Abstract
From a mathematical and physical point of view it is very important that we can define on principal bundles certain fields, known as connection 1-forms. At least locally (after a choice of local gauge) we can interpret connection 1-forms as fields on spacetime (the base manifold) with values in the Lie algebra of the gauge group. These fields are often called gauge fields and correspond in the associated quantum field theory to gauge bosons.
In gauge theories, additional matter fields, like fermions or scalars, can be introduced using associated vector bundles. The crucial point is that connections (the gauge fields) define a covariant derivative on these associated vector bundles, leading to a coupling between gauge fields and matter fields (if the matter fields are charged, i.e. the vector bundles are associated to a non-trivial representation of the gauge group). In a gauge-invariant Lagrangian this results in terms of order higher than two in the matter and gauge fields, which are interpreted as interactions between the corresponding particles.
In non-abelian gauge theories, like quantum chromodynamics (QCD), there are also terms in the Lagrangian of order higher than two in the gauge fields themselves. This implies a direct interaction between gauge bosons (the gluons in QCD) that does not occur in abelian gauge theories like quantum electrodynamics (QED).
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We said in the introduction to Chap. 4 that principal bundles and associated vector bundles are the stage for gauge theories. From a mathematical and physical point of view it is very important that we can define on principal bundles certain fields, known as connection 1-forms. At least locally (after a choice of local gauge) we can interpret connection 1-forms as fields on spacetime (the base manifold) with values in the Lie algebra of the gauge group. These fields are often called gauge fields and correspond in the associated quantum field theory to gauge bosons. Every connection 1-form A defines a curvature 2-form F which can be identified with the field strength of the gauge field. Connection and curvature can be seen as generalizations of the classical potential A μ and field strength F μν in electromagnetism, which is a U(1)-gauge theory, to possibly non-abelian gauge groups.
Pure gauge theory, also known as Yang–Mills theory , involves only the gauge field A and its curvature F. Additional matter fields, like fermions or scalars, can be introduced using associated vector bundles. The crucial point is that connections (the gauge fields) define a covariant derivative on these associated vector bundles, leading to a coupling between gauge fields and matter fields (if the matter fields are charged, i.e. the vector bundles are associated to a non-trivial representation of the gauge group). In a gauge-invariant Lagrangian this results in terms of order higher than two in the matter and gauge fields, which are interpreted as interactions between the corresponding particles.
In non-abelian gauge theories, like quantum chromodynamics (QCD) , there are also terms in the Lagrangian of order higher than two in the gauge fields themselves, coming from a quadratic term in the curvature that appears in the Yang–Mills Lagrangian . This implies a direct interaction between gauge bosons (the gluons in QCD) that does not occur in abelian gauge theories like quantum electrodynamics (QED) . The difficulties that are still present nowadays in trying to understand the quantum version of non-abelian gauge theories, like quantum chromodynamics, can ultimately be traced back to this interaction between gauge bosons.
Although in this book we are mainly interested in applications of gauge theories to physics, gauge theories are also very influential in pure mathematics, for example, the Donaldson and Seiberg–Witten theories of 4-manifolds and Chern–Simons theory of 3-manifolds (see Exercise 5.15.16 for an introduction to the Chern–Simons action).
5.1 Distributions and Connections
Definition 5.1.1
A distribution on a manifold M is a vector subbundle of the tangent bundle TM.
This notion of distributions is not related to the concept of distributions in analysis. Connections on principal bundles P, sometimes also called Ehresmann connections, are defined as certain distributions on the total space of the principal bundle.
5.1.1 The Vertical Tangent Bundle
We first want to show that on the total space of every principal bundle there is a canonical vertical bundle. Let
be a principal G-bundle. For a point x ∈ M we have the fibre
over x, which is an embedded submanifold of P. Let p ∈ P x be a point in the fibre.
Definition 5.1.2
The vertical tangent space V p of the total space P in the point p is the tangent space T p (P x ) to the fibre.
Proposition 5.1.3 (Vertical Tangent Bundle)
For all p ∈ P the vertical tangent space has the following properties:
-
1.
V p = ker D p π.
-
2.
The map
$$\displaystyle\begin{array}{rcl} \phi _{{\ast}}: \mathfrak{g}& \longrightarrow & V _{p} {}\\ X& \longmapsto & \tilde{X}_{p}, {}\\ \end{array}$$where \(\tilde{X}\) is the fundamental vector field associated to \(X \in \mathfrak{g}\) , determined by the G-action on P, is a vector space isomorphism between \(\mathfrak{g}\) and V p .
-
3.
The set of all vertical tangent spaces V p for p ∈ P forms a smooth distribution on P, called the vertical tangent bundle V. Its rank is equal to the dimension of G. The distribution V is globally trivial as a vector bundle, with trivialization given by
$$\displaystyle\begin{array}{rcl} P \times \mathfrak{g}& \longrightarrow & V {}\\ (\,p,X)& \longmapsto & \tilde{X}_{p}. {}\\ \end{array}$$ -
4.
The vertical tangent bundle is right-invariant , i.e.
$$\displaystyle{ r_{g{\ast}}\left (V _{p}\right ) = V _{p\cdot g}\quad \forall g \in G. }$$
Proof
-
1.
We have
$$\displaystyle{ D_{p}\pi (Y ) = 0\quad \forall Y \in V _{p}, }$$because we can write Y as the tangent vector to a curve in P x , which maps under π to the constant point x ∈ M. Hence
$$\displaystyle{ V _{p} \subset \mathrm{ ker}\,D_{p}\pi. }$$Since π: P → M is a submersion, it follows from the Regular Value Theorem A.1.32 that
$$\displaystyle\begin{array}{rcl} \mathrm{dim}\,\mathrm{ker}\,D_{p}\pi & =& \mathrm{dim}\,P -\mathrm{ dim}\,M {}\\ & =& \mathrm{dim}\,G {}\\ & =& \mathrm{dim}\,P_{x}. {}\\ \end{array}$$This implies the claim.
-
2.
It is clear that this map has image in V p and considering dimensions it suffices to show that the map is injective. This follows from Proposition 3.4.3.
-
3.
This is clear by 2.
-
4.
This follows because according to Proposition 3.4.6 \(r_{g{\ast}}(\tilde{X}) =\tilde{ Y }\), with \(Y =\mathrm{ Ad}_{g^{-1}}X \in \mathfrak{g}\).
□
5.1.2 Ehresmann Connections
Let P → M be a principal G-bundle.
Definition 5.1.4
A horizontal tangent space in p ∈ P is a subspace H p of T p P complementary to the vertical tangent space V p , so that
Note that horizontal tangent spaces are not defined uniquely (if the dimensions of G and M are positive).
The following should be clear:
Proposition 5.1.5
Let H p be a horizontal tangent space at p ∈ P, π( p) = x. Then
is a vector space isomorphism.
Definition 5.1.6
Let H be a distribution on P consisting of horizontal tangent spaces. Then H is called an Ehresmann connection or a connection on P if it is right-invariant, i.e.
The distribution H is also called horizontal tangent bundle given by the connection.
Right-invariance of an Ehresmann connection means that along a fibre P x the horizontal subspaces are mutually “parallel” (with respect to right translation along the fibre). In particular, all H p along a fibre P x are determined by fixing a single \(H_{p_{0}}\) for some p 0 ∈ P x , since the G-action is transitive on the fibres of P. Right-invariance of a connection can also be seen as a symmetry property: The right action of the gauge group G on P induces a natural right action on TP and Ehresmann connections are invariant under this action.
Example 5.1.7 (Connections on the Trivial Bundle)
Let
be the trivial principal G-bundle. Then the vertical subspaces are given by
We can choose
It is clear that this defines a horizontal subspace complementary to the vertical subspace. Furthermore, the collection of all of these horizontal subspaces are right-invariant and hence define a connection on the trivial bundle, called the canonical flat connection .
It can also be shown that every non-trivial principal bundle has a connection (see Exercise 5.15.1 for a proof in the case of compact structure groups).
Notice that connections are not unique (if dimM, dimG ≥ 1), not even in the case of trivial principal bundles (all connections that appear in the Standard Model over Minkowski spacetime, for example, are defined on trivial principal bundles).
5.2 Connection 1-Forms
In this section we study an equivalent description of connections using differential forms.
5.2.1 Basic Definitions
Recall that we defined in Sect. 3.5.1 the notion of differential forms on a manifold with values in a vector space. We now need this notion to define so-called connection 1-forms.
Definition 5.2.1
A connection 1-form or connection on a principal G-bundle π: P → M is a 1-form \(A \in \varOmega ^{1}(P,\mathfrak{g})\) on the total space P with the following properties:
-
1.
\(r_{g}^{{\ast}}A =\mathrm{ Ad}_{g^{-1}} \circ A\) for all g ∈ G.
-
2.
\(A\left (\tilde{X}\right ) = X\) for all \(X \in \mathfrak{g}\), where \(\tilde{X}\) is the fundamental vector field associated to X.
A connection 1-form is also called a gauge field on P.
At a point p ∈ P, a connection 1-form is thus a linear map
Recall that \(\mathrm{Ad}_{g^{-1}}\) is a linear isomorphism of \(\mathfrak{g}\) onto itself. This shows that the composition \(\mathrm{Ad}_{g^{-1}} \circ A\) is well-defined and again an element of \(\varOmega ^{1}(P,\mathfrak{g})\).
We want to show that the notion of connection 1-forms is completely equivalent to the notion of Ehresmann connections on a principal bundle as defined in Sect. 5.1.2.
Theorem 5.2.2 (Correspondence Between Ehresmann Connections and Connection 1-Forms)
There is a bijective correspondence between Ehresmann connections on a principal G-bundle π: P → M and connection 1-forms:
-
1.
Let H be an Ehresmann connection on P. Then
$$\displaystyle{ A_{p}\left (\tilde{X}_{p} + Y _{p}\right ) = X, }$$for p ∈ P, \(X \in \mathfrak{g}\) and Y p ∈ H p , defines a connection 1-form A on P.
-
2.
Let \(A \in \varOmega ^{1}(P,\mathfrak{g})\) be a connection 1-form on P. Then
$$\displaystyle{ H_{p} =\mathrm{ ker}\,A_{p} }$$defines an Ehresmann connection H on P.
Proof
-
1.
We have to verify the conditions defining a connection 1-form. It is clear that
$$\displaystyle{ A\left (\tilde{X}\right ) = X\quad \forall X \in \mathfrak{g}. }$$We want to calculate r g ∗ A. We have shown in Proposition 3.4.6 that \(r_{g{\ast}}\left (\tilde{X}\right ) =\tilde{ Z}\), where \(Z =\mathrm{ Ad}_{g^{-1}}X\). Note that r g∗ Y p is horizontal if Y p is horizontal by the definition of Ehresmann connections. Therefore
$$\displaystyle\begin{array}{rcl} (r_{g}^{{\ast}}A)_{ p}\left (\tilde{X}_{p} + Y _{p}\right )& =& A_{p\cdot g}\left (\tilde{Z}_{p\cdot g} + r_{g{\ast}}Y _{p}\right ) {}\\ & =& Z {}\\ & =& \mathrm{Ad}_{g^{-1}} \circ A_{p}\left (\tilde{X}_{p} + Y _{p}\right ). {}\\ \end{array}$$This implies the claim.
-
2.
We have to verify that H is a horizontal right-invariant distribution on P. We first show that H is a subbundle of TP: using a basis {T a } for the Lie algebra \(\mathfrak{g}\) we can write A = ∑ a A a T a , where A a ∈ Ω 1(P) are real-valued 1-forms. Since \(A(\tilde{X}) = X\) for all \(X \in \mathfrak{g}\), it follows that \(A_{a}(\tilde{T}_{b}) =\delta _{ab}\). In particular, the 1-forms A a are linearly independent in each point p ∈ P. Let g be a Riemannian metric on P and Z a the vector fields g-dual to the 1-forms A a . The {Z a } are linearly independent and span a subbundle ζ of TP of rank \(\dim \mathfrak{g}\). It follows that H is the g-orthogonal complement of ζ in TP and hence a distribution.
To verify that H is horizontal, we first show that H p ∩ V p = {0}: Let Y ∈ ker A p ∩ V p . Then Y is equal to a fundamental vector, hence \(Y =\tilde{ X}_{p}\) for some \(X \in \mathfrak{g}\). But then
$$\displaystyle{ 0 = A_{p}(Y ) = X, }$$hence Y = 0.
Furthermore, the 1-form A p is surjective onto \(\mathfrak{g}\), hence
$$\displaystyle{ \mathrm{dim}\,\mathrm{ker}\,A_{p} =\mathrm{ dim}\,T_{p}P -\mathrm{ dim}\,\mathfrak{g} =\mathrm{ dim}\,T_{p}P -\mathrm{ dim}\,V _{p}. }$$Thus T p P = ker A p ⊕ V p and H p is horizontal. To check that H is right-invariant, let Y ∈ H p . Then
$$\displaystyle\begin{array}{rcl} A_{p\cdot g}(r_{g{\ast}}Y )& =& (r_{g}^{{\ast}}A)_{ p}(Y ) {}\\ & =& \mathrm{Ad}_{g^{-1}}(A_{p}(Y )) {}\\ & =& 0. {}\\ \end{array}$$This shows r g∗ Y ∈ H p⋅ g and hence the claim.
□
Example 5.2.3
Let G be a Lie group and H ⊂ G a closed subgroup. By Theorem 4.2.15
is an H-principal bundle. Suppose there exists a vector subspace \(\mathfrak{m} \subset \mathfrak{g}\) such that
The homogeneous space is then called reductive . In this situation we can define a canonical connection 1-form A on the bundle G → G∕H. See Exercise 5.15.6 for details.
We describe another explicit example of a connection 1-form in the following subsection.
5.2.2∗A Connection 1-Form on the Hopf Bundle S 3 → S 2
In this subsection we follow reference [14, Example 3.3]. We consider the Hopf bundle
and think of S 1 as the unit circle in \(\mathbb{C}\) with Lie algebra \(i\mathbb{R}\). If \(Y = iy \in i\mathbb{R}\), then exp(Y ) = e iy ∈ S 1. We also think of S 3 as the unit sphere in \(\mathbb{C}^{2}\) with tangent spaces
We define 1-forms \(\alpha _{j},\bar{\alpha }_{j} \in \varOmega ^{1}(S^{3}, \mathbb{C})\) by
Proposition 5.2.4 (Connection 1-Form on the Hopf Bundle)
The 1-form A on S 3 , given by
has values in \(i\mathbb{R}\) and is a connection 1-form for the Hopf bundle.
Proof
It is clear that A has values in \(i\mathbb{R}\), since \(\bar{A} = -A\). We check the defining properties of connection 1-forms. Since S 1 is abelian, we have \(\mathrm{Ad}_{g^{-1}} =\mathrm{ Id}\) for all g ∈ S 1. We therefore have to show that
We fix a tangent vector \(X \in T_{(z_{0},z_{1})}S^{3}\), given by the velocity vector
of a suitable curve in S 3. Then
Hence
where we used that \(g\bar{g} = 1\).
We also have to show that
for all Y in the Lie algebra of S 1. Let Y = iy, with \(y \in \mathbb{R}\). Then the associated fundamental vector field is given by
This implies
since (z 0, z 1) ∈ S 3. □
See Exercise 7.9.9 for a generalization of this construction to the Hopf bundle S 7 → S 4 with structure group SU(2).
5.3 Gauge Transformations
Let π: P → M be a principal G-bundle.
Definition 5.3.1
A (global) gauge transformation is a bundle automorphism of P, i.e. a diffeomorphism f: P → P which preserves the fibres of P and is G-equivariant:
-
1.
π ∘ f = π.
-
2.
f( p ⋅ g) = f( p) ⋅ g for all p ∈ P and g ∈ G.
Under composition of diffeomorphisms the set of all gauge transformations forms a group that we denote by \(\mathbb{G}(P)\) or Aut(P). A local gauge transformation is a bundle automorphism on the principal G-bundle π: P U → U, where U ⊂ M is an open set.
We sometimes prefer to call gauge transformations in this sense bundle automorphisms and leave the name gauge transformations to physical gauge transformations that we introduce later. Notice that whether a bundle automorphism f is global or local is not related to the question of whether f is constant or non-constant in some sense. We will later call gauge transformations rigid if they are constant in a specific way.
Depending on the context one sometimes calls G or \(\mathbb{G}(P) =\mathrm{ Aut}(P)\) the gauge group of the principal bundle P. The group \(\mathbb{G}(P)\) is infinite-dimensional if both the dimensions of M and G are at least 1.
The group Aut(P) of bundle automorphisms is one of the places in differential geometry where an infinite-dimensional group appears naturally. Gauge theories, which are field theories invariant under all gauge transformations, in this regard have the huge symmetry group Aut(P). The diffeomorphism group Diff(M) of spacetime M plays a comparable role in general relativity .
5.3.1 Bundle Automorphisms as G-Valued Maps on P
We would like to give another, equivalent description of bundle automorphisms (we follow [14, Sect. 3.5]).
Definition 5.3.2
We denote by C ∞(P, G)G the following set of maps from P to G:
where \(c_{g^{-1}}\) is conjugation by g −1. This set is a group under pointwise multiplication:
The neutral element is given by the constant map on P with value e ∈ G.
Proposition 5.3.3 (Correspondence Between Bundle Automorphisms and G-Valued Maps)
The map
with σ f defined by
is a well-defined group isomorphism. We can therefore identify the group of bundle automorphisms \(\mathbb{G}(P)\) with \(\mathbb{C}^{\infty }(P,G)^{G}\).
Proof
Since f( p) is in the same fibre as p, there exists a unique g ∈ G such that f( p) = p ⋅ g. This g we call σ f ( p).
We first have to show that σ f is an element of \(\mathbb{C}^{\infty }(P,G)^{G}\): It is not difficult to show that σ f is a smooth map from P to G. We have
This implies that
and thus \(\sigma _{f} \in \mathbb{ C}^{\infty }(P,G)^{G}\).
The inverse of the map above is given by
with f σ defined by
We only have to show that f σ is a bundle automorphism. It is clear that f σ ( p) is in the same fibre as p, hence f σ is a bundle map. It is easy to check that f σ is G-equivariant and that \(f_{\sigma }^{-1} = f_{\sigma ^{-1}}\), hence f σ is a diffeomorphism. Thus \(f_{\sigma } \in \mathbb{ G}(P)\).
Finally, we can check that σ f′∘f = σ f′ ⋅ σ f , hence the map defines a group isomorphism between \(\mathbb{G}(P)\) and \(\mathbb{C}^{\infty }(P,G)^{G}\). □
In the special case when the structure group G is abelian we have a simpler description.
Proposition 5.3.4 (Bundle Automorphisms for Abelian Structure Groups)
If the Lie group G is abelian, then there is a group isomorphism
where \(\mathbb{C}^{\infty }(M,G)\) denotes the set of smooth maps from M to G (a group under pointwise multiplication) and σ τ is defined by
where π: P → M is the projection.
Proof
This is Exercise 5.15.2. □
Corollary 5.3.5
For principal T n -bundles P → M there is an isomorphism of the group of bundle automorphisms \(\mathbb{G}(P)\) with the group of smooth maps from M to T n.
5.3.2 Physical Gauge Transformations
In physics, gauge transformations are often defined as maps on the base manifold M to the structure group G, even for non-abelian Lie groups G. We discuss the relation of this notion to our definition of gauge transformations as bundle automorphisms.
Definition 5.3.6
Let π: P → M be a principal G-bundle. A physical gauge transformation is a smooth map τ: U → G, defined on an open subset U ⊂ M. The set of all physical gauge transformations on U forms a group \(\mathbb{C}^{\infty }(U,G)\) with pointwise multiplication. A rigid physical gauge transformation is a constant map τ: U → G. The rigid physical gauge transformations form a group isomorphic to G.
Proposition 5.3.7 (Physical Gauge Transformations and Bundle Automorphisms)
Let s: U → P be a local section. Then s defines a group isomorphism
The inverse of this map is given by
where
Proof
The proof is left as an exercise. □
The upshot is that after a choice of local gauge s on U we can identify local bundle automorphisms on the principal G-bundle P U → U with physical gauge transformations on U.
5.3.3 The Action of Bundle Automorphisms on Associated Vector Bundles
Bundle automorphisms on a principal bundle have the important property that they act on every associated vector bundle. Let π P : P → M be a principal G-bundle and π E : E = P × ρ V → M an associated vector bundle.
Theorem 5.3.8 (Action of Bundle Automorphisms on Associated Bundles)
The group of bundle automorphisms of the principal bundle acts on the associated vector bundle through bundle isomorphisms via
Proof
We only have to show that the action is well-defined: If \(\left [\,p',v'\right ] = [\,p,v]\), then p′ = p ⋅ g and v′ = ρ(g)−1 v for some g ∈ G, so that
□
We can also describe this action in the language of physics:
Theorem 5.3.9 (Action of Physical Gauge Transformations on Associated Bundles)
Let s: U → P be a local gauge and Φ: U → E a local section. We write the section with respect to the local gauge as
where ϕ: U → V is a smooth map. Suppose f is a local bundle automorphism of P over U and τ f : U → G the associated physical gauge transformation. Then
Proof
This is a simple calculation. □
As a consequence the action of a local bundle automorphism on a local section Φ of E is given by the action of the physical gauge transformation on the vector-valued map ϕ. In physics one writes the action of a physical gauge transformations τ: U → G on a field ϕ: U → V as
The more general notion of bundle automorphism above has the advantage that it also works for non-trivial principal bundles and associated vector bundles, independent of the choice of (local) gauge.
Remark 5.3.10
There is a simple, but profound, difference between gauge theories and general relativity (Edward Witten [150] attributes this insight to Bryce DeWitt). In gauge theories the group of symmetries, the gauge group \(\mathbb{G}(P)\), acts through bundle automorphisms, i.e. it preserves all points on the base manifold M. This is related to the fact that gauge theories describe local interactions (the interactions occur in single spacetime points). In general relativity, however, the group of symmetries, the diffeomorphism group Diff(M) , acts by moving points around in M. If the diffeomorphism invariance holds in quantum gravity on the level of Green’s functions (correlators) , then they must be constant, in striking contrast to the behaviour of Green’s functions in Poincaré invariant quantum field theories .
It is nowadays thought that gravity cannot be described by a local quantum field theory of point particles and that a theory of quantum gravity must be fundamentally non-local. This leads to alternatives such as string theory , where the graviton and other particles are no longer 0-dimensional point particles, but 1-dimensional strings.
5.4 Local Connection 1-Forms and Gauge Transformations
Let π: P → M be a principal G-bundle and \(A \in \varOmega ^{1}(P,\mathfrak{g})\) a connection 1-form. It is very useful to consider the following notion.
Definition 5.4.1
Let s: U → P be a local gauge of the principal bundle on an open subset U ⊂ M. Then we define the local connection 1-form (or local gauge field ) \(A_{s} \in \varOmega ^{1}(U,\mathfrak{g})\), determined by s, by
The local connection 1-form is thus defined on an open subset in the base manifold M and can be considered as a “field on spacetime” in the usual sense. If we have a manifold chart on U and {∂ μ } μ = 1, …, n are the local coordinate basis vector fields on U, we set
We can also choose in addition a basis {e a } of the Lie algebra \(\mathfrak{g}\) and then expand
The real-valued fields \(A_{\mu }^{a} \in \mathbb{ C}^{\infty }(U, \mathbb{R})\) and the corresponding real-valued 1-forms A s a ∈ Ω 1(U) are called (local) gauge boson fields .
A principal bundle can have many local gauges and it is interesting to determine how the local connection 1-forms transform as we change the local gauge. Let s i : U i → P and s j : U j → P be local gauges with U i ∩ U j ≠ ∅. Recall from the proof of Proposition 4.7.11 that
where
is the smooth transition function between the associated local trivializations. We can consider g ji as a physical gauge transformation between the local gauges s i and s j .
We have local connection 1-forms
We want to calculate the relation between A i and A j . Recall that the Maurer–Cartan form \(\mu _{G} \in \varOmega ^{1}(G,\mathfrak{g})\) was defined as
for v ∈ T g G. We set
Then we have:
Theorem 5.4.2 (Transformation of Local Gauge Fields Under Changes of Gauge)
With the notation above, the local connection 1-forms transform as
on U i ∩ U j . If \(G \subset \mathrm{ GL}(n, \mathbb{K})\) is a matrix Lie group, then
where ⋅ denotes matrix multiplication, g ji −1 denotes the inverse in G and dg ji is the differential of each component of the function \(g_{ji}: U_{i} \cap U_{j} \rightarrow G \subset \mathbb{K}^{n\times n}\) . In particular, if G is abelian, then
Proof
Let x ∈ U i ∩ U j and Z ∈ T x M. We set
With the group action
we calculate by Proposition 3.5.4 and the chain rule
Therefore, by the defining properties of a connection 1-form A,
To prove the second claim recall from Proposition 2.1.48 that for a matrix Lie group
for all g ∈ G and \(a \in \mathfrak{g}\), and μ G (v) = g −1 ⋅ v for v ∈ T g G, hence
□
Remark 5.4.3
In physics one considers connection 1-forms usually only in the local sense as \(\mathfrak{g}\)-valued 1-forms A i on open subset U i of M together with the transformation rule given by Theorem 5.4.2. The mathematical concept of connections on principal bundles clarifies the invariant geometric object behind this transformation principle.
A very similar argument implies the following global statement:
Theorem 5.4.4 (Transformation of Connections Under Bundle Automorphisms)
Let P → M be a principal bundle and \(A \in \varOmega ^{1}(P,\mathfrak{g})\) a connection 1-form on P. Suppose that \(f \in \mathbb{ G}(P)\) is a global bundle automorphism. Then f ∗ A is a connection 1-form on P and
Proof
This is Exercise 5.15.3. □
Theorem 5.4.4 corresponds to the “active” point of view for gauge transformations (symmetries are related to the behaviour under certain bundle automorphisms), while Theorem 5.4.2 corresponds to the “passive” point of view (symmetries are implicit in the behaviour under coordinate transformations).
5.5 Curvature
5.5.1 Curvature 2-Forms
Let π: P → M be a principal G-bundle and \(A \in \varOmega ^{1}(P,\mathfrak{g})\) a connection 1-form on P. Let H be the associated horizontal vector bundle, defined as the kernel of A. We have
and set
for the projection onto the horizontal vector bundle.
Definition 5.5.1
The 2-form \(F \in \varOmega ^{2}(P,\mathfrak{g})\), defined by
is called the curvature 2-form or curvature of the connection A. We sometimes write F A to emphasize the dependence on A.
Here are some simple properties of the curvature.
Proposition 5.5.2
The following identities hold:
-
1.
\(r_{g}^{{\ast}}F =\mathrm{ Ad}_{g^{-1}} \circ F\) for all g ∈ G.
-
2.
\(\tilde{X}\lrcorner F = 0\) for all \(X \in \mathfrak{g}\) , where \(\tilde{X}\lrcorner\) denotes insertion of the vector field \(\tilde{X}\).
Proof
-
1.
Note that
$$\displaystyle\begin{array}{rcl} r_{g{\ast}}H_{p}& =& H_{p\cdot g}, {}\\ r_{g{\ast}}V _{p}& =& V _{p\cdot g}. {}\\ \end{array}$$Hence
$$\displaystyle{ \pi _{H} \circ r_{g{\ast}} = r_{g{\ast}}\circ \pi _{H} }$$on T p P, since both sides evaluated on X = X h + X v ∈ T p P, where X h is horizontal and X v is vertical, are equal to r g∗(X h ). We now calculate for vectors X, Y ∈ T p P:
$$\displaystyle\begin{array}{rcl} (r_{g}^{{\ast}}F)_{ p}(X,Y )& =& dA(\pi _{H} \circ r_{g{\ast}}(X),\pi _{H} \circ r_{g{\ast}}(Y )) {}\\ & =& (r_{g}^{{\ast}}dA)(\pi _{ H}(X),\pi _{H}(Y )) {}\\ & =& d\left (\mathrm{Ad}_{g^{-1}} \circ A\right )(\pi _{H}(X),\pi _{H}(Y )) {}\\ & =& \mathrm{Ad}_{g^{-1}} \circ F(X,Y ). {}\\ \end{array}$$ -
2.
This is clear, because \(\pi _{H}(\tilde{X}) = 0\).
□
5.5.2 The Structure Equation
Definition 5.5.3
Let P be a manifold and \(\mathfrak{g}\) a Lie algebra. For \(\eta \in \varOmega ^{k}(P,\mathfrak{g})\) and \(\phi \in \varOmega ^{l}(P,\mathfrak{g})\) we define \([\eta,\phi ] \in \varOmega ^{k+l}(P,\mathfrak{g})\) by
where the commutators on the right are the commutators in the Lie algebra \(\mathfrak{g}\). In the literature one also finds the notation η ∧ϕ or [η ∧ϕ] for [η, ϕ].
If we expand in a vector space basis {T a } for the Lie algebra \(\mathfrak{g}\)
with η a, ϕ a standard real-valued k- and l-forms, then the definition is equivalent to
Most of the time we need the definition only for 1-forms \(\eta,\phi \in \varOmega ^{1}(P,\mathfrak{g})\), where we have
and
We can now state the following important formula for the curvature 2-form.
Theorem 5.5.4 (Structure Equation)
The curvature form F of a connection form A satisfies
We need the following lemma:
Lemma 5.5.5
Let \(X =\tilde{ V }\) be a fundamental vector field and Y a horizontal vector field on P. Then the commutator [X, Y ] is horizontal.
Proof
The flow of X is given by ϕ t = r exp(tV ). This implies by Theorem A.1.46
since Y p⋅ exp(tV ) ∈ H p⋅ exp(tV ) and ϕ −t∗ preserves the horizontal tangent bundle. □
We can now prove Theorem 5.5.4.
Proof
We check the formula by inserting X, Y ∈ T p P on both sides of the equation, where we distinguish the following three cases:
-
1.
Both X and Y are vertical: Then X and Y are fundamental vectors,
$$\displaystyle\begin{array}{rcl} X& =& \tilde{V }_{p}, {}\\ Y & =& \tilde{W}_{p}, {}\\ \end{array}$$for certain elements \(V,W \in \mathfrak{g}\). We get
$$\displaystyle{ F(X,Y ) = dA(\pi _{H}(X),\pi _{H}(Y )) = 0. }$$On the other hand we have
$$\displaystyle{ \frac{1} {2}[A,A](X,Y ) = [A(X),A(Y )] = [V,W]. }$$The differential dA of a 1-form A is given according to Proposition A.2.22 by
$$\displaystyle{ dA(X,Y ) = L_{X}(A(Y )) - L_{Y }(A(X)) - A([X,Y ]), }$$where we extend the vectors X and Y to vector fields in a neighbourhood of p. If we choose the extension by the fundamental vector fields \(\tilde{V }\) and \(\tilde{W}\), then
$$\displaystyle\begin{array}{rcl} dA(X,Y )& =& L_{X}(W) - L_{Y }(V ) - [V,W] {}\\ & =& -[V,W], {}\\ \end{array}$$since V and W are constant maps from P to \(\mathfrak{g}\) and we used that \([\tilde{V },\tilde{W}] =\widetilde{ [V,W]}\) according to Proposition 3.4.4. This implies the claim.
-
2.
Both X and Y are horizontal: Then
$$\displaystyle{ F(X,Y ) = dA(X,Y ) }$$and
$$\displaystyle{ \frac{1} {2}[A,A](X,Y ) = [A(X),A(Y )] = [0,0] = 0. }$$This implies the claim.
-
3.
X is vertical and Y is horizontal: Then \(X =\tilde{ V }_{p}\) for some \(V \in \mathfrak{g}\). We have
$$\displaystyle{ F(X,Y ) = dA(\pi _{H}(X),\pi _{H}(Y )) = dA(0,Y ) = 0 }$$and
$$\displaystyle{ \frac{1} {2}[A,A](X,Y ) = [A(X),A(Y )] = [V,0] = 0. }$$Furthermore,
$$\displaystyle\begin{array}{rcl} dA(X,Y )& =& L_{\tilde{V }}(A(Y )) - L_{Y }(V ) - A([\tilde{V },Y ]) {}\\ & =& -A([\tilde{V },Y ]) {}\\ & =& 0 {}\\ \end{array}$$since \([\tilde{V },Y ]\) is horizontal by Lemma 5.5.5. This implies the claim.
□
The structure equation is very useful when we want to calculate the curvature of a given connection.
5.5.3 The Bianchi Identity
Let F be the curvature 2-form of a connection A. Then dF is a 3-form on P with values in the Lie algebra \(\mathfrak{g}\). We want to consider the situation where we insert in all three arguments of dF a vector in the horizontal subbundle H defined by A.
Theorem 5.5.6 (Bianchi Identity (First Form))
The differential dF of the curvature 2-form vanishes on H × H × H.
Proof
We use the following formula for the differential of a 2-form η on P, see Proposition A.2.22:
for all vector fields X, Y, Z on P. By the structure equation we have \(F = dA + \frac{1} {2}[A,A]\) so that
We set \(\eta = \frac{1} {2}[A,A]\). Then
for all X, Y, Z ∈ T p P. We have \(V \lrcorner \eta \equiv 0\) if V is a horizontal vector field, since
for an arbitrary vector field W on P. This implies the claim, because we can assume that X, Y, Z are horizontal in the neighbourhood of p ∈ P. □
5.6 Local Curvature 2-Forms
Let A be a connection 1-form on the principal bundle P and s: U → P a local section (local gauge), defined on an open subset U ⊂ M. We then defined the local connection 1-form (or local gauge field) \(A_{s} \in \varOmega ^{1}(U,\mathfrak{g})\) by
Similarly we define:
Definition 5.6.1
The local curvature 2-form (or local field strength ) \(F_{s} \in \varOmega ^{2}(U,\mathfrak{g})\), determined by s, is defined by
If we have a manifold chart on U and {∂ μ } are local coordinate basis vector fields on U, we set
If we choose in addition a basis {e a } of the Lie algebra \(\mathfrak{g}\), we can expand the local field strength as
Proposition 5.6.2 (Local Structure Equation)
The local field strength can be calculated as
and
If the structure group G is abelian, then F s = dA s and
Proof
We calculate
Here we used that
which is easy to verify. This implies the first formula. The second formula follows from
Here we used that [∂ μ , ∂ ν ] = 0, because the basis vector fields {∂ μ } come from a chart on U. □
In physics, the quadratic term [A μ , A ν ] in the expression for F μν (leading to cubic and quartic terms in the Yang–Mills Lagrangian , see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field A μ . The quadratic term in the curvature is only present if the gauge group G is non-abelian , like G = SU(3) in quantum chromodynamics (QCD) , but not if G is abelian , like G = U(1) in quantum electrodynamics (QED) .
This explains why gluons , the gauge bosons of QCD, interact directly with each other, while photons , the gauge bosons of QED, do not. It is also the reason for phenomena in QCD such as colour confinement (at low energies) and asymptotic freedom (at high energies).
We would like to determine how the local field strength transforms under local gauge transformations . Let s i : U i → P and s j : U j → P be local gauges with U i ∩ U j ≠ ∅ and associated local curvature 2-forms F i , F j . The local gauge transformation
is defined by
We then have:
Theorem 5.6.3
The local curvature 2-forms transform as
on U i ∩ U j . If G is a matrix Lie group, then
Proof
Recall from the proof of Theorem 5.4.2 that
for a vector V ∈ T x M. Hence we get for V, W ∈ T x M, since F vanishes if a vertical vector is inserted:
□
Corollary 5.6.4
If G is abelian , then F s is independent of the choice of local gauge and hence determines a well-defined, global, closed 2-form \(F_{M} \in \varOmega ^{2}(M,\mathfrak{g})\).
Proof
In the abelian case the curvature defines a global form on M by Theorem 5.6.3. It remains to check that F M is closed. In a local gauge s we have according to the local structure equation
Since G is abelian, we have [A s , A s ] = 0. We conclude that
□
Remark 5.6.5
Note one important point about this corollary: Locally we have F s = dA s if G is abelian, hence F s is locally exact. However, the 2-form F M in general is not globally exact, because A s does not define a global 1-form on M (there is a change of A s under changes of local gauge even if the structure group is abelian, see Theorem 5.4.2).
5.6.1∗The Curvature 2-Form of the Connection on the Hopf Bundle S 3 → S 2
We consider the connection 1-form A on the Hopf bundle from Sect. 5.2.2 and continue to use the same notation (we follow [14, Example 3.10]).
Proposition 5.6.6
The curvature of the connection 1-form A on the Hopf bundle is given by
Proof
Since S 1 is abelian, we have [A, A] = 0, hence F A = dA. The claim follows once we have shown that
since then also \(d\alpha _{i} = 0 = d\bar{\alpha }_{i}\). This is clear from the definition of α i and \(\bar{\alpha }_{i}\): for example, if
with curves γ 0, γ 1, then
□
According to Corollary 5.6.4 the curvature F A determines a well-defined, global, closed 2-form \(F_{S^{2}}\) on S 2, where
for any local gauge s: U → S 3 on an open subset U ⊂ S 2. We want to determine the 2-form \(F_{S^{2}}\). Consider the open subset
together with the chart map
We consider a 2-form \(\tilde{F}\) on \(\mathbb{C}\), defined by
Proposition 5.6.7
The 2-form \(F_{S^{2}}\) is given on U 1 ⊂ S 2 by
Proof
It suffices to show that
because then
for all local gauges s: U 1 → S 3. To prove the formula note that
This implies for (z 0, z 1) ∈ S 3
We have \(\bar{z}_{0}z_{0} +\bar{ z}_{1}z_{1} = 1\), hence
This implies
Therefore
□
Proposition 5.6.8
For the connection on the Hopf bundle the following equation holds:
Proof
Since
(i.e. \(S^{2} = \mathbb{C}\mathbb{P}^{1}\) is the one-point compactification of \(\mathbb{C}\)) we can calculate
□
The form \(\psi _{1}^{{\ast}}\tilde{F}\) extends to a well-defined 2-form on all of \(\mathbb{C}\mathbb{P}^{1}\), which is equal to \(F_{S^{2}}\). The 2-form
is known as the Fubini–Study form of \(\mathbb{C}\mathbb{P}^{1}\). It is related to the standard volume form ω std on S 2 of area 4π by
Remark 5.6.9
We can define for any principal S 1-bundle P → M over a manifold M the first Chern class or Euler class as
This is a real cohomology class in H dR 2(M). It turns out that this class does not depend on the choice of connection 1-form on P (even though the 2-form F M does). In the case of the Hopf bundle we have
5.7∗Generalized Electric and Magnetic Fields on Minkowski Spacetime of Dimension 4
For the following notion from physics see, for example, [100]. Suppose π: P → M is a principal G-bundle and M is \(\mathbb{R}^{4}\) with Minkowski metric η of signature (+, −, −, −) (a similar construction works locally on any four-dimensional Lorentz manifold) . Let x 0, x 1, x 2, x 3 be global coordinates in an inertial frame with coordinate vector fields satisfying
We also write for the coordinates
Suppose A is a connection 1-form on P with curvature F. Let s: M → P be a global gauge and
as above. We write
and we have the local structure equation
The 4 × 4-matrix (F μν ) comes from a 2-form on M and is skew-symmetric.
Definition 5.7.1
The generalized electric and magnetic field , determined by the connection, the choice of gauge and the inertial frame, are the \(\mathfrak{g}\)-valued functions
defined by
Equivalently,
where ε ijk is totally antisymmetric with ε 123 = 1. We could expand the generalized electric and magnetic fields further in a basis for the Lie algebra \(\mathfrak{g}\).
For quantum electrodynamics (QED) with G = U(1) these are the standard real-valued electric and magnetic fields (after choosing a basis for \(\mathfrak{u}(1)\cong \mathbb{R}\)). In this situation the electric and magnetic fields do not depend on the choice of gauge according to Corollary 5.6.4, because G is abelian (the gauge field A s does depend on the choice of gauge).
For G = SU(n), in particular G = SU(3) corresponding to quantum chromodynamics (QCD) , these \(\mathfrak{g}\)-valued fields are also called chromo-electric and chromo-magnetic fields (or colour-electric and colour-magnetic fields). They describe the field strength of the gluon field corresponding to the connection 1-form A μ .
5.8 Parallel Transport
Connections define an important concept: parallel transport in principal and associated vector bundles. The notion of parallel transport also leads to the concept of covariant derivative on associated vector bundles.
Let π: P → M be a principal G-bundle with a connection A. We want to lift curves in M to horizontal curves in P, which are defined in the following way (by a curve we always mean in this and the following sections a smooth curve).
Definition 5.8.1
A curve γ ∗: I → P is called a horizontal lift of a curve γ: I → M, defined on an interval I, if:
-
1.
π ∘γ ∗ = γ
-
2.
the velocity vectors \(\dot{\gamma }^{{\ast}}(t)\) are horizontal, i.e. elements of \(H_{\gamma ^{{\ast}}(t)}\), for all t ∈ I.
The following theorem says that a horizontal lift of a curve in the base manifold always exists and is unique once the starting point has been given.
Theorem 5.8.2 (Existence and Uniqueness of Horizontal Lifts of Curves)
Let γ: [a, b] → M be a curve with γ(a) = x. Let p be a point in the fibre P x . Then there exists a unique horizontal lift γ p ∗ of γ with γ p ∗(a) = p.
Proof
Since P is locally trivial, there exists some lift δ of γ with δ(a) = p (one could also argue that the pullback of the bundle P under the map γ is trivial, because [a, b] is contractible). We want to find a map g: [a, b] → G such that
is horizontal. We will determine g(t) as the solution of a differential equation.
The curve γ ∗(t) will be horizontal if
We can calculate \(\dot{\gamma }^{{\ast}}(t)\) with Proposition 3.5.4:
Hence
We conclude that g(t) has to be the solution of the differential equation
with g(0) = e. This is the integral curve in the Lie group G through e of the time-dependent right-invariant vector field on G, corresponding to the Lie algebra element \(-A(\dot{\delta }(t)) \in \mathfrak{g}\). Such an integral curve on the interval [a, b] exists by Theorem 1.7.18. An explicit solution for g(t) in the case of a linear Lie group G can also be found in Proposition 5.10.4. □
Definition 5.8.3
Let γ: [a, b] → M be a curve in M. The map
is called parallel transport in the principal bundle P along γ with respect to the connection A. See Fig. 5.1.
Parallel transport
Theorem 5.8.4 (Properties of Parallel Transport)
Let P be a principal bundle with connection A.
-
1.
Parallel transport Π γ A is a smooth map between the fibres P γ(a) and P γ(b) and does not depend on the parametrization of the curve γ.
-
2.
Let γ be a curve in M from x to y and γ′ a curve from y to z. Denote the concatenation by γ ∗γ′, where γ comes first. Then
$$\displaystyle{ \varPi _{\gamma {\ast}\gamma '}^{A} =\varPi _{ \gamma '}^{A} \circ \varPi _{\gamma }^{A}. }$$ -
3.
If γ − denotes the curve γ traversed backwards, then
$$\displaystyle{ \varPi _{\gamma ^{-}}^{A} = \left (\varPi _{\gamma }^{A}\right )^{-1}. }$$In particular, parallel transport is a diffeomorphism between the fibres.
-
4.
Parallel transport is G-equivariant: The following identity holds:
$$\displaystyle{ \varPi _{\gamma }^{A} \circ r_{ g} = r_{g} \circ \varPi _{\gamma }^{A}\quad \forall g \in G. }$$
Proof
Properties 1–3 follow from the theory of ordinary differential equations. We only prove 4: let γ be a curve from x to y in M and p ∈ P x . Let γ p ∗ be the horizontal lift of γ to p. For g ∈ G consider the curve r g ∘γ p ∗. This curve starts at p ⋅ g and projects to γ. Furthermore, it is horizontal, because r g∗ maps horizontal vectors to horizontal vectors by the definition of connections. It follows that r g ∘γ p ∗ is equal to γ p⋅ g ∗. We get
□
Since parallel transport does not depend on the parametrization of the curve γ, we will often assume that γ is defined on the interval [0, 1].
5.9 The Covariant Derivative on Associated Vector Bundles
So far we have considered connections on principal bundles. Associated vector bundles play an important role in gauge theory, because matter fields are sections of such bundles. It turns out that connections on principal bundles define so-called covariant derivatives on all associated vector bundles (this will explain the third row in the diagram at the beginning of Sect. 4.7). These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories.
We first want to define the notion of parallel transport in associated vector bundles. Let P → M be a principal G-bundle with a connection A, ρ: G → GL(V ) a representation on a \(\mathbb{K}\)-vector space V (\(\mathbb{K} = \mathbb{R}, \mathbb{C}\)) and E = P × ρ V the associated vector bundle.
Theorem 5.9.1
For a curve γ: [0, 1] → M the map
is well-defined and a linear isomorphism. This map is called parallel transport in the associated vector bundle E along the curve γ with respect to the connection A.
Proof
We first show that Π γ E, A is well-defined, independent of the choice of representative [ p, v]. Suppose that
Then there exists an element g ∈ G such that
Part 4 of Theorem 5.8.4 then implies
Hence the map Π γ E, A is well-defined. It is then also clear that Π γ E, A is a linear isomorphism. □
Let Φ be a section of E, x ∈ M a point and X ∈ T x M a tangent vector. We want to define a covariant derivative as follows: choose an arbitrary curve γ: (−ε, ε) → M with
For each t ∈ (−ε, ε) parallel transport the vector Φ(γ(t)) ∈ E γ(t) back to E x along γ. Then take the derivative in t = 0 of the resulting curve in the fibre E x , giving an element in E x . More formally, we set
Here γ t denotes the restriction of the curve γ starting at time 0 and ending at time t, for t ∈ (−ε, ε).
We want to prove the following formula.
Theorem 5.9.2
Let s: U → P be a local gauge, A s = s ∗ A and ϕ: U → V the map with Φ = [s, ϕ]. Then the vector D(Φ, γ, x, A) ∈ E x is given by
Proof
We have
Let q(t) be the uniquely determined smooth curve in the fibre P x such that
Write
with a uniquely determined smooth curve g(t) in G. Then
For t = 0 we have
hence
This implies
It follows that
It remains to calculate \(\rho _{{\ast}}(\dot{g}(0))\). We have
and
Since the curve \(\varPi _{\gamma _{t}}^{A}(s(x))\) is horizontal with respect to A, we get
However,
hence
by the definition of connection 1-form. It follows that
and thus the claim. □
The theorem implies that D(Φ, γ, x, A) depends only on the tangent vector X and not on the curve γ itself. We can therefore set:
Definition 5.9.3
Let Φ be a section of an associated vector bundle E and \(X \in \mathfrak{X}(M)\) a vector field on M. Then the covariant derivative ∇ X A Φ is the section of E defined by
where γ is any curve through x and tangent to X x . The covariant derivative is a map
The fact that ∇A Φ is a smooth 1-form in Ω 1(M, E) for every Φ ∈ Γ(E) is clear from the local formula.
We often write in a local gauge s: U → P, with Φ = [s, ϕ], the covariant derivative as
where
i.e.
Here are some properties of the covariant derivative.
Proposition 5.9.4 (Properties of Covariant Derivative)
The map ∇A is \(\mathbb{K}\) -linear in both entries and satisfies
for all smooth functions \(f \in \mathbb{ C}^{\infty }(M, \mathbb{R})\) and the Leibniz rule
for all smooth functions \(\lambda \in \mathbb{ C}^{\infty }(M, \mathbb{K})\).
Proof
\(\mathbb{K}\)-linearity of ∇A and function linearity in X is clear. Let \(\lambda: U \rightarrow \mathbb{K}\) be a smooth function. Then
Here we used the product rule for functions to the vector space V multiplied by functions to the scalars \(\mathbb{K}\). □
Remark 5.9.5
If {∂ μ } are local basis vector fields on U, we get
where
In physics the covariant derivative is typically written in this form and acts on functions ϕ on U with values in the vector space V, determined by sections Φ in E and the local gauge s. In mathematics the covariant derivative acts directly on the sections of the vector bundle. We denote both operators by ∇A (it will be clear from the context which operator is meant).
From a physics point of view it is important that the second summand A μ ϕ in the covariant derivative is non-linear (quadratic) in the fields A μ and ϕ. This non-linearity, called minimal coupling , leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by A μ and the particles described by the field ϕ.
Notice the crucial role played by the representation ρ: It is not only needed to define the associated vector bundle E, but also to define the covariant derivative ∇A. The gauge field A can act on maps with values in V (or sections of E) only if V carries a representation ρ of the gauge group G. If the representation
is non-trivial and hence the coupling between the gauge field A μ and the field ϕ is (potentially) non-trivial, then the particles corresponding to ϕ are called charged (charged particles are affected by the gauge field). In Chaps. 8 and 9 we will discuss in some detail the representations that appear in the description of matter particles in the Standard Model and in Grand Unified Theories.
Figure 5.2 shows the Feynman diagrams for the cubic and quartic terms which appear in the Klein–Gordon Lagrangian in Eq. (7.3), representing the interaction between a gauge field A and a charged scalar field described locally by a map ϕ with values in V.
Feynman diagrams for interaction between gauge field and charged scalar
Remark 5.9.6
In physics the covariant derivative is often defined (without referring to parallel transport) by the local formula
where
One then has to show that this definition is independent of the choice of local gauge: Suppose s′: U → P is another local gauge. Then there exists a smooth physical gauge transformation g: U → G such that
We have
with
Furthermore,
where
It follows that
A lengthy calculation (if done in this abstract setting) then shows that
is equal to
It is often important to consider covariant derivatives compatible with a bundle metric on E. The natural bundle metrics constructed in Proposition 4.7.12 are compatible with covariant derivatives.
Proposition 5.9.7 (Natural Bundle Metrics Are Compatible with Covariant Derivatives)
Let 〈⋅ , ⋅ 〉 V be a G-invariant scalar product on the vector space V and 〈⋅ , ⋅ 〉 E the induced bundle metric on the associated vector bundle E = P × ρ V. Then the covariant derivative associated to a connection A is compatible with the bundle metric in the sense that
for all sections Φ, Φ′ of E and all vector fields X on M.
Proof
Since the scalar product on V is G-invariant, the map ρ ∗ induced by the representation satisfies
for all \(\alpha \in \mathfrak{g}\) and ϕ, ϕ′ ∈ V; see Proposition 2.1.37. This implies:
□
5.10∗Parallel Transport and Path-Ordered Exponentials
We derive in this section a formula that is used in physics to calculate the parallel transport on principal bundles. The following arguments are outlined in [103]. Recall that for the proof of Theorem 5.8.2 concerning the existence of a horizontal lift γ ∗ of a curve γ: [0, 1] → M, where
we had to solve the differential equation
with g(0) = e, where δ is some lift of γ and g: [0, 1] → G is a map with
There is a nice way to write the solution g(t) explicitly, at least if G is a matrix Lie group and γ is contained in an open set over which the principal bundle is trivial.
Suppose that the curve γ is contained in an open set U ⊂ M, so that P U is trivial over U. Let s: U → P be a local gauge with s(γ(0)) = p. We can choose
We then have to solve
Suppose that \(G \subset \mathrm{ GL}(n, \mathbb{K})\) is a linear group, i.e. a closed Lie subgroup. Then the differential equation can be written as
We write this as
where \(f: [0,1] \rightarrow \mathfrak{g}\) is a smooth map determined by γ(t), independent of g.
5.10.1 Path-Ordered Exponentials
Let G be a linear group with Lie algebra \(\mathfrak{g}\).
Definition 5.10.1
For a smooth map \(f: [0,1] \rightarrow \mathfrak{g}\) we define for all t ∈ [0, 1] the following matrices in \(\mathrm{Mat}(n \times n, \mathbb{K})\):
where in the definition of P n ( f, t)
The following is easy to show:
Lemma 5.10.2
For all n ≥ 2 the integral
where
evaluates to
Considering
with respect to a matrix norm on \(\mathrm{Mat}(n \times n, \mathbb{K})\) it follows that:
Proposition 5.10.3
The series
converges for every t ∈ [0, 1] and defines a smooth map
We write
and call this the path-ordered exponential of the function f.
Path-ordered exponentials are useful, because they define solutions to the ordinary differential equation we are interested in.
Proposition 5.10.4 (Path-Ordered Exponential Defines Solution of ODE)
Consider a smooth map \(f: [0,1] \rightarrow \mathfrak{g}\) and define a smooth map
by
Then
In particular, g is a map
Proof
It is clear that g(0) = I n . Calculating the derivative with respect to t we get
This implies the first claim.
To prove the claim that g takes values in G, note that
defines a time-dependent vector field X on \(\mathrm{Mat}(n \times n, \mathbb{K})\), which is right-invariant in the sense that
The smooth map
is an integral curve of the vector field X through the unit matrix I n . Let
be the integral curve through I n of the restriction of the right-invariant vector field X to G, given by Theorem 1.7.18 (the vector field X is tangent to G, because f takes values in the Lie algebra \(\mathfrak{g}\)). Then uniqueness of the solution to ordinary differential equations shows that g ≡ h, hence g takes values in G. □
5.10.2 Explicit Formula for Parallel Transport
Returning to the situation before Sect. 5.10.1, we can write the curve γ(t) in a chart on U with coordinates x μ as γ(t) = x μ(t). Then
The solution to this differential equation is
where γ t denotes the restriction of the curve γ to [0, t]. In particular,
We therefore get:
Theorem 5.10.5 (Parallel Transport Expressed with Path-Ordered Exponential)
Let P → M be a principal bundle with matrix structure group G. Suppose that γ: [0, 1] → M is a curve inside an open set U ⊂ M over which P U is trivial. Let p ∈ P γ(0) be a point and s: U → P a local gauge, such that s(γ(0)) = p. Suppose s(γ(1)) = q. Then the parallel transport of p can be written as
5.11∗Holonomy and Wilson Loops
We saw that the induced parallel transport on associated vector bundles can be used to define covariant derivatives. We want to explain another concept where parallel transport on associated vector bundles appears in physics (we follow the definition in [47]).
Suppose γ: [0, 1] → M is a closed curve in M (a loop) with γ(0) = γ(1) = x. Then parallel transport Π γ E, A is a linear isomorphism of the fibre E x to itself.
Definition 5.11.1
We call the isomorphism Π γ E, A of E x the holonomy Hol γ, x E(A) of the loop γ in the basepoint x with respect to the connection A.
We can express the holonomy using path-ordered exponentials.
Proposition 5.11.2 (Holonomy Expressed with Path-Ordered Exponential)
Suppose that G is a matrix Lie group and the loop γ is contained in an open set U ⊂ M over which P is trivial and s: U → P is a local gauge. Then s determines an isomorphism
and with respect to this isomorphism
Proof
This is Exercise 5.15.8. □
We want to understand how the holonomy changes if we choose a different base point on the curve γ.
Lemma 5.11.3
Let y be another point on the closed curve γ and σ the part of γ from x to y. Then
where we abbreviate Π σ = Π σ E, A.
Proof
Let σ′ be the remaining part of γ from y to x. Then γ = σ ∗σ′ and
This implies the claim. □
Therefore the following map is well-defined.
Definition 5.11.4
The Wilson operator or Wilson loop is the map W γ E that associates to a connection A and a loop γ the number
where tr denotes trace, x is any point on γ, and the second formula holds if G is a matrix Lie group and γ is inside a trivializing open set U for P.
Proposition 5.11.5 (Wilson Loops Are Gauge Invariant)
The Wilson loop is invariant under all bundle automorphisms of P:
Proof
This is Exercise 5.15.9. □
In quantum field theory, the gauge field A μ is a function on spacetime with values in the operators on the Hilbert state space V (if we ignore for the moment questions of whether this operator is well-defined and issues of regularization). The formula
shows that the Wilson loop W γ E(A) is a gauge invariant operator on this Hilbert space.
5.12 The Exterior Covariant Derivative
In Sect. 5.9 we defined a covariant derivative
We can think of this map as a generalization of the differential
In fact, the differential d can be identified with the covariant derivative on the trivial line bundle over M induced from the trivial connection. The differential d can be uniquely extended in the standard way to an exterior derivative
by demanding that ddf = 0 for all functions \(f \in \mathbb{ C}^{\infty }(M)\) and
for all α ∈ Ω k(M) and β ∈ Ω l(M). This differential satisfies d ∘ d = 0 on all forms, see Exercise 5.15.11. Because of this property, the de Rham cohomology
is well-defined for all k.
We want to show that we can extend the covariant derivative in a similar way to an exterior covariant derivative
This exterior covariant derivative, however, in general does not satisfy d A ∘ d A = 0. The non-vanishing of d A ∘ d A is precisely measured by the curvature of A, see Exercise 5.15.12.
The constructions in this section work for general covariant derivatives on vector bundles, which are defined as follows.
Definition 5.12.1
Let E → M be a \(\mathbb{K}\)-vector bundle. Then a covariant derivative ∇ on E is a \(\mathbb{K}\)-linear map
such that
for all smooth functions \(f \in \mathbb{ C}^{\infty }(M, \mathbb{R})\) and the Leibniz rule
holds for all smooth functions \(\lambda \in \mathbb{ C}^{\infty }(M, \mathbb{K})\) and sections e ∈ Γ(E).
We need the following wedge product.
Definition 5.12.2
There is a well-defined wedge product
between standard differential forms (with values in \(\mathbb{K}\)) and differential forms with values in E.
To explain this definition, we only have to see that the standard definition of the wedge product works in this case. The standard definition involves the sum over products of the two differential forms after we inserted a permutation of the vectors (cf. Definition A.2.5). In the standard case we get the product between two scalars in \(\mathbb{K}\), while here we get the product between a scalar in \(\mathbb{K}\) and a vector in E, which is still well defined.
To define the exterior covariant derivative, let ω be an element of Ω k(M, E). We choose a local basis e 1, …e r of E over an open set U ⊂ M. Then ω can be written as
with uniquely defined k-forms ω i ∈ Ω k(U) (with values in \(\mathbb{K}\)).
Definition 5.12.3
Let ∇ be a covariant derivative on a vector bundle E. Then we define the exterior covariant derivative or covariant differential
by
If ∇ = ∇A is the covariant derivative on an associated vector bundle determined by a connection A on a principal bundle, we write d A = d ∇.
Lemma 5.12.4
The definition of d ∇ is independent of the choice of local basis {e i } for E.
Proof
Let {e i ′} be another local basis of E over U. Then there exist unique functions \(C_{ji} \in \mathbb{ C}^{\infty }(U, \mathbb{K})\) with
The matrix C with entries C ji is invertible. Let C −1 be the inverse matrix with entries C lj −1 and define
Then
We calculate
But
This implies the claim. □
The first part of the next proposition follows immediately from the definition by considering a local basis {e i } for E. The second part is clear.
Proposition 5.12.5
The exterior covariant derivative d ∇ satisfies
for all ω, ω′ ∈ Ω k(M, E), σ ∈ Ω k(M) and e ∈ Γ(E). Furthermore, we have on Γ(E) = Ω 0(M, E)
so that the exterior covariant derivative d ∇ is an extension of the covariant derivative ∇.
We want to show the following formula:
Proposition 5.12.6 (Leibniz Formula for Exterior Covariant Derivative)
The exterior covariant derivative d ∇ satisfies
for all σ ∈ Ω k(M) and ω ∈ Ω l(M, E).
Note that this reduces in the case of a 0-form ω with values in E to the second formula in Proposition 5.12.5, because d ∇ on Ω 0(M, E) = Γ(E) is equal to ∇.
Proof
We write
with a local basis {e i } of E over U and ω i ∈ Ω l(U). Then
and
□
Remark 5.12.7
Contrary to the case of the standard exterior derivative d, it can be shown that d ∇ in general has square
The non-vanishing of d ∇ 2 is related to the curvature F ∇ of the covariant derivative ∇ (see Exercise 5.15.12).
We finally want to derive a local formula for the exterior covariant derivative d A in the case of an associated vector bundle. Let P → M be a principal G-bundle, ρ: G → GL(V ) a representation and E = P × ρ V the associated vector bundle. Let A be a connection 1-form on P.
Definition 5.12.8
We define the wedge product
by expanding ω = ∑ i = 1 n ω i ⊗ v i in an arbitrary basis {v i } for V and setting
This is independent of the choice of basis {v i } for V.
Let s: U → P be a local gauge. With respect to the local gauge a form σ ∈ Ω l(M, E) defines a form σ s ∈ Ω l(M, V ). We get:
Theorem 5.12.9
With respect to a local gauge s: U → P we can write
for all ω ∈ Ω k(M, E).
Proof
Choose a basis v 1, …, v n for V. This determines a local frame e 1, …, e n for E by setting e i = [s, v i ]. If we expand a form σ ∈ Ω l(M, E) as
then
We write
and calculate
which implies
□
5.13 Forms with Values in Ad(P)
Recall that connections are 1-forms on the total space of a principal bundle P with values in the Lie algebra \(\mathfrak{g}\). We now want to show that the difference between two connections can be understood as a field on the base manifold M with values in the vector bundle Ad(P). We then get a better understanding of why gauge bosons in physics are said to transform under the adjoint representation.
Let π P : P → M be a principal G-bundle. We then have the vector space \(\varOmega ^{k}(P,\mathfrak{g})\) of k-forms on P with values in the Lie algebra \(\mathfrak{g}\). We want to consider a certain vector subspace of this vector space (we follow [14, Chap. 3]).
Definition 5.13.1
Let \(\omega \in \varOmega ^{k}(P,\mathfrak{g})\) be a k-form on P with values in the Lie algebra \(\mathfrak{g}\). We call ω
-
1.
horizontal if for all p ∈ P
$$\displaystyle{ \omega _{p}(X_{1},\ldots,X_{k}) = 0 }$$whenever at least one of the vectors X i ∈ T p P is vertical.
-
2.
of type Ad if
$$\displaystyle{ r_{g}^{{\ast}}\omega =\mathrm{ Ad}_{ g^{-1}}\circ \omega }$$for all g ∈ G.
We denote the set of horizontal k-forms of type Ad on P with values in \(\mathfrak{g}\) by
which is clearly a real vector space (usually infinite-dimensional).
This notion is useful for the following reason.
Proposition 5.13.2
Let P → M be a principal G-bundle.
-
1.
Suppose \(A,A' \in \varOmega ^{1}(P,\mathfrak{g})\) are connection 1-forms on P. Then
$$\displaystyle{ A' - A \in \varOmega _{\mathrm{hor}}^{1}(P,\mathfrak{g})^{\mathrm{Ad}(P)}. }$$Moreover, if ω is an arbitrary element in \(\varOmega _{\mathrm{hor}}^{1}(P,\mathfrak{g})^{\mathrm{Ad}(P)}\) , then A + ω is a connection on P.
-
2.
The curvature F of a connection A on P is an element of \(\varOmega _{\mathrm{hor}}^{2}(P,\mathfrak{g})^{\mathrm{Ad}}\).
Proof
This follows immediately from the defining properties of connections and the curvature. □
Corollary 5.13.3
The set of connection 1-forms on P is an affine space over the vector space \(\varOmega _{\mathrm{hor}}^{1}(P,\mathfrak{g})^{\mathrm{Ad}(P)}\) . A base point is given by any connection 1-form on P.
It is sometimes useful to have a different description of the vector space of horizontal k-forms of type Ad on P. Recall that we defined in Example 4.7.17 the adjoint bundle
which is the real vector bundle associated to the principal bundle P via the adjoint representation \(\mathrm{Ad}: G \rightarrow \mathrm{ GL}(\mathfrak{g})\).
Theorem 5.13.4
The vector space \(\varOmega _{\mathrm{hor}}^{k}(P,\mathfrak{g})^{\mathrm{Ad}}\) is canonically isomorphic to the vector space Ω k(M, Ad(P)).
Proof
We define a map
as follows: Let \(\bar{\omega }\) be an element of \(\varOmega _{\mathrm{hor}}^{k}(P,\mathfrak{g})^{\mathrm{Ad}}\). Then we define \(\omega =\varLambda (\bar{\omega })\) by
where
-
x ∈ M and p ∈ P are arbitrary with π P ( p) = x.
-
X i ∈ T x M and Y i ∈ T p P are arbitrary with π P∗(Y i ) = X i .
We first show that ω is well-defined. For fixed p ∈ P the definition is independent of the choice of vectors Y i : If Y i ′ are a different set of vectors with π P∗(Y i ′) = X i , then
hence Y i ′− Y i is vertical. Since \(\bar{\omega }\) is horizontal, we get
We now show independence of the choice of p in the fibre P x : Let p′ be another point in P with π P ( p′) = x. Then p′ = p ⋅ g −1 for some g ∈ G. Let Y 1, …, Y k be vectors in T p′ P. We calculate
where we set Z i = r g∗ Y i . We have
This proves independence of the choice of p.
We prove that the form ω is smooth: Let s: U → P be a local gauge and X 1, …, X k smooth vector fields on U. Then
Hence ω ∈ Ω k(M, Ad(P)).
This shows that the map Λ is well-defined and it is clearly linear. It remains to show that Λ is bijective: Let ω ∈ Ω k(M, Ad(P)) and define \(\bar{\omega }\in \varOmega ^{k}(P,\mathfrak{g})\) by
Then \(\bar{\omega }\in \varOmega _{\mathrm{hor}}^{k}(P,\mathfrak{g})^{\mathrm{Ad}}\) and \(\varLambda (\bar{\omega }) =\omega\). □
As a consequence, we get the following statement about connection 1-forms and curvature 2-forms.
Corollary 5.13.5 (Connections, Curvature and Forms with Values in Ad(P))
Let P → M be a principal G-bundle.
-
1.
The difference of two connection 1-forms on P can be identified with an element of Ω 1(M, Ad(P)). The set of all connections on P is an affine space over Ω 1(M, Ad(P)).
-
2.
The curvature F A of a connection A on P can be identified with an element F M A of Ω 2(M, Ad(P)).
The notation F M A generalizes the notation in Corollary 5.6.4, because for an abelian structure group G the adjoint bundle Ad(P) is trivial and F M A has values in \(\mathfrak{g}\).
In quantum field theory, particles in general are described as excitations of a given vacuum field. In the case of a gauge field the vacuum field is a certain specific connection 1-form A 0 on the principal bundle (the form A 0 ≡ 0 is not a connection). Strictly speaking, gauge bosons , the excitations of the gauge field , should then be described classically by the difference A − A 0, where A is some other connection 1-form, and not by the field A itself. By Corollary 5.13.5 this difference can be identified with a 1-form on spacetime M with values in Ad(P). In physics this fact is expressed by saying that gauge bosons, the differences A μ − A μ 0, are fields on spacetime that transform in the adjoint representation of G under gauge transformations.
5.14∗A Second and Third Version of the Bianchi Identity
Let P → M be a principal G-bundle and A a connection 1-form with curvature F A. According to Exercise 5.15.14 we can state the Bianchi identity in the following equivalent form:
Theorem 5.14.1 (Bianchi Identity (Second Form))
The curvature F A ∈ Ω 2 \((P,\mathfrak{g})\) satisfies
for any connection A on P.
In Sect. 5.13 we saw that the curvature F A can be identified with an element F M A in Ω 2(M, Ad(P)). On the other hand the connection A defines an exterior covariant derivative d A on the associated bundle Ad(P):
We can then write the Bianchi identity in a third equivalent form:
Theorem 5.14.2 (Bianchi Identity (Third Form))
The curvature F M A ∈ Ω 2 (M, Ad(P)) satisfies
for any connection A on P.
Proof
This is an immediate consequence of Theorem 5.12.9 and Theorem 5.14.1. □
5.15 Exercises for Chap. 5
5.15.1. Let G be a compact Lie group.
-
1.
Suppose P × G → P is a right-action of G on a manifold P. Prove that P has a G-invariant Riemannian metric.
-
2.
Prove that every principal G-bundle π: P → M has a connection.
5.15.2. Suppose that π: P → M is a principal G-bundle where the Lie group G is abelian. Show that the following map is a group isomorphism
where \(\mathbb{C}^{\infty }(M,G)\) denotes the set of smooth maps from M to G (a group under pointwise multiplication) and σ τ is defined by
5.15.3. Let P → M be a principal bundle and \(A \in \varOmega ^{1}(P,\mathfrak{g})\) a connection 1-form on P. Suppose that \(f \in \mathbb{ G}(P)\) is a global bundle automorphism. Prove that f ∗ A is a connection 1-form on P and
5.15.4. Let P → M be a principal G-bundle with a connection 1-form \(A \in \varOmega ^{1}(P,\mathfrak{g})\) and curvature 2-form \(F \in \varOmega ^{2}(P,\mathfrak{g})\). Let X and Y be horizontal vector fields on P with respect to the Ehresmann connection H defined by A.
-
1.
Show that F(X, Y ) = −A([X, Y ]).
-
2.
Prove that the curvature F vanishes identically if and only if the distribution H is integrable , i.e. [X, Y ] is a horizontal vector field for all horizontal vector fields X, Y on P.
-
3.
Suppose that M is connected and simply connected (π 1(M) = 1) and the curvature F vanishes identically. Prove that P is trivial and there exists a global gauge s: M → P such that A s = s ∗ A ≡ 0.
5.15.5. Let G be a Lie group. Then G acts by right-multiplication on the right of G:
Since the action is simply transitive, it follows that this defines a principal G-bundle over the manifold consisting of one point:
-
1.
Show that the Maurer–Cartan form \(\mu _{G} \in \varOmega ^{1}(G,\mathfrak{g})\) is a connection on this principal bundle and that it is the only one.
-
2.
Determine the curvature of the connection μ G . What is the interpretation of the structure equation?
5.15.6. Let G be a Lie group and H ⊂ G a closed subgroup. By Theorem 4.2.15
is an H-principal bundle. We assume that there exists a vector subspace \(\mathfrak{m} \subset \mathfrak{g}\) such that
i.e. the homogeneous space is reductive.
-
1.
Consider \(A =\mathrm{ pr}_{\mathfrak{h}} \circ \mu _{G} \in \varOmega ^{1}(G,\mathfrak{h})\). Prove that A is a connection 1-form on G → G∕H.
-
2.
Show that the vertical and horizontal subspaces (defined by the connection A) at a point g ∈ G are given by \(L_{g{\ast}}\mathfrak{h}\) and \(L_{g{\ast}}\mathfrak{m}\).
-
3.
Prove that the curvature of the connection A is given by
$$\displaystyle{ F = -\frac{1} {2}\mathrm{pr}_{\mathfrak{h}} \circ \left [\mathrm{pr}_{\mathrm{m}} \circ \mu _{G},\mathrm{pr}_{\mathrm{m}} \circ \mu _{G}\right ] \in \varOmega ^{2}(G,\mathfrak{h}) }$$(the commutator on the right is taken in \(\mathfrak{g}\)).
5.15.7. Recall from Exercise 4.8.7 that the Hopf right action of S 1 on S 3 induces a right action of \(S^{1}/\mathbb{Z}_{p}\cong S^{1}\) on \(S^{3}/\mathbb{Z}_{p} = L(\,p,1)\) and the lens space L( p, 1) thus has the structure of a principal circle bundle over S 2:
Prove that the connection A on the Hopf bundle defined in Sect. 5.2.2 induces a connection A′ on L( p, 1) → S 2. Determine the relation between the global curvature 2-form \(F_{S^{2}}^{{\prime}}\) of this connection and the curvature 2-form \(F_{S^{2}}\) of the Hopf connection, as well as the integral
5.15.8. Suppose that P → M is a principal G-bundle with a matrix Lie group G, E an associated vector bundle, A a connection on P and γ a loop in M. Suppose that the loop γ is inside an open set U ⊂ M over which P is trivial and s: U → P is a local gauge. Then s determines an isomorphism
Prove that with respect to this isomorphism
5.15.9. Let P → M be a principal G-bundle with a connection A and \(f \in \mathbb{ G}\) a bundle automorphism. Suppose γ: [0, 1] → M is a curve in M.
-
1.
Show that parallel transport with respect to the connection f ∗ A is given by
$$\displaystyle{ \varPi _{\gamma }^{f^{{\ast}}A } = f^{-1} \circ \varPi _{\gamma }^{A} \circ f. }$$ -
2.
Let E → M be a vector bundle associated to P and suppose that γ is a closed curve in M. Show that the Wilson loop is invariant under bundle automorphisms of P:
$$\displaystyle{ W_{\gamma }^{E}(\,f^{{\ast}}A) = W_{\gamma }^{E}(A)\quad \forall f \in \mathbb{ G}(P). }$$
5.15.10. Let
be the Hopf bundle with the connection A defined in Sect. 5.2.2. Let σ denote the equator in \(S^{2}\cong \mathbb{C}\mathbb{P}^{1}\), starting and ending at the point \([1: 1] \in \mathbb{C}\mathbb{P}^{1}\).
-
1.
Show that σ can be parametrized as
$$\displaystyle\begin{array}{rcl} \sigma: [0,2\pi ]& \longrightarrow & \mathbb{C}\mathbb{P}^{1} {}\\ t& \longmapsto & \left [1: e^{it}\right ] {}\\ \end{array}$$for a suitable identification of S 2 with \(\mathbb{C}\mathbb{P}^{1}\).
-
2.
Determine the horizontal lift σ ∗: [0, 2π] → S 3 of σ with respect to the connection A, starting at \(\frac{1} {\sqrt{2}}(1,1)\).
-
3.
Let γ k → S 2 be the complex line bundle associated to the Hopf bundle via the representation
$$\displaystyle\begin{array}{rcl} \rho _{k}: S^{1}& \longrightarrow & \mathrm{U}(1) {}\\ z& \longmapsto & z^{k} {}\\ \end{array}$$as in Example 4.7.16. Determine the Wilson loop \(W_{\sigma }^{\gamma ^{k} }(A)\).
5.15.11. Define the differential
by demanding that
-
\(d: \mathbb{C}^{\infty }(M) \rightarrow \varOmega ^{1}(M)\) is the standard differential of functions
-
ddf = 0 for all \(f \in \mathbb{ C}^{\infty }(M)\)
-
d(α ∧β) = dα ∧β + (−1)k α ∧ dβ for all α ∈ Ω k(M) and β ∈ Ω l(M).
Prove that ddω = 0 for all ω ∈ Ω k(M) and all k.
5.15.12. Let E → M be a vector bundle with a covariant derivative ∇. We define the curvature of ∇ by
where \(X,Y \in \mathfrak{X}(M)\) and Φ ∈ Γ(E).
-
1.
Show that F ∇(X, Y )Φ is function linear in each argument X, Y, Φ. The curvature thus defines an element
$$\displaystyle{ F^{\nabla }\in \varOmega ^{2}(M,\mathrm{End}(E)), }$$where End(E) denotes the endomorphism bundle of E over M, whose fibre End(E) x over x ∈ M is given by End(E x ).
-
2.
Show that, as 2-forms with value in E,
$$\displaystyle{ d_{\nabla }d_{\nabla }\varPhi = F^{\nabla }\varPhi }$$for all sections Φ ∈ Γ(E).
-
3.
Define a wedge product
$$\displaystyle{ \wedge: \varOmega ^{k}(M,\mathrm{End}(E)) \times \varOmega ^{l}(M,E)\longrightarrow \varOmega ^{k+l}(M,E) }$$by writing ω ∈ Ω l(M, E) with a local frame {e i } for E over U as ω = ∑ i = 1 n ω i ⊗ e i with ω i ∈ Ω l(U) and setting for α ∈ Ω k(M, End(E))
$$\displaystyle{ \alpha \wedge \omega =\sum _{ i=1}^{n}\alpha (e_{ i}) \wedge \omega _{i} }$$(this definition is independent of choices). Prove that
$$\displaystyle{ d_{\nabla }d_{\nabla }\omega = F^{\nabla }\wedge \omega }$$for all ω ∈ Ω l(M, E).
5.15.13. Let P → M be a principal G-bundle and E = P × ρ V → M an associated vector bundle. We fix a connection A on P with induced covariant derivative ∇A on E. Let F ∇ be the curvature of ∇A defined in Exercise 5.15.12. Suppose that s: U → P is a local gauge and write a section Φ of E locally as Φ = [s, ϕ] with v: U → V. Show that
where F s = s ∗ F and F is the curvature of A on P.
5.15.14. Suppose that P is a manifold and \(\mathfrak{g}\) a Lie algebra. Consider forms \(\phi \in \varOmega ^{1}(P,\mathfrak{g})\), \(\omega \in \varOmega ^{k}(P,\mathfrak{g})\) and \(\tau \in \varOmega ^{l}(P,\mathfrak{g})\). Prove the following identities:
Derive as an application the following second form of the Bianchi identity
where F A is the curvature of a connection 1-form A on a principal bundle P → M.
5.15.15. This exercise is a preparation for Exercise 5.15.16. Suppose that P is a manifold and \(\mathfrak{g}\) a Lie algebra with a scalar product 〈⋅ , ⋅ 〉 which is skew-symmetric with respect to the adjoint representation ad. Let \(\omega \in \varOmega ^{k}(P,\mathfrak{g})\) and \(\tau \in \varOmega ^{l}(P,\mathfrak{g})\). We define a real-valued form 〈ω, τ〉 ∈ Ω k+l(P) by
Consider also a 1-form \(\phi \in \varOmega ^{1}(P,\mathfrak{g})\). Prove the following identities:
5.15.16 (From [ 52 ]). We use the notation from Exercise 5.15.15. Suppose that P → M is a principal G-bundle over a manifold M and 〈⋅ , ⋅ 〉 an Ad-invariant scalar product on the Lie algebra \(\mathfrak{g}\). Let A be a connection 1-form on P with curvature F. We define the Chern–Simons form α(A) ∈ Ω 3(P) by
-
1.
Prove that dα = 〈F, F〉.
-
2.
Let \(f \in \mathbb{ G}(P)\) be a bundle automorphism with induced map σ f : P → G and set ϕ = σ f ∗ μ G . Prove that the Chern–Simons form changes under bundle automorphisms as
$$\displaystyle{ f^{{\ast}}\alpha =\alpha +d\langle \mathrm{Ad}_{\sigma _{ f}^{-1}}A,\phi \rangle -\frac{1} {6}\langle \phi,[\phi,\phi ]\rangle. }$$ -
3.
Show that the form
$$\displaystyle{ \tau _{G} = -\frac{1} {6}\langle \mu _{G},[\mu _{G},\mu _{G}]\rangle \in \varOmega ^{3}(G) }$$is closed.
-
4.
Suppose that M is a closed oriented 3-manifold, P → M a trivial G-bundle and τ G represents an integral class in \(H^{3}(G; \mathbb{R})\). If s: M → P is a global gauge, define the Chern–Simons action by
$$\displaystyle{ S_{M}(s,A) =\int _{M}s^{{\ast}}\alpha (A). }$$Prove that modulo \(\mathbb{Z}\) the number \(S_{M}(A)\,=\,S_{M}(s,A)\,\in \,\mathbb{R}/\mathbb{Z}\) is independent of the choice of global gauge s.
Remark
Notice that the Chern–Simons action is purely topological, i.e. does not depend on the choice of a metric on M. This leads to the concept of topological quantum field theories (TQFT). Similar Chern–Simons terms appear in many places in physics, for example, in the actions of supergravity and in the actions for D-branes in string theory.
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Hamilton, M.J.D. (2017). Chapter 5 Connections and Curvature. In: Mathematical Gauge Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-68439-0_5
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