Nakahara M. Geometry, Topology and Physics
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Now the definition of covariant derivative is in order. Let s( p) be a section
of E. Along a curve γ :[0, 1]→M we have s(t) =[( ˜γ(t), η(t))],where ˜γ(t) is
an arbitrary horizontal lift of γ(t). The covariant derivative of s(t) along γ(t) at
p
0
= γ(0) is defined by
∇
X
s ≡
˜γ(0),
d
dt
η(γ (t))
t=0
(10.49)
where X is the tangent vector to γ(t) at p
0
. For the covariant derivative to be
really intrinsic, it should not depend on the extra information, that is the special
horizontal lift. Let ˜γ
(t) =˜γ(t)a (a ∈ G) be another horizontal lift of γ .If ˜γ
(t)
is chosen to be the horizontal lift, we have a representative [( ˜γ
(t), a
−1
η(t))].
The covariant derivative is now given by
˜γ
(0),
d
dt
{a
−1
η(t)}
t=0
=
˜γ
(0)a
−1
,
d
dt
η(t)
t=0
which agrees with (10.49). Hence, ∇
X
s depends only on the tangent vector X
and the sections s ∈ (M, E) and not on the horizontal lift ˜γ(t). Our definition
depends only on a curve γ and a connection and not on local trivializations. The
local form of the covariant derivative is useful in practical computations and will
be given later.
So far we have defined the covariant derivative at a point p
0
= γ(0).It
is clear that if X is a vector field, ∇
X
maps a section s to a new section ∇
X
s,
hence ∇
X
is regarded as a map (M, E) → (M, E). To be more precise, take
X ∈
(M) whose value at p is X
p
∈ T
p
M. There is a curve γ(t) such that
γ(0) = p and its tangent at p is X
p
. Then any horizontal lift ˜γ(t) of γ enables
us to compute the covariant derivative ∇
X
s|
p
≡∇
X
p
s. We also define a map
∇:(M, E) → (M, E) ⊗
1
(M) by
∇s(X ) ≡∇
X
sX∈ (M) s ∈ (M, E). (10.50)
Exercise 10.8. Show that
∇
X
(a
1
s
1
+ a
2
s
2
) = a
1
∇
X
s
1
+ a
2
∇
X
s
2
(10.51a)
∇(a
1
s
1
+ a
2
s
2
) = a
1
∇s
1
+ a
2
∇s
2
(10.51b)
∇
(a
1
X
1
+a
2
X
2
)
s = a
1
∇
X
1
s + a
2
∇
X
2
s (10.51c)
∇
X
( fs) = X[ f ]s + f ∇
X
s (10.51d)
∇( fs) = (d f )s + f ∇s (10.51e)
∇
fX
s = f ∇
X
s (10.51f)
where a
i
∈ , s, s
∈ (M, E) and f ∈ (M).
10.4.2 A local expression for the covariant derivative
In practical computations it is convenient to have a local coordinate representation
of the covariant derivative. Let P(M, G) be a G bundle and E = P ×
ρ
G be an
associate vector bundle. Take a local section σ
i
∈ (U
i
, P) and employ the
canonical trivialization σ
i
( p) = φ
i
( p, e).Letγ :[0, 1]→M be a curve in U
i
and ˜γ its horizontal lift, which is written as
˜γ(t) = σ
i
(t)g
i
(t) (10.52)
where g
i
(t) ≡ g
i
(γ (t)) ∈ G. Take a section e
α
( p) ≡[(σ
i
( p), e
α
0
)] of E,where
e
α
0
is the αth basis vector of V ; (e
α
0
)
β
= (δ
α
)
β
.Wehave
e
α
(t) =[( ˜γ(t)g
i
(t)
−1
, e
α
0
)]=[( ˜γ(t), g
i
(t)
−1
e
α
0
)]. (10.53)
Note that g
i
(t)
−1
acts on e
α
0
to compensate for the change of basis along γ .The
covariant derivative of e
α
is then given by
∇
X
e
α
=
˜γ(0),
d
dt
{g
i
(t)
−1
e
α
0
}
t=0
=
˜γ(0), −g
i
(t)
−1
d
dt
g
i
(t)
g
i
(t)
−1
e
α
0
t=0
=[( ˜γ(0)g
i
(0)
−1
,
i
(X )e
α
0
)] (10.54)
where (10.13b) has been used. From (10.54) we find the local expression,
∇
X
e
α
=[(σ
i
(0),
i
(X )e
α
0
)]. (10.55)
Let
i
=
iµ
dx
µ
=
iµ
α
β
dx
µ
where
iµ
α
β
≡
iµ
γ
(T
γ
)
α
β
. The second
entry of (10.55) is
i
(X )e
α
0
=
dx
µ
dt
e
β
0
iµ
β
γ
δ
α
γ
=
dx
µ
dt
iµ
β
α
e
β
0
.
Substituting this into (10.55), we finally have
∇
X
e
α
=
σ
i
(0),
dx
µ
dt
iµ
β
α
e
β
0
=
dx
µ
dt
iµ
β
α
e
β
(10.56a)
or
∇e
α
=
i
β
α
e
β
.(10.56b)
In particular, for a coordinate curve x
µ
,wehave
∇
∂/∂x
µ
e
α
=
iµ
β
α
e
β
. (10.57)
It is remarkable that a connection
on a principal bundle P completely specifies
the covariant derivative on an associated bundle E (modulo representations).
Exercise 10.9. Let s( p) =[(σ
i
( p), ξ
i
( p))]=ξ
i
α
( p)e
α
be a general section of E,
where ξ
i
( p) = ξ
i
α
( p)e
α
0
. Use the results of exercise 10.8 to verify that
∇
X
s =
σ
i
(0),
dξ
i
dt
+
i
(X )ξ
i
t=0
=
dx
µ
dt
∂ξ
i
α
∂x
µ
+
iµ
α
β
ξ
i
β
e
α
.
(10.58)
By construction, the covariant derivative is independent of the local
trivialization. This is also observed from the local form of ∇
X
s.Letσ
i
( p) and
σ
j
( p) be local sections on overlapping charts U
i
and U
j
.OnU
i
∩ U
j
,wehave
σ
j
( p) = σ
i
( p)t
ij
( p).Inthei-trivialization, the covariant derivative is
∇
X
s =
σ
i
(0),
dξ
i
dt
+
i
(X )ξ
i
t=0
=
σ
j
(0) ·t
−1
ij
,
d
dt
(t
ij
ξ
j
) +
i
(X )t
ij
ξ
j
t=0
=
σ
j
(0),
dξ
j
dt
+
j
(X )ξ
j
t=0
(10.59)
where use has been made of the condition (10.9). The last line of (10.59) is ∇
X
s
expressed in the j-trivialization.
We have found that the covariant derivative defined by (10.49) is independent
of the horizontal lift as well as the local section. The gauge potential
i
transforms under the change of local trivialization so that ∇
X
s is a well-defined
section of E.Inthissense,∇
X
is the most natural derivative on an associated
vector bundle, which is compatible with the connection on the principal bundle
P.
Example 10.4. Let us recover the results obtained in section 7.2. Let FM be a
frame bundle over M and let TM be its associated bundle. We note FM =
P(M, GL(m,
)) and TM = FM×
ρ
m
,wherem = dim M and ρ is the m ×m
matrix representation of GL(m,
).Elementsof (m, ) are m × m matrices.
Let us rewrite the local connection form
i
as
α
µβ
dx
µ
. We then find that
∇
∂/∂x
µ
e
α
=[(σ
i
(0),
µ
e
α
0
)]=
β
µα
e
β
(10.60)
which should be compared with (7.14). For a general section (vector field),
s( p) =[(σ
i
( p), X
i
( p))]=X
i
α
( p)e
α
,wefind
∇
∂/∂x
µ
s =
∂
∂x
µ
X
i
α
+
α
µβ
X
β
e
α
(10.61)
which reproduces the result of section 7.2. It is evident that the roles played by the
indices α, β and µ in
α
µβ
are very different in their characters; µ is the
1
(M)
index while α and β are the
(m, ) indices.
Example 10.5. Let us consider the U(1) gauge field coupled to a complex scalar
field φ. The relevant fibre bundles are the U(1) bundle P(M, U(1)) and the
associated bundle E = P ×
ρ
where ρ is the natural identification of an element
of U(1) with a complex number. The local expression for ω is
i
=
iµ
dx
µ
,
where
iµ
=
i
(∂/∂ x
µ
) is the vector potential of Maxwell’s theory. Let γ be
acurveinM with tangent vector X at γ(0). Take a local section σ
i
and express
a horizontal lift ˜γ of γ as ˜γ(t) = σ
i
(t)e
iϕ(t)
.If1∈ is taken to be the basis
vector, the basis section is
e =[(σ
i
( p), 1)].
Let φ(p) =[(σ
i
( p), ( p))]=(p)e ( : M → ) be a section of E,which
is identified with a complex scalar field. With respect to ˜γ(t), the section is given
by
φ(t) = (t)[( ˜γ(t), U (t)
−1
)] (10.62)
where U(t) = e
iϕ(t)
. The covariant derivative of φ along γ is
∇
X
φ =
d
dt
[( ˜γ(0), U (0)
−1
)]+(0)[( ˜γ(0), U (0)
−1
i
(X ) · 1)]
=
d
dt
+
iµ
dx
µ
dt
e = X
µ
∂
∂x
µ
+
iµ
e. (10.63)
Example 10.6. Let us consider the SU(2) Yang–Mills theory on M. The relevant
bundles are the SU(2) bundle P(M, SU(2)) and its associated bundle E =
P ×
ρ
2
, where we have taken the two-dimensional representation. The gauge
potential on a chart U
i
is
i
=
iµ
dx
µ
= A
iµ
α
σ
α
2i
dx
µ
(10.64)
where σ
α
/2i are generators of SU(2), σ
α
being the Pauli matrices. Let e
α
0
(α = 1, 2) be basis vectors of
2
and consider sections
e
α
( p) ≡[(σ
i
( p), e
α
0
)] (10.65)
where σ
i
( p) defines a canonical trivialization of P over U
i
.Letφ(p) =
[(σ
i
( p),
α
( p)e
α
0
)] beasectionofE over M. Along a horizontal lift ˜γ(t) =
σ
i
( p)U(t), U(t) ∈ SU(2),wehave
φ(t) =[( ˜γ(t), U(t)
−1
α
(t)e
α
0
)]. (10.66)
The covariant derivative of φ along X = d/dt is
∇
X
φ =
˜γ(0), U (0)
−1
d
α
(0)
dt
e
α
0
+[( ˜γ(0), U (0)
−1
i
(X )
α
β
β
(0)e
α
0
)]
= X
µ
∂
α
∂x
µ
+
iµ
α
β
β
e
α
(10.67)
where (10.13b) has been used to obtain the last equality.
Exercise 10.10. Let us consider an associated adjoint bundle E
= P ×
Ad
where
the action of G on
is the adjoint action V → Ad
g
V = g
−1
Vg, V ∈ and
g ∈ G. Take a local section σ
i
∈ (U
i
, P) such that ˜γ(t) = σ
i
(t)g(t).Takea
section s( p) =[(σ
i
( p), V ( p))] on E ,whereV ( p) = V
α
( p)T
α
, {T
α
} being the
basis of
. Define the covariant derivative
X
s by
X
s ≡
˜γ(0),
d
dt
{Ad
g(t )
−1
V (t)}
t=0
. (10.68a)
Show that
X
s =
σ
i
(0),
dV (t)
dt
+[
i
(X ), V (t)]
t=0
= X
µ
∂V
α
∂x
µ
+ f
βγ
α
iµ
β
V
γ
[(σ
i
(0), T
α
)]. (10.68b)
10.4.3 Curvature rederived
The covariant derivative ∇
X
s defines an operator ∇:(M, E) → (M, E ⊗
1
(M)) by (10.50). More generally, the action of ∇ on a vector-valued p-form
s ⊗ η, η ∈
p
(M),isdefinedby
∇(s ⊗ η) ≡ (∇s) ∧ η + s ⊗ dη. (10.69)
Let U
i
be a chart of M and σ
i
a section of P over U
i
. We take the canonical local
trivialization over U
i
. We now prove
∇∇e
α
= e
β
⊗
i
β
α
(10.70)
where e
α
=[(σ
i
, e
α
0
)]∈(U
i
, E). In fact, by straightforward computation, we
find
∇∇e
α
=∇(e
β
⊗
i
β
α
) =∇e
β
∧
i
β
α
+ e
β
⊗ d
i
β
α
= e
β
⊗ (d
i
β
α
+
i
β
γ
∧
i
γ
α
) = e
β
⊗
i
β
α
.
Exercise 10.11. Let s( p) = ξ
α
( p)e
α
( p) be a section of E. Show that
∇∇s = e
α
⊗
i
α
β
ξ
β
. (10.71)
10.4.4 A connection which preserves the inner product
Let E
π
−→ M be a vector bundle with a positive-definite symmetric inner product
whose action is defined at each point p ∈ M by
g
p
: π
−1
( p) ⊗ π
−1
( p) → . (10.72)
Then g is said to define a Riemannian structure on E. A connection ∇ is called
a metric connection if it preserves the inner product,
d [g(s, s
)]=g(∇s, s
) + g(s, ∇s
). (10.73)
In particular, if we take s = e
α
, s
= e
β
and set g(e
α
, e
β
) = g
αβ
,wefind
dg
αβ
=
i
γ
α
g
γβ
+
i
γ
β
g
αγ
. (10.74)
This should be compared with (7.30b). If E = TM and, moreover, the torsion-
free condition is imposed, our connection reduces to the Levi-Civita connection
of the Riemannian geometry.
Given an inner product, we may take an orthonormal frame {ˆe
α
} such that
g( ˆe
α
, ˆe
β
) = δ
αβ
. The structure group G is taken to be O(k), k being the dimension
of the fibre. The Lie algebra
(k) is a vector space of skew symmetric matrices
and the connection one-form ω satisfies
ω
α
β
=−ω
β
α
. (10.75)
Theorem 10.6. Let E be a vector bundle with inner product g and let ∇ be the
covariant derivative associated with the orthonormal frame. Then ∇ is a metric
connection.
Proof.Sinceg is bilinear, it suffices to show that
d[g(s, s
)]=g(∇s, s
) + g(s, ∇s
)
for s = f ˆe
α
and s
= f
ˆe
β
where f, f
∈ (M). In fact, the LHS is
d[g( f ˆe
α
, f
ˆe
β
)]=d[ ff
δ
αβ
]=d ( ff
)δ
αβ
while the RHS is
g(∇ f ˆe
α
, f
ˆe
β
) + g( f ˆe
α
, ∇ f
ˆe
β
)
= g(d f ˆe
α
+ f ˆe
γ
ω
γ
α
, f
ˆe
β
) + g( f ˆe
α
, d f
ˆe
β
+ f
ˆe
γ
ω
γ
β
)
= d ff
δ
αβ
+ ff
ω
γ
α
δ
γβ
+ f d f
δ
αβ
+ ff
ω
γ
β
δ
αγ
= d( ff
)δ
αβ
where (10.75) has been used to obtain the final equality.
10.4.5 Holomorphic vector bundles and Hermitian inner products
Definition 10.6. Let E and M be complex manifolds and π : E → M a
holomorphic surjection. The manifold E is a holomorphic vector bundle if the
following axioms are fulfilled.
(i) The typical fibre is
k
and the structure group is GL(k, ).
(ii) The local trivialization φ
i
: U
i
×
k
→ π
−1
(U
i
) is a biholomorphism.
(iii) The transition function t
ij
: U
i
∩U
j
→ G = GL(k, ) is a holomorphic
map.
For example, let M be a complex manifold with dim
M = m.The
holomorphic tangent bundle TM
+
≡
A
p∈M
T
p
M
+
is a holomorphic vector
bundle. The typical fibre is
m
and the local basis is {∂/∂z
µ
}.
Let h be an inner product on a holomorphic vector bundle whose action at
p ∈ M is h
p
: π
−1
( p) × π
−1
( p) → . The most natural inner product is a
Hermitian structure which satisfies:
(i) h
p
(u, av + bw) = ah
p
(u,v)+ bh
p
(u,w),foru,v,w ∈ π
−1
( p), a, b ∈ ;
(ii) h
p
(u,v) = h
p
(v, u), u,v ∈ π
−1
( p);
(iii) h
p
(u, u) ≥ 0, h
p
(u, u) = 0 if and only if u = φ
i
( p, 0);and
(iv) h(s
1
, s
2
) ∈ (M) for s
1
, s
2
∈ (M, E).
A set of sections {ˆe
1
,...,ˆe
k
} is a unitary frame if
h(ˆe
i
, ˆe
j
) = δ
ij
. (10.76)
The unitary frame bundle LM is not a holomorphic vector bundle since the
structure group U(m) is not a complex manifold.
Given a Hermitian structure h, we define a connection which is compatible
with h.TheHermitian connection ∇ is a linear map (M, E) → (M, E ⊗
T
∗
M ) which satisfies:
(i) ∇( fs) = (d f )s + f ∇s, f ∈
(M) , s ∈ (M, E);
(ii) d [h(s
1
, s
2
)]=h(∇s
1
, s
2
) + h(s
1
, ∇s
2
);and
(iii) according to the destination, we separate the action of ∇ as ∇s = Ds +
¯
Ds,
Ds (
¯
Ds) being a (1, 0)-form ((0, 1)-form) valued section. We demand that
¯
D =
¯
∂.
It can be shown that given E and a Hermitian metric h, there exists a unique
Hermitian connection ∇. The curvature is defined from the Hermitian connection.
Let {ˆe
1
,...,ˆe
k
} be a unitary frame and define the local connection form
β
α
by
∇ˆe
α
=ˆe
β
β
α
. (10.77)
The field strength is defined by
≡ d + ∧ . (10.78)
We verify that
∇∇ˆe
α
=∇(ˆe
β
β
α
) =ˆe
β
β
α
. (10.79)
We prove that both
and are skew Hermitian:
¯
β
α
+
α
β
= h(∇ˆe
α
, ˆe
β
) + h(ˆe
α
, ∇ˆe
β
) = dh ( ˆe
α
, ˆe
β
) = dδ
αβ
= 0
β
α
+
¯
α
β
= d
β
α
+
β
γ
∧
γ
α
+ d
¯
α
β
+
¯
α
γ
∧
¯
γ
α
= d (
β
α
−
β
α
) +
β
γ
∧
γ
α
+
γ
α
∧
α
γ
= 0.
Thus, we have shown that
α
β
=−
¯
β
α
β
α
=−
¯
α
β
. (10.80)
Next we show that
is a (1, 1)-form. Let {ˆe
α
} be a unitary frame. cannot
have a component of bidegree-(0, 2) since
ˆe
β
β
α
=∇∇ˆe
α
= (D +
¯
∂)(D +
¯
∂)ˆe
α
= DDˆe
α
+ (D
¯
∂ +
¯
∂D) ˆe
α
.
It follows from
β
α
=−
¯
α
β
that
¯
has no component of bidegree-(0, 2), and,
hence,
has no component of bidegree-(2, 0) either. Thus
β
α
is a two-form of
bidegree-(1, 1).
10.5 Gauge theories
As we have remarked several times, a gauge potential can be regarded as a local
expression for a connection in a principal bundle. The Yang–Mills field strength is
then identified with the local form of the curvature associated with the connection.
We summarize here the relevant aspects of gauge theories from the geometrical
viewpoint.
10.5.1 U(1) gauge theory
Maxwell’s theory of electromagnetism is described by the U(1) gauge group. U(1)
is Abelian and one dimensional, hence we omit all the group indices α,β,...
and put the structure constants f
αβ
γ
= 0. Suppose the base space M is a four-
dimensional Minkowski spacetime. From corollary 9.1, we find that the U(1)
bundle P is trivial, namely P =
4
× U(1) and a single local trivialization over
M is required. The gauge potential is simply
=
µ
dx
µ
. (10.81)
Our gauge potential
differs from the usual vector potential A by the Lie algebra
factor i:
µ
= iA
µ
. The field strength is
=d . (10.82a)
In components, we have
µν
= ∂
ν
/∂x
µ
− ∂
µ
/∂x
ν
.(10.82b)
satisfies the Bianchi identity,
d
= ∧ − ∧ = 0. (10.83a)
This should be expected from the outset since
is exact, = d ; and hence
closed, d
= d
2
= 0. In components, we have
∂
λ µν
+ ∂
ν λµ
+ ∂
µ νλ
= 0.(10.83b)
If we identify the components
µν
≡ iF
µν
with the electric field E and the
magnetic field B as
E
i
= F
i0
, B
i
=
1
2
ijk
F
jk
(i, j, k = 1, 2, 3) (10.84)
(10.83b) reduces to two of Maxwell’s equations,
∇×E +
∂ B
∂t
= 0 ∇·B = 0.(10.83c)
These equations are geometrical rather than dynamical. To find the dynamics, we
have to specify the action. The Maxwell action
M
[ ] is a functional of and
is given by
M
[ ]≡
1
4
4
µν
µν
d
4
x =−
1
4
4
F
µν
F
µν
d
4
x. (10.85a)
Exercise 10.12. (a) Let ∗
µν
≡
1
2
κλ
ε
κλµν
be the dual of
µν
. Show that
M
[ ]=−
1
4
4
∧∗ .(10.85b)
(b) Use (10.84) to show that
−
1
4
F
µν
F
µν
=
1
2
(E
2
− B
2
). (10.86)
Show also that
F
µν
∗ F
µν
= B · E. (10.87)
By the variation of
M
[ ] with respect to
µ
, we obtain the equation of
motion,
∂
µ
µν
= 0. (10.88a)
We find this equation is reduced to the second set of Maxwell’s equations (in the
vacuum):
∇·E = 0 ∇×B −
∂ E
∂t
= 0.(10.88b)
10.5.2 The Dirac magnetic monopole
We have studied Maxwell’s theory of electromagnetism defined on
4
.The
triviality of the base space makes the U(1) bundle trivial. Poincar´e’s lemma
ensures that the field strength
is globally exact: = d . It is interesting to
extend our analysis to U(1) bundles over a non-trivial base space. We assume
everything is independent of time for simplicity.
The Dirac monopole is defined in
3
with the origin O removed.
3
−{0}
and S
2
are of the same homotopy type and the relevant bundle is a U(1) bundle
P(S
2
, U(1)). S
2
is covered by two charts
U
N
≡{(θ, φ)|0 ≤ θ ≤
1
2
π + } U
S
≡{(θ, φ)|
1
2
π − ≤ θ ≤ π}
where θ and φ are polar coordinates. Let ω be an Ehresmann connection on P.
Take a local section σ
N
(σ
S
)onU
N
(U
S
) and define the local gauge potentials
N
= σ
∗
N
ω
S
= σ
∗
S
ω.
We take
N
and
S
to be of the Wu–Yang form (section 1.9),
N
= ig(1 − cos θ)dφ
S
=−ig(1 + cos θ)dφ (10.89)
where g is the strength of the monopole.
Let t
NS
be the transition function defined on the equator U
N
∩ U
S
. t
NS
defines a map from S
1
(equator) to U(1) (structure group), which is classified
by π
1
(U(1)) = , see example 9.7. Let us write
t
NS
(φ) = exp[iϕ(φ)] (ϕ : S
1
→ ). (10.90)
The gauge potentials
N
and
S
are related on U
N
∩U
S
by
N
= t
−1
NS
S
t
NS
+ t
−1
NS
dt
NS
=
S
+ idϕ. (10.91)
For the gauge potentials (10.89), we find
dϕ =−i(
N
−
S
) = 2g dφ.
While φ runs from 0 to 2π around the equator, ϕ(φ) takes the range
ϕ ≡
dϕ =
2π
0
2g dφ = 4πg. (10.92)
For t
NS
to be defined uniquely, ϕ must be a multiple of 2π,
ϕ/2π = 2g ∈
(10.93)
which is the quantization condition of the magnetic monopole. The integer 2g
represents the homotopy class to which this bundle belongs. This number is also
obtained by considering F
N
= dA
N
and F
S
= dA
S
(
N
= iF
N
etc). The total
flux is
=
S
2
B ·dS =
U
N
dA
N
+
U
S
dA
S
=
S
1
A
N
−
S
1
A
S
= 2g
2π
0
dφ = 4πg. (10.94)
Thus, the curvature, that is the pair of the field strengths dA
N
and dA
S
,
characterizes the twisting of the bundle. We discuss this further in chapter 11.