a canonical way
of producing a principal
G
-bundle out of it, and these procedures are mutual
inverses.
Moreover, this gives a correspondence between local trivializations of
P → M
and local trivializations of E → M.
Note that since each fiber of
P → M
is a group, and trivializations are
required to respect this group structure, a local trivialization is in fact equivalent
to a local section of the bundle, where we set the section to be the identity.
Proof. If the expression
P ×
G
V → M
makes any sense to you, then this proves the first part directly. Otherwise,
just note that both a principal
G
-bundle and a
G
-bundle with fiber
V
can be
specified just by the transition functions, which do not make any reference to
what the fibers look like.
The proof is actually slightly less trivial than this, because the same vector
bundle can have be many choices of trivializing covers, which gives us different
transition functions. While these different transition functions patch to give the
same vector bundle, by assumption, it is not immediate that they must give the
same principal G-bundle as well, or vice versa.
The way to fix this problem is to figure out explicitly when two collection of
transition functions give the same vector bundle or principal bundle, and the
answer is that this is true if they are cohomologous. Thus, the precise statement
needed to prove this is that both principal
G
-bundle and
G
-bundles with fiber
V
biject naturally with the first
ˇ
Cech cohomology group of
M
with coefficients
in G.
We now get to the physics part of the story. To specify a gauge theory
with gauge group
G
, we supplement our universe
M
with a principal
G
-bundle
π
:
P → M
. In QED, the gauge group is U(1), and in QCD, the gauge group is
SU(3). In the standard model, for some unknown reason, the gauge group is
G = SU(3) ×SU(2) × U(1).
Normally, a field with values in a vector space
V
is is given by a smooth map
φ
:
M → V
. To do gauge coupling, we pick a representation
ρ
:
G → V
, and
then form the associated bundle to
P → M
. Then a field is now a section of
this associated bundle.
Example.
In Maxwell theory, we have
G
= U(1). A complex scalar field is a
function
φ
:
G → C
. The vector space
C
has a very natural action of U(1) given
by left-multiplication.
We pick our universe to be
M
=
R
4
, and then the bundle is trivial. However,
we can consider two different trivializations defined on the whole of
M
. Then
we have a transition function
t
:
M →
U(1), say
t
(
x
) =
e
iθ(x)
. Then under this
change of trivialization, the field would transform as
φ(x) 7→ e
iθ(x)
φ(x).
This is just the usual gauge transformation we’ve seen all the time!
Example.
Recall that a vector bundle
E → M
with fiber
R
n
is naturally a
GL
(
n
)-bundle. In this case, there is a rather concrete description of the principal
GL(n)-bundle that gives rise to E.
At each point
x ∈ M
, we let
Fr
(
E
x
) be the set of all ordered bases of
E
x
.
We can biject this set with
GL
(
n
) as follows — we first fix a basis
{e
i
}
of
E
x
.
Then given any other basis
{f
i
}
, there is a unique matrix in
GL
(
n
) that sends
{e
i
}
to
{f
i
}
. This gives us a map
Fr
(
E
x
)
→ GL
(
n
), which is easily seen to be a
bijection. This gives a topology on Fr(E
x
).
The map constructed above obviously depends on the basis
e
i
chosen. Indeed,
changing the
e
i
corresponds to multiplying
t
on the right by some element
of
GL
(
n
). However, we see that at least the induced smooth structure on
Fr
(
E
x
) is well-defined, since right-multiplication by an element of
GL
(
n
) is a
diffeomorphism.
We can now consider the disjoint union of all such
Fr
(
E
x
). To make this
into a principal
GL
(
n
)-bundle, we need to construct local trivializations. Given
any trivialization of
E
on some neighbourhood
U
, we have a smooth choice
of basis on each fiber, since we have bijected the fiber with
R
n
, and
R
n
has
a standard basis. Thus, performing the above procedure, we have a choice of
bijection between
Fr
(
E
x
) between
GL
(
n
). If we pick a different trivialization,
then the choice of bijection differs by some right-multiplication.
This is almost a principal
GL
(
n
)-bundle, but it isn’t quite so — to obtain
a principal
GL
(
n
)-bundle, we want the transition functions to be given by left
multiplication. To solve this problem, when we identified
Fr
(
E
x
) with
GL
(
n
)
back then, we should have sent
{f
i
}
to the inverse of the matrix that sends
{e
i
}
to {f
i
}.
In fact, we should have expected this. Recall from linear algebra that under
a change of basis, if the coordinates of elements transform by
A
, then the basis
themselves transform by
A
−1
. So if we want our principal
GL
(
n
)-bundle to have
the same transition functions as
E
, we need this inverse. One can now readily
check that this has the same transition functions as
E
. This bundle is known as
the frame bundle, and is denoted Fr(E).
Note that specifying trivializations already gives a smooth structure on
π
:
Fr
(
E
)
→ M
. Indeed, on each local trivialization on
U
, we have a bijection
between
π
−1
(
U
) and
U × GL
(
n
), and this gives a chart on
π
−1
(
U
). The fact
that transition functions are given by smooth maps
U → GL
(
n
) ensures the
different charts are compatible.
Recall that we previously said there is a bijection between a section of a
principal
G
-bundle and a trivialization of the associated bundle. This is very
clearly true in this case — a section is literally a choice of basis on each fiber!
Connection
Let’s go back to the general picture of vector bundles, and forget about the
structure group
G
for the moment. Consider a general vector bundle
π
:
E → M
,
and a section
s
:
M → E
. We would like to be able to talk about derivatives of
this section. However, the “obvious” attempt looking like
s(x + ε) − s(x)
|ε|
doesn’t really make sense, because
s
(
x
+
ε
) and
s
(
x
) live in different vector
spaces, namely E
x+ε
and E
x
.
We have encountered this problem in General Relativity already, where we
realized the “obvious” derivative of, say, a vector field on the universe doesn’t
make sense. We figured that what we needed was a connection, and it turns
out the metric on
M
gives us a canonical choice of the connection, namely the
Levi-Civita connection.
We can still formulate the notion of a connection for a general vector bundle,
but this time, there isn’t a canonical choice of connection. It is some additional
data we have to supply.
Before we meet the full, abstract definition, we first look at an example of a
connection.
Example.
Consider a trivial bundle
M × V → M
. Then the space of sections
Ω
0
M
(
V
) is canonically isomorphic to the space of maps
M → V
. This we know
how to differentiate. There is a canonical map d : Ω
0
M
(
V
)
→
Ω
1
M
(
V
) sending
f 7→ df, where for any vector X ∈ T
p
M, we have
df(X) =
∂f
∂X
∈ V.
This is a one-form with values in
V
(or
M × V
) because it takes in a vector
X
and returns an element of V .
In coordinates, we can write this as
df =
∂f
∂x
µ
dx
µ
.
We can now define a connection:
Definition
(Connection)
.
A connection is a linear map
∇
: Ω
0
M
(
E
)
→
Ω
1
M
(
E
)
satisfying
(i) Linearity:
∇(α
1
s
1
+ α
2
s
2
) = α
1
(∇s
1
) + α
2
(∇s
2
)
for all s
1
, s
2
∈ Ω
0
M
(E) and α
1
, α
2
constants.
(ii) Leibnitz property:
∇(fs) = (df)s + f(∇S)
for all
s ∈
Ω
0
M
(
E
) and
f ∈ C
∞
(
M
), where, d
f
is the usual exterior
derivative of a function, given in local coordinates by
df =
∂f
∂x
µ
dx
µ
.
Given a vector field
V
on
M
, the covariant derivative of a section in the direction
of V is the map
∇
V
: Ω
0
M
(E) → Ω
0
M
(E)
defined by
∇
V
s = V y∇s = V
µ
∇
µ
s.
In more physics settings, the connection is usually written as D
µ
.
Consider any two connections
∇, ∇
0
. Their difference is not another connec-
tion. Instead, for any f ∈ C
∞
(M) and s ∈ Ω
0
M
(E), we have
(∇
0
− ∇)(fs) = f(∇
0
− ∇)(s).
So in fact the difference is a map Ω
0
M
(
E
)
→
Ω
1
M
(
E
) that is linear over functions
in
C
∞
(
M
). Equivalently, it is some element of Ω
1
M
(
End
(
E
)), i.e. some matrix-
valued 1-form A
µ
(x) ∈ End(E
x
).
In particular, consider any open set
U ⊆ M
equipped with a local trivializa-
tion. Then after picking the trivialization, we can treat the bundle on
U
as a
trivial one, and our previous example showed that we had a “trivial” connection
given by d. Then any other connection ∇ can be expressed as
∇s = ds + As
for some
A ∈
Ω
1
U
(
End
(
V
)), where the particular
A
depends on our trivialization.
This is called the connection 1-form, or the gauge field. In the case where
E
is
the tangent bundle, this is also known as the Christoffel symbols.
This was all on a general vector bundle. But the case we are really interested
in is a
G
-bundle. Of course, we can still say the same words as above, as any
G
-bundle is also a vector bundle. But can we tell a bit more about how the
connection looks like? We recall that specifying a
G
-bundle with fiber
V
is
equivalent to specifying a principal
G
-bundle. What we would like is to have
some notion of “connection on a principal G-bundle”.
Theorem.
There exists a notion of a connection on a principal
G
-bundle.
Locally on a trivializing neighbourhood
U
, the connection 1-form is an element
A
µ
(x) ∈ Ω
1
U
(g), where g is the Lie algebra of G.
Every connection on a principal
G
-bundle induces a connection on any asso-
ciated vector bundle. On local trivializations, the connection on the associated
vector bundle has the “same” connection 1-form
A
µ
(
x
), where
A
µ
(
x
) is regarded
as an element of End(V ) by the action of G on the vector space.
Readers interested in the precise construction of a connection on a principal
G
-bundle should consult a textbook on differential geometry. Our previous work
doesn’t apply because G is not a vector space.
It is useful to know how the connection transforms under a change of local
trivialization. For simplicity of notation, suppose we are working on two trivial-
izations on the same open set
U
, with the transition map given by
g
:
U → G
.
We write
A
and
A
0
for the connection 1-forms on the two trivializations. Then
for a section s expressed in the first coordinates, we have
g · (ds + As) = (d + A
0
)(g · s).
So we find that
A
0
= gAg
−1
− gd(g
−1
).
This expression makes sense if
G
is a matrix Lie group, and so we can canonically
identify both
G
and
g
as subsets of
GL
(
n, R
) for some
n
. Then we know what
it means to multiply them together. For a more general Lie group, we have to
replace the first term by the adjoint representation of
G
on
g
, and the second by
the Maurer–Cartan form.
Example.
In the U(1) case,our transition functions
g
αβ
are just multiplication
by complex numbers. So if g = e
−iλ
, then we have
A
β
= −gdg
−1
+ gA
α
g
−1
= −gdg
−1
+ A
α
= i(dλ − iA
α
).
Note that since U(1) is abelian, the conjugation by
g
has no effect on
A
. This is
one of the reasons why non-abelian gauge theory is simple.
Minimal coupling
So how do we do gauge coupling? Suppose we had an “ordinary” scalar field
ψ : M → C on our manifold, and we have the usual action
S[ψ] =
Z
1
2
|∂ψ|
2
+
1
2
m
2
ψ
2
+ ··· .
We previously said that to do gauge theory, we pick a representation of our
gauge group
G
= U(1) on
C
, which we can take to be the “obvious” action by
multiplication. Then given a principal U(1)-bundle
P → M
, we can form the
associated vector bundle, and now our field is a section of this bundle.
But how about the action? As mentioned, the
∂ψ
term no longer makes
sense. But in the presence of a connection, we can just replace
∂
with
∇
! Now,
the action is given by
S[ψ] =
Z
1
2
|∇ψ|
2
+
1
2
m
2
ψ
2
+ ··· .
This is known as minimal coupling.
At this point, one should note that the machinery of principal
G
-bundles
was necessary. We only ever have one principal
G
-bundle
P → M
, and a single
connection on it. If we have multiple fields, then we use the same connection on
all of them, via the mechanism of associated bundles. Physically, this is important
— this means different charged particles couple to the same electromagnetic field!
This wouldn’t be possible if we only worked with vector bundles; we wouldn’t
be able to compare the connections on different vector bundles.
Curvature
Given a connection
∇
, we can extend it to a map Ω
p
M
(
E
)
→
Ω
p+1
M
(
E
) by requiring
it to satisfy the conditions
∇(α
1
s
1
+ α
2
s
2
) = α
1
(∇s
1
) + α
2
(∇s
2
),
∇(ω ∧ s) = (dω) ∧ s + (−1)
deg ω
ω ∧ ∇s.
whenever ω ∈ Ω
q
(M) and s ∈ Ω
p−q
M
(E).
We can think of
∇
as a “covariant generalization” of the de Rham operator
d. However, where the ordinary d is nilpotent, i.e. d
2
= 0, here this is not
necessarily the case.
What kind of object is ∇
2
, then? We can compute
∇
2
(ω ∧ s) = ∇(dω ∧ s + (−1)
q
ω ∧ ∇s)
= d
2
ω ∧ s + (−1)
q+1
dω ∧ ∇s + (−1)
q
dω ∧ ∇s + (−1)
2q
ω ∧ ∇
2
s
= ω ∧ ∇
2
s.
So we find that
∇
2
is in fact a map Ω
q
M
(
E
)
→
Ω
q+2
M
(
E
) that is linear over any
forms! Specializing to the case of q = 0 only, we can write ∇
2
as
∇
2
(s) = F
∇
s,
for some
F
∇
∈
Ω
2
M
(
End
(
E
)). It is an easy exercise to check that the same
formula works for all q. In local coordinates, we can write
F
∇
=
1
2
(F
µν
(x))
a
b
dx
µ
∧ dx
ν
.
Since we have
∇s = ds + As,
we find that
∇
2
s = ∇(ds + As) = d
2
s + d(As) + A(ds + As) = (dA + A ∧ A)s.
Note that by
A ∧ A
, we mean, locally in coordinates,
A
a
b
∧ A
b
c
, which is still a
form with values in End(V ).
Thus, locally, we have
F = dA + A ∧A
= (∂
µ
A
ν
+ A
µ
A
ν
) dx
µ
∧ dx
ν
=
1
2
(∂
µ
A
ν
− ∂
ν
A
µ
+ A
µ
A
ν
− A
ν
A
µ
) dx
µ
∧ dx
ν
=
1
2
(∂
µ
A
ν
− ∂
ν
A
µ
+ [A
µ
, A
ν
]) dx
µ
∧ dx
ν
Of course, when in the case of U(1) theory, the bracket vanishes, and this is
just the usual field strength tensor. Unsurprisingly, this is what will go into the
Lagrangian for a general gauge theory.
Crucially, in the non-abelian theory, the bracket term is there, and is non-zero.
This is important. Our
F
is no longer linear in
A
. When we do Yang–Mills later,
the action will contain a
F
2
term, and then expanding this out will give
A
3
and
A
4
terms. This causes interaction of the gauge field with itself, even without
the presence of matter!
We end by noting a seemingly-innocent identity. Note that we can compute
∇
3
s = ∇(∇
2
s) = ∇(F s) = (∇F ) ∧s + F ∧ (∇s).
On the other hand, we also have
∇
3
s = ∇
2
(∇s) = F ∧ ∇s.
These two ways of thinking about
∇
3
must be consistent. This implies we have
∇(F
∇
) ≡ 0.
This is known as the Bianchi identity.