Solitons in High Energy Physics

LU TP 17-24 May 2017

Edvin Olofsson

Department of Astronomy and Theoretical Physics, Lund University

Bachelor thesis supervised by Johan Bijnens Abstract

Solitons are solutions to classical field equations that are particle like, meaning they are localized in space, have definite energy and can be stable against deformations into the vacuum. In this thesis we investigate under which circumstances such solutions may exist and what effects their existence could have. It will turn out that topological considerations will play a large role in the question of existence and stability, and that results depend on the number of spatial dimensions in which the theory is defined. Mixed in with the more general results are several examples of solitons, like sine-Gordon kinks, vortices and magnetic monopoles, that serve the purpose of illustrating the ideas behind the arguments as well as showing what effects the existence of solitons may have. A large part of the thesis is dedicated to showing the equivalence of the sine-Gordon model and the massive Thirring model. The equivalence of the two theories will make it possible to connect the strong coupling regime of one theory to the weak coupling regime of the other, and vice versa. It will also be possible to connect the fermion number of the Thirring model to the topological charge of the sine-Gordon model. Populärvetenskaplig beskrivning

Partikelfysik ärden del av fysiken som beskriver de allra minsta best˚andsdelarnai v˚ar värld. De ekvationer som användsföratt beskriva dessa sm˚apartiklar kan vara mycket sv˚araatt lösa,det finns faktiskt ett pris p˚aen miljon dollar till den som lyckas lösaett visst problem inom partikelfysik. Föratt underlättaalla beräkningar som ska utföras, s˚ahar fysiker tagit fram n˚agraregler som man kan följa. Till varje händelseförloppsom ska undersökas s˚aritar man upp diagram som ska representera vad som händer.Fr˚anett s˚adant diagram g˚ardet sedan att räknap˚asannolikheten föratt just en s˚adan reaktion ska ske. Detta ärvad som kallas störningsteori. Till en och samma reaktion s˚akan det finnas väldigtm˚anga diagram att h˚allareda p˚a. Som tur ärbrukar det inte vara nödvändigt att bry sig om alla diagram, eftersom de allra flesta representerar händelsersom ärväldigt osannolika. Det ärdärförofta okej att bara bry sig om de f˚adiagram som bidrar med mest sannolikhet. Med hjälpav dessa verktyg har fysiker idag lyckats beskriva det mesta av den materia som vi stöterp˚ahärp˚a jorden. Problemet med störningsteoriärdock att det bara fungerar om krafterna mellan partiklarna ärs˚asvaga att man kan anse att kraften endast utgören liten störningför partiklarna. Det ärdärförnödvändigtatt ibland försöka lösasina ekvationer utan att användastörningsteori,föratt se om det dyker upp n˚agotsom man annars skulle missa. Det härprojektet har undersökten viss typ av partiklar som kallas solitoner som det normalt inte tas hänsyntill inom störningsteori.En intressant förutsägelsesom hörihop med solitoner äratt de kan ge upphov till partiklar med magnetisk laddning iställetför endast elektrisk laddning.

2 Contents

1 Introduction 4

2 The Sine-Gordon and Massive Thirring Model Equivalence 4 2.1 Calculations in classical ﬁeld theory ...... 5 2.2 Normal ordering of the SG Hamiltonian ...... 7 2.3 Variational computation ...... 11 2.4 Equivalence of sine-Gordon and Thirring model ...... 12 2.5 Consequences of the SG/Thirring equivalence ...... 14

3 Solitons in Higher Dimensions 15 3.1 The global U(1) vortex and Derrick’s theorem ...... 16 3.2 U(1) vortices with gauge symmetry ...... 20 3.3 Adding Fermions ...... 23 3.4 Solitons from the perspective of homotopy ...... 26

4 Magnetic Monopoles 28

5 Conclusions 31

A Vortex and monopole equations near origin 31

B Calculation of fermion equations 32

C Some deﬁnitions related to homotopy 33

D Calculating the energy density for the monopole 33

3 1 Introduction

When working with quantum field theories one often identifies the classical wave solutions with the elementary particles of the theory. However, some classical field theories already have particle like solutions in the form of solitons, which are solutions to the field equations that are localized in space and whose energy remains the same as time passes. It might therefore be interesting to see what effects solitons can have on a theory. We will find that the existence of solitons is very dependent on the number of spatial dimensions, D, in which the theory is defined. In this thesis we will almost exclusively consider the cases of one, two or three spatial dimensions, with higher dimensions being discussed briefly in sections 3.1 and 3.4. The thesis is divided into three main sections where the results are presented and discussed, after which follows a concluding section where the thesis is summarized and some outlook is provided. Section 2 starts with a discussion about solitons in one spatial dimension, with the sine-Gordon model as an example, and then moves on to prove the equivalence of the sine-Gordon model and the massive Thirring model. After the proof, we consider some of the physical consequences of the equivalence, such as how charges and couplings are related between the two models. Section 2 is based on chapter 2 in [1] and work done by Coleman [2]. In section 3, we discuss some criteria that need to be fulfilled in order for solitons to exist in more than one spatial dimension. Specifically we consider Derrick’s theorem in section 3.1 which states that no solitons should exist for a certain class of theories, while section 3.4 is dedicated to the connection between the topology of the space of vacua for a scalar potential V (φ) and the topology of spatial infinity. Throughout section 3 we will consider the example of vortices in D = 2, with an excursion to D = 3 in section 3.3. The topics discussed in this section are covered in chapters 3 and 4 of [1]. Section 4 will deal with magnetic monopoles in D = 3 and is based on the material in chapter 5 of [1]. The appendix contains some derivations and definitions that were left out of the main thesis. As is standard practice in particle physics, Greek letters like µ, ν will denote the Lorentz indices 0,1,2,..., while Roman letters like i, j denote Euclidean indices 1,2,... . ijk... is the completely antisymmetric tensor on n indices defined such that 123...n = 1. We will make use of the Einstein summation convention where repeated indices indicate summation. Natural unitsh ¯ = c = 1 are employed throughout the text.

2 The Sine-Gordon and Massive Thirring Model Equiv- alence

The purpose of this section is to first find solitons in the sine-Gordon model and then derive the equivalence of the sine-Gordon and the massive Thirring models, which will be done by comparing the perturbation series of the two theories to all orders and showing that they are equal, provided certain identifications are made between the theories. The sine-Gordon Model, SG, is a two dimensional scalar field theory defined by the

4 Lagrangian density 1 α L = ∂ φ∂µφ + 0 cos(βφ) + γ , (2.1) SG 2 µ β2 0

where α0, β and γ0 are real, positive parameters. The massless Thirring model is described by the Lagrangian 1 L = ψiγ ∂µψ − gjµj , (2.2) Th µ 2 µ with the current jµ = ψγµψ. The massive Thirring model is obtained by adding the term 0 −m σ to LTh, where σ = Zψψ. When we do perturbation theory for the Thirring model, we will consider the mass term as a perturbation to the massless theory.

2.1 Calculations in classical ﬁeld theory In classical ﬁeld theory, the Euler-Lagrange equation for a Lagrangian with a potential V (φ) is ∂2φ ∂2φ ∂V (φ) − + = 0, (2.3) ∂t2 ∂x2 ∂φ where the static case is d2φ dV (φ) = . (2.4) dx2 dφ dφ It is possible to rewrite this equation, by multiplying with dx , in the following form " # d2φ dφ dV (φ) dφ d 1 dφ2 0 = − = − V (φ) . (2.5) dx2 dx dφ dx dx 2 dx

An integration then yields dφ2 = 2V (φ) + C, (2.6) dx where C is some constant. If V = 0 at its global minima and φ approaches a constant vacuum value at x = ±∞, then C = 0 and dφ = ±p2V (φ). (2.7) dx Solving for x, we get Z φ dφ0 x = ± . (2.8) p 0 φ0 2V (φ ) If x should vary continuously between ±∞, φ must connect two adjacent minima of V in a monotone fashion, since there the integral diverges. The sine-Gordon potential has N2π minima at φ = β = Nv, where N is an integer. Substituting the sine-Gordon potential

5 into (2.8), integrating and then solving for φ, we ﬁnd the solution that connects the Nth minimum to the one above: 2v √ φ(x) = Nv + arctan(exp( α (x − x ))), (2.9) π 0 0

where x0 determines where the soliton is localized. A solution of this type is called a kink. By exchanging the sign in front of the arctan, solutions called anti-kinks, connecting the the Nth minima to the N − 1th are obtained. From equation (2.7), it is possible to calculate the energy of the kink. ! Z ∞ 1 dφ2 Z ∞ E = dx + V (φ) = 2 dxV (φ) = −∞ 2 dx −∞ (2.10) Z φ(∞) dφ−1 Z φ(∞) = dφ V (φ) = dφp2V (φ) φ(−∞) dx φ(−∞)

Plugging in the solution (2.9) and the expression for V (φ) into equation (2.10), and using the periodicity of V , we arrive at s Z v 2α0 2π Ekink = dφ 2 1 − cos φ . (2.11) 0 β v

√ 8 α0 Evaluating the integral results in Ekink = β2 . It should be noted here that equation (2.4) is also a condition for φ to be a stationary point of the energy, since in the static case the integrand in (2.10) is just −LSG. This will be used to derive the ﬁeld equations of the 2-D vortices in section 3. The topological current is deﬁned by 1 jµ = µν∂ φ, (2.12) top 2v ν

µν where is the two index Levi-Civita tensor. Deﬁning the topological charge, Qtop as the spatial integral of the timelike component of jµ results in Z 1 Z 1 Q = dxj0 = dx∂ φ = (φ(∞) − φ(−∞)) . (2.13) top 2v 1 2v

The topological charge of the kink in (2.9) therefore is +1 while the anti-kinks have topological charge -1. The topological current is different from the currents defined in Noether’s theorem in the sense that it does not correspond to a continuous symmetry of the La- grangian. Since changing the values of φ at ±∞ would require both infinite time and energy, (2.13) shows that topological charge is conserved. Therefore, we should expect the sine-Gordon kink to be a stable solution.

6 2.2 Normal ordering of the SG Hamiltonian Here the sine-Gordon model will be quantized, in order for us to compare its perturbation expansion with that of the Thirring model. Most of the work done in the rest of section 2 will be based on the paper by Coleman [2]. The quantization amounts to expanding the ﬁelds in terms of creation and annihilation operators that satisfy canonical commutation relations. For a free scalar ﬁeld of mass m we have in the Heisenberg picture, where operators are time dependent,

Z dp 1 1/2 φ(x) = [a e−ipx + a†eipx]. (2.14) 2π 2E(p, m) p p

† Here, the operator ap annihilates a particle with x-momentum p while ap creates one. E(p, m) = pp2 + m2 is just the regular relativistic expression for the energy of a particle. In the exponents of the exponentials, the product px is to be understood as the product between the vectors pµ = (E, p) and xµ = (t, x), µ = 0, 1. The creation and annihilation operators satisfy the following commutation relations

h † i 0 h † † i ap, ap0 = 2πδ(p − p ), [ap, ap0 ] = ap, ap0 = 0, (2.15)

where δ(p) is the Dirac delta function. In order to be able to find the relationship between the sine-Gordon model and the Thirring model, it will be necessary to work with the Hamiltonian density, instead of the Lagrangian. First we define the momentum density as ∂L π = . (2.16) ∂φ˙ In terms of a creation and annihilation operators, the momentum density for a free field will be Z dp E(p, m)1/2 π(x) = ∂ φ = −i [a e−ipx − a†eipx]. (2.17) 0 2π 2 p p The Hamiltonian (density) is defined as

H = φπ˙ − L (2.18) and the sine-Gordon Hamiltonian is therefore 1 α 1 1 α H = φπ˙ − ∂ φ∂µφ − 0 cos(βφ) − γ = (∂ φ)2 − (∂ φ)2 + (∂ φ)2 − 0 cos(βφ) − γ = 2 µ β2 0 0 2 0 2 1 β2 0 1 1 α 1 1 α = (∂ φ)2 + (∂ φ)2 − 0 cos(βφ) − γ = π2 + (∂ φ)2 − 0 cos(βφ) − γ . 2 0 2 1 β2 0 2 2 1 β2 0 (2.19)

Before we can do any perturbative calculations, it is necessary to normal order the Hamiltonian, which amounts to rearranging creation and annihilation operators so that in

7 any product containing such operators, all annihilation operators are to the right of the † † creation operators. For example if Napap0 means the normal ordered version of apap0 , then † † Napap0 = ap0 ap. Throughout the text, a subscript on N will be used to indicate which mass is used to deﬁne φ during normal ordering, since it will be necessary to consider two diﬀerent masses, m and µ, in our calculations. Starting with the momentum density, the square of (2.17) becomes the following integral

Z 0 0 1/2 2 dpdp E(p, m)E(p , m) −ipx † ipx −ip0x † ip0x π = − [a e − a e ][a 0 e − a 0 e ] = (2π)2 4 p p p p Z 0 0 1/2 dpdp E(p, m)E(p , m) −i(p+p0)x † −i(p−p0)x = − [a a 0 e − a a 0 e − (2π)2 4 p p p p Z 0 0 1/2 † i(p−p0)x † † i(p+p0)x dpdp E(p, m)E(p , m) −i(p+p0)x − a a 0 e + a a 0 e ] = − [a a 0 e − p p p p (2π)2 4 p p † 0 −i(p−p0)x † i(p−p0)x † † i(p+p0)x − (apap0 + 2πδ(p − p ))e − apap0 e + apap0 e ] (2.20) where the commutation relations (2.15) have been used in the last equality. Doing the same for ∂1φ results in

Z 0 1/2 2 dpdp 0 1 −i(p+p0)x (∂ φ) = − pp [a a 0 e − 1 (2π)2 4E(p, m)E(p0, m) p p (2.21) † 0 −i(p−p0)x † i(p−p0)x † † i(p+p0)x − (apap0 + 2πδ(p − p ))e − apap0 e + apap0 e ]. If we deﬁne 1 1 H = π2 + (∂ φ)2, (2.22) 0 2 2 1 then (2.20) and (2.21) imply

Z 0 0 0 dpdp E(p, m)E(p , m) + pp 0 −i(p−p0)x H0 = NmH0 + δ(p − p )e = 8π (E(p, m)E(p0, m))1/2 (2.23) Z dp E(p, m)2 + p2 = N H + = N H + E (m). m 0 8π E(p, m) m 0 0

The integral Z dp E(p, m)2 + p2 Z dp 2p2 + m2 E0(m) = = (2.24) 8π E(p, m) 8π pp2 + m2 diverges unless a cutoﬀ is introduced, since the integrand does not approach zero for large p. Choosing a symmetric cutoﬀ Λ we get,

Z Λ 2 2 √ dp 2p + m 2 E0(m, Λ) = = 2Λ Λ + m . (2.25) p 2 2 −Λ 8π p + m

8 The divergence can be handled by a redefinition of γ0, so that the quantity γ = γ0 −E0(m) is finite in the limit where Λ → ∞. If the normal ordering was done with respect to a different mass µ, the result would be

H0 = NµH0 + E0(µ). (2.26)

The two diﬀerent orderings are related by

NmH0 = NµH0 + E0(µ) − E0(m). (2.27)

Computing the diﬀerence E0(µ) − E0(m) shows that it is ﬁnite in the limit Λ → ∞,

√ 2 2 2 2 2Λ p 2 2 2 2 2Λ µ − m µ − m E0(µ) − E0(m) = Λ + µ − Λ + m = √ → . 8π 8π pΛ2 + µ2 + Λ2 + m2 8π (2.28) The parameter µ is introduced to handle divergences that arise from the fact that there is no clear mass term in the SG Lagrangian, which means for instance that the deﬁnitions of the ﬁeld operators diverge. In the perturbative calculations of section 2.4, µ will take the role of a mass that will later be sent to zero. It now remains to normal order the cosine term of the Hamiltonian, and for that purpose we will need the formula Z Z 1 Z exp i J(x)φ(x)d2x = N exp i J(x)φ(x)d2x exp − J(x)∆(x − y)J(y)d2xd2y , m 2 (2.29) where J(x) is some function of the spacetime coordinates and ∆(x − y) is the commutator

∆(x − y) = φ+(x), φ−(y) , (2.30)

with φ+ containing all the annihilation operators of φ and φ− all the creation operators. This can be done using the general expression

A B A+B+ 1 [A,B] e e = e 2 , (2.31)

which holds when both A and B commute with the commutator, see chapter 19 of [3]. Normal ordering of exp i R J(x)φ(x)d2x corresponds to taking A = i R dx2J(x)φ−(x) and R 2 + 1 [B,A] B = i dx J(x)φ (x) in (2.31). If both sides are multiplied by e 2 , (2.29) is obtained. For our purposes, it will be necessary to evaluate the commutator explicitly for spacelike x. Writing the ﬁelds in terms of ladder operators and using the commutation relations

9 results in

Z 0 1/2 + − dpdp 1 † i(p0y−px) † i(p0y−px) φ (x), φ (y) = a a 0 e − a 0 a e = 8π2 E(p, m)E(p0, m) p p p p Z 0 1/2 dpdp 1 h † i i(p0y−px) = a , a 0 e = 8π2 E(p, m)E(p0, m) p p 0 1/2 Z dpdp 1 0 = 2πδ(p − p0)ei(p y−px) = 8π2 E(p, m)E(p0, m) Z dp eip(y−x) = . 4π pp2 + m2 (2.32)

The commutator is therefore a function of z = y − x. For spacelike z, it is always possible to make a Lorentz-transformation so that the timelike component is zero, which means that the exponent depends only on the spacelike components of z and p. We also see that the integral diverges for z = 0, which will be dealt with later when we consider small z. It is allowed to make a Lorentz transformations since dp/E is Lorentz invariant. Rewriting the exponential in terms of sine and cosine, we find that the sine integral vanishes since the rest of the integrand is an even function. The final result is 1 Z ∞ cos pz 2 Z ∞ cos pz 1 ∆(z) = dp = dp = 2K0(m|z|) (2.33) p 2 2 p 2 2 4π −∞ p + m 4π 0 p + m 4π where Kν are the modified Bessel functions of the second kind[4][5].The first few terms in the series representation of K0 are, section 11.5 in [6],

2 K0(x) = − ln x − γ + ln 2 + O(x ), (2.34) where γ is the Euler-Mascheroni constant. ∆(z) can therefore be written as 1 ∆(z) = − ln cm2z2, (2.35) 4π for small z. The constants in (2.34) have been collected into ln c. To bypass the divergence of ∆, we make the replacement

1 1 1 m2 ∆(z) → ∆(z, Λ) = − ln cm2z2 + ln cΛ2z2 = − ln . (2.36) 4π 4π 4π Λ2

Like with the normal ordering of H0, Λ is used as a cutoﬀ in this context. If we take J = βδ(y − x), then (2.29) gives

2 m2 β /8π eiβφ = N eiβφ. (2.37) Λ2 m

10 Deﬁning α as 2 m2 β /8π α = α (2.38) 0 Λ2 results in α0 cos βφ = αNm cos βφ. (2.39)

α will be a ﬁnite parameter after renormalization of α0 to remove the dependence on Λ. The result of a normal ordering with respect to the parameter µ would then be

2 µ2 β /8π α cos βφ = α N cos βφ (2.40) 0 0 Λ2 µ

leading to the identiﬁcation

2 2 2 m2 β /8π µ2 β /8π µ2 β /8π N cos βφ = N cos βφ ⇔ N cos βφ = N cos βφ, Λ2 m Λ2 µ m m2 µ (2.41) which will be used in the following two sections. Using (2.23) and (2.39), the normal ordered version of (2.19) is

H = NmH0 − αNm cos βφ − γ. (2.42)

2.3 Variational computation It is possible to draw some conclusions about the allowed values of parameters in the model by doing variational calculations with respect to µ. Deﬁne trial vacuum states such that a(p, µ) |0, µi = 0. The result of normal ordering (2.19) with respect to µ, by using (2.27) and (2.41), is

2 ! α µ2 β /8π 1 H = N H − cos βφ − γ + µ2 − m2 . (2.43) µ 0 β2 m2 8π

The vacuum expectation value is therefore

2 ! α µ2 β /8π 1 h0, µ| H |0, µi = h0, µ| N H − cos βφ − γ + µ2 − m2 |0, µi = µ 0 β2 m2 8π

2 α µ2 β /8π 1 1 = − h0, µ| N 1 − β2φ2 + ... |0, µi − γ + µ2 − m2 = β2 m2 µ 2 8π 2 α µ2 β /8π 1 = − − γ + µ2 − m2 . β2 m2 8π (2.44)

11 Equation (2.44) implies that, for β2 > 8π, the vacuum expectation value approaches −∞ when µ → ∞, since the terms involving µ can be rewritten as 1 α 2 µ2 − µ2β /8π−2 → −∞. (2.45) 8π β2m2 Therefore, by the variational method, the energy density of the ground state is unbounded below which is unphysical.

2.4 Equivalence of sine-Gordon and Thirring model Having normal ordered the Hamiltonian in section 2.2 and performed some variational computations in section 2.3, it is time to establish the correspondence between the sine-Gordon model and the massive Thirring model. The equivalence between the two models is found by calculating the vacuum expectation value of products of the interaction Hamiltonians HI, which contains the interaction terms of H, evaluated at diﬀerent spacetime points,

h0 |T ΠiHI (xi)| 0i. (2.46)

The T operator denotes time ordering, which is similar to normal ordering, but here operators evaluated at earlier times are to the right of operators evaluated at later times. In the case of the SG model, HI corresponds to the cosine term of H. These expectation values are called Green’s functions and they determine the amplitude for a particle to propagate from one spacetime point to another. Before the Green’s functions for the SG model are calculated, it is useful to calculate more general Green’s functions. The SG Green’s functions are then special cases of the more general ones. These more general objects are of the form

iβiφ(xi) h0, µ T ΠiNme 0, µi, (2.47) where µ is a small mass added to the theory to handle divergences and that will be sent to zero at the end of the calculations. The index on the βi:s indicates that the β:s can depend on xi. In (2.47), the process of time ordering only amounts to a permutation of the xi which 0 0 0 means that there is no loss in generality in assuming that x1 ≥ x2 ≥ x3 .... If the xi:s are not already in time order, then after time ordering of the operators we can always relabel the points so that the above inequality holds. The operator product is therefore

2 P 2 2 βi /8π 2 i βi /8π iβ φ(x ) iβ φ(x ) µ iβ φ(x ) µ iβ φ− iβ φ+ T Π N e i i = Π N e i i = Π N e i i = Π e i i e i i , i m i m i m2 µ m2 i (2.48) where in the second step (2.41) was used and φi is shorthand for φ(xi). Using the formula

eAeB = eBeAe[A,B], (2.49)

12 iβ φ+ it is possible to normal order the entire product. A given e j j will need to be commuted iβ φ− with all the e i i for which j > i. The result is therefore

P 2 2 i βi /8π µ −β β φ+,φ− T Π N eiβi(xi) = Π N eiβiφi · Π e i j [ i j ] = i m m2 i µ i>j (2.50) P β2/8π µ2 i i βiβj = Π N eiβiφi · Π cµ2(x − x )2 4π . m2 i µ i>j i j

Since the exponentials have been normal ordered, they will just contribute a factor of 1, which comes from the ﬁrst term of the series expansion, to the matrix elements (2.47). This leads to

P β2/8π µ2 i i βiβj iβi(xi) 2 2 4π h0, µ T ΠiNme 0, µi = Πi>j cµ (xi − xj) (2.51) m2

2 P β /4π The ﬁnal expression is proportional to µ( i i) , which can be seen by using the relation P 2 P 2 P ( i pi) = i pi + i>j 2pipj. This means that, in the limit µ → 0, the matrix elements will be zero unless the βi sum to zero. It is now time to carry out the calculations for the actual SG Green’s functions. As stated in the beginning of this section, the interaction terms of this model are contained in α α H = −N cos βφ = − (A − A ), (2.52) I m β2 2β2 + − ±iβφ where A± = Nme . As a consequence of the arguments below eq. (2.51), only Green’s functions involving an even equal number of A+ and A− can be nonzero. It is therefore enough to calculate matrix elements of the form

n h0 |T Πi A+(xi)A−(yi)| 0i, (2.53) where the limit µ → 0 has already been taken. Using (2.51) we arrive at

2 2 4 2 2 β /4π n Πi>j [c m (xi − xj) (yi − yj) ] h0 |T Πi A+(xi)A−(yi)| 0i = 2 . (2.54) 2 2 β /4π Πij [cm (xi − yj) ] Klaiber was able to show, [7], that in the massless Thirring Model the following relation holds for the Green’s functions

2n 4 2 2 1+b/π n 1 Πi>j [M (xi − xj) (yi − yj) ] h0 |T Πi σ+(xi)σ−(yi)| 0i = , (2.55) 2 2 1+b/π 2 Πij [M (xi − yj) ]

g where M is an arbitrary mass and b = − 1+g/π . The σ± are deﬁned as 1 σ = Zψ(1 ± γ )ψ. (2.56) ± 2 5 13 Comparing (2.54) and (2.55) shows that the Green’s functions of the two theories are equal provided the following identiﬁcations are made

σ± = −A± M 2 = cm2 (2.57) 1 β2 = . 1 + g/π 4π

2.5 Consequences of the SG/Thirring equivalence By considering two particular commutators,

µ [j (x), σ±(y)] and [∂νφ(x),A±(y)] , (2.58) and using the identiﬁcations made in (2.57), it is possible to connect the topological current (2.12) with jµ. Klaiber showed, [7], that in the massless Thirring model the following commutation relation holds

µ µν −1 µν [j (x), ψ(y)] = − g + (1 + g/π) γ5 ψ(y)Dν(x − y), (2.59) where Dµ(x − y) = ∂µ[φ(x), φ(y)]. Only the commutators involving σ+ and A+ will be calculated explicitly, since the other two are calculated in the same manner.

µ 1 µ [j (x), σ+(y)] = Z j , ψ(1 + γ5)ψ = 2 (2.60) 1 = Z jµ, ψ ψ + ψ [jµ, ψ] + jµ, ψ γ ψ + ψ [jµ, γ ψ] 2 5 5 Where the operator identity [A, BC] = [A, B]C +B[A, C] has been used. The commutators that involve ψ can be calculated directly from (2.59), while those involving ψ require the application of the identity µ µ † j , ψ = − [j , γ0ψ] . (2.61) Another application of (2.59), and using that gµν is symmetric and µν is antisymmetric, results in µ µν −1 µν j , ψ = g ψ − (1 + g/π) ψγ5 Dν(x − y). (2.62) The end result is

µ −1 µν [j (x), σ±(y)] = ∓2(1 + g/π) σ±Dν(x − y). (2.63)

Moving on to the commutator involving scalar ﬁelds, we can immediately write

[∂νφ(x),A+(y)] = ∂ν [φ(x),A+(y)] , (2.64)

14 since the gradient is with respect to x and not y. Remembering the deﬁnitions of φ+ and φ− from section 2.2 we can calculate the commutator above as

iβφ− iβφ+ iβφ− iβφ+ iβφ− iβφ+ [φ(x),A+(y)] = φ(x), e e = φ(x), e e + e φ(x), e =

iβφ− iβφ+ iβφ− iβφ+ = iβ [φ+(x), φ−(y)] e e + iβ [φ−(x), φ+(y)] e e = (2.65)

iβφ− iβφ+ = iβ [φ(x), φ(y)] e e = iβ [φ(x), φ(y)] A+(y), where the second line follows from expanding the exponentials as power series and using the formula [A, Bn] = nBn−1 [A, B] . (2.66) from chapter 19 in [3]. Plugging this into (2.64) yields

[∂νφ(x),A±(y)] = ±iβDνA±. (2.67)

It is now possible to make the identiﬁcation 2 β jµ = − µν∂ φ = − µν∂ φ = −2jµ , (2.68) β(1 + g/π) ν 2π ν top

µ with jtop deﬁned as in (2.12). This result for jµ combined with the identiﬁcations (2.57) makes it possible to connect a theory with only bosons, the sine-Gordon model, with a theory with only fermions, the massive Thirring model. From (2.57), we see that strong coupling in one theory corresponds to weak coupling in the other, which can be exploited when doing calculations by, for instance, recasting a theory about fermions into bosonic form when dealing with the strong coupling regime of the fermionic theory. The relations between the currents also make it possible to connect the fermion number, with the topological charge.

3 Solitons in Higher Dimensions

What made the existence of solitons in one spatial dimension possible, was the fact that the scalar field could take on different vacuum values at spatial infinity, making it impossible to deform the field configuration to a uniform vacuum solution. In one dimension, spatial infinity is represented by the two points ±∞ so only two distinct vacua are required in order for solitons to exist. In higher dimensions this requirement must be modified to reflect the topological characteristics of spatial infinity. For instance, with two spatial dimensions spatial infinity is described by a circle with radius r = ∞. If we still want the field to approach different vacuum values at different parts of spatial infinity and be continuous, we arrive at the conclusion that there must be a continuous family of vacua for solitons to exist. As will be discussed in more detail, it will also be necessary to consider gauge theories, since the coupling terms between the scalar field and the gauge field that come from the covariant derivative will be needed to balance the spatial gradient terms of the scalar field,

15 which would otherwise lead to infinite energy. The coupling between the gauge field and the scalar field will mean that the resulting field equations will not be possible to solve analytically, meaning that a lot of the information about the nature of solitons in higher dimensions comes from numerical studies, analysis of asymptotic behavior and qualitative reasoning. The derivations and discussions in this section follow chapters 3 and 4 in [1].

3.1 The global U(1) vortex and Derrick’s theorem As a first attempt to find a soliton in two spatial dimensions, we can consider a complex scalar field iα(x) φ(x) = φ1(x) + iφ2(x) = ρ(x)e , (3.1) with the Lagrangian 1 λ µ2 2 L = (∂ φ)† (∂µφ) − |φ|2 − . (3.2) 2 µ 4 λ µ and λ are both real, positive parameters. The theory posses a global U(1) symmetry, since the Lagrangian is invariant under the transformation φ → eiξφ, where ξ is a real number. The potential λ µ2 2 V (φ) = |φ|2 − (3.3) 4 λ has a continuous set of minima at r µ2 |φ| = v = . (3.4) λ The discussion earlier implies that it might therefore be possible for solitons to exist within this theory. In two spatial dimensions, we can define the vorticity of a scalar field as 1 I 1 I N[C] = dl∇(argφ) = dl∇α (3.5) 2π C 2π C where the contour C is the circle at r = ∞. N[C] counts how many rotations the phase of φ goes through during one loop of C. If φ is to be single-valued, then N[C] must be an integer since the line integral must be an integer multiple of 2π. The quantization of N means that a configuration with one value of N cannot be continuously deformed into a configuration with another value. If we consider C to be the circle at infinity, then the fact that φ is single-valued again means that N must be an integer. The exception to this is when φ has a zero on C, because then α is not well defined. Since the result of going around a loop enclosing a zero of φ must be an integer, we can see that if a loop not enclosing a particular zero is deformed to do so, the difference in N between the two loops must be the N that is found when encircling just the zero in question. If we call N[L]i the result of going around an infinitesimal loop L encircling the i:th zero of φ, then starting with N[C] = N for C at infinity and contracting C to a point

16 P that is not a zero of φ shows that i N[L]i = N. Thus, any configuration with nonzero vorticity must have at least one zero. Since a uniform vacuum solution must clearly have N = 0, it is reasonable to assume that a stable soliton in two dimensions should have N 6= 0 to ensure that it cannot be continuously deformed to the vacuum. The Euler Lagrange equations for φ1 and φ2 with the Lagrangian (3.2) are ∂L ∂L ∂ − = ∂µ∂ φ + λφ φ2 + φ2 − v2 = 0 µ ∂(∂ φ ) ∂φ µ 1 1 1 2 µ 1 1 (3.6) ∂L ∂L µ 2 2 2 ∂µ − = ∂ ∂µφ2 + λφ2 φ1 + φ2 − v = 0, ∂(∂µφ2) ∂φ2 which can be combined into the single equation ∂2 − ∇2 φ + λφ |φ|2 − v2 = 0. (3.7) ∂t2 The static version becomes ∇2φ − λφ(|φ|2 − v2) = 0. (3.8) To look for solutions to (3.8), we can try to make an ansatz that possesses symmetries that simplify the equation. A completely rotationally invariant ansatz, where φ is a function of only r = px2 + y2, would not lead to solitons since the field would have the same value everywhere at infinity. Another possible ansatz is

φ = f(r)eiθ, (3.9)

where θ is the polar angle. This is rotationally invariant in the sense that a rotation can be completely compensated by a U(1) transformation. If we also demand that φ should be invariant under a reflection through the x-axis combined with a complex conjugation we find φ(r, θ) = φ∗(r, −θ) = f ∗(r)(e−iθ)∗ = f ∗(r)eiθ, (3.10) which means that f must be a real function. Inserting the ansatz into (3.8) gives the following equation for f d2f 1 df f + − − λf(f 2 − v2) = 0. (3.11) dr2 r dr r2 This equation has the form of Bessel’s equation with an added nonlinear term. The non- linearity of the equation makes finding a solution difficult, so trying to find numerical solutions is a better option. In order to ensure that φ is well defined at the origin, we require f(0) = 0 since there eiθ is not defined. A more detailed discussion about the behavior of f near r = 0 is provided in appendix A. Since we are trying to find solutions that approach a vacuum value at r = ∞, the second boundary condition for f becomes f(∞) = v. With these boundary conditions, we can see from (3.5) that N = 1. A numerical solution to (3.11) for λ = v = 1, generated using standard MATLAB functions, is presented in figure 1.

17 Figure 1: Numerical solution for the vortex with global U(1) symmetry.

The solution presented here has a problem in that the energy is in fact inﬁnite. To see why this is the case we examine the form of the energy Z 1 E = d2x (∇φ)2 + V (φ) . (3.12) 2 Inserting our ansatz and changing to polar coordinates results in ! ! Z 1 ∂f 2 f 2 λ E = dθdrr + + f 2 − v22 . (3.13) 2 ∂r r2 4

As r becomes large, the term f 2/r2, which comes from the angular part of the gradient, will approach v2/r2 meaning that the extra factor of r that comes from the integration element will make the energy diverge. This is in fact a consequence of a result known as Derrick’s theorem [8], which is presented below. At first we will consider a theory in D spatial dimensions involving one or more scalar fields φa with a Lagrangian of the form 1 L = G (φ)∂ φ ∂µφ − V (φ) (3.14) 2 ab µ a b where G(φ) is a positive definite matrix for all φ and the potential has a global minimum at V (φmin) = 0. The global U(1) theory discussed in this section is an example of such a theory with D = 2. When considering static solutions the energy is given by Z D E = − d xL = Ik[φ] + IV [φ], (3.15) where 1 Z I [φ] = dDxG (φ)∂ φ ∂ φ k 2 ab i a i b Z (3.16) D IV [φ] = d xV (φ).

18 A finite energy static solution should be a stationary point of E. If a solution is stationary when we consider all possible field configurations, it must also be stationary if we consider any subset of configurations to which it belongs. If we assume that φ is a solution, we can then consider variations of this solution that amount to a rescaling of coordinates

fλ(x) = φ(λx), (3.17)

where λ is a real number. The energy of such a conﬁguration is 1 Z Z E(λ) = I [f ] + I [f ] = dDxG (φ(λx))∂ φ (λx)∂ φ (λx) + dDxV (φ(λx)). (3.18) k λ V λ 2 ab i a i b

Changing variables to y = λx, the volume element will contribute a factor of λ−D to both Ik and IV while each derivative factor in Ik will contribute a factor λ. We therefore have

2−D −D E(λ) = λ Ik φ + λ IV φ . (3.19)

The assumption that f1 = φ(x) is a solution requires that λ = 1 is a stationary point of E(λ). We should therefore have

dE 1−D −(D+1) 0 = = (2 − D)λ Ik − Dλ IV = (2 − D)Ik φ − DIV φ , (3.20) dλ λ=1 λ=1 or (D − 2)Ik φ + DIV φ = 0. (3.21) For D = 1, this can be seen as a generalization of the relation (2.6). Since both integrals Ik and IV are positive, D = 2 implies IV (φ) = 0 which means that any φ must be equal to the vacuum everywhere, and so no solutions with nonzero vorticity can exist since those require φ to have at least one zero. D ≥ 3 requires Ik = IV = 0, so φ must identically equal one of the vacuum values, which is uninteresting. Since the above theories with global symmetries do not have solitons in higher dimensions, we can try to see if theories with gauge symmetries are better alternatives. We consider a gauge theory, that can be Abelian or non-Abelian, with a Lagrangian 1 1 L = − trF F µν + (D φ)† (Dµφ) − V (φ) (3.22) 2 µν 2 µ where Fµν = ∂µAν − ∂νAν + ig [Aµ,Aν] (3.23) is the ﬁeld strength. The covariant derivative is

Dµ = ∂µ + igAµ, (3.24)

g is the coupling strength, the gauge ﬁelds are

a a Aµ = AµT (3.25)

19 and T a are the generators of the symmetry. We will only consider static solutions in a gauge such that A0 = 0, i.e. F0j = 0 which means there is no electric ﬁeld. This is what we will work with in this section and in the section about magnetic monopoles, section 4. The energy of such a solution is given by

E = IF [A] + Ik[φ, A] + IV [φ] (3.26)

with 1 Z I [A] = dDxtrF F F 2 ij ij 1 Z I [φ, A] = dDx (D φ)† (D φ) (3.27) k 2 i j Z D IV [φ] = d xV (φ).

Given a solution φ, Ai, we deﬁne a rescaling such that φ is transformed like in (3.17) and Ai is transformed like giλ(x) = λAi(λx). (3.28)

The reason for multiplying A by λ is to ensure that the result of the transformation on Fµν and Dµ is to just multiply them by powers of λ. By the same reasoning as in the global case, we arrive at an analog of (3.21), namely (D − 4)IF A + (D − 2)Ik φ, A + DIV φ = 0 (3.29)

This equation allows for the possibility of having both the scalar and gauge ﬁelds be nontrivial for D = 1, 2, 3. If D = 4, it might be possible to have nontrivial gauge ﬁelds, so long as φ = 0 and V (0) = 0. For D ≥ 5 we only expect trivial solutions since the integrals and their prefactors are all positive.

3.2 U(1) vortices with gauge symmetry In the previous section we saw that it might be possible for solitons to exist in two spatial dimensions, provided we are working with a local gauge theory. This section will argue for the existence of ﬁnite energy solitons with nonzero vorticity in a local U(1) theory. The Lagrangian under consideration is

1 1 λ µ2 2 L = − F F µν + (D φ)† (Dµφ) − |φ|2 − , (3.30) 4 µν 2 µ 4 λ

with F = ∂ A − ∂ A µν µ ν ν µ (3.31) Dµ = ∂µ + ieAµ.

20 Since the potential is just (3.3), this theory has the same family of possible vacuum expectation values as the global vortex. If one were to expand√ φ around such a vacuum, one would ﬁnd that the theory has a scalar with mass mφ = 2µ and a vector boson of mass mA = ev. These masses will later play a role in the long range behavior of φ and Aµ. From standard particle physics we know that L is invariant under

φ → eieΛ(x)φ (3.32) Aµ → Aµ − ∂µΛ(x).

A quick calculation also shows that under the same transformation 1 I 1 I e I N[C] → N 0[C] = dl∇(argφ) = dl∇(α + eΛ(x)) = N[C] + dl∇Λ(x) 2π C 2π C 2π C (3.33) i.e. that N[C] is also invariant, provided Λ is well behaved at inﬁnity. Trying to ﬁnd a N = 1 vortex, we make the ansatz

φ = veiθf(evr) a(evr) A = jkrˆk (3.34) j er A0 = 0, where f and a are real functions andr î = xi/r. We see that due to the factor eiθ, N = 1 and f(0) must be zero in order for φ to be well defined at the origin. a(0) = 0 is necessary to avoid singularities in Aj. After calculating the necessary partial derivatives of (3.34), one finds that

2 da2 F F = ij ij e2r2 dr (3.35) df 2 v2(1 − a)2f 2 (D φ)† (D φ) = v2 + . i j dr r2

Plugging into (3.26), changing to polar coordinates and introducing u = evr, we ﬁnd ! Z 1 da2 df 2 (1 − a)2f 2 λ E = πv2 duu + + + 1 − f 22 = πv2G(λ/2e2). u2 du du u2 2e2 (3.36) The Euler-Lagrange equations for a and f become

2 d a 1 da 2 0 = 2 − + (1 − a)f du u du (3.37) d2f 1 df (1 − a)2 λ 0 = + − f + f 1 − f 2 . du2 u du u2 e2

21 Figure 2: Numerical solutions of (3.37) for diﬀerent values of λ/e2. On the left λ/e2 = 1/4, while on the right λ/e2 = 10. The solid and dashed lines represent f and a respectively.

We have already found the boundary conditions at u = 0. If IV should be finite, then f(∞) = 1 so that φ approaches vacuum values. Inserting this into the second equation shows that a(∞) = 1. Two numerical solutions for different values of λ/2e2 are provided in figure 2. The behavior of f and a near u = 0 is explored in appendix A. As figure 2 shows, the behavior of f seems to have a large dependence on λ/2e2 com- pared to a, which seems largely unchanged between the two plots. The reason for this can be understood by considering the asymptotic behavior of both functions, which is done by linearizing (3.37). For large u, we can expand both functions around 1

a = 1 − (3.38) f = 1 − δ,

and insert into the ﬁrst equation of (3.37), keeping only linear terms. The resulting equation is 1 00 − 0 − = 0. (3.39) u This equation has a solution in terms of Bessel functions [5]

AuJ1(iu) + BuY1(iu). (3.40)

Asymptotically these Bessel functions take the form

1/2 u−i3/4π −u+i3/4π 1/2 u−i3/4π −u+i3/4π J1(iu) ∝ u e + e ,Y1(iu) ∝ −iu e − e , (3.41)

where only the ﬁrst term in the asymptotic expansion of each function has been kept. Since should approach 0 for large u, we require that A = iB which leads to √ √ ∼ ue−u ∼ re−mAr. (3.42)

22 Linearizing the second equation then leads to 1 2λ δ00 + δ0 − δ = 0. (3.43) u e2 Similar arguments as above result in √ √ e− 2λu/e e− 2λvr e−mφr δ ∼ √ ∼ √ = √ . (3.44) u r r

2 However, the last result is only valid if 2mA > mφ, λ/e < 2, since otherwise the nonlinear term 2/u2, that is neglected after linearizing, does not fall faster than any of the terms in (3.43). δ must now match the behavior of this term, which goes like e−2mAr/r. Using the guess that δ = Ce−2u/u, and neglecting any terms in the derivatives of δ that fall faster than this, shows that δ ∼ e−2u/u. In summary, we have for δ

e−mφr δ ∼ √ , λ/e2 < 2 r (3.45) e−2mAr δ ∼ , λ/e2 > 2. r This dependence on λ/e2 has an eﬀect on the interactions between vortices. The force associated with Aµ is the B ﬁeld, B = F12 since all other components of Fµν are zero, which will behave like 1 da e−mAr B = ∂ A − ∂ A = ∼ √ . (3.46) 1 2 2 1 er dr r

When mA < mφ this force will dominate over the interaction mediated by φ, while for mA > mφ the roles are reversed. If we consider the force between two vortices with the same charge, then φ will give an attractive force while B will give a repulsive force, the sign of the force depending on whether the mediating boson has even or odd spin, see section 3.4 [1].

3.3 Adding Fermions Having found a soliton in two dimensions we will now consider what happens when a fermion is included in the theory. First we can extend our vortex solution in two space dimensions into three and assume that the vortex has no z dependence. Such a solution will then be interpreted as a string of vortices along the z-axis. When D = 3, a fermion is described by a four component spinor ψ that is the sum of two chirality eigenspinors, ψR and ψL with,

5 γ ψR = ψR 5 (3.47) γ ψL = −ψL.

23 We add the Lagrangian

µ µ ∗ LF = iψRγ DµψR + iψLγ DµψL − gφψLψR − gφ ψRψL, (3.48)

5 to (3.30). Since ψR and ψL are eigenspinors of γ ,we have

iβγ5 iβ iβγ5 −iβ e ψR = e ψR, e ψL = e ψL. (3.49)

Our Lagrangian is therefore invariant under the chiral transformation

ψ → eiβγ5 ψ (3.50) φ → e−i2βφ.

Including mass terms, e.g. mψψ, breaks this symmetry, since something like mψLψR is not invariant. It should also be noted that in order for LF to be invariant under U(1), φ must have twice the charge of ψ The Dirac equations for ψR and ψL that follow from (3.48) are

∂LF ∂LF ∗ µ ∂µ − = gφ ψL − iγ DµψR = 0 ∂ ∂µψ ∂ψ R R (3.51) ∂LF ∂LF µ ∂µ − = gφψR − iγ DµψL = 0. ∂ ∂µψL ∂ψL In trying to look for solutions to these equations we ﬁrst make the assumption that there should be no t or z dependence. We can also consider ψR and ψL to only depend on the distance r from the z-axis and for them to be eigenspinors of iγ1γ2

1 2 iγ γ ψL = ψL 1 2 (3.52) iγ γ ψR = −ψR.

With these assumptions and with φ and Aµ taking their local vortex form, (3.51) simpliﬁes to, see appendix B, a ∂ + ψ − igvfγ1ψ = 0 r 2r R L a (3.53) ∂ + ψ − igvfγ1ψ = 0. r 2r L R We see that the equations decouple if

1 1 ψL = −iγ ψR, ψR = −iγ ψL, (3.54) leading to a ∂ + ψ + gvfψ = 0, (3.55) r 2r L L

24 and a similar equation for ψR. (3.55) is solved by

Z r 0 0 0 a(r ) ψL = A exp − dr gvf(r ) + 0 χ (3.56) 0 2r where A is a constant and χ is constant left handed spinor with iγ1γ2χ = χ. A simple way to introduce t and z dependence for the fermion ﬁeld is to multiply our existing solutions ψR and ψL by a function b(z, t)

0 0 ψR = b(z, t)ψR, ψL = b(z, t)ψL. (3.57)

Inserting into the second equation of (3.51) and using that ψR solves the part that does not involve t, z leads to 0 3 i γ ∂0 + γ ∂3 b(z, t)ψL = 0. (3.58)

Since ψL satisﬁes (3.56), we have

0 3 γ ∂0 + γ ∂3 b(z, t)χ = 0. (3.59)

Using that χ is left handed and iγ1γ2χ = χ, in conjunction with the anticommutation relations for gamma matrices results in

−χ = γ5χ = iγ0γ1γ2γ3χ = −iγ0γ1γ3γ2χ = iγ0γ3γ1γ2χ = γ0γ3χ. (3.60)

Using (γ0)2 = 1 we arrive at γ0χ = −γ3χ and (3.59) becomes

(∂0 − ∂3) b(z, t) = 0. (3.61)

This has solutions b(z, t) = h(z + t). Speciﬁcally we can have

b = e−iω(t+z), (3.62) which represents particles moving at the speed of light in the negative z direction. If we introduce a second fermion field η with the opposite charge of ψ, an interesting effect will appear. The opposite charge of η will have the effect of changing the minus sign in (3.59) to a plus sign, meaning that it will have solutions traveling in the positive z- direction. The opposite sign comes from the fact that the signs in (3.52) should be reversed in order for ηL to obey an equation similar to (3.55). If we take the U(1) that has been discussed until now to not be that of electromagnetism, but instead of some other force, and let ψ and η have the same charge, say e, under true electromagnetism then we can construct an interesting situation. If we apply an electric field in the negative z-direction then this will increase the momentum of ψ states but decrease the momentum of η states. If we work in the Dirac sea picture when all negative energy states are occupied for both ψ and η before the field has been applied. When the field is turned on, negative energy ψ states will be promoted to positive energy states, thus creating a particle with charge e moving along the negative z-direction. For η, unoccupied

25 positive energy states will become unoccupied negative energy states, creating hole states that correspond to antiparticles with charge −e moving in the positive z-direction. Since we are creating particles and antiparticles, we are not violating charge conservation, but there is a net current since oppositely charged particles are moving in opposite directions. We should expect the fermions to remain bound to the string of vortices along the z-axis if their energies satisfy E < gv, which is the fermion mass when working in a pure vacuum background, f = v and a = 0. If the electric field is removed, particles satisfying this condition should persist since they cannot leave the string and all energy states below are occupied. Therefore, the current remains even after the field is switched off, which means that the string acts as a superconductor.

3.4 Solitons from the perspective of homotopy In order to get a more complete picture of solitons in higher dimension we can introduce the concept of homotopy. Essentially, homotopy will provide a more rigorous framework in which it is possible to discuss solitons as a consequence of the topology of the vacuum manifold M, the space of vacua. A simple way to think of homotopy is that it describes continuous maps from a sphere in D dimensions to another manifold. In our case we will consider maps where the target manifold is M and the sphere represents a parametrization of spatial infinity. The aim of this section is to try to convey the basic ideas, the interested reader is referred to appendix C for some more definitions of the concepts and to chapter 4 of [1] for a more in depth discussion. In two dimensions we will consider a closed loop on the vacuum manifold. We are interested in what is known as the fundamental group, or the first homotopy group of M, π1 (M). The product that defines this group is the combination of two loops that share the same base point (start and end point) into a new loop that shares a base point with the original loops. Strictly speaking the group elements are called homotopy classes, where each homotopy class consists of all the loops sharing a base point that can be continuously deformed into one another without leaving M during the process. Two such loops are said to be homotopic and the map between them is called a homotopy. The base point must remain fixed during the deformation process otherwise the paths are not homotopic but freely homotopic. If we consider our M to be a circular disk with a hole in the middle like in figure 3, then the two loops A and B are homotopic while C is homotopic to neither A nor B since any deformation that would take C to either of the other two loops would have to pass through the hole. The trivial loop, which is when the whole circle is mapped to single point, will be a part of the homotopy class that forms the identity element of the fundemental group, since the product of any loop with the trivial loop returns the original loop. If we consider an M that is simply connected, i.e. every loop on M can be contracted to a single point, then π1(M) must be the trivial group, the group with only one element, since every loop is homotopic to the trivial loop and thus there is only one homotopy class. The vortex that we have discussed above provides a simple example that illustrates the connection between solitons and homotopy. The vacuum manifold, defined by the equation

26 Figure 3: Illustration of homotopy. A and B can be continuously deformed into one another and are thus homotopic, while C can not be deformed to A or B without passing the hole and is thus not in the same homotopy class as A and B.

V (φ) = 0, is the circle S1. If we assume that φ takes on a vacuum value at every point at spatial infinity, then as one goes around the circle r = ∞ one will automatically trace out a loop on M at the same time. For a uniform vacuum solution, we will trace out the trivial loop, while a solution like (3.34) will trace out a loop that goes one turn around S1. These two loops live in different homotopy classes since they cannot be deformed into one 1 1 another. This is a result of S not being simply connected, since in fact π1(S ) = Z the group of integers under addition, which can be seen by considering the product between going around the circle n times with a loop going around m times, with n, m counted positive if the loop is counter clockwise and negative if it is clockwise. More generally, we can consider a theory in two spatial dimensions with a continuous symmetry group G, whose action takes some collection of scalar fields φi to the different minima of some potential V (φ), that is broken by the vacuum expectation value of the fields to a subgroup H. H is the set of elements in G whose action leave the vacuum expectation value of the scalar fields invariant. We define the factor group or quotient group G/H as the group that is obtained when we consider g1, g2 ∈ G to be the same element if g2 = g1h, h ∈ H. Any such g1, g2 will have the same effect on the vacuum and so we can associate each element of G/H to a minima of V, or simply M = G/H, (3.63) where the equality is to be understood as saying that the parameter space that describes G/H is the same as the vacuum manifold. The result states that we should only expect solitons if π1(G/H) is nontrivial since otherwise it should be possible to deform any configuration to the vacuum, the trivial loop. In the case of vortices, G = U(1) while H is the trivial group leading to G/H = G. Because U(1) can be described by a circle, we have 1 π1(G/H) = π1(U(1)) = π1(S ) = Z. (3.64)

If we work in D spatial dimensions, then the correct object to consider is πD−1(M), where n πn(M) is similar to π1, but instead of loops we are considering maps from S , the n-sphere, to M. The reason is, as mentioned in the beginning of this section, that spatial inﬁnity in D spatial dimensions takes the form of a D − 1 sphere with r = ∞.

27 4 Magnetic Monopoles

In classical electromagnetism without sources, Maxwell’s equations can take the form ∂ ∇ × E~ + iB~ − i E~ + iB~ = 0 ∂t (4.1) ∇ · E~ + iB~ = 0

which is unaﬀected by the transformation E~ + iB~ → eiα E~ + iB~ . (4.2)

When we include sources in the equations this symmetry between the electric and magnetic fields is broken since in the standard form of Maxwell’s equations only electric charges and currents appear. If we want to restore the broken symmetry, a reasonable approach might be including sources of magnetic charge in the theory. Such magnetic charges are often called magnetic monopoles. In analogy with the electric point charge, we might try and find magnetic charges that generate a Coulomb field Q g B~ = m rˆ = rˆ (4.3) 4πr2 r2

where Qm represents magnetic charge and in g we have absorbed the 4π factor. Since the B-field will no longer be globally divergenceless, it can not be written as the curl of a vector potential in all of space. However, if we assume that the sources are confined to finite regions, then it might be possible to write the field as a curl outside any regions containing monopoles. In fact, a potential like g Ai = −ij3rˆj (4.4) z + r does produce a field of the form (4.3), except for the singularity on the negative z-axis. It can be shown, section 5.1 of [1], that the location of this string of singularities, called the Dirac String, is gauge-dependent and should not have physical meaning. The fact that these vector potentials have this kind of string singularities actually implies that electric and magnetic charges should be quantized. By taking into account the Aharonov-Bohm effect, that the vector potential can affect the phase of the wave function of an electrically charged particle, see section 2.7 of [9], it is possible to show that the wave function of a particle with electric charge q should be multiplied by a factor e−i4πqg, section 5.1 [1]. If the wave function is to be single valued then we must have n qg = (4.5) 2 where n is an integer. This means that g and q can be written as

g = kgmin, q = lqmin, (4.6)

28 with k, l integers and gminqmin = 1/2. (4.5) is known as the Dirac quantization condition. Now that we have discussed why magnetic monopoles might be interesting to study, it is time to present a model where they appear. The original work was performed by ’t Hooft [10] and Polyakov [11], and the monopole that we will ﬁnd is known as the ’t Hooft-Polyakov monopole. The Lagrangian of the model is 1 2 1 L = D φ~ − F~ 2 − V φ~ , (4.7) 2 µ 4 µν with φ~ a triplet of scalar ﬁelds transforming under SU(2) and the potential

µ2 λ 2 λv4 V φ~ = − φ~2 + φ~2 + (4.8) 2 4 4 which has global minima at φ~2 = µ2/λ = v2 that break the symmetry to U(1). The covariant derivative acts like ~ ~ ~ ~ Dµφ = ∂µφ + eAµ × φ, (4.9) while the ﬁeld strength is given by ~ ~ ~ ~ ~ Fµν = ∂µAν − ∂νAµ + eAµ × Aν. (4.10) The ansatz that will be used is φa =r ˆah(r) 1 − u(r) Aa = aimrˆm (4.11) i er a A0 = 0. This gauge is known as hedgehog gauge since the scalar triplet is always parallel to the radial direction in SU(2) space. Since this is a static ansatz, the energy is again given by Z E = − d3xL (4.12)

Inserting (4.11) into (4.7) and using the properties of ijk, see chapter 26 of [3], leads to ! Z u0 2 (1 − u2)2 1 λr2 E = 4π dr + + u2h2 + r2(h0)2 + (h2 − v2)2 . (4.13) e 2e2r2 2 4

For a detailed derivation see appendix D. The Lagrange equations that follow from (4.13) are 2 2 d u u(u − 1) 2 2 2 − 2 − e uh = 0 dr r (4.14) d2h 2 dh 2u2h + − + λ(v2 − h2)h = 0. dr2 r dr r2

29 a To avoid singularities in Ai at the origin, we must have u(0) = 1 while h(0) = 0 is required to have a well defined φ~ there sincer ˆ is not defined at the origin. The r dependence of u and h near r = 0 is found in appendix A. u(∞) = 0 and h(∞) = v are required for the energy of the solution to be finite. An asymptotic analysis similar to that in section 3.2 leads to the equation δ 0 = δ00 + − e2v2δ ≈ δ00 − e2v2δ (4.15) r2 where δ = u is the expansion of u. In order to be consistent with the boundary conditions we must have δ ∼ e−evr. For h = v + the equation is 2 0 = 00 + 0 − 2λv2 (4.16) r 0 where all nonlinear terms have been√ neglected. If we consider the term to also be negligible then the solution is ∼ e− 2λvr, which is consistent with that approximation. For λ/e2 > 2 neglecting the δ2/r2 term in the equation for is a poor approximation and reasoning similar to that in section 3.2 leads to ∼ e−2evr/r2. Defining a magnetic field 1 Ba = − ijkF a (4.17) i 2 jk it is possible to write, doing similar computations as those that led to (4.13), 1 − u2 u0 Ba =r ârî − (δia − rârî) . (4.18) i er2 er a a a i 1 Disregarding terms that fall exponentially fast, Bi becomes Bi =r ˆ rˆ er2 , which looks like the coulomb field of a magnetic monopole carrying charge Qm = 4π/e. Strictly speaking this quantity should form a dot product with φˆ = φ/~ |φ| so that we pick out the components that lie in the unbroken U(1) group that leaves the vacuum invariant, which is formed by the rotations in SU(2) space around the axis defined by the vacuum expectation value of φ~. This U(1) group is then interpreted as that of regular electromagnetism, which means that we indeed have a magnetic monopole. After a rescaling s = evr, (4.13) becomes 4πv Z (1 − u2)2 1 λs2 4πv E = ds (u0)2 + + u2h2 + s2(h0)2 + (h2 − v2)2 = C(λ/e2). e 2s2 2 4e2 e (4.19) Figure 4 shows a plot of the function C for different values of λ/e2 showing that it takes values roughly between 1 and 1.7. The integral was calculated by solving the equations for h and u numerically and then integrating the resulting solutions. A rough estimate of the energy can be found by viewing the energy density as completely due to the asymptotic form of B outside some sphere of radius R, and constantly equal to λv4 coming from the constant term in V , since both h and 1 − u should be close to zero for sufficiently small R. Integrating to find the total energy and minimizing with respect to R leads to 1 8πv R ≈ ev , which in turn leads to E ≈ 3e , which is of the same order as the values that were calculated numerically.

30 Figure 4: Plot of the monopole energy integral for diﬀerent values of λ/e2.

5 Conclusions

We have seen that solitons are deeply connected to the topology of the vacuum manifold and spatial inﬁnity. From Derrick’s theorem, section 3.1, we know that we should consider theories with local symmetries if we want to have solitons. In this thesis we have only considered one theory with magnetic monopoles, and only with scalar and gauge ﬁelds. It is therefore natural to ask if other theories contain monopoles and what happens when fermions are included in the theory. We have also mainly been interested in the existence of solitons, not so much in how they interact with each other or with other particles that may be included in the theory. Each of the theories with solitons that we have studied has an interesting feature. For the sine-Gordon we have the equivalence with the massive Thirring model, for the vortices we have the superconducting string and for the ’t Hooft-Polyakov monopole we had the existence of a magnetic monopole.

Acknowledgements

I would like to thank my supervisor Johan Bijnens for all the help that he has provided during the project. I also want to thank Erik Linn´erfor being a good oﬃcemate and for the interesting discussions we had about our projects.

A Vortex and monopole equations near origin

If we linearize (3.11) around r = 0 we arrive at

d2f 1 df 1 + + λv2 − f = 0. (A.1) dr2 r dr r2

31 √ This equation has a solution proportional to the Bessel function J1( λvr) which behaves like, up to some constant factor, J1 ∼ r for small r [5]. For the vortex with gauge symmetry the linearized equations near u = 0 are

d2a 1 da − = 0 du2 u du (A.2) d2f 1 df λ 1 + + − f. dr2 r dr e2 r2

√ 2 λ 2 Solving the equations results in a ∝ u , f ∝ J1( e u) and therefore a ∼ u , f ∼ u near u = 0. Linearizing (4.14) near r = 0, with u = 1 − δ, leads to

d2δ 2δ − = 0 dr2 r2 (A.3) d2h 2 dh 2 + + λv2 − h = 0. dr2 r dr r2 √ 2 −1 2 2 The solutions are δ ∝ r , h ∝ r J2( λv r), which means that δ ∼ r , h ∼ r for small r [5]. Combining these results with the boundary conditions stated in the main text ensures that all fields are well defined and continuous at the origin and that the energy densities in (3.13), (3.36) and (4.13) are finite there as well.

B Calculation of fermion equations

Here we go through the steps between equations (3.51) and (3.53). Since we are assuming that our solutions should be independent of z and t, we have

1 2 i γ D1 + γ D2 ψL − gφψR = 0 (B.1)

The choices in (3.52) together with (γ1)2 = −1 leads to

1 (D1 + iD2)ψL − igφγ φR = 0. (B.2)

Inserting the vortex ﬁelds gives ya xa ∂ + i + i∂ + ψ − igvfeiθγ1ψ = 0 (B.3) x 2r2 y 2r2 L R which after some trigonometry will yield (3.53). The other equation can be handled in the same way, kepping in mind that ψR should have the opposite charge in the covariant derivative.

32 C Some deﬁnitions related to homotopy

When we talk about a loop we mean a continuous map f(t) from 0 ≤ t ≤ 1 to M such that f(0) = f(1) = x0. The product between two loops is deﬁned as ( f(2t) 0 ≤ t ≤ 1/2 (f ◦ g)(t) = . (C.1) g(2t − 1) 1/2 ≤ t ≤ 2

Two loops f and g sharing a base point x0 are homotopic if there exists a continuous map k(s, t), 0 ≤ s, t ≤ 1 such that k(0, t) = f(t) k(1, t) = g(t) (C.2)

k(s, 0) = x0

When we are talking about π1 (M), we should technically be referring also to a base point on M since all our deﬁnitions so far make reference to one. However if M is connected, i.e. any two points on M can be connected by a continuous path, then it is okay to drop the reference to the base point when discussing π1 since the fundamental group at one point can always be mapped to the fundamental group at another point by connecting the two points by a path.

D Calculating the energy density for the monopole

By making use of identities such as ijkilm = δjlδkm − δjmδkl, ijkijm = 2δkm, δii = 3 (D.1) which hold in D = 3 and by utilizing the antisymmetry of ijk we can make the calculation 2 ~ 2 ~ of Fµν and Dµφ much simpler since both quantities contain vector products whose a abc b c ~ components can be written as (A × B) = A B and because, with our ansatz, the Aµ field contains an ijk. We start with the field strength F~ 2 = F a F a = ∂ Aa − ∂ Aa + eabcAb Ac ∂ Aa − ∂νAa + eadeAd Ae = µν µν µν µ ν ν µ µ ν µ ν µ µ ν (D.2) a a a a 2 abc ade b c d e a abc b c = 2∂µAν∂µAν − 2∂µAν∂νAµ + e AµAνAµAν + 4e∂µAν AµAν. Plugging in the ansatz and using (D.1) should then yield a result after some algebra. Starting with the term without derivatives we find abc ade b c d e bd ce be cd b c d e b c b e b c c b Ai AjAi Aj = δ δ − δ δ Ai AjAi Aj = Ai AjAi Ac − Ai AjAi Ac = 1 − u4 = bimbincjkcjlrˆmrˆnrˆkrˆl − bimbjncjkcilrˆmrˆnrˆkrˆl = er 1 − u4 1 − u4 = 4δmnδkl − δmnδkl + δknδml rˆmrˆnrˆkrˆl = 2 . er er (D.3)

33 Doing the same for the other terms results in (1 − u)u0r (1 − u)2 (u0r + 2(1 − u))2 ∂ Aa∂ Aa = −4 − 2 + 2 i j i j (er2)2 (er2)2 (er2)2 (1 − u)u0r (1 − u)2 ∂ Aa∂ Aa = 4 + 2 (D.4) i j j i (er2)2 (er2)2 (1 − u)3 abc∂ AaAb Ac = −2 . µ ν µ ν e3r4 So the ﬁeld strength term becomes 1 1 1 F a F a = (u0)2 + (1 − u2)2. (D.5) 4 ij ij e2r2 2e2r4 Next we will do the corresponding calculations for the covariant derivative.

a a a abc b c a ade d e Diφ Diφ = ∂iφ + e Ai φ ∂iφ + e Ai φ = a a a abc b c 2 abc ade b c d e (D.6) = ∂iφ ∂iφ + 2e∂iφ Ai φ + e Ai φ Ai φ . The first term is xah xah h h h h ∂ φa∂ φa = ∂ ∂ = δai +r îrâ h − δai +r îrâ h − = i i i r i r r r r r h2 h0h h2 h0h h0h h2 h0h h2 h2 = 3 + − + + (h0)2 − − − + = 2 + (h0)2. r2 r r2 r r r2 r r2 r2 (D.7) The middle one is h h 1 − u 2e∂ φaabcAbφc = 2e δai +r îrâ h − abcbimrˆmrˆc h = i i r r er 1 − u h h = −2h δai +r îrâ h − δaiδcm − δamδci rˆmrˆc = r r r (D.8) 1 − u h h = −2h δai +r îrâ h − δai − rârî = r r r 1 − u h h h h 1 − u = −2h 3 + h0 − − − h0 + = −4h2 . r r r r r r2 The last term is 2 abc ade b c d e 2 bd ce be cd b c d e 2 b b c c b c c b e Ai φ Ai φ = e δ δ − δ δ Ai φ Ai φ = e Ai Ai φ φ − Ai Ai φ φ = (1 − u)2 (1 − u)2 (D.9) = e2 bimbilrˆmrˆl − bimcilrˆmrˆlrˆbrˆc h2 = 2h2 . (er)2 r2 The contribution from the covariant derivative is therefore 1 (1 − u)2 h2 1 1 − u D φaD φa = h2 + + (h0)2 − 2h2 . (D.10) 2 i i r2 r2 2 r2 34 References

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