## Tesi etd-06192009-191334 |

Tipo di tesi

Tesi di dottorato di ricerca

Autore

VINCI, WALTER

URN

etd-06192009-191334

Titolo

Classical Moduli Spaces of Non-Abelian Vortices

Settore scientifico disciplinare

FIS/02

Corso di studi

FISICA

Commissione

**Relatore**Prof. Konishi, Kenichi

Parole chiave

- soliton
- non-perturbative effect
- dualities
- confinement
- supersymmetric gauge theories
- Vortex

Data inizio appello

2009-06-22

Disponibilità

parziale

Data di rilascio

2049-06-22

Riassunto analitico

At the beginning of the mid seventies, mainly thanks to the discovery of asymptotic freedom, people started to believe that

QCD could be a good model to describe strong interactions. After that, an impressive amount of experimental data was

collected, which confirmed the predictions of QCD. To recognize QCD as a correct model was a breakthrough in the

study of strong interactions, but this discovery forced physicist to face a difficult problem: the

understanding of strongly coupled gauge theories. In fact, no one knows exactly how to use QCD to explain the most elementary

experimental observation for strong interactions: color confinement.

While we can always rely on perturbation theory to extract predictions from a weakly coupled gauge theory, there are no

general techniques to obtain quantitative results for the physics of strongly coupled theories. Nevertheless, physicists identified many non-perturbative

mechanisms, and their effects, which generically occur in gauge theories. Maybe one of the first and most known example

is the resolution of the so-called $U(1)$ problem. 't Hooft showed how instantons contribute to the exceedingly large mass of the $\eta, \eta'$ particles. \cite{'tHooft:1976fv}. Instantons are a particular type of

solitons, extended classical configurations which can give relevant contributions to the functional integral. It is today widely

believed that solitons play a crucial role in the dynamics of non-Abelian gauge theories. This is true

for the microscopic dynamics as well as for the evolution of the universe (cosmic defects).

Another remarkable example of the importance of solitons in the non-perturbative dynamics of gauge theories is the 't Hooft- Mandelstam mechanism of confinement: dual superconductivity \cite{'tHooft:1981ht}. The QCD vacuum is a kind of dual superconductor, in which electric and magnetic charges are exchanged. If magnetic monopoles condense in the vacuum, the electric sources are

confined by electric flux tubes, just like in an ordinary superconductor magnetic charges would be confined by

Abrikosov-Nielsen-Olesen vortices (Meissner effect). This idea is supported by experimental observations like

Regge trajectories and lattice simulations which show a linear quark-antiquark potential.

This mechanism is very simple, but its actual occurrence has still to be demonstrated: it involves solitonic

objects, like magnetic monopoles and vortices, which appear in a phase of the gauge theory which is usually strongly

coupled. For this reason, this picture of confinement remained just a qualitative sketch

for many years.

In the nineties, Seiberg and Witten reached a striking goal: they found the exact low-energy effective action of

supersymmetric gauge theories with extended supersymmetry (${\cal N} =2$) \cite{Seiberg:1994rs,Seiberg:1994aj}. To

reach this result they made use of all known properties of supersymmetric gauge theories ({\it i.e.} non-renormalization

theorems, holomorphicity of superpotentials, appearance of moduli space of vacua), as well as general properties of gauge

theories ({\it i.e.} perturbative and instantonic effects). In their study the electromagnetic duality plays a crucial role.

Using their result it is possible to rigourously demonstrate the presence of a mechanism of dual superconductivity in supersymmetric gauge theories. The appearance of this mechanism in close supersymmetric relatives of QCD

strengthened the idea that dual superconductivity could be the real mechanism of color confinement chosen by Nature. Even

though the first models studied with these techniques were very interesting, they had some crucial qualitative

differences with respect to QCD. One of the most important is the excessive richness of the hadronic spectrum. In fact,

the generic quantum vacuum of an ${\cal N} =2$ $SU(N)$ gauge, undergoes dynamical Abelianization. This means

that the gauge group is dynamically broken to $U(1)^{N-1}$. Thus, an infinite tower of mesons, for each of the $N-1$

abelian strings, exist. This is clearly not the case of QCD.

We can significantly improve this picture considering ${\cal N} =2$ theories with fundamental matter fields. In these

theories, for some values of the bare masses of the matter fields, there exist the so-called $r$-vacua, where the low

energy physics is described by a dual non-Abelian $SU(r)$ gauge theory \cite{Carlino:2000uk}. In these vacua there appear massless degrees of

freedom which can be identified as quantum monopoles. When we add a mass perturbation, these monopoles condense,

providing a dual mechanism of non-Abelian Meissner effect. It is possible to study this mechanism in a semi-classical

regime \cite{Auzzi:2003fs}. If we have a sufficient number of flavors, and we take the bare masses of the squarks to be large enough (much

bigger than the strong coupling scale of the theory), we can get a system with the following hierarchical symmetry

breaking pattern:

\[

SU(N+1) \,\,\,{\stackrel {v_{1}} {\longrightarrow}} \,\,\, SU(N)\times U(1) \,\,\,{\stackrel {v_{2}} {\longrightarrow}} \,\,\,

\emptyset.

\]

The theory is weakly coupled at all scales, and the hierarchy is huge: $v_1 \gg v_2$. Topological arguments show that

at the scale $v_1$ the theory supports monopoles while at lower scales, $v_2$, it supports vortices. At the same time,

if one considers the theory as a whole, from low to high energies, one can see that there are neither truly stable monopoles, nor vortices. This

argument shows that a vortex can exist only if it ends on a monopole-antimonopole pair, thus monopoles are confined. It

is important to stress that this is a non-Abelian generalization of the Meissner effect. Both monopoles and vortices

are truly non-Abelian objects. It is also important to observe that, thanks to supersymmetry, this model is fully

reliable also at the quantum level. Notice that we now have only one $U(1)$ factor that provides the existence of

vortices at low energies. This leads to a significant reduction of the meson spectrum in comparison to models in which

dynamical Abelianization takes place.

Eventually, we want to study the model when strong coupling effects take place. We can reach this regime by taking very small bare masses: monopoles become massless and condense. The Meissner effect should now take place in the dual theory and the original

electric degrees of freedom are confined. Along the path to the strong coupling regime, we can recall the famous

holomorphic dependence of physical quantities with respect to the couplings that is a fundamental property of

supersymmetric gauge theories. We may thus learn a lot about the strong coupling regime, and about a mechanism of

confinement which can be relevant for QCD, studying the semiclassical regime. Furthermore, one finds that non-Abelian

monopoles and vortices are deeply connected, so that we can learn much about one object by studying the other, even at the

semiclassical level. This observation is important if we recall that non-Abelian monopoles are still regarded by many as mysterious objects. They posses non-normalizable zero modes which make the study of their quantum properties subtle.

According to the well-known GNO conjecture, non-Abelian monopoles should form, at the quantum level, multiplets of

a {\it dual} gauge group. Because of the aforementioned difficulties, it is not easy to check this conjecture explicitly.

Non-Abelian vortices attracted much attention also thanks to another interesting property: they provide a map between theories in two and

four dimensions. It has been well-known for many years that two dimensional sigma models and four dimensional gauge theories

share many features, including their non-perturbative ones. The progresses in the study of supersymmetric theories made

it possible to prove strong quantitative relations: the BPS spectrum of the mass deformed two-dimensional ${\cal N}=

(2,2)$ $\textbf{CP}^{N-1}$ sigma-model coincides with the BPS spectrum of four-dimensional ${\cal N}=2$ $SU(N)$

supersymmetric QCD. This relation is not a coincidence: the two-dimensional sigma model emerges exactly as the effective theory which lives on the vortex worldsheet which appears in the higgs phase of the four-dimensional gauge theory. In some

sense, the vortex theory must ``reproduce'' the physics of the bulk theory. Many close relations have been found also for

theories with less supersymmetry.

Is thus hard to underestimate the importance of non-Abelian vortices in the strongly coupled dynamics of supersymmetric

gauge theories. We believe that some of the non-perturbative mechanism in which non-Abelian vortices play crucial role

can be of relevance also for non-supersymmetric theories like QCD.

This Ph.D. thesis is devoted to the description of our latest efforts to improve the present knowledge about non-Abelian vortices. We shall mainly be focusing on the study of supersymmetric solitons. Supersymmetry is a sufficient property to make a soliton ``BPS'' (Bogomoln'yi-Prasad-Sommerfeld saturated). This property drastically simplifies the study of these objects. First of all, the equations of motion reduce to first-order partial differential equations, instead of being of the second order. Secondly, the tension of a BPS soliton is completely given by boundary terms which depends only on the topology of the configurations, thus becoming proportional to an integer which identifies a topological winding. This property has an important consequence: the energy of composite configurations of solitons does not depend on their relative separations; there are no net static forces between them. This also implies the existence of a continuous set of configurations, degenerate in energy, which all together form the so-called ``moduli space`` of solutions. The knowledge of the moduli space and its properties is crucial to understand the low-energy physics of the vortex. Generically, in fact, this physics will be described by a two-dimensional non-linear sigma model, whose target space is given by the moduli space. This sigma model will describe the relevant classical and quantum physics of the vortex. Usually one determines only the bosonic moduli and degrees of freedom. Supersymmetry is sufficient for allowing us to recover the fermionic sector, if needed.

We will use the so-called ``moduli-matrix formalism'', and

the equivalent ``Kahler-quotient construction'' to systematically explore the bosonic zero modes and the corresponding

bosonic sector of a wide range of vortex configurations. A $U(N_{c})$ gauge

theory with $N_{f} \geq N_c$ flavour in the fundamental representation will be extensively studied as a theoretical laboratory. We shall choose its parameters so that it could be

embedded into ${\cal N}=2$, SQCD, and it will admit a degenerate set of vortex solutions.

The thesis is organized as follows.

In Part I, the model will be introduced and the basic construction of non-Abelian vortices will be discussed. We will review some of the most important efforts to use non-Abelian vortices for a better understanding of the 4-d dynamics of non-Abelian gauge theories.

In Part II, I will review our contribution to the study of the moduli space of non-Abelian vortices. In Chapter 4 configurations of composite vortices \cite{Eto:2006cx} will be studied. In Chapter 5 we will study the effects on moduli space structure when an arbitrary number of flavors is included \cite{Eto:2007yv}. In these cases there appear the so-called ``semi-local'' vortices. The connection with lump solutions is discussed. This knowledge will be used in Chapter 6 to study the interactions of vortices, when some non-BPS corrections are added to our model \cite{Auzzi:2007iv}, \cite{Auzzi:2007wj}. The same non-BPS corrections are also introduced in models with several flavors in Chapter 7, to discuss the stability of semi-local vortices \cite{Auzzi:2008wm}. A dual model of confinement for $SO(N)$ gauge theories is discussed in Chapter 8 \cite{Eto:2006dx}, while we study the problem of the reconnection of cosmic strings in Chapter 9 \cite{Eto:2006db}.

In Part III, we will make use of a more general derivation of the vortex moduli spaces, which allow us to construct non-Abelian vortices in a set of non-Abelian gauge theories with a generic gauge group. In Chapter 10 we derive this generalized construction, using the deep connection between vortices and lumps in related sigma models \cite{Eto:2008yi}, \cite{Vinci:2008hd}. In Chapter 11 this construction is used for a detailed study of vortices in non-Abelian gauge theories with orthogonal ($SO$) and symplectic ($USp$) gauge groups \cite{Eto:2009bg}. These new solutions provide semiclassical hints for the understanding of GNO duality from the vortex side, as will be discussed in Chapter 12.

% In the third chapter we will describe our progress in the study of semilocal vortices. The term ``semilocal''

% was invented for string-like objects in abelian Higgs models with more than one Higgs

%field \cite{Vachaspati:1991dz}, where a global (flavor) symmetry group is present in addition to the local (gauge)

%symmetry group. In our non Abelian context, semilocal will refer to the case $\NF

%>\NC$. The study of models with a high number of flavours is interesting per se. Here we stress that the existence of the

% r-vacua with non Abelian gauge symmetry usually require $\NF > 2 r$, thus the existence of semilocal vortices.

%Semilocal abelian vortices are known to exhibit peculiar properties, which are very different from those of the usual

%Abrikosov-Nielsen-Olesen (ANO) vortices \cite{Abrikosov:1956sx,Nielsen:1973cs}.

In the last chapter we conclude and discuss possible future developments of this work.

QCD could be a good model to describe strong interactions. After that, an impressive amount of experimental data was

collected, which confirmed the predictions of QCD. To recognize QCD as a correct model was a breakthrough in the

study of strong interactions, but this discovery forced physicist to face a difficult problem: the

understanding of strongly coupled gauge theories. In fact, no one knows exactly how to use QCD to explain the most elementary

experimental observation for strong interactions: color confinement.

While we can always rely on perturbation theory to extract predictions from a weakly coupled gauge theory, there are no

general techniques to obtain quantitative results for the physics of strongly coupled theories. Nevertheless, physicists identified many non-perturbative

mechanisms, and their effects, which generically occur in gauge theories. Maybe one of the first and most known example

is the resolution of the so-called $U(1)$ problem. 't Hooft showed how instantons contribute to the exceedingly large mass of the $\eta, \eta'$ particles. \cite{'tHooft:1976fv}. Instantons are a particular type of

solitons, extended classical configurations which can give relevant contributions to the functional integral. It is today widely

believed that solitons play a crucial role in the dynamics of non-Abelian gauge theories. This is true

for the microscopic dynamics as well as for the evolution of the universe (cosmic defects).

Another remarkable example of the importance of solitons in the non-perturbative dynamics of gauge theories is the 't Hooft- Mandelstam mechanism of confinement: dual superconductivity \cite{'tHooft:1981ht}. The QCD vacuum is a kind of dual superconductor, in which electric and magnetic charges are exchanged. If magnetic monopoles condense in the vacuum, the electric sources are

confined by electric flux tubes, just like in an ordinary superconductor magnetic charges would be confined by

Abrikosov-Nielsen-Olesen vortices (Meissner effect). This idea is supported by experimental observations like

Regge trajectories and lattice simulations which show a linear quark-antiquark potential.

This mechanism is very simple, but its actual occurrence has still to be demonstrated: it involves solitonic

objects, like magnetic monopoles and vortices, which appear in a phase of the gauge theory which is usually strongly

coupled. For this reason, this picture of confinement remained just a qualitative sketch

for many years.

In the nineties, Seiberg and Witten reached a striking goal: they found the exact low-energy effective action of

supersymmetric gauge theories with extended supersymmetry (${\cal N} =2$) \cite{Seiberg:1994rs,Seiberg:1994aj}. To

reach this result they made use of all known properties of supersymmetric gauge theories ({\it i.e.} non-renormalization

theorems, holomorphicity of superpotentials, appearance of moduli space of vacua), as well as general properties of gauge

theories ({\it i.e.} perturbative and instantonic effects). In their study the electromagnetic duality plays a crucial role.

Using their result it is possible to rigourously demonstrate the presence of a mechanism of dual superconductivity in supersymmetric gauge theories. The appearance of this mechanism in close supersymmetric relatives of QCD

strengthened the idea that dual superconductivity could be the real mechanism of color confinement chosen by Nature. Even

though the first models studied with these techniques were very interesting, they had some crucial qualitative

differences with respect to QCD. One of the most important is the excessive richness of the hadronic spectrum. In fact,

the generic quantum vacuum of an ${\cal N} =2$ $SU(N)$ gauge, undergoes dynamical Abelianization. This means

that the gauge group is dynamically broken to $U(1)^{N-1}$. Thus, an infinite tower of mesons, for each of the $N-1$

abelian strings, exist. This is clearly not the case of QCD.

We can significantly improve this picture considering ${\cal N} =2$ theories with fundamental matter fields. In these

theories, for some values of the bare masses of the matter fields, there exist the so-called $r$-vacua, where the low

energy physics is described by a dual non-Abelian $SU(r)$ gauge theory \cite{Carlino:2000uk}. In these vacua there appear massless degrees of

freedom which can be identified as quantum monopoles. When we add a mass perturbation, these monopoles condense,

providing a dual mechanism of non-Abelian Meissner effect. It is possible to study this mechanism in a semi-classical

regime \cite{Auzzi:2003fs}. If we have a sufficient number of flavors, and we take the bare masses of the squarks to be large enough (much

bigger than the strong coupling scale of the theory), we can get a system with the following hierarchical symmetry

breaking pattern:

\[

SU(N+1) \,\,\,{\stackrel {v_{1}} {\longrightarrow}} \,\,\, SU(N)\times U(1) \,\,\,{\stackrel {v_{2}} {\longrightarrow}} \,\,\,

\emptyset.

\]

The theory is weakly coupled at all scales, and the hierarchy is huge: $v_1 \gg v_2$. Topological arguments show that

at the scale $v_1$ the theory supports monopoles while at lower scales, $v_2$, it supports vortices. At the same time,

if one considers the theory as a whole, from low to high energies, one can see that there are neither truly stable monopoles, nor vortices. This

argument shows that a vortex can exist only if it ends on a monopole-antimonopole pair, thus monopoles are confined. It

is important to stress that this is a non-Abelian generalization of the Meissner effect. Both monopoles and vortices

are truly non-Abelian objects. It is also important to observe that, thanks to supersymmetry, this model is fully

reliable also at the quantum level. Notice that we now have only one $U(1)$ factor that provides the existence of

vortices at low energies. This leads to a significant reduction of the meson spectrum in comparison to models in which

dynamical Abelianization takes place.

Eventually, we want to study the model when strong coupling effects take place. We can reach this regime by taking very small bare masses: monopoles become massless and condense. The Meissner effect should now take place in the dual theory and the original

electric degrees of freedom are confined. Along the path to the strong coupling regime, we can recall the famous

holomorphic dependence of physical quantities with respect to the couplings that is a fundamental property of

supersymmetric gauge theories. We may thus learn a lot about the strong coupling regime, and about a mechanism of

confinement which can be relevant for QCD, studying the semiclassical regime. Furthermore, one finds that non-Abelian

monopoles and vortices are deeply connected, so that we can learn much about one object by studying the other, even at the

semiclassical level. This observation is important if we recall that non-Abelian monopoles are still regarded by many as mysterious objects. They posses non-normalizable zero modes which make the study of their quantum properties subtle.

According to the well-known GNO conjecture, non-Abelian monopoles should form, at the quantum level, multiplets of

a {\it dual} gauge group. Because of the aforementioned difficulties, it is not easy to check this conjecture explicitly.

Non-Abelian vortices attracted much attention also thanks to another interesting property: they provide a map between theories in two and

four dimensions. It has been well-known for many years that two dimensional sigma models and four dimensional gauge theories

share many features, including their non-perturbative ones. The progresses in the study of supersymmetric theories made

it possible to prove strong quantitative relations: the BPS spectrum of the mass deformed two-dimensional ${\cal N}=

(2,2)$ $\textbf{CP}^{N-1}$ sigma-model coincides with the BPS spectrum of four-dimensional ${\cal N}=2$ $SU(N)$

supersymmetric QCD. This relation is not a coincidence: the two-dimensional sigma model emerges exactly as the effective theory which lives on the vortex worldsheet which appears in the higgs phase of the four-dimensional gauge theory. In some

sense, the vortex theory must ``reproduce'' the physics of the bulk theory. Many close relations have been found also for

theories with less supersymmetry.

Is thus hard to underestimate the importance of non-Abelian vortices in the strongly coupled dynamics of supersymmetric

gauge theories. We believe that some of the non-perturbative mechanism in which non-Abelian vortices play crucial role

can be of relevance also for non-supersymmetric theories like QCD.

This Ph.D. thesis is devoted to the description of our latest efforts to improve the present knowledge about non-Abelian vortices. We shall mainly be focusing on the study of supersymmetric solitons. Supersymmetry is a sufficient property to make a soliton ``BPS'' (Bogomoln'yi-Prasad-Sommerfeld saturated). This property drastically simplifies the study of these objects. First of all, the equations of motion reduce to first-order partial differential equations, instead of being of the second order. Secondly, the tension of a BPS soliton is completely given by boundary terms which depends only on the topology of the configurations, thus becoming proportional to an integer which identifies a topological winding. This property has an important consequence: the energy of composite configurations of solitons does not depend on their relative separations; there are no net static forces between them. This also implies the existence of a continuous set of configurations, degenerate in energy, which all together form the so-called ``moduli space`` of solutions. The knowledge of the moduli space and its properties is crucial to understand the low-energy physics of the vortex. Generically, in fact, this physics will be described by a two-dimensional non-linear sigma model, whose target space is given by the moduli space. This sigma model will describe the relevant classical and quantum physics of the vortex. Usually one determines only the bosonic moduli and degrees of freedom. Supersymmetry is sufficient for allowing us to recover the fermionic sector, if needed.

We will use the so-called ``moduli-matrix formalism'', and

the equivalent ``Kahler-quotient construction'' to systematically explore the bosonic zero modes and the corresponding

bosonic sector of a wide range of vortex configurations. A $U(N_{c})$ gauge

theory with $N_{f} \geq N_c$ flavour in the fundamental representation will be extensively studied as a theoretical laboratory. We shall choose its parameters so that it could be

embedded into ${\cal N}=2$, SQCD, and it will admit a degenerate set of vortex solutions.

The thesis is organized as follows.

In Part I, the model will be introduced and the basic construction of non-Abelian vortices will be discussed. We will review some of the most important efforts to use non-Abelian vortices for a better understanding of the 4-d dynamics of non-Abelian gauge theories.

In Part II, I will review our contribution to the study of the moduli space of non-Abelian vortices. In Chapter 4 configurations of composite vortices \cite{Eto:2006cx} will be studied. In Chapter 5 we will study the effects on moduli space structure when an arbitrary number of flavors is included \cite{Eto:2007yv}. In these cases there appear the so-called ``semi-local'' vortices. The connection with lump solutions is discussed. This knowledge will be used in Chapter 6 to study the interactions of vortices, when some non-BPS corrections are added to our model \cite{Auzzi:2007iv}, \cite{Auzzi:2007wj}. The same non-BPS corrections are also introduced in models with several flavors in Chapter 7, to discuss the stability of semi-local vortices \cite{Auzzi:2008wm}. A dual model of confinement for $SO(N)$ gauge theories is discussed in Chapter 8 \cite{Eto:2006dx}, while we study the problem of the reconnection of cosmic strings in Chapter 9 \cite{Eto:2006db}.

In Part III, we will make use of a more general derivation of the vortex moduli spaces, which allow us to construct non-Abelian vortices in a set of non-Abelian gauge theories with a generic gauge group. In Chapter 10 we derive this generalized construction, using the deep connection between vortices and lumps in related sigma models \cite{Eto:2008yi}, \cite{Vinci:2008hd}. In Chapter 11 this construction is used for a detailed study of vortices in non-Abelian gauge theories with orthogonal ($SO$) and symplectic ($USp$) gauge groups \cite{Eto:2009bg}. These new solutions provide semiclassical hints for the understanding of GNO duality from the vortex side, as will be discussed in Chapter 12.

% In the third chapter we will describe our progress in the study of semilocal vortices. The term ``semilocal''

% was invented for string-like objects in abelian Higgs models with more than one Higgs

%field \cite{Vachaspati:1991dz}, where a global (flavor) symmetry group is present in addition to the local (gauge)

%symmetry group. In our non Abelian context, semilocal will refer to the case $\NF

%>\NC$. The study of models with a high number of flavours is interesting per se. Here we stress that the existence of the

% r-vacua with non Abelian gauge symmetry usually require $\NF > 2 r$, thus the existence of semilocal vortices.

%Semilocal abelian vortices are known to exhibit peculiar properties, which are very different from those of the usual

%Abrikosov-Nielsen-Olesen (ANO) vortices \cite{Abrikosov:1956sx,Nielsen:1973cs}.

In the last chapter we conclude and discuss possible future developments of this work.

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