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Notes on Gauge

Leonardo Almeida Lessa Abril de 2019

Contents

1 Motivating Examples 2 1.1 Classical ...... 2 1.2 Electrodynamics ...... 5 1.3 Dirac Monopole ...... 6 1.3.1 Magnetic Charges ...... 6 1.3.2 Electromagnetic in .8 1.3.3 QM and the Dirac Monopole ...... 9

2 Fibre Bundles 11 2.1 Motivation: Tangent ...... 11 2.2 Fibre Bundles ...... 13 2.2.1 Triviality of Bundles ...... 16 2.3 Principal Bundles ...... 17 2.3.1 Associated ...... 18

3 Connections in Fibre Bundles 20 3.1 Fundamental Vector ...... 20 3.2 ...... 22 3.3 ...... 22 3.4 ...... 24 3.5 Local Forms ...... 24 3.5.1 Gauge Potential ...... 24 3.5.2 Field Strength ...... 27 3.6 Covariant ...... 28

4 The Examples from Another View 30 4.1 QED and Yang-Mills Gauge Theories ...... 30 4.2 Dirac Monopole ...... 31

Here are some notes on gauge theories. We begin by discussing some examples from and their common features. Then, we detour to the of fibre bundles and connections on principal bundles, which form the mathematical of . Finally, we review our previous examples in this new framework.

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

1 Motivating Examples

The notion of gauge appears when we have more degrees of freedom in the description of the theory than there is in the physics of the problem. Sometimes, the underlying mathematical objects, confined to our abstraction of the world, are redundant and unphysical, even though derived results are unambiguous. We artificially distinguish quantities that are physically equivalent. Thus, the of our theory have to be symmetric upon gauge transforma- tions in these equivalent quantities. They possess gauge described by gauge groups, similarly to the familiar symmetries of , but qual- itatively different. A gauge transformation changes our description of the modeled system while a spacetime symmetry changes the modeled system itself, although not modifying the physical laws. Before presenting the mathematical details, we will now discuss some examples. Our goal now is to create a clear image of what we mean by a gauge 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 are determined by Maxwell’s equations. In natural (Lorentz-Heaviside) units, by which we set c = 0 = µ0 = ~ = 1, Maxwell’s equations can be written as

∂B ∇ × E = − , ∇ · B = 0, (1.1) ∂t ∂E ∇ · E = ρ, ∇ × B = + J. (1.2) ∂t Poincaré’s Lemma applied to the homogeneous equations (1.1) tells us there exists at least locally1 a field V and a vector field A, also known as potentials, satisfying (1.1) Exercise 1.1 3 In R , Poincaré’s ∂A Lemma amounts to E = − − ∇V, (1.3) ∂t ∇ × v = 0 ⇒ v = ∇f, B = ∇ × A. (1.4) ∇ · v = 0 ⇒ v = ∇ × u, for some locally defined real Although we are guaranteed of their existence, the potentials V and A function f and vector field u. are far from unique. For every arbitrary function Λ, we may transform the Adapt this to prove equations (1.3) and (1.4). 1We will see later that the Dirac monopole is an example where the vector field A cannot be defined in the whole . This has to do with the topology of the problem and has interesting consequences.

2 1.1 Classical Electromagnetism Leonardo A. Lessa potentials by ∂Λ V 0 = V − , (1.5) ∂t A0 = A + ∇Λ, (1.6) but still get the same electric E0 = E and magnetic fields B0 = B(1.2). Since Exercise 1.2 E and B control the dynamics of charged particles via Lorentz force, we Verify this arrive at the same physics after this transformation. Indeed, this is a case of 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 J µ = (ρ, J), which transform like a four-vector under Lorentz transformations. Similarly, the electric and magnetic fields can be grouped into a antisymmetric (0, 2)- Fµν , called the Faraday tensor or electromagnetic field strength tensor. The components of Fµν in a reference frame where the electric and magnetic fields at some are E = (Ex,Ey,Ez) and B = (Bx,By,Bz) are, in notation,   0 Ex Ey Ez −Ex 0 −Bz By  Fµν =   . (1.8) −Ey Bz 0 −Bx −Ez −By Bx 0 We immediately see that the transformation laws of the electric and magnetic fields 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 Faraday tensor 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 − ∂ν Aµ, (1.9) where we have used Einstein notation to sum repeated indices and lowered the potential with the Minkowski metric ηµν = diag(−1, +1, +1, +1) as such:

ν Aµ = ηµν A = (−V, A). Furthermore, the gauge transformation of (1.6) and (1.5) simplifies to

0 Aµ = Aµ + ∂µΛ. (1.10) Since the electric and magnetic fields remain the same under a gauge transformation, the same happens with Fµν . Thus, the Faraday tensor is gauge . Accordingly, we can write Maxwell’s equations in terms of Fµν and they are also gauge invariant:

µν ∂µ(∗F ) = 0, (1.11) µν ν ∂µF = J , (1.12)

3 1.1 Classical Electromagnetism Leonardo A. Lessa

where ∗F is the dual field strength tensor, defined by ∗F µν = 1 µναβF (1.3). Exercise 1.3 2 αβ Interpret equation (1.11) know- We can solve equations for V and A and work with them, but the final ing that the dual field strength observable results have to be gauge independent. This is not a negative ∗F µν is F µν with E and B swapped. aspect of our theory, quite the contrary. The labor of dealing with non- gauge-invariant quantities like Aµ is compensated by the manifest Lorentz invariance of the theory. Unitarity and locality are also manifest when we keep 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 the case of Electromagnetism, we can set up a differential equation for Λ so as the potentials have some property we want, as long as it doesn’t contradict Maxwell’s equations.

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

For example, we may choose the Lorenz Gauge ∂ Aµ = 0(1.4) to arrive Exercise 1.4 µ What differential equation does at a wave equation for the potential, Λ have to satisfy?

ν (1.12) µν J = ∂µF , µ ν ν µ = ∂µ(∂ A − ∂ A ), µ ν = ∂µ∂ A . (1.13)

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

4 1.2 Leonardo A. Lessa

1.2 Quantum Electrodynamics Having already seen how classical electromagnetism has gauge symmetry, it may not be clear what is the corresponding gauge . In Quantum Elec- trodynamics (QED), however, the gauge group appears naturally. In fact, we won’t even quantize anything, (fortunately!). The only “new” concept that comes from the quantum world is . If the reader is unfamiliar with 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 is controlled by the following Lagrangian 1 L = − F F µν − A J µ. (1.14) EM 4 µν µ More precisely, by the Euler-Lagrange equations associated to L (1.5) EM Exercise 1.5 ∂L ∂ ∂L Why is (1.11) automatically sat- isfied? − ν = 0. (1.15) ∂Aµ ∂x ∂(∂ν Aµ)

One may argue that LEM is not gauge invariant, because of the term µ R 4 AµJ , and it is true. However, the SEM = LEMd x 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: Z Z µ 4 µ 4 − AµJ d x → − (Aµ + ∂µΛ)J d x, M M Z Z µ 4 µ µ 4 = − AµJ d x − [∂µ(ΛJ ) − Λ∂µJ ]d x, M M Z Z = − AµJ µd4x − ΛJ µd3x, M ∂M Z = − AµJ µd4x, M where in the third line we used that J µ is conserved and in the fourth, that the 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 gauge invariant theory. In QED, the field of and their antiparticles, the positrons, is coupled with the electromagnetic field of , the force carriers, in a quite natural way. First, we begin by writing the free Lagrangian for a Dirac ψ, µ LDirac = ψ(iγ ∂µ − m)ψ. (1.16) 1 It describes a fermion (spin 2 particle) with mass m by a Dirac spinor (1.6) field ψ , which has four complex components ψα, α ∈ {0, 1, 2, 3}, and 2 Exercise 1.6 transforms under an irreducible representation of the Lorentz group . Electrons are charged particles. The Lagrangian LDirac has a global internal symmetry, Then why does (1.16) not have the elementary constant ψ(x) → eieλψ(x), ψ(x) → ψ(x)e−ieλ, (1.17) e in it?

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].

5 1.3 Dirac Monopole Leonardo A. Lessa

where global means λ ∈ R is independent of the spacetime point x and internal means the transformation doesn’t change the point x where the fields are evaluated. Since λ is arbitrary and ψ is multiplied by a pure phase eieλ, our gauge group is U(1), complex numbers of modulus one endowed with complex multiplication. Let’s now see what happens if we require this symmetry to be local. With that, 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

µ µ LDirac = ψ(iγ ∂µ − m)ψ → ψ[iγ (∂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 ∇µ by

∇µ = ∂µ + ieAµ (1.19) then we can modify our Lagrangian as

0 µ LDirac = ψ(iγ ∇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, 1 µν we add the electromagnetic Lagrangian term − 4 Fµν F to (1.20) to get our final QED Lagrangian 1 L = − F F µν + ψ(iγµ∇ − m)ψ (1.21) QED 4 µν µ

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

LQED = LDirac + LEM, 1 = ψ(iγµ∂ − m)ψ − F F µν − A J µ, µ 4 µν µ with J µ = eψγµψ.(1.7) Exercise 1.7 Prove that the conserved We did not justify why is it necessary that we have local gauge invariance Noether current associated with (1.18) instead of the weaker assumption of global gauge invariance (1.17). the global symmetry (1.17) of One a posteriori reason is that, from this assumption, the electromagnetic µ the Lagrangian LDirac is J . field and its coupling to the Dirac field follow naturally. A more thorough discussion about the need for local gauge invariance can be found in [7].

1.3 Dirac Monopole 1.3.1 Magnetic Charges The interplay between topology, geometry and gauge theory will become apparent once we introduce the notion of fibre bundles. Before that, the

6 1.3 Dirac Monopole Leonardo A. Lessa example of the Dirac illustrates the far-reaching conse- quences of the geometry of spacetime. µ We include a magnetic four-current JB = (ρB, JB) in Maxwell’s homo- µ µ geneous equation (1.11) in the same way the electric four-current J ≡ JE appears in (1.12):

µν ν ∂µ(∗F ) = JB, µν ν ∂µF = JE.

In terms of E and B fields,

∂B ∇ × E = − − J , ∇ · B = ρ , ∂t B B ∂E ∇ · E = ρ , ∇ × B = + J . E ∂t E Consider a point magnetic charge of strength g in the origin. Its magnetic 3 charge density is ρB(r) = gδ (r), so, similarly to the electrostatic case, the magnetic field produced is g ˆr B = . (1.22) 4π r2 We used the then homogeneous Maxwell’s equations (1.1) to derive the potentials V and A satisfying (1.3) and (1.4). Since they are not homo- geneous if magnetic charges are present, we cannot have globally defined potentials as before. If we could, then B = ∇ × A would imply ∇ · B = 0 everywhere. However, in the case of a point magnetic charge, there is a AN 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 spherical coordinates (r, θ, φ), AN is given by

g 1 − cos θ AN(r) = ˆe , (1.23) 4π r sin θ φ which is well-defined for r ∈ R3 \ S, with S = {(x, y, z) ∈ R3|z ≤ 0, x = y = 0}, a region called the . Not only is AN singular at the origin, where the monopole sits, but also in a line starting on the origin and going to infinity in the −z direction.(1.8) For r ∈ R3 \ S, Exercise 1.8 Why is AN not well-defined on 1 ∂ 1 ∂ ∇ × AN(r) = (AN sin θ)ˆr − (rAN )θ,ˆ S? r sin θ ∂θ φ r ∂r φ g ˆr = , 4π r2 = B(r).

Can we also cover the S region? We know we cannot do this with just one vector potential. However, we can imitate AN by defining AS on R3 \S0, with S0 = −S = {(x, y, z) ∈ R3|z ≥ 0, x = y = 0},

g 1 + cos θ AS(r) = − ˆe , (1.24) 4π r sin θ φ

7 1.3 Dirac Monopole Leonardo A. Lessa

Figure 2: Illustration of the magnetic field B produced by a magnetic monopole 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 this scene.

You may have noticed that the superscripts N and S mean north and south hemispheres, where the vector potentials AN and AS are well-defined. Let us pause for a moment to ponder the geometric meaning of AN and AS. Outside the origin there is no magnetic charge, thus we can use the usual Maxwell’s equations (1.1) and (1.2) to find find our solutions. Indeed, for r 6= 0, we can find vector potentials for the magnetic field. But because we are now solving Maxwell’s equations in R3 \0, a non simply connect subset of R3, 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 connection with the geometry of the underlying space M will be made precise when we study connections ( 3) and the cohomology group H2(M) (Section 4.2).

1.3.2 Electromagnetic Interaction in Quantum Mechanics When we join Quantum Mechanics with the Dirac monopole, we get a surprising result: the of electrical charge. Before doing so, 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 coordinate patches to cover S2.

8 1.3 Dirac Monopole Leonardo A. Lessa

Without electromagnetism, Schrödinger’s equation for a particle of ψ(t, r) in a potential V (r) is (~ = 1) ∂ψ Hψ = i , with ∂t |p|2 H = + V (r). 2m The quantum-mechanical momentum is p = −i∇, a space derivative. Imitating what we did for the electromagnetic coupling of the Dirac field, we will use equation (1.19) of the covariant derivative to redefine our momentum as p → p + qA. This redefinition of the momentum to account for electromagnetic interaction is called . It couples a charged particle with charge q (with no spin) to the electromagnetic field. Although A is not gauge invariant, this coupling gives gauge invariant physical results! To see this, let us pick a wave function ψ(t, r) which is a solution to the Schrödinger’s equation in a region with vector potential A. Its Hamiltonian is 1 H = (p + qA)2 + V (r). (1.25) 2m If we do a gauge transformation A → A + ∇Λ, the Hamiltonian H changes and so does its eigenfunction ψ(t, r) → ψ˜(t, r), which can be expressed in terms of the old solution ψ by(1.9) Exercise 1.9 −ieΛ(r) Verify this by calculating the ef- ψ˜(t, r) := e ψ(t, r). (1.26) fect of the operator (p + eA + e∇Λ) acting on ψ˜. We know from Quantum Mechanics that global complex phases multiply- ing the wave function are not physical, since all measurable quantities come from taking the squared of the absolute value of inner products. With this in mind, we might think the complex phase factor in (1.26) can be removed without repercussions. Quite the contrary, this complex factor is responsible for a plethora of counterintuive effects. They all have in common the ap- pearance of a gauge invariant measurable quantity in terms of a not gauge invariant quantity, like A. One of these effects is what we will discuss now: when we interact a quantum particle with a magnetic monopole. The other is the Aharanov-Bohm effect, by which we can detect the changes in phase of 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. The domains of definition of the vector potentials AN and AS intersect each other in R3 \ (S ∪ S0), which is the whole space minus the z line. In this region 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

9 1.3 Dirac Monopole Leonardo A. Lessa transformation. To see if this is the case, we calculate the difference g 1 AN − AS = ˆe , 2π r sin θ φ = ∇(gφ/2π).

Mixed feelings. The expression gφ/2π is not a function, for a 2π rotation in 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 now Λ = gφ/2π as our gauge transformation between AS and AN, postponing a more rigorous discussion.4 Here comes the main argument. If we have a quantum particle in a space with a magnetic monopole at the origin, then its wave function ψ in the 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 and ψ into −ieΛ −i qg φ ψ˜(r) = e ψ = e 2π ψ(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

−i eg φ −i qg (φ+2π) e 2π = e 2π , which is equivalent to qg = n ∈ Z. (1.28) 2π In words, If there exists a magnetic monopole with strength g, then electrical charges are quantized. Moreover, magnetic charges also come in discrete amounts by equation (1.28). From the of , we know every charged 1 body has a charge which is a integer multiple of 3 e, where e is the elementary charge. Most particles (elementary or not) have charges that are integer 1 multiples of e, such as electrons, , photons. The 3 includes , 2 which come in a family with charge + 3 e – up, charm and top – and a family 1 with charge − 3 e – down, strange and bottom.

4If the reader is unsettled, we anticipate the solution. We will treat the vector potential g as a one-form, so this gauge transformation is really a one-form 2π dφ, which is well- defined. 5We look for wave functions in the equator to stay away from the Dirac strings S and S0, so both AS and AN are well-defined

10 Leonardo A. Lessa

2 Fibre Bundles

We 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 principal bundles, , etc, many of which we will treat here. The main refer- ence for the theory of principal bundles and connections is Kobayashi and Nomizu’s “Foundations of Diferential Geometry” [4]. For the introduction of fibre bundles, we will use Steenrod’s “The Topology of Fibre Bundles” [9]. For our first example of fibre bundle, the , we assume the reader has some familiarity with the theory of . Although this example is independent of the subsequent sections, it already has many features we want to explore in the general theory.

2.1 Motivation: Tangent Bundle One very useful prototypical example of a fibre bundle is the tangent bundle TM of a M of dim(M) = n. It is the disjoint union of the tangent TpM, p ∈ M, as such G [ TM := TpM ≡ (p, TpM). p∈M p∈M

3 If M is a two-dimensional surface of R , one can imagine TpM is a plane tangent to a point p ∈ M of the surface, and TM is the set of all those planes, separated by which points they are tangent to. The tangent bundle TM is itself a manifold of dimension 2n, called the total space if viewed as a fibre bundle. The charts of TM are described by a chart of M, the base space, and an open set of the vector space TpM. This arrangement may seem like TM is the product space M ×V , with V an n-dimensional vector space, isomorphic to TpM, ∀p ∈ M, but this is not the case for every manifold M. In fact, if TM ' M × V (is trivial), then M is a . For the spheres Sm, only the m = 1, 3 and 7 ones are parallelizable. 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 2.1 The tangent bundle TM is not necessarily a direct product, but it is Do these numbers remind you of locally a direct product or, in other words, locally trivial. More precisely, for something? each chart (U, xi) of M, we can consider the restricted bundle TU, treating the coordinate neighbourhood U as a manifold. From the definition of TM, 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 ∈ M and a vector V ∈ TpU, which can be written as

µ ∂ V = V µ , ∂x p

1 n n ∂ n for (V ,...,V ) ∈ R , since { ∂xα p}α=1 is a base for TpU. Naturally, we n can define a diffeomorphism φU : TU → U × R by

1 n φU (u) = (p, (V ,...,V )),

11 2.1 Motivation: Tangent Bundle Leonardo A. Lessa

Figure 3: Illustration of the tangent bundle TM. A vector v ∈ TpM in the fiber TpM is projected to p ∈ M by π. which is precisely what we meant by saying TM is locally a direct product space. We will call φU the local trivialization map. Another common feature of fibre bundles is the projection to the base space. In the case of TM, we can define π : TM → M simply by π(u) = p, −1 −1 where u = (p, V ) ∈ TM. Clearly, TU = π (U) and TpM = π (p), the latter 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- −1 n −1 n alizations φU : π (U) → U × R and φV : π (V ) → V × R are connected, just as the coordinates xµ : U → R and yν : V → R are related by −1 ν ν ∂(y ◦ x ) µ y (p) = µ x (p), ∂x x(p)

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

µ ∂ V = Vx µ , ∂x p ν µ ∂y ∂ = Vx µ µ , ∂x ∂y p

ν ∂ = Vy ν , ∂y p

ν ∂yν µ so Vy = ∂xµ Vx and thus the local trivializations φU and φV are related by

−1 µ −1 ν −1 ∂yν µ φU (p, Vx ) = φV (p, Vy ) = φV (p, ∂xµ Vx ).

ν 6 ν ∂y µ We will often shorten the notation and write y = ∂xµ x when the point p ∈ U ∩V ⊆ M is implicit from the context.

12 2.2 Fibre Bundles Leonardo A. Lessa

ν ∂yν The matrix G µ = ∂xµ is nonsingular (det(G) 6= 0) because we demand that y ◦ x−1 be a diffeomorphism for M to be smooth. It is this matrix that mixes the components of the vector V when we change coordinates. The set of all real nonsingular matrices of order n is called the GL(n, R).(2.2) Thus, we say the structure group of TM is GL(n, R) and ν Exercise 2.2 G µ is a transition function between charts. Given a coordinate chart (U, xi) The bold terms above are the essential components of a fibre bundle, and a matrix G ∈ GL(n, R), which are beautifully exemplified by the tangent bundle TM. Other cor- can we create another coordinate j related concepts will appear throughout this presentation, such as cross chart (U, y ) such that the tran- sition function from x to y is section and connection, generalizations of vector field and parallel trans- given by G? This type of ques- portation, respectively. tion is related to the problem of reducing the structure group of a (See Sec. 2.3). 2.2 Fibre Bundles With 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 G called the structure group, which acts freely on F on the left,

(vi) a open covering {Ui}i∈I of M indexed by I, consisting of open neigh- bourhoods Ui,

−1 7 (vii) a φi : Ui × F → π (Ui) for each i ∈ I, called a local trivialization. satisfying the following relations:

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

−1 (ii’) Each local trivialization φi : Ui × F → π (Ui) is constrained by the projection 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, then −1 we require that the transition function tij(p) := φi,p ◦ φj,p : F → F coincides with the operation of an element of G on F . Furthermore, we require tij : Ui ∩ Uj → G, the map from p to the corresponding group element tij(p) of the transition function – also denoted by tij by abuse of notation – to be continuous. 7To comply with the notation of Nakahara’s and Steenrod’s books, we switched domain with 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),

(2.3) where here we treated tij(p) ∈ G as an element of G acting of f ∈ F . Exercise 2.3 −1 Prove that It will also be useful to define a function pi : π (Ui) → F that as- signs the elements of the bundle to their fibre representatives via the local tij tjk = tik trivialization, such that, for u ∈ E, tii = e −1 −1 tij = [tji] pi := [φi,π(u)] (2.1)

Analogous to the definition of manifolds by equivalences of atlases, we start with an open cover {Ui} to define a coordinate bundle to, then, form a 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 local trivializations – are equivalent if (E, π, M, F, G, {Ui} ∪ {Vj}, {φi} ∪ {ψj}) is a coordinate bundle (2.4).A fibre bundle, denoted by (E, π, M, F, G), E →π Exercise 2.4 Convince yourself that this M or simply E, when the rest of the structure is implicit, is an equivalence amounts to requiring compati- class of coordinate bundles by the relation defined above. bility between local trivializa- tions φi and ψj as we did in Just as with manifolds, we think about fibre bundles in terms of their condition (iii’) of Definition 2.1. representatives – the coordinate bundles – and prove theorems about these which are invariant by the 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 topological requisites with smooth ones. More specifically, all the topological spaces E, M, F and G are smooth manifolds, with G being a , and all continuous (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 sends points p ∈ M of the base space to an element of its fibre X(p) ∈ Fp. In other words, π ◦ X = id. The set of all sections from M to E is Γ(M,E), or simply 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 can construct 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 trivialization is 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 action of 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 structure group 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 wavefunction and the Dirac field are sections in a vector bundle associated to a principal bundle (See Section 2.3). Example 2.3. Principal bundle. One way to define principal bundles is to say they are fibre bundles whose typical fibre F is equal to the structure group 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 acts freely on E on the right, and then define the base space M to be the quotient space of the action of G on E

M = P/G.

Furthermore, we require E to be locally trivial in the sense that there exists −1 an open covering {Ui} of M and diffeomorphisms φi : Ui × G → π (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 local trivializations: Theorem 2.1. Let G be a Lie Group acting freely on a manifold F , M a 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 = tki π and tii = e. Then, there exists a fibre bundle E → M with transition functions {tij} and coordinate neighbourhoods {Ui}. Proof. First, we define G X := Ui × F i∈I This is our prototype for the total space. We still need to connect the fibres from different intersecting parts of the union. To do this, we assign a equivalence relation between points (p, fi) ∈ Ui × F and (q, fj) ∈ Uj × F of X: (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 by π([(p, f)]) = p and the local trivialization associated to a open neighbour- hood U is defined by φ (p, f ) = [(p, f )].(2.5) i i i i Exercise 2.5 With this in mind, we can play with cylinders and Möbius strips: I glossed over some details, like what is the topology of X and if π is continuous and so on. If you Example 2.4. Cylinder and Möbius strip. What are the possible (topo- care about those, try for yourself 1 logical) fibre bundles with base spaces M = S and fibres F = [−1, 1], the to complete them! All the details unit interval? One natural candidate is the cylinder M × F . For us to use are done in [9]. Theorem 2.1, it suffices to find an open covering {Ui} of S1, a structure group G acting on F and a set of transition functions {tij}. 1 2 1 The circumference S ⊂ R is naturally covered by UN = S \{(0, −1)} 1 and US = S \{(0, 1)}. Their intersection has two connected components 1 UN ∩ US = U+ ∪ U−, defined by U± := {(x, y) ∈ S | ± x ≥ 0}. Thus, we have 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 on U−. Hence, the simplest cases are if G has one element, the trivial group G = {e}, or two elements, the binary numbers group G = Z2. In the former case, we can have G = {e} acting on F with the identity, and then 1 tSN = id and E = S × [−1, 1] is the cylinder. In the latter case, we can have G = Z2 = {0, 1} acting on F by

0 · f = f 1 · f = −f,

and then

tSN |U+ = 0 tSN |U− = 1. This results in the fibre bundle of the Möbius band!

0 Definition 2.4. Let E →π M and E0 →π M 0 be fibre bundles. A smooth 0 map f : E → E is a if it maps each fibre Fp ⊂ E onto the 0 0 corresponding fibre Ff(p) ⊂ E .

2.2.1 Triviality of Bundles One way to determine the triviality of a fibre bundle is if the base space is contractible to a point. Before enunciating the exact theorem for this, we need to introduce bundle maps and pullback bundles. A bundle map f : E → E0 induces a map f¯ : M → M 0 on the base manifolds, since f preserves the fibres.(2.6) Exercise 2.6 Prove that f¯◦π = π0 ◦f. Express this in terms of a commuting di- Lemma 2.1. Two coordinate bundles are equivalent if, and only if, they have agram. the same base space and there exists a diffeomorphic bundle map between them.

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, we ∗ can 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 map ∗ f : N → M, the total space of the pullback bundle (f E, π1,N,F,G) is defined 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 is −1 just π1(p, u) = p; the fibre is the same of E, as π1 (p) = (p, Ff(p)) ' F ; ∗ given an open covering of M {Ui}, the open neighbourhoods of f E are 0 −1 Ui = f (Ui); and the local trivializations are

0 0 −1 0 φi : Ui × F → π1 (Ui ), 0 φi(p, f) = (p, φi(f(p), f)).

There is a natural bundle map π2 :(p, u) 7→ u between the pullback bundle 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 theorem.

16 2.3 Principal Bundles Leonardo A. Lessa

Definition 2.6. Two maps f, g : X0 → X between topological spaces X and X0 are said to be homotopic if there exists a map F : X0 × [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 homotopic 8 ∗ ∗ maps from a Cσ space N to M, then the pullback bundles f E and g E are equivalent. Proof. See Steenrod [9], Theorem 11.4.

If M is contractible to a point p0 ∈ M, then id : M → M is homotopic ∗ ∗ to the constant map e0 : M → {p0} ⊂ M. Since id E = E and e0E = M × F (2.7), then Theorem 2.2 says Exercise 2.7 Corollary 2.1. The bundle E →π M is trivial if M is contractible to a point. If this is not clear, try to prove that the pullback of E via e0 is the same as the pullback of 2.3 Principal Bundles {p0} × F via e0, since it is the constant map. Principal 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) where F = G, also called the G-bundle over M, denoted only by (P, π, M, G). G acts trivially on the fibre. Roughly, if G is the gauge group of a physical theory, the G-bundle will be the space of all possible gauge choices. Now, back to the mathematical details. Since the transition functions act on the fibres on the left, then the right action of G on the fibres F = G is independent of the local trivialization and thus G can act on the P bundle itself. The action of a ∈ G on a bundle element u ∈ P is defined locally via −1 −1 a trivialization φi : Ui × G → π (Ui), where p = π(u) ∈ Ui. If φi (u) = (p, gi), then we define ua by

ua := φi(p, gia). Now, we prove that this definition is independent of the local trivialization chosen. 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 argument and is transitive on fibres:

∀u1, u2 ∈ Fp, ∃a ∈ G, u1 = u2a, and free:(2.8) Exercise 2.8 8 A Cσ-space is a normal locally compact manifold that admits a countable open cov- Prove that the action of G on E ering. is transitive and free.

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 bundle P , we can construct a local trivialization defining

−1 φσ : U × G → π (U),

φσ(p, g) = σ(p)g.

This map we have constructed is invertible since the right action of G on E is 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 2.9 Prove that the transition func- Definition 2.8. The canonical local trivialization associated to local tion tij : Ui ∩ Uj → G that connects the trivializations de- section σ ∈ Γ(U, P ) is a map g : P → G such that fined by σi ∈ Γ(Ui,P ) and σj ∈ Γ(Uj ,P ) satisfies σj (p) = ∀u ∈ U, u = σ(p)g(u). σi(p)tij (p).

We will see later that the choice of a cross section in a G-bundle is interpreted as gauge fixing. Note that, as with the local gauge invariance of QED encountered in Section 1.2, the cross section can vary along the base space 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 that G 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 to be 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 useful notation 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 principle behind 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 principal bundle. 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 trivialization −1 −1 φE : U × V → πE (U) on E using the trivialization φ : U × G → π (U) on P such that the following diagram commutes

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

18 2.3 Principal Bundles Leonardo A. Lessa

id ×ρ−1 U × G × V U × V

φ×id φE

−1 [] −1 π (U) × V πE (U) More directly, we can define(2.10)(2.11) Exercise 2.10 Show that this definition guaran- φE(u, v) := [(φ(u, e), v)] tees the preceding diagram com- mutes. Conversely, given a vector bundle (E, πE,M,V,G), we can employ The- Exercise 2.11 orem 2.1 to define an associated principal bundle (P, π, M, G). To do this, What are the local trivializations we just employ the same open covering, base space and transitions functions of E = P ×G V ? from E →π M, and require that G acts on itself, the fibre, by the usual group operation.(2.12) Exercise 2.12 In our previous examples, the physical states that underwent gauge trans- Convince yourself that the vec- formations were maps from a spacetime M to a vector space V (2.13). The tor bundle associated to the gauge freedom made the vector representation of a physical state highly non- principal bundle we constructed unique. Thus, a one-to-one association of a physical state to a mathematical here is the original bundle we started with. object has to gauge out this unphysical freedom. This is the main idea be- Exercise 2.13 hind the construction of the associated vector bundle. We append the vector Identify M and V of Sections 1.2, space V to the principal bundle P and take the quotient by the right action 1.3.1 and 1.3.2. defined in (2.3.1), which is just a way to gauge transform the vector v via the representation 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 of TM is GL(n, R), where n = dim(M), then LM is locally a product space Ui × GL(n, R), welded by the transition functions tij : Ui ∩ Uj → GL(n, R). In matrix notation, we have

µ ν (p, X α ) ∈ Ui × GL(n, R), (p, Y β ) ∈ Uj × GL(n, R), µ µ µ ν ∂x ν X α = (tij) ν Y α = ν Y α . ∂y p As the notation used above may have already hinted, we can also view n the frame bundle LM as the collection of vector bases {Xα}α=1 ⊂ TpM, called frames. More specifically, given a base point p ∈ M, we can define n the frame space LpM as the set of all frames {Xα}α=1 ⊂ TpM and then G LM = LpM. p∈M

Given a chart (U, x) of M, with p ∈ U, each frame {Xα} ∈ LpM is asso- µ ciated to a non-singular matrix X α ∈ GL(n, R) whose columns are the x coordinates of X : (2.14) α Exercise 2.14 µ  | | |  Why is X α non-singular? µ X α = X1 X2 ...Xn . | | |

19 Leonardo A. Lessa

The manifold structure of LM is analogous to the tangent bundle one L – if U is an open neighbourhood of (p, {Xα}) ∈ LM whose projection µ µ to M fits inside a chart set U, then we map (p, {Xα}) 7→ (x (p),X α ) ∈ 2 Rn+n . The bundle structure of (LM, π, M, GL(n, R)) is then equivalent to the construction via association with TM.(2.15) Exercise 2.15 α What the right action of G β ∈ GL(n, R) does to frames in LM? 3 Connections in Fibre Bundles

3.1 Fundamental Before jumping into the theory of connections, we briefly review Lie groups and Lie algebras and prove some technical theorems that will be of good use in later proofs. If the reader is used to Lie groups and don’t want to dive into the details of proofs, then they can skip this section. The g of the Lie group G is the algebra of all left-invariant vector fields A ∈ X(G), (Lg)∗A|h = A|gh, under the operation of the Lie bracket. Alternatively, it is the Lie algebra TeG, since any vector a ∈ TeG induces a left-invariant vector field via A|g = (Lg)∗a.

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

# d ∞ A |u(f) = f(u exp(tA)) , f ∈ C (P ). dt t=0

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

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

# # 1 # # [A ,B ]|u = lim [(Ra )∗(B |ua ) − B |u], t→0 t −t t 1 # = lim [(Ra ◦ σua )∗(B|e) − σu∗(B |e)], t→0 t −t t by the definitions,

−1 Ra−t ◦ σuat (c) = uatcat = σuRa−t Lat ,

for any c ∈ G. Since B is left-invariant, then (Ra−t ◦σuat )∗(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

# # 1 # [A ,B ]u = lim [σu∗Ra ∗(B|a ) − σu∗(B |e)], t→0 t −t t 1 # = σu∗ lim [Ra ∗(B|a ) − (B |e)], t→0 t −t t # = σu∗([A, B]e) = [A, B] .

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

Definition 3.2. The vertical subspace VuP is the subspace of TuP tan- 11 gent to the fibre Gp , 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 on P , and the dimension of g is the same as of the fibre Gp ' G, the association # A 7→ A is an isomorphism of Lie algebras g and VuP . In other words, VuP is 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 the pushforward map ad(a) := (Ad(a)) .(3.2) ∗ Exercise 3.2 For G a matrix group, ad(a) = Ad(a), if we view g as a set of matrices Prove that ad(a) = (Ra−1 )∗. with the same order as G’s.(3.3). Since we will work with matrix group in −1 Exercise 3.3 our examples from physics, then we will often write ad(a)A = aAa . Prove this 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→ −1 φ ◦ φt ◦ φ generates φ∗X. Proof. Let p ∈ M, q = φ−1p and f ∈ C∞(M). The flow of the 1-parameter −1 group of local transformations φ ◦ φt ◦ φ generates the following vector field

d −1 d −1 (φ ◦ φt ◦ φ ) (f) = f(φ ◦ φt ◦ φ p), dt p dt d = f(φ[φ (q)]), dt t = φ∗(X|q),

= (φ∗X)|p.

−1 Thus, φ ◦ φt ◦ φ 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, by # −1 −1 Proposition 3.1, (Ra)∗A is generated by the flow RaRat Ra = Ra ata = −1 RAd(a )at . Again, since t 7→ at generates the vector field A, then t 7→ −1 −1 −1 Ad(a )at generates Ad(a )∗A = ad(a )A, as desired. 11 −1 The fibre Gp = π (p) at p ∈ M is a closed submanifold of P .

21 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 TuP , u ∈ P , into the vertical subspace VuP and a horizontal subspace HuP such that

(i) TuP = HuP ⊕ VuP for all u ∈ P

(ii) () A smooth vector field X ∈ X(P ) is decomposed into H V a horizontal part X |u ∈ HuP and a vertical part X |u ∈ VuP as X = XH + XV .

(iii) HugP = Rg∗(HuP ) for every u ∈ P and g ∈ G. The preferred way to generate this separation is by a ω ∈ g ⊗ Ω1(P ), which is a Lie-algebra-valued one form on P .(3.4) Besides Exercise 3.4 being an element of this tensor product space ω can also be viewed as a Given a basis {Tα} of g, prove there exist one-forms {ωα} such map that smoothly associates to each u ∈ P a linear transformation ωu : P α ∞ that ω = α Tα ⊗ ω . TuP → g, or as a ω : X(P ) → C (M, g) between modules over C∞(M) functions. With this last form, it is more clear what the connection does to vectors fields X ∈ X(P ). It projects X|u ∈ TuP to VuP , with ker(ωu) = HuP . Now, let us present a proper definition. Definition 3.4. Connection form or 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) Rgω = 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 , so there is no loss in generality for considering ω ∈ g ⊗ Ω1(P ) instead of the former geometrical definition.(3.5)(3.6) Exercise 3.5 Prove that

Rg∗HuP = HugP 3.3 Parallel Transport for HuP = ker(ωu). With a connection, we can distinguish horizontal spaces in the tangent Exercise 3.6 spaces of the principal bundle. This enables us to lift a curve γ : [0, 1] → M How would you define XH and V in the base space to a curve γ˜ : [0, 1] → P in the principal bundle such that X given ω? 0 γ˜ (t) ∈ Hγ(t)P . Intuitively, γ specifies the direction of γ˜ between fibers, and the connection tells how the elevation changes along fibers.

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

0 (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 that π(u) = γ(0). Then there exists an unique horizontal lift γ˜ : [0, 1] → M of γ such 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 other horizontal 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 a horizontal lift of γ (See exercise 3.5). If γˆ is another horizontal lift of γ, then there exists a g ∈ G such that γˆ(0) =γ ˜(0)g. From the uniqueness of the horizontal lift given a starting point u =γ ˆ(0) =γg ˜ ∈ P , then we have γˆ =γg ˜ Given a curve γ on M with endpoints γ(0) = p and γ(1) = q, we can now construct a map Γ(γ): Gp → Gq that parallel transports elements u ∈ Gp of the fibre at p to elements Γ(γ)(u) of the fibre at Gq via a horizontal lift starting at u. (3.7) Exercise 3.7 There is a connection between parallel transport and the fundamental Using Corollary 3.1, prove that group π1(M) of the base space M. To see this, we first note how Γ behaves under composition of curves. Γ(γ) ◦ Rg = Rg ◦ Γ(γ) Theorem 3.4. Let α : [0, 1] → M and β : [0, 1] → M be curves satisfying α(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 some (3.8) τγ (u) ∈ G . If we span all loops γ based on p ∈ M, we get the holonomy (3.9) (3.10)(3.11) Exercise 3.8 group What is the analogous of Theo- rem 3.4 to τ ? Hol = {τ (u)|γ is a loop based on p = π(u)}. γ u γ Exercise 3.9 Analogously, we can define the restricted holonomy group restricting to count Prove that Holu is a subgroup of G using Theorem 3.4. only the contractible curves: Exercise 3.10 0 Using exercise 3.7, prove that Holu = {τγ (u)|γ is contractible to p = π(u)}. −1 Holug = g Hug. These groups are related to the fundamental group π1(M, p) based on p by the following theorem Thus, all holonomy groups Holu with the same base point p = Theorem 3.5. There is a epimorphism (surjective homomorphism) Φ: π(u) are isomorphic. 0 Exercise 3.11 π1(M, p) → Holu / Hol , with p = π(u). u If M is connected, prove that Hol is the same up to a conju- Proof. For each equivalence class of curves α ∈ π1(M, p), we pick a repre- p 0 gation: sentative curve γ ∈ α and assign Φ(α) = τγ / Holu. This map is independent −1 of the representative γ since if we had picked another one γˆ ∈ α, then Holq = τγ Holp τγ ,

0 0 where γ is a curve from p to q. τγˆ(u)/ Holu = τγ (u) · τγ−1γˆ(u)/ Holu, 0 = τγˆ(u)/ Holu,

23 3.4 Curvature Leonardo A. Lessa since γˆ−1γ is contractible. That the map Φ is a epimorphism comes from Theorem 3.4 and the definition of Holu.

3.4 Curvature In differential geometry, the measures the de- gree to which a vector parallel transported in a loop fails to come back to itself. We have seen that there is also a notion of parallel transport in gauge theories, so it is natural to think that there exists a notion of curvature dependent only on the connection of a principal bundle.

Definition 3.6. The Covariant Derivative of a vector-valued k−form φ ∈ V ⊗ Ωk(P ) is a vector-valued (k + 1)−form Dφ ∈ V ⊗ Ωk+1(P ) defined by H H Dφ(X1,...,Xk+1) := (dP φ)(X1 ,...,Xk+1), H where dP is the that acts on the k−forms of φ and Xi are the horizontal parts of the vectors Xi ∈ X(P ). Definition 3.7. The Curvature two-form Ω ∈ g ⊗ Ω2(P ) is the covariant derivative of the connection one-form ω:

Ω := Dω.

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

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

Holu0 , 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 Forms 3.5.1 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 the connection form ω ∈ g ⊗ Ω1(P ).

Definition 3.8. Given a connection form ω ∈ g ⊗ Ω1(P ) and a local section σ ∈ Γ(U, P ), the (local) gauge potential is the Lie-algebra-value one-form A ∈ 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 be globally defined. It is clear now that this corresponds to the fact that we cannot 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 values in the Lie algebra g. It is easier to do calculations in the base manifold M than in the principal bundle P , since the latter can have an unknown twisted topology, 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 than that, what is the compatibility condition between gauge potentials Ai and Aj 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 theorems below 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 ⊂ M and a local section σ ∈ Γ(U, P ), there exists a local connection one-form ω ∈ g × Ω1(π−1(U)) such that A = σ∗ω. The connection is given by

−1 ∗ −1 ω = g (π A)g + g dP g, where g is the canonical local trivialization associated to σ (See Definition 2.8).(3.12) Exercise 3.12 Prove that π∗A# = 0. What is For the proof of this theorem, we refer to Theorem 10.1 of [1]. π∗ doing?

1 Theorem 3.8. Two local gauge potentials Ai ∈ g × Ω (Ui) and Aj ∈ g × 1 Ω (Uj) arise from the same connection form if, and only if, they satisfy

−1 −1 Aj = tij Aitij + tij 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 and σj∗X are connected by

−1 # σj∗X = Rtij ∗(σi∗X) + (tij 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 ⊗ 1 Ω (U) associated to σ1 ∈ Γ(U, P ) changes via a gauge transformation σ2 := σ (p) · g(p), where g : U → G, as(3.13) 1 Exercise 3.13 Use Lemma 3.1 to prove the −1 −1 A2 = g A1g + g dg. transformation law of the gauge potential. If G is abelian, then we have simply

−1 A2 = A1 + g dg.

12If the reader wants to see the general versions of Theorems 3.7 and 3.8, in which it is not 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 gauge transformations are of the form g(p) = e−iΛ(p). Then, the gauge potential transforms as

A0 = A + eiΛ(−idΛ)e−iΛ, = A − idΛ.

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

0 Aµ = 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 theory as described in this chapter.

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

˜ρ −1ρ α β −1ρ λ A σ = txx˜ αA βtxx˜ σ + txx˜ λdtxx˜ σ ∂x˜ρ ∂xβ ∂x˜ρ ∂xλ  = Aα + d , ∂xα ∂x˜σ β ∂xλ ∂x˜σ

α ∂xα ˜ρ ˜ρ ν for txx˜ β = ∂x˜β If we write the gauge potentials as A σ = A νσd˜x and α α µ α ∂xµ ν A β = A µβdx = A µβ ∂x˜ν d˜x , then in the x˜ components, the equation above becomes

∂x˜ρ ∂xβ ∂xµ ∂x˜ρ ∂2xλ A˜ρ = Aα + , νσ ∂xα ∂x˜σ ∂x˜ν µβ ∂xλ ∂x˜ν ∂x˜σ which is the same transformation law of Christoffel symbols, if we assign α α A µβ = Γ µβ. In section 3.6, we will see a stronger reason for the validity of 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˜ locally on 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 we did 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 local cross section σ ∈ Γ(U, P ), which sets our local trivialization, and satisfies σ(γ(0)) = u. Calling σ(t) = σ(γ(t)), for t ∈ V in a neighbourhood of t = 0, then there exists a function g : V → G such that γ˜(t) = σ(t)g(t) and g(0) = e. Since g(t) is much more tractable than γ˜, we will now construct an 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 0 0 0 (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)), 0 −1 0 # = ω(Rg(t)∗(σ∗γ ) + [g(t) dg(γ )] ), dg(t) = g(t)−1ω(σ γ0)g(t) + g(t)−1 . ∗ dt

0 ∗ 0 0 Since ω(σ∗γ ) = σ ω(γ ) = A(γ ), then

dg(t) = −A(γ0)g(t), (3.1) dt whose formal solution is

 Z t   Z  g(t) = P exp − A(γ0)dt = P exp − A 0 γ 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 section σ can be defined. Using the compactness of the interval [0, 1], we only need finitely made of these sets to cover the entire curve. Finally, to go from open set to the other, we just transform the initial point of the horizontal lift as γ˜(0) → γ˜(0)g(1). The uniqueness of γ˜ is guaranteed by the uniqueness of the solution of the ODE (3.1). (3.14) Exercise 3.14 Prove that the path-ordered exponential transforms under 3.5.2 Field Strength gauge transformations by conju- gation. Thus, the trace of g(t) Similarly to the connection one-form, we can pullback the curvature two- is gauge invariant. It is called form on P via a local section σ ∈ Γ(U, P ) to a two-form on M: the in the context of QFT. Definition 3.9. Given a curvature two-form Ω = Dω ∈ g ⊗ Ω2(P ) and a local section σ ∈ Γ(U, P ), the (local) field strength is the Lie-algebra-value two-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 = T ⊗ Aα, we define(3.15) α Exercise 3.15 Prove all the relations in equa- 1 A∧A := (T T )⊗(Aα ∧Aβ) = f γ T ⊗(Aα ∧Aβ) = [T ,T ]⊗(Aα ⊗Aβ). tion (3.3). α β 2 αβ γ α β (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 ) = dP ω(X,Y ) + [ω(X), ω(Y )],

in other words, Ω and ω are related by

Ω = dP ω + 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 1 µ ν (3.2). In components F = 2 Fµν dx ∧ dx , we have

Fµν = ∂µAν − ∂ν Aµ + [Aµ, Aν ], (3.4)

where each component Fµν or Aµ is a function of type U → g. If we pick a γ basis {Tα} of g with fαβ , we can expand the equation even further with

α Fµν = Fµν Tα, β Aµ = Aµ Tβ, such that (3.4) becomes an equation with real functions defined on U.

γ γ γ γ α β Fµν = ∂µAν − ∂ν Aµ + fαβ Aµ Aν (3.5) In electromagnetism, the gauge group is U(1). This group has a one- α dimensional Lie Algebra, so we can drop the algebra indices on Aµ and α Fµν . Another consequence of this is that U(1) is an , so [Aµ, Aν ] = 0. Finally, equation (3.5) becomes Fµν = ∂µAν − ∂ν Aµ with Fµν naturally interpreted as the Faraday tensor. Under a gauge transformation, the field strength changes under conjuga- tion:

2 Theorem 3.10. The field strength F1 ∈ g ⊗ Ω (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 3.16 −1 Use Example 3.1 to prove the F2 = g F1g. transformation law of the field strength. If G is abelian, then the field strength does not change at all! This explains the gauge invariance of the electric and magnetic fields. In the case where G is a matrix group, we can form gauge invariant quantities taking the traces, which are inherently invariant under conjugation. For example, a gauge invariant scalar field used in the Yang-Mills Lagrangian (See Section 4.1) is Tr{F F µν }(3.17) Exercise 3.17 µν µν Prove Tr{Fµν F } 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 notion of covariant derivative (See equation (1.19)), which included the four po- tential and made the action gauge invariant. We know that physical states are 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 the associated 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 space is 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 (physical state) 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] → M be a curve on M with a horizontal lift γ˜. The horizontal lift implies the existence 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 only depends on the curve γ.(3.18) Exercise 3.18 The covariant derivative of a section s is just the infinitesimal version of Use Corollary 3.1 to prove that parallel transport Definition 3.10 is independent of the horizontal lift. Definition 3.11. Let s ∈ Γ(M,E) be a section on E and γ : [0, 1] → M a curve on M. The covariant derivative of s along γ at p = γ(0) is   d ∇X s := γ˜(0), η(γ(t)) , dt t=0 where X = γ0(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 = γ0(0).(3.19) Exercise 3.19 The definition of parallel transport and of covariant derivative are in Prove this. 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 derivative can be viewed as a linear map ∇ : X(M) ⊗ Γ(M,E) → Γ(M,E), treating X(M) as a 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 the ∇(fs) = (df)s + f∇s. Now let us see what is the local form of the covariant derivative, by local we mean writing everything in terms of a local section σ ∈ Γ(U, P ):

γ˜(t) = σ(γ(t))g(t), V eα(p) = [(σ(p), eα )],

29 Leonardo A. Lessa

V where g(t) := g(γ(t)) is the canonical local trivialization and {eα } is a basis for V . We only need to calculate the covariant of the form ∇∂/∂xµ eα, since any section s ∈ Γ(M,E) is written as

s(p) = [(σ(p), ξ(p))], α V = [(σ(p), ξ (p)eα )], α α = ξ (p)eα ≡ s (p)eα,

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

α ∇X s = ∇Xµ∂/∂xµ (s eα),  α  µ ∂s α = X e + s ∇ µ e . ∂xµ α ∂/∂x α

V −1 V Since eα(γ(t)) = [(σ(γ(t)), eα )] = [(˜γ(t), g(t) eα )], then the covariant ∂ 0 derivative of eα in the direction of ∂xµ = γ (0) is   d −1 V ∇∂/∂xµ eα = γ˜(0), g(t) eα , dt t=0   −1 dg(t) −1 V = γ˜(0), −g(0) g(0) eα , dt t=0 −1 V = [(˜γ(0)g(0) , Aµeα )],

µ where we have used equation (3.1) and that A = Aµdx . Bear in mind that Aµ here is in reality the representation ρ(Aµ). In matrix notation with V α α respect to the basis eα , we write Aµ β := ρ(Aµ) β, so

β V β ∇∂/∂xµ = [(σ(0), Aµ αeβ )] = Aµ αeβ.

Finally,  ∂sβ  (∇ s)β = Xµ + A β sα . (3.6) X ∂xµ µ α

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). The principle bundle with this gauge group and a connection in it forms the electromagnetic part of the theory, with the four-potential and the field strength tensor being local expressions for the connection and the curvature, respectively. The 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 the vector space C4. If instead we were interested in non-relativistic quantum mechanics with electromagnetic , then the matter field would be 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 gauge invariance intrinsically, since a physical field is a section on the that has, at each point of the base space, many representations in the original vector space, corresponding to gauge transformation. However so, these representations do not change the section itself. Because of this, the section removes gauge redundancy at the cost of a definition based on harder to work with conjugacy classes. Furthermore, we can parallel trans- port the matter fields, which have noticeable physical consequences, like the Aharanov-Bohm effect. If we allow the gauge group G to be non-abelian, then we still get the terms involving the structure constants. This case is not just some purely mathematical generalization, but it in fact happens in Quantum Chromo- dynamics (QCD), the theory of strong interactions. In QCD, the gauge group is SU(3), with gauge potential corresponding to the fields, and the matter fields are the , 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∧A is now quadratic in the gauge potential. This has an impact in the dynamics µν of QCD, since its Lagrangian contains a term proportional to Tr[Fµν F ], which is now quartic with A. This means that the quantum has an vertex (interaction) with 4 . This contrasts with QED, that does not exhibit -photon interaction at low . The proper quantum theory of QCD is much more complicated than this, but gauge theory 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 has profound consequences, one of them being the quantization of the . Also, we noted that this solution is essentially topological, since we can exclude the point of the space where the magnetic monopole sits and thus analyze the vacuum solutions to Maxwell’s equations. We will see later how we can expect to have nontrivial solution to this problem just from topological properties of the base space M = R3 \{0}, but first we will connect 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 that the associated gauge theory will have the quantum mechanical gauge group G = U(1). Since gauge theory predicts the Faraday tensor F = dA is a closed form, then it does not accommodate magnetic charges, so we are forced to consider our base space to be M = R3 \{0}, meaning the magnetic charge is at the origin. From Theorem 2.2, we know that fibre bundles with homotopic base spaces are equivalent, thus we can simplify our base space to M = S2, the sphere. The sphere cannot be covered by a single coordinate map, and this reflects on the fact that we cannot use a single gauge potential A to describe the solution to the magnetic monopole problem on the whole sphere. We already knew 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 (See equations (1.23) and (1.24)) g AN = i (1 − cos θ)dφ, (4.1) 4π g AS = −i (1 + cos θ)dφ, (4.2) 4π

where we used the correspondence A = ieA .(4.1) For these potentials to be Exercise 4.1 j j Prove that originated from the same connection form, they have to satisfy the following ∂ relation on the intersection (See Theorem 3.8) ˆeφ = , ∂φ N −1 S −1 A = t A tNS + t dtNS, and thus, transforming into a NS NS one-form via the metric, we have the association where tNS = exp[iϕ(φ)] : HN ∩HS → U(1) is the transition function between 1 the hemispheres, which we consider to be a function only of the azimuthal ˆe → dφ. r sin θ φ angle φ since the strip HN ∩ HS can be made as small as we want. From the explicit expressions (4.1) and (4.2) and the fact that the gauge group U(1) Remember that ˆeφ is normal- ized. is abelian, we have

−1 idϕ(φ) = tNSdtNS, = AN − AS, eg = i dφ. 2π

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

Z 2π 2πn = ϕ(2π) − ϕ(0) = dϕ = eg, 0 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 describing the transition function tNS = exp[iϕ(φ)], which is essentially defined on the equator S1. Thus, an equivalent problem is the classification of continuous 1 functions ϕ : S → U(1), that is, what is the fundamental group π1(U(1))? And we know how to answer that! Since U(1) is homotopically equivalent 1 to S , 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 \{0} that 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 and a Faraday tensor F = dA, an exact one-form. Moreover, vacuum Maxwell’s equations say that F is a closed form, that is, dF = 0. If we are trying to search for solutions that do not come from electrical sources, then we want to find classify all closed two-forms F that are not exact, which is exactly what the cohomology group is! More specifically, the second cohomology group of the base space M = R3 \{0}, calculated as H2(M) = H2(S2) = R, a non-trivial group! Note that we cannot conclude anything related to charge quantization 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. . 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 , and all that. 1989. Chap. 1, pp. 9–21. [6] N. N. Bogoliubov and D. V. Shirkov. Introduction to the Theory of Quantized Fields. 3rd ed. John Wilet & Sons, Inc., 1980. Chap. 1, pp. 51– 63. [7] Gerard ’t Hooft. The Conceptual Basis of . 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 in Mathematics and Physics v. 14. Princeton University Press, 1999.

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