Non-Abelian strings in the Higgs phase

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Universit`

a degli Studi di Pisa

DIPARTIMENTO DI FISICA E. FERMI Corso di Laurea Magistrale in Fisica

Tesi di laurea magistrale

Non-Abelian Strings in the Higgs Phase

Candidato:

Vittorio Asero

Relatore:

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Introduction

In this thesis it will be proposed the analysis of dierent physical objects known under the name of solitons.

The rst study of a soliton is attributed to John Scott Russell in 1834, who tried to describe the motion of a solitary wave along a canal.

In general, solitons are said to be stable and localized solutions of classical equations of motion. In this scheme, an important feature is the non-linearity of the equations involved, that is often fundamental to prevent the dispersion of a wave.

With the advent of quantum eld theory, solitons theory was soon recognized as a powerful method to better understand some particular solutions with localized energy density, that actually behave like particles. The stability of such solutions can be guaranteed by the topological behaviour of the elds, therefore we usually refer to this objects as topological solitons.

In Chapter 1 we will introduce the basic concepts of topology and homotopy theory, starting from the denition of the continuous deformation of a map. Moreover, it will be applied the concept of homotopy groups and its role in solitons theory. We observe how particular boundary conditions for the elds are crucial for the stability of the solutions, which are often labelled by an invariant integer N, the topological charge.

As an initial physical overview we will consider a general action of n scalar elds and will observe how, imposing specic boundary conditions at inn-ity for the elds, is a necessary condition (but not sucient) to have stable solutions. The other important check is the Derrick's theorem, a scaling ar-gument said to be a "no-go theorem", because it establishes if solitons can exist within a theory. This is achieved if the energy, after the rescaling, con-verges.

Nevertheless, eld theory gives us a coherent way to evade the Derrick's the-orem and admits solitons. These procedures will be illustrate in the last part of Chapter 1 with some simple and explanatory examples.

Initially we will focus on strings-vortices, the main subject of this work, in-troducing the string-vortex of Abrikosov, Nielsen and Olesen (ANO).

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The rst model of this soliton-object is due to Abrikosov in 1957 [1] in the context of superconductivity, within a theory of a single complex scalar eld. To have stable solutions and evade the Derrick's theorem we can add a gauge eld with a U(1) symmetry, through which the system undergoes a sponta-neous symmetry breaking: the Ginzburg-Landau mechanism.

This phenomenon provides a characteristic length scale to the electro-magnetic eld (the photon acquires mass) and, below a critical temperature, a super-conductor in such a eld deects the magnetic eld because of the Meissner eect.

In 2 + 1 dimensions we refer to these stable solutions as Abelian vortices, and they actually behave like particles, while in 3 + 1 dimensions they are characterized by an innite magnetic ux tube, similar to a string, squeezed into the superconductor by the Meissner eect.

After the introduction of the Higgs mechanism, the Abrikosov vortex-string was extended by Nielsen and Olesen to particle physics and it had many other extensions and applications in string-eld theory [2].

We will then treat the Q-balls case, non-topologial solitons that were rst introduced by Coleman [9]. They are non topological in the sense that they are not characterized by elds with specic boundary conditions at innity, but they are still a good prototype of charged solitons.

Another way to evade the Derrick's theorem is encountered in the F addeev-Skyrme soliton example, in which we add terms with higher derivatives in the elds (the Skyreme term) that make the energy convergent. It is basically a sigma-model and the solitons that emerge are often called Hopfions or knotted solitons, because they are characterized by a topological invariant that takes the role of the topological charge called Hopf charge.

As a last example we will briey overview the concept of instantons and instanton size, introduced for the rst time by Belavin, Polyakov, Schwartz and Tyupkin. They are soliton-like solutions to the action of a pure Yang-Mills theory in the Euclidean space-time. This last request is essential to ensure the convergence of the action, that is conformal, i.e. invariant by rescaling, so it trivially satises the Derrick's theorem. Instantons are con-sidered to be a powerful method to better understand the non-perturbative aspects of non-Abelian theories as Quantum-Chromo-Dynamics (QCD) and its vacua structure.

In Chapter 2 we will deal specically with other examples of strings in eld theory, that dier from the ANO-strings case in their dierent content of elds and symmetries, making them non-Abelian.

The interest for non-Abelian theories is naturally related to the huge connec-tions and discoveries in theoretical and particle physics in the past century, that led to the Standard Model of the fundamental interactions.

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Some of the most interesting challenges are related to QCD, the theory of quarks, in which we can nd phenomena as confinement . For this pos-tulate, a single quark cannot be observed and is bonded to interact with another quark (at least) by the gluon eld. The rst attempt to clarify this fact was done by 't Hooft, Mandlestam and Nambu in the '70s of the last century, who hypothesized that a quark-antiquark pair could be considered as a soliton object. In this picture the two quarks can be viewed as conned by the chromo-electric eld, that, as in superconductivity theory, squeezed and can form a vortex-ux tube: the vortex-antivortex production is thus called dual-Meissner eect.

However, the question about connement is still open, even if recent devel-opments in the theoretical treatment of this phenomenon was given in [3]. Within this scheme, the matter and gauge elds have a pure non-Abelian symmetry. These are vortex-string type solutions as in the ANO case, but we will illustrate the most salient aspect of this model (with a simple SU(2) symmetry), which is the introduction of the orientational moduli fields. As a residue of the original non-Abelian symmetry, we will show that this mod-uli elds appear as observables in the eective world-sheet action, making the theory actually a sigma-model.

The rest of Chapter 4 is dedicated to another model in string-eld theory, due to Witten: superconducting strings [4].

This model can be viewed as an intermediate step between the pure Abelian ANO-strings and the non-Abelian ones cited above. It is characterized by two complex scalar elds and two Abelian gauge elds, therefore the symme-try group of the theory is U(1) × U(1), and is still characterized by solutions of the vortex-string type. More specically, one of the two matter elds (cou-pled with its gauge eld) is responsible for the formation of the vortex-string, while the other one, with its gauge partner, will condensate in the string core, creating a charged current.

The Witten model inspired huge research in dierent branches of physics, from superconductivity and condensed matter to astrophysics and cosmol-ogy. In this context we will focus our attention to cosmic strings, which are supposed to be vortex-string solitons with astronomical dimensions, orig-inated by some primordial symmetry braking (Kibble mechanism). We will briey introduce the cosmological scenario in which cosmic strings could have been formed in the past, and will meet for the rst time a new hypotheti-cal string-like object hypotheti-called vorton. Vortons were introduced by Davis and Shellard in the '80s of the last century [5], as a topological extension of the Witten model, actually compactifying a string to create a loop. In doing this, we will observe how the current in the string core contributes to the angular momentum preventing the loop to collapse and become singular (a point).

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Such objects, if they are stable, could be determinant for the problem of dark matter, because of their huge dimensions and hypothetical mass.

As a last example of strings we will review the case of semi-local strings, in which the number of matter elds is dierent from the number of gauge elds. We will also examine the most remarkable facts and the analogies with the other models, showing the linking between this model and the Faddeev-Skyrme model cited above.

In Chapter 3 we will treat in deep the argument of spinning solitons and their stability [6]. Many of the models introduced in Chapter 2 admit a natural extension, as in the vortons case, involving string-like solitons with a ring structure. Starting from the simple example of a rotating Q-ball, we will then approach to string-like objects, discussing more extensively the vortons case and the stability problem. Stability is crucial for the existence of soli-tons, but is dicult to establish because of equations complexity involved, that must be solved almost numerically. In the past, most of the work in this context has been done considering particular limits of the Witten model (the global limitor the sigma model limit). Only a few years ago, it was shown in [7] the rst numerical prove of vorton stability in the full gauged theory. As a last example of spinning soliton we will evaluate a recent attempt made by Gorsky, Shifman and Yung [8] (GSY) to construct a semi-local Abelian string of the vorton type. In this section we will clarify the linking between semi-local strings and Hopons, and make particular attention on the theoret-ical set-up used to establish the error committed using dierent coordinates and approximate solutions.

In Chapter 4 we will try to deal with the general case of a non-Abelian string in a vorton-like state. Using the tools exposed in the previous chapters we will rst study the eective world-sheet action presented in [3], noting that the energy and the angular momentum of the model satisfy, as the GSY soli-ton case, the so called Regge trajectories.

The rest of the chapter will be dedicated to the solution of the problem in 4 dimensions, with the aim to nd the correct prole functions for the elds and lay the foundation for a future numerical work. This will be designed to investigate the minimal number of elds to obtain a pure non-Abelian vor-ton; to study the possible implications in eld theory and, as in the vorton case, their role in cosmology as possible constituents of dark matter.

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Chapter 1 Topological solitons in eld

theories

1.1 Topological solitons

The concept of soliton was used for the rst time to describe stable solitary waves. In eld theory solitons usually emerge as localized solutions (with non-trivial topology) of non-linear dierential equations. We briey intro-duce in this chapter the basic concepts of topology and homotopy groups and their important role in soliton theory.

1.1.1 Introduction to topology

In topology, the homotopy theory is used to describe the relationships be-tween continuous maps dened on manifolds. In order to label dierent equivalence classes between maps, the homotopy theory gives a rigorous def-inition of deformation of maps. In this context, two maps are equivalent if one can be deformed into the other.

For the general theory we refer to the text [13] in Bibliography.

Let X and Y be two dierentiable manifolds, with x0 ∈ X and y0 ∈ Y. We

dene F as the family of maps f between X and Y for which f(x0) = y0.

The points x0 and y0 are called base points.

Two functions f, g ∈ F are said to be homotopic if there exists a continuous function

H(x, t) : X × [0, 1] → Y, (1.1.1)

with t ∈ [0, 1], such that H(x, 0) = f(x) and H(x, 1) = g(x).

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following properties are trivially veried:

• It is reexive: for every f ∈ F the function H(x, t) = f(x) deforms f into itself.

• It is symmetric: if H(x, t) deforms f into g, H(x, 1−t) deforms g into f.

• It is transitive: if H(x, t) deforms f into g and I(x, t) deforms g into h, the function L(x, t) dened as

L(x, t) = ( H(x, 2t) if 0 ≤ t ≤ 1 2 I(x, 2t − 1) if 1 2 ≤ t ≤ 1 (1.1.2) deforms f into h.

This equivalence relation labels on F distinct equivalence classes called ho-motopy classes.

If the space X is taken to be the n-th sphere Sn, the set of equivalence classes

F : Sn_{→ Y} _{has the structure of a group, and its called π}

n(Y ), the n-th

ho-motopy group, with n ≥ 1. In fact, it is demonstrable that can be dened an operation of compositions between functions whose result is still an element of the group.

We show this for n = 1.

For two distinct functions f(θ), g(θ) ∈ F : S1 _{→ Y}, with base points

0 ∈ [0, 2π] and y0 ∈ Y, we dene the composition function h(θ) as:

h(θ) = (

f(2θ) if 0 < θ < π

g(2θ − 2π) if π < θ < 2π (1.1.3) θ = 0 is still a base point, because h(0) = 0, so it is an element of π1(Y ).

The trivial map f(θ) = y0 is the identity.

Thus, for every f(θ) exists the inverse f−1_{(θ) = f (2π − θ)}_{for which applying}

(1.1.3) we get the identity.

As a rst example of homotopy theory we can take the theory of continuous maps dened from S1 _{to the real plane without the origin:}

F : S1

→ R2_/{0} _(1.1.4)

As before, we choose as base points θ = 0 ∈ [0, 2π] and (x0, y0) ∈ R2

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Any closed curve that does not encircle the origin can be continuously de-formed into the constant map, so it represents the trivial identity element. Closed curves that encircle the origin are dierent in general, because they cannot be deformed into the identity and it becomes relevant if they wind the origin clockwise or anticlockwise. In this theory it can be established that:

π1(R2/{0}) = Z (1.1.5)

The integer value represents the number of times that the curve encircles the origin. The sign is the orientation of the winding.

With this simple example we can see that the homotopy theory can gives us information even about the target manifold Y and its topology. In this case Y is connected, but not simply connected, in such case the π1(Y ) would be

just the identity, i.e. the trivial map. Another simple computation is the π1(S1).

A map f(θ) : S1 _{→ S}1 based on 0 (f(0) = 0), with θ ∈ [0, 2π], must be

continuous in this interval. This requires that f(2π) = 2πk, with k ∈ Z. Even in this example we obtain:

π1(S1) = Z (1.1.6)

Thus, we can extend this result to the Lie group U(1), that is omeomorphic to S1 _{and follows that:}

π1(U (1)) = Z. (1.1.7)

As will be discussed, this result is crucial for the formation of strings: the main topological defects treated in this work. As an extension to the n > 1 case, we only mention that, without proving:

πn(Sn) = πn(U (N )) = Z. (1.1.8)

1.1.2 Topological degree

As we described before for particular target manifolds, the computing of some simple homotopy groups can be an easy task. For a general manifold this cannot be so simple, but homotopy theory gives us a powerful tool in dealing with this calculation. Let us briey introduce this method, starting with the denition of the topological degree.

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This method works with manifolds of the same dimension, so let X and Y be two manifolds of dimension n and f : X → Y be a continuous based map. We also need that f is dierentiable everywhere with continuous derivatives. Let Ω be an n-form on Y with

Z

Y

Ω = 1. (1.1.9)

The topological degree of f is thus dened as the integral on X of the pull-back of Ω through f:

deg f = Z

X

f∗(Ω). (1.1.10)

Moreover, if we introduce the coordinates x in X and y in Y , the map f can be expressed as y(x). So the explicit form of the the volume Ω is:

Ω = Ω(y)dy1∧ dy2 ∧ ... ∧ dyn, (1.1.11) and the pull-back of Ω is:

f∗(Ω) = Ω(y(x))∂y 1_(x) ∂xj dx j_∧ ∂y2(x) ∂xk dx k_{∧ ... ∧}∂yn(x) ∂xi dx i = Ω(y(x))det ∂y i_(x) ∂xj dx1∧ dx2_{∧ ... ∧ dx}n = Ω(y(x))det (J(x)) dx1_{∧ dx}2_{∧ ... ∧ dx}n_. _(1.1.12)

J(x) in the last line represent the Jacobian of the transformation. This is an n-form over X, so it can be integrated over the manifold. In practice we integrate the form (1.1.11) as many times as the number of times the map y(x)covers the manifold Y . It can be shown that the topological degree is always an integer and actually is a topological invariant.

Let us prove this result for the π1(S1) that we have calculated before. As

1-form we choose:

Ω = 1

2πdθ. (1.1.13)

As a general map we take a function θ(θ0

). The topological degree is then: deg θ = 1 2π Z 2π 0 dθ dθ0dθ 0 = 1 2π[θ(2π) − θ(0)] = k k ∈ Z. (1.1.14) As expected, we obtain again the winding number.

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1.2 Field theory and topology

In this section we will apply some topological concepts to classical eld the-ories.

We start to consider a generic action S of n elds φ = (φ1, φ2, ...φn) and

a generic potential V (φ1, φ2, ...φn) in Minkowski space-time with d spatial

dimensions: S = Z dd+1x h1 2∂µφk∂ µ φk− V (φ1, φ2, ...φn) i , (1.2.1) with k = 1, .., n.

Here, φ can be thought, in topological terms, as a map:

φ(t, ~x) : M → Y, (1.2.2)

from the Minkowski space to the target space Y , that in the case under examination is Rn_.

In the static conguration (neglecting the time dimension), we can write the energy of the theory as:

E = Z ddxh1 2∂iφk∂iφk+ V (φ1, φ2, ...φn) i , (1.2.3) with i = 1, .., d.

The relation (1.2.2) is then restricted to:

φ(~x) : Rd_{→ Y.} _(1.2.4)

Nevertheless, as we will show in the instanton example (1.3.4), the time de-pendence could still be treated as a continuous deformation of maps.

Let us consider now a sub-manifold H ⊂ Y , in which the potential V (φ1, ..., φn)

takes its minimum value VH = 0. Then, a necessary (but not sucient)

con-dition to have nite energy, is that one in which elds must approach to a nite value at innity. From the topological point of view we are actually dening the limit of our elds in the boundary of Rn, that is the sphere Sd−1

∞ .

Therefore, our maps can be thought as:

φ∞: S∞d−1→ H. (1.2.5)

In this way, dierent maps with dierent behaviour at innity can be labelled in the same homotopy class if their maps at innity are homotopic.

The relevant homotopy group is then:

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Any conguration, apart from the trivial one (i.e. any constant map), that minimizes the energy is called topological soliton.

It is useful to introduce at this point another tool that is often used in topology and goes under the name of compactication.

Assuming, for example, to have the real plane R2. In the compactication

picture, we can think R2 _{as a 2-sphere S}2_{, in which each innity in every}

direction in the plane corresponds to a single point on the sphere (e.g. the north-pole). When we impose to a map φ : Rd_{→ Y} to have a nite limit at

innity we are actually considering that: lim

|x|→∞φ(x) = φ0, (1.2.7)

and therefore, we are compactifying Rd to Sd. The topological classes of the

theory can be thought as:

πd(Y ) (1.2.8)

If dim(Y ) = d the theory has more advantages because of the topological degree dened above and other computing facilitations.

1.2.1 Derrick's theorem

From the previous paragraph it is clear that a fundamental request to have solitons in a general eld theory is given by the non triviality of the relevant homotopy group involved.

Another important check to guarantee the presence of solitons in a theory is the Derrick's theorem. It is considered to be a "no go theorem", because it provides a way to determine if there are stable solutions or not. Anyhow, as we will see, eld theory suggests a coherent way to evade the Derrick's theorem and admit solitons.

Taking into account the scalar theory (1.2.1), we rescale the coordinates x by a factor λ > 0: x → λx. The elds φ will be transformed as:

φ(x) → φ(λx) (1.2.9)

The energy can be written as a sum of n contributions of the polynomial form En = R ddx[(∂φ)nφm], when the important contribution is only given

by the power order of the derivative of the eld. After the rescaling of the coordinates, a single element En will be transformed as:

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The sum of these contributions will be E[λ], and it will be proportional to λn−d_.

The Derrick's theorem establishes that a soliton solution is possible only if there is a minimum to this energy functional.

As an example of a possible application of the theorem (and a possible way to evade it) we consider a generic theory in d = 1, 2, 3 dimensions with an energy of the form:

E = E0+ E2, (1.2.11)

therefore, a theory with a term of the form (∂φ)2_{, which can be identied}

with a kinetic term, and another one with zero derivatives in φ that takes the role of the potential.

After the rescaling x → λx:

E[λ] = λ2−d_E

2+ λ−dE0. (1.2.12)

For each dimension:

E[λ] =      λE2+ λ−1E0 for d = 1 E2+ λ−2E0 for d = 2 λ−1E2+ λ−3E0 for d = 3

In d = 1 we see that the theorem is satised because the minimum can be achieved if λ = pE0/E2 and soliton can exists.

In two and three dimensions we are unable to nd a minimum for the energy. In two dimensions, for example, the derivative of the energy is −2λ−3_E

0,

and, apart from the trivial case of the vacuum E = 0, this term is never zero. A similar result is obtained in three dimensions.

A possible way to evade the theorem in d = 2, 3 and obtain solitons is to add terms with more than two derivatives (for example E4). In this way, it can

be shown that the minimum is achievable.

Another way to evade the theorem, as often used in eld theory, is the intro-duction of a gauge eld A(x).

After the usual rescaling, it will be transformed as:

A(x) → λA(λx). (1.2.13)

The derivative ∂ will be replaced by the covariant derivative D, that it trans-forms as:

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and there will be also a contribution due to the eld strength tensor F , that after the rescaling will be:

F → λ2F. (1.2.15)

As a result, F behaves as a E4 term after the rescaling, giving a natural

way to evade the theorem in more than one spatial dimension and stabilize solutions, admitting solitons in a general eld theory.

1.3 First examples of solitons

In this section we will review some basic, but instructive, examples of solitons in eld theory, starting from the ANO vortex-string and continuing with an overview of Q-balls, the Faddeev-Skyrme model and Instantons.

1.3.1 The Abrikosov-Nielsen-Olesen string

The global limit

The simplest example in which strings (or vortices) emerge as topological objects is a theory of one complex scalar eld with a U(1) symmetry. The rst vortex soliton-like solution was given in 1957 by Abrikosov in the context of superconductivity [1], and then, after the discovery of the Higgs' mechanism in particle physics, Nielsen and Olesen extended the Abrikosov vortex to high energy physics [2]. The Lagrangian of this model is:

L = ∂µφ∂µφ − U(|φ|), (1.3.1)

where the potential U(|φ|) is of the type:

U(|φ|) = λ(|φ|2− v2₎2_. _(1.3.2)

This typical "Mexican hat" potential guarantees that the vacuum is reached when |φ| = v, but the phase of the eld is free to rotate around the vacuum manifold, which in this circumstance is a circle.

If we use polar coordinates in the xy plane (g. 1.1) we can say that:

φ(r, α) = veinα at r → ∞. (1.3.3)

This eld conguration is called a vortex. It is characterized by the "winding number" n, which is an integer and counts the number of times that the eld encircles the vacuum manifold. More specically, we can map the abstract

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Figure 1.1: Polar coordinates in the xy plane, r = px2_{+ y}2_._[21]

circle of vacuums into a spacial one. The integer n labels topologically dis-tinct maps, characterizing the number of windings of the eld and it can be positive, negative or zero, because we are dealing with orientable maps. So we can express these maps using the homotopy groups. The rst homotopy group of U(1) (that is the symmetry group of the circle S1) is Z:

π1(U (1)) = Z. (1.3.4)

The non-trivial value of the rst homotopy group establishes the existence of topological defects in the theory. However, the global symmetry makes the energy divergent when we integrate in the all space.

In the xy plane (i, j = 1, 2) we have ∂iφ ∝ inφ∂iα = −inφεij xj

r2 (εij is the

Levi-Civita tensor in two dimensions), so for the vortex energy (or the string tension in D = 4), evaluated in the eld conguration (1.3.3), we nd a logarithmic divergence: E = Z d2x[∂iφ∂iφ+ U (|φ|)] −→ 2πv2n2 Z _dr r → ∞. (1.3.5)

The small-r divergence can be cured setting φ = 0 at r = 0, but for the large-r divergence we need to add a gauge eld.

Gauging the ANO string

The Lagrangian of the model is obtained from the previous one (1.3.1) adding a U(1) gauge eld (a photon) and substituting the partial derivative with the

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covariant derivative:

∂µφ → Dµφ = (∂µ− ineeAµ)φ, (1.3.6)

where ne is the electric charge of the eld φ in units of e, the charge of the

electron. Thus we have: L = − 1 4e2FµνF µν_{+ D} µφDµφ − U(|φ|). (1.3.7)

The rst term is the kinetic term of the photon, Fµνis the gauge eld strength

tensor:

Fµν = ∂µAν − ∂νAµ, (1.3.8)

and the potential U(|φ|) is the same of Eq. (1.3.2), which allows the Higgs mechanism in the vacuum of the theory.

After the Higgs mechanism the photon acquires mass mV =

√

2neev, and

perturbing the scalar eld around the vacuum, setting φ(x) = v + η(x)_√ 2 , we

observe that even the eld η, that is the Higgs boson, takes a non-zero value for the mass: mH = 2

√ λv. The energy in this model is:

E[φ(~x), ~A(~x)] = Z d2x 1 4e2FijFij + DiφDiφ+ U (|φ|) , (1.3.9)

and for the niteness of the integral in the vacuum conguration (1.3.3) we require that the potential ~A satises the following conditions:

• Fij = 0 at innity (pure gauge).

• Diφ goes to zero fast enough to guarantee the convergence of the term

R d2_xD iφDiφ.

If ~A= 0 at |x| → ∞ the term R d2_xD

iφDiφ →R d2x∂iφ∂iφ, and the integral

diverges as in the previous case.

Thus we have to choose the gauge eld in such a way to cancel the divergence caused by the phase α. It can be shown that this conguration is achieved if we set: Ai = n ne ∂iα= − n ne εij xj r2 at r → ∞. (1.3.10)

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The introduction of the gauge eld (1.3.10) allows an integral representation of the winding number n:

n= ne 2π I |x|→∞ dxiAi = ne 2π Z d2x B. (1.3.11)

The second equality is obtained using the Stokes' theorem and B is the magnetic eld, dened as follows:

B = F12=

1 2ε

ij_F

ij. (1.3.12)

Thanks to (1.3.11) we can interpret the winding number as a quantity pro-portional to the ux of the magnetic eld carried by the string in its core. An important parameter of the theory is the scalar coupling λ, because, as mentioned before, it determines the mass of the Higgs boson. The ratio of

mV

mH is used in the theory of superconductivity and it tells us how is the

in-teraction between strings (vortices). If mV > mH the strings attract each

other (type I superconductors). Otherwise, if mV < mH the strings repel

each other (type II superconductors). When mV = mH, that corresponds to:

λ= n

2 ee2

2 , (1.3.13)

strings do not interact. This special case denes a critical limit, in which the interaction vanishes, and it can be studied with a method that takes the name of Bogomol'nyi completion.

Critical string: BPS equations

In D = 4 dimensions the relation (1.3.9) can be interpreted as the string tension T . In the limit (1.3.13) we can rewrite T as follows:

T = Z d2x 1 4e2FijFij + DiφDiφ+ n2_ee2 2 (|φ| 2 − v2₎2 (1.3.14) = Z d2x ( 1 2 1 eB + nee(|φ| 2_{− v}2₎ 2 + |(D1+ iD2)φ|2 ) (1.3.15) + 2πv2n, (1.3.16)

that is the Bogomol'nyi completion.

When the integral (1.3.15) in the Bogomol'nyi completion vanishes the ten-sion has its minimum value:

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Therefore, this limit is achieved when the following Bogomol'nyi-Parasad-Sommereld (BPS) equations are satised:

B+ nee2(|φ| 2

− v2_{) = 0,} _(D

1+ iD2)φ = 0. (1.3.18)

To nd a solution to these equations we need a suitable ansatz for the elds φ and Ai. We introduce two new prole functions, ϕ(r) and f(r), and we

write the elds (xing the winding number n = 1) as:

φ(x) = vϕ(r)eiα_, _(1.3.19) Ai(x) = − 1 ne εij xj r2[1 − f (r)]. (1.3.20)

It is also convenient to set a dimensionless parameter for distance, ρ:

ρ= neevr. (1.3.21)

With this ansatz the equations (1.3.18) become: −1 ρ df dρ + ϕ 2_{− 1 = 0,} _ρdϕ dρ − f ϕ = 0, (1.3.22)

that can be solved using the following boundary conditions for the elds, at large distances:

ϕ(∞) = 1, f(∞) = 0, (1.3.23)

and in the string core:

ϕ(0) = 0, f(0) = 1. (1.3.24)

A solution to (2.1.22) can be found numerically (g. 1.2), and we note that at large distances the prole functions behave as follows:

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Figure 1.2: Prole functions ϕ(ρ) and f(ρ) for the ANO-string in the BPS limit. [21]

1.3.2 Q-balls

Q-balls were rst introduced by Coleman [9]. They are non-topological soli-tons, in the sense that they are stable bound state solutions whose boundary condition at innity is the same as that for the physical vacuum state [12]. Nevertheless, it is an important model because it shares many features with vortices and can be treated as one of the simplest spinning-soliton solution, as we will see in Chapter 3.

We start with a theory in D = 3 + 1 dimensions with only one complex eld Φ.

LQ = ∂µΦ∗∂µΦ − U (|Φ|). (1.3.26)

The eld equations are:

∂µ∂µΦ +

∂U

∂|Φ|2Φ = 0, (1.3.27)

and the energy-momentum tensor of the theory will be:

Tµν = ∂µΦ∗∂νΦ + ∂νΦ∂µΦ∗− gµνLQ. (1.3.28)

The global U(1) symmetry, under which the theory is invariant:

Φ → Φeiα, (1.3.29)

causes the conservation of the Noether charge: Q= i

Z

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The potential should have its minimum at U(0) = 0 and it must also satisfy the following condition:

ω2₋= min |Φ| U(|Φ|) |Φ|2 < ω 2 += 1 2 ∂2U ∂|Φ|2 _Φ=0, (1.3.31)

where ω+ represents the mass M of the eld.

To satisfy this condition it is necessary that U contains terms with the power of the eld |Φ| major than four. A possible ansatz could be:

U(|Φ|) = λ|Φ|2(|Φ|4− a|Φ|2 + b), (1.3.32) with λ, a, b positive constants.

In this picture we obtain:

ω₊2 = λb ω2₋= ω2 + 1 −a 2 4b . (1.3.33)

Derrik's theorem can be evaded in this context assuming a time dependence for the eld Φ, thus we can rewrite it with the following ansatz:

Φ = φ(~x)eiωt_, _(1.3.34)

for which the Noether charge becomes: Q= 2ω

Z

d3xφ2 ≡ 2ωN . (1.3.35)

The full Lagrangian, obtained integrating L over d3_x can be written in this

form:

L= Z

d3xL = ω2_{N − E}

0− E2, (1.3.36)

with E0 = R d3x(U ) and E2 = R d3x(∇φ)2. Rescaling the coordinates x →

λx the Lagrangian transforms as:

L → λ3(ω2_{N − E}

0) − λE2. (1.3.37)

The virial theorem for this conguration (λ = 1) is: 3ω2_{N = 3E}

0+ E2, (1.3.38)

and it imposes E0 6= 0 and E2 6= 0 for a general ω 6= 0.

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the equation of motion of the theory (1.3.27), which in the presence of ω can be rewritten as:

(∇2_{+ ω}2_{)φ =} ∂U

∂φ2φ. (1.3.39)

An equivalent way to nd Q-balls solution is to minimize the energy, which from the energy-momentum tensor (1.3.28) is computed as:

E =

Z

d3x(T00) = ω2N + E0+ E2 =

Q2

4N + E0+ E2. (1.3.40) Using (1.3.38) the energy can also be written as:

E = ωQ + 2

3E2. (1.3.41)

Imposing δE = 0 at xed N (thus even at xed Q) we nd: ω2 = ∂(E0+ E2)

∂N . (1.3.42)

The simplest ansatz that we can imagine to solve the equation of motion (1.3.27) is to impose, as the vortex case, a radial dependence of the eld:

Φ = eiωt_f(r). _(1.3.43) Evaluating into (1.3.27) we nd: f00+2 rf 0 + ω2f = 1 2 ∂U ∂f, (1.3.44)

in which the prime index denote the derivative with respect to r. We require to have nite energy, therefore, the eld must approach to zero at innity. For large r we assume that:

f ∼ exp[−1 r

√

M2_{− ω}2_r]. (1.3.45)

Solutions to this equation are functions that can be labelled by an integer number n, that counts the number of nodes of the function f(r)(Figure 1.3). As n increases also increases the energy of the Q-ball.

We can give a simple interpretation to this set of functions looking at the equation (1.3.44) in a dierent way. We write it as:

f02+ Uef f(f ) = ˜E −4

Z r

0

f02dr

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Figure 1.3: Left: the amplitude f(r) for the n = 0, 1, 2 spherically symmetric Q-balls. Right: the eective potential Uef f for three values of ω. [6]

where Uef f is an eective potential (Figure 1.3):

Uef f(f ) = ω2f2− U (f ), (1.3.47)

and ˜E is the total energy.

This model describes a particle that moves with friction in a one dimensional potential Uef f. So the nodes of the functions f(r) represent the number of

times that the particle oscillates between the two valley around the local minimum B (right graphics in Figure 1.3). In Figure 1.3 ω− and ω+ are

the extremes of the condition (1.3.31), while the red curve is given by an intermediate value of ω.

1.3.3 Faddeev-Skyrme model

We introduce in this section the Faddeev-Skyrme model, due to Faddeev ([10],[11]). It is basically a sigma model in 3+1 dimensions, in which there is an adding of an extra term (called the Skyrme term), which contains higher derivatives of the elds, and that stabilizes the solution. The model describes three scalar elds

n ≡ na

= (n1, n2, n3) (1.3.48)

under the condition:

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The eld n is then identied as a vector that spans a sphere S2 _{and the}

symmetry group of the theory it is clearly O(3). The Lagrangian of the theory is given by:

L(n) = 1 32π2(∂ µ_{n · ∂} µn − FµνFµν), (1.3.50) where Fµν is: Fµν = 1 2n · ∂µn × ∂νn ≡ 1 2ε abc_na_∂ µnb∂µnc, (1.3.51)

and εabc is the Levi-Civita symbol in three dimensions.

The equation of motion is:

∂µ∂µn + ∂µFµν(n × ∂νn) = (n · ∂µ∂µn)n. (1.3.52)

In the static limit, the energy of the system is: E(n) = 1

32π2

Z

d3x(∂in)2+ (Fik)2 ≡ E2+ E4, (1.3.53)

where i and k are spatial indices, i.e. i, k = 1, ..., 3.

Under the usual rescaling x → λx, the two terms of the energy transform as:

E2 → λE2, (1.3.54)

E4 →

E4

λ . (1.3.55)

For λ = 1 we have stationary energy only if:

E2 = E4. (1.3.56)

This shows that the adding of the Skyrme term E4 is crucial in order to evade

the Derrick's theorem and to have a stable solution in this model. Any static eld conguration n(~x) can be viewed as a map:

n(~x) : R3 _{→ S}2 _(1.3.57)

Even in this case, to avoid nite energies we must impose that the eld approaches a constant value at innity. In the S2 _{space this can be expressed}

in an analogous way to the compactication procedure introduced in section 1.2: all points at innity of R3 are mapped to one point on S2. In this

context we can choose to identify innity with the north pole of the sphere. In formulas:

lim

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Therefore, this condition makes evident the fact that there is a residual O(2) symmetry, due to rotations around the third axis of the internal space S2

(that is the axis that connects the north and the south pole):

n1+ in2 _{→ (n}1_{+ in}2_)eiα _n3 _{→ n}3_. _(1.3.59)

The Hopf charge

From the condition (1.3.58) is clear that we can compactify R3 _{into S}3 _and

consider a general map:

n(~x) : S3 _{→ S}2_. _(1.3.60)

As a result, the relevant homotopy group of the theory will be π2(S3), that

is (we write it without proving):

π2(S3) = Z (1.3.61)

The topological invariant integer of the theory, that we indicate as H(n) ∈ π2(S3), is called the Hopf charge ([15]), and it can be expressed, as will briey

be described, in an integral form.

Starting from the denition of Fµν given in (1.3.51) we can dene a 2-form

in the static picture as follows: F = 1

2Fikdx

i_{∧ dx}k_, _(1.3.62)

which is a closed form, i.e.:

dF = 0. (1.3.63)

It can be shown that, for a closed form, is always possible to nd a vector potential A = Akxk, for which:

F = dA. (1.3.64)

With the introduction of the vector potential Ak the Hopf charge can be

written in the following way:

H(n) = 1

8π2

Z

εijkAiFjk. (1.3.65)

It was proved in [11] that the minimal symmetry for which this integral is non null is the O(2) symmetry: spherically symmetric solutions are ruled out of the theory against axially symmetric ones.

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To better understand the meaning of the Hopf charge we start from the condition (1.3.59) and write an ansatz for our elds in cylindrical coordinates (ρ,z,φ) :

n3 = cos Θ(ρ), n1+ in2 _{= sin Θ(ρ)e}i(pz+nφ) _(1.3.66)

Remembering the ANO vortex-string case, we can identify n ∈ Z as a vortex winding number. The phase of the total angle, however, is even along the z axis, caused by the boost p.

In this picture, we can imagine periodic conditions in z, actually compacti-fying the z dimension into a loop of lenght L, writing:

pL= 2πm, m ∈ Z (1.3.67)

The integer m is interpreted as another winding number, and counts the number of windings around the loop, while n counts the winding along the loop.

As we can see from the Figure 1.4 is possible to identify the pre-images of any two points in the target space S2 _{as two "knotted" loops.}

With this ansatz, it is demonstrable that the Hopf charge results:

Figure 1.4: Preimage loops of any two points on the target space S2_{. [6]}

H(n) = nm, (1.3.68)

and it can be interpreted as the number of linking between the two loops (in Figure 1.4 it is shown the simplest H = 1 case).

Due to these features we often refer to this type of solitons as Hopons or knot solitons.

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1.3.4 BPST Instanton: an overview

In the previous examples we have always considered static congurations. Instantons are particular solitons that have nite action even in time, al-though it is necessary, in order to capture this behaviour, to perform a so called Wick rotation and go in Euclidean space-time:

t → −iτ. (1.3.69)

Instantons are characteristics of non-Abelian gauge theories and rstly pro-posed by Belavin, Polyakov, Schwartz and Tyupkin (BPST), but their name is due to 't Hooft.

In simple words, in quantum mechanics an instanton is supposed to be a quasi-particle that it manifests when a eld passes from two distinct vacua of the theory by tunnel eect ([25], [27]).

They historically had great interest in QCD and the understanding of its vacuum structure, but in this section we limit ourselves to consider a pure SU(2) Yang-Mills theory in four Euclidean dimensions.

The action of this model is simply: SY M = − 1 2g Z d4xET r[FµνEF E µν], (1.3.70)

where the index E means that we are in Euclidean space-time. In what follows we will omit this index, bearing in mind that we are considering Euclidean quantities.

The equation of motion is:

DµFµν, (1.3.71)

where Dµ is the covariant derivative in the adjoint representation of SU(2):

DµFµν = ∂µFµν− ig[Aµ, Fµν], (1.3.72)

and Aµ is a non-Abelian gauge eld dened from R4 to the Lie algebra of

SU(2), that we indicate with su(2) ([24]):

Aµ: R4 → su(2). (1.3.73)

Introducing the eld ∗F , dened by the use of the Levi-Civita tensor in four dimensions:

∗Fµν =

1

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we can rewrite the action, as in the ANO-string case, in a BPS way: SY M = − 1 4g2 Z d4xT r[(Fµν ∓ ∗Fµν)(Fµν ± ∗Fµν)] ±2T r[Fµν ∗ Fνν] . (1.3.75)

If the rst line of (1.3.75) is zero, i.e.:

Fµν = ± ∗ Fµν, (1.3.76)

we nd that the action is limited from below because of the non-vanishing second line. We obtain:

SY M ≥ 8π2|N |, (1.3.77) with: N = − 1 16π2_g2 Z d4x T r[Fµν ∗ Fµν] ∈ Z. (1.3.78)

To have a nite action we must require that the gauge eld at innity ap-proaches to a pure gauge form:

Aµ→ iU−1∂µU, (1.3.79)

with U(x) ∈ SU(2).

These maps are thus dened from the ∂R4 _{to SU(2), that can be both}

com-pactied in S3.

The relevant homotopy group of the theory is then:

π3(SU (2)) = π3(S3) = Z, (1.3.80)

and the integer (1.3.78), that is a topological charge, coincides this time with the topological degree of the theory.

Solutions to equation (1.3.76) are called instantons (with the plus sign) and anti-instantons (with the minus sign). Since the theory is invariant under rescaling (it is said to be conformal), there is no needing to check the Derrick's theorem: solitons can naturally exist in pure Yang-Mills theories.

We conclude this overview giving the form of one of the possible ansatz for the instanton.

Following [27], we write the gauge eld as: Aµ=

1

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where ρ is a function of the space-time coordinates and σµν is a tensor dened

by:

σi0 = σi, σij = εijkσk.

We report the solution for the N = 1 instanton, that is: ρ(x) = 1 + λ

2

|x − a|2, (1.3.82)

where a is a four-vector and is interpreted as the position of the instanton, while the positive term λ represents its size.

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Chapter 2 Vortices and strings

In this chapter we will deal more extensively the concept of strings/vortices: topological objects that can be found as soliton-like solutions in particular eld theories. In (2+1)-dimensions we usually refer to this objects as vortices, which are dened by a mass and actually behave like particles. In (3+1) dimensions we have to deal with string-like objects, characterized by a tension (energy per unit length).

In section 1.3.1 we have already met the simplest example of strings: the ANO string, which was a theory of only one complex eld and actually possessed a pure Abelian behaviour. In the next sections we will add another gauge eld and another scalar eld to our theory and we will focus on the U(1) × U(1) theory (superconducting strings) as well as the SU(2)×U(1) theory, in which we can see strings that are truly non-Abelian.

We will also introduce the concept of semi-local strings, in which the number of scalar elds is not equal to the number of gauge elds. This intermediate model, between the Witten model and the pure non-Abelian one, has many interesting features and will be useful to understand some of these concepts in Chapter 3.

2.1 Non-Abelian strings

We are going to introduce in this section a non-Abelian version of topological strings that emerge in particular non-Abelian gauge theories and, as we will see, are characterized also by "orientational" moduli, describing the possi-bility of the non-Abelian gauge eld to rotate in its "internal" non-Abelian group.

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consider a model with the following Lagrangian: L = − 1 4g2 1 FµνFµν − 1 4g2 2 Fa µνF µν a + (DµφA)∗(DµφA) + g 2 2 2 φ∗_Aτ a 2φ A 2 + g 2 1 8 (φ ∗ A)(φA) − 2v2 2 . (2.1.1)

It has two gauge bosons, a U(1) with coupling g1 and an SU(2) with coupling

g2. Fµν is the same as (1.3.8), while the non-Abelian eld strength is:

Fa

µν = ∂µAaν − ∂νAµa− εabcAbµAcν, (2.1.2)

where εabc is the Levi-Civita tensor in three dimensions (the structure

con-stants of SU(2)).

The matter sector consists of two scalar elds (A = 1, 2) in the doublet rep-resentation of SU(2)gauge, and the covariant derivative is (τa are the Pauli

matrices): Dµφ= ∂µφ − i 2Aµφ − i 2A a µτ a φ. (2.1.3)

This equation shows that the U(1) charges of the scalar elds are 1 2.

The second line of the equation (2.1.1) represents the potential. The coe-cients are tuned to guarantee the BPS limit, in which there is no interaction between strings and the equations can be rearranged with the Bogomol'nyi completion.

The model at hand has an SU(2) × U(1) gauge symmetry, but there is also another global SU(2) symmetry, that becomes explicit if we unify the two matter elds φA _as:

Φ =φ

11 _φ12

φ12 _φ22

, (2.1.4)

where the rst index refers to the SU(2)gauge and the second to the avor

group (it is the old index A).

With this replacement the Lagrangian (2.1.1) takes the form:

L = − 1 4g2 1 FµνFµν − 1 4g2 2 Fa µνF µν a + T r(DµΦ)†(DµΦ) − U (Φ†,Φ). (2.1.5)

The potential U(Φ†_,_Φ)_{, in this new formalism, can be rewritten as:}

U(Φ†,Φ) = g 2 2 2T r Φ†τ a 2 Φ T r Φ†τ a 2 Φ +g 2 1 8 T r[Φ † Φ] − 2v22 , (2.1.6)

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and it reaches its minimum (the vacuum conguration) when: Φ0 = v1 0 0 1 , (Aa µ)0 = 0, (Aµ)0 = 0. (2.1.7)

Therefore, the vacuum of the model is invariant under a global combined color-avor SU(2), that we indicate as SU(2)C+F (color-avor locking):

Φ → Φ0 = U†ΦU, (2.1.8)

with U ∈ SU(2)C+F.

In the following paragraphs we will extensively use this symmetry, which has an important role for the denition of a general non-Abelian string solution. We will use the freedom to rotate elds in this internal subgroup to show the new feature of this model: orientational moduli.

For the moment, we better describe how elds transform under the original U(2)gauge = SU (2) × U (1) symmetry, which, as we have shown, acts beside

the SU(2)f lavor group.

In this context, we can compact the gauge elds into a single U(2) eld, dened as: aµ= Aµ1 2 + A a µ τa 2 , (2.1.9)

a useful form that will be used in Chapter 4.

The U(2)gauge symmetry now acts on elds in the following way:

Φ → Φ0 = UCΦ (2.1.10)

aµ→ a0µ = UCaµUC−1+ i(∂µUC)UC−1, (2.1.11)

where UC ∈ U (2) is a general matrix of the form (the C sux stays for

"color"): UC = exp h a(x)1 2 + ω a (x)τ a 2 i (2.1.12) and a(x) and ωa_(x)_{are functions of space-time coordinates.}

Moreover, the SU(2)F symmetry acts on Φ as:

Φ → ΦUF, (2.1.13)

with UF ∈ SU (2)F.

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the original U(2)gauge× SU (2)F down to a diagonal subgroup, the so called SU(2)C+F, for which: Φ → Φ0 = UCΦUF, (2.1.14) aµ → a0µ= UCaµUc−1, (2.1.15) with U† C = UF ∈ SU (2)C+F.

We remark the fact that this is an exact symmetry of the theory that will have particular importance in the course of this treatment.

In the next paragraph we will examine the solitons content of this model, initially observing that it still admits an Abelian ANO-string type in its solution. Studying the dierences between the pure Abelian case we will then examine the genuine non-Abelian behaviour.

2.1.1 Elementary non-Abelian strings: (1,0) and (0,1)

strings

If we consider a vanishing non-Abelian gauge eld in our theory, we can still have topological defects restored by the Abelian gauge eld, π1(U (1)) = Z,

and the same behaviour of the ANO-string case. The only dierence is that the tension is doubled, because of the two avors of the scalar elds:

TAN O= 4πv2. (2.1.16)

It must be a more fundamental string with half tension, and we can hypoth-esize that this occurs in the non-Abelian case, allowing only one of the two scalar elds to wind around the string axis. Since π1(SU (2)) is trivial (all

the closed paths into a sphere can be contracted in a point) we could expect that even π1(SU (2) × U (1)) is trivial, but although it is not, because we

can always combine, for example, the Z2 center of SU(2) with −1 ∈ U(1),

obtaining a string-like solution: π1

SU (2) × U (1) Z2

= Z. (2.1.17)

This is achieved if we choose the following form for the elds: Φ(x) = v exp iα(x) 2 (1 ± τ 3₎ at |x| → ∞, (2.1.18) Ai = −εij xj r2, A 3 i = ∓εij xj r2. (2.1.19)

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From (2.1.18) it's clear that only one of the two avors can wind.

If we perform the Bogomol'nyi completion we can rewrite the string tension as: T = Z d2x ( 1 2g2 2 F₁₂a + g 2 2 2T r(Φ † τaΦ) 2 + 1 2g2 1 F12+ g2 1 2T r(Φ † Φ) − v2 2 +(D1+ iD2)φA ∗ (D1+ iD2)φA + v2F12 ) , (2.1.20)

and when it is BPS-saturated we nd that: T± = v2 Z d2xF12 = v2 I |x|→∞ dxiAi = 2πv2, (2.1.21)

and the elementary string tension is recovered. The ± subscript indicates the two possible choices of winding in the ansatz (2.1.18) and (2.1.19). The string in which winds only the rst avor it is called (1,0)-string. Otherwise, if we let wind the second avor we have to deal with the (0,1)-string. The two elementary strings are related by a Z2 symmetry, under which they

interchange.

The vanishing of the integrals in (2.1.20) denes the so called BPS equations, that we will soon examine in this non-Abelian theory. For the moment we underline the fact, known from the ANO-string theory, that solitons in the BPS limit do not interact.

2.1.2 BPS equations

From the Bogomol'nyi completion (2.1.20) we derive the rst-order BPS equations, that are:

Fa 12+ g2 2 2(φ ∗ Aτ a_φA_{) = 0,} _a_{= 1, 2, 3} F12+ g₁2 2( φA 2 − 2v2_{) = 0,} _(2.1.22) (D1+ iD2)φA= 0.

To obtain the (1,0) (or the (0,1)) string we set A1

µ= A2µ = 0, with only A3µas

a non vanishing term. Referring to the boundary conditions (2.1.18)-(2.1.19) we also choose a diagonal form for the eld Φ, setting the o-diagonal terms to zero .

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Figure 2.1: Prole functions ϕ1(r)(lower curve) and ϕ2(r)(upper curve) for

the (1,0)-string. [21]

for the (1,0)-string with winding number n = 1), introducing new prole functions: Φ(x) = ve iα_ϕ 1(r) 0 0 ϕ2(r) , A3 i = −εij xj r2[1 − f3(r)], (2.1.23) Ai = −εij xj r2[1 − f (r)].

Therefore, the equations (2.1.22) become: rdϕ1 dr − 1 2(f + f3)ϕ1 = 0, rdϕ2 dr − 1 2(f − f3)ϕ2 = 0, −1 r df dr + g₁2v2 2 (ϕ 2 1+ ϕ 2 2− 2) = 0, −1 r df3 dr + g2 2v2 2 (ϕ 2 1− ϕ 2 2) = 0, (2.1.24)

to be solved within the following boundary conditions: f(0) = 1, f(∞) = 0, f3(0) = 1, f3(∞) = 0,

ϕ1(0) = 0, ϕ1(∞) = 1,

ϕ2(∞) = 1.

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We immediately note that there is no condition for ϕ2(0). The value of the

eld that does not wind can be nite in the string core. The BPS equations (2.1.24) were solved numerically ([3]), here we show (Figures 2.1 and 2.2) the graphics of the prole functions obtained in [21], observing that the prole function ϕ2 is actually non zero in the string core.

Figure 2.2: Prole functions f3(r)(lower curve) and f(r) (upper curve) for

the (1,0)-string. [21]

2.1.3 A non-Abelian ansatz: orientational moduli

In the ansatz (2.1.23) we note that the SU(2)C+F invariance of the model

is lost. The system is now invariant only under rotations around the third axis of SU(2). To restore the symmetry of the vacuum (2.1.8) we rotate the string ansatz (2.1.23) in the SU(2) space, with the following transformation:

Φ → Φ0 = U ΦU−1_, _with _U _{= e}₂iωa_τa

∈ SU (2). (2.1.26) In the asymptotic limit |x| → ∞ for the (1,0)-string, the introduction of the "moduli matrix" U leads to:

UΦU−1 = U veiα2 (1+τ 3₎ϕ₁(r) 0 0 ϕ2(r) ! U−1 = veiα2 (1+U τ3U −1₎ Uϕ1(r) 0 0 ϕ2(r) U−1 = veiα2 (1+S a_τa₎ Uϕ1(r) 0 0 ϕ2(r) U−1, (2.1.27)

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where Sa _{is a moduli vector dened by:}

Saτa= U τ3_U−1

, SaSa = 1. (2.1.28)

Therefore, the ansatz (2.1.23) for a straight string centred at the origin be-comes: Φ(x) = veiα2(1+S a_τa₎ Uϕ1(r) 0 0 ϕ2(r) U−1, Aa i(x) = −S a_ε ij xj r2[1 − f3(r)], (2.1.29) Ai(x) = −εij xj r2[1 − f (r)], where: Uϕ1(r) 0 0 ϕ2(r) U−1 = ϕ1(r) + ϕ2(r) 2 + S a_τaϕ1(r) − ϕ2(r) 2 , (2.1.30)

and ϕ1, ϕ2, f and f3 are the prole functions solutions of the BPS equations

(2.1.24).

Figure 2.3: The color-magnetic ux is directed along Sa_{. [21]}

The moduli vectors Sa _{actually parametrize the coset SU(2)/U(1) = S}2_{, and}

it can be shown that they give the SU(2)-orientation to the color-magnetic ux Fa

12 (g. 2.3).

This quantity can be dened in a gauge-invariant way with an explicit ~S dependence. Remembering that: ˜ Fa i = 1 2εijkF a jk, (2.1.31)

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we can write: ˜ Fa 3 = 1 v2T r Φ ˜Fb 3 τb 2 Φ † τa = −Sa(ϕ21+ ϕ22) 2 1 r df3 dr. (2.1.32)

In this context is clear that the orientational moduli determine the orientation of the non-Abelian color-magnetic ux, which results aligned with the vector Sa, thus, with a general orientation in its internal space.

The occurrence of this new degree of freedom is what makes this kind of strings eectively non-Abelian.

In next sections we will observe how these moduli elds enter in the eective string description, actually introducing the world-sheet action. This will be done supposing an adiabatic dependence in z and t for the vector ~S and integrating out the two dimensions transverse to the string, leaving an action that depends only on z and t.

For the time being, we conclude reporting another important ansatz, used in the so called singular gauge.

Such a gauge will be useful for the derivation of the world-sheet action cited above, and is dened when there is no winding for the Φ eld at innity. In this way, the ansatz for a straight innite string centred at the origin reduces to: Φ(x) = vUϕ1(r) 0 0 ϕ2(r) U−1, Aa_i(x) = Saεij xj r2f3(r), (2.1.33) Ai(x) = εij xj r2f(r).

This ansatz will often simplify the following calculations and will be used as a prototype extension when we will try to dene dierent topological strings in other coordinates.

In particular we will concentrate our work to strings forming a loop-ring soliton, in which this ansatz for an innite string will be viewed, with some particular assumptions, as an approximate solution.

2.1.4 The string world-sheet

To obtain the world-sheet theory we promote the translational moduli x0

and y0 and the orientational moduli Sa to moduli elds, depending on xp =

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(2.1.33) in d2_x

⊥, with x⊥ = {x, y}.

We immediately note that, because of the {t, z} dependence of Sa_{, the {t, z}}

components of the non-Abelian eld no longer vanish, thus we need to enlarge the ansatz (2.1.33) in the singular gauge to include the term Ap(x), that is:

Ap(x) = −i(∂pU)U−1ρ(r), (2.1.34)

with ρ(r) as a new prole function. It can be shown that:

−i(∂pU)U−1 = − 1 2τ a_ε abcSb∂pSc, (2.1.35) so we obtain: Aa_p = −εabcSb∂pScρ(r). (2.1.36)

With this new term we have to consider also new terms in the original La-grangian. One of this is the component Fa

pi, that is no longer zero. In this

picture it results: F_pia = ∂pSaεij xj r2f3(1 − ρ) + ε abc_Sb_∂ pSc∂iρ(r). (2.1.37)

With the replacement: ∂i = ∂ ∂xi → ∂r ∂xi ∂ ∂r = xi r d dr, (2.1.38)

due to the polar coordinate change in the xy plane and the only radial de-pendence for the prole function, we rewrite the equation (2.1.37) as:

F_pia = ∂pSaεij xj r2f3(1 − ρ) + ε abc_Sb_∂ pSc xi r dρ(r) dr . (2.1.39)

As a consequence of this term, we can already dene the boundary limit for the prole function ρ(r) in r = 0 in order to have nite-energy solution, that is:

ρ(0) = 1. (2.1.40)

We can now perform the integration of the original action in the xy plane. Adding the kinetic term:

(DpΦ)†DpΦ = 1 4(∂pS a₎2h_(ϕ 1− ϕ2)2 1 − ρ 2 2 + ρ 2 4 (ϕ1+ ϕ2) 2i = 1 4(∂pS a₎2h_(ϕ 1− ϕ2)2(1 − ρ) + ρ2 2(ϕ 2 1 + ϕ 2 2) i , (2.1.41)

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the 2D-action that emerges after the integration is: S = Z dzdt T 2(∂px⊥) 2₊ β 2(∂pS a₎2 , (2.1.42)

where the term β is: β = 2π g2 2 Z ∞ 0 rdr ( dρ dr 2 + 1 r2f 2 3(1 − ρ) 2 + g2 ρ 2 2(ϕ 2 1+ ϕ 2 1) + (1 − ρ)(ϕ1− ϕ2)2 ) . (2.1.43)

Minimizing the above integral with respect to ρ we obtain a second order equation: −d 2_ρ dr2 − 1 r dρ dr − 1 r2f 2 3(1 − ρ) + g2 2 2(ϕ 2 1+ ϕ 2 2)ρ − g2 2 2(ϕ1− ϕ2) 2 _{= 0,} _(2.1.44)

in which we have to impose the other boundary condition for ρ at r → ∞. For the above integral to converge we must have:

ρ(∞) = 0. (2.1.45)

The equation (2.1.44) has solution:

ρ= 1 − ϕ1 ϕ2

. (2.1.46)

With this value of ρ we nd that β is actually a constant: β = 2π

g2 2

. (2.1.47)

Analyzing the rst term of the action (2.1.42) we recognize the structure of the Nambu-Goto string action, that takes the form ([40], [41]):

SN G = Z dzdtT 2(∂px⊥) 2 _{= −T} Z dzdt q −det(gpq) = −T Z dzdt (2.1.48) gpq is the induced metric and has determinant equal to one in at space-time.

Furthermore, the second term in the action (2.1.42) represents the O(3)-sigma model, and it can be rewritten using the stereographic projection, in which the coordinates of Sa _{on the S}2_{-sphere are projected back into the}

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Figure 2.4: Stereographic projection of Sa_{. [21]}

Using a complex eld φ = φ1 + iφ2 (g. 2.4) dened with the following

transformations: S1 = 2φ1 1 + φ2 1+ φ22 , S2 = 2φ2 1 + φ2 1+ φ22 , S3 = 1 − φ 2 1− φ22 1 + φ2 1+ φ22 , (2.1.49) the world-sheet string action becomes:

S = Z dzdt −T + 4π g2 2 ∂pφ∂pφ (1 + φφ)2 . (2.1.50)

This action will be the starting point to discuss loop-string solutions, showing that is a good approximation when we suppose a big ring radius compared to the string thickness.

Moreover, the determinant of the induced metric, which is related to the Jacobian of the coordinates transformation, will have particular importance in the following discussion, in which we will often consider curvilinear (but not curved, i.e. still in at space-time) coordinates.

For the time being, we will give some more details about the action (2.1.50), showing that admits a new type of soliton solutions, similar to instantons introduced in Chapter 1: the so called CP(1) lumps.

This kind of solutions will play an important role also in semi-local string models, introduced at the end of this Chapter.

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2.1.5 Instantons in the CP(1) model

An important feature of the CP (1) model is that it also admits instantons. This can be shown considering the action (2.1.50) (without the tension term) in Euclidean space-time.

It can be rewritten in a BPS way: SE = 4π g2 Z d2x[(∂pφ ∓ iεpq∂qφ)(∂pφ ± iεpq∂qφ) ∓ 2iεpq∂pφ∂qφ](1 + φφ)−2, (2.1.51)

where the second line represents the integral over a total derivatives, thus it is a topological charge.

The relevant homotopy group of the theory is:

π1(SU (2)/U (1)) = π1(U (1)) = Z (2.1.52)

The minimal action is achieved, as usual, if terms in the integral in the rst line of (2.1.51) vanish.

This is obtained when:

∂pφ ± iεpq∂qφ = 0, (2.1.53)

under which we have:

SE ≥

4πQE

g2 , (2.1.54)

where QE is the expression of the topological charge, that is:

QE = − i 2π Z d2xεpq∂µφ∂νφ (1 + φφ)2 . (2.1.55)

As in the Yang-Mills instanton case, the plus sign in (2.1.53) is for the in-stanton and the minus is for the anti-inin-stanton.

The ansatz for the QE = 1 instanton is given by:

φ(z) = a

z − b, (2.1.56)

where z is the complex variable z = x1+ ix2, b is a complex parameter too,

and as in the Yang-Mills case, it represents the center of the instanton. The last term a is related to the instanton size, and it can be shown that it has the general following form:

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We interpret the meaning of the α term as follows.

The SU(2) symmetry of the model is spontaneously broken down to U(1) by a particular choice of the vacuum state (the north pole of the target space sphere). While the vacuum is invariant under rotations around the vertical axis in the target space, the instanton solution is not.

In the treatment of semi-local strings we will discuss the principal conse-quences of the introduction of the instanton size |ρ| and the phase angle α, used as fundamental parameters even for the compactied version: the GSY soliton (at the end of Chapter 3).

2.2 Superconducting strings

In the context of string-eld theories it is useful to cite another example in which strings appear as superconducting wires. The rst model of supercon-ducting strings was proposed by E. Witten in 1985 in a pioneer work ([4]) that soon found a wide range of applicability in many physics' branches like astrophysics, with the introduction of the concept of cosmic strings (objects with astronomical dimensions that could be signicant to the dark matter problem), and condensed matter physics ([33],[38]).

2.2.1 The Witten model

The model describes two Abelian gauge bosons (U(1) × U(1)) and two scalar elds. With a particular choice of the potential term we will be able to construct a string-like solution characterized by a charged scalar eld in the string core. The Lagrangian is:

L = ˜Dµφ ˜Dµφ+ DµσDµσ − U(φ, σ) − 1 4F˜µνF˜ µν₋ 1 4FµνF µν_, _(2.2.1) where: ˜ Fµν = ∂µA˜ν− ∂νA˜µ, Fµν = ∂µAν − ∂νAµ, (2.2.2)

are the Abelian elds strengths. The covariant derivatives are dened by: ˜

Dµφ= ∂µφ − ig ˜Aµφ, Dµσ= ∂µσ − ieAµσ, (2.2.3)

with g and e as coupling constants. The potential term is:

U(φ, σ) = λφ 4 (|φ| 2_{− η}2 φ) 2₊λσ 4 (|σ| 2_{− η}2 σ) 2_{+ β|φ|}2_|η|2 . (2.2.4)

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For large values of β, only one of the two U(1) symmetries, that we choose to be the ˜U(1) symmetry of the ˜Aµ gauge eld, is broken in the vacuum.

The choice of the ˜U(1) instead of the other one, that we indicate as U(1)Q,

is ensured by the constraint λφηφ4 > λσησ4.

Therefore, the vacuum of the model is reached when |φ| = ηφ and |σ| = 0

and drives the σ eld to have mass: m2_σ = βη2 φ− 1 2λση 2 σ. (2.2.5)

It may be energetically favourable to have a non zero value of |σ| and a broken U(1)Q in the core of the string. For a string lying in the z axis we can write

the equation of motion for the eld σ in its stable conguration (σ = 0) and look for solution with small perturbation of the type σ(xµ) = |σ(x, y)|eiωt,

thus: (−∂2 x− ∂ 2 y)|σ| + V (r)|σ| = ω 2_|σ|, _(2.2.6) where V (r) = β|φ(r)|2 −1 2λση 2 σ and r = px2+ y2.

Depending on the value of the parameters used in the theory we can have dierent distributions of the condensate σ inside the string, which results charged.

A non-vanishing current along the string is fundamental for the treatment of specic solutions, which dene new topological objects called vortons: soli-tons with a closed-string structure. In the world-sheet low energy theory they are characterized by angular momentum, that stabilizes the object against the string tension.

However, we will extensively talk about vorton solutions in the next chap-ter, where will be discussed in detail dierent general solutions to spinning solitons. For the moment we will limit ourselves to obtain an expression of the string world-sheet of the Witten model and give some basic knowledge about cosmic strings.

2.2.2 The world-sheet eective action

In deriving the world-sheet eective action for a straight string we assume the following ansatz for the eld σ:

σ(t, x, y, z) = eiθ(z,t)_{σ(x, y),} _(2.2.7)

where θ is a slowly-varying function of the same order of the vortex thickness. We make also the same assumptions for Aµ, and we consider for the moment

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for the φ eld an analogous behaviour of the Abelian ANO case, i.e. the Nambu-Goto action.

Therefore, we substitute the ansatz (2.2.7) in (2.2.1) and we integrate over the transverse dimensions x and y, obtaining:

S = Σ Z dtdz(∂pθ+ eAp)2, (2.2.8) where Σ is: Σ = Z dxdy |σ|2, (2.2.9) and p = t, z.

However, this is not the best way to express the eective world-sheet action (due to the squared of the expression). Following Witten ([4]) we can rewrite this action using the current of the theory.

From (2.2.8) we compute the electromagnetic current density jµ_:

jµ = − δS δAµ

= Jµ_{(z, t)δ(x)δ(y).} _(2.2.10)

Dirac deltas impose Jx = Jy = 0.

The two other currents can be expressed as:

Jp = −2Σe(∂pθ+ eAp), (2.2.11)

and they are conserved currents in the world-sheet, thus:

∂pJp = 0. (2.2.12)

This is in practice the Euler-Lagrange equation for θ.

The total current through the cross section in this assumption is directed all along the z direction:

J = Jz. (2.2.13)

The current, moreover, can always be written as a derivative of a new scalar eld, said ϕ, conned to live into the world-sheet:

Jp = qεpq_∂

qϕ, (2.2.14)

where εpq _{is the Levi-Civita tensor in two dimensions and q is a rescaled}

charge:

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Comparing the two expressions for the current (2.2.11) and (2.2.14) we ob-tain:

∂pϕ= −qεpq_(∂

qθ+ eAq), (2.2.16)

which derived again with respect to xp _{gives the equation of motion for ϕ:}

∂p∂pϕ= −qεpq∂pAq. (2.2.17)

In his work, Witten shows that the equation of motion (2.2.17) and the equations of currents can be derived from the following Lagrangian:

S = Z dzdt 1 2(∂pϕ) 2_{− qA} pεpq∂qϕ , (2.2.18)

that is equivalent to (2.2.8), but easier to manage.

The last term represents an interaction term, and, as we already describes, in the full world-sheet action must be added the Nambu-Goto contribution.

2.2.3 Cosmic strings

As mentioned before, strings can emerge in cosmological context as topo-logical defects originated by phase transitions in the early Universe. If our theories about the "Big Bang" and ination are correct, such objects could have been formed during the cooling of the Universe and could play a central role in the explanation of the anisotropy of the cosmic microwave background, formation of galaxies and dark matter ([17]). We will briey introduce the cosmological scenario in which cosmic strings are considered and show the mechanism through which the formation takes place, that is often called the Kibble mechanism.

Cosmological scenario Big Bang and ination

Our knowledge about the early Universe is based on two observational facts: the red-shift of galaxies, suggesting that the Universe is expanding, and the cosmic microwave background, that is compatible with the Big Bang theory ([28], [20]). In 1926 Hubble observed the red-shift of far galaxies, from which he elaborated the law that goes under his name and relates the recession velocity of the galaxies to their distances in a proportional way: