Notes on Gauge Theories

Leonardo Almeida Lessa

Abril de 2019

Contents

1 Motivating Examples 21.1 Classical Electromagnetism . . . . . . . . . . . . . . . . . . . 21.2 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . 51.3 Dirac Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Magnetic Charges . . . . . . . . . . . . . . . . . . . . 61.3.2 Electromagnetic Interaction in Quantum Mechanics . 81.3.3 QM and the Dirac Monopole . . . . . . . . . . . . . . 9

2 Fibre Bundles 112.1 Motivation: Tangent Bundle . . . . . . . . . . . . . . . . . . . 112.2 Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Triviality of Bundles . . . . . . . . . . . . . . . . . . . 162.3 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Associated Vector Bundle . . . . . . . . . . . . . . . . 18

3 Connections in Fibre Bundles 203.1 Fundamental Vector Field . . . . . . . . . . . . . . . . . . . . 203.2 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Local Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5.1 Gauge Potential . . . . . . . . . . . . . . . . . . . . . 243.5.2 Field Strength . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . 28

4 The Examples from Another View 304.1 QED and Yang-Mills Gauge Theories . . . . . . . . . . . . . . 304.2 Dirac Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Here are some notes on gauge theories. We begin by discussing someexamples from Physics and their common features. Then, we detour to thetheory of fibre bundles and connections on principal bundles, which formthe mathematical basis of gauge theory. Finally, we review our previousexamples in this new framework.

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Leonardo A. Lessa

I personally find fascinating that many phenomena that appear uncorre-lated at first can be explained in an unified way. In the case of gauge theory,we see another intersection of Geometry and Topology with Physics.

The text is heavily based on “Geometry Topology and Physics”, by MikioNakahara [1]. Other references are included at the end.

1 Motivating Examples

The notion of gauge appears when we have more degrees of freedomin the description of the theory than there is in the physics of the problem.Sometimes, the underlying mathematical objects, confined to our abstractionof the world, are redundant and unphysical, even though derived observableresults are unambiguous.

We artificially distinguish quantities that are physically equivalent. Thus,the observables of our theory have to be symmetric upon gauge transforma-tions in these equivalent quantities. They possess gauge symmetry describedby gauge groups, similarly to the familiar symmetries of spacetime, but qual-itatively different. A gauge transformation changes our description of themodeled system while a spacetime symmetry changes the modeled systemitself, although not modifying the physical laws.

Before presenting the mathematical details, we will now discuss someexamples. Our goal now is to create a clear image of what we mean by agauge theory, to then formalize it in more generality.

1.1 Classical Electromagnetism

The first contact one may have with gauge transformations is with Elec-tromagnetism, whose dynamics are determined by Maxwell’s equations. Innatural (Lorentz-Heaviside) units, by which we set c = ε0 = µ0 = ~ = 1,Maxwell’s equations can be written as

∇×E = −∂B∂t, ∇ ·B = 0, (1.1)

∇ ·E = ρ, ∇×B =∂E

∂t+ J. (1.2)

Poincaré’s Lemma applied to the homogeneous equations (1.1) tells usthere exists at least locally1 a scalar field V and a vector field A, also knownas potentials, satisfying (1.1)

Exercise 1.1In R3 vector calculus, Poincaré’sLemma amounts to

∇× v = 0 ⇒ v =∇f,∇ · v = 0 ⇒ v =∇× u,

for some locally defined realfunction f and vector field u.Adapt this to prove equations(1.3) and (1.4).

E = −∂A∂t−∇V, (1.3)

B =∇×A. (1.4)

Although we are guaranteed of their existence, the potentials V and Aare far from unique. For every arbitrary function Λ, we may transform the

1We will see later that the Dirac monopole is an example where the vector field Acannot be defined in the whole space. This has to do with the topology of the problemand has interesting consequences.

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1.1 Classical Electromagnetism Leonardo A. Lessa

potentials by

V ′ = V − ∂Λ

∂t, (1.5)

A′ = A +∇Λ, (1.6)

but still get the same electric E′ = E and magnetic fields B′ = B(1.2). SinceExercise 1.2Verify thisE and B control the dynamics of charged particles via Lorentz force, we

arrive at the same physics after this transformation. Indeed, this is a caseof gauge transformation.

In the manifestly covariant formulation of Electromagnetism [2], the po-tentials V and A are part of a (four-)potential

Aµ = (V,A), (1.7)

as well as the four-current Jµ = (ρ,J), which transform like a four-vectorunder Lorentz transformations. Similarly, the electric and magnetic fieldscan be grouped into a antisymmetric (0, 2)-tensor Fµν , called the Faradaytensor or electromagnetic field strength tensor. The components of Fµν ina reference frame where the electric and magnetic fields at some point areE = (Ex, Ey, Ez) and B = (Bx, By, Bz) are, in matrix notation,

Fµν =

0 Ex Ey Ez−Ex 0 −Bz By−Ey Bz 0 −Bx−Ez −By Bx 0

. (1.8)

We immediately see that the transformation laws of the electric and magneticfields under change of reference frame are not as simple as with the four-potential Aµ, since they are mixed up in the coordinates of the Faradaytensor Fµν .

Equations (1.3) and (1.4) can be written in terms of the four-potential(1.7) and the Faraday tensor (1.8) via

Fµν = ∂µAν − ∂νAµ, (1.9)

where we have used Einstein notation to sum repeated indices and loweredthe potential with the Minkowski metric ηµν = diag(−1,+1,+1,+1) as such:

Aµ = ηµνAν = (−V,A).

Furthermore, the gauge transformation of (1.6) and (1.5) simplifies to

A′µ = Aµ + ∂µΛ. (1.10)

Since the electric and magnetic fields remain the same under a gaugetransformation, the same happens with Fµν . Thus, the Faraday tensor isgauge invariant. Accordingly, we can write Maxwell’s equations in terms ofFµν and they are also gauge invariant:

∂µ(∗Fµν) = 0, (1.11)∂µF

µν = Jν , (1.12)

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1.1 Classical Electromagnetism Leonardo A. Lessa

where ∗F is the dual field strength tensor, defined by ∗Fµν = 12εµναβFαβ

(1.3).Exercise 1.3Interpret equation (1.11) know-ing that the dual field strength∗Fµν is Fµν with E and Bswapped.

We can solve equations for V and A and work with them, but the finalobservable results have to be gauge independent. This is not a negativeaspect of our theory, quite the contrary. The labor of dealing with non-gauge-invariant quantities like Aµ is compensated by the manifest Lorentzinvariance of the theory. Unitarity and locality are also manifest when wekeep this redundancy in the quantum realm (See [8]).

Even fixing the gauge is a useful idea. It is often the case that simplifi-cation in the calculation occurs when we choose a particular gauge. In thecase of Electromagnetism, we can set up a differential equation for Λ so asthe potentials have some property we want, as long as it doesn’t contradictMaxwell’s equations.

Figure 1: Illustration of the space of all potentials Aµ. A point in this gaugeis connected to other points (dashed lines) by gauge transformations, andto fix a gauge is to choose independent representatives among those (boldline).

For example, we may choose the Lorenz Gauge ∂µAµ = 0(1.4) to arriveExercise 1.4What differential equation doesΛ have to satisfy?

at a wave equation for the potential,

Jν(1.12)

= ∂µFµν ,

= ∂µ(∂µAν − ∂νAµ),

= ∂µ∂µAν . (1.13)

Although the potential Aµ has four components, by solving (1.13), weget two independent solutions, corresponding to the possible polarizationsof a electromagnetic wave. This is also a manifestation of gauge symmetry.

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1.2 Quantum Electrodynamics Leonardo A. Lessa

1.2 Quantum ElectrodynamicsHaving already seen how classical electromagnetism has gauge symmetry,

it may not be clear what is the corresponding gauge group. In Quantum Elec-trodynamics (QED), however, the gauge group appears naturally. In fact,we won’t even quantize anything, (fortunately!). The only “new” conceptthat comes from the quantum world is spinors. If the reader is unfamiliarwith this formalism, we recommend [6].

The main idea is to express our previous results in the Lagrangian for-malism. The potential Aµ(x) is our dynamical field, whose dynamics iscontrolled by the following Lagrangian

LEM = −1

4FµνF

µν −AµJµ. (1.14)

More precisely, by the Euler-Lagrange equations associated to LEM(1.5)

Exercise 1.5Why is (1.11) automatically sat-isfied?

∂L∂Aµ

− ∂

∂xν∂L

∂(∂νAµ)= 0. (1.15)

One may argue that LEM is not gauge invariant, because of the termAµJ

µ, and it is true. However, the action SEM =∫LEMd4x is gauge in-

variant, because the gauge transformation of SEM gives off a surface term,which goes to zero as we integrate over the entire spacetime M = R4:

−∫M

AµJµd4x→ −

∫M

(Aµ + ∂µΛ)Jµd4x,

= −∫M

AµJµd4x−

∫M

[∂µ(ΛJµ)− Λ∂µJµ]d4x,

= −∫M

AµJµd4x−∫∂M

ΛJµd3x,

= −∫M

AµJµd4x,

where in the third line we used that Jµ is conserved and in the fourth, thatthe integrand goes to zero at infinity. Since the Euler-Lagrange equations(1.15) come from the variation of the action, then we still have a gaugeinvariant theory.

In QED, the fermion field of electrons and their antiparticles, the positrons,is coupled with the electromagnetic field of photons, the force carriers, in aquite natural way. First, we begin by writing the free Lagrangian for a Diracspinor ψ,

LDirac = ψ(iγµ∂µ −m)ψ. (1.16)

It describes a fermion (spin 12 particle) with mass m by a Dirac spinor

field ψ(1.6), which has four complex components ψα, α ∈ 0, 1, 2, 3, and Exercise 1.6Electrons are charged particles.Then why does (1.16) not havethe elementary charge constante in it?

transforms under an irreducible representation of the Lorentz group2.The Lagrangian LDirac has a global internal symmetry,

ψ(x)→ eieλψ(x), ψ(x)→ ψ(x)e−ieλ, (1.17)2Technically, ψ transforms under a projective representation of the Lorentz group,

which is a proper representation of the double cover group SL(2,C). See [3, 5].

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1.3 Dirac Monopole Leonardo A. Lessa

where global means λ ∈ R is independent of the spacetime point x andinternal means the transformation doesn’t change the point x where thefields are evaluated. Since λ is arbitrary and ψ is multiplied by a pure phaseeieλ, our gauge group is U(1), complex numbers of modulus one endowedwith complex multiplication.

Let’s now see what happens if we require this symmetry to be local. Withthat, the gauge transformation (1.17) becomes

ψ(x)→ e−ieλ(x)ψ(x), ψ(x)→ ψ(x)eieλ(x), (1.18)

This comes at a price, since LDirac is no longer gauge invariant, but trans-forms as

LDirac = ψ(iγµ∂µ −m)ψ → ψ[iγµ(∂µ − ie∂µλ)−m]ψ,

which looks similar to the gauge transformation (1.10) for the electromag-netic potential Aµ. In fact, if we define a covariant derivative ∇µ by

∇µ = ∂µ + ieAµ (1.19)

then we can modify our Lagrangian as

L′Dirac = ψ(iγµ∇µ −m)ψ. (1.20)

so we have a theory that is invariant under the local gauge transformations

ψ(x)→ e−ieλ(x)ψ(x), ψ(x)→ ψ(x)eieλ(x),

Aµ → Aµ + ∂µλ,

and couples the electromagnetic field Aµ to the fermionic field ψ. Finally,we add the electromagnetic Lagrangian term − 1

4FµνFµν to (1.20) to get our

final QED Lagrangian

LQED = −1

4FµνF

µν + ψ(iγµ∇µ −m)ψ (1.21)

In view of LEM from (1.14), we may also rearrange LQED as

LQED = LDirac + LEM,

= ψ(iγµ∂µ −m)ψ − 1

4FµνF

µν −AµJµ,

with Jµ = eψγµψ.(1.7)

Exercise 1.7Prove that the conservedNoether current associated withthe global symmetry (1.17) ofthe Lagrangian LDirac is Jµ.

We did not justify why is it necessary that we have local gauge invariance(1.18) instead of the weaker assumption of global gauge invariance (1.17).One a posteriori reason is that, from this assumption, the electromagneticfield and its coupling to the Dirac field follow naturally. A more thoroughdiscussion about the need for local gauge invariance can be found in [7].

1.3 Dirac Monopole1.3.1 Magnetic Charges

The interplay between topology, geometry and gauge theory will becomeapparent once we introduce the notion of fibre bundles. Before that, the

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1.3 Dirac Monopole Leonardo A. Lessa

example of the Dirac magnetic monopole illustrates the far-reaching conse-quences of the geometry of spacetime.

We include a magnetic four-current JµB = (ρB,JB) in Maxwell’s homo-geneous equation (1.11) in the same way the electric four-current Jµ ≡ JµEappears in (1.12):

∂µ(∗Fµν) = JνB,

∂µFµν = JνE.

In terms of E and B fields,

∇×E = −∂B∂t− JB, ∇ ·B = ρB,

∇ ·E = ρE, ∇×B =∂E

∂t+ JE.

Consider a point magnetic charge of strength g in the origin. Its magneticcharge density is ρB(r) = gδ3(r), so, similarly to the electrostatic case, themagnetic field produced is

B =g

4π

r

r2. (1.22)

We used the then homogeneous Maxwell’s equations (1.1) to derive thepotentials V and A satisfying (1.3) and (1.4). Since they are not homo-geneous if magnetic charges are present, we cannot have globally definedpotentials as before. If we could, then B = ∇×A would imply ∇ ·B = 0everywhere.

However, in the case of a point magnetic charge, there is a vector potentialAN whose curl almost equals the magnetic field (1.22). By “almost”, we mean∇×AN(r) = B(r) for r ∈ R3 \ S, for a “small set” S ⊂ R3. In sphericalcoordinates (r, θ, φ), AN is given by

AN(r) =g

4π

1− cos θ

r sin θeφ, (1.23)

which is well-defined for r ∈ R3 \ S, with S = (x, y, z) ∈ R3|z ≤ 0, x = y =0, a region called the Dirac string. Not only is AN singular at the origin,where the monopole sits, but also in a line starting on the origin and goingto infinity in the −z direction.(1.8) For r ∈ R3 \ S,

Exercise 1.8Why is AN not well-defined onS?∇×AN(r) =

1

r sin θ

∂

∂θ(ANφ sin θ)r− 1

r

∂

∂r(rANφ )θ,

=g

4π

r

r2,

= B(r).

Can we also cover the S region? We know we cannot do this with justone vector potential. However, we can imitate AN by defining AS on R3\S′,with S′ = −S = (x, y, z) ∈ R3|z ≥ 0, x = y = 0,

AS(r) = − g

4π

1 + cos θ

r sin θeφ, (1.24)

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1.3 Dirac Monopole Leonardo A. Lessa

Figure 2: Illustration of the magnetic field B produced by a magneticmonopole g. The ray S is called the Dirac String, where the vector po-tential AN is not defined. I thank Gabriel Solis for helping me render thisscene.

You may have noticed that the superscripts N and S mean north and southhemispheres, where the vector potentials AN and AS are well-defined.

Let us pause for a moment to ponder the geometric meaning of AN andAS. Outside the origin there is no magnetic charge, thus we can use the usualMaxwell’s equations (1.1) and (1.2) to find find our solutions. Indeed, forr 6= 0, we can find vector potentials for the magnetic field. But because weare now solving Maxwell’s equations in R3\0, a non simply connect subset ofR3, Poincaré’s lemma does not guarantee a globally defined vector potential,and so we need at least two vector fields to cover all R3 \ 03. Effectively,the magnetic monopole is altering our space topology. This close connectionwith the geometry of the underlying space M will be made precise when westudy connections (Section 3) and the cohomology group H2(M) (Section4.2).

1.3.2 Electromagnetic Interaction in Quantum Mechanics

When we join Quantum Mechanics with the Dirac monopole, we get asurprising result: the quantization of electrical charge. Before doingso, let’s see how electromagnetism enters in Quantum Mechanics.

3We will see this is closely related to the fact that we need at least to coordinatepatches to cover S2.

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1.3 Dirac Monopole Leonardo A. Lessa

Without electromagnetism, Schrödinger’s equation for a particle of wavefunction ψ(t, r) in a potential V (r) is (~ = 1)

Hψ = i∂ψ

∂t, with

H =|p|2

2m+ V (r).

The quantum-mechanical momentum is p = −i∇, a space derivative.Imitating what we did for the electromagnetic coupling of the Dirac field, wewill use equation (1.19) of the covariant derivative to redefine our momentumas

p→ p + qA.

This redefinition of the momentum to account for electromagnetic interactionis called minimal coupling. It couples a charged particle with charge q (withno spin) to the electromagnetic field.

Although A is not gauge invariant, this coupling gives gauge invariantphysical results! To see this, let us pick a wave function ψ(t, r) which is asolution to the Schrödinger’s equation in a region with vector potential A.Its Hamiltonian is

H =1

2m(p + qA)2 + V (r). (1.25)

If we do a gauge transformation A→ A+∇Λ, the Hamiltonian H changesand so does its eigenfunction ψ(t, r) → ψ(t, r), which can be expressed interms of the old solution ψ by(1.9)

Exercise 1.9Verify this by calculating the ef-fect of the operator (p + eA +e∇Λ) acting on ψ.

ψ(t, r) := e−ieΛ(r)ψ(t, r). (1.26)

We know from Quantum Mechanics that global complex phases multiply-ing the wave function are not physical, since all measurable quantities comefrom taking the squared of the absolute value of inner products. With thisin mind, we might think the complex phase factor in (1.26) can be removedwithout repercussions. Quite the contrary, this complex factor is responsiblefor a plethora of counterintuive effects. They all have in common the ap-pearance of a gauge invariant measurable quantity in terms of a not gaugeinvariant quantity, like A. One of these effects is what we will discuss now:when we interact a quantum particle with a magnetic monopole. The otheris the Aharanov-Bohm effect, by which we can detect the changes in phaseof a wave function in a B = 0 (but A 6= 0) environment by interferometry.

1.3.3 QM and the Dirac Monopole

We will keep the complex phase of (1.26) and see where it leads us to.Remember that the electromagnetic field around a magnetic charge of

strength g can be described by two vector potentials, AN and AS. Thedomains of definition of the vector potentials AN and AS intersect eachother in R3 \ (S ∪ S′), which is the whole space minus the z line. In thisregion of intersection, both vector potentials have to give the same electro-magnetic fields. One way to achieve this is if they are equal up to a gauge

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1.3 Dirac Monopole Leonardo A. Lessa

transformation. To see if this is the case, we calculate the difference

AN −AS =g

2π

1

r sin θeφ,

=∇(gφ/2π).

Mixed feelings. The expression gφ/2π is not a function, for a 2π rotationin the angle φ doesn’t change a point in space, but changes the expression.On the other side, it is painful to not be able to say that AN and AS

are connected by a gauge transformation. In either way, will treat for nowΛ = gφ/2π as our gauge transformation between AS and AN, postponing amore rigorous discussion.4

Here comes the main argument. If we have a quantum particle in aspace with a magnetic monopole at the origin, then its wave function ψ inthe equator5 may be the solution of the Hamiltonian with minimal coupling(1.25) and A = AS.

Applying a gauge transformation Λ = gφ/2π will turn AS into AN andψ into

ψ(r) = e−ieΛψ = e−iqg2πφψ(r). (1.27)

Here, we demand ψ(r) is a well-defined wave function, and so is single-valued.This is equivalent to demanding that the complex phase from (1.27) satisfy

e−ieg2πφ = e−i

qg2π (φ+2π),

which is equivalent toqg

2π= n ∈ Z. (1.28)

In words, If there exists a magnetic monopole with strength g,then electrical charges are quantized. Moreover, magnetic charges alsocome in discrete amounts by equation (1.28).

From the Standard Model of Particle Physics, we know every chargedbody has a charge which is a integer multiple of 1

3e, where e is the elementarycharge. Most particles (elementary or not) have charges that are integermultiples of e, such as electrons, protons, photons. The 1

3 includes quarks,which come in a family with charge + 2

3e – up, charm and top – and a familywith charge − 1

3e – down, strange and bottom.

4If the reader is unsettled, we anticipate the solution. We will treat the vector potentialas a one-form, so this gauge transformation is really a one-form g

2πdφ, which is well-

defined.5We look for wave functions in the equator to stay away from the Dirac strings S and

S′, so both AS and AN are well-defined

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Leonardo A. Lessa

2 Fibre BundlesWe now introduce the concept of fibre bundle, the backbone of all the

mathematical formalism for gauge theory. Fibre bundles have a special inter-est in itself as a mathematical construct and its subtopics include principalbundles, holonomy, etc, many of which we will treat here. The main refer-ence for the theory of principal bundles and connections is Kobayashi andNomizu’s “Foundations of Diferential Geometry” [4]. For the introduction offibre bundles, we will use Steenrod’s “The Topology of Fibre Bundles” [9].

For our first example of fibre bundle, the tangent bundle, we assumethe reader has some familiarity with the theory of manifolds. Althoughthis example is independent of the subsequent sections, it already has manyfeatures we want to explore in the general theory.

2.1 Motivation: Tangent BundleOne very useful prototypical example of a fibre bundle is the tangent

bundle TM of a manifold M of dimension dim(M) = n. It is the disjointunion of the tangent spaces TpM , p ∈M , as such

TM :=⊔p∈M

TpM ≡⋃p∈M

(p, TpM).

If M is a two-dimensional surface of R3, one can imagine TpM is a planetangent to a point p ∈ M of the surface, and TM is the set of all thoseplanes, separated by which points they are tangent to.

The tangent bundle TM is itself a manifold of dimension 2n, called thetotal space if viewed as a fibre bundle. The charts of TM are described bya chart of M , the base space, and an open set of the vector space TpM .This arrangement may seem like TM is the product spaceM×V , with V ann-dimensional vector space, isomorphic to TpM , ∀p ∈M , but this is not thecase for every manifold M . In fact, if TM 'M ×V (is trivial), then M is aparallelizable manifold. For the spheres Sm, only them = 1, 3 and 7 ones areparallelizable. There is a entire area of study dedicated to the non-triviality,or twisting, of bundles, called Chern-Weil theory of Characteristic Classes[1](2.1)

Exercise 2.1Do these numbers remind you ofsomething?

The tangent bundle TM is not necessarily a direct product, but it islocally a direct product or, in other words, locally trivial. More precisely, foreach chart (U, xi) of M , we can consider the restricted bundle TU , treatingthe coordinate neighbourhood U as a manifold. From the definition ofTM , we know that in the local coordinates x = (x1, . . . , xn) : U → Rn of U ,every element u = (p, V ) ∈ TU can be decomposed in its base point p ∈ Mand a vector V ∈ TpU , which can be written as

V = V µ∂

∂xµ

∣∣∣∣p

,

for (V 1, . . . , V n) ∈ Rn, since ∂∂xα

∣∣pnα=1 is a base for TpU . Naturally, we

can define a diffeomorphism φU : TU → U × Rn by

φU (u) = (p, (V 1, . . . , V n)),

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2.1 Motivation: Tangent Bundle Leonardo A. Lessa

Figure 3: Illustration of the tangent bundle TM . A vector v ∈ TpM in thefiber TpM is projected to p ∈M by π.

which is precisely what we meant by saying TM is locally a direct productspace. We will call φU the local trivialization map.

Another common feature of fibre bundles is the projection to the basespace. In the case of TM , we can define π : TM →M simply by π(u) = p,where u = (p, V ) ∈ TM . Clearly, TU = π−1(U) and TpM = π−1(p), thelatter called the fibre of the bundle. All fibres are isomorphic to one another,but are associated to different points in the base manifold M .

Given another open chart (V, yi) of M , one may ask how the local trivi-alizations φU : π−1(U)→ U ×Rn and φV : π−1(V )→ V ×Rn are connected,just as the coordinates xµ : U → R and yν : V → R are related by

yν(p) =∂(y x−1)ν

∂xµ

∣∣∣∣x(p)

xµ(p),

for p ∈ U ∩ V . 6 Likewise, the vector components of V ∈ TpM satisfy

V = V µx∂

∂xµ

∣∣∣∣p

,

= V µx∂yν

∂xµ∂

∂yµ

∣∣∣∣p

,

= V νy∂

∂yν

∣∣∣∣p

,

so V νy = ∂yν

∂xµVµx and thus the local trivializations φU and φV are related by

φ−1U (p, V µx ) = φ−1

V (p, V νy ) = φ−1V (p, ∂y

ν

∂xµVµx ).

6We will often shorten the notation and write yν = ∂yν

∂xµxµ when the point p ∈ U∩V ⊆

M is implicit from the context.

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2.2 Fibre Bundles Leonardo A. Lessa

The matrix Gνµ = ∂yν

∂xµ is nonsingular (det(G) 6= 0) because we demandthat y x−1 be a diffeomorphism for M to be smooth. It is this matrix thatmixes the components of the vector V when we change coordinates. The setof all real nonsingular matrices of order n is called the general linear groupGL(n,R).(2.2) Thus, we say the structure group of TM is GL(n,R) and

Exercise 2.2Given a coordinate chart (U, xi)and a matrix G ∈ GL(n,R),can we create another coordinatechart (U, yj) such that the tran-sition function from x to y isgiven by G? This type of ques-tion is related to the problem ofreducing the structure group of aprincipal bundle (See Sec. 2.3).

Gνµ is a transition function between charts.The bold terms above are the essential components of a fibre bundle,

which are beautifully exemplified by the tangent bundle TM . Other cor-related concepts will appear throughout this presentation, such as crosssection and connection, generalizations of vector field and parallel trans-portation, respectively.

2.2 Fibre BundlesWith the elements highlighted in Section 2.1 for TM , we enunciate the

definition of a coordinate bundle, and then of a fibre bundle, following Steen-rod [9].

Definition 2.1. A coordinate bundle (E, π,M,F,G, Ui, φi) consists of

(i) A topological space E called the total space or bundle space,

(ii) a topological space M called the base space,

(iii) a topological space F called the fibre,

(iv) a continuous surjection π : E →M called the projection,

(v) a topological group G called the structure group, which acts freelyon F on the left,

(vi) a open covering Uii∈I ofM indexed by I, consisting of open neigh-bourhoods Ui,

(vii) a homeomorphism φi : Ui × F → π−1(Ui)7 for each i ∈ I, called a

local trivialization.

satisfying the following relations:

(i’) For each p ∈ M , the inverse image π−1(p) =: Fp is homeomorphic tothe fibre F .

(ii’) Each local trivialization φi : Ui × F → π−1(Ui) is constrained by theprojection as such: π φi (p, f) = p ∈ Ui.

(iii’) If we define φi,p : F → Fp as φi,p(f) = φi(p, f) for p ∈ Ui ∩ Uj , thenwe require that the transition function tij(p) := φ−1

i,p φj,p : F → Fcoincides with the operation of an element of G on F . Furthermore,we require tij : Ui ∩ Uj → G, the map from p to the correspondinggroup element tij(p) of the transition function – also denoted by tij byabuse of notation – to be continuous.

7To comply with the notation of Nakahara’s and Steenrod’s books, we switched domainwith the codomain of the local trivializations from our previous definition in Section 2.1.This change is not restrictive since the trivializations are invertible.

13

2.2 Fibre Bundles Leonardo A. Lessa

The transition function connects the trivializations by

φj(p, f) = φi(p, tij(p)f),

where here we treated tij(p) ∈ G as an element of G acting of f ∈ F . (2.3)

Exercise 2.3Prove that

tijtjk = tik

tii = e

tij = [tji]−1

It will also be useful to define a function pi : π−1(Ui) → F that as-signs the elements of the bundle to their fibre representatives via the localtrivialization, such that, for u ∈ E,

pi := [φi,π(u)]−1 (2.1)

Analogous to the definition of manifolds by equivalences of atlases, westart with an open cover Ui to define a coordinate bundle to, then, forma fibre bundle out of a equivalence condition between coordinate bundles:

Definition 2.2. Two coordinate bundles (E, π,M,F,G, Ui, φi) and(E, π,M,F,G, Vj, ψj) – differing only on the open covers and the localtrivializations – are equivalent if (E, π,M,F,G, Ui ∪ Vj, φi ∪ ψj) isa coordinate bundle (2.4). A fibre bundle, denoted by (E, π,M,F,G), E π→

Exercise 2.4Convince yourself that thisamounts to requiring compati-bility between local trivializa-tions φi and ψj as we did incondition (iii’) of Definition 2.1.

M or simply E, when the rest of the structure is implicit, is an equivalenceclass of coordinate bundles by the relation defined above.

Just as with manifolds, we think about fibre bundles in terms of theirrepresentatives – the coordinate bundles – and prove theorems about thesewhich are invariant by the equivalence relation introduced in Definition 2.2.Thus, hereafter we will not distinguish the equivalence class with its repre-sentatives, hopefully not confusing the reader in the process.

In the following, we will only work with smooth fibre bundles, whose defi-nition is the same as the fibre bundle one, but we strengthen the topologicalrequisites with smooth ones. More specifically, all the topological spacesE, M , F and G are smooth manifolds, with G being a Lie group, and allcontinuous (homeomorphic) functions are smooth (diffeomorphic).

The generalization of a vector field X ∈ X(M) is a section X ∈ Γ(M,E),defined by

Definition 2.3. A (cross) section X : M → E is a smooth map that sendspoints p ∈ M of the base space to an element of its fibre X(p) ∈ Fp. Inother words, π X = id. The set of all sections from M to E is Γ(M,E), orsimply Γ(E).

A local section is a section X ∈ Γ(U,E) restricted to a open set U ⊂M .

Example 2.1. Product bundle. For M and F topological spaces, we canconstruct the product bundle (E = M × F, π,M,F,G) such that π = pr1 :M × F → M and G = e is the trivial group, as every local trivializationis the identity.

Example 2.2. Vector bundle. A vector bundle is a fibre bundle (E, π,M, V,G)where the fibre is a vector space V . Because of this, it is common the actionof G on V to be a linear representation, i.e. an action by linear operators.

The tangent bundle TM is an example of a vector bundle with structuregroup G = GL(n,R), as shown in Section 2.1. Other vector bundles will

14

2.2 Fibre Bundles Leonardo A. Lessa

come along our way. For example, the quantum mechanical wavefunctionand the Dirac field are sections in a vector bundle associated to a principalbundle (See Section 2.3).

Example 2.3. Principal bundle. One way to define principal bundles isto say they are fibre bundles whose typical fibre F is equal to the structuregroup G. We will see the consequences of this in Section 2.3.

Another way is to start with a manifold E and a Lie group G that actsfreely on E on the right, and then define the base spaceM to be the quotientspace of the action of G on E

M = P/G.

Furthermore, we require E to be locally trivial in the sense that there existsan open covering Ui of M and diffeomorphisms φi : Ui × G → π−1(Ui)such that pi(ua) = pi(u)a, where pi is defined by (2.1).

Can we loosen the requirements to construct a fibre bundle via Defi-nitions 2.1 and 2.2? Fortunately, we can spare the specification of localtrivializations:

Theorem 2.1. Let G be a Lie Group acting freely on a manifold F , Ma manifold, Ui an open covering of M and tij a set of functions tij :Ui ∩Uj → G for each intersecting open sets Ui and Uj satisfying tkjtji = tkiand tii = e. Then, there exists a fibre bundle E

π→ M with transitionfunctions tij and coordinate neighbourhoods Ui.

Proof. First, we defineX :=

⊔i∈I

Ui × F

This is our prototype for the total space. We still need to connect thefibres from different intersecting parts of the union. To do this, we assign aequivalence relation between points (p, fi) ∈ Ui × F and (q, fj) ∈ Uj × F ofX:

(p, fi) ∼ (q, fj)⇐⇒ p = q and fi = tij(p)fj .

Now, we can define our total space to be E = X/ ∼, the quotient space,formed by equivalence classes. Naturally, our projection π is defined byπ([(p, f)]) = p and the local trivialization associated to a open neighbour-hood Ui is defined by φi(p, fi) = [(p, fi)].(2.5)

Exercise 2.5I glossed over some details, likewhat is the topology of X and ifπ is continuous and so on. If youcare about those, try for yourselfto complete them! All the detailsare done in [9].

With this in mind, we can play with cylinders and Möbius strips:

Example 2.4. Cylinder and Möbius strip. What are the possible (topo-logical) fibre bundles with base spaces M = S1 and fibres F = [−1, 1], theunit interval? One natural candidate is the cylinder M × F . For us to useTheorem 2.1, it suffices to find an open covering Ui of S1, a structuregroup G acting on F and a set of transition functions tij.

The circumference S1 ⊂ R2 is naturally covered by UN = S1 \ (0,−1)and US = S1 \ (0, 1). Their intersection has two connected componentsUN ∩ US = U+ ∪ U−, defined by U± := (x, y) ∈ S1| ± x ≥ 0. Thus, wehave only one continuous transition function tSN : U+ ∪ U− → G.

15

2.2 Fibre Bundles Leonardo A. Lessa

If we assume G is discrete, then tSN has to be constant on U+ and onU−. Hence, the simplest cases are if G has one element, the trivial groupG = e, or two elements, the binary numbers group G = Z2. In theformer case, we can have G = e acting on F with the identity, and thentSN = id and E = S1 × [−1, 1] is the cylinder. In the latter case, we canhave G = Z2 = 0, 1 acting on F by

0 · f = f 1 · f = −f,

and thentSN |U+ = 0 tSN |U− = 1.

This results in the fibre bundle of the Möbius band!

Definition 2.4. Let E π→ M and E′π′→ M ′ be fibre bundles. A smooth

map f : E → E′ is a bundle map if it maps each fibre Fp ⊂ E onto thecorresponding fibre F ′f(p) ⊂ E

′.

2.2.1 Triviality of Bundles

One way to determine the triviality of a fibre bundle is if the base spaceis contractible to a point. Before enunciating the exact theorem for this, weneed to introduce bundle maps and pullback bundles.

A bundle map f : E → E′ induces a map f : M → M ′ on the basemanifolds, since f preserves the fibres.(2.6)

Exercise 2.6Prove that f π = π′f . Expressthis in terms of a commuting di-agram.

Lemma 2.1. Two coordinate bundles are equivalent if, and only if, they havethe same base space and there exists a diffeomorphic bundle map betweenthem.

Proof. See Steenrod [9], Lemmas 2.6, 2.7 and 2.8.

Given a fibre bundle (E, π,M,F,G) and a smooth map f : N → M , wecan construct a bundle (f∗E, π1, N, F,G) over N , called the pullback bundle.

Definition 2.5. Given a fibre bundle (E, π,M,F,G) and a smooth mapf : N → M , the total space of the pullback bundle (f∗E, π1, N, F,G) isdefined as

f∗E = (p, u) ∈ N × E|f(p) = π(u),

a closed subspace of the manifold N × E; the projection π1 : f∗E → N isjust π1(p, u) = p; the fibre is the same of E, as π−1

1 (p) = (p, Ff(p)) ' F ;given an open covering of M Ui, the open neighbourhoods of f∗E areU ′i = f−1(Ui); and the local trivializations are

φ′i : U ′i × F → π−11 (U ′i),

φ′i(p, f) = (p, φi(f(p), f)).

There is a natural bundle map π2 : (p, u) 7→ u between the pullbackbundle and the original bundle. If M = N and f = id, then, by Lemma 2.1,f∗E ' E.

We are now ready to enunciate the homotopy theorem.

16

2.3 Principal Bundles Leonardo A. Lessa

Definition 2.6. Two maps f, g : X ′ → X between topological spaces Xand X ′ are said to be homotopic if there exists a map F : X ′× [0, 1]→ X,called a homotopy, satisfying F (p, 0) = f(p) and F (p, 1) = g(p).

Theorem 2.2. Let E π→ M be a fibre bundle and f and g be homotopicmaps from a Cσ8 space N to M , then the pullback bundles f∗E and g∗E areequivalent.

Proof. See Steenrod [9], Theorem 11.4.

If M is contractible to a point p0 ∈ M , then id : M → M is homotopicto the constant map e0 : M → p0 ⊂ M . Since id∗E = E and e∗0E =M × F (2.7), then Theorem 2.2 says

Exercise 2.7If this is not clear, try to provethat the pullback of E via e0is the same as the pullback ofp0 × F via e0, since it is theconstant map.

Corollary 2.1. The bundle E π→M is trivial if M is contractible to a point.

2.3 Principal BundlesPrincipal bundles will be crucial in the mathematical formalism of gauge

theory. One definition of principal bundles was given in Example 2.3

Definition 2.7. A principal bundle is a fibre bundle (P, π,M,F,G) whereF = G, also called the G-bundle over M , denoted only by (P, π,M,G). Gacts trivially on the fibre.

Roughly, if G is the gauge group of a physical theory, the G-bundle willbe the space of all possible gauge choices. Now, back to the mathematicaldetails.

Since the transition functions act on the fibres on the left, then the rightaction of G on the fibres F = G is independent of the local trivialization andthus G can act on the P bundle itself.

The action of a ∈ G on a bundle element u ∈ P is defined locally viaa trivialization φi : Ui × G → π−1(Ui), where p = π(u) ∈ Ui. If φ−1

i (u) =(p, gi), then we define ua by

ua := φi(p, gia).

Now, we prove that this definition is independent of the local trivializationchosen. Indeed, if p = π(u) ∈ Ui ∩ Uj , then

ua = φi(p, gia),

= φj(p, tji(gia)),

= φj(p, (tjigi)a),

= φj(p, gja).

The action of G on E does not change the base point of the argumentand is transitive on fibres:

∀u1, u2 ∈ Fp, ∃a ∈ G, u1 = u2a,

and free:(2.8)

Exercise 2.8Prove that the action of G on Eis transitive and free.

8A Cσ-space is a normal locally compact manifold that admits a countable open cov-ering.

17

2.3 Principal Bundles Leonardo A. Lessa

(∃u ∈ E, ua = u) ⇒ a = e.

Given a local cross section σ ∈ Γ(U,P ) over U ⊂M in a principal bundleP , we can construct a local trivialization defining

φσ : U ×G→ π−1(U),

φσ(p, g) = σ(p)g.

This map we have constructed is invertible since the right action of G on Eis transitive: for every point u ∈ Fp, there exists g(u) ∈ G such that

u = σ(p)g = φσ(p, g)

This is so important that we have a name for g(u):(2.9)

Exercise 2.9Prove that the transition func-tion tij : Ui ∩ Uj → G thatconnects the trivializations de-fined by σi ∈ Γ(Ui, P ) andσj ∈ Γ(Uj , P ) satisfies σj(p) =σi(p)tij(p).

Definition 2.8. The canonical local trivialization associated to localsection σ ∈ Γ(U,P ) is a map g : P → G such that

∀u ∈ U, u = σ(p)g(u).

We will see later that the choice of a cross section in a G-bundle isinterpreted as gauge fixing. Note that, as with the local gauge invariance ofQED encountered in Section 1.2, the cross section can vary along the basespace M – our spacetime.

2.3.1 Associated Vector Bundle

We are going to show a method to construct a vector bundle (E, πE ,M, V,G)(See Example 2.2) given a G-bundle (P, π,M,G) and a vector space V thatG acts on by a linear representation ρ : G→ GL(V ).9

The group G acts on the right on the manifold P × V as follows

(u, v) · g := (ug, ρ−1(g)v).

We then define the total space of the vector bundle associated to P π→M tobe the quotient of P ×V by the right action of G, denoted by E := P ×G V .If we denote the action of G on V by juxtaposition gv := ρ(g)v, then a usefulnotation for the elements of E is, instead of the standard equivalence class[(u, v)] ∈ E, we denote simply by uv, where u ∈ P and v ∈ V . The principlebehind this notation is because, for every g ∈ G, we have

uv = [(u, v)] = [(ug, ρ−1(g)v)] = (ug)(g−1v).

The bundle structure of E follows naturally from its associated principalbundle. Namely, we define a projection πE : E →M by πE(uv) := π(u) and,for every open neighborhood U ⊂ M of P , we define a local trivializationφE : U × V → π−1

E (U) on E using the trivialization φ : U ×G→ π−1(U) onP such that the following diagram commutes

9Given a manifold F , We may construct a fibre bundle with fibre F associated to aprincipal bundle. Since the important cases will be with vector spaces, we anticipate andonly work with them.

18

2.3 Principal Bundles Leonardo A. Lessa

U ×G× V U × V

π−1(U)× V π−1E (U)

id×ρ−1

φ×id φE

[ ]

More directly, we can define(2.10)(2.11)

Exercise 2.10Show that this definition guaran-tees the preceding diagram com-mutes.Exercise 2.11What are the local trivializationsof E = P ×G V ?

φE(u, v) := [(φ(u, e), v)]

Conversely, given a vector bundle (E, πE ,M, V,G), we can employ The-orem 2.1 to define an associated principal bundle (P, π,M,G). To do this,we just employ the same open covering, base space and transitions functionsfrom E

π→M , and require that G acts on itself, the fibre, by the usual groupoperation.(2.12)

Exercise 2.12Convince yourself that the vec-tor bundle associated to theprincipal bundle we constructedhere is the original bundle westarted with.

In our previous examples, the physical states that underwent gauge trans-formations were maps from a spacetime M to a vector space V (2.13). The

Exercise 2.13IdentifyM and V of Sections 1.2,1.3.1 and 1.3.2.

gauge freedom made the vector representation of a physical state highly non-unique. Thus, a one-to-one association of a physical state to a mathematicalobject has to gauge out this unphysical freedom. This is the main idea be-hind the construction of the associated vector bundle. We append the vectorspace V to the principal bundle P and take the quotient by the right actiondefined in (2.3.1), which is just a way to gauge transform the vector v via therepresentation ρ, but also leaving a trace of the transformation in u 7→ ug.Thus, physical states are sections of E, elements of Γ(M,E).

Example 2.5. Frame Bundle. The associated vector bundle of the tan-gent bundle TM is the frame bundle LM . Since the structure group ofTM is GL(n,R), where n = dim(M), then LM is locally a product spaceUi ×GL(n,R), welded by the transition functions tij : Ui ∩ Uj → GL(n,R).In matrix notation, we have

(p,Xµα ) ∈ Ui ×GL(n,R), (p, Y νβ ) ∈ Uj ×GL(n,R),

Xµα = (tij)

µνY

να =

∂xµ

∂yν

∣∣∣∣p

Y να .

As the notation used above may have already hinted, we can also viewthe frame bundle LM as the collection of vector bases Xαnα=1 ⊂ TpM ,called frames. More specifically, given a base point p ∈ M , we can definethe frame space LpM as the set of all frames Xαnα=1 ⊂ TpM and then

LM =⊔p∈M

LpM.

Given a chart (U, x) of M , with p ∈ U , each frame Xα ∈ LpM is asso-ciated to a non-singular matrix Xµ

α ∈ GL(n,R) whose columns are the xcoordinates of Xα: (2.14)

Exercise 2.14Why is Xµ

α non-singular?

Xµα =

| | |X1 X2 . . . Xn

| | |

.19

Leonardo A. Lessa

The manifold structure of LM is analogous to the tangent bundle one– if UL is an open neighbourhood of (p, Xα) ∈ LM whose projectionto M fits inside a chart set U , then we map (p, Xα) 7→ (xµ(p), Xµ

α ) ∈Rn+n2

. The bundle structure of (LM,π,M,GL(n,R)) is then equivalent tothe construction via association with TM .(2.15)

Exercise 2.15What the right action of Gαβ ∈GL(n,R) does to frames in LM?

3 Connections in Fibre Bundles

3.1 Fundamental Vector Field

Before jumping into the theory of connections, we briefly review Liegroups and Lie algebras and prove some technical theorems that will beof good use in later proofs. If the reader is used to Lie groups and don’twant to dive into the details of proofs, then they can skip this section.

The Lie algebra g of the Lie group G is the algebra of all left-invariantvector fields A ∈ X(G), (Lg)∗A|h = A|gh, under the operation of the Liebracket. Alternatively, it is the Lie algebra TeG, since any vector a ∈ TeGinduces a left-invariant vector field via A|g = (Lg)∗a.

Definition 3.1. The fundamental vector field A# ∈ X(P ) generated byA ∈ g is the vector field associated to the flow t 7→ u exp(tA)10. In otherwords,

A#|u(f) =d

dtf(u exp(tA))

∣∣∣∣t=0

, f ∈ C∞(P ).

Theorem 3.1. The mapping σ : g → X(P ) which sends A to A# is ahomomorphism between Lie algebras. Furthermore, A# does not vanish atany point of P for A 6= 0.

Proof. We first show a more intrinsic way to define σ. For every u ∈ P , wedefine σu : G → P by σu(g) = ug = Rgu. Thus, σA|u = (σu)∗(A|e), inaccordance with Definition 3.1.(3.1)

Exercise 3.1Prove this To prove that σ is a homomorphism, we need to show that [A,B]# =

[A#, B#]. Since the flow t 7→ Rat , with at = exp(tA), generates A#, then

[A#, B#]|u = limt→0

1

t[(Ra−t)∗(B

#|uat)−B#|u],

= limt→0

1

t[(Ra−t σuat)∗(B|e)− σu∗(B#|e)],

by the definitions,

Ra−t σuat(c) = uatca−1t = σuRa−tLat ,

for any c ∈ G. SinceB is left-invariant, then (Ra−tσuat)∗(B|e) = σu∗Ra−t∗(B|at)

10For us, a flow is a 1-parameter group of local transformations. In the case of A#,they are global. See definition in [4].

20

3.1 Fundamental Vector Field Leonardo A. Lessa

and

[A#, B#]u = limt→0

1

t[σu∗Ra−t∗(B|at)− σu∗(B#|e)],

= σu∗ limt→0

1

t[Ra−t∗(B|at)− (B#|e)],

= σu∗([A,B]e) = [A,B]#.

Finally, if A# vanished at, say, u ∈ P , then Rat leaves u fixed for everyt ∈ R (WHY?). Since G acts freely on P , then at = e for every t ∈ R, whichimplies A = 0.

Definition 3.2. The vertical subspace VuP is the subspace of TuP tan-gent to the fibre Gp11, where p = π(u).

Since the flow t 7→ u exp(tA) is entirely contained in Gp, with p = π(u),then A#|u ∈ VuP . Moreover, as G acts freely on P , A# never vanishes onP , and the dimension of g is the same as of the fibre Gp ' G, the associationA 7→ A# is an isomorphism of Lie algebras g and VuP . In other words, VuPis generated by A#, for A ∈ g.

Definition 3.3. Let Ad(a) : G→ G be the adjoint map for a ∈ G, Ad(a)g =aga−1. Then the adjoint representation ad : G→ Aut(g) of G in g is thepushforward map ad(a) := (Ad(a))∗.(3.2)

Exercise 3.2Prove that ad(a) = (Ra−1 )∗.For G a matrix group, ad(a) = Ad(a), if we view g as a set of matrices

with the same order as G’s.(3.3). Since we will work with matrix group inExercise 3.3Prove thisour examples from physics, then we will often write ad(a)A = aAa−1.

Proposition 3.1. Let φ : M → M be a transformation of a manifold M .If the flow t 7→ φt generates a vector field X ∈ X(P ), then the flow t 7→φ φt φ−1 generates φ∗X.

Proof. Let p ∈M , q = φ−1p and f ∈ C∞(M). The flow of the 1-parametergroup of local transformations φ φt φ−1 generates the following vectorfield

d

dt(φ φt φ−1)

∣∣∣∣p

(f) =d

dtf(φ φt φ−1p),

=d

dtf(φ[φt(q)]),

= φ∗(X|q),= (φ∗X)|p.

Thus, φ φt φ−1 generates φ∗X.

Theorem 3.2. If A# corresponds to A ∈ g, then (Ra)∗A# corresponds to

ad(a−1)A ∈ g.

Proof. Since A# is generated by the flow Rat , where at = exp(tA), then, byProposition 3.1, (Ra)∗A

# is generated by the flow RaRatRa−1 = Ra−1ata =RAd(a−1)at . Again, since t 7→ at generates the vector field A, then t 7→Ad(a−1)at generates Ad(a−1)∗A = ad(a−1)A, as desired.

11The fibre Gp = π−1(p) at p ∈M is a closed submanifold of P .

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3.2 Connection Leonardo A. Lessa

3.2 Connection

The geometric interpretation of a connection on a principal bundle P π→M is a smooth separation of each tangent space TuP , u ∈ P , into the verticalsubspace VuP and a horizontal subspace HuP such that

(i) TuP = HuP ⊕ VuP for all u ∈ P

(ii) (smoothness) A smooth vector field X ∈ X(P ) is decomposed intoa horizontal part XH |u ∈ HuP and a vertical part XV |u ∈ VuP asX = XH +XV .

(iii) HugP = Rg∗(HuP ) for every u ∈ P and g ∈ G.

The preferred way to generate this separation is by a connection formω ∈ g ⊗ Ω1(P ), which is a Lie-algebra-valued one form on P .(3.4) Besides

Exercise 3.4Given a basis Tα of g, provethere exist one-forms ωα suchthat ω =

∑α Tα ⊗ ωα.

being an element of this tensor product space ω can also be viewed as amap that smoothly associates to each u ∈ P a linear transformation ωu :TuP → g, or as a linear map ω : X(P ) → C∞(M, g) between modules overC∞(M) functions. With this last form, it is more clear what the connectiondoes to vectors fields X ∈ X(P ). It projects X|u ∈ TuP to VuP , withker(ωu) = HuP . Now, let us present a proper definition.

Definition 3.4. Connection form or Ehresmann connection is a one-form ω with values on g, denoted by ω ∈ g⊗ Ω1(P ), satisfying

(i) ω(A#) = A, for every A ∈ g,

(ii) R∗gω = adg−1 ω, for every g ∈ G. That is, for every X ∈ TuP ,ωug(Rg∗X) = adg−1(ωu(X)).

There is a one-to-one correspondence between separations of type TuP =HuP ⊕ VuP and projectors ωu with im(ωu) = VuP and ker(ωu) = HuP , sothere is no loss in generality for considering ω ∈ g ⊗ Ω1(P ) instead of theformer geometrical definition.(3.5)(3.6)

Exercise 3.5Prove that

Rg∗HuP = HugP

for HuP = ker(ωu).Exercise 3.6How would you define XH andXV given ω?

3.3 Parallel Transport

With a connection, we can distinguish horizontal spaces in the tangentspaces of the principal bundle. This enables us to lift a curve γ : [0, 1]→Min the base space to a curve γ : [0, 1]→ P in the principal bundle such thatγ′(t) ∈ Hγ(t)P . Intuitively, γ specifies the direction of γ between fibers, andthe connection tells how the elevation changes along fibers.

Definition 3.5. Let γ : [0, 1] → M be a curve on M . A horizontal liftγ : [0, 1]→ P of γ is a curve on P such that

(i) γ′(t) ∈ Hγ(t)P ,

(ii) π γ = γ.

The theorem for existence and uniqueness of a horizontal lift is the fol-lowing

22

3.3 Parallel Transport Leonardo A. Lessa

Theorem 3.3. Let γ : [0, 1] → M be a curve on M , and u ∈ P such thatπ(u) = γ(0). Then there exists an unique horizontal lift γ : [0, 1]→ M of γsuch that γ(0) = u.

We postpone the proof to Section 3.5

Corollary 3.1. If γ is a horizontal lift of a curve γ on M , then any otherhorizontal lift of γ is γg, for some g ∈ G.

Proof. Let γ be a horizontal lift of γ. Then for every g ∈ G, γg is also ahorizontal lift of γ (See exercise 3.5). If γ is another horizontal lift of γ,then there exists a g ∈ G such that γ(0) = γ(0)g. From the uniqueness ofthe horizontal lift given a starting point u = γ(0) = γg ∈ P , then we haveγ = γg

Given a curve γ onM with endpoints γ(0) = p and γ(1) = q, we can nowconstruct a map Γ(γ) : Gp → Gq that parallel transports elements u ∈ Gpof the fibre at p to elements Γ(γ)(u) of the fibre at Gq via a horizontal liftstarting at u. (3.7)

Exercise 3.7Using Corollary 3.1, prove that

Γ(γ) Rg = Rg Γ(γ)

There is a connection between parallel transport and the fundamentalgroup π1(M) of the base space M . To see this, we first note how Γ behavesunder composition of curves.

Theorem 3.4. Let α : [0, 1] → M and β : [0, 1] → M be curves satisfyingα(1) = β(0) and α ∗ β : [0, 1]→M be their composition, then

Γ(α ∗ β) = Γ(α) Γ(β),

Γ(α−1) = Γ(α)−1

What happens when γ is a loop? If the endpoints of γ are the same,then Γ(γ) maps the fiber at p to itself. Thus, Γ(γ)(u) = u · τγ(u) for someτγ(u) ∈ G(3.8). If we span all loops γ based on p ∈M , we get the holonomy

Exercise 3.8What is the analogous of Theo-rem 3.4 to τγ?

group (3.9) (3.10)(3.11)

Exercise 3.9Prove that Holu is a subgroup ofG using Theorem 3.4.Exercise 3.10Using exercise 3.7, prove that

Holug = g−1Hug.

Thus, all holonomy groups Holuwith the same base point p =π(u) are isomorphic.Exercise 3.11If M is connected, prove thatHolp is the same up to a conju-gation:

Holq = τγ Holp τ−1γ ,

where γ is a curve from p to q.

Holu = τγ(u)|γ is a loop based on p = π(u).

Analogously, we can define the restricted holonomy group restricting to countonly the contractible curves:

Hol0u = τγ(u)|γ is contractible to p = π(u).

These groups are related to the fundamental group π1(M,p) based on pby the following theorem

Theorem 3.5. There is a epimorphism (surjective homomorphism) Φ :π1(M,p)→ Holu /Hol0u, with p = π(u).

Proof. For each equivalence class of curves α ∈ π1(M,p), we pick a repre-sentative curve γ ∈ α and assign Φ(α) = τγ/Hol0u. This map is independentof the representative γ since if we had picked another one γ ∈ α, then

τγ(u)/Hol0u = τγ(u) · τγ−1γ(u)/Hol0u,

= τγ(u)/Hol0u,

23

3.4 Curvature Leonardo A. Lessa

since γ−1γ is contractible.That the map Φ is a epimorphism comes from Theorem 3.4 and the

definition of Holu.

3.4 CurvatureIn differential geometry, the Riemann curvature tensor measures the de-

gree to which a vector parallel transported in a loop fails to come back toitself. We have seen that there is also a notion of parallel transport in gaugetheories, so it is natural to think that there exists a notion of curvaturedependent only on the connection of a principal bundle.

Definition 3.6. The Covariant Derivative of a vector-valued k−formφ ∈ V ⊗Ωk(P ) is a vector-valued (k + 1)−form Dφ ∈ V ⊗Ωk+1(P ) definedby

Dφ(X1, . . . , Xk+1) := (dPφ)(XH1 , . . . , X

Hk+1),

where dP is the exterior derivative that acts on the k−forms of φ and XHi

are the horizontal parts of the vectors Xi ∈ X(P ).

Definition 3.7. The Curvature two-form Ω ∈ g⊗Ω2(P ) is the covariantderivative of the connection one-form ω:

Ω := Dω.

The connection of the curvature form to the non-commutativity of par-allel transport is summarized in the following theorem, by Ambrose andSinger.

Theorem 3.6 (Ambrose-Singer). The Lie Algebra of the holonomy groupHolu0

, u0 ∈ P , is equal to the subalgebra of g spanned by elements Ωu(X,Y ),for all X,Y ∈ HuP and all u connected to u0 by a horizontal lift.

For the proof, see Theorem 8.1, page 89 of [4].

3.5 Local Forms3.5.1 Gauge Potential

Let σ ∈ Γ(U,P ) be a local section defined on a open neighbourhood U .Like a choice of gauge, we can use σ to introduce a local expression of theconnection form ω ∈ g⊗ Ω1(P ).

Definition 3.8. Given a connection form ω ∈ g⊗Ω1(P ) and a local sectionσ ∈ Γ(U,P ), the (local) gauge potential is the Lie-algebra-value one-formA ∈ g⊗ Ω1(U) on U , given by

A := σ∗ω.

In our examples with electromagnetism, A is actually the four-potential.We saw in the Dirac monopole case that there are cases where A cannot beglobally defined. It is clear now that this corresponds to the fact that wecannot always find a section σ on the whole base manifold M .

24

3.5 Local Forms Leonardo A. Lessa

Note that A is a one-form over U ⊂ M , the base manifold, with valuesin the Lie algebra g. It is easier to do calculations in the base manifold Mthan in the principal bundle P , since the latter can have an unknown twistedtopology, when the former is usually well known from the start.

Given that A is better to work with, can we go from A to ω? More thanthat, what is the compatibility condition between gauge potentials Ai andAj that arise from two different sections σi ∈ Γ(Ui, P ) and σj ∈ Γ(Uj , P )defined in intersecting open neighbourhoods Ui and Uj? The two theoremsbelow answer these questions if G is a matrix group.12

Theorem 3.7. Given a g-valued one form A ∈ g⊗Ω1(U) defined on U ⊂Mand a local section σ ∈ Γ(U,P ), there exists a local connection one-formω ∈ g× Ω1(π−1(U)) such that A = σ∗ω. The connection is given by

ω = g−1(π∗A)g + g−1dP g,

where g is the canonical local trivialization associated to σ (See Definition2.8).(3.12)

Exercise 3.12Prove that π∗A# = 0. What isπ∗ doing?For the proof of this theorem, we refer to Theorem 10.1 of [1].

Theorem 3.8. Two local gauge potentials Ai ∈ g × Ω1(Ui) and Aj ∈ g ×Ω1(Uj) arise from the same connection form if, and only if, they satisfy

Aj = t−1ij Aitij + t−1

ij dtij .

Proof. Before we prove the theorem, we need the following technical lemma.

Lemma 3.1. Let P (M,G) be a principal bundle and σi ∈ Γ(Ui, P ), σj ∈Γ(Uj , P ) be local sections, where Ui ∩ Uj 6= ∅. For X ∈ TpM , σi∗X andσj∗X are connected by

σj∗X = Rtij∗(σi∗X) + (t−1ij dtij(X))#.

For the proof of this lemma, we refer to Lemma 10.1 of [1].

Example 3.1. Gauge Transformation. The gauge potential A1 ∈ g ⊗Ω1(U) associated to σ1 ∈ Γ(U,P ) changes via a gauge transformation σ2 :=σ1(p) · g(p), where g : U → G, as(3.13)

Exercise 3.13Use Lemma 3.1 to prove thetransformation law of the gaugepotential.

A2 = g−1A1g + g−1dg.

If G is abelian, then we have simply

A2 = A1 + g−1dg.

12If the reader wants to see the general versions of Theorems 3.7 and 3.8, in which it isnot assumed that G is a matrix group, see Proposition 1.4, page 66 of [4].

25

3.5 Local Forms Leonardo A. Lessa

Example 3.2. U(1) Gauge Transformation. If G = U(1), then gaugetransformations are of the form g(p) = e−iΛ(p). Then, the gauge potentialtransforms as

A′ = A+ eiΛ(−idΛ)e−iΛ,

= A− idΛ.

If we define A := −iA, then, in components,

A′µ = Aµ + ∂µΛ,

the same transformation law of the electromagnetic four-potential (See Sec-tion 1.2). This is a good hint that electromagnetism is a U(1) gauge theoryas described in this chapter.

Example 3.3. Frame transformation. The Lie algebra of GL(n,R) isMat(n,R), the set of real square matrices of order n. Thus, the gaugepotentials of LM , the frame bundle, are matrix-valued one-forms Aαβ ∈Mat(n,R) ⊗ Ω1(U). Let (U, x) and (V, x) be intersecting charts on M andAαβ and (A)ρσ be their respective local gauge potentials of a common con-nection. Then Theorem 3.8 tells us they are connected by

Aρσ = t−1xx

ραAαβt

βxx σ + t−1

xxρλdt λ

xx σ

=∂xρ

∂xα∂xβ

∂xσAαβ +

∂xρ

∂xλd

(∂xλ

∂xσ

),

for t αxx β = ∂xα

∂xβIf we write the gauge potentials as Aρσ = Aρνσdxν and

Aαβ = Aαµβdxµ = Aαµβ ∂xµ

∂xν dxν , then in the x components, the equationabove becomes

Aρνσ =∂xρ

∂xα∂xβ

∂xσ∂xµ

∂xνAαµβ +

∂xρ

∂xλ∂2xλ

∂xν∂xσ,

which is the same transformation law of Christoffel symbols, if we assignAαµβ = Γαµβ . In section 3.6, we will see a stronger reason for the validityof this association.

We are now capable of proving Theorem 3.3

Proof. We will use the compactness of the interval [0, 1] to construct γ locallyon open sets Uα, to then cover [0, 1] with finitely many of Uα.

We first want to define γ(t) in a neighbourhood of t = 0. Suppose wedid that. From Condition (ii) of Definition 3.5, we know that γ(t) ∈ Gγ(t) =π−1(γ(t)). Thus, we take an open neighbourhood U of γ(t) with a localcross section σ ∈ Γ(U,P ), which sets our local trivialization, and satisfiesσ(γ(0)) = u. Calling σ(t) = σ(γ(t)), for t ∈ V in a neighbourhood oft = 0, then there exists a function g : V → G such that γ(t) = σ(t)g(t) andg(0) = e. Since g(t) is much more tractable than γ, we will now constructan equation for g(t) to find γ.

26

3.5 Local Forms Leonardo A. Lessa

The equation will be a differential equation, and will come from Condition(i), which tells us that ω(γ′) = 0. Since γ′ = γ∗γ

′ if we treat γ(t) = σ(t)g(t)as a section in Γ(im(γ), P ), then, by Lemma 3.1,

0 = ω(γ(t)),

= ω(Rg(t)∗(σ∗γ′) + [g(t)−1dg(γ′)]#),

= g(t)−1ω(σ∗γ′)g(t) + g(t)−1 dg(t)

dt.

Since ω(σ∗γ′) = σ∗ω(γ′) = A(γ′), then

dg(t)

dt= −A(γ′)g(t), (3.1)

whose formal solution is

g(t) = P exp

(−∫ t

0

A(γ′)dt

)= P exp

(−∫γ

A)

where P, the path-ordering symbol, indicates that the product of the inte-grals in the exponential function expansion is time-ordered.

This solution is valid for each open neighbourhood U where a local sectionσ can be defined. Using the compactness of the interval [0, 1], we only needfinitely made of these sets to cover the entire curve. Finally, to go from openset to the other, we just transform the initial point of the horizontal lift asγ(0) → γ(0)g(1). The uniqueness of γ is guaranteed by the uniqueness ofthe solution of the ODE (3.1). (3.14)

Exercise 3.14Prove that the path-orderedexponential transforms undergauge transformations by conju-gation. Thus, the trace of g(t)is gauge invariant. It is calledthe Wilson loop in the contextof QFT.

3.5.2 Field Strength

Similarly to the connection one-form, we can pullback the curvature two-form on P via a local section σ ∈ Γ(U,P ) to a two-form on M :

Definition 3.9. Given a curvature two-form Ω = Dω ∈ g ⊗ Ω2(P ) and alocal section σ ∈ Γ(U,P ), the (local) field strength is the Lie-algebra-valuetwo-form F ∈ g⊗ Ω2(U) on U , given by

F := σ∗Ω.

The field strength can be expressed in terms of the gauge potential A by

F = dA+A ∧A, (3.2)

where for A = Tα ⊗Aα, we define(3.15)

Exercise 3.15Prove all the relations in equa-tion (3.3).A∧A := (TαTβ)⊗(Aα∧Aβ) =

1

2f γαβ Tγ⊗(Aα∧Aβ) = [Tα, Tβ ]⊗(Aα⊗Aβ).

(3.3)To prove this, we first need to relate Ω to ω with the exterior differenti-

ation explicit:

27

3.6 Covariant Derivative Leonardo A. Lessa

Theorem 3.9 (Cartan’s structure equation). Let X,Y ∈ TuP , then

Ω(X,Y ) = dPω(X,Y ) + [ω(X), ω(Y )],

in other words, Ω and ω are related by

Ω = dPω + ω ∧ ω

For the proof of this Theorem, we refer to Theorem 10.3 of [1].Applying the pullback σ∗ to Cartan’s structure equation, we get equation

(3.2). In components F = 12Fµνdxµ ∧ dxν , we have

Fµν = ∂µAν − ∂νAµ + [Aµ,Aν ], (3.4)

where each component Fµν or Aµ is a function of type U → g. If we pick abasis Tα of g with structure constants f γ

αβ , we can expand the equationeven further with

Fµν = F αµν Tα,

Aµ = A βµ Tβ ,

such that (3.4) becomes an equation with real functions defined on U .

F γµν = ∂µA

γν − ∂νA γ

µ + f γαβ A α

µ A βν (3.5)

In electromagnetism, the gauge group is U(1). This group has a one-dimensional Lie Algebra, so we can drop the algebra indices on A α

µ andF αµν . Another consequence of this is that U(1) is an abelian group, so

[Aµ,Aν ] = 0. Finally, equation (3.5) becomes Fµν = ∂µAν − ∂νAµ with Fµνnaturally interpreted as the Faraday tensor.

Under a gauge transformation, the field strength changes under conjuga-tion:

Theorem 3.10. The field strength F1 ∈ g ⊗ Ω2(U) associated to σ1 ∈Γ(U,P ) changes via a gauge transformation σ2 := σ1(p) · g(p), where g :U → G, as(3.16)

Exercise 3.16Use Example 3.1 to prove thetransformation law of the fieldstrength.

F2 = g−1F1g.

If G is abelian, then the field strength does not change at all! Thisexplains the gauge invariance of the electric and magnetic fields. In the casewhere G is a matrix group, we can form gauge invariant quantities takingthe traces, which are inherently invariant under conjugation. For example, agauge invariant scalar field used in the Yang-Mills Lagrangian (See Section4.1) is TrFµνFµν(3.17)

Exercise 3.17Prove TrFµνFµν is gauge in-variant. 3.6 Covariant Derivative

As we saw in Section 1.2, we may need to modify the notion of deriva-tive to include local gauge invariance. For this, we introduced the notionof covariant derivative (See equation (1.19)), which included the four po-tential and made the action gauge invariant. We know that physical statesare sections on an associated vector bundle, so we need to define a notion

28

3.6 Covariant Derivative Leonardo A. Lessa

of derivative on these sections that take account of the connection in theassociated principal bundle.

First, let us remember the structure of the associated vector bundle. If(P, π,M,G) is a principle bundle, then we can construct a vector bundle(E, πE ,M, V,G) for any vector space V upon which G acts. The total spaceis E made of equivalence classes [(u, v)], with u ∈ P and v ∈ V , and [(u, v)] =[(ug, g−1v)].

Given a section s ∈ Γ(M,E) on the associated vector bundle (physicalstate) and a local section σ ∈ Γ(U,P ) on the principal bundle (gauge choice),then s can be represented by a vector-valued function ξ : U → V , where

s(p) = [(σ(p), ξ(p))].

Now, we define the parallel transport of a section.

Definition 3.10. Let s ∈ Γ(M,E) be a section on E and γ : [0, 1] → Mbe a curve on M with a horizontal lift γ. The horizontal lift implies theexistence of η : [0, 1]→ V such that

s(γ(t)) = [(γ(t), η(t))].

Then s is said to be parallel transported along γ if η is constant.

This definition is independent of the particular horizontal lift and onlydepends on the curve γ.(3.18)

Exercise 3.18Use Corollary 3.1 to prove thatDefinition 3.10 is independent ofthe horizontal lift.

The covariant derivative of a section s is just the infinitesimal version ofparallel transport

Definition 3.11. Let s ∈ Γ(M,E) be a section on E and γ : [0, 1] → M acurve on M . The covariant derivative of s along γ at p = γ(0) is

∇Xs :=

[(γ(0),

d

dtη(γ(t))

∣∣∣∣t=0

)],

where X = γ′(0), with the same notation from Definition 3.10.

The covariant derivative is also independent of the horizontal lift. In fact,it only depends on the vector X = γ′(0).(3.19)

Exercise 3.19Prove this.The definition of parallel transport and of covariant derivative are in

harmony with the ones from differential geometry if P = LM .If X ∈ X(M) is a vector field, then we naturally have a map p 7→

∇X(p)s ∈ Fp, which is again a section on E. Thus, the covariant derivativecan be viewed as a linear map ∇ : X(M) ⊗ Γ(M,E) → Γ(M,E), treatingX(M) as a module over the space of C∞ functions and Γ(M,E) as a vec-tor space over R. We do not have C∞ linearity on Γ(M,E) because of theproduct rule

∇(fs) = (df)s+ f∇s.

Now let us see what is the local form of the covariant derivative, by localwe mean writing everything in terms of a local section σ ∈ Γ(U,P ):

γ(t) = σ(γ(t))g(t),

eα(p) = [(σ(p), eVα )],

29

Leonardo A. Lessa

where g(t) := g(γ(t)) is the canonical local trivialization and eVα is abasis for V . We only need to calculate the covariant derivatives of the form∇∂/∂xµeα, since any section s ∈ Γ(M,E) is written as

s(p) = [(σ(p), ξ(p))],

= [(σ(p), ξα(p)eVα )],

= ξα(p)eα ≡ sα(p)eα,

and any X ∈ X(M) is written as X = Xµ ∂∂xµ , so

∇Xs = ∇Xµ∂/∂xµ(sαeα),

= Xµ

(∂sα

∂xµeα + sα∇∂/∂xµeα

).

Since eα(γ(t)) = [(σ(γ(t)), eVα )] = [(γ(t), g(t)−1eVα )], then the covariantderivative of eα in the direction of ∂

∂xµ = γ′(0) is

∇∂/∂xµeα =

[(γ(0),

d

dtg(t)−1eVα

∣∣∣∣t=0

)],

=

[(γ(0),−g(0)−1 dg(t)

dt

∣∣∣∣t=0

g(0)−1eVα

)],

= [(γ(0)g(0)−1,AµeVα )],

where we have used equation (3.1) and that A = Aµdxµ. Bear in mindthat Aµ here is in reality the representation ρ(Aµ). In matrix notation withrespect to the basis eVα , we write A α

µ β := ρ(Aµ)αβ , so

∇∂/∂xµ = [(σ(0),A βµ αe

Vβ )] = A β

µ αeβ .

Finally,

(∇Xs)β = Xµ

(∂sβ

∂xµ+A β

µ αsα

). (3.6)

4 The Examples from Another View

4.1 QED and Yang-Mills Gauge Theories

As we have seen, QED is a gauge theory with gauge group U(1). Theprinciple bundle with this gauge group and a connection in it forms theelectromagnetic part of the theory, with the four-potential and the fieldstrength tensor being local expressions for the connection and the curvature,respectively.

The matter field part comes from the associated vector bundle. In QED,we have charged particles called electrons and positrons, which classically –and not worrying about gauge invariance – are just fields with values on thevector space C4. If instead we were interested in non-relativistic quantummechanics with electromagnetic interactions, then the matter field wouldbe a wave function, a complex field. In either case, we can transport the

30

4.2 Dirac Monopole Leonardo A. Lessa

concepts from the principle bundle to a associated vector bundle via a repre-sentation on the vector space in question. This approach incorporates gaugeinvariance intrinsically, since a physical field is a section on the associatedbundle that has, at each point of the base space, many representations inthe original vector space, corresponding to gauge transformation. Howeverso, these representations do not change the section itself. Because of this,the section removes gauge redundancy at the cost of a definition based onharder to work with conjugacy classes. Furthermore, we can parallel trans-port the matter fields, which have noticeable physical consequences, like theAharanov-Bohm effect.

If we allow the gauge group G to be non-abelian, then we still get theterms involving the structure constants. This case is not just some purelymathematical generalization, but it in fact happens in Quantum Chromo-dynamics (QCD), the theory of strong interactions. In QCD, the gaugegroup is SU(3), with gauge potential corresponding to the gluon fields, andthe matter fields are the quark, which are composed of three Dirac fields,corresponding to the three colors they have.

With SU(3) being a non-abelian group, the field strength F = dA+A∧Ais now quadratic in the gauge potential. This has an impact in the dynamicsof QCD, since its Lagrangian contains a term proportional to Tr[FµνFµν ],which is now quartic with A. This means that the quantum perturbationtheory has an vertex (interaction) with 4 gluons. This contrasts with QED,that does not exhibit photon-photon interaction at low energies. The properquantum theory of QCD is much more complicated than this, but gaugetheory by itself already predicts a variety of complex phenomena.

4.2 Dirac Monopole

We saw in Section 1.3 that the existence of a single magnetic charge hasprofound consequences, one of them being the quantization of the electriccharge. Also, we noted that this solution is essentially topological, since wecan exclude the point of the space where the magnetic monopole sits andthus analyze the vacuum solutions to Maxwell’s equations. We will see laterhow we can expect to have nontrivial solution to this problem just fromtopological properties of the base space M = R3 \ 0, but first we willconnect the arguments given earlier with Gauge Theory.

The interesting effects of the Dirac monopole came from coupling non-relativistic quantum mechanics with electromagnetism. This suggests thatthe associated gauge theory will have the quantum mechanical gauge groupG = U(1). Since gauge theory predicts the Faraday tensor F = dA is aclosed form, then it does not accommodate magnetic charges, so we areforced to consider our base space to be M = R3 \0, meaning the magneticcharge is at the origin. From Theorem 2.2, we know that fibre bundles withhomotopic base spaces are equivalent, thus we can simplify our base spaceto M = S2, the sphere.

The sphere cannot be covered by a single coordinate map, and this reflectson the fact that we cannot use a single gauge potential A to describe thesolution to the magnetic monopole problem on the whole sphere. We alreadyknew this from our explicit vector potential solutions, which had a singularity

31

4.2 Dirac Monopole Leonardo A. Lessa

on a ray starting at the monopole, called Dirac string.For simplicity, let us cover the sphere with the north and south hemi-

spheres, plus small strips around the equator on which these open neigh-bourhoods intersect. In spherical coordinates,

HN = (θ, φ) | 0 ≤ θ < π/2 + ε, 0 ≤ φ < 2π,HS = (θ, φ) | π/2− ε ≤ θ < π, 0 ≤ φ < 2π.

As before, we can define a gauge potential on each of hemispheres as (Seeequations (1.23) and (1.24))

AN = ig

4π(1− cos θ)dφ, (4.1)

AS = −i g4π

(1 + cos θ)dφ, (4.2)

where we used the correspondence Aj = ieAj .(4.1) For these potentials to beExercise 4.1Prove that

eφ =∂

∂φ,

and thus, transforming into aone-form via the metric, we havethe association

1

r sin θeφ → dφ.

Remember that eφ is normal-ized.

originated from the same connection form, they have to satisfy the followingrelation on the intersection (See Theorem 3.8)

AN = t−1NSA

StNS + t−1NSdtNS ,

where tNS = exp[iϕ(φ)] : HN∩HS → U(1) is the transition function betweenthe hemispheres, which we consider to be a function only of the azimuthalangle φ since the strip HN ∩HS can be made as small as we want. From theexplicit expressions (4.1) and (4.2) and the fact that the gauge group U(1)is abelian, we have

idϕ(φ) = t−1NSdtNS ,

= AN −AS ,

= ieg

2πdφ.

For tNS = exp[iϕ(φ)] to be well defined, the phase ϕ : S1 → U(1) has tochange by an integer multiple of 2π:

2πn = ϕ(2π)− ϕ(0) =

∫ 2π

0

dϕ = eg,

from which we get our familiar charge quantization equation.A deeper topological reason why charge quantization follows from the

existence of a magnetic monopole comes from analyzing the transition func-tion tNS . The different charges comes from the different ways of describingthe transition function tNS = exp[iϕ(φ)], which is essentially defined on theequator S1. Thus, an equivalent problem is the classification of continuousfunctions ϕ : S1 → U(1), that is, what is the fundamental group π1(U(1))?And we know how to answer that! Since U(1) is homotopically equivalentto S1, then π1(U(1)) = π1(U(1)) = Z, the integer additive group.

An even earlier prediction we could have made is the existence of nontriv-ial vacuum solutions to Maxwell’s equations on the punctured space R3 \0that do not come from electrical charges or currents. We already know that

32

4.2 Dirac Monopole Leonardo A. Lessa

electrical charges generate fields described by a global gauge potential A anda Faraday tensor F = dA, an exact one-form. Moreover, vacuum Maxwell’sequations say that F is a closed form, that is, dF = 0. If we are trying tosearch for solutions that do not come from electrical sources, then we want tofind classify all closed two-forms F that are not exact, which is exactly whatthe cohomology group is! More specifically, the second cohomology groupof the base space M = R3 \ 0, calculated as H2(M) = H2(S2) = R, anon-trivial group! Note that we cannot conclude anything related to chargequantization because we have not touched any aspect of gauge theory.

33

REFERENCES Leonardo A. Lessa

References[1] Mikio Nakahara. Geometry, Topology and Physics. 2nd ed. Institute of

Physics Publishing, 2003. Chap. 1, 9, 10.

[2] John David Jackson. Classical electrodynamics. 3rd ed. New York, NY:Wiley, 1999. Chap. 11.

[3] Robert M. Wald. General Relativity. The University of Chicago Press,1984. Chap. 13, pp. 342–347.

[4] S. Kobayashi and K. Nomizu. Foundations of Differential Geometry.Vol. 1. Wiley Classics Library. Wiley-Interscience, 1963. Chap. 1,2.

[5] R. F. Streater and A. S. Wightman. PCT, spin and statistics, and allthat. 1989. Chap. 1, pp. 9–21.

[6] N. N. Bogoliubov and D. V. Shirkov. Introduction to the Theory ofQuantized Fields. 3rd ed. JohnWilet & Sons, Inc., 1980. Chap. 1, pp. 51–63.

[7] Gerard ’t Hooft. The Conceptual Basis of Quantum Field Theory. June 8,2016. url: http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf (visited on 04/14/2019).

[8] David Tong. Gauge Theory. 2018. url: http://www.damtp.cam.ac.uk/user/tong/gaugetheory/1em.pdf (visited on 04/16/2019).

[9] N. Steenrod. The Topology of Fibre Bundles. Princeton Landmarks inMathematics and Physics v. 14. Princeton University Press, 1999.

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