Universita degli Studi di Pisa

Facolta di Scienze Matematiche Fisiche e Naturali

Doctorate of Philosophy in

Physics

Classical Moduli Spacesof

Non-Abelian Vortices

Ph.D Thesis

Walter Vinci

Supervisor:

Prof. K. Konishi

Academic years 2006-2008

Contents

Introduction 2

I A Brief Introduction 7

1 Construction and Properties 8

1.1 Basic set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 The vortex solution . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Effective world-sheet theory . . . . . . . . . . . . . . . . . . . 14

1.4 Non-Abelian vortices and 4-D gauge dynamics . . . . . . . . . 16

2 Confinement and Dualities in 4-d Gauge Theories 20

2.1 SU(N + 1) model with hierarchical symmetry breaking . . . . 24

2.2 Vortex-monopole complex . . . . . . . . . . . . . . . . . . . . 26

2.3 Transformation properties of monopoles . . . . . . . . . . . . 30

2.4 Non-Abelian duality requires an exact flavor symmetry . . . . 32

3 The Moduli Space of U(N) Non-Abelian Vortices 33

3.1 The moduli matrix formalism . . . . . . . . . . . . . . . . . . 33

3.2 Kahler quotient construction . . . . . . . . . . . . . . . . . . . 37

3.3 Fundamental vortex revised . . . . . . . . . . . . . . . . . . . 39

II Vortices in U(N) Gauge Theories 42

4 Non-Abelian vortices of higher winding numbers 43

4.1 k = 2 Vortices in U(2) Gauge Theory . . . . . . . . . . . . . . 43

4.1.1 SU(2) transformation law of co-axial k = 2 vortices . . 48

4.2 k = 2 Vortices in U(N) Gauge Theory . . . . . . . . . . . . . 51

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

ii

5 Semi-local strings and lumps 575.1 Moduli spaces and quotients . . . . . . . . . . . . . . . . . . . 585.2 The lump limit . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 645.4 Non-normalizable modes . . . . . . . . . . . . . . . . . . . . . 685.5 Dynamics of the effective k = 1 vortex theory . . . . . . . . . 725.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Interactions of Non-BPS Non-Abelian Vortices 796.1 Type I non-Abelian supersymmetric superconductors . . . . . 80

6.1.1 Spectrum of the Theory . . . . . . . . . . . . . . . . . 816.1.2 (p, k) Coincident Vortices . . . . . . . . . . . . . . . . 836.1.3 Generic Coincident Vortices . . . . . . . . . . . . . . . 876.1.4 Vortex Interactions at Large Distance . . . . . . . . . . 936.1.5 Effective worldsheet theory . . . . . . . . . . . . . . . . 976.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 Static interactions . . . . . . . . . . . . . . . . . . . . . . . . . 1026.2.1 Theoretical set-up . . . . . . . . . . . . . . . . . . . . . 1036.2.2 Non-Abelian vortices in the fine-tuned model . . . . . . 1046.2.3 Vortex interaction in the fine-tuned model . . . . . . . 1086.2.4 Vortices with generic couplings . . . . . . . . . . . . . 1176.2.5 Interaction at large vortex separation . . . . . . . . . . 1236.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 129

7 Stability of non-Abelian semi-local vortices 1317.1 Abelian Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.2 Non-Abelian Vortices . . . . . . . . . . . . . . . . . . . . . . . 133

7.2.1 Fluctuation Analysis . . . . . . . . . . . . . . . . . . . 1357.3 Meta-stability versus Absolute Stability . . . . . . . . . . . . . 1407.4 Low-energy Effective Theory . . . . . . . . . . . . . . . . . . . 1427.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8 A Dual Model of Confinement for SO(N) Gauge Theories 1448.1 Maximal SU factor; SO(5)→ U(2)→ 1 . . . . . . . . . . . . 1458.2 SO(2N + 1)→ SU(r)× U(1)N−r−1 → 1 (r < N) . . . . . . . 1498.3 Other symmetry breaking patterns and GNOW duality . . . . 1528.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9 Universal Reconnection of non-Abelian Cosmic Strings 1559.1 Reconnection of non-Abelian local strings . . . . . . . . . . . . 1569.2 Reconnection of semi-local strings . . . . . . . . . . . . . . . . 159

iii

9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

III Vortices in Arbitrary Gauge Theories 162

10 Vortices and Lump 16310.1 Model and BPS Vortex Equations . . . . . . . . . . . . . . . . 16310.2 The Strong Coupling Limit . . . . . . . . . . . . . . . . . . . . 16510.3 Vortex-Lump Correspondence . . . . . . . . . . . . . . . . . . 16610.4 A Generalized Rational Map Construction for Lumps . . . . . 16710.5 Moduli Space of Vortices . . . . . . . . . . . . . . . . . . . . . 16910.6 Local (ANO-like) Vortices . . . . . . . . . . . . . . . . . . . . 171

11 Orthogonal and Symplectic Vortices 173

12 Semi-classical hints of GNOW Duality from vortex side 174

Conclusions and Outlook 175

Acknowledgements 179

Bibliography 181

1

Introduction

At the begin of the mid seventies, mainly thanks to the discovery of asymp-totic freedom, people started to believe that QCD could be a good model todescribe strong interactions. After that, a striking collection of experimentaldata was collected, which confirmed the predictions of QCD. To recognizeQCD as a correct model was a breakthrough in the understanding of stronginteractions, but this discovery forced physicist to face an difficult problem:the study of strongly coupled gauge theories. In fact, no one knows howto use QCD to explain the most elementary experimental observations forstrong interactions: color confinement.

While we can always rely on perturbation theory to extract predictionsfrom a weakly coupled gauge theory, there are no general techniques to ex-tract exact 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 mostknown example is the resolution of the so called U(1) problem. ’t Hooftsolved the problem by taking into account the instantonic contributions tothe QCD functional integral [1]. Instantons are particular type of solitons,extended classical configurations which can give relevant contributions to thepath integral. It is thus widely believed that solitons play crucial role in thedynamics of non-Abelian gauge theories. This is true for the microscopicdynamics as well as for the evolution of the universe (cosmic defect).

Another, striking, example of the importance of solitons in the non-perturbative dynamics of gauge theories is the ’t Hooft and Mandelstammechanism of confinement called dual superconductivity [2]: the QCD vac-uum is a kind of dual superconductor, in which the role of electric and mag-netic charges is exchanged. If magnetic monopoles condense in the vacuum,the electric sources are confined by tubes of electric flux, just like in anordinary superconductor magnetic charges would be confined by Abrikosov-Nielsen-Olesen vortices (Meissner effect). This idea is strongly supportedby experimental observations like Regge trajectories and lattice simulationswhich show a linear quark-antiquark potential.

2

This mechanism is very simple, but it is prohibitive to demonstrate itseffective occurrence: it involves solitonic objects, like magnetic monopolesand vortices, which appear in a phase of the gauge theory which is usu-ally strongly coupled. Far these reasons this picture of confinement in nonAbelian gauge theories remained just a qualitative sketch for many years.

In the nineties, Seiberg and Witten reached a striking goal: they couldfind the exact low energy effective action of supersymmetric gauge theo-ries with extended supersymmetry (N = 2) [3, 4]. To reach their resultthey collected together properties of supersymmetric gauge theories like nonrenormalization theorems, holomorphicity of superpotentials, appearance ofmoduli space of vacua, with general properties of gauge theories like perturba-tive and instantonic effects. In their study the electro-magnetic duality playsa crucial role. Using their result it is possible to rigourously demonstrate theactual occurrence of dual superconductivity in a large class of supersymmetricgauge theories. The appearance of this mechanism in close supersymmetricrelatives of QCD strengthened the idea that dual superconductivity could bethe actual mechanism of color confinement chosen by Nature. Even thoughthe first models studied with these techniques were very interesting, theyhad some crucial qualitative differences with respect to QCD. One of themost important is the excessive richness of the hadronic spectrum. In fact,the generic quantum vacuum of an N = 2 SU(N) gauge, always undergoesdynamical abelianization. This means that the gauge group is dynamicallybroken to U(1)N−1. Thus, an infinite tower of mesons, for each of the N − 1abelian strings, exists. This is clearly not the case of QCD.

We can significantly improve this picture considering N = 2 theorieswith fundamental matter fields. In these theories, for some values of thebare masses of the matter fields, there exist the so called r-vacua, where thelow energy physics is described by a dual non Abelian SU(r) gauge theory.In this vacua ther appear massless degrees of freedom which can be identifiedas quantum monopoles. When we add a mass perturbation, these monopolecondense, providing a dual mechanism of non-Abelian Meissner effect. It ispossible to study this mechanism in a semi-classical regime. If we have asufficient number of flavor, and we take the bare mass of the squarks to behigh enough (much bigger than the strong coupling scale of the theory), wecan get a system with the following hierarchical symmetry braking pattern:

SU(N + 1)v1−→ SU(N)× U(1)

v2−→ ∅.

The theory is weakly coupled at all scales, and the hierarchy is huge: v1 v2. Topological arguments show that at the scale v1 the theory supportsmonopoles while at lower scales, v2, it supports vortices. At the same time,

3

if one consider the theory at every scale, one can see that there are no trulystable monopoles, neither vortices. This argument show that a vortex canexist only if it ends on a monopole-antimonopole pair, thus monopoles areconfined. It is important to stress that this is a non-Abelian generalizationof the Meissner effect. Both monopoles and vortices are truly non-Abelianobjects. It is also important to observe that, thank to supersymmetry, thismodel is fully reliable also at the quantum level. Notice that we now haveonly one U(1) factor that provides the existence of vortices at low energies.This leads to a significant reduction of the meson spectrum with respect tomodels in which dynamical abelianization take place.

Eventually, we want to study the model when strong coupling effects takeplace, taking the bare masses very small: monopoles becomes massless andcondense. The Meissner effect now should take place in the dual theory andthe original electric degrees of freedom are confined. Along the path to thestrong coupling regime, we can recall the famous holomorphic dependence ofphysical quantities with respect to the couplings that is a fundamental prop-erty of supersymmetric gauge theories. We can thus learn a lot about thestrong coupling regime, and about a mechanism of confinement which can berelevant for QCD, studying the semiclassical one. Furthermore, one finds thatnon-Abelian monopoles and vortices are deeply connected, so that we canlearn much of one object studying the other, even at the semiclassical level.This observation is important if we consider that non-Abelian monopoles arestill mysterious object. They usually possess non-normalizable modes whichmake the study of their quantum properties subtle. According to the wellknown GNOW conjecture, non-Abelian monopoles should form, at the quan-tum level, multiplets of a dual gauge group. Because of the aforementioneddifficulties, it is difficult to check explicitly this conjecture.

Non Abelian vortices attracted much attention also for another interest-ing property: they provide a map between theories in two and four dimen-sions. It has been well known for many years that two dimensional sigmamodels and four dimensional gauge theories share many features, includedthe non-perturbative ones. The progresses in the study of supersymmetrictheories made possible to prove indeed strong quantitative relations: theBPS spectrum of the mass deformed two-dimensional N = (2, 2) CPN−1

sigma-model coincides with the BPS spectrum of four-dimensional N = 2SU(N) supersymmetric QCD. This relation is not a coincidence: the two-dimensional sigma model is exactly the effective theory which lives on thevortex worldsheet which appears in the higgs phase of the four-dimensionalgauge theory. In some sense, the vortex theory must “reproduce” the physicsof the bulk theory. Many close relations have been found also for theorieswith less supersymmetry.

4

Is thus hard to underestimate the importance of non-Abelian vortices inthe strongly coupled dynamics of supersymmetric gauge theories. We believethat some of the non-perturbative mechanism in which non-Abelian vorticesplay crucial role can be of relevance also for non-supersymmetric theories likeQCD.

This Ph.D. Thesis is devoted to describe our efforts to improve the presentknowledge about non-Abelian vortices. We will be mainly focused on thestudy of supersymmetric solitons. Supersymmetry is a sufficient propertyto make a soliton “BPS” (Bogomoln’yi-Prasad-Sommerfeld). This propertydrastically simplify the study of these objects. First of all, the equations ofmotion reduce to first order partial differential equations, instead of being ofthe second order. Secondly, the tension of a BPS soliton is completely givenby boundary terms which depends only by the topology of the configura-tions, becoming thus proportional to an integer which identify a topologicalwinding. This property has an important consequence: the energy of com-posite configurations of solitons does not depend on their relative separations,thus there are no net static forces between them. This also implies the ex-istence of a continuous set of configurations, degenerate in energy, which alltogether form the so-called “moduli space“ of solutions. The knowledge ofthe moduli space and its properties is crucial to understand the low-energyphysics of the vortex. Generically, in fact, this physics will be described by atwo dimensional non linear sigma model, whose target space is given by themoduli space. This sigma model will describe the relevant classical and quan-tum physics of the vortex. Usually one determines only the bosonic moduliand degrees of freedom. Supersymmetry is in fact sufficient to recover thefermionic sector, if needed.

We will use the so called “moduli-matrix formalism”, and the equivalent“Kahler-quotient construction” to systematically explore the bosonic zeromodes and the corresponding bosonic sector of a wide range of vortex con-figurations. We will consider, as laboratory model, a U(Nc) gauge theorywith Nf ≥ Nc flavour in the fundamental representation. We will choose itsparameters so that it could be embedded into an N = 2 QCD, and it willadmit a degenerate set of vortex solutions.

In Part I we will introduce the model and the basic construction of non-Abelian vortices. We will review some of the most important efforts to usenon-Abelian vortices for a better understanding of the 4-d dynamics of non-Abelian gauge theories.

In Part II we will describe our results about the properties of the modulispace of non-Abelian vortices in gauge theories with unitary gauge group,U(Nc). In Chapter 4 we will study configurations of composite vortices, im-portant in the problem of interactions. In Chapter 5 we will study the effect

5

on moduli space structure when an arbitrary number of flavor is added. Inthese case there appear the so-called “semi-local” vortices. The connectionswith lump solutions are also discussed. We will use these knowledge in Chap-ter 6 to study the interactions of vortices, when some non-BPS correctionare added to our model. The same non-BPS correction are also introducedin models with several flavors in Chapter 7, to discuss the stability of semi-local vortices. The knowledge about the moduli space of composite vorticesis used to discuss a dual model of confinement in SO(N) gauge theories inChapter 8, and to solve the problem of the reconnection of cosmic strings inChapter 9.

In Part III, we will introduce a more general derivation of the modulispace, which will enable us to construct non-Abelian vortices in a set ofnon-Abelian gauge theories with a generic gauge group. In Chapter 10 wewill derive this generalized construction, using the deep connection betweenvortices and lumps in related sigma models. In Chapter 11 we will use theconstruction for a detailed study of vortices in non-Abelian gauge theorieswith orthogonal and symplectic gauge groups. These new vortex solutionsprovide semiclassical hints for the emergence of a GNOW duality, which willbe discussed in Chapter 12.

In the last chapter we will conclude and discuss, future, possible devel-opment of this PhD thesis work.

6

Part I

A Brief Introduction

7

Chapter 1

Construction and Properties

1.1 Basic set-up

The basic model for the construction of non-Abelian vortices is an N = 2supersymmetric U(Nc) gauge theory with Nf = Nc = N hypermultiplets(quarks) Qf , Qf (f = 1, . . . , N). The field content is as follows. There are aU(1) gauge field Aµ and SU(N) gauge fields Aiµ (where i = 1, · · · , N2 − 1),together with the Weyl fermion superpartners: (λ1

α, λ2α) and (λ1i

α , λ2iα ), where

α is the spinorial index. The N = 2 vector multiplet includes also a complexsinglet a and a complex field in the adjoint representation ai. The matterhypermultiplets contain N pair of complex fields, (Qf , Q

†f ), together with

the fermionic superpartners, (ψf , ψ†f ), all in the fundamental representation.

The superpotential is completely determined by supersymmetry:

W =1√2

[Qf (a0 + aiτ

i +√

2mf )Qf +WFI(a0)], (1.1.1)

where

WFI = −N2ξ a0. (1.1.2)

mf are the (bare) quark masses. ξ is the F -term Fayet-Iliopoulos (FI) pa-rameter and can take an arbitrary value . This term is equivalent to thestandard D-term FI term [5]1 . As well-known, it does not break any su-persymmetry [11, 15]. This kind of system naturally arises in the N = 2,

1In the terminology used in Davis et al. [6] in the discussion of the Abelianvortices in supersymmetric models, this model corresponds to an F model while themodels of [7, 8, 9, 10], which we will also consider in the next Sections, correspondto a D model. The two models are equivalent: they are related by an SUR(2)transformation [11, 12, 13, 14].

8

SU(N+1) SQCD (see Sec. 2 for more discussions on this), with SUSY softlybroken down to N = 1 with a mass term for the adjoint fields of the formW = κTrΦ2 [3].

The bosonic part of the Lagrangian is [16] (we use the same symbols forthe scalars as for the corresponding superfields):

L = − 1

4g2(F i

µν)2 − 1

4e2(F 0

µν)2 +

1

g2|Dµai|2 +

1

e2|∂µa0|2

+ (DµQf )†DµQf +DµQf (DµQf )

† − V (Q, Q, ai, a0) , (1.1.3)

where e is the U(1) gauge coupling and g is the SU(N) gauge coupling. Thecovariant derivatives and field strengths, respectively, are defined by

Dµ(Qf , Q

∗f

)=

(∂µ − iAiµ

τ i

2− i

2A0µ

)(Qf , Q

∗f

), Dµai = ∂µai + εijkAjµak,

F iµν = ∂µA

iν − ∂νAiµ + εijkAjµA

kν , Fµν = ∂µA

0ν − ∂νA0

µ , (1.1.4)

where the generators are normalized as

Tr (τaτ b) = 2δab . (1.1.5)

The potential V is the sum of the following D and F terms

V = V1 + V2 + V3 + V4 ;

V1 =g2

8

(2

g2εijkajak + Trf [Q

†τ iQ]− Trf [QτiQ†]

)2

;

V2 =e2

8

(Trf [Q

†Q]− Trf [QQ†])2

;

V3 =g2

2

∣∣∣Trf [QτiQ]∣∣∣2 +

e2

2

∣∣∣∣Trf [QQ]− N

2ξ

∣∣∣∣2 ;

V4 =1

2

Nf∑f=1

∣∣∣(a0 + τ iai +√

2mf )Qf

∣∣∣2 +1

2

Nf∑f=1

∣∣∣(a0 + τ iai +√

2mf )Q†f

∣∣∣2 .

(1.1.6)

Trf denotes a trace over the flavor indices. For generic, unequal quark masses,

M = diag(m1,m2, . . . ,mN), (1.1.7)

the vacuum expectation value of the adjoint field:

〈a0 + aiτ i〉 = −√

2

m1 0 · · · 00 m2 · · · 0

0 0. . . 0

0 0 · · · mN

, (1.1.8)

9

breaks the gauge group down to U(1)N .Non-Abelian vortices are constructed in the most symmetric situation in

which the squark multiplets have the same bare masses. The value of thecommon mass can be chosen to be equal to zero, after a proper shift of theadjoint scalars:

mf = 0 . (1.1.9)

At the classical level, the theory as a unique vacuum. Up to gauge and flavorrotations, we can choose the following VEV for the scalar fields

Q = Q† =

√ξ

21N , a0 = 0 , a = 0 , (1.1.10)

where a ≡ aiτi/2. The gauge symmetry is completely broken by the squarks

expectation value. The vacuum is invariant under the following global color-flavor locked rotations (Uc ∈ SU(N), Uf ∈ SU(N)):

Q→ UcQU†f , Q→ U †f QUc , a→ UcaU

†c , Fµν → UcFµνU

†c .

(1.1.11)

Let us briefly consider the main quantum effects. If all bare masses aretaken to be zero, theory (1.1.3) is asymptotically free, and the running of itsgauge coupling in terms of the energy scale µ is given by:

8π

g2(µ)= N log

µ

ΛSU(N)

. (1.1.12)

The theory thus necessarily runs in a strong coupling regime at energies belowthan the dynamically generated scale ΛSU(N). Nevertheless, it is possible toextract exact informations about this regime. Using the well-known Seiberg-Witten solution of N = 2 gauge theories [3, 4] it is possible to show thatstrong coupling effects would dynamically break gauge symmetry down toU(1)N . There are two possible ways to prevent the flowing of the theoryto this strong coupling regime. One is to take different masses, so that∆m ΛSU(N). In this case, the gauge symmetry is semiclassically brokendown to U(1)N . The various U(1) sector are then infrared free. Anothertechnique is to keep equal masses but taking a large FI term ξ ΛSU(N).In fact, the theory will enter the Higgs phase at the energy scales equal tothe masses of the gauge and scalar fields (∼ g

√ξ). Because of the mass

gap, all the degrees of freedom are frozen below this scale, and the gaugecoupling will stop its running at lower energies. We will consider this secondway to study non-Abelian vortices, from now on, because it preserves the full

10

SU(Nf ) flavor symmetry (on the existence of such a symmetry relies the veryexistence of non-Abelian vortices)2. To be more precise, in the next Sectionwe will construct reliable semiclassical vortex solutions in a weakly coupledtheory, provided we choose:

ξN/2 ΛNSU(N) = ξN/2 exp

(− 8π2

g2(ξ)

). (1.1.13)

1.2 The vortex solution

The model that we considered in the previous Section admits BPS vortexsolutions, e.g., supersymmetric, string-like, classical configurations. This canbe easily understood if we notice that the theory (1.1.3) in the chosen vacuum(6.1.4) undergoes a gauge symmetry breaking which has the correct topologythat supports vortices:

U(N)ξ−→ 1 π1(U(N)) = Z . (1.2.1)

These kind of solutions have been thoroughly studied in literature [15, 11,17, 16, 7, 18, 19].

It turns out that for a semiclassical vortex solution, the adjoint fieldsare identically zero, and Q = Q†. Using this in the Lagrangian (1.1.3) wecan rewrite the full action after a Bogomol’nyi completion [20]. For staticsolutions, like those we are looking for, it reduce to the expression for theenergy (tension) of the string:

T =

∫d2x

1

2

[1

gF iij + gTrf [Q

†τ iQ]εij

]2

+

+

[1

eF 0ij + e

(Trf [Q

†Q]− N

2ξ

)εij

]2

+∣∣DiQA + iεijDj QA

∣∣2 +NξF 0ij

≥ Nξ

∫d2xF 0

ij = 2πξκ (1.2.2)

where κ is the quantized Abelian magnetic flux, which gives the lower boundfor the vortex tension. This bound (Bogomol’ny bound) is saturated by

2Studying the case with large mass differences gives precious informations too,in particular when one consider quantum aspects. We will discuss this case inSection (1.4)

11

solutions of the first order BPS equations for the vortex:

F i12 + g2Trf [Q

†τ iQ] = 0, i = 1, · · · , N2 − 1;

F 012 + e2

(Trf [Q

†Q]− N

2ξ

)= 0;

D1Qf + iD2Qf = 0, f = 1, . . . , N. (1.2.3)

The fundamental (κ = 1) vortex is explicitly constructed by the use ofthe following standard Ansatz, which gives a static, cylindrically invariantsolution [16]:

Q =

φ0(r)eiθ 0 . . . 0

0 φ1(r) . . . 0

0 0. . . 0

0 0 . . . φ1(r)

,

Ai = Aαiτα

2= − 1

N

1 0 . . . 00 1 . . . 0

0 0. . . 0

0 0 . . . −(N − 1)

εijxjr2

[1− f1(r)] ,

A0i = − 1

N

εijxjr2

[1− f0(r)] . (1.2.4)

By plugging the ansatz (6.1.10) into Eq. (1.2.3), we get the following set offirst order differential equations for the four unknown functions φ0, φ1, f0, f1.

rd

drφ0(r)− 1

N(f0(r) + (N − 1)f1(r))φ0(r) = 0,

rd

drφ1(r)− 1

N(f0(r)− f1(r))φ1(r) = 0,

−1

r

d

drf1(r) +

e2N

4

(φ0(r)2 + (N − 1)φ1(r)2 −Nξ

)= 0,

−1

r

d

drf3(r) +

g2

2

(φ0(r)2 − φ1(r)2

)= 0, (1.2.5)

which must be solved with the boundary conditions:

f1(0) = 1, f0(0) = 1, f1(∞) = 0, f0(∞) = 0,

φ0(∞) = 1, φ1(∞) = 1 (1.2.6)

12

and the following behavior for small r

φ0 ∝ O(r), φ1 ∝ O(1). (1.2.7)

A numerical solution is shown in Fig. (1.1). The tension saturate the BPSbound :

TBPS = 2πξ . (1.2.8)

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1.2

Figure 1.1: Profile functions for a vortex in a U(2) gauge theory in terms of theradial direction r: f0 (red), f1 (dashed red), φ0 (blue), φ1 (dashed blue). Thenumerical values are taken as ξ = 2, e = 2, g = 1

.

A fundamental property of non-Abelian vortices is the possession of acontinuum set of degenerate solutions which represent the arbitrary orien-tation of the flux of the vortex inside the non-Abelian gauge group [16, 7].This set of solution is called “moduli space”. The study of the moduli spaceis the main tool to understand the physics of non-Abelian vortices, and itwill be the main subject of all our forthcoming discussions. To determinethe moduli space in the case considered here, let us start from the solutionof Eq. 6.1.10 and then apply to it the SU(N)c+f symmetry (7.2.3) of thevacuum. This symmetry is partially broken by the vortex solution, with anSU(N − 1)× U(1) subgroup which leave the solution unchanged. Thus, weget for the moduli space of solutions

CPN−1 =SU(N)

SU(N − 1)× U(1). (1.2.9)

13

Notice that the existence of such a degeneracy is not related to the BPS sat-uration of the solitons. On the contrary, it represent zero modes interpretedas massless goldstone bosons which come from the breaking of the globalcolor-flavor symmetry of the vacuum.

1.3 Effective world-sheet theory

In the previous Section we have derived the moduli space for non-Abelianvortices, which is given by the projective complex space CPN−1. The effectivetheory is thus a non linear sigma model with that target space. Let us thusstudy this effective theory in more detail [16, 7, 9, 8, 21].

Let us introduce the following parametrization for CPN−1:

1

N

Uc+f

1 0 . . . 00 1 . . . 0

0 0. . . 0

0 0 . . . −(N − 1)

U †c+f

l

p

= −nln∗p +1

Nδlp (1.3.1)

where nl is a complex vector in the fundamental representation of SU(N),with k = 1, · · · , N and n∗l n

l = 1. Also, the phase of n is redundant and mustbe gauged away. We are ready to write the full set of solutions for a genericorientation nk:

A0i =

1

N

εijxjr2

[1− f0],

A =

(n · n∗ − 1

N

)εijxjr2

[1− f1],

Q = Q† =1

N[(N − 1)φ1 + φ0] + (φ0 − φ1)

(n · n∗ − 1

N

)(1.3.2)

The presence of orientational zero-modes on the non-Abelian string leadsto a non-trivial interacting physics, when we consider the low-energy effectivetheory on the string background. The basic procedure to derive the effectiveaction is to promote the vector nl to be a slowly varying function of the 1+1dimensional world-sheet coordinates: nl(xi), i = 0, 3. As soon as we considernon-static solutions, we have to generalize the Ansatz of Eq. 1.3.2 for thegauge fields, in order to include the now non-zero fields Ai, i = 0, 3 ([16, 9]):

Ai = −i[∂in · n∗ − n · ∂in∗ − 2n · n∗(n∗∂in)]ρ(r) (1.3.3)

14

where ρ(r) is a function which must be determined with a variational pro-cedure. After some long calculations, one gets the following action for theorientational modes:

S1+1eff = 2β

∫dt dz

(∂in

∗∂in) + (n∗∂in)2

(1.3.4)

which is the well-known action for a bosonic CPN−1 non-linear sigma model.The two-dimensional coupling β is related to the four-dimensional non-Abeliangauge coupling g by the relation:

β =2π

g2. (1.3.5)

The action above includes only the bosonic zero modes, parameterized bythe complex vector nl. The true effective action includes also fermionic zero-modes, which can be completely determined by the use of supersymmetry.The complete effective theory is the N = (2, 2) supersymmetric extension ofthe CPN−1 sigma model.

Let us briefly discuss some of the quantum aspects of the supersymmetricCPN−1 sigma model obtained [22]. Again, the most important quantumeffect is the running of the coupling constant β, which is asymptotically free.It is given by

4πβ(E) = N logE

Λσ

, (1.3.6)

where Λσ is a strong coupling scale dynamically generated. The running ofthe couplings implies that we have to specify a scale at which the relation(1.3.5) is satisfied. It turns out that the correct scale is given by g

√ξ. This

is the scale defined by the mass of the SU(N) gauge bosons in the Higgsphase, where we constructed the vortex. It also roughly gives the inverse ofthe width of the vortex itself. This scale length has to interpreted as an UVcut-off which limit the validity of the effective action (1.3.4). In fact, thegauge coupling runs down to this scale and it freeze. A t lower scales thesigma model coupling take its place, starting the running. The matching ofthe running couplings at this scales

4πβ(g√ξ) = N log

g√ξ

Λσ

=8π

g2(g√ξ)

= N logg√ξ

ΛSU(N)

(1.3.7)

gives the following important result:

ΛNSU(N) = ΛN

σ (1.3.8)

15

There are several ways to study the strong coupling regime of the CPN−1

non-linear sigma model. It can be solved exactly in the limit of large N , forexample [23]. It also possible to work out a “mirror” description in terms ofan equivalent, weakly coupled theory [24, 25, 26, 27]. The main result, in allthe approaches, thanks to the supersymmetric nature, is the computation ofthe exact superpotential, which is non-perturbatively generated by instan-tonic effects. The minimization of this potential gives the exact quantumvacua of the model. In general, it turns out that the supersymmetric CPN−1

non-linear sigma model has N isolated vacua, which all posses a mass gap. Itis important to say that in these quantum vacua, the expectation value of theorientational field is zero: 〈nl〉 = 0. This can be considered as a consequenceof a well-known theorem which forbids the existence of Goldstone bosons intwo-dimensional theories [28].

We have already encountered many “correspondences” between the 4-dimensional gauge theory and the 2-dimensional vortex theory. In the nextsection we will briefly review the physical picture which explain why this isnot a “coincidence”.

1.4 Non-Abelian vortices and 4-D gauge dy-

namics

Let us explore in more detail the physical intuition which lies behind thecorrespondences found in the previous Section between the bulk theory andthe effective vortex theory, following Ref. [29]. To this end, we start addingbig mass differences to the bulk theory, e.g. we consider ∆mij ΛSU(N). Wekeep the theory on its Coulomb phase, by setting the FI term to zero: ξ = 0.As already said, the bulk theory is now weakly coupled, and undergoes thefollowing symmetry breaking pattern:

U(N)∆mij−→ U(1)N . (1.4.1)

This breaking supports ’t Hooft-Polyakov monopoles which are characterizedby the second homotopy group

π2

(U(N)/U(1)N

)= Z

N−1. (1.4.2)

There are thus N−1 distinct monopoles, and each of them can be constructedin the semiclassical limit. Their masses are exactly determined (as alwaysfor BPS solitons) by boundary terms proportional to the quantized magneticflux:

Mmon =4π

g2∆mij. (1.4.3)

16

Almost free monopole

<<!

"!1/2

<m < "1/2

Confined monopole,quasiclassical regime

!!1

m 0

Confined monopole,highly quantum regime

>> "1/2m

"!1/2

Figure 1.2: Confined monopoles.

Let us now turn on the FI term. This will put the theory on the Higgsbranch. Vortices now will be present in the Higgs vacuum. In the presence ofnon-zero mass differences, vortices loose their orientational moduli becausethe SU(N)c+f symmetry is broken down to U(1)N−1. In this case we obtainN distinct vortices which are to be considered as different embedding ofan abelian (ANO-like) vortex. These vortices must be present, and mustconfine the magnetic flux of the monopoles.This picture is confirmed by thefact that the fluxes of monopoles and vortices coincide in various differentmodels [30, 31, 32, 33, 34, 35, 36, 37, 12, 38]. As soon as ∆mij

√ξ, the

situation is nicely showed in the upper picture of Figure (1.2): the size ofthe monopoles (∼ ∆m−1

ij ) is much smaller than the transverse size of the

vortex (√ξ−1

). Monopoles are almost unconfined. If we increase the value ofthe FI term, the size of vortices will shrink, and eventually will be so smallthat the attached monopoles will be squeezed inside the vortex. Monopolesare now strongly confined (left-bottom picture of Figure 1.2). However, thetheory is still in a weakly coupled regime. We can study this configurationfrom the point of view of the CPN−1 sigma model: if the mass differences aremuch smaller than the gauge bosons masses, the effective theory is reliable(∆mij

√ξ). The effective lagrangian (1.3.4) will be deformed by the

introduction of non-zero mass differences. This deformation was found in

17

Refs. [9, 8]. and correspond to the inclusion of the following superpotential:

VCPN−1 = 2β

∑l

|∆ml|2|nl|2 −∣∣∣∣∣∑

l

∆ml|nl|2∣∣∣∣∣2 , (1.4.4)

with ∆ml = ml − 1N

∑lml. This deformation corresponds to the inclusion

of the so-called twisted masses in the two-dimensional sigma model [39, 40,41, 42].

The potential admits N distinct minima, which correspond to N isolatedsemiclassical vacua. This implies the existence of N−1 kinks (domain walls inone dimension) which interpolate between two of them. Again, it is possibleto exactly calculate the mass of these kinks, which turns out to be:

Mkink = 2β∆mij. (1.4.5)

The dependence of the mass of the kinks in terms of the twisted masses isthe same as that of the unconfined monopoles in Eq. (1.4.3). This has anatural explanation if we interpret the kinks as the sigma model descriptionof the monopoles confined on the string. This interpretation is strengthenedby the observation that both expression (1.4.3) and (1.4.5) do not dependby the FI term ξ (there are standard argument in supersymmetric theorieswhich forbids the dependence of BPS quantities, like the masses of BPSsolitons, by the FI term [11, 15]). The identification between free monopolesin the Coulomb phase and kinks on the string in the Higgs phase leads tothe relation:

MCoulombmon = MHiggs

mon = Mkink. (1.4.6)

As it is easy to check, this relation is equivalent to the previously men-tioned relation between the two and four dimensional couplings Eq. (1.3.5).One may now go to the strong coupling regime, turning all the masses off:∆mij Λ (right-bottom picture of Figure 1.2). In this regime the semiclas-sical description of both monopoles and kinks is not reliable. Nonetheless, itis possible to determine their exact masses:

Mmon =2

πΛSU(N), Mkink =

2

πΛσ. (1.4.7)

Equating the two values above, we again obtain equivalence between thestrong coupling scales: ΛSU(N) = Λσ

The physical rationale which lies behind the correspondence of monopolesand kinks can be generalized for all the BPS states of the two theories. It

18

was shown that the full BPS spectrum of the four dimensional gauge theory,at the root of the baryonic branch, coincides with that of the supersymmetricCPN−1 sigma model [39, 43]. We interpret the states of the effective theory asstates of the bulk theory which are confined on the string. This interpretationwork for the solitonic sector for the perturbative one too. The gauge bosonsare massless in the Coulomb phase. When we enter the Higgs phase, all ofthem acquire non zero masses. But in the presence of a vortex the gaugesymmetry is (partially) restored at the core vortex. Thus, gauge bosons mayshow themselves as massless excitation confined inside the vortex. They arein fact interpreted as the orientational modes of the CPN−1 sigma model.These statements can be formalized as follows. First of all we consider analternative description of the vortex theory. The CPN−1 of (1.3.4) is obtainedby taking the strong coupling limit of the following U(1), N = 2, gaugetheory with N chiral hypermultiplets (l = 1, . . . , N) [22]:

S1+1eff =

∫dt dz

2β|Dµnl|2 +

1

4e21+1

F 2µν +

1

e21+1

|∂µσ|2 +

+ 4β|σ − ∆ml√2|2|nl|2 + 2e2

1+1β2(|nl|2 − 1)2

, (1.4.8)

where we have included non-zero masses. As we take the strong couplinglimit, the photon and the σ field become massive and can be integratedout. At the same time the condition |nl|2 − 1 = 0 is implemented, andgives us a CPN−1 as moduli space of vacua of the theory. The quantitieswe are interested in, like masses of BPS states, do not depend by the twodimensional gauge coupling e1+1. Thus we are allowed to study the lineargauged sigma model (1.4.8) at finite coupling, which is much easier. Theimportant result found in Ref. [39, 43] is that the exact BPS spectrum of thetheory (1.4.8) (vortex theory) coincides with that of the theory (1.1.3) (bulktheory), once the bare masses of the hypermultiplets and the strong couplingscales of the two models are identified. The correspondence holds both atthe semiclassical and quantum level. At the time of its discovery, this resultseemed really puzzling. As we briefly review discussed, non-Abelian vorticesprovides the fundamental connection ring between the two theories.

19

Chapter 2

Confinement and Dualities in4-d Gauge Theories

A system in which the gauge symmetry is spontaneously broken

G〈v1〉6=0−→ H (2.0.1)

where H is some non-Abelian subgroup of G, possesses a set of regular mag-netic monopole solutions in the semi-classical approximation, which are nat-ural generalizations of the ’t Hooft-Polyakov monopoles [44, 45] found inthe system G = SO(3), H = U(1). A straightforward generalization of theDirac’s quantization condition leads to the GNOW (Goddard-Nuyts-Olive-E.Weinberg) conjecture, i.e., that they form a multiplet of the group H, dualof H. The group H is generated by the dual root vectors

α∗ =α

α · α, (2.0.2)

where α are the non-vanishing roots of H [46]. There are however well-known difficulties in such an interpretation. The first concerns the topologicalobstruction discussed in [47, 48, 49, 50]: in the presence of the classicalmonopole background, it is not possible to define a globally well-defined setof generators isomorphic to H. As a consequence, no “colored dyons” exist.In the simplest example of a system with the symmetry breaking,

SU(3)〈v1〉6=0−→ SU(2)× U(1), (2.0.3)

this means that no monopoles exist which carry the quantum number, e.g.,

(2, 1∗) (2.0.4)

20

where the asterisk indicates the dual, magnetic U(1) charge.

The second can be regarded as the infinitesimal version of the same diffi-culty: certain bosonic zero-modes around the monopole solution, correspond-ing to the H gauge transformations, are non-normalizable (behaving as r−1/2

asymptotically). Thus the standard procedure of semiclassical quantizationleading to the H multiplet of the monopoles does not work. Some progresson the check of GNO duality along this orthodox approach has been reportednevertheless in [51] for N = 4 supersymmetric gauge theories, which howeverrequires the consideration of particular multi-monopole systems neutral withrespect to the non-Abelian group (more precisely, non-Abelian part of) H.

Both of these difficulties concern the transformation properties of themonopoles under the subgroup H, while the truly relevant question is howthey transform under the dual group, H. As field transformation groups,H and H are relatively non-local; the latter should look like a non-localtransformation group in the original, electric description.

Another related question concerns the multiplicity of the monopoles; takeagain the case of the system with breaking pattern Eq. (2.0.3). One mightargue that there is only one monopole, as all the degenerate solutions arerelated by the unbroken gauge group H = SU(2).1 Or one might say thatthere are two monopoles as, according to the semiclassical GNO classification,they are supposed to belong to a doublet of the dual SU(2) group. In short,what is the multiplicity of the monopoles? The answer to this question needsa deeper understanding of the quantum properties of non-Abelian monopoles.Clearly the very concept of the dual gauge group or dual gauge transformationmust also be better understood. In attempting to gain such an improvedinsight on the nature of these objects, we are naturally led to several generalconsiderations.

The first is the fact when H and H groups are non-Abelian the dynamicsof the system should enter the problem in an essential way. It should not besurprising if the understanding of the concept of non-Abelian duality requireda full quantum mechanical treatment of the system.

For instance, the non-AbelianH interactions can become strongly-coupledat low energies and can break itself dynamically. This indeed occurs in pureN = 2 super Yang-Mills theories (i.e., theories without quark hypermulti-plets), where the exact quantum mechanical result is known in terms of theSeiberg-Witten curves [52, 52, 53, 54, 55, 56, 57]. Consider for instance, apure N = 2, SU(N + 1) gauge theory. Even though partial breaking, e.g.,

1This interpretation however encounters the difficulties mentioned above. Alsothere are cases in which degenerate monopoles occur, which are not simply relatedby the group H, see below.

21

SU(N + 1)→ SU(N)× U(1) looks perfectly possible semi-classically, in anappropriate region of classical degenerate vacua, no such vacua exist quan-tum mechanically. In all vacua the light monopoles are Abelian, the effective,magnetic gauge group being U(1)N .

Generally speaking, the concept of a dual group multiplet is well-definedonly when H interactions are weak (or, at worst, conformal). This howevermeans that one must study the original, electric theory in the regime ofstrong coupling, which would usually make the task of finding out whathappens in the system at low energies exceedingly difficult. Fortunately,in N = 2 supersymmetric gauge theories, the exact Seiberg-Witten curvesdescribe the fully quantum mechanical consequences of the strong-interactiondynamics in terms of weakly-coupled dual magnetic variables. And this ishow we know that the non-Abelian monopoles do exist in fully quantumtheories [58]: in the so-called r-vacua of softly broken N = 2 SQCD, thelight monopoles interact as a point-like particle in a fundamental multipletr of the effective, dual SU(r) gauge group. In the system of the type Eq.(2.0.3) with appropriate number of quark multiplets (Nf ≥ 4), we know thatlight magnetic monopoles carrying the non-Abelian quantum number

(2∗, 1∗) (2.0.5)

under the dual SU(2)× U(1) appear in the low-energy effective action (cfr.Eq. (2.0.4)). The distinction between H and H is crucial here.

In general N = 2 SQCD with Nf flavors, light non-Abelian monopoles

with SU(r) dual gauge group appear for r ≤ Nf2

only. Such a limit clearlyreflects the dynamical properties of the soliton monopoles under renormaliza-tion group: the effective low-energy gauge group must be either infrared freeor conformal invariant, in order for the monopoles to emerge as recognizablelow-energy degrees of freedom [59, 60].

A closely related point concerns the phase of the system. Even if thereis an ample evidence for the non-Abelian monopoles, as explained above,we might still wish to understand them in terms of something more famil-iar, such as semiclassical ’t Hooft-Polyakov solitons. An analogous questioncan be (and should be) asked about the Seiberg’s “dual quarks” in N = 1SQCD [61]. Actually, the latter can be interpreted as the GNOW monopolesbecoming light due to the dynamics, at least in SU(N) theories [62, 63].

Dynamics of the system is thus a crucial ingredient: if the dual groupwere in Higgs phase, the multiplet structure among the monopoles would getlost, generally. Therefore one must study the dual (H) system in confinementphase.But then, according to the standard electromagnetic duality argument,one must analyze the electric system in Higgs phase. The monopoles will

22

appear confined by the confining strings which are nothing but the vorticesin the H system in Higgs phase.

We are thus led to study the system with a hierarchical symmetry break-ing,

G〈v1〉6=0−→ H

〈v2〉6=0−→ 1, (2.0.6)

where

|〈v1〉| |〈v2〉|, (2.0.7)

instead of the original system Eq. (2.0.1). The smaller VEV breaks H com-pletely. Also, in order for the degeneracy among the monopoles not to bebroken by the breaking at the scale |〈v2〉|, we assume that some global color-flavor diagonal group

Hc+f ⊂ Hcolor ⊗Gf (2.0.8)

remains unbroken.It is hardly possible to emphasize the importance of the role of the mass-

less flavors too much. This manifests in several different aspects.

(i) In order that H must be non-asymptotically free, there must be sufficientnumber of massless flavors: otherwise, H interactions would becomestrong at low energies and H group can break itself dynamically;

(ii) The physics of the r vacua [59, 60, 64] indeed shows that the non-Abelian

dual group SU(r) appear only for r ≤ Nf2

. This limit can be understoodfrom the renormalization group: in order for a non-trivial r vacuum toexist, there must be at least 2 r massless flavors in the fundamentaltheory;

(iii) Non-Abelian vortices [7, 8, 16], which as we shall see are closely relatedto the concept of non-Abelian monopoles, require a flavor group. Thenon-Abelian flux moduli arise as a result of an exact, unbroken color-flavor diagonal symmetry of the system, broken by individual solitonvortex.

In this Chapter we will review the idea that the properties of non-Abelianmonopoles related to the dual group (such as multiplicities and transforma-tion properties) can be inferred by the study of the vortices which form ourmonopole-vortex complex [36].

23

2.1 SU(N + 1) model with hierarchical sym-

metry breaking

We consider now a concrete model in which the set-up discussed in the pre-vious section is explicitly realized. It is similar to that we studied in the pre-vious Chapter for the construction of vortices (1.1.3). Actually, it containsthe vortex theory as a low energy approximation. It is the standard N = 2SQCD with Nf quark hypermultiplets, with gauge symmetry SU(N + 1),which is broken at a much larger mass scale as

SU(N + 1)v1 6=0−→ SU(N)× U(1)

ZN

. (2.1.1)

The remaining gauge symmetry is then completely broken at a lower massscale.

Clearly one can attempt a similar embedding of the vortex theory ina larger gauge group broken at some higher mass scale, in the context of anon-supersymmetric model, even though in such a case the potential must bejudiciously chosen and the dynamical stability of the scenario would have tobe carefully monitored. Here we have chosen to study the softly broken N =2 SQCD for concreteness, and above all because the dynamical properties ofthis model are well understood: this will provide us with a non-trivial checkof our results. Another motivation is purely of convenience: it gives a definitepotential with desired properties.2

The bosonic sector of our model is given by the following Lagrangian

L = − 1

4g2(F i

µν)2 +

1

g2|Dµai|2 + (DµQf )

†DµQf +DµQf (DµQf )†

− V (Q, Q, ai, a0) , (2.1.2)

where the potential is basically given by the non-Abelian part of the ex-pressions in Eq. 1.1.6. The only difference comes from the superpotential. Inthe present case, in fact, we add a mass term for the adjoint chiral multiplet,which softly breaks the supersymmetry to N = 1.

Wµ =1

2µ aiai (2.1.3)

2 Recent developments [65, 66, 67, 68, 69, 70, 71] allow us actually to considersystems of this sort within a much wider class of N = 1 supersymmetric models,whose infrared properties are very much under control. We stick ourselves to thestandard N = 2 SQCD, however, for concreteness.

24

which changes V3 in Eq. 1.1.6 as

V3 =g2

2

∣∣∣Trf [QτiQ] + µai

∣∣∣2 , (2.1.4)

In order to keep the hierarchy of the gauge symmetry breaking scales, Eq.(2.0.7), we choose the masses such that

m1 = . . . = mNf = m, (2.1.5)

m µ Λ. (2.1.6)

Although the theory described by the above Lagrangian has many degeneratevacua, we are interested in the vacuum [64]

〈aiτ i〉 = −√

2

m 0 0 0

0. . .

......

0 . . . m 00 . . . 0 −N m

; (2.1.7)

Q = Q† =

d 0 0 0 . . .

0. . . 0

... . . .0 0 d 0 . . .0 . . . 0 0 . . .

, d =

√(N + 1)µm√

2. (2.1.8)

This is a particular case of the so-called r vacuum, with r = N . Althoughsuch a vacuum certainly exists classically, the existence of the quantum r = Nvacuum in this theory requires Nf ≥ 2N , which we shall assume.3

To start with, ignore the smaller squark VEV, Eq. (2.1.8). As π2(G/H) ∼π1(H) = π1(SU(N) × U(1)) = Z, the symmetry breaking Eq. (2.1.7) givesrise to regular magnetic monopoles with mass of order of O(v1

g) (v1 ∼ m),

3 This might appear to be a rather tight condition as the original theory losesasymptotic freedom for Nf ≥ 2N +2. This is not so. An analogous discussion canbe made by considering the breaking SU(N)→ SU(r)×U(1)N−r. In this case thecondition for the quantum non-Abelian vacuum is 2N > Nf ≥ 2 r, which is a muchlooser condition. Also, although the corresponding U(N) theory Eq. (1.1.3) withsuch a number of flavor has semi-local strings [72, 73, 74, 75, 6], these moduli arenot directly related to the discussion about monopoles and dual gauge symmetry,which is our interest in this Chapter. We will study the fate of semi-local moduliin Section (7).

25

whose continuous transformation property is our main concern here. Thesemiclassical formulas for their mass and fluxes are well known [46, 37] andwill not be repeated here.

At scales much lower than v1 = m but still neglecting the smaller squarkVEV v2 = d ∼

√(N + 1)µm v1, the theory reduces to an SU(N) ×

U(1) gauge theory with Nf light quarks qi, qi (the first N components of the

original quark multiplets Qi, Qi). By integrating out the massive fields, after

a suitable redefinition of fields, the effective Lagrangian valid between thetwo mass scales is precisely the vortex model (1.1.3) that we introduced inChapter (1). where the FI term is given by

ξ =√

2µm(N + 1) (2.1.9)

The splitting of the coupling constant into two different couplings is ac-complished by the renormalization group flow, under which the Abelian andthe non-Abelian couplings run differently towards the infrared. To exactlyget the model (1.1.3), we have to ignore contributions to the action which arequadratic in the mass parameter µ. Within this approximation, N = 2 su-persymmetry is unbroken. The presence of higher order corrections, however,makes the low-energy vortices not strictly BPS (and this will be importantin the discussion of Section (7) about their stability).

2.2 Vortex-monopole complex

The fact that there must be a continuous set of monopoles, which transformunder the color-flavor Gc+f group, follows from the following exact homotopysequence

· · · → π2(G)→ π2(G/H)→ π1(H)f→ π1(G)→ · · · , (2.2.1)

applied to our systems with a hierarchical symmetry breaking, Eq. (2.0.6),with an exact unbroken symmetry, Eq. (2.0.8). π2(G) = 1 for any Lie group,and π1(G) depends on the group considered. Eq. (2.2.1) was earlier used toobtain the relation between the regular, soliton monopoles (represented byπ2(G/H)) and the singular Dirac monopoles, present if π1(G) is non-trivial.The isomorphism

π1(G) ∼ π1(H)/π2(G/H) (2.2.2)

implied by Eq. (2.2.1) shows that among the magnetic monopole configura-tions Aai (x) classified according to π1(H) [76], the regular monopoles corre-spond to the kernel of the map f : π1(H)→ π1(G) [77].

26

When the homotopy sequence Eq. (2.2.1) is applied to a system withhierarchical breaking, in which H is completely broken at low energies,

Gv1−→ H

v2−→ 1,

it allows an interesting re-interpretation. π1(H) classifies the quantized fluxof the vortices in the low-energy H theory in Higgs phase. Vice versa, thehigh-energy theory (in which the small VEV is negligible) has ’t Hooft-Polyakov monopoles quantized according to π2(G/H). However, there issomething of a puzzle: when the small VEV’s are taken into account, whichbreak the “unbroken” gauge group completely, these monopoles must disap-pear somehow. A related puzzle is that the low-energy vortices with π1(H)flux, would have to disappear in a theory where π1(G) is trivial.

What happens is that the massive monopoles are confined by the vorticesand disappear from the spectrum; on the other hand, the vortices of thelow-energy theory end at the heavy monopoles once the latter are taken intoaccount, having mass large but not infinite Fig. (2.2). The low-energy vor-tices become unstable also through heavy monopole pair productions whichbreak the vortices in the middle (albeit with small, tunneling rates [78]),which is really the same thing. Note that, even if the effect of such stringbreaking is neglected, a monopole-vortex-antimonopole configuration is nottopologically stable anyway: its energy would become smaller if the stringbecomes shorter (so such a composite, generally, will get shorter and shorterand eventually disappear).

In the case G = SU(N + 1), H = SU(N)×U(1)ZN we have a trivial π1(G), so

π2

(SU(N + 1)

U(N)

)= π2(CPN) ∼ π1(U(N)) = Z : (2.2.3)

each non-trivial element of π1(U(N)) is associated with a non-trivial element

of π2(SU(N+1)U(N)

). Each vortex confines a regular monopole. The monopoletransformation properties follow from those of the vortices, as will be moreconcretely studied in the next section.

In theories with a non-trivial π1(G) such as SO(N), the application ofthese ideas is slightly subtle: these points will be discussed in Chapter (8).

In all cases, as long as the group H is completely broken at low energiesand because π2(G) = 1 always, none of the vortices (if π1(G) = 1) andmonopoles are truly stable, as static configurations. They can be only ap-proximately so, in an effective theory valid in respective regions (v1 ' ∞ orv2 ' 0).

However, this does not mean that a monopole-vortex-antimonopole com-posite configuration cannot be dynamically stabilized, or that they are not

27

Figure 2.1: A pictorial representation of the exact homotopy sequence, Eq. (2.2.1),with the leftmost figure corresponding to π2(G/H).

relevant as a physical configuration. A rotation can stabilize easily sucha configuration dynamically, except that it will have a small non-vanishingprobability for decay through a monopole-pair production, if such a decay isallowed kinematically.

After all, we believe that the real-world mesons are quark-string-antiquarkbound states of this sort, the endpoints rotating almost with a speed of light!An excited meson can and indeed do decay through quark pair productionsinto two lighter mesons (or sometimes to a baryon-antibaryon pair, if allowedkinematically and by quantum numbers). Only the lightest mesons are trulystable. The same occurs with our monopole-vortex-antimonopole configura-tions. The lightest such systems, after the rotation modes are appropriatelyquantized, are truly stable bound states of solitons, even though they mightnot be stable as static, semiclassical configurations.

Our model is thus a reasonably faithful (dual) model of the quark con-finement in QCD.

A related point, more specific to the supersymmetric models we considerhere as a concrete testing ground, is the fact that monopoles in the high-energy theory and vortices in the low-energy theory, are both BPS saturated.It is crucial in our argument that they are both BPS only approximately;they are almost BPS but not exactly.4 They are unstable in the full theory.But the fact that there exists a limit (of a large ratio of the mass scales,v1v2→ ∞) in which these solitons become exactly BPS and stable, means

that the magnetic flux through the surface of a small sphere surrounding themonopole and the vortex magnetic flux through a plane perpendicular to the

4 The importance of almost BPS soliton configurations have also been empha-sized by Strassler [63].

28

Monopole Moduli Vortex Moduli~ CPN-1

SU(N)

1(H)2(G/H)

Figure 2.2: The non-trivial vortex moduli implies a corresponding moduli ofmonopoles.

vortex axis, must match exactly. These questions (the flux matching) havebeen discussed extensively already in [36].

Our argument, applied to the simplest case, G = SO(3), and H = U(1),is precisely the one adopted by ’t Hooft in his pioneering paper [44, 45] toargue that there must be a regular monopole of charge two (with respect tothe Dirac’s minimum unit): as the vortex of winding number k = 2 mustbe trivial in the full theory (π1(SO(3)) = Z2), such a vortex must end ata regular monopole. What is new here, as compared to the case discussedby ’t Hooft [44, 45] is that now the unbroken group H is non-Abelian andthat the low-energy vortices carry continuous, non-Abelian flux moduli. Themonopoles appearing as the endpoints of such vortices must carry the samecontinuous moduli Fig. (2.2).

The fact that the vortices of the low-energy theory are BPS saturated(which allows us to analyze their moduli and transformation properties el-egantly, as discussed in the next section), while in the full theory there arecorrections which make them non BPS (and unstable), could cause some con-

29

cern. Actually, the rigor of our argument is not affected by those terms whichcan be treated as perturbation. The attributes characterized by integers suchas the transformation property of certain configurations as a multiplet of anon-Abelian group which is an exact symmetry group of the full theory, can-not receive renormalization. This is similar to the current algebra relationsof Gell-Mann which are not renormalized. Conserved vector current (CVC)of Feynman and Gell-Mann [79] also hinges upon an analogous situation.5

The results obtained in the BPS limit (in the limit v2/v1 → 0) are thus validat any finite values of v2/v1.

2.3 Transformation properties of monopoles

The concepts such as the low-energy BPS vortices or the high-energy BPSmonopole solutions are thus only approximate: their explicit forms are validonly in the lowest-order approximation, in the respective kinematical regions.Nevertheless, there is a property of the system which is exact and does notdepend on any approximation: the full system has an exact, global SU(N)c+fsymmetry, which is neither broken by the interactions nor by both sets ofVEVs, v1 and v2. This symmetry is broken by individual soliton vortex,endowing the latter with non-Abelian orientational moduli, analogous to thetranslational zero-modes of a kink. As we have already emphasized, thevortex breaks the color-flavor symmetry as

SU(N)c+f → SU(N − 1)× U(1), (2.3.1)

leading to the moduli space of the minimum vortices which is

M' CPN−1 =SU(N)

SU(N − 1)× U(1). (2.3.2)

The fact that this moduli coincides with the moduli of the quantum statesof an N -state quantum mechanical system, is a first hint that the monopolesappearing at the endpoint of a vortex, transform as a fundamental multipletN of a group SU(N).

The moduli space of the vortices is described by the complex vector nl

which we introduced in (6.1.11). It transforms like a fundamental vector ofthe SU(N)c+f symmetry

n→ Uc+f n (2.3.3)

5The absence of “colored dyons” [47, 48, 49, 50] mentioned earlier can also beinterpreted in this manner.

30

If we remember the constraint |n|2 = 1 and the phase identification n ∼ eiφn,we get the parametrization of the moduli space CPN−1. The points on thisspace represent all possible k = 1 vortices. Note that points on the space ofa quantum mechanical N-state system,

|Ψ〉 =N∑i=1

ai|ψi〉, ai ∼ λ ai, λ ∈ C, (2.3.4)

can be put in one-to-one correspondence with the points of a CPN−1.

The fact that the vortices (seen as solitons of the low-energy approxima-tion) transform as in the N representation of SU(N)c+f , implies that thereexist a set of monopoles which transform accordingly, as N . The existence ofsuch a set follows from the exact SU(N)c+f symmetry of the theory, brokenby the individual monopole-vortex configuration. Such a kind of consider-ations can answer questions like the one we pose at the beginning of thisChapter, about the quantum multiplicity of non-Abelian monopoles. Notethat in our derivation of continuous transformations of the monopoles, theexplicit, semiclassical form of the latter is not utilized.

A subtle point is that in the high-energy approximation, and to lowestorder of such an approximation, the semiclassical monopoles are just certainnon-trivial field configurations involving a(x) and Ai(x) fields, and thereforeapparently transform under the color part of SU(N)C+F only. When the fullmonopole-vortex configuration a(x), Ai(x), q(x) Fig. (2.2) are considered,however, only the combined color-flavor diagonal transformations keep theenergy of the configuration invariant. In other words, the monopole trans-formations must be regarded as part of more complicated transformationsinvolving flavor, when higher order effects in O(v1

v2) are taken into account.6

And this means that the transformations are among physically distinctstates, as the vortex moduli describe obviously physically distinct vortices[16].

6 Another independent effect due to the massless flavors is that of Jackiw-Rebbi [80]: due to the normalizable zero-modes of the fermions, the semi-classicalmonopole is converted to some irreducible multiplet of monopoles in the flavorgroup SU(Nf ). The “clouds” of the fermion fields surrounding the monopole havean extension of O( 1

v1), which is much smaller than the distance scales associated

with the infrared effects discussed here and should be regarded as distinct effects.

31

2.4 Non-Abelian duality requires an exact fla-

vor symmetry

In the N = 2 supersymmetric QCD, the presence of massless flavor and theexact color-flavor diagonal symmetry is fundamental for the emergence of thedual (non-Abelian) gauge transformations. It is well known in fact that thecontinuous non-Abelian vortex flux moduli - hence the non-Abelian vortex- disappear as soon as non-zero mass differences mi − mj are introduced.7

Also in order for the SU(N)C+F color-flavor symmetry not to be destroyedby the gauge dynamics itself, it is necessary to have the number of flavorssuch that Nf ≥ 2N . These points have been emphasized already in the firstpaper on the subject [16].

It is illuminating that the same phenomenon can be seen in the fullyquantum behavior of the theory of Section (2.1), in another regime,

µ,mi ∼ Λ (2.4.1)

(cfr. Eq. (2.1.6)). Indeed, this model was analyzed thoroughly in this regimein [64]. The so-called r vacua with the low-energy effective SU(r)×U(1)N+1−r

gauge symmetry emerges in the equal mass limit mi → m in which theglobal symmetry group SU(Nf ) × U(1) of the underlying theory becomeexact. When the bare quark masses are almost equal but distinct, the theorypossesses a group of

(Nfr

)nearby vacua, each of which is an Abelian U(1)N

theory, with N massless Abelian magnetic monopole pairs. The jump fromthe U(1)N to SU(r) × U(1)N+1−r theory in the exact SU(Nf ) limit mightappear a discontinuous change of physics, but is not so. What happensis that the range of validity of Abelian description in each Abelian vacuum,neglecting the light monopoles and gauge bosons (including massless particlesof the neighboring vacua, and other light particles which fill up a larger gaugemultiplet in the limit the vacua coalesce), gradually tends to zero as the vacuacollide. The non-Abelian, enhanced gauge symmetry of course only emergesin the strictly degenerate limit, in which the underlying theory has an exactSU(Nf ) global symmetry.

7 Such an alignment of the vacuum with the bare mass parameters is charac-teristic of supersymmetric theories, familiar also in the N = 1 SQCD [83]. In realQCD we do not expect such a strict alignment.

32

Chapter 3

Moduli Space of Non-AbelianVortices

In this Chapter we will review the construction of the most general non-Abelian vortex solution. A composite state of several BPS vortices developsa complicate moduli space of degenerate solutions. This property is a conse-quence of the BPS saturation, which implies that the static forces betweenBPS solitons are exactly zero. The full moduli space, for a generic number ofvortices, was first obtained in certain D-brane configuration in string theory[7, 8], and later in a field theory framework [10] as well.

3.1 The moduli matrix formalism

To study the most general vortex solution, it is convenient to slightly changenotations. First of all we will make an SU(2)R rotation on the vortex theory(1.1.3) to align the FI term along the D-term direction. After the rotation,the VEVS of the squark fields become:

〈H〉 = ξ, 〈H†〉 = 0, with

(H

H†

)= UR

(Q

Q†

), (3.1.1)

while the VEV of the rotated adjoint fields still remain zero. It is easy toshow that the fields with zero VEV, like the adjoint fields and the anti-chiralsquark multiplet H are trivial (equal to their VEVS) for every BPS vortexconfiguration. To study the classical properties of vortices, it thus suffices to

33

consider a truncated bosonic sector of theory (1.1.3):

L = − 1

4e2F 0µνF

0µν − 1

4g2F aµνF

aµν + (DµHf )†DµHf

−e2

2

∣∣∣∣H†f t0Hf −v2

√2Nc

∣∣∣∣2 − g2

2|H†f taHf |2 . (3.1.2)

We can consider, for a general discussion, a generic number of flavors: f =1, . . . , Nf with Nf ≥ Nc ≡ N To obtain the expression above from (1.1.3)one has to conveniently redefine the gauge coupling e, the abelian fields A0

µ

and a and the FI term. The abelian generator is now normalized as

t0 =1√2Nc

1Nc , (3.1.3)

while field strength and covariant derivative are given by

Fµν = ∂µWν − ∂νWµ + i [Wµ,Wν ] , DµH = (∂µ + iWµ) H. (3.1.4)

in terms of the gauge fields Wµ. The vacuum equation H H† = v2 1Nc impliesthat H has maximal rank and so, after quotienting out the gauge equivalentconfigurations, H defines a Grassmannian manifold. Therefore the modelhas a continuous Higgs branch

VHiggs = GrNf ,Nc 'SU(Nf)

SU(Nc)× SU(Nf −Nc)× U(1)(3.1.5)

and no Coulomb vacuum.1 The gauge symmetry is completely broken whilean exact global SU(Nc)C+F color-flavor symmetry remains unbroken. Wedefine a dual theory by the same Lagrangian (3.1.2) with different gaugegroup U(Nc) (Nc ≡ Nf −Nc) and the same number of flavors. From the lastexpression of (3.1.5) the Higgs branch of the dual theory is obviously identicalto that of the original theory. We refer to this duality as “Seiberg-like dual”,or simply “Seiberg dual”. In the Hanany-Witten type D-brane realization[87] this duality can be understood as exchange of two NS5-branes, while theoriginal Seiberg duality in N = 1 theory can be understood by this procedurewith one NS5-brane rotated.

1 In the context of N = 2 supersymmetry the Higgs branch is the cotangentbundle over the Grassmannian manifold, T ∗GrNf ,Nc [84, 85] obtained as a hyper-Kahler quotient [86].

34

The Bogomol’nyi completion for independent configurations reads in thisnotation:

T =

∫d2x

[1

2e2

∣∣∣∣F 012 − e2

(H†At

0HA −v2

√2N

)∣∣∣∣2+ 4 |DzH|2 +

1

2g2

∣∣∣F a12 − g2H†At

aHA

∣∣∣2 − v2

√2N

F 012

]≥ − v2

√2N

∫d2xF 0

12 , (3.1.6)

where a complex coordinate z ≡ x1 + ix2 has been introduced. We obtainthe BPS vortex equations

DzH = 0 , (3.1.7)

F 012 −

e2

√2N

(tr (HH†)− v2

)= 0 , (3.1.8)

F a12t

a − g2

2

[HH† − 1N

Ntr (HH†)

]= 0 . (3.1.9)

To simplify the algebra, we take the gauge couplings to be equal e = g2.The BPS equations can be casted in a simple matrix form:

DzH = 0, F12 −g2

2

(v2 1NC

−H H†)

= 0. (3.1.10)

Actually these equations possess a continuous moduli space of solutions.Since every solution is characterized by a topological (winding) number val-ued in π1(U(Nc)) = Z, the moduli space is divided into topological sec-tors. Upon having introduced the complex parametrization of the plane,z = x1 + ix2, coordinates on the moduli space are conveniently collected ina unique mathematical object H0(z), a holomorphic Nc × Nf matrix calledthe moduli matrix which is defined by [81]

H = S−1(z, z)H0(z), W1 + iW2 = −2 i S−1(z, z) ∂zS(z, z) (3.1.11)

where S(z, z) is an Nc × Nc invertible matrix. The elements of H0(z) arepolynomials in z whose coefficients are good coordinates (in the sense of[88]) on the moduli space. A configuration has winding number k if

detH0H†0 = O(|z|2k) (3.1.12)

2In the next Chapters we will generalize some discussions to the case of differentcouplings.

35

for large z. The tension of the corresponding vortex configuration is givenby Eq. 3.1.63:

T = 2πv2 k

N(3.1.13)

From the definition (3.1.11) one sees that H0(z) and S(z, z) are onlydetermined up to the so-called V -equivalence given by

H0(z)→ V (z)H0(z), S−1(z, z)→ S−1(z, z)V (z)−1 (3.1.14)

where V (z) is a GL(Nc,C) matrix whose elements are polynomials in z.Eq. (3.1.12) implies that if one fixes the winding number k, then detV mustbe constant.

The first BPS equation is automatically solved by the ansatz (3.1.11),while the second can be rewritten as [81]

∂z(Ω−1∂zΩ) =

g2

4

(v21NC

− Ω−1H0H†0

), (3.1.15)

where Ω ≡ S(z, z)S†(z, z). We will refer to Eq. (3.1.15) as the master equa-tion, and assume that it has a unique solution with a boundary conditionΩ → v−2H0H

†0. This assumption has only been proven in the abelian case

and for vortices on compact Riemann surfaces; however, there are argumentsthat it extends to general vortices on C (see [75] for a more detailed discus-sion).

The moduli space of the vortex equations Eq. (3.1.10) is obtained as thequotient space Mtotal = H0(z)/GL(N,C). This space is infinite dimen-sional and can be decomposed into topological sectors according to the vortexnumber k. The k-th topological sectorMN,k, the moduli space of k vortices,is determined by the condition that detH0(z) is of order zk:

MN,k 'H0(z)

∣∣∣ detH0(z) = O(zk)/V (z). (3.1.16)

Notice that the rank of H0 gets reduced by one at vortex positions ziwhen all the vortices are separated, zi 6= zj for i 6= j. A constant vectordefined by

H0(z = zi)~φi = 0 (3.1.17)

3The tension of the ANO vortex is, in this notation 2πk, which differs from(1.2.8) by a factor of 1/N . The difference is simply due to the different normal-ization a the U(1) generator

36

is associated with each component vortex at z = zi. An overall constantof ~φi cannot be determined from Eq. (3.1.17) so we should introduce anequivalence relation “∼”, given by

~φi ∼ λ ~φi, with λ ∈ C∗. (3.1.18)

Thus, each vector ~φi takes a value in the projective space

CPN−1 = SU(N)/[SU(N − 1)× U(1)]. (3.1.19)

This space can be understood as a space parameterized by Nambu-Goldstonemodes associated with the symmetry breaking,

SU(N)G+F → U(1)× SU(N − 1), (3.1.20)

caused by the presence of a vortex [7, 8, 16, 89, 10, 75]. We call φi theorientational vector.

3.2 Kahler quotient construction

Following [75], it is possible to organize the moduli into a Kahler quotient[86]. First, let us write the moduli matrix as

H0(z) = (D(z),Q(z)) (3.2.1)

where D(z) is an Nc×Nc matrix and Q(z) is a Nc×Nc matrix (Nc ≡ Nf−Nc).Defining

P (z) ≡ det D(z), (3.2.2)

we set degP (z) = k, whereas all other minor determinants of H0(z) havedegree at most k − 1: this guarantees that Eq. (3.1.12) is satisfied and thewinding number is equal to k. One can show that only a subset of theminor determinants are independent, namely P (z) and the determinants ofthe minor matrices obtained by substituting the r-th column of D(z), r =1, . . . , Nc, with the A-th column of Q(z), A = 1, . . . , Nc. The matrix havingthese minor determinants as its (r A) elements is denoted by F(z),

FrA =Nc∑k=1

QkA (Cof D)rk = P (z)Nc∑k=1

QkA (D−1)rk,

F(z) = P (z) D−1 Q(z). (3.2.3)

37

Consider the equation

D(z)~φ(z) = ~J(z)P (z) = 0 mod P (z) (3.2.4)

for ~φ modulo P (z), where the components of ~φ are polynomials at most of

degree zk−1. We find k linearly independent such vectors ~φi(z), i = 1, . . . , k,

each of which is a solution with a suitable ~Ji(z). In matrix form,

D(z)Φ(z) = J(z)P (z) = 0 mod P (z) (3.2.5)

where Φ(z) and J(z) are Nc × k matrices made of the ~φi and the ~Jirespectively. We are naturally free to choose a basis in the linear space ofsolutions to Eq. (3.2.4) using the equivalence relation

Φ(z) ∼ Φ(z)V−1, V ∈ GL(k,C). (3.2.6)

Since Φ(z) has the maximal rank, the ~φi being linearly independent, theGL(k,C) action is free4

Φ(z) = Φ(z)V−1 ⇒ V−1 = 1k. (3.2.7)

Now consider the product z ~φi(z), whose degree is in general less than orequal to k. The polynomial division by P (z) leads to a constant quotient

and a remainder that must be a linear combination of ~φi, as it must satisfyEq. (3.2.4). In matrix form, these are summarized as

zΦ(z) = Φ(z)Z + ΨP (z). (3.2.8)

By multiplying D(z)/P (z) from the left and using Eq.(3.2.5), we also find,

zJ(z) = J(z)Z + D(z)Ψ. (3.2.9)

This defines uniquely the constant matrices Z and Ψ, of sizes k × k andNc × k respectively. They enjoy an equivalence relation due to (3.2.6)

(Z, Ψ) ∼(V ZV−1, ΨV−1

), V ∈ GL(k,C), (3.2.10)

where the GL(k,C) action is free (Eq. (3.2.7)). Eigenvalues of Z describek positions of vortices, and thus there is an equality P (z) = det D(z) =det(z−Z). Roughly speaking, each column of Ψ parametrizes an orientationin CPNc−1 of each corresponding vortex. This would be the whole story in

4A group G is said to act freely on a space M , if for any point x ∈M , g x = x(g ∈ G) implies g = 1.

38

the local case, but in the semi-local case there are extra moduli coming fromQ(z).

Using the relation

D(z)F(z) = Q(z)P (z) (3.2.11)

which follows from Eq. (3.2.3) and the condition deg F(z)rA ≤ k−1 one findsthat the columns of F(z) are linear combinations of those of Φ(z),

F(z) = Φ(z)Ψ, (3.2.12)

where Ψ is a k×Nc constant matrix. Comparing Eq. (3.2.5) and Eq. (3.2.11)we find

Q(z) = J(z)Ψ. (3.2.13)

As Φ(z) is defined modulo equivalence relation Eq. (3.2.6), it follows that Ψis also defined up to

Ψ ∼ V Ψ, V ∈ GL(k,C). (3.2.14)

Having identified the moduli space parameters with the space defined byEqs. (3.2.10) and (3.2.14), it is straightforward to identify the same spaceas the Higgs branch of a (1+1)-dimensional, N = 1 supersymmetric, non-Abelian U(k) gauge theory, with Nc fundamental hypermultiplets Ψ, Nc

anti-fundamental multiplets Ψ, and an adjoint multipet Z. The theory hasalso a positive FI terms, which put the theory on the Higgs Branch. TheD-term constraint is given by:

D = [Z†,Z] + Ψ†Ψ− ΨΨ† − r = 0. (3.2.15)

The constraint, as usual for a supersymmetric theory, can be considered asa fixing condition for the complexified part of the gauge symmetry. The 2-dimensional FI term is related to the 4-dimensional gauge coupling by therelation:

r =2π

g2(3.2.16)

3.3 Fundamental vortex revised

Let us first discuss again single non-Abelian vortex in U(2) gauge theory,within the moduli matrix formalism. We will obtain again the results ofSection (1.2).

39

The condition on the moduli matrix H0 is detH0 = O(z). Modulo V -equivalence relation Eq. (3.1.14), the moduli matrix can be brought to oneof the following two forms [89]:

H(1,0)0 (z) =

(z − z0 0−b′ 1

), H

(0,1)0 (z) =

(1 −b0 z − z0

)(3.3.1)

with b, b′ and z0 complex parameters. Here z0 gives the position moduliwhereas b and b′ give the orientational moduli as we see below. The twomatrices in Eq. (3.3.1) describe the same single vortex configuration but intwo different patches of the moduli space. Let us denote them U (1,0) = z0, b

′and U (0,1) = z0, b. The transition function between these patches is given,except for the point b′ = 0 in the patch U (1,0) and b = 0 in U (0,1), by aV -transformation of the form [89]

V =

(0 −1/b′

b′ z − z0

)∈ GL(2,C). (3.3.2)

This yields the transition function

b =1

b′, (b, b′ 6= 0). (3.3.3)

b′ and b are seen to be the two patches of a CP 1, leading to the conclusionthat the moduli space of the single non-Abelian vortex is

MN=2,k=1 ' C×CP 1, (3.3.4)

where the first factor C corresponds to the position z0 of the vortex.The same conclusion can be reached from the orientation vector, defined

by H0(z = z0) ~φ = 0: ~φ is given by

~φ ∼(

1b′

)∼(b1

). (3.3.5)

We see that the components of ~φ are the homogeneous coordinates of CP 1;b, b′ are the inhomogeneous coordinates.5

The individual vortex breaks the color-flavor diagonal symmetry SU(2)c+f ,so that it transforms nontrivially under it. The transformation property ofthe vortex moduli parameters can be conveniently studied by the SU(2)fflavor transformations on the moduli matrix, as the color transformations

5Similarly, in the case of U(N) gauge theory, the components of ~φ correspondto the homogeneous coordinates of CPN−1.

40

acting from the left can be regarded as a V transformation. The flavor sym-metry acts on H0 as H0 → H0 U with U ∈ SU(2)f . A general SU(2) matrix

U =

(u v−v∗ u∗

), (3.3.6)

with u, v ∈ C satisfying |u|2 + |v|2 = 1, acts for instance on H(0,1)0 in

Eq. (3.3.1) as

H(0,1)0 (z) → H

(0,1)0 (z)U =

(u+ v∗ b −u∗ b+ v−v∗ (z − z0) u∗ (z − z0)

). (3.3.7)

The right hand side should be pulled back to the form H(0,1)0 in Eq. (3.3.1)

by using an appropriate V -transformation. This can be achieved by

VU H(0,1)0 (z)U =

(1 −u∗ b−v

v∗ b+u

0 z − z0

),

VU =

((u+ v∗ b)−1 0v∗ (z − z0) u+ v∗ b

)∈ GL(2,C). (3.3.8)

The SU(2)c+f transformation law of b is then

b→ u∗ b− vv∗ b+ u

, (3.3.9)

which is the standard SU(2) transformation law of the inhomogeneous coor-dinate of CP 1.6

In terms of the orientational vector ~φ in Eq. (3.3.5), the transformationlaw Eq. (3.3.9) can be derived more straightforwardly. According to the

definition Eq. (3.1.17), ~φ is transformed in the fundamental representationof SU(2)f :

~φ→ U †~φ, U ∈ SU(2)F. (3.3.10)

6 The coordinate b is invariant (more precisely the orientational vector re-ceives a global phase) under the U(1) transformations generated by n · ~σ/2,n = 1

(1+|b|2)(2< b, 2= b, |b|2 − 1). This implies the coset structure CP 1 ' SU(2)

U(1) .

41

Part II

Vortices in U(N) GaugeTheories

42

Chapter 4

Non-Abelian vortices of higherwinding numbers

The moduli subspace of k = 2 axially symmetric vortices was studied bytwo groups: Hashimoto and Tong [90] and Auzzi, Shifman and Yung [91].The former concluded that it is CP 2 by using the brane construction [7, 8]whereas the latter found CP 2/Z2 by using a field theoretical construction.This discrepancy is crucial when one discusses the interactions of vorticesbecause the latter contains an orbifold singularity (see Chapter (9) for adiscussion about the reconnection of cosmic strings). In this Section westudy the moduli space of k = 2 axially symmetric non-Abelian vortices ofU(N) gauge theories, by using the method of moduli matrix. For simplicitywe will take here Nf = Nc = N . We definitely solve the discrepancy betweenthe two results.

4.1 k = 2 Vortices in U(2) Gauge Theory

Configurations of k = 2 vortices at arbitrary positions are given by themoduli matrix whose determinant has degree two, detH0 = O(z2). By usingV -transformations Eq. (3.1.14) the moduli matrix satisfying this conditioncan be brought into one of the following three forms [10, 75]

H(2,0)0 =

(z2 − α′ z − β′ 0−a′ z − b′ 1

), H

(1,1)0 =

(z − φ −η−η z − φ

),

H(0,2)0 =

(1 −a z − b0 z2 − α z − β

). (4.1.1)

These define the three patches U (2,0) = a′, b′, α′, β′, U (1,1) = φ, φ, η, η,U (0,2) = a, b, α, β of the moduli space MN=2,k=2. The transition from

43

U (1,1) to U (0,2) is given via the V -transformation V =(

0 −1/ηη z−φ

):

a =1

η, b = − φ

η, α = φ+ φ, β = η η − φ φ. (4.1.2)

Similarly the transition from U (2,0) to U (0,2) is given by

V =

(−a′2

a′2 β′−a′ b′ α′−b′2−a′ z+a′ α′+b′

a′2 β′−a′ b′ α′−b′2

a′ z+b′ z2−α′ z−β′

), (4.1.3)

which yields

a =a′

a′2 β′ − a′ b′ α′ − b′2 , b = − b′ + a′ α′

a′2 β′ − a′ b′ α′ − b′2 , α = α′, β = β′.

(4.1.4)

Finally those between U (1,1) and U (2,0) are given by the composition of thetransformations Eq. (4.1.2) and Eq. (4.1.4). Let us now discuss the modulispace of k = 2 vortices separately for the cases where the two vortex centersare 1) distinct (z1 6= z2), and 2) coincident (z1 = z2).

1) At the vortex positions zi the orientational vectors are determined by

Eq. (3.1.17). The orientational vectors ~φi (i = 1, 2) are then obtained by

~φi ∼(azi + b

1

)∼(zi − φη

)∼(

ηzi − φ

)∼(

1a′zi + b′

). (4.1.5)

The two (i = 1, 2) parameters defined by bi ≡ a zi + b (or b′i ≡ a′ zi +b′) parameterize the two different CP 1’s separately. Conversely if the twovortices are separated z1 6= z2, then moduli parameters a, b are described byb1, b2 with positions of vortices z1, z2 as

a =b1 − b2

z1 − z2

, b =b2 z1 − b1 z2

z1 − z2

, α = z1 + z2, β = −z1 z2, (4.1.6)

(and similar relations for the primed variables). Thus in the case of sep-arated vortices b1, b2, z1, z2 (b′1, b′2, z1, z2) can be taken as appropriatecoordinates of the moduli space, instead of a, b, α, β (a′, b′, α′, β′). Thetransition functions are also obtained by applying the equivalence relationEq. (3.1.18) to Eq. (4.1.5), for instance,

bi =1

b′ i(bi, b

′i 6= 0). (4.1.7)

It can be shown that these are equivalent to Eqs. (4.1.4) by use of Eq. (4.1.6)and analogous relation for the primed parameters. The coordinates in U (1,1)

44

are also the orientational moduli. If we take b1, b′2, z1, z2 as a set of inde-

pendent moduli and substitute Eq. (4.1.6) and b2 = 1/b′2 to Eq. (4.1.2), thenwe obtain, for b1 b

′2 6= 1

φ =z2 − b1 b

′2z1

1− b1 b′2, η =

z1 − z2

1− b1 b′2b1, φ =

z1 − b1 b′2 z2

1− b1 b′2, η = − z1 − z2

1− b1 b′2b′2.

(4.1.8)

It can be seen that the representation Eq. (4.1.6) implies that U (0,2) andU (2,0) are suitable for describing the situation when two orientational moduliare parallel or nearby. On the other hand, Eq. (4.1.8) implies that U (1,1) issuitable to describe the situation when orientational moduli are orthogonalor close to such a situation, while not adequate for describing a parallelset. Therefore, the moduli space for two separated vortices are completelydescribed by the positions and the two orientational moduli b1, b2, (b′1, b

′2):

the moduli space for the composite vortices in this case is given by [90, 10]

Mseparatedk=2,N=2 '

(C×CP 1

)2/S2, (4.1.9)

where S2 permutes the centers and orientations of the two vortices.

2) We now focus on coincident (co-axial) vortices (z1 = z2), with themoduli space denoted by

MN=2,k=2 ≡MN=2,k=2

∣∣z1=z2

. (4.1.10)

As an overall translational moduli is trivial, we set z1 = z2 = 0 without lossof generality. According to Eqs. (4.1.6) and Eq. (4.1.8), all points in themoduli space tend to the origin of U (1,1) in the limit of z2 → z1, as long asb1 and b2 take different values. A more careful treatment is needed in thiscase. In terms of the moduli matrix, the condition of coincidence is given bydetH0(z) = z2. We have

α = 0, β = 0 ,φ = −φ, φ φ− η η = 0

and α′ = 0, β′ = 0 ,(4.1.11)

in U (2,0), U (1,1) and U (0,2), respectively. MN=2,k=2 is covered by the reducedpatches U (2,0), U (1,1) and U (0,2), defined by the moduli matrices

H(2,0)0 =

(z2 0

−a′ z − b′ 1

), H

(1,1)0 =

(z − φ −η−η z + φ

),

H(0,2)0 =

(1 −a z − b0 z2

). (4.1.12)

45

The following constraint exists among the coordinates in U (1,1):

φ2 + η η = 0. (4.1.13)

The transition functions between U (0,2) and U (1,1) are given by

a =1

η, b =

φ

η= −η

φ, (4.1.14)

and those between U (0,2) and U (2,0) by

a = − a′

b′2, b =

1

b′. (4.1.15)

All the patches defined by Eq. (4.1.12) are parameterized by two in-dependent complex parameters. The reduced patches U (2,0) and U (0,2) arelocally isomorphic to C2 with a, b or a′, b′ being good coordinates. However,U (1,1) suffers from the constraint Eq. (4.1.13) which gives the A1-type (Z2)orbifold singularity at the origin and therefore U (1,1) ' C2/Z2 locally (SeeEq. (4.1.18), below). Note that the moduli matrix H0(z) is proportional tothe unit matrix at the singularity: H0(z) = z 12. This implies that config-urations of the physical fields (H and F12) are also proportional to the unitmatrix where the global symmetry SU(2)G+F is fully recovered at that sin-gularity. The full gauge symmetry is also recovered at the core of coincidentvortices.

Remark: A brief comment on the orientational vectors. We could extract apart of moduli in the moduli matrix as the orientational vector at z = 0, as in thecase of separated vortices discussed above:

~φ ∼(

1b′

)∼(

η−φ

)∼(φη

)∼(b1

). (4.1.16)

From the identification ~φ ∼ λ ~φ (λ ∈ C∗) with the transition functions givenin Eqs. (4.1.14) and Eq. (4.1.15), we find that the orientational moduli againparameterizes CP 1. However, the orientational vectors in Eq. (4.1.16) are notsufficient to pick up all the moduli parameters in the moduli matrix H0. Forinstance a is lost in the U (0,2) patch. It is even ill-defined at the singular point, asH

(1,1)0 (z = 0) = 0.

To clarify the whole structure of the space MN=2,k=2, let us define newcoordinates, solving the constraint Eq. (4.1.13)

X Y ≡ −φ, X2 ≡ η, Y 2 ≡ −η. (4.1.17)

This clarifies the structure of the singularity at the origin. The coordinates(X, Y ) describe the patch U (1,1) correctly modulo Z2 identification

(X, Y ) ∼ (−X,−Y ). (4.1.18)

46

(a, b) (a′, b′) (X, Y )

(a, b) = ** (−a′/b′2, 1/b′) (−1/Y 2, X/Y )

(a′, b′) = (−a/b2, 1/b) ** (1/X2, Y/X)

(X, Y ) = (±ib/√a,±i1/√a) (±1/√a′,±b′/

√a′) **

Table 4.1: Transition functions between the three patches U (2,0), U (1,1) and U (0,2).

Using the transition functions Eq. (4.1.14) and Eq. (4.1.15), the three lo-cal domains are patched together as in Table (4.1). In terms of the newcoordinates (X, Y ), the orientational vector defined at z = 0 is given by

~φ ∼(

1b′

)∼(XY

)∼(b1

)(4.1.19)

with ~φ ∼ λ ~φ (λ ∈ C∗). This equivalence relation recovers the transitionfunctions between b, b′ and (X, Y ) in the Table (4.1). These are coordinateson the CP 1 as was mentioned above. But this CP 1 is only a subspace of themoduli space MN=2,k=2.

The full space MN=2,k=2 can be made visible by attaching the remainingparameters a, a′ to CP 1. We arrange the moduli parameters in the threepatches U (2,0), U (1,1) and U (0,2) as a′

1b′

∼ 1

XY

∼ −ab

1

, (4.1.20)

respectively, with the equivalence relation “∼”, defined by φ0

φ1

φ1

∼ λ2 φ0

λφ1

λφ1

with λ ∈ C∗. (4.1.21)

All the transition functions in Table (4.1) are then nicely reproduced. Theequivalence relation Eq. (4.1.21) defines a weighted complex projective spacewith the weights (2, 1, 1). We thus conclude that the moduli space for the co-incident (coaxial) k = 2 non-Abelian vortices is a weighted projective space,

MN=2,k=2 ' WCP 2(2,1,1). (4.1.22)

While the complex projective spaces with common weights, CP n, aresmooth, weighted projective spaces have singularities. In fact, we have

47

shown that U (1,1) ' C2/Z2, and it has a conical singularity at the ori-gin by (1, X, Y ) ∼ (1,−X,−Y ), whose existence was first pointed out byASY [91]. The origin of the conical singularity can be seen clearly from theequivalence relation Eq. (4.1.21). As mentioned above the transition func-tions in Table (4.1) are reproduced via the equivalence relation Eq. (4.1.21).In fact, one finds that λ = 1

Xgives (λ2, λX, λ Y ) = (a′, 1, b′) and λ = 1

Y

gives (λ2, λX, λ Y ) = (−a, b, 1). Note that λ in the equivalence relationEq. (4.1.21) is completely fixed in the patches U (2,0) and U (0,2) given inEq. (4.1.20). However, in the middle patch (1, X, Y ) we still have a free-dom λ = −1 which leaves the first component 1 untouched, but changes(1, X, Y )→ (1,−X,−Y ).

The relation between our result and that in [91] becomes clear by definingξ2 ≡ φ0 (ξ = ±√φ0). Now the parameters (ξ, φ1, φ1) have a common weightλ, so they can be regarded as the homogeneous coordinates of CP 2. But onemust identify ξ ∼ −ξ clearly, and this leads to the Z2 quotient (ξ, φ1, φ1) ∼(ξ,−φ1,−φ1). Therefore our moduli space can also be rewritten as

MN=2,k=2 ' CP 2/Z2 (4.1.23)

reproducing the result of [91]. Such a Z2 equivalence, however, does notchange the homotopy of MN=2,k=2: it remains CP 2 [90]. This is analogousto an (x, y) ∼ (−x,−y) equivalence relation (with real x, y) introduced in onelocal coordinate system of CP 1 (a sphere), which leads to a sphere with twoconic singularities (a rugby ball, or a lemon) instead of the original smoothsphere. 1

4.1.1 SU(2) transformation law of co-axial k = 2 vor-tices

The complex projective space CP 2 ' SU(3)SU(2)×U(1)

with the Fubini-Study metric

has an SU(3) isometry. On the other hand, the weighted projective spaceWCP 2

(2,1,1) can have an SU(2) isometry at most due to the difference of the

weights Eq. (4.1.21). This matches with the fact that we have only SU(2)G+F

symmetry acting on the moduli space. In this Section we investigate theSU(2)G+F transformation laws of the moduli for the co-axial two vortices, aswas done for the fundamental vortex in Section (3.3).

Let us start with the patch U (0,2), with the moduli matrix

H(0,2)0 =

(1 −a z − b0 z2

)(4.1.24)

1For instance, it is easily seen that MN=2,k=2 ' CP 2/Z2 remains simply con-nected. The higher homotopy groups cannot change by a discrete fibration [92].

48

An SU(2) matrix U like Eq. (3.3.6) acts on the above H0 from the right,

H(0,2)0 → H

(0,2)0 U =

(v∗ a z + u+ v∗ b −u∗ a z + v − u∗ b−v∗ z2 u∗ z2

). (4.1.25)

A V -transformation V = V1 V2, where

V1 =

(− v∗ a(u+v∗ b)2

0

0 1

), V2 =

(z − u+v∗ b

v∗ aa

v∗ z2 v∗ a z + u+ v∗ b

)(4.1.26)

brings the result back to the upper right triangle form,

H(0,2)0 → H

(0,2)0 U ∼ V H

(0,2)0 U =

(1 − a

(u+v∗ b)2z + v−u∗ b

u+v∗ b

0 z2

). (4.1.27)

The SU(2) transformation laws of the parameters a, b are then

a→ a

(v∗ b+ u)2 , b→ u∗ b− vv∗ b+ u

. (4.1.28)

As in the previous Section, b can be regarded as an inhomogeneous coordinateof CP 1; in fact, b is invariant under a U(1) subgroup and this means that b

parameterizes CP 1 ' SU(2)U(1)

. On the other hand, the transformation law ofthe parameter a can be re-written as

a→[d

db

(u∗ b− vv∗ b+ u

)]a, (4.1.29)

showing that the parameter a is a tangent vector on the base space CP 1

parameterized by b. This is very natural. First recall the situation for sep-arated vortices. The moduli parameters are extracted from their positionsand their orientations, defined at the vortex centers. However, once the vor-tices overlap exactly (zi → zj), the positions and the orientations only donot have enough information. When l(≤ k) vortices are coincident, we need1, 2, · · · , l − 1 derivatives at the coincident point in order to extract all theinformation. They define how vortices approach each other (bi → bj) [10].

Some SU(2) action sends the points in the patch U (0,2) to where a betterdescription is in the patch U (2,0), and vice versa. Compare Eq. (4.1.28) withu = 0, v = i, with Eq. (4.1.15). This shows indeed that

U (0,2) ∪ U (2,0) ' TCP 1. (4.1.30)

Next consider the patch U (1,1) with

H(1,1)0 =

(z − φ −η−η z + φ

), φ2 + η η = 0. (4.1.31)

49

|!0|2

|!1|2

WCP 2(2,1,1)

CP 2

(1, 1) patch

(2, 0) patch

(0, 2) patch

singularity

a

b

(X1, X2, X3)

11/2

1

O

a!

b!

Figure 4.1: Toric diagram of WCP 2(2,1,1) and their three patches U (2,0), U (1,1)

and U (0,2). The diagram is drawn under a gauge fixing condition (called theD term constraint)

∑2a=0 q

a|φa|2 = 1 where U(1)C charges are qa = (2, 1, 1)for WCP 2

(2,1,1) while qa = (1, 1, 1) for the ordinary CP 2. The triangle with

the broken line and O (without singularity) denotes the ordinary CP 2.

It is convenient to rewrite this as

H(1,1)0 = z 12 − ~X · ~σ (4.1.32)

where ~σ are the Pauli matrices and

φ ≡ X3, η ≡ X1 − iX2, η ≡ X1 + iX2. (4.1.33)

X1, X2, X3 are then complex coordinates with a constraint X21 +X2

2 +X23 = 0.

To keep the form Eq. (4.1.32) under SU(2)F transformation, we perform a

V -transformation with V = U †: H(1,1)0 → U †H(1,1)

0 U . Equivalently, we studythe transformation property of the vortex under SU(2)G+F. We find

~X · ~σ → U †(~X · ~σ

)U, (4.1.34)

that is, the vector ~X transforms as an adjoint (triplet) representation, except

at ~X = 0. This last point - singular point of WCP 2(2,1,1) - or the origin of

the patch U (1,1), is a fixed point of SU(2) (a singlet). Note also that the

50

transition functions between the patches U (0,2) and U (1,1) are given by

X3 =b

a, X1 − iX2 = −b

2

a, X1 + iX2 =

1

a. (4.1.35)

The patch U (1,1) does not cover points at “infinity”, namely the subspacedefined by a = 0 in the patch U (0,2). That submanifold is nothing but CP 1

parameterized by b which is an edge of WCP 2(2,1,1). See Fig. (4.1). One can

verify that the transformation law for a, b in Eq. (4.1.28) and that for φ, η, ηin Eq. (4.1.34) are consistent through the transition function Eq. (4.1.35).These results confirm those in [91].

4.2 k = 2 Vortices in U(N) Gauge Theory

In this Section the composition of two non-Abelian vortices in a U(N) gaugetheory is systematically investigated. Up to now we made use of the directform of the moduli matrix H0(z) for studying the moduli space structure.Another method for studying the latter will be developed and used to deter-mine the moduli space below.

Let Z and Ψ be k by k and N by k constant complex matrices, respec-tively. We consider the GL(k,C) action defined by

Z→ V ZV−1, Ψ→ ΨV−1, V ∈ GL(k,C). (4.2.1)

It was shown in [75] that the moduli spaceMN,k of k vortices can be writtenas the Kahler quotient [86] defined by

MN,k ' Z,Ψ //GL(k,C), (4.2.2)

where GL(k,C) action is free on these matrices.2 The moduli space MN,k

given by the moduli matrix H0(z) in Eq. (3.1.16) and hence by the com-plex Kahler quotient in Eq. (4.2.2) is identical to that obtained by use ofthe D-brane construction by Hanany-Tong [7, 8]. The concrete correspon-dence between them is obtained by fixing the imaginary part of GL(k,C) in

2 The relation between the moduli matrix H0(z) and the two matrices (Z,Ψ)is given by the following ADHM-like equation

∇†L = 0, det(z − Z) = detH0(z), (4.2.3)

where L† ≡ (H0(z),J(z)) and ∇† ≡(−Ψ†, z − Z†

). Here J(z) is N by k matrix

whose elements are holomorphic function of z. (Z,Ψ) and J(z) can be uniquelydetermined from a given H0(z) [75].

51

Eq. (4.2.2) by the moment map [Z†,Z] + Ψ†Ψ:

MN,k ' (Z,Ψ) | [Z†,Z] + Ψ†Ψ ∝ 1k /U(k), (4.2.4)

where Z and Ψ are again in the adjoint and fundamental representations ofU(k) group, respectively.

We now use the Kahler quotient construction to generalize the discussionto general N . The authors in [90] and [91] used the expression Eq. (4.2.4) butEq. (4.2.2) is easier to deal with. Let us discuss the moduli space of co-axialvortices in terms of Eq. (4.2.2). A subspace of the moduli space MN,k forcoincident vortices at the origin of the x1-x2 plane is given by putting theconstraint det(z − Z) = zk, that is,

Tr (Zn) = 0, for n = 1, 2, · · · k. (4.2.5)

To understand the subspace clearly we need to solve the above constraint bytaking appropriate coordinates with k2 − k complex parameters.

In the case of k = 2, U(N) vortices, the constraints Eq. (4.2.5) are equiv-alent to the constraint Eq. (4.1.13) for the N = 2 case. Z can be solved inthis case as

Z = ε v vT , ε =

(0 1−1 0

), (4.2.6)

with v a column two-vector with complex components. The fact that theabove form of Z transforms as in the adjoint representation under SL(2,C) ⊂GL(2,C) means that v is a fundamental representation of SL(2,C) since 2and 2∗ is equivalent in the k = 2 case. Let us define a complex 2 by N + 1matrix by

M =(

ΨT v). (4.2.7)

The GL(2,C) = SL(2,C) × C∗ action with elements S ∈ SL(2,C) andλ ∈ C∗ read

M → SM,(

ΨT v)→(λΨT v

). (4.2.8)

The quotient by GL(2,C) results in a kind of complex Grassmannian mani-fold whose C∗ action has weights (1, · · · , 1︸ ︷︷ ︸

N

, 0). The moduli subspace of coin-

cident two vortices in U(N) gauge theory is therefore found to be a weightedGrassmannian manifold,

MN,k=2|coincident ' WGr(1,··· ,1,0)N+1,2 . (4.2.9)

52

We note again that the elements S = −12 and λ = −1 acting as(ΨT v

)→(

ΨT −v)

(4.2.10)

is precisely the Z2 action which gives orbifold singularities. Note that, al-though the ordinary complex Grassmannian manifoldGrN+1,2 ' SU(N+1)

SU(N−1)×SU(2)×U(1)

naturally enjoys an SU(N + 1) isometry, the weighted Grassmannian mani-

fold WGr(1,··· ,1,0)N+1,2 can have an SU(N) isometry at most, due to the difference

of U(1)C charges. This is consistent with the existence of the SU(N)G+F

symmetry acting on the moduli space in the U(N) case. In cases of N > 2the orbifold singularities are not isolated points but form a submanifoldgiven by v = 0, which is the ordinary complex Grassmannian manifoldGrN,2 ⊂ WGr

(1,··· ,1,0)N+1,2 reflecting the SU(N)G+F symmetry.

As an illustration consider again the case of U(2) theory. 2 by 2 matricesZ and Ψ correspond to the moduli space of k = 2 non-Abelian vortices inthe U(2) gauge theory with the equivalence relation Eq. (4.2.2). The doubleco-axial vortices are described by det (z12 − Z) = z2. This can be rewrittenas Tr Z = Tr Z2 = 0. These conditions are easily solved and we find thatthese vortices are described by the following two 2 by 2 matrices Z and Ψ:

Z = ε v vT =

(v1 v2 v2

2

−v21 −v1 v2

), Ψ =

(ψ11 ψ12

ψ21 ψ22

)(4.2.11)

where vT = (v1, v2) and ε = ( 0 1−1 0 ). These obey the equivalence relation

GL(2,C) given in Eq. (3.2.10). At this stage the matter would become simpleif we consider v rather than Z. Since Z is in the adjoint representation ofGL(2,C), v is in the fundamental representation of SL(2,C). Notice that vis not charged under the overall U(1)C ⊂ GL(2,C).

It is natural to define k (= 2) by N + 1 (= 2 + 1) matrix

M =(ΨT , v

)=

(ψ11 ψ21 v1

ψ12 ψ22 v2

). (4.2.12)

This matrix M transforms under GL(2,C) = U(1)C × SL(2,C) as follows

M =(ΨT , v

)∼(S λΨT , S v

)(4.2.13)

where S ∈ SL(2,C) and λ ∈ U(1)C. If the vector v had a charge 1 underU(1)C (that is, v → λ v), the above identification would correspond to thecomplex Grassmannian Gr3,2 ' M/GL(2,C) which is same as CP 2. Butsince v is not charged under U(1)C, the manifold is not a Grassmannian.Eq. (4.2.13) is an example of a weighted Grassmannian manifold and we

53

denote it by WGr(1,1,0)3,2 . Here the numbers (1, 1, 0) denote the U(1)C-charges

(the weights) of columns of M .

We choose appropriate GL(2,C) matrices to obtain various patches onthe moduli space for the composite vortices. Let us define the 2 by 2 minorsM[ij] and their determinants as

M[ij] =

(Mi1 Mj1

Mi2 Mj2

), τij = detM[ij]. (4.2.14)

There are 3 minors M[12], M[23] and M[13]. Using the GL(2,C), one of themcan be brought to identity. So one has 3 patches as follows.

• M[23] = 12 patch.First act S = ( τ23 0

0 1 )M−1[23] to the matrix M in Eq. (4.2.12), and after

that by λ = τ−123 :

M →(

τ13 τ23 0− τ12τ23

0 1

)→(

τ13τ23

1 0

− τ12τ223

0 1

). (4.2.15)

• M[12] = 12 patch.

First one acts S = (τ12)1/2M−1[12] to the matrix M in Eq. (4.2.12), and

after that then by λ = (τ12)−12 :

M →( √

τ12 0 −τ23√τ12

0√τ12

τ13√τ12

)→(

1 0 −τ23√τ12

0 1 τ13√τ12

). (4.2.16)

Notice that the element S = −12 with λ = −1, has not been fixed,thus one has a Z2 symmetry in this patch(

− τ23√τ12

τ13√τ12

)∼(

τ23√τ12

− τ13√τ12

)(4.2.17)

• M[13] = 12 patch.Act first by S = ( τ13 0

0 1 )M−1[13] to the matrix M in Eq. (4.2.12), and then

by λ = τ−113 :

M →(τ13 τ23 00 τ12

τ131

)→(

1 τ23τ13

0

0 τ12τ213

1

)(4.2.18)

54

The corresponding matrices ( ΨZ ) for the above three patches are summarized

as follows τ13τ23

−τ12τ223

1 00 10 0

,

1 00 1

−τ23τ13τ12

τ213

τ12−τ2

23

τ12

τ23τ13τ12

,

1 0τ23τ13

τ12τ213

0 10 0

. (4.2.19)

The leftmost one corresponds to the M[23] = 12 patch, the middle to theM[12] = 12 and the rightmost one to the M[13] = 12 patch. These must be

identified respectively with the patches: U (0,2), U (1,1) and U (2,0). The concreteidentification is(

τ12τ213τ23τ13

)=

(−ab

),

(τ13√τ12τ23√τ12

)=

(XY

),

(τ12τ223τ13τ23

)=

(a′

b′

).(4.2.20)

Now we are ready to understand a little mysterious relation Eq. (4.1.21)which gave us the weighted complex projective space WCP 2

(2,1,1). Ordinary

complex Grassmannian Gr3,2 is known to be equivalent to CP 2 and theweighted cases are quite analogous. Because all the parameters in Eq. (4.2.19)are functions of the determinants of the minorsM[12], M[23] andM[13], it wouldbe natural to consider that the manifold is naturally parameterized by them.In particular the origin of the weighted equivalence relation Eq. (4.1.21)becomes clear since τij are invariant under SL(2,C) while transforming underU(1)C as φ0

φ1

φ1

=

τ12

τ13

τ23

∼ λ2 τ12

λ τ13

λ τ23

. (4.2.21)

This is nothing but Eq. (4.1.21). The U (0,2) patch is obtained by fixing with

λ = τ−1/212 , the U (1,1) patch by λ = τ

−1/223 and the U (2,0) patch by λ = τ

−1/213 .

Thus

MN=2,k=2 ' WCP 2(2,1,1) ' WGr

(1,1,0)3,2 . (4.2.22)

4.3 Summary

The moduli subspace of two co-axial vortices in the U(N) gauge theories withN flavors, is found to be a weighted Grassmannian manifold, Eq. (4.2.9). Inthe case of U(2) gauge theory, it reduces to a weighted projective space

55

WCP 2(2,1,1) ' CP 2/Z2 (CP 2 homotopically), in agreement with the known

results [90, 91]. This space contains a Z2 orbifold (conic) singularity at theorigin of the (1, 1) patch. In the case of U(N) gauge theory, it containssingularities along a subspace which is GrN,2.

The presence of this kind of orbifold singularities is a general featureof weighted Grassmannian manifolds. This fact implies the necessity to re-consider the reconnection of non-Abelian vortices. This issue was studiedconsidering the moduli space of k = 2 co-axial vortices smooth everywhere[90]. We saw that this is not the case and we need to reconsider the problem.We will solve it in Chapter (9).

56

Chapter 5

Semi-local strings and lumps

Semi-local vortices interpolate between ANO vortices and sigma model lumps[74, 93], to which they reduce in two different limits. As usual their totalmagnetic flux is quantized in terms of their topological charge, however themagnetic field of a semi-local vortex does not decay exponentially in theradial direction, instead it falls off according to a power law. Moreover thetransverse size of the flux tube is not fixed but becomes a modulus. Thisfeature gives rise to questions about the stability of these objects; in [74](see also [94]) it is argued that they are stable if the quartic coupling in thepotential is less than or equal to the critical (BPS) value, i.e., if the mass ofthe scalar is less than or equal to the mass of the photon. It is possible tostudy their dynamics [95] in the moduli space approximation [96] and alsoin the lump limit [97]. It turns out that, in general, the fluctuations of somezero-modes corresponding to the global size of the configuration actually costinfinite energy. These zero-modes have to be fixed to make moduli spacedynamics meaningful.

It is natural to ask what emerges for semi-local non-Abelian vortices.The number of zero modes, namely the dimension of the moduli space, ofwinding number k was calculated to be 2kNF in [7, 8]. The problem of(non-)normalizability of these zero modes, or the construction of the effec-tive theory, has been considered in detail for N = 2 supersymmetric U(2)gauge theories with Nf = 3, 4 in [6] by Shifman and Yung. They found BPSsolutions for single non-Abelian semi-local vortices and then they used sym-metry arguments to develop an effective theory on the vortex worldsheet.They noted that single semi-local vortices have only non-normalizable zero-modes except for the position modulus: not only the size modulus, but alsothe orientational moduli undergo this pathology, which is somewhat moresurprising. This behavior is manifest because the effective worldsheet theoryis a two dimensional sigma model, whose target space has a divergent metric

57

unless an infrared regulator is provided. In this respect the geodesic motionon the moduli space seems essentially frozen, in contrast with the local case.

The aim of this Chapter is to use the moduli matrix approach to study inmore detail semi-local vortices (a first application of this method to semi-localstrings is found in [81]). Our considerations here apply to vortex configura-tions of generic winding number k in any U(Nc) gauge theory with Nf flavorsand the critical quartic coupling, in the case of Nf > Nc.

5.1 Moduli spaces and quotients

To discuss the moduli space of semi-local vortices, it is much more convenientto consider the Kahler quotient construction. According to this constructionreviewed in Section (3.2), for configurations of winding number k in a U(Nc)gauge theory with Nf fundamental flavors, the moduli are conveniently col-lected into the triplet (Z,Ψ, Ψ), where Z is a k × k, Ψ an Nc × k and Ψ ak × Nc complex matrix. They are defined modulo the GL(k,C) equivalencerelation (

Z,Ψ, Ψ)∼(VZV−1,ΨV−1,VΨ

), V ∈ GL(k,C). (5.1.1)

The GL(k,C) action is free on the set Z,Ψ, Ψ, in fact it is even free onthe subset Z,Ψ. This is enough to define a good Kahler quotient [86] and,indeed, the k-vortex moduli space turns out to be

MNc,Nf ;k = (Z,Ψ, Ψ) : GL(k,C) free on (Z,Ψ)/GL(k,C). (5.1.2)

Let us, instead, consider the quotient

MNc,Nf ;k ≡ Z,Ψ, Ψ/GL(k,C) (5.1.3)

where the GL(k,C) acts freely. Now, while any free action of a compactgroup produces a reasonable quotient, this is not always the case for anon-compact group, like GL(k,C). The corresponding quotient can indeedpresent some pathologies: in particular it becomes typically non-Hausdorff[22]. The absence of the Hausdorff property may appear to be just a math-ematical detail but it is actually crucial to the physics. As is well known,in certain kinematical regimes, the dynamics of solitons (and vortices amongthem) can be described by geodesic motion on their moduli space. If thismoduli space is non-Hausdorff, two distinct points may happen to lie at zerorelative distance, in such a way that a geodesic can end at, or simply touch,

58

both of them at once. This is physically meaningless because two differ-ent points in the moduli space correspond to two distinguishable physicalconfigurations.

In general, it is possible to “regularize” a non-Hausdorff quotient space(i.e., to make it Hausdorff) by removing some points (GL(k,C) orbits forus). This can be done in more than one way; indeed, as intuition suggests,if two distinct points do not have disjoint neighborhoods, one could removeeither one point or the other.

Let us describe this phenomenon from another point of view. It is possibleto associate to the quotient Eq. (5.1.3) a moment map D,

D = [Z†,Z] + Ψ†Ψ− ΨΨ† − r. (5.1.4)

Setting D = 0, which corresponds to fixing the imaginary part of the gaugegroup GL(k,C) = U(k)C, and further dividing by the real part, which isU(k), leads to the symplectic quotient

Z,Ψ, Ψ|D = 0/U(k). (5.1.5)

Now, the symplectic quotient depends on the value of r in Eq. (5.1.4). Inparticular, its topology is related to the sign of r. There are three cases:r > 0, r < 0 and r = 0, which represent three possible regularizations of thespace (5.1.3), i.e., three possible ways to obtain Hausdorff spaces. In fact,if we choose r = 0, the point (Z,Ψ, Ψ) = (0, 0, 0), which would be a fixedpoint of U(k), will be an element of (5.1.5). This point would have to beexcluded by hand. It corresponds to a small lump singularity as discussedbelow. The choice r 6= 0 guarantees a non-singular space automatically.

A large class of examples of such quotients consists of weighted projectivespaces. Consider for instance the simple example WCP 1

(1,−1). This is the

space y1, y2/C∗ defined by the equivalence relations (y1, y2) ∼ (λy1, λ−1y2),

λ ∈ C∗. After removing the origin (0, 0) the remaining sick points are (0, y2)and (y1, 0), which are each in every open neighborhood that contains theother. The two possible regularizations are:

i) WCP 1(1,−1) = (y1, y2)|y1 6= 0/C∗

Introducing the moment mapD = |y1|2−|y2|2−r with r > 0, WCP 1(1,−1)

is seen to be equivalent to the symplectic U(1) quotient D = 0/U(1).We have introduced a notation in which the underlined coordinates arethe ones which cannot all vanish. This is because D restricted to asingle C∗ orbit,

D(λ) = |λ|2|y1|2 − |λ|−2|y2|2 − r, (5.1.6)

59

is a monotonic function of |λ|; for |λ| → +∞, D goes to +∞ andfor |λ| → 0 it goes to −∞ or −r if y2 6= 0 or y2 = 0, respectively.This implies that there is a unique value of |λ| which gives D = 0,unambiguously fixing the imaginary part of U(1)C = C∗.

Note that the C∗ action is free on the set y1, as the point y1 = 0 isexcluded.

ii) WCP 1(1,−1) = (y1, y2)|y2 6= 0/C∗

This is, instead, equivalent to the symplectic U(1) quotient obtainedby setting r < 0 in the moment map of i).

Now the C∗ action is free on the set y2.

Both i) and ii) turn out to be isomorphic to C, but, as mentioned above, twodifferent regularizations of a complex quotient lead in general to differentspaces. Note that in the case with r = 0, the fixed point (0, 0) will be asolution of (5.1.6) and the resulting space will be a singular conifold. Thisconifold is resolved into a regular space by setting r 6= 0. Similar phenomenaoccur in the general case of (5.1.5).

Based on considerations similar to those above, we are led to claim that,for any k, the k-vortex moduli spaces of two Seiberg-like dual theories (inthe sense of Section 3.1) correspond to the two different regularization of theparent space in Eq. (5.1.3); these regularized spaces appear after a symplecticreduction as the quotients Eq. (5.1.5) with r > 0 and r < 0 respectively. In-deed, in the Seiberg dual theory the representations of the moduli Z,Ψ, Ψunder GL(k,C) are replaced by their complex conjugates, which is formallyequivalent to flipping the sign of the Fayet-Iliopoulos parameter in Eq. (5.1.4).

In fact, adding the condition that GL(k,C) is free on the subset Z,Ψ(resp. Z, Ψ) to Eq. (5.1.3), as imposed by the moduli matrix constructionof Section (3.1), turns out to be equivalent to selecting the specific regular-ization corresponding to the symplectic quotient (5.1.5) with r > 0 (resp.r < 0), that was first found in brane theory [7, 8]. The quotient

MNC,Nf ;k = (Z,Ψ, Ψ) : GL(k,C) free on (Z,Ψ)/GL(k,C). (5.1.7)

is isomorphic to the symplectic quotient

(Z,Ψ, Ψ) : D = [Z†,Z] + Ψ†Ψ− ΨΨ† − r = 0/U(k). (5.1.8)

with r > 0.Obviously, an analogous result holds with the following substitutions:

1. Nc → Nc (= Nf −Nc)

60

2. GL(k,C) free on (Z,Ψ)→ GL(k,C) free on (Z, Ψ)

3. r > 0→ r < 0.

5.2 The lump limit

Lumps are well-known objects, arising as static finite energy configurations ofcodimension two in non-linear sigma models (see, for example, [98]). Lumpstypically are partially characterized by a size modulus, which in particularimplies that the set of lump solutions is closed with respect to finite rescaling.Formally one would also include the solution with vanishing size modulus,but a physical configuration of zero width (and an infinitely spiked energydensity) makes no sense and must be discarded. Such limiting situations areknown as small lump singularities, and they actually represent singularitiesin the moduli space of lumps, which is then geodesically incomplete. Incontrast, semi-local vortices do not present this kind of pathology becausethey have a minimum size equal to the ANO radius 1/(gv).

The situation is particularly clear for CP 1 lumps (related to our modelwith Nc = 1 and Nf = 2). In fact, the set of lump configurations in the twodimensional CP 1 sigma model was found to be in one-to-one correspondencewith the set of holomorphic rational maps of the type [99]

R(z) =p(z)

q(z)(5.2.1)

where p(z) and q(z) are two polynomials with no common factors and deg p <deg q. The topological charge of the configuration π2(CP 1) = Z is given bydeg q. It is clear that, in order to define a fixed topological sector of thelump moduli space, one must consider the space of pairs of polynomialsE = (p(z), q(z)) of appropriate degree, subject to the constraint that theresultant is non-vanishing,

Res [p(z), q(z)] 6= 0, or, |p(z)|2 + |q(z)|2 6= 0 ∀z. (5.2.2)

This condition excludes the singular points at which p(z) and q(z) share acommon factor, implying that the corresponding state is not physical1.

In the limit of infinite gauge coupling, g2 → ∞, our model Eq. (3.1.2)reduces to a sigma model whose target space is the Higgs branch VHiggs =

1These considerations can be extended also to general CPn lumps using holo-morphic rational maps.

61

GrNf ,Nc . On the other hand, semi-local vortices therein reduce, upon com-pactification of the z plane, to Grassmannian lumps [7, 8, 100], which aretopologically supported by π2(GrNf ,Nc) = Z. A rational map in this case is

extended to holomorphic NC × NC matrix given by

R(z) ≡ 1

D(z)Q(z) =

F(z)

P (z)= Ψ

1kz − Z

Ψ, (5.2.3)

which is invariant under the V -transformation (3.1.14) and gives a holomor-phic map from S2 to GrNf ,Nc . Since GrNf ,Nc = GrNf ,Nf−Nc , the Seiberg dualtheory of Eq. (3.1.2) is the same Grassmannian sigma model in the dualinfinite gauge coupling limit, g2 →∞; moreover, its semi-local vortex config-uration tends to the same lump solutions. Actually, the extended rationalmap in the last form in Eq.(5.2.3), which is obtained by using Eq.(3.2.13) andEq.(3.2.9), is manifestly invariant under the Seiberg-like duality. Of coursethe two dual limits cannot physically co-exist: here we are interested in themathematical correspondences among moduli spaces of topological string-likeobjects of two dual theories in the various limits.2

In the end we expect the two dual vortex moduli spaces to be deformedand/or modified in the (respective) infinite gauge coupling limits in such away that they reduce to the same moduli space of Grassmannian lumps.Indeed, in the lump limit, the master equation (3.1.15) can be solved alge-braically by

Ω(z, z) = v−2H0(z)H†0(z). (5.2.4)

detH0H†0 must be non-vanishing in order to have non-singular configurations.

A set of parameters for which detH0H†0 = 0 at some point on the z plane

corresponds to a small lump singularity which must be discarded. In terms ofR(z), such unphysical singularities can be avoided by means of the constraint

∀z : |P (z)|2 det(1NC

+ R(z)R†(z))6= 0, (5.2.5)

that is nothing but the generalization of Eq. (5.2.2), to which it correctlyreduces for Nc = 1 and Nf = 2. This nicely completes the extension of theholomorphic rational map approach to Grassmannian lumps.

It is possible to show that the “lump” condition detH0H†0 6= 0 is equiv-

alent to statement that (Z, Ψ) is GL(k,C) free, so that the moduli space of

2 A Seiberg-like dual pair of solitons was previously found for domain wallsolutions [101, 102] and was then nicely understood in a D-brane configurationsby exchanging of positions of two NS5-branes along one direction [103].

62

k-lumps is given by:

MlumpNc,Nf ;k

=

(Z,Ψ, Ψ) : GL(k,C) free on (Z,Ψ) and (Z, Ψ)/GL(k,C)

= MNc,Nf ;k ∩MNc,Nf ;k. (5.2.6)

Namely, the moduli space of k-lumps is the intersection of the moduli spaceof k-vortices in one theory with that of the Seiberg dual. The physical in-terpretation is the same as for singularities in the CP 1 lump moduli space.Increasing g2 the semi-local vortex moduli space is deformed and approachesthat of Grassmannian lumps and, in the infinite coupling limit, it only de-velops small lump singularities, as expected. These singularities correspondexactly to the presence of local vortices, whose sizes, 1/(gv), shrink to zeroin the infinite gauge coupling limit. The same occurs in the Seiberg dualtheory. What is left after the removal of the singular points is nothing butthe intersection of the two vortex moduli spaces. In other words, the modulispace of semi-local vortices in each dual theory is given by the same modulispace of lumps in which we “blow-up” the small lump singularities with theinsertion of the local vortex moduli subspace of the respective theory. Fromthese considerations it is easy to convince ourselves that (5.2.6) is correct:taking the intersection in (5.2.6) eliminates the local vortices of both dualtheories, leaving us with the moduli space of lumps.

In terms of the symplectic quotient (5.1.8), one must identify the param-eter r with the gauge coupling of the four dimensional gauge theory as inEq. 3.2.16 r = 2π/g2. As one may expect, the lump limit is seen to be for-mally equivalent to taking the limit r → 0. This limit is singular, in fact itdevelops singularities that correspond to the already mentioned small lumpsingularities. We can write for the moduli space of lumps:

MlumpNc,Nf ;k

= (Z,Ψ, Ψ) :

: D = [Z†,Z] + Ψ†Ψ− ΨΨ† = 0, U(k) free /U(k),

(5.2.7)

were we have excluded “by hand” the small lump singularities by consideringonly points for which the U(k) action is free3.

Coming back to our example WCP 1(1,−1), from Section 5.1, we see that

we must take away both of the pathological points that spoil the Hausdorffproperty, instead of only one. The net result is the intersection of the tworegularized spaces WCP 1

(1,−1) and WCP 1(1,−1), e.g. C∗. This is also the space

that we obtain if we eliminate the singularity of the conifold, in agreementwith the general statement Eq. (5.2.7).

3A free quotient of a compact group is always smooth.

63

The moduli space duality and the lump limit are then summarized by thefollowing “diamond” diagram:

MNc,Nf ;k

''OOOOOOOOOOO

wwoooooooooooo

MNc,Nf ;k

g2→∞ ''OOOOOOOOOOOSeiberg duality //oo MNc,Nf ;k

g2→∞wwooooooooooo

MlumpNc,Nf ;k

5.3 Some examples

In this Section we consider the topological sector k = 1, which consists offundamental (single) semi-local vortices and lumps. The basic mathematicalobjects in this case are the weighted projective spaces with both positive andnegative weights. For these we adopt the notation WCP n−1[Qw1

1 , . . . , Qwll ],

where the Qi, i = 1, . . . , l(≤ n), represents the weight and wi the number ofhomogeneous coordinates carrying that weight; clearly

∑li=1wi = n.

This particular kind of toric variety plays a fundamental role in gaugedsigma models in two dimensions [22] and their solitons [93]. As was notedin [22], when the set of weights includes both positive and negative integers(recall that multiplying all of the weights by a common integer number hasno effect), the space is non-Hausdorff. There are two possible regularizations(in the sense of Section (5.1)), which correspond to eliminating the subspacewhere either all positively charged or all negatively charged coordinates van-ish.

Looking at Eq. (5.1.1), it is easy to see that Z “decouples”, in the sensethat the GL(1,C) = U(1)C = C∗ acts trivially on it; indeed(

Z,Ψ, Ψ)∼(Z, λ−1Ψ, λΨ

), λ ∈ C∗. (5.3.1)

Although we shall keep the same notation for the moduli spaces, they willbe intended as the internal moduli spaces from now on, as the positionmoduli Z ∈ C always factorize. Given this, we identify MNc,Nf ;1 with

WCPNf−1[1Nc ,−1Nc ]. We can regularize (in the sense of Section (5.1)) thisspace by insisting that Ψ 6= 0. We indicate this space with the followingnotation:

MNc,Nf ;1 = O(−1)⊕Nc → CPNc−1 ≡ WCPNf−1[1Nc ,−1Nc ] (5.3.2)

64

where O(−1) stands for the universal line bundle4. Analogously, the dualregularization is obtained by imposing Ψ 6= 0:

MNc,Nf ;1= O(−1)⊕Nc → CP Nc−1 ≡ WCPNf−1[1Nc ,−1Nc ]. (5.3.3)

Note that when Nc = Nc these spaces become non-compact (local) Calabi-Yau manifolds, which corresponds to the fact that just in this case the con-formal bound for a four-dimensional U(Nc) with N = 2 supersymmetry issaturated.

In the lump limit one must take Ψ, Ψ 6= 0:

MlumpNc,Nf ;1

= (CNc)∗ n CPNc−1

' (CNc)∗ n CP Nc−1 ' (CNc)∗ ⊕ (CNc)∗/C∗,(5.3.4)

with F n B denoting a fiber bundle with F a fiber and B a base. The C∗

acts with charges +1 and −1 on (CNc)∗ and (CNc)∗ respectively. If we define

MlumpNc,Nf ;1

≡ WCPNf−1[1Nc ,−1Nc ], (5.3.5)

we can summarize the situation with the following diamond diagram:

WCPNf−1[1Nc ,−1Nc ]

**UUUUUUUUUUUUUUUU

ttiiiiiiiiiiiiiiii

WCPNf−1[1Nc ,−1Nc ]

g2→∞ **UUUUUUUUUUUUUUUUSeiberg duality //oo WCPNf−1[1Nc ,−1Nc ]

g2→∞ttiiiiiiiiiiiiiiii

WCPNf−1[1Nc ,−1Nc ]

Let us consider some concrete examples.

• Nc = 1, Nf = 2

This is a self-dual system. From M1,2;1 = WCP 1[1,−1] one findsM1,2;1 = C for both the original and the dual theory. The modulispace of lumps is obtained by removing the small lump singularity andit is Mlump

1,2;1 = C∗. Explicitly, from the moduli matrix

H0 = (z − z0, b) ⇔

Z,Ψ, Ψ

= z0, 1, b , (5.3.6)

4The fiber of the universal line bundle at each point in CPn−1 is the line thatit represents in Cn.

65

one finds the solution (5.2.4)

Ω = |z − z0|2 + |b|2 (5.3.7)

and the non-vanishing condition is b 6= 0 (consider the point z = z0).Removing the point b = 0 from the vortex moduli space C one obtainsthe lump moduli space C∗. In summary:

WCP 1[1,−1]

&&LLLLLLLLLLL

xxrrrrrrrrrrr

C

g2→∞ &&MMMMMMMMMMMMM Seiberg duality //oo C

g2→∞xxqqqqqqqqqqqqq

C∗

• Nc = 2, Nf = 3 dual to Nc = 1, Nf = 3

We have now the “parent” moduli space, M2,3;1 = WCP 2[1, 1,−1].The two dual regularizations are M2,3;1 = C2, namely the blow-up ofthe origin of C2 with an S2 ' CP 1, and M1,3;1 = C2. The lump limit

is Mlump2,3;1 = (C2)∗, the two-dimensional complex vector space minus

the origin.

All of the moduli spaces can be found using the moduli matrix. In thelump limit, the general solution for the original theory leads

H0 =

(1 b 00 z − z0 c

)⇔

Z,Ψ, Ψ

=

z0,

(−b1

), c

,

(5.3.8)

det Ω|z=z0 = |c|2(1 + |b|2) (5.3.9)

and so the determinant vanishes, indicating a small lump singularity,on the blown-up 2-sphere c = 0. Removing this 2-sphere from thevortex moduli space C2 one is left with the lump moduli space (C2)∗.In order to cover the whole moduli space C2, together with that inEq. (5.3.8), one needs another patch for the moduli matrix. In the caseof the dual theory

H0 = (z − z0, b, c) ⇔

Z,Ψ, Ψ

=z0, 1,

(b, c)

, (5.3.10)

Ω|z=z0 = |b|2 + |c|2 (5.3.11)

66

and so the determinant vanishes at the point b = c = 0 ∈ C2. Thediamond diagram is

WCP 2[1, 1,−1]

&&NNNNNNNNNNNN

xxpppppppppppp

C2

g2→∞ &&NNNNNNNNNNNN Seiberg duality //oo C2

g2→∞xxpppppppppppp

(C2)∗

• Nc, Nf = Nc + 1 dual to Nc = 1, Nf

This is a generalization of the previous two examples. The parent spaceis MNc,Nc+1;1 = WCPNc [1Nc ,−1]. On one side we have MNc,Nc+1;1 =WCPNc [1Nc ,−1] = CNc , which is CNc with the origin blown up intoa CPNc−1, while on the other side the dual moduli space is simplyM1,Nc+1;1 = WCPNc [1Nc ,−1] = CNc . In the lump limit we are left

with MlumpNc,Nc+1;1 = (CNc)∗ since in the original theory

H0 =

(1Nc−1 b 0

0 z − z0 c

)⇔

Z,Ψ, Ψ

=

z0,

(−b1

), c

,

(5.3.12)

det Ω|z=z0 = |c|2(1 + |b|2) (5.3.13)

and so the small lump singularity is the blown-up CPNc−1 at c = 0 inthe vortex moduli space. Here b is a column (Nc − 1)-vector. For thedual theory

H0 = (z − z0, b) ⇔

Z,Ψ, Ψ

=z0, 1, b

, (5.3.14)

Ω|z=z0 = |b|2 (5.3.15)

which identifies the small lump singularity with the point |b| = 0 ∈CNc . Here b is a row Nc-vector. These moduli spaces are summarizedby the diamond diagram

WCPNc [1Nc ,−1]

''OOOOOOOOOOOO

wwoooooooooooo

CNc

g2→∞ ''OOOOOOOOOOOO Seiberg duality //oo CNc

g2→∞wwoooooooooooo

(CNc)∗

67

• Nc = 2, Nf = 4

This theory is again self-dual. The parent space is M2,4;1 = WCP 3[1, 1,−1,−1],which yields M2,4;1 = O(−1) ⊕ O(−1) → CP 1, namely the resolved

conifold [104]. The moduli space of lumps is Mlump2,4;1 = (C2)∗ n CP 1.

Indeed

H0 =

(1 b 0 00 z − z0 c d

)⇔

Z,Ψ, Ψ

=

z0,

(−b1

), (c, d)

,

(5.3.16)

and from the non-vanishing condition

det Ω|z=z0 = (1 + |b|2)(|c|2 + |d|2) = |Ψ|2|Ψ|2 6= 0 (5.3.17)

we recognize (c, d) as coordinates of (C2)∗ and b as the inhomogeneouscoordinate of the base CP 1. Therefore one removes the CP 1 at c =d = 0, that is Ψ = 0. In the dual theory, on the other hand, the rolesof Ψ and Ψ are interchanged and so one instead removes the CP 1 atΨ = 0, which is related by a flop transition to the CP 1 of the previousmoduli space. In the end

WCP 3[1, 1,−1,−1]

**VVVVVVVVVVVVVVVVV

ttiiiiiiiiiiiiiiiii

O(−1)⊕O(−1)→ CP 1

g2→∞ **VVVVVVVVVVVVVVVVVSeiberg duality //oo O(−1)⊕O(−1)→ CP 1

g2→∞ttiiiiiiiiiiiiiiiii

(C2)∗ n CP 1

It is suggestive to note that similar topological transitions of the typedescried above occur within the non-commutative vortex moduli space asthe non-commutativity parameter is varied [7, 8].

5.4 Non-normalizable modes

The effective theory on the vortex worldsheet is obtained via the usual proce-dure [96] of promoting the moduli to slowly varying fields on the worldsheet[16, 75, 6]. It turns out to be a two dimensional sigma model whose Kahlerpotential can be calculated from the moduli matrix [75, 82] :

K = Tr

∫d2z(v2 log Ω + Ω−1H0H

†0 +O(1/g2)

). (5.4.1)

68

This formula with explicit expression of the third term was first obtainedafter tedious calculation in terms of component fields [75], but the derivationhas been drastically simplified by using superfields [82]. By virtue of trans-lational symmetry, it is possible to show that the center-of mass parameteris always decoupled [88] and, specifically, it appears with an ordinary kineticterm whose coefficient is proportional to the total tension. The center-of-mass is a free field.

Let us concentrate on the other moduli. Analyzing the divergences ofthe Kahler potential, one can establish which moduli among 2kNf have aninfinite kinetic term in the Lagrangian and are non-normalizable. Fluctu-ations of those moduli are frozen, as well as their motion in the geodesicapproximation, while the evolution of the rest of the moduli will be allowed.Very recently it has been found that all modes become normalizable whenwe couple ore semi-local vortices to gravity [105].

The divergent terms of the Kahler potential can come only from integra-tions around the boundary |z| = L (L is a suitable infra-red cutoff), since Ωis assumed to be invertible and smooth. Remembering that Ω → v−2H0H

†0

for large z, the divergent terms can be calculated keeping only the first termin Eq. (5.4.1):

v2

∫ |z|=Ld2z log det(H0H

†0) ∼ v2

∫ |z|=Ld2z log det

(D−1H0(D−1H0)†

)= v2

∫ |z|=Ld2z log det

(1NC

+

∣∣∣∣Ψ 1

z − ZΨ

∣∣∣∣2)

= v2

∫ |z|=Ld2z

[1

|z|2 Tr∣∣∣ΨΨ

∣∣∣2 +O(|z|−3)

]= 2πv2 logLTr

∣∣∣ΨΨ∣∣∣2 + const.+O(L−1)

(5.4.2)

where we used a Kahler transformationK → K+f+f ∗, with f = v2∫d2z log det D−1.

Equation (5.4.2) means that the elements of ΨΨ are non-normalizable andshould be fixed. The number of non-normalizable parameters crucially de-pends on the rank of ΨΨ:

r ≡ rank(ΨΨ

)≤ min

(k,NC, NC

)≡ j. (5.4.3)

In the following we calculate the number of non-normalizable moduli whenthe above inequality is saturated, r = j. This happens for generic points ofthe moduli space. It follows that for particular submanifolds of the modulispace when r < j the number of normalizable parameters is enhanced.

69

Using the global symmetry SU(NC)C+F × SU(NC)F, we can always fix

ΨΨ to have the following form:

ΨΨ =

(Λr 00 0

)(5.4.4)

where Λr = diag(λ1, · · · , λr) with positive real parameters λi > 0. Notethat this symmetry of the vacuum is generally broken by the vortex andso it generates moduli for our solution. But the corresponding moduli arenon-normalizable, so that we will not count them in the following.

To proceed further we have to distinguish two cases:

• k ≤ min(NC, NC)

In this case the saturation of the inequality (5.4.3) means r = k, and the

matrices Ψ and Ψ have the following block-wise form (suffixes indicatedimensions of blocks):

Ψ =

(A[k×k]

B[(Nc−k)×k]

), Ψ = (C[k×k], D[k×(Nc−k)]), (5.4.5)

from which we find AC = Λk. Because detΛk 6= 0⇒ detA 6= 0, we cancompletely fix GL(k,C) by taking A = 1k. Thus, we obtain:

Ψ =

(1k0

), Ψ = (Λk,0). (5.4.6)

The corresponding moduli matrix is:

H0 =

(z1k − Z 0 Λk 0

0 1NC−k 0 0

). (5.4.7)

From the above we find that the normalizable moduli are all containedin the k × k matrix Z, so that:

dimMnormNC,NF;k = 2k2. (5.4.8)

From here it is easy to see that fundamental semi-local vortices, k = 1,always have only 2 real moduli, corresponding to the position on theplane. Orientation moduli are instead non-normalizable, independentlyof Nc and Nf . This behavior is very different from the local case,Nc = Nf .

70

• k ≥ min(NC, NC)

We assume NC ≤ NC without loss of generality (The results for NC ≥NC are obtained using Seiberg duality NC ↔ NC). Thus the saturation

of Eq. (5.4.3) leads k = NC. For Ψ and Ψ we have the following blockform:

Ψ =(A[Nc×Nc], B[Nc×(k−Nc)]

), Ψ =

(C[Nc×Nc] D[Nc×(Nc−Nc)]

E[(k−Nc)×Nc] F[(k−Nc)×(Nc−Nc)]

).

(5.4.9)

We see that Ψ must have rank equal to Nc so that can be always putin the following form:

Ψ = (1NC,0) (5.4.10)

via a GL(k,C) transformation. Thus Ψ and the remaining GL(k,C)symmetry are

Ψ =

(ΛNC

0E F

),

(1NC

0G H

)∈ GL(k,C). (5.4.11)

Here E can be fixed to be zero using G. We obtain:

Ψ = (1NC,0) , Ψ =

(ΛNC

00 T

), Z =

(X Y

Y W

), (5.4.12)

with the remaining u ∈ GL(k −NC,C) action:

X → X, Y → Y u, Y → u−1Y , W → u−1Wu, T → u−1T.

(5.4.13)

The normalizable parameters are contained in the Z and T matrices,from which we subtract the (k−NC)2 parameters of the remaining GLaction:

dimMnormNC,NF;k = 2

(k2 + (k −NC)(NC −NC)− (k −NC)2

)= 2

((NC + NC)k −NCNC

). (5.4.14)

The formula obtained is clearly symmetric in Nc and Nc. It is inter-esting to note, for instance, that k = Nc vortices in the U(Nc) theoryalways have 2N2

c real, normalizable moduli for any Nf ; they roughlycorrespond to the 2Nc positions in the plane and the 2Nc(Nc − 1) rel-ative sizes and orientations in the color-flavor space.

71

-10-5

05

10-10

-5

0

5

10

00.20.40.60.8

0-5

05

-10-5

05

10-10

-5

0

5

10

00.20.40.60.8

0-5

05

-10-5

05

10-10

-5

0

5

10

00.0050.01

0.015

0-5

05

(a) Lump 1 position modulus (b) Lump 2 position modulus (c) Relative orientation

Figure 5.1: Wave functions of some normalizable moduli fields for k = 2 lumps inNc = 2, Nf = 4 model. Position moduli look localized around the correspondingvortex, while relative orientation modulus lies in between.

5.5 Dynamics of the effective k = 1 vortex

theory

The presence of non-normalizable modes has a remarkable consequence inthe low-energy effective description of the vortex. As we have seen, thesemodes must be fixed, they are not dynamical. Even more remarkable are theconsequences of the presence of non-normalizable modes with the physicaldimension of a length, such as the size moduli. In this case the derivative ex-pansion in the effective action will contain, generically, powers of λ ∂, whereλ is the size moduli and ∂ is a derivative with respect to a worldsheet co-ordinate. Furthermore we must consider λ has an ultraviolet cut-off in theeffective theory on the vortex [6].

It is practically impossible to evaluate expression (5.4.1), analyticallyor even numerically, in the general case, but one can hope to do it in someparticular simple examples. In this Section we will derive the complete Kahlerpotential for a single non-Abelian semi-local vortex.

Using the set of coordinates defined by the moduli matrix formalism fora single non-Abelian semi-local vortex:

H0 =

(1Nc−1 b 0

0 z − z0 c

), (5.5.1)

we can determine the most general expression for the Kahler potential, com-patible with the SU(Nc)C+F×SU(Nc)F×U(1) isometry of the vacuum. HereNc− 1 column vector b and Nc row vector c are moduli parameters. To thisend we have to find the transformation properties of b and c under this

72

symmetries. The moduli matrix (5.5.1) transforms as:

δH0 = −H0 u+ v(u, z)H0, Tru = Tr v = 0, u† = −u

u =

ΛNc−1 + i λ1NC−1 v 0−v† −i(NC − 1)λ 0

0 0 ΛNc

, (5.5.2)

where for simplicity u is an infinitesimal SU(Nc)C+F×SU(Nc)F×U(1) trans-formation, and v(u, z) is an infinitesimal V -transformation that pulls backthe matrix H0 into the standard form of Eq. (5.5.1). After some calculationswe find the following transformation properties for the moduli parameters:

δb = ΛNc−1 · b + iNc λb− v − (v† · b)b,

δc = −i(Nc − 1)λ c + (v† · b) c− c · ΛNc, (5.5.3)

from which one can infer:

δ log(1 + |b|2

)= −(v† · b) + c.c., δ log |c|2 = (v† · b) + c.c.,

δ((1 + |b|2)|c|2

)= 0. (5.5.4)

These relations can be explained if we note that (ci, cib) (with arbitrary i,i = Nc for instance) transforms like a fundamental of SU(Nc)C+F while c asa fundamental of SU(Nc)F

5.Since the moduli parameters are zero modes related to the symmetry

breaking of SU(Nc)C+F × SU(Nc)F × U(1), the low energy action should beinvariant under the symmetry. In other words, the Kahler potential shouldbe written in terms of invariants under the transformation (up to Kahlertransformation). The most general expression for the Kahler potential, upto Kahler transformations, is thus given by

K(z0,b, c) = A|z0|2 + F (|a|2) +B log(1 + |b|2), (5.5.5)

where A and B are constants while F (|a|2) is an unknown function of theinvariant combination |a|2 ≡ (1 + |b|2)|c|2. Note that a term log |c|2 wouldalso be invariant, but can be absorbed by a redefinition of F (|a|2) and B. Theconstants and the function are determined as follows. First of all, z0 is thecenter of mass, so it is decoupled from any other modulus and its coefficientA equals half of the vortex mass, A = πv2. Next let us consider the functionF (|a|2). Now, if one fixes the orientational parameters to some constant, e.g.

5One can verify this property using the transformation laws of b and c. It is di-rectly connected with the property of lump moduli spaces expressed by Eq. (5.3.4)

73

b = 0 (|a|2 = |c|2), a non-Abelian vortex becomes simply an embedding ofan abelian vortex into a larger gauge group, therefore the Kahler potentialin (5.5.5) must reduce to that of an abelian semi-local vortex:

K(z0, 0, c) = Kabelian semi−local(z0, c) = π v2 |z0|2 + F (|c|2). (5.5.6)

It is important that the function F (|a|2) is independent of Nc(≥ 1), becausethe solution for Nc = 1 can be embedded into those for Nc > 1. Furthermore,F (|a|2), written in term of the moduli parameters defined by the modulimatrix and defined as an integral over the configurations, should be smootheverywhere. In particular, in the limit |a|2 → 0 it must be unique andequal just to that of the ANO vortex. (A numerical result for F (|a|2) withg = v2 = 1 and L = 103 is shown in Fig. (5.2)). In this limit, which can beachieved letting c → 0, the vortex reduces to a local vortex, and also theKahler potential should reduce to that of a local vortex. B is thus the Kahlerclass of the non-Abelian vortex, B = 4π/g2, as was found in Ref. [89]:

K(z0,b, 0) = Knon−Abelian local(z0,b) = πv2|z0|2 +4π

g2log(1 + |b|2).

(5.5.7)

This fixes the constants in (5.5.5). Therefore we find that the Kahler poten-tial is determined uniquely in terms of that of an abelian semi-local vortex:

K(z0,b, c) = Kabelian semi−local(z0, |a|) +4π

g2log(1 + |b|2) (5.5.8)

The function F (|a|2), being the Kahler potential for a single abelian vortex,can be computed numerically. Furthermore it is possible to find analyticallythe following behavior:

F (|a|2) ∼

π v2 log (g2v2 L2α−1)× |a|2 for |a| 1

g√v2

π v2 |a|2(

log L2

|a|2 + 1)

+ const. for |a| 1gv.

, (5.5.9)

where α is some unknown constant of O(1). The behavior at small a issimply a consequence of the smoothness of the Kahler potential at a = 0and the cut-off dependence (5.4.2). Since the coefficient of the |a|2 term isO(L2), this term is dominant even for a medium region of |a| as shown inFig. (5.2)(a). The analytic form at large a can be related to the expressionof the potential in the strong coupling limit as we will show in Eq. (5.5.19)below. It is very interesting to note that we used a very simple function to

74

2 4 6 8 10!a!2000

25003000350040004500

2 4 6 8 10!a!40

50

60

(a) Kahler potential (b) Kahler metric

Figure 5.2: (a) The red dots are numerical computations of F (|a|2), while the blueline is an interpolation done with the function shown in (5.5.10). (b) The red lineis the Kahler metric ∂a,aKabelian semi−local(z0, a) for an abelian semi-local vortex atfinite gauge coupling and is obtained from the interpolation function in (a), whilethe blue dashed line is the metric in the infinite gauge coupling limit (lump limit).The cut-off has been set to a very big value, L = 103, and g = v2 = 1.

interpolate the numerical results:

F (|a|)interp = const + π v2

(|a|2 +

α

g2v2

)(log

L2

|a|2 + αg2v2

+ 1

).

(5.5.10)

This function gives a very good interpolation with only one relevant freeparameter, α, which appears to be of order one. This interpolating function isnothing but the Kahler potential in the lump limit (Eq. (5.5.19)), regularizedat a = 0 by the introduction of a sort of UV cut-off: ρeff = α/g2v2. Thisappears to be a very nice, though empirical, way to show that local vorticesact as regularizers for small lump singularities.

Remarkably, in the case of a single vortex, the large-size limit is com-pletely equivalent to the strong coupling limit. To see this, note that therelevant quantity that triggers both limit is the following ratio:

R =ρ

ρloc

, ρloc ≡ 1/(gv), (5.5.11)

where ρ is the physical size of the semi-local vortex, and ρloc is the typicalsize of a local vortex. When the size moduli λ vanish, the physical size ρshrink to ρloc, so that R ≥ 1. In the strong coupling limit, ρloc → 0, andR → ∞. The lump limit is thus really defined by the limit R → ∞, whichcan be achieved also at finite gauge coupling, just considering the limit in

75

which the physical size is very big: ρ ∼ λ→∞. The lump Kahler potential(see Eq. (5.5.12)) is thus also a good approximation at finite gauge coupling,provided that we restrict to solutions with a very big size. In fact one cansee from Figure (5.2(b)) that this approximation is very good also for smallsize6.

In the lump limit the expression (5.4.1) can be calculated analytically:

K ∼ v2Tr

∫d2z log Ω = v2

∫d2z log(det Ω) ∼ v2

∫d2z log(detH0H

†0).

(5.5.12)

Let us consider two particular, dual, examples: k = 1, Nc = 2, Nc = 1 (Nf =3). In this case all moduli are non-normalizable except for position moduli,so we find no nontrivial dynamics on the vortex. Nonetheless one may studythe dynamics of these moduli by providing an infrared cut-off in (5.4.1).Furthermore, according to the above discussion, we can study the large sizelimit, in order to have an effective theory which can be derived analytically.The expression (5.5.12) is thus the Kahler potential for a Gr2,3 (Gr1,3) lumpwith topological charge k = 1. In the abelian theory we find, from (5.3.10):

KNc=1,Nf=3 = v2

∫|z|≤L

d2z log(|z − z0|2 + |b|2 + |c|2). (5.5.13)

If we set z0 = 0 for simplicity, this integral is easily performed:

KNc=1,Nf=3 = v2π(|b|2 + |c|2) log

(L2

|b|2 + |c|2

)+ v2π(|b|2 + |c|2) +O(L−1),

(5.5.14)

where we omit divergent terms that do not depend on the moduli. Thecorresponding metric is:

LNc=1,Nf=3 = v2π(|∂µb|2 + |∂µc|2) logL2

(|b|2 + |c|2)+O(L0). (5.5.15)

Note that we have obtained a conformally flat metric on C2 = R4, whichmight be expected given the U(1) × SU(2)F isometry that acts on the pa-rameters b and c.

6Presumably Eq. (5.5.12) should give a good approximation to the Kahler po-tential also in the general case with several vortices, provided that we considersolutions with big typical sizes (to this end we should not consider, for example,configurations such that detH0H

†0 vanishes at some vortex point).

76

Now consider the non-Abelian theory with Nc = 2. The Kahler potentialis given by:

KNc=2,Nf=3 = v2π|c|2(1 + |b|2) logL2

|c|2(1 + |b|2)+O(L0), (5.5.16)

where we have used the moduli matrix coordinates defined in (5.3.8). Notethat this potential is consistent with the general expression we gave in (5.5.8),in the lump limit, up to logarithmic accuracy. The SU(2)c+f ×U(1) symme-try of the theory, which leaves the quantity |c|2(1 + |b|2) invariant is againmanifest. The metric that follows from this potential is:

LNc=2,Nf=3 = v2 π[|c|2|∂µb|2 + (1 + |b|2)|∂µc|2 +

+ (c b† ∂µc†∂µb+ c. c.)

]log

L2

|c|2(1 + |b|2). (5.5.17)

The expressions (5.5.15) and (5.5.17) are related by the following change ofcoordinates:

c = c, b = c b (|c|2(1 + |b|2) = |c|2 + |b|2 6= 0, c 6= 0). (5.5.18)

The regularized metric of a semi-local vortex is a conformally flat metricof C2 (modulo a change of coordinates). This is valid, in the large size limit,for both dual theories: Nc = 2, Nc = 1, Nf = 3. This is related to the factthat in both dual theories the semi-local vortex reduce to the same object, aGr2,3 = Gr1,3 = CP 2 lump.

The effective action in the lump limit, for generic Nc and Nc can be foundfrom the following Kahler potential:

KNc,Nf = v2π|c|2(1 + |b|2) logL2

|c|2(1 + |b|2)+ v2π |c|2(1 + |b|2),

(5.5.19)

where b and c are vectors length Nc − 1 and Nc respectively7.

5.6 Summary

We analyzed the moduli space of semi-local non-Abelian vortices both atthe kinematical and at the dynamical level. First we provided an unambigu-ous smooth parametrization of the moduli space, and studied its topological

7Alternatively, vectors b and c of dimensions Nc − 1 and Nc can be used, asalready emphasized.

77

structure (without the metric). The moduli metric is defined only for nor-malizable moduli which parameterize a subspace inside the whole modulispace of dimension 2kNF. We used supersymmetry to derive an effectiveaction for the system of k vortices and showed that, even though a singlesemi-local vortex always has only non-normalizable moduli [6], higher wind-ing configurations admit normalizable moduli, which roughly correspond torelative sizes and orientations in the internal space, as first noted in [88].This means that, upon fixing the non-normalizable global moduli, the anal-ysis of the geodesic motion on the moduli space becomes meaningful. Alongthe way we discovered an interesting relationship between the vortex modulispaces of a Seiberg-like dual pair of theories at fixed winding number k: theyboth descend, by means of two alternative regularizations, from the same“parent space”, which is a non-Hausdorff space defined in terms of a certainholomorphic quotient. As a result they are guaranteed to be birationallyequivalent and in fact they are related by a geometric transition. Moreover,in the limit of infinite gauge coupling we saw how they reduce to the samemoduli space of k Grassmannian sigma model lumps, as expected on generalgrounds. We also found the normalizable moduli enhancement on specialsubmanifolds: the number of normalizable moduli can change depending onthe point of moduli space we are dealing with. For instance, the orienta-tional moduli CPNc−1 of a k = 1 vortex are non-normalizable unless thesize modulus vanishes. However they become normalizable in the limit ofvanishing size modulus where the semi-local vortex shrinks to a local vortex(with physically non-zero size).

78

Chapter 6

Interactions of Non-BPSNon-Abelian Vortices

So far, all our discussions about non-Abelian vortices were focused on theBPS limit [20] (a single non-Abelian non-BPS vortex configuration is dis-cussed in [106, 107, 108, 109, 110, 111]). No forces arise among BPS vortices,because there is a nice balance between the repulsive forces mediated by thevector particles and attractive forces mediated by the scalar particles. In thisparticular limit, the solutions to the equations of motion develop a full mod-uli space of solutions [96], which was the subject of our study in the previousSection. However, once the balance between the attractive force and the re-pulsive force is lost, the moduli space disappears. Alternatively we can thinkthat an effective potential is generated on this moduli space. It is well knownthat ANO vortices in the Type I system feel an attractive force while those inthe Type II model feel a repulsive force [20, 112, 113, 114, 115]. In condensedmatter physics, it is also known that Type II vortices form the so-calledAbrikosov lattice, [116] due to the repulsive force between them. Further-more, lattice simulations give some evidence of the presence of a (marginal)Type II superconductivity in QCD [117, 118].

We are interested in studying interactions between non-Abelian vorticeswhich are non-BPS. In non-supersymmetric theories, BPS configurations areobtained with fine-tuned values of the couplings. If supersymmetry exists inthe real world, it is surely broken at a low energy scale; therefore non-BPSvortices are more natural than BPS ones.

79

6.1 Type I non-Abelian supersymmetric su-

perconductors

A breakthrough in this context was the Seiberg-Witten solution [3] of N = 2super Yang-Mills theories; they found massless monopoles at strong coupling.Adding a smallN = 2 breaking mass term for the adjoint field, the monopolescondense creating dual vortex strings which carry a chromoelectric flux. Thedetails of confinement in the Seiberg-Witten scenario are indeed quite differ-ent from QCD. The SU(Nc) gauge symmetry is spontaneously broken brokento U(1)Nc−1 by the expectation value of an adjoint field and the strings ofthe theory carry an Abelian U(1)Nc−1 charge. A careful examination showsthat the “hadronic” spectrum is much richer than that of QCD [119, 15].

As we have already noticed, the very existence of the non-Abelian vortexis related to the presence of a color-flavor mixed global symmetry. As a con-sequence, non-Abelian vortices can be studied in many different theoreticalsettings; indeed it is possible to start with a N = 1 theory [120, 121] or evenwith a non-supersymmetric theory [21]. The details of the effective 1 + 1dimensional sigma model are different due to different number of fermionsand various amount of supersymmetry. Also the number of quantum vacuais different, for example there are N vacua in the N = 2 case [122, 9, 7, 8]and just one vacuum in the non-supersymmetric case [21].

In this Section, we study the impact on the vortices of the N = 2 vortexmodel (1.1.3) of some mass terms η0, η3 for the adjoint fields, which breakthe extended supersymmetry. For concreteness, we will discuss the caseNc = Nf = 2. The vortex with winding number one is not anymore a BPSobject, but still has a CP 1 moduli space. On the other hand the physicsfor vortices with higher winding numbers is very different: almost all theflat directions in the moduli space are lifted by the parameters η0, η3. Theforce between two vortices is not as simple as in an Abelian superconductor,where we have attraction for Type I vortices and repulsion for Type II ones[20, 113]. There is a non-trivial dependence on the orientations of the twovortices in the internal space.

Even if the force between two vortices in our model is not attractive forall values of the vortex orientations ~n1, ~n2, we have a close resemblance withType I Abelian vortices: we find that the scalars of the theory are lighterthan the vector bosons. Hence if we consider two well separated vortices, wehave that the prevailing part of the interaction is mediated by scalars andnot by vectors. Moreover, for ~n1 = ~n2 the force is always attractive. We havealso found evidences that the configurations which minimize the energy arealways given by two coincident vortices, just as in the Type I Abelian case.

80

For these reasons we call these objects ”non-Abelian Type I vortices”.

6.1.1 Spectrum of the Theory

We start from our N = 2 supersymmetric U(2) gauge theory with Nf = 2hypermultiplets (1.1.3), but we consider the following superpotential instead:

W =1√2

[Qf (a0 + aiτ i +

√2mf )Qf +W0(a0) +W3(aiτ i)

], (6.1.1)

where the terms W0,3 are of the form

W0 =N

2

(−ξa0 + η0a

20

), W3(aiτ i) =

N

2η3a

iai. (6.1.2)

Here we have introduced two real positive mass parameters η0 and η3 for theadjoint scalars which break N = 2 SUSY to N = 1. These terms changeonly the V3 piece of the F-term which we have already written in Eq. 1.1.6

V3 =g2

2

∣∣∣Trf [QτiQ] +Nη3a

iai∣∣∣2 +

e2

2

∣∣∣∣Trf [QQ]− N

2ξ +Nη0a0

∣∣∣∣2 , (6.1.3)

The vacuum of the theory is not changed by the parameters η0,3:

Q = Q =

√ξ

2

(1 00 1

), a = 0, ab = 0. (6.1.4)

For η0 6= 0, the theory has also another classical vacuum:

Q = Q = 0, a =ξ

2η0

, ak = 0, (6.1.5)

which “runs away” at infinity for η0 = 0. In what follows, we consider thevacuum (6.1.4) and the non-Abelian vortices therein.

The masses of the gauge bosons can easily be read of the Lagrangian:

M2U(1) = ξe2, M2

SU(2) = ξg2.

The masses of the scalars are given by the eigenvalues M2i of the mass matrix

(calculated in the vacuum (6.1.4)):

M =1

2

∂2V

∂si∂sj, (6.1.6)

81

where we denote by sk (k = 1, 2, · · · , 24) the real scalar fields of the theory.The calculation is a bit tedious but quite straightforward (a very similar sit-uation is discussed in Ref. [120], in the case of a D-term Fayet-Iliopoulos).First of all, there are four zero eigenvalues, which correspond to the scalarparticles eaten by the Higgs mechanism. There is one real scalar with massMS0 = MU(1) which is in the same N = 1 multiplet as the U(1) massivephoton and moreover there are also three scalars with a mass MT0 = MSU(2)

in the same multiplet as the non-Abelian vector field. The other mass eigen-values are given by

M2S1,S2 = ξe2 + e4η2

0 ±√

2ξη20e

6 + e8η40 ; (6.1.7)

M2T1,T2 = ξg2 + g4η2

3 ±√

2ξη23g

6 + g8η43 ,

where the upper sign is for MS1,T1 and the lower sign is for MS2,T2. MS1 andMS2 have multiplicity 2; MT1 and MT2 have multiplicity 6. The mass of thefermions is obviously the same as the mass of the bosonic degrees of freedom,because of the unbroken N = 1 supersymmetry.

Note that for η0 = η3 = 0 (which is also discussed in Ref. [123]), the massdegeneracy of the spectrum is bigger (the particles fit into N = 2 hypermul-tiplets). The parameter η0 affects only the masses of the particles which arein the same N = 2 hypermultiplets as the U(1) vector field; η3 affects themass of the particles which are in the non-Abelian vector hypermultiplet.The case of N = 2 SQED was studied in Ref.[11]; the results are very similarto our U(1) subsector.

In the limit η0e √ξ, we find M2

S1 ≈ 2e4η20 and M2

S2 ≈ ξ2/(4η20). In

a similar way, if η3g √ξ the masses become M2

T1 ≈ 2g4η23 and M2

T2 ≈ξ2/(4η2

3). The particles with masses MS1 and MT1 (which in this limit cor-respond to the fields a and ak) become very massive and decouple from thelow energy physics.

For η0e√ξ and η3g

√ξ nonetheless we find

M2S1,S2 = ξe2 ±

√ξη0e

3, M2T1,T2 = ξg2 ±

√ξη3g

3. (6.1.8)

Some of the scalars become slightly heavier and some slightly lighter.

The mass eigenvectors take a quite complicated form for small η0,3, withat non-trivial mixing between Q, Q and a, ak. On the contrary they are quitesimple for large η0,3, because the fields a, ak decouple from the low energyphysics. The effective Lagrangian for large η0,3 is discussed in Appendix A.

82

6.1.2 (p, k) Coincident Vortices

Second order equations

In this Section, we will study some special solutions representing coincidentvortices that live in an Abelian subgroup of the fields of the theory. Thesesolutions are parameterized by two positive integers (p, k); the topological Zwinding number is given by w = p + k. This kind of solution gives us themost general vortex with winding w = 1 up to a colour-flavour rotation1.

Due to the symmetry between Q and Q†, for the vortex solution wecan consistently set Q = Q† 2. With this assumption the Euler-Lagrangeequations of the theory are:

∂µFµν0 = −ie2

(Q†fDνQf − (DνQf )

†Qf

),

DµF µνi = −ig2

(Q†fτiDνQf − (DνQf )

†τiQf

)− εijk

(aj(Dνak)† + ajDνak

),

DµDµQ = −1

2

δV

δQ†, ∂µ∂µa0 = −e2 δV

δa0

, DµDµai = −g2 δV

δai.

(6.1.9)

We make the following axial symmetric ansatz (which in the BPS limit re-duces to the one of Ref. [16]):

Q =

(φ0e

piϕ 00 φ1e

kiϕ

),

A3i = −εijxj

r2[(p− k)− f1(r)], A0

i = −εijxjr2

[(p+ k)− f0(r)],

a0 = λ0(r), a3 = λ3(r), a1 = a2 = 0. (6.1.10)

Notice that the adjoint fields a, ai are non-trivial in the non-BPS model,whereas they vanish everywhere when η0,3 are zero (BPS).

Let us introduce the S2 coordinate ni, with i = 1, 2, 3 and |~n| = 1:

niτ i = Uτ 3U †, (6.1.11)

where U ∈ SU(2)c+f . Using this parametrization we can write down theexpression for a w = 1 vortex with generic orientation ni:

A0i = −εijxj

r2[1− f0], Aii = −εijxj

r2[1− f1]ni, a0 = λ0, a

i = niλ3,

1For higher winding, it follows from the moduli matrix formalism that it is notthe most general solutions, at least in the BPS case.

2This is equivalent to the assumption that H ≡ 0 in the D-term model (3.1.2).

83

Q = Q† =φ0e

Iϕ + φ1

21 +

φ0eIϕ − φ1

2τ ini. (6.1.12)

It is easy to see that the (p, k) vortex, when p 6= k partially break thesymmetry in Eq. (7.2.3); as a consequence, this object has some internalzero modes associated to this breaking. In fact, the vortex leaves a U(1)subgroup of SU(2)c+f unbroken, so that zero modes parameterize a CP 1 =SU(2)/U(1) = S2. If p = k the vortex is Abelian, it breaks completelySU(2)c+f , and no moduli space is generated.

The energy with respect to φ0,1, f0,1 and λ0,3 is expressed as

E = 2π

∫rdr

(f ′20

2e2r2+

f ′21

2g2r2+λ′20e2

+λ′23g2

+ 2(φ′20 + φ′21 )+

+(φ2

0 + φ21)(f 2

0 + f 21 ) + 2f1f0(φ2

0 − φ21)

2r2+e2

2(φ2

0 + φ21 − ξ + 2η0λ0)2+

+g2

2(φ2

0 − φ21 + 2η3λ3)2 + ((λ0 + λ3)φ0)2 + ((λ0 − λ3)φ1)2

). (6.1.13)

We have to minimize this expression with the appropriate boundary condi-tions for each (p, k):

f1(0) = p− k, f0(0) = p+ k, f1(∞) = 0, f0(∞) = 0.

φ0(∞) = 1, φ1(∞) = 1, λ0(∞) = 0, λ3(∞) = 0. (6.1.14)

We also find for small r:

φ0 ∝ O(rp), φ1 ∝ O(rk), λ0 ∝ O(1), λ3 ∝ O(1). (6.1.15)

The Euler-Lagrange equations obtained are:

f ′′0r− f ′0r2

=e2

r(f1(φ2

0 − φ21) + f0(φ2

0 + φ21)) ;

f ′′1r− f ′1r2

=g2

r(f1(φ2

0 + φ21) + f0(φ2

0 − φ21)) ;

φ′′0 +φ′0r− φ0(f0 + f1)2

4r2=

=φ0 ((λ0 + λ3)2 + e2(φ2

0 + φ21 − ξ + 2η0λ0) + g2(φ2

0 − φ21 + 2η3λ3))

2;

φ′′1 +φ′1r− φ1(f0 − f1)2

4r2=

84

=φ1 ((λ0 − λ3)2 + e2(φ2

0 + φ21 − ξ + 2η0λ0)− g2(φ2

0 − φ21 + 2η3λ3))

2;

λ′′0 +λ′0r

=e2 ((a0 + a3)φ2

0 + (a0 − a3)φ21 + e4η0(φ2

0 + φ21 − ξ + 2η0λ0))

2;

λ′′3 +λ′3r

=g2 ((a0 + a3)φ2

0 − (a0 − a3)φ21 + g4η3(φ2

0 − φ21 + 2η3λ3))

2.(6.1.16)

It is easy to check that these equations can be obtained substituting theansatz (6.1.10) in Eqs. (6.1.9). This shows that the ansatz is consistent.

In the following, we will concentrate our effort on the study of the sectorswith topological winding 1 and 2; in other words, we will discuss the (1, 0),the (1, 1) and the (2, 0) vortices. In the BPS limit (η0 = η3 = 0) the tensionis proportional to the topological winding number (T(1,0) = 2πξ, T(1,1) =T(2,0) = 4πξ). For non-BPS solutions, η0,3 6= 0, we find that the tension isalways less than the BPS limit. This is because the non-BPS terms in thetension formula Eqs. (6.1.13) do not give any contribution if we put the BPSsolutions into the expression (λ0,3 are identically zero for the BPS solutions).The non-BPS solutions will of course be a true minimum or saddle point ofthe energy functional, so that their energy will be smaller than that of theBPS configurations3.

For fixed ξ, the tension of the (1, 1) vortex is a function of only e, η0,because for this vortex f1 = 0 and φ0 = φ1. This is clearly explained bythe fact that the (1, 1) vortex is completely Abelian. On the other hand, thetension of the (1, 0) and of the (2, 0) vortex is a non-trivial function of all theparameters e, g, η0,3.

If we take g = e and η3 = η0 the vortex becomes easier to study. In thiscase we can use a more convenient basis for the gauge field, which is just thesum and the difference of A0

µ and A3µ. The potential V also factorize, and

takes the form V = V1(φ0)+V2(φ1). Each diagonal component of Q does notinteract with the other ones, and can be treated as an Abelian vortex. Forthe (1, 0) vortex we can use the simple ansatz4

Q =

(φeiϕ 0

0√ξ/2

), (6.1.17)

while for the (1, 1) and for the (2, 0) vortices we can use

Q =

(φ(r1)eiϕ1 0

0 φ(r2)eiϕ2

), Q =

(φe2iϕ 0

0√ξ/2

). (6.1.18)

3This also has a clear resemblance to the Abelian case, where Type I vorticeshave a smaller energy with respect to the BPS case.

4 Notice that we cannot impose φ1 =√ξ/2 even for (1, 0) vortex in the generic

models.

85

The system reduces to the Abelian vortex studied in Ref. [11]. The tensionof the (1, 1) vortex is exactly twice the tension of the (1, 0) one. In eachof the U(1) factors, we have Type I superconductivity. Since the two U(1)subgroups are decoupled, the (1, 0) and (0, 1) vortices do not interact. Fur-thermore, the tension of the (2, 0) vortex is less than twice the tension of the(1, 0) vortex.

Numerical Solutions

At generic e0,3, η0,3 Eqs. (6.1.16) have been solved numerically. It is a littlesubtle to solve this system of ordinary differential equations directly. Thedifficulties basically arise because there are many equations; there are alsosubtleties in defining the boundary conditions at∞, because, in general, thefields which appear in our ansatz do not correspond to mass eigenstates. Inorder to perform the numerics we found that the method of relaxation isvery effective. We add an auxiliary time dependence to the profile functions~u = (f0, f1, φ0, φ1, λ0, λ3) . At t = 0 we start with some arbitrary functionsuj(r, 0)The choice of these initial conditions is crucial to find convergence.The evolution in t is then given by:

∂uj∂t

= Ej. (6.1.19)

If the solution converges with time to a static configuration, then at final timewe have obtained a solution of the equations Ej = 0, which are equivalentto Eqs. (6.1.16). The results for (p, k) = (1, 0), (2, 0), (1, 1) are shown in Fig.(7.1).

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12 14

0.5

1

1.5

2

2 4 6 8 10 12 14

0.5

1

1.5

2

Figure 6.1: Profile functions f0 (solid black), f1 (solid blue), φ0 (long dashes,black), φ1 (long dashes, blue), λ0 (short dashes, black), λ3 (short dashed,blue) forthe numerical values ξ = 2, e = 1/4, g = 1/2, η0 = η3 = 1. In the left panel areshown the profiles for the (1, 0) vortex, in the middle, the ones for the (2, 0) andin the right, the ones for the (1, 1). Note that in this last case f1 = λ3 = 0 andφ0 = φ1.

It is interesting to compare numerical result for the tension 2T1,0, T2,0 andT1,1 of the 2 × (1, 0), (2, 0) and (1, 1) vortices, respectively (see Fig. (6.2)).

86

We have always found that T2,0 < 2T1,0. This is consistent with the fact thatat large separation distance, the force between two vortices with the samecolour-flavour orientation is always attractive (we will discuss this aspect inSect. 4). As can be checked in Fig. (6.2), three different regimes have beenfound for T1,1: T1,1 < T2,0 < 2T1,0 or T2,0 < T1,1 < 2T1,0 or T2,0 < 2T1,0 < T1,1.

0.2 0.4 0.6 0.8 1 1.2 1.4

1.2

1.4

1.6

1.8

0.2 0.4 0.6 0.8 1 1.2 1.4

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Figure 6.2: T1,1 (long dashes), T2,0 (short dashes) and 2T1,0 (solid) for differentvalues of 0 < ω < π/2, where η3 = η sinω, η0 = η cosω. The tension of theBPS 2-vortex is normalized to TBPS = 2. In the left figure the numerical valuese = g = 1/2, η = 4, ξ = 2 are used; in the right figure e = 1/2, g = 1/4, η = 4,ξ = 2).

If η0 = 0, η3 6= 0, the tension of the (1, 1) vortex is found to be the sameas in the BPS case. The tensions of the 2 × (1, 0) and of the (2, 0) vortexare strictly less than that of the BPS vortices (η3 = 0). Hence in this caseT2,0 < 2T1,0 < T1,1. On the contrary, if η3 = 0, η0 6= 0, we have foundT1,1 < T2,0 < 2T1,0 for all the numerical values of the couplings that we haveinvestigated.

6.1.3 Generic Coincident Vortices

The BPS case

The calculation of the number of dimensions of the k-vortex moduli spacein U(Nc) N = 2 gauge theory with Nf = Nc = N hypermultiplets hasbeen reviewed in Section (3.1). The result is 2kN . Thus for η0 = η3 = 0,the moduli space of a 2-vortex configuration is a manifold with eight realdimensions. Two of these dimensions are associated with the global positionof the system; other 2 coordinates are associated with the relative position Rof the two elementary vortices. The remaining 4 coordinates are associatedwith the orientation of the system in the colour-flavour space. In this Section,we will write an ansatz for the case of coincident vortices (R = 0), and we

87

will show that it is non-trivially consistent with the second order equationsof the theory.

Modulo a global SU(2) rotation we can parameterize a subset of themoduli space with the angle α between ~n1 and ~n2. The expression for Q = Q†

is

Q =

( − cos α2e2iϕκ1 sin α

2eiϕκ2

− sin α2eiϕκ3 − cos α

2κ4

). (6.1.20)

The ansatz for the gauge fields is:

A0(i) = −εijxj

r2(2− f0), A3

(i) = −εijxjr2

((1 + cosα)− f1),

A1(i) = −εijxj

r2(sinα)(cosϕ)(1− g), A2

(i) = +εijxjr2

(sinα)(sinϕ)(1− g).

(6.1.21)

We have introduced here the profile functions κ1(r), κ2(r), κ3(r), κ4(r) for thesquark scalars and f0(r), f1(r), g(r) for the gauge field. For r → ∞ all thegauge profile functions vanish and all the squark ones go to the value

√ξ/2.

The boundary conditions at r → 0 are:

f(r) = 2 +O(r2), f1(r) = (1 + cosα) +O(r2), g(r) = 1 +O(r3),

κ1(r)→ O(r2), κ2(r)→ O(r), κ3(r)→ O(r), κ4(r)→ O(1) .

For the (2, 0) vortex we have:

α = 0, φ0 = κ1, φ1 = κ4,

while for the (1, 1) vortex (after a simple diagonalization):

α = π, φ0 = κ2 = φ1 = κ3.

For the BPS vortex it is simpler to consider first order equations; butwe are interested in understanding what is happening for η0,3 6= 0. Thuswe will write the equations in a form that can be easily generalized to anon-BPS setting. This will also give the possibility to check our equationsand numerical results, just comparing the result for the tensions against theexact Bogomol’nyi bound; for completeness, we provide the first order BPSequations in Appendix B. The energy density due to the kinetic part of thegauge field is:

Sg =f ′0

2

2 r2 e2+

f ′21

2 r2 g2+

sin2 α g′2

2 r2 g2. (6.1.22)

88

The part due to the kinetic energy of squark is:

SQ = cos2 α

2(κ′21 + κ′24 ) + sin2 α

2(κ′22 + κ′23 )+

+ cos2 α

2

((1− cosα + f0 + f1)2κ2

1

4r2+

(1− cosα− f0 + f1)2κ24

4r2

)+

+ sin2 α

2

((1 + cosα− f0 − f1)2κ2

2

4r2+

(1 + cosα + f0 − f1)2κ23

4r2

)+

+(1− g)2 sin2 α

4r4

(cos2 α

2(κ2

1 + κ24) + sin2 α

2(κ2

2 + κ23))−

− (1− g) sin2 α

2r4((1 + f0)κ1κ3 + (1− f0)κ2κ4) . (6.1.23)

The part due to the potential reads:

VBPS =e2

2

(cos2 α

2(κ1

2 + κ42) + sin2 α

2(κ2

2 + κ32)− ξ

)2

+

+g2

2

(cos2 α

2(κ1

2 − κ42) + sin2 α

2(κ2

2 − κ32))2

+ sin2 α (κ1 κ3 − κ2 κ4)2

.

(6.1.24)

The total energy is given by:

E = 2π

∫rdr(Sg + 2SQ + VBPS). (6.1.25)

It is straightforward to write the Euler-Lagrange equations for this energydensity, which are a system of seven second order equations, one for eachprofile function; for brevity we will not show them explicitly. We have solvedthis system numerically. In Fig. (6.3) it is shown an example of the solution.The tension is found to be equal to TBPS = 4πξ with an excellent precisionfor every α; this is a good numerical check for the solution obtained.

An analytical check of the ansatz can also be found substituting Eqs.(6.1.20) and (6.1.21) into the Euler-Lagrange equations (6.1.9). With thisapproach we find a system of the same seven second order equations withthe following first order expression:

K = g2r2(κ3κ′1 − κ1κ

′3 + κ2κ

′4 − κ4κ

′2)− (1− g)f ′1 − (f1 − cosα)g′ = 0.

89

2.5 5 7.5 10 12.5 15 17.5

0.2

0.4

0.6

0.8

1

2.5 5 7.5 10 12.5 15 17.5

0.5

1

1.5

2

Figure 6.3: Vortex profile functions for α = π/3. In the left panel there are κ1

(solid), κ2 (long dashes), κ3 (short dashes), κ4 (dots); in the right panel there aref0 (solid), f1 (long dashes), g (short dashes). The following numerical values havebeen used: ξ = 2, e = 1/4, g = 1/2).

This seems to be a paradox, because this is a system of eight differentialequation with seven unknown functions. Actually, everything is consistent,because using the seven second order equations we can show the followingproperty:

dK

dr=K

r, (6.1.26)

which shows that K is linear in r. From the boundary conditions of theprofile functions, we find that the coefficient of this linear function has to bezero. This shows that our ansatz is consistent with the equations of motion.

The non-BPS case: η0, η3 6= 0

For η0, η3 6= 0 the (1, 1) and the (2, 0) vortices are still solutions to the equa-tions of motion; so these field configurations are extremal points of the energywhich can be local minima or saddle points. For generic values of the pa-rameters, we find T(1,1) 6= T(2,0), so the continuous moduli space interpolatingbetween these two particular solutions disappears. For small values of η0, η3

we expect that the low energy physics of these solitons is described by aneffective potential of the moduli space. In this Section, we will estimate thispotential numerically for generic values of α.

A constraint on this potential comes from the BPS limit at η0, η3 = 0.In this case, a continuous family of degenerate solutions exists, with tensionT = 4πξ. If we insert these solutions into the energy density for η0, η3 6= 0, theenergy of these field configurations does not change. However, the solutionsto the second order equations have energies which are less than this value.

90

This sets an upper bound:

T (α) ≤ TBPS = 4πξ. (6.1.27)

There is an obvious invariance of the equations:

α→ −α.

Indeed, if we expand around α = 0 + δ or α = π + δ we find that the linearorder in δ is zero and that the first non-trivial correction to the tension isO(δ2). This shows that solutions with α = 0, π correspond to local minimaor maxima of the tension. In order to find which of the two alternativesholds, an explicit calculation is needed.

In order to compute the potential of the vortex moduli space, we gener-alize the ansatz that we have used for the solutions in the BPS case, usingthe same expressions for the gauge fields, Q and the following expression forthe adjoint fields:

a0 = λ0(r), a3 = λ3(r),

a1 = (sinα)x1

rλ12(r), a2 = (sinα)

−x2

rλ12(r), (6.1.28)

where we have introduced the profile functions λ0, λ3, λ12, with the followingboundary conditions:

λ0(∞) = 0, λ3(∞) = 0, λ12(∞) = 0, (6.1.29)

and the following r → 0 behaviour:

λ0 ∝ O(1), λ3 ∝ O(1), λ12 ∝ O(r). (6.1.30)

This ansatz is suggested by the expression we get for these adjoint fields inthe limit of large η0, η3, where we can integrate these fields out (see Appendix(??)). In the following we replace these expressions in the action and findsecond order equations for the profile functions for generic α. These fieldconfigurations at α 6= 0, π are not solutions to the full equations of motion,Eqs. (6.1.9); they are just functional generalizations of the BPS solutions.We use these profiles as reasonable test functions to compute the effectivemoduli space potential.

The kinetic energy of the adjoint scalars is:

Sa =λ0′2

e2+λ3′2 + sin(α)2 λ12

′2

g2+

sin(α)2

r2g2

(1− g)2 λ3

2+

91

2.5 5 7.5 10 12.5 15 17.5

0.2

0.4

0.6

0.8

1

1.2

2.5 5 7.5 10 12.5 15 17.5

0.5

1

1.5

2

2.5 5 7.5 10 12.5 15 17.5

0.02

0.04

0.06

0.08

0.1

0.12

Figure 6.4: Vortex profiles for α = π/3 and ξ = 2, e = 1/2, g = 1/4, η0 = η3 = 1.In the left panel there are κ1 (solid), κ2 (long dashes), κ3 (short dashes), κ4 (dots);in the middle panel f0 (solid), f1 (long dashes), g (short dashes); in the right panelλ0 (solid), λ3 (long dashes), λ12 (short dashes).

0.5 1 1.5 2 2.5 3

1.92

1.94

1.96

1.98

0.5 1 1.5 2 2.5 3

1.885

1.89

1.895

1.905

0.5 1 1.5 2 2.5 3

1.885

1.89

1.895

1.905

1.91

Figure 6.5: Vortex tension as function of α in the topological winding 2 sector; thetension of the BPS 2-vortex is normalized to TBPS = 2. In the left panel: ξ = 2,e = 1/2, g = 1/2, η0 = 0.1, η3 = 1. In the middle panel: ξ = 2, e = 1/2, g = 1/2,η0 = 1, η3 = 0.1. In the right panel: ξ = 2, e = 1/2, g = 1/4, η0 = η3 = 1.

+(f1 − cos(α))2 λ122 + 2λ3 λ12 (cosα− f1) (g − 1)

. (6.1.31)

The potential term is:

V =e2

2

(cos(

α

2)2

(κ12 + κ4

2) + sin(α

2)2

(κ22 + κ3

2) + 2 η0 λ0 − ξ)2

+

+g2

2

(cos(

α

2)2

(κ12 − κ4

2) + sin(α

2)2

(κ22 − κ3

2) + 2 η3 λ3

)2

+

+g2

2sin(α)2 (κ1 κ3 − κ2 κ4 + 2 η3 λ12)2 + 2 sin(α)2 λ0 λ12 (κ1 κ3 − κ2 κ4)+

+

(sin(

α

2)2

κ32 + cos(

α

2)2

κ42

) ((λ0 − λ3)2 + sin(α)2 λ12

2)

+

+

(cos(

α

2)2

κ12 + sin(

α

2)2

κ22

)((λ0 + λ3)2 + sin(α)2 λ12

2). (6.1.32)

92

The energy is the sum of all pieces:

E = 2π

∫rdr(Sg + 2SQ + Sa + V ). (6.1.33)

The system of ten second order differential equations which we obtainis quite complicated, but can still be solved numerically. The qualitativeplot of the profile functions is similar to the BPS case (see Fig. (6.3) andFig. (6.4)). The tension is a non-trivial function on the coordinate α, whichgives us an effective potential of the moduli space. We solved this equationsnumerically for different values of the couplings e, g, η0, η3 and we have foundthree different regimes (see Fig. (6.5)). The tension can have a maximumat α = 0 and a minimum at α = π; we also find the opposite situation inwhich there is a minimum at α = 0 and a maximum at α = π. The thirdalternative is that both α = 0, π are local minima of the tension, with one ofthem a metastable minimum. We never obtain a minimum at α 6= 0, π.

6.1.4 Vortex Interactions at Large Distance

Vortex profiles at large distance

At large distance r from the center of the vortex, the equations can be lin-earized and solved analytically. Let us introduce the following notation:

φ0 =√ξ/2 + δφ0, φ1 =

√ξ/2 + δφ1, ~v = (δφ0, δφ1, λ0, λ3). (6.1.34)

The following linear differential equations can be written:

~v′′(r) +~v′(r)

r−W~v(r) = 0, (6.1.35)

where the matrix W is given by:

W =

ξ(e2 + g2)/2 ξ(e2 − g2)/2√

ξ2e2η0

√ξ2g2η3

ξ(e2 − g2)/2 ξ(e2 + g2)/2√

ξ2e2η0 −

√ξ2g2η3

√2ξe4η0

√2ξe4η0 2η2

0e4 + ξe2 0

√2ξg4η3 −√2ξg4η3 0 2η2

3g4 + ξg2

.(6.1.36)

The eigenvalues of the matrix W are in direct correspondence with some ofthe scalar spectrum of the theory (see Eq. (6.1.7)):

w1,2 = M2S1,S2 = ξe2 + e4η2

0 ±√

2ξη20e

6 + e8η40, (6.1.37)

93

w3,4 = M2T1,T2 = ξg2 + g4η2

3 ±√

2ξη23g

6 + g8η43.

The corresponding eigenvectors are:

~v1,2 =

(−e4η0

2 ±√

2ξe6η02 + e8η0

4

2√

2ξe4η0,−e4η0

2 ±√

2ξe6η02 + e8η0

4

2√

2 ξ e4 η0, 1, 0

),

~v3,4 =

(−g4η3

2 ±√

2ξg6η32 + g8η3

4

2√

2ξg4η3,g4η3

2 ∓√

2ξg6η32 + g8 η3

4

2√

2 ξ g4 η3, 0, 1

).

Note that ~vk are also eigenvectors of the mass matrix defined in Eq. (6.1.6).The solutions to these equations which are zero at infinity are given by

the modified Bessel function:

~v(r) =∑

k=1,··· ,4bk~vkK0(

√wkr), (6.1.38)

where bk are appropriate constants which can be found solving the completedifferential equation also at small r. For large x we can use:

K0(x) ≈√

π

2xe−x. (6.1.39)

The asymptotic solutions for the scalar profiles read:

~v(r) ≈∑

k=1,··· ,4bk~vk

√π

2√wkr

e−√wkr. (6.1.40)

The large r equations for f1 and f0 are:

f ′′0 −f ′0r− ξe2f0 = 0, f ′′1 −

f ′1r− ξg2f1 = 0. (6.1.41)

This leads to the following asymptotic expression in terms of Bessel functions:

f0,1 = c0,1rK1((e, g)√ξr) ∝ √re−((e,g)

√ξ)r, (6.1.42)

where c0, c1 are constants which should be determined by the original 2ndorder differential equations. Note that there is the identity: K1(r) = −K ′0(r).

Static vortex potential

The next step is to reproduce the vortex asymptotic interactions in the ef-fective linear theory by coupling the low-energy degrees of freedom to an

94

effective scalar density ρ and an effective vector current jµ. We shall extendan approach used in Refs. [115, 17].

First of all, we have to discuss the bosonic particle spectrum of the theory.There is a massive U(1) vector and a massive SU(2) vector; then in principlethere are 12 complex scalar fields (Q, Q, A, Ai). For the vortex solution wehave used the ansatz Q = Q†, so in order to discuss the vortex interactionswe can neglect the modes that break this condition5. There are 16 real fields,4 of which are eaten by the Higgs mechanism; finally we have 12 physicalscalars. We have already calculated the masses of these particles in Section(6.1.1); at low energy and at low coupling we can write a free theory whichdescribes the infrared physics:

Lfree =1

4e2(F 0

µν)2 +

ξ

2A0µA

0µ +1

4g2(F i

µν)2 +

ξ

2AiµA

iµ+ (6.1.43)

+1

2(∂µSl)

2 +1

2(∂µT

il )

2 +M2

Sl

2S2l +

M2T l

2(T il )

2.

This effective Lagrangian contains three real scalar fields, Sl=0,1,2, which areSU(2) singlets and three real scalar fields, T il = 0, 1, 2, which are SU(2)triplets. These scalars correspond to the appropriate eigenvectors of themass matrix in Eq. (6.1.6). The index i is an SU(2) triplet index; this SU(2)group corresponds to the SU(2)c+f in the full theory.

In order to include external sources (vortices) in this effective Lagrangian,we need the following effective terms:

Lsource = ρSlSl + ρiT lTil + j0µA0µ + jiµAiµ. (6.1.44)

The corresponding wave equations are:

(+M2Sl)Sl = ρSl, (+M2

T l)Til = ρiT l, (6.1.45)

(+ e2ξ)A0µ = jµ, (+ g2ξ)Aiµ = jiµ.

On the other hand, for the (1, 0) vortex with orientation ni, we have thefollowing asymptotic profiles, converted into the singular real Q gauge:

S0 = 0, S1 = b1K0(MS1r) S2 = b2K0(MS2r), (6.1.46)

T i0 = 0, T i1 = b3niK0(MT1r) T i2 = b4n

iK0(MT2r),

~A0 = −c0(z ∧∇K0(e√ξr)), ~Ai = −c3n

i(z ∧∇K0(g√ξr)),

5If we wish to include these extra modes in the low energy theory, we need onlyto promote the real fields S1, S2, T

k1 , T

k2 in Eq. (6.1.43) to complex fields.

95

where ∇ is only the ordinary gradient and not the covariant derivative asin the other Sections. We need the following mathematical identity for the2 + 1 dimensional Laplacian of K0 in term of Dirac’s δ function:

(−∆ +M2)K0(Mr) = 2πδ(~r). (6.1.47)

The following expressions are found for the scalar densities corresponding toa vortex placed at the position ~x and having orientation ni:

ρS0 = 0, ρS1 = 2πb1δ(~x), ρS2 = 2πb2δ(~x), (6.1.48)

ρT0 = 0, ρT1 = 2πb3niδ(~x), ρT2 = 2πb4n

iδ(~x).

In a similar way we obtain the following expressions for the currents:

~j = −2πc0z ∧∇δ(~x), ~ji = −2πc3niz ∧∇δ(~x). (6.1.49)

Using these expressions, it is straightforward to compute the static inter-vortex potential between two vortices with orientations ~n1 and ~n2 at distanceR:

U = 2π(c2

0K0(e√ξR)− b2

1K0(MS1R)− b22K0(MS2R)+ (6.1.50)

+(~n1 · ~n2)(c23K0(g

√ξR)− b2

3K0(MT1R)− b24K0(MT2R))

).

In the BPS case (η0 = η3 = 0) this potential is exactly zero, because we haveMS1 = MS2 = e

√ξ, MT1 = MT2 = g

√ξ and c2

0 = b21 + b2

2, c23 = b2

3 + b24.

If η3, η0 6= 0, we find that at large distance the particle with lowest massis the one which dominates the interaction. We have always the followinginequalities:

MS2 < e√ξ < MS1, MT2 < g

√ξ < MT1. (6.1.51)

Thus if MS2 < MT2 then we have:

U ≈ −2πb22K0(MS2R) ≈ −2πb2

2

√π

2MS2Re−MS2R, (6.1.52)

which gives an always attractive force. On the other hand, if MT2 < MS2:

U ≈ −2πb24K0(MT2R)(~n1 · ~n2) ≈ −2πb2

4

√π

2MT2Re−MT2R(~n1 · ~n2),(6.1.53)

which gives attraction for ~n1 = ~n2 and repulsion for ~n1 = −~n2.

96

A very peculiar thing happens for e = g and η0 = η3 6= 0. For thesefine-tuned values of the couplings MS2 = MT2 = M2 and b2 = b4, so theeffective vortex potential has the form:

U ≈ −2πb22K0(M2R)(1 + ~n1 · ~n2) ≈ −2πb2

2

√π

2M2Re−M2R(1 + ~n1 · ~n2),(6.1.54)

which gives a flat potential for ~n1 = −~n2. This is consistent with the factthat in this limit the (1, 0) and the (0, 1) vortices do not interact becausethey are completely decoupled (see the argument below Eq. (6.1.18)). Thisbehaviour is similar to the one found in Ref. [124, 125] for global non-Abelianvortices.

6.1.5 Effective worldsheet theory

Single vortex

It is useful in the following to use the singular gauge in which the squarksfields at r → ∞ tend to a fixed VEV and do not wind. In this gauge, theansatz (6.1.12) for the single vortex reduces to

A0i =

εijxjr2

f0, Aii =

εijxjr2

f1ni, a0 = λ0, a

i = niλ3, (6.1.55)

Q =φ0 + φ1

21 +

φ0 − φ1

2τ ini.

We will assume that the orientational coordinates ~n are functions of thestring worldsheet coordinates. ~n becomes a field of a 1+1 dimensional sigmamodel. This effective theory has no potential due to the fact that the ni

parameterize some zero modes; in the following we will compute the kineticterm. For the gauge field components A0,3, we will use the approach reviewedin Section (1.3). With the ansatz:

Ai = −1

2(τaεabcnb∂inc)ρ(r), i = 0, 3, (6.1.56)

the field strength components Fki with k = 0, 3 and i = 1, 2 are not zero anymore:

Fki =1

2∂kn

aτaεijxjr2f1[1− ρ(r)] +

1

2(τaεabcnb∂knc)

xir

d

drρ(r). (6.1.57)

Substituting this expression into the kinetic term for the gauge field andfor the squark fields, we obtain a simple generalization of the BPS casediscussed in Refs. [16, 9, 82]:

S1+1 =β

2

∫dtdz(∂jn

i)2, (6.1.58)

97

where:

β =2π

g2

∫rdr

ρ′2 +

f 21 (1− ρ)2

r2+ λ2

3(1− ρ)2+ (6.1.59)

+g2(φ20 + φ2

1)ρ2

2+ (1− ρ)(φ0 − φ1)2

.

We have to solve the Euler-Lagrange equations for ρ(r), with ρ(0) = 1 andρ(r → ∞) = 0. In the BPS case, where λ3 is trivially 0, we can showfrom the equations of motion [9] that ρ = 1 − φ0/φ1 and that β = 2π/g2

(see also [82, 89]); in the general case η3, η0 6= 0 there is not such powerfulanalytical result, here we have to solve the equations for ρ numerically andthen calculate β.

In the BPS case we have additional fermionic zero modes, associated withthe unbroken supercharges; at small η0, η3 6= 0 these modes should be stillpresent, but they will not be described anymore by the fermionic sector of asupersymmetric effective theory in 1+1 dimensions. We will not discuss thisaspect in this paper and we will leave it as a problem for further investigation.

The color-flavor modes of the (2, 0) vortex are very similar to the (1, 0)ones: both the vortices have a CP 1 moduli space and the value of β canbe determined using Eq. (6.1.59). For the (1, 1) vortex, nevertheless, thesemodes are just trivial because all the profile functions are proportional to theidentity matrix.

Two well separated vortices

A proper description of the system has to take into account also the quantumaspects of the sigma model physics. Let us consider two vortices with internalorientations ~n1, ~n2. The relative distance between them can be promoted toa complex field R; the global position of the system, on the other hand,decouples from the other the degrees of freedom. If the distance of thetwo vortices is large (|R| → ∞), we expect that the effective world-volumedescription of the bosonic degrees of freedom is:

S =

∫dtdz

β

2(∂kn

a1)2 +

β

2(∂kn

a2)2 + T |∂kR|2 + vs(|R|) + vt(|R|)~n1 · ~n2

,

where:

vs(|R|) = −2πb22

√π

2MS2|R|e−MS2|R|,

vt(|R|) = −2πb24

√π

2MT2|R|e−MT2|R|,

98

where T is the tension of a single vortex. This description is good only forlarge values of the VEV of the field R; at R = 0 the internal degrees offreedom are no longer described by CP 1 ×CP 1, but by the space CP 2/Z2

(see Chapter (4)). Moreover, the expression used for the potential is goodonly for large vortex separation.

If we keep the VEV of R fixed (which physically corresponds to keep thedistance of the two vortices fixed with some external device), the effectivedescription is given by two CP 1 sigma models with a small interaction termof the form c ~n1 · ~n2.

6.1.6 Summary

For Abelian Type I superconductors, the force between two vortices with thesame winding number is always attractive. This is true at large and at smalldistances, as shown by numerical calculations in Ref. [113]. In the modeldiscussed here, for η0, η3 > 0, the masses of some of the scalars fields arealways found to be less than the mass of the corresponding vector boson. Inthis sense we can think the system as a generalization of the Abelian Type Isuperconductor. However, here is an important difference: the force betweentwo vortices is not always attractive; there is a non-trivial dependence on thecoupling, the relative internal orientation and the distance.

We have studied the problem in two different limits: large vortices sepa-ration and coincident vortices. For large separations we have computed theleading potential analytically; the behavior at large distance is dominatedby the particle with the lowest mass Mlow. There are two main alternatives,which hold for different values of the couplings:

U(R) ∝

−√

12MS2R

e−MS2R for Mlow = MS2, Type I

−(~n1 · ~n2)√

12MT2R

e−MT2R for Mlow = MT2, Type I∗(6.1.60)

where MS2, MT2 are the masses of the scalars in Eq. (6.1.7). In order todistinguish these regimes, we call them Type I and Type I∗; for Type I∗

vortices the sign of the asymptotic force depend on ~n1 · ~n2. For the fine-tuned values e = g and η0 = η3 6= 0, the relation MS2 = MT2 = M2 holds,and the effective vortex potential has the form:

U(R) ∝ −(1 + ~n1 · ~n2)

√1

2M2Re−M2R, (6.1.61)

which gives a flat potential for ~n1 = −~n2.For coincident vortices we have found two stationary solutions of the

equations of motion, the (1, 1) and the (2, 0) vortices, and we computed their

99

tensions numerically. The results are shown in Fig. (6.2); both the casesT1,1 > T2,0 or T2,0 > T1,1 are possible for different values of the coupling. Themoduli space interpolating between these solutions at η0 = η3 = 0 disappearsfor non-zero values of one of these parameters (see Fig. (6.5)).

It is interesting to match the data of the two complementary approaches.Let us for simplicity consider the case of parallel (~n1 = ~n2) and anti-parallel(~n1 = −~n2) vortices. In the case of parallel vortices at large separationdistance, the force is always attractive; also from numerical calculations wefind T2,0 < 2T1,0 for all the values of the coupling that we have analyzed. Wehave not made the calculation for arbitrary distances, but we think that theabove are a good evidence for the fact that the force between two parallelvortices is always attractive in our model.

2 4 6 8 10

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

2 4 6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

Figure 6.6: Qualitative plot of the vortex potential as function of the vortexdistance for ~n1 = −~n2 and e = g. For η0 > η3 we have attraction (left); for η0 < η3

we have repulsion (right). With the choice η0 = η3 there is no net classical force.

For anti-parallel vortices, on the other hand, the situation is more com-plicated. At large distances there is attraction if MT2 > MS2 and repulsionif MT2 < MS2. For the choice e = g and η0 = η3, the relation MT2 = MS2

holds. For these particular values the two diagonal U(1) factors decouple:there is no net classical force between the (1, 0) and the (0, 1) force for arbi-trary distance (this configuration is not stable, because if we allow ~n1, ~n2 tovary, we have that (2, 0) vortex has a lower energy). Indeed, if we keep e = g,we obtain MT2 > MS2, T1,1 < 2T1,0 for η0 > η3 and MT2 < MS2, T1,1 > 2T1,0

for η0 < η3. This is a good evidence that for η0 > η3 we have an attractiveforce and for η0 < η3 we have a repulsive one (see Fig. (6.6)).

If we relax the condition e = g, there are situations in which at largedistance there is attraction (because MT2 > MS2) and also we get T1,1 >2T1,0, as is shown in Fig. (6.7) to the left: this means that there is a criticaldistance, in which there is a minimum of the inter-vortex potential for anti-parallel vortices (if we allow the vortex orientation to flip, probably it will not

100

2 4 6 8 10

-0.2

0.2

0.4

2 4 6 8 10

-0.4

-0.2

0.2

Figure 6.7: For e 6= g the qualitative plot of the vortex potential as function ofthe vortex distance for ~n1 = −~n2 can have maxima or minima for non-zero vortexseparation.

be a minimum any more, because we still have T2,0 < 2T1,0). An exampleof this situation can be obtained with the couplings ξ = 2, e = 1/4, g =1/2, η0 = 0, η3 = 4. Moreover we can obtain MT2 < MS2 and T1,1 < 2T1,0,which means that there is a critical distance at which there is a maximum ofthe inter-vortex potential (see Fig. (6.7) on the right). An example of thissituation can be obtained with the couplings ξ = 2, e = 1/2, g = 1/4, η0 =3.5, η3 = 2.

It is interesting that at R = 0 there are two different regimes, whichdepend on the values of the coupling, with very different properties. Thephysics of the (2, 0) vortex is described by a bosonic CP 1 sigma model; the(1, 1) vortex on the other hand is an Abelian vortex with no internal degreesof freedom. For some values of the coupling we have an evidence that bothvortices are local minima of the tension (see Fig. (6.5)): one of the two ismetastable (indeed for some fine tuned values of η0,3 we have that both thevortices have the same tension).

A model with metastable vortices at weak coupling has already beenstudied in Ref. [78]. This behavior is reminiscent of SU(N) Yang-Mills,where for each topological n-ality we can have different string tensions foreach representation of the Wilson Loop. In each topological sector thereis just one stable string, corresponding to the antisymmetric representation;there are evidences that the strings with other representations are metastablestrings, at least in the large N limit (see Ref. [126] for a discussion).

In this Section we have considered non-Abelian non-BPS vortices in anN = 1 supersymmetric model. In the next Section, we will discuss a simpler,non-supersymmetric model, in which we can have both Type I and Type IInon-Abelian superconductivity.

101

6.2 Static interactions

In the previous Section, we have discussed these aspects in an N = 2 the-ory with an adjoint mass term which breaks the extended supersymmetry,and we have found a natural non-Abelian generalization of Type I supercon-ductors. Even if the force between two non-Abelian vortices is not alwaysattractive, we have found a close resemblance with Type I Abelian vortices:the lightest field of the theory is a scalar field. So if we put two vortices atlarge distance, the prevailing part of the interaction is mediated by the scalarparticles and not by vector particles. Moreover, if the two vortices have thesame orientation in the internal moduli space, the force is always attractive.In this Section we study the same problem in another theoretical setting: anextension of the Abelian-Higgs model with arbitrary scalar couplings whichis generically incompatible with the BPS limit. The simplest extension inthis direction is a theory in which there are just two mass scales; the massof the vector bosons and the mass of the scalars. There is one parameter λwhich controls the ratio of the two mass scales. We find that λ < 1 leadsto an attractive force as a usual Abelian Type I, while for λ > 1 a repul-sive force works, similarly to the usual Abelian Type II. There is no forcebetween vortices with opposite CPN−1 orientation. For λ < 1 (Type I) thisconfiguration is unstable and the true minimum of the potential correspondsto two coincident vortices with the same orientation. For λ > 1 (Type II)this configuration is stable; in other words a part of the moduli space corre-sponding to the relative distance between vortices with opposite orientationssurvives the non-BPS perturbation.

However, in more general theories where the masses of the Abelian andnon-Abelian degrees of freedom are different, we find a more complicatedpicture. There are four mass scales, the masses of the U(1) and of theSU(N) vector bosons, the masses of the scalars in the adjoint representationand the singlet of SU(N)C+F. At large distance, the interaction between twovortices is dominated by the particle with the lightest mass. So if we keepthe four masses as generic parameters, at large vortex separation we findfour different regimes that we call Type I, Type II, Type I∗ and Type II∗. Inthe last two categories repulsive and attractive interactions depend on therelative orientation. We study also numerically the interactions among twovortices at any separation with arbitrary orientations, and find that shortdistance forces also have rich qualitative features depending both on therelative orientations and the relative distance.

102

6.2.1 Theoretical set-up

A fine-tuned model

Our natural starting point is the D-term vortex theory of (3.1.2):

L = Tr

[− 1

2g2FµνF

µν +DµH(DµH)† − λ2 g2

4

(v21N −HH†

)2]. (6.2.1)

Here, for simplicity we take the same gauge coupling g for both the U(1) andSU(N) groups and we have introduced a new parameter λ. We have onlythree couplings (g, λ, v). The vacuum of the model is the usual one:

HH† = v21N . (6.2.2)

As already stated, it breaks completely the U(N) gauge symmetry, but aglobal color-flavor locking symmetry SU(N)c+f is preserved

H → UcHUf , Uc = U †f , Uc ∈ SU(N)c, Uf ∈ SU(N)f . (6.2.3)

The trace part TrH is a singlet under the color-flavor group and the tracelessparts are in the adjoint representation.

We have two mass scales, one for the vector bosons and the other for thescalar bosons. The U(1) and the SU(N) gauge vector bosons have both thesame mass

MU(1) = MSU(N) = g v. (6.2.4)

The masses of the scalars are given by the eigenvalues of the mass matrix.We start with 2N2 real scalar fields in H: N2 of them are eaten by the gaugebosons (the Higgs mechanism) and the other N2 (one singlet and the restadjoint) have same masses

Ms = Mad = λ g v. (6.2.5)

When we choose the critical coupling λ = 1 (BPS), the mass of these scalarsis the same as the mass of the gauge bosons and the Lagrangian allows anN = 2 supersymmetric extension. The BPS vortices saturating the BPSenergy bound admit infinitely degenerate set of solutions.

Models with general couplings

Together with the fine-tuned model (6.2.1) we consider the more generalmodel (3.1.2) keeping different gauge couplings, e for the U(1) part and g

103

for the SU(N) part. The more general scalar potential is written here in adifferent fashion:

L = Tr

[− 1

2g2FµνF

µν − 1

2e2fµνf

µν +DµH(DµH)†]− V, (6.2.6)

where we have defined Fµν =∑N2−1

A=1 FAµνtA and fµν = F 0

µνt0 with Tr (tAtB) =

δAB/2 and t0 = 1/√

2N 6. The scalar potential is:

V =λ2gg

2

2

N2−1∑A=1

(H i†tAHi

)2+λ2ee

2

4N

(H i†Hi −Nv2

)2

=λ2gg

2

4Tr X2 +

λ2ee

2

4Tr(X0t0 − v21N

)2, (6.2.7)

where

X ≡ HH† = X0t0 +N2−1∑A=1

XAtA, X ≡N2−1∑A=1

XAtA = 2N2−1∑A=1

(H i†tAHi

)tA.

The Lagrangian has the same symmetries as the previous fine-tuned model(6.2.1). The potential in Eq. (6.2.7) is the most general gauge invariantquartic potential which can be built with the matter content of the theory.The U(1) and the SU(N) vector bosons have different masses

MU(1) = e v, MSU(N) = g v. (6.2.8)

Moreover, the singlet part of H has a mass Ms different from that of theadjoint part Mad

Ms = λe e v, Mad = λg g v. (6.2.9)

When we take equal couplings, g = e and λ ≡ λe = λg, the scalar potential

reduces to the simple potential Vg=e = λ2g2

4Tr (X − v21N)

2. For the critical

values λe = λg = 1, the Lagrangian allows an N = 2 supersymmetric exten-sion and then the model admits BPS vortices which saturate the BPS energybound.

6.2.2 Non-Abelian vortices in the fine-tuned model

Vortex equations

Let us consider the fine-tuned model (6.2.1). For convenience, we make thefollowing rescaling of fields and coordinates:

H → vH, Wµ → gvWµ, xµ →xµgv. (6.2.10)

6 In the case g = e it is more compact to use Fµν = Fµν + fµν .

104

The Lagrangian then in Eq. (6.2.1) takes the form

L =Lg2v4

= Tr

[−1

2FµνF

µν +DµH(DµH)† − λ2

4

(1N −HH†

)2],

(6.2.11)

and the masses of vector and scalar bosons are rescaled to

MU(1) = MSU(N) = 1, Ms = Mad = λ. (6.2.12)

As explained in the introduction, the model with λ < 1 (λ > 1) in theAbelian case (N = 1) is called Type I (Type II) and the forces betweenvortices are attractive (repulsive). At the critical coupling λ = 1, there areno forces between vortices, so that multiple vortices stably coexist.

In order to construct non-BPS non-Abelian vortex solutions, we have tosolve the following 2nd order differential equations, derived from the La-grangian (6.2.1),

DµF µν − i

2

[H(DνH)† − (DνH)H†

]= 0, (6.2.13)

DµDµH +λ2

4

(1−HH†

)H = 0. (6.2.14)

From now on, we restrict ourselves to static configurations depending onlyon the coordinates x1, x2. Here we introduce a complex notation

z = x1 + ix2, ∂ =∂1 − i∂2

2, W =

W1 − iW2

2, D =

D1 − iD2

2= ∂ + iW.

The equation of motions are of course not gauge invariant but covariant. Itis possible to rewrite these equations in terms of gauge invariant quantities,instead of dealing with the original fields H and Wµ. For this purpose, wecan use the same techniques which enabled us to write the master equation(3.1.15) within the moduli matrix formalism. Rewriting our fields as follows

W (z, z) = −iS−1(z, z)∂S(z, z), H(z, z) = S−1(z, z)H(z, z), (6.2.15)

where S takes values in GL(N,C) and it is in the fundamental representationof U(N) while the gauge singlet H is an N ×N complex matrix. There is anequivalence relation (S, H) ∼ (V (z)S, V (z)H), where V (z) is a holomorphicGL(N,C) matrix with respect to z, because different elements in the sameequivalence class give us the same physical fields as in Eq. (6.2.15). In order towrite down the equations of motion (6.2.13) and (6.2.14) in a gauge invariantfashion, we introduce a gauge invariant quantity

Ω(z, z) ≡ S(z, z)S(z, z)†. (6.2.16)

105

With respect to the gauge invariant objects Ω and H, the equations (6.2.13)and (6.2.14) are written in the following form

4∂2(Ω∂Ω−1

)− H∂

(H†Ω−1

)+ ∂HH†Ω−1 = 0, (6.2.17)

Ω∂(

Ω−1∂H)

+ ∂(

Ω∂(

Ω−1H))

+λ2

4

(Ω− HH†

)Ω−1H = 0. (6.2.18)

Notice that Eq. (6.2.17) is a 3rd order differential equation. This is the pricewe have to pay in order to write down the equations of motion in terms ofgauge invariant quantities. These equations must be solved with the followingboundary conditions for k vortices:

det H → zk, Ω→ HH†, as z →∞. (6.2.19)

The field strength is given by

F12 = 2S−1∂(Ω∂Ω−1

)S. (6.2.20)

Notice that Eq. (6.2.17) is invariant under the SU(N) flavor symmetry whileEq. (6.2.18) is covariant. This leads to Nambu-Goldstone zero modes forvortex solutions.

BPS Limit

To see the relation with the BPS equations, let us take a holomorphic functionH with respect to z as

H = H0(z). (6.2.21)

Then the equations (6.2.17) and (6.2.18) reduce to

∂[4∂(Ω∂Ω−1

)−H0H

†0Ω−1

]= 0, (6.2.22)

∂(Ω∂Ω−1

)+λ2

4

(1−H0H

†0Ω−1

)= 0. (6.2.23)

These two equations are consistent only in the BPS limit λ = 1. The equa-tion (6.2.23) is the master equation for the BPS non-Abelian vortex and theholomorphic matrix H0(z) is the moduli matrix for BPS vortices. For anygiven moduli matrix H0(z), given the corresponding solution to the masterequation, the physical fields Wµ and H are obtained via Eq. (6.2.15). All thecomplex parameters contained in the moduli matrix are moduli of the BPSvortices. For example, the position of the vortices can be read from the mod-uli matrix as zeros of its determinant detH0(zi) = 0. Furthermore, the num-ber of vortices (the units of magnetic flux of the configuration) corresponds to

106

the degree of detH0(z) as a polynomial with respect to z. The classificationof the moduli matrix for the BPS vortices is given in Ref. [10, 75].

From now on, we consider a U(2) gauge theory (N = 2), which is theminimal model for non-Abelian vortices. As described in Section (3.3), theminimal winding BPS vortex is described by two moduli matrices

H(1,0)0 =

(z − z0 0−b′ 1

), H

(0,1)0 =

(1 −b0 z − z0

), (6.2.24)

and the orientational vectors are

~φ (1,0) =

(1b′

)∼ ~φ (0,1) =

(b1

). (6.2.25)

Here “∼” stands for an identification up to complex non zero factors: ~φ ∼ λ~φ,λ ∈ C∗. If we start with ~φ = (1, 0)T , we can recover ~φ = (1, b′)T by use ofthe color-flavor rotation as

~φ = U †F~φ ⇔

(10

)→(α∗ −ββ∗ α

)(10

)∼(

1β∗/α∗

)(6.2.26)

with |α|2 + |β|2 = 1. Thus we identify b′ and β∗/α∗. In what follows, wewill call two non-Abelian vortices with equal orientational vectors parallel,while when they have orthogonal orientational vectors we will call them anti-parallel. The reader must keep in mind that vortices are always parallel inreal space. Throughout this paper, we use the words parallel and anti-parallelonly referring to the internal orientational vectors.

Generic configurations of two vortices at arbitrary positions and witharbitrary orientations are described by the moduli matrices introduced inChapter (4)

H(1,1)0 =

(z − φ −η−η z − φ

), H

(2,0)0 =

(z2 − α′z − β′ 0−a′z − b′ 1

). (6.2.27)

The superscripts label patches covering the moduli space. One more patchsimilar to (2, 0) is needed to complete the full moduli space. The positions ofthe vortices are the roots of z2

i −(φ+φ)zi+φφ−ηη = z2i −α′zi−β′ = 0. By us-

ing translational symmetry we can set z1+z2 = 0 (φ+φ = α′ = 0) without loss

of generality. The orientation vectors are ~φ(1,1)

1 = (η, z1 − φ)T and ~φ(1,1)

2 =

(η, z2 − φ)T for the (1, 1) patch, while they are ~φ(2,0)

1 = (1, a′z1 + b′)T and~φ

(2,0)2 = (1, a′z2 + b′)T for the (2, 0) patch. Overall complex factor does not

have physical meaning, so that each vector takes value on CP 1. We candescribe anti-parallel vortices only in the (1,1) patch when η = η, because of

107

~φ†1~φ1 = 0. On the other hand, we can describe parallel vortices only in the(2, 0) patch when a′z1 + b′ = a′z2 + b′.

For convenience, let us take a special subspace where η = 0 in the (1, 1)patch:

H(1,1)0 red ≡

(z − z0 −η

0 z + z0

), z0 = z1 = −z2 = φ = −φ. (6.2.28)

One can always recover generic points in Eq. (6.2.27) using flavor rotations.The parameters (z0, η) and (β, a′, b′) are related by the following relations:β′ = z2

0 , a′ = 1/η and b′ = −z0/η. The orientational vectors are then of the

form

~φ(1,1)

1

∣∣z=z0

=

(10

), ~φ

(1,1)2

∣∣z=−z0 =

(η−2z0

). (6.2.29)

When the vortices are coincident, however, the rank of the moduli matrixat the vortex position reduces. In this case we can no longer define twoindependent orientations for each vortex but we can only define an overallorientation. In fact, one can see that when z0 = 0, the two orientations inEq. (6.2.29) are both equal to ~φ (1,1) = (1, 0)T . We cannot really give tothe parameter η an exact physical meaning of a relative orientation betweentwo coincident vortices. It is better, in this case, to consider this parametermerely as an internal degree of freedom of the composite vortex. Whenwe take correctly into account both the parameter η and the global flavourrotations that we previously factorized out, we recover the full moduli spacefor coincident vortices, which is WCP 2

(2,1,1) ' CP 2/Z2 [90, 91, 127].The definitions of the position and orientation of a single vortex can be

rigorously extended to the non-BPS case by merely replacing H0(z) withH0(z, z). For configurations with several vortices, all the flat directions thatare not related to Goldstone modes or translational symmetries will disap-pear. It is possible to use these definitions as constraints on the H0(z, z)matrix to fix positions and orientations. Then our formalism allows us tostudy the static interactions of non-BPS configurations.

6.2.3 Vortex interaction in the fine-tuned model

We now concentrate on the fine-tuned model (6.2.11). We will first calculatethe masses of a special class of non-BPS coincident vortices. Then we willderive an effective potential for coincident almost BPS vortices but withgeneric value of the internal modulus parameter. Finally, we will computean effective potential for two almost BPS vortices at any distance and withany relative orientations.

108

(k1, k2) coincident vortices

The minimal winding solution in the non-Abelian gauge theory is a mereembedding of the ANO solution into the non-Abelian theory. This is obviousalso from the moduli matrix view point. In fact, in the non-Abelian modulimatrix (6.2.24) we can put b (or b′) to zero with a global flavour rotation.Henceforth we can recognize the moduli matrix for the single ANO vortex:HANO

0 (z) = z − z0 as the only non-trivial element of the moduli matrix.This kind of embedding is also useful to investigate a simple non-BPS

configuration. Let us start with the moduli matrix for a configuration of kcoincident vortices. Since we have an axial symmetry around the k coincidentvortices, we can make the following reasonable ansatz for Ω and H

Ω(0,1) =

(1 00 w(r)

), H(0,1) =

(1 00 f(r)zk

). (6.2.30)

Note that f(r) = 1 means H(0,1) = H(0,1)0 (z) which is nothing but the BPS

solution. We will call the multiple vortex which is generated by the ansatzin Eq. (6.2.30) “(0, k)-vortex”. In terms of the two fields w(r) = eY (r) andf(r) the equations (6.2.17) and (6.2.18) reduce to the following form

Y ′′ +1

rY ′ − 2

f

(f ′′ +

1 + 2k

rf ′ − Y ′f ′

)− λ2

(1− r2kf 2e−Y

)= 0,

Y ′′′ +1

rY ′′ − 1

r2Y ′ + e−Y r2k−1f 2 (2k − rY ′) = 0. (6.2.31)

The boundary conditions are

Y → 2k log r, Y ′ → 2k/r, f → 0 (r →∞);

Y ′ → 0, f ′ → 0 (r → 0). (6.2.32)

Although it is quite impossible to solve these differential equation analyti-cally, we can solve them numerically. The results for the single vortex (k = 1)are shown in Fig. (6.8), where we used several different values of λ .

When k ≥ 2 it is possible that the ansatz (6.2.30) does not give thetrue solution (minimum of the energy) of the equations of motion (6.2.17)and (6.2.18). This is because there could be repulsive forces between thevortices. With ansatz (6.2.30) we fix the positions of all the vortices at theorigin by hand. The reduced equations (6.2.31) are nevertheless still usefulto investigate the interactions between two vortices. The results are listedin Table (6.1). For λ = 1, the masses are identical to integer values, upto 10−5 order, which are nothing but the winding number of the vortices.Furthermore, our numerical results for generic λ are in perfect agreement

109

2 4 6 8 10r

!2

!1

1

2

3

4

5Y

2 4 6 8 10r

0.250.50.751

1.251.51.752f

Figure 6.8: Numerical plots of Y (left) and f (right) for the single vortex: blackfor λ = 1, red for λ = 1.7, blue for λ = 0.5. The broken line is 2 log r.

λ k = 1 k = 2 k = 30.6 0.81305 1.52625 2.212050.7 0.86440 1.65337 2.421010.8 0.91231 1.77407 2.621150.9 0.95737 1.88936 2.813821 1.00000 2.00000 3.00000

1.1 1.04053 2.10655 3.180451.2 1.07922 2.20944 3.355751.3 1.11626 2.30905 3.526391.4 1.15182 2.40566 3.69276

Table 6.1: Numerical value for the masses of coincident vortices.

with the numerical value for ANO vortices obtained about 30 years ago byJacobs and Rebbi [113]. As mentioned, this happens because the (0, k)-vortexis obtained by embedding of the k ANO vortices.

There is another type of composite configuration which can easily beanalyzed numerically. These configurations are generated by the followingansatz for Ω and H

Ω(1,1) =

(w1(r) 0

0 w2(r)

), H(1,1) =

(f1(r)zk1 0

0 f2(r)zk2

).

(6.2.33)

This ansatz corresponds to a configuration with k1 composite vortices whichwind in the first diagonal U(1) subgroup of U(2) and with k2 coincident vor-tices that wind the second diagonal U(1) subgroup. The two sets of vortices

110

can be considered each as embedded ANO vortices for the two decoupledAbelian subgroups. We refer to these decoupled non-Abelian vortices as a“(k1, k2)-vortex”. The mass of a (k1, k2)-vortex is thus the sum of the massof the (k1, 0)-vortex and that of the (0, k2)-vortex. For example, the massof (1, 1)-vortex is double of the mass of the (0, 1)-vortex listed in the firstcolumn of Table (6.1). As in the previous case, we get the minima of theenergy under the constraint that the vortices are coincident.

(a) type I (! < 1)

! = 1: 2! single vortex

(b) type II (! > 1)

(0, 2)-vortex

(1, 1)-vortex

(1, 1)-vortex

(0, 2)-vortex

Figure 6.9: Spectrum of the (0, 2) and (1, 1) coincident vortices.

Because our fine-tuned non-Abelian model is a simple extension of theAbelian-Higgs model, we expect similar behavior for the interactions. Actu-ally we have only one parameter λ. Thus we will call the non-Abelian vorticesfor λ < 1 Type I, while they will be called Type II for λ > 1. From Fig.(6.9), in which is summarized the relevant data of Table (6.1), we can arguewhich kind of interaction appears between two non-Abelian vortices. In theType I case, the (0, 2)-vortex is energetically preferred to the (1, 1)-vortex,while in Type II case the (1, 1)-vortex is preferred. If the two vortices areseparated sufficiently, we can ignore any interaction between them. Regard-less of their orientations, the mass of two well separated vortices is twicethat of the single vortex. This mass is equal to the mass of the (1, 1)-vortex.Furthermore, it seems that the two separated anti-parallel vortices do notinteract, and the energy does not depend on the relative distance. For theType I case, Fig. (6.9) suggests that the configuration with (2, 0)-vortices isthe true minimum of the system. This means that attractive forces appearsbetween vortices with different orientations. An attractive force also worksin the internal space, which aligns the orientations. In the Type II case, itseems that we do not have an isolated minimum of the energy. In fact, allthe anti-parallel configurations with arbitrary distance have the same valueof the energy. In the following Sections we will confirm the picture we have

111

outlined here.

Effective potential for coincident vortices

The dynamics of BPS solitons can be investigated by the so-called moduliapproximation [96]. The effective action is a massless non-linear sigma modelwhose target space is the moduli space. The sigma model is obtained byplugging a BPS solution into the original Lagrangian and promoting themoduli parameters to massless fields, then picking up quadratic terms in thederivatives with respect to the vortex world-volume coordinates

L =

∫dx1dx2 L

[Hsol(ϕi(t, x

3)),W µsol(ϕi(t, x

3))]λ=1

, (6.2.34)

where ϕi represents the set of moduli parameters (η, η, φ, φ) or (α′, β′, a′, b′)contained in the moduli matrix (6.2.27).

If the coupling constant λ is close to the BPS limit λ = 1, we can stilluse the moduli approximation, to investigate dynamics of the non-BPS non-Abelian vortices by adding a potential of order |1− λ2| 1 to the masslesssigma model

L =

∫dx1dx2 L

[Hsol(ϕi(t, x

3)),W µsol(ϕi(t, x

3))]λ=1− V (ϕi). (6.2.35)

We shall now calculate the effective potential V (ϕi) using the method sug-gested by Hindmarsh, who calculated this effective potential for non-BPSsemi-local vortex in the Abelian-Higgs model [128].

First we write the Lagrangian (6.2.11) in the following way

L = LBPS +(λ2 − 1)

4

(1N −HH†

)2. (6.2.36)

We get non-BPS corrections of order O(λ2 − 1) by putting BPS solutionsinto Eq. (6.2.36). This is because the first term is minimized by the BPSsolution, while the second one is already a term of order O(λ2 − 1). Theenergy functional thus takes the following form

E =E

2πv2= 2 +

(λ2 − 1)

8π

∫dx1dx2 Tr

(1−HBPS(ϕi)H

†BPS(ϕi)

)2

(6.2.37)

whereHBPS(ϕi) stands for the BPS solution generated by the moduli matricesin Eq. (6.2.27). The first term corresponds to the mass of two BPS vortices

112

and the second term is the deviation from the BPS solutions which is nothingbut the effective potential we want.

In this Section we consider the effective potential on the moduli spaceof coincident vortices. To this end, it suffices to consider only the followingmatrices

H(1,1)0 =

(z −η0 z

), H

(2,0)0 =

(z2 0−a′z 1

). (6.2.38)

The parameters η and a′ are related by η = 1/a′. The effective potential onthe moduli space for two coincident vortices is thus 7

V (|η|)2πv2

≡ (λ2 − 1)

8π

∫dx1dx2 Tr

(1−HBPS(|η|)H†BPS(|η|)

)2

≡ (λ2 − 1)V(|η|), (6.2.39)

where we have defined a reduced effective potential V which is independentof λ. To evaluate this effective potential, we need to solve the BPS equationsfor a composite state of two non-Abelian vortices with an intermediate valueof η. Such numerical solutions found in [91]. We propose here anotherreliable technique for the numerics, which needs much less algebraic efforts.In the moduli matrix formalism, what we should solve is only the master

2 4 6 8 10 12r

!6

!4

!2

2

4

2 4 6 8 10 12r

0.2

0.4

0.6

0.8

1

Magnetic Flux

Figure 6.10: Left: Numerical plots of Y1 (red), Y2 (green) and Y3 (blue) for η = 3.The black line is Y1 = Y3 for η = 0. Right: Magnetic flux TrF12 with η = 0 (red),η = 3 (green) and η =∞ (a′ = 0) (blue).

equation (6.2.23) for Ω with λ = 1 and H(z, z) = H0(z). Because of the axialsymmetry of the composite vortex and the boundary condition at infinity:

Ω→ H0(z)H†0(z), (6.2.40)

7 The potential depends only on |η| because the phase of η can be absorbed bythe flavor symmetry. The same holds for the coordinate a′

113

we can make a simple ansatz for Ω. For example in the patch (1, 1) we canwrite

Ω(1,1) =

(w1(r) −ηe−iθw2(r)

−ηeiθw2(r) w3(r)

). (6.2.41)

The advantage of the moduli matrix formalism is that only three functionswi(r) are needed and the formalism itself is gauge invariant. Plugging theansatz into Eq. (6.2.23), after some algebra we get the following differentialequations

Y ′′1 +1

rY ′1 +

|η|2r2

(1 + rY ′1 − rY ′2)2

|η|2 − eY1+Y3−2Y2= 1− e−Y1(r2 + |η|2);

Y ′′2 +1

rY ′2 −

1

r2

(1 + rY ′1 − rY ′2)(1 + rY ′2 − rY ′3)

1− |η|2e−Y1−Y3+2Y2= 1− e−Y2r;

Y ′′3 +1

rY ′3 +

|η|2r2

(1 + rY ′2 − rY ′3)2

|η|2 − eY1+Y3−2Y2= 1− e−Y3r2, (6.2.42)

where we have redefined the fields as wi(r) = eYi(r) with i = 1, 2, 3. We solvenumerically these differential equations using a simple relaxation method,see Fig. (6.10).

The effective potential can be obtained by plugging numerical solutionsinto Eq. (6.2.39). The result is shown in Fig. (6.11). When we consider

5 10 15 20!!!

0.42

0.44

0.46

0.48

0.52

0.54

V

0.01 0.02 0.03 0.04!a’!

0.5405

0.5415

0.542

0.5425

V

Figure 6.11: Numerical plots of the effective reduced potential V(|η|). In thesecond plot a′ = 1/η.

configurations with big values of |η|, it is better to switch to the other patchand use the variable a′. We can still make a similar ansatz for Ω, leading tosimple differential equations like those in Eqs. (6.2.42). In the left of Fig.(6.11) we show a numerical plot of the reduced effective potential V from|η| = 0 (|a′| = ∞) to |η| = 20. In the right of Fig. (6.11) we show another

114

plot from |a′| = 0 (|η| = ∞) to |a′| = 1/20. To give an estimate of therange of validity of our approximation we can compare the results obtainedin this Section with the numerical integrations obtained for (2, 0)-vorticesand also (1, 1)-vortices. The comparison is made in Table (6.2) from whichwe can argue that the effective potential approximation gives result with anaccuracy around 10% for the range of values 0.7 < λ < 1.15.

λ (2, 0)num (2, 0)eff (1, 1)num (1, 1)eff

0.6 1.52625 1.65280 1.62611 1.734400.7 1.65337 1.72332 1.72880 1.788350.9 1.88936 1.89692 1.91473 1.921150.95 1.94523 1.94711 1.95793 1.95954

1 2.00000 2.00000 2.00000 2.000001.05 2.05376 2.05561 2.04102 2.042541.1 2.10655 2.11393 2.08106 2.087151.15 2.15843 2.17496 2.12018 2.133841.2 2.20944 2.23870 2.15843 2.18260

Table 6.2: Numerical value for the masses of coincident vortices. (2, 0)num is forthe numerical results while (2, 0)eff is for our approximation using the effectivepotential.

In the Type II case ((λ2 − 1) > 0) the effective potential has the samequalitative behavior as showed in the figure. As we expected, it has a min-imum at |η| = 0. This matches the previous result that the (1, 1)-vortex isenergetically preferred to the (2, 0)-vortex. In the Type I case ((λ2− 1) < 0)the shape of the effective potential can be obtained just by flipping the over-all sign of the effective potential of the Type II case. Then the effectivepotential always takes a negative value, which is consistent with the factthat the masses of the Type I vortices are less than that of the BPS vortices,see the middle column in Table (6.1). Contrary to the Type II case, theType I potential has a minimum at |a′| = 0 (|η| =∞). This means that the(2, 0)-vortex is preferred with respect to the (1, 1) vortex.

Interaction at generic vortex separation

In this subsection we go on investigating the interactions of non-Abelianvortices in the U(2) gauge group at generic distances. As in Sect. (6.2.3) wewill use the moduli space approximation, considering only small deviationsfrom the BPS case. The generic configurations are described by the modulimatrices in Eq. (6.2.27). We will consider here only the reduced (1, 1) patch

115

defined in Eq. (6.2.28). By putting the two vortices on the real axis we canreduce z0 to a real parameter d. Furthermore, by the flavor symmetry, wecan freely put η = 0 and suppress the phase of η. The relevant configurationswill be described by the following moduli matrix:

H(1,1)0 red =

(z − d −η

0 z + d

), (6.2.43)

where 2d is the relative distance and η the relative orientation.

-10 -5 5 10d

0.2

0.4

0.6

0.8

1

-10 -5 5 10d

0.2

0.4

0.6

0.8

1

Figure 6.12: The magnetic flux TrF12 of a configuration of 2 BPS vortices. Thefigure shows slices including the centers of the vortex. (red, green, blue) correspondto η = (0, 4,∞) while (solid, small broken, wide broken) lines to d = (0, 1, 4). Theleft panel shows the fine-tuned model with e = g and the right shows the modelwith e = 2g.

Now let us study the effective potential as function of η and d. We firstneed the numerical solution to the BPS master equation for two vorticeswith any relative distance and orientation. Unlike the computation for thecoincident vortices, we do not have an axial symmetry. We can no longer re-duce the problem to one spatial dimension by making an appropriate ansatz.Nevertheless the moduli matrix formalism is a powerful tool also for the nu-merical calculations. The master equation is a 2 by 2 hermitian matrix, so itincludes four real 2nd order partial differential equations. Despite the greatcomplexity of this system of coupled equations, the relaxation method is veryeffective to solve the problem. We show several numerical solutions in Fig.(6.12). As before, once we get the numerical solution to the BPS equations,the effective potential is obtained by plugging them into Eq. (6.2.37).

The numerical plot of the reduced effective potential V is shown in Fig.(6.13). The effective potential for the Type II case has the same shape, upto a small positive factor (λ2− 1). The potential forms a hill whose top is at

116

(d, |η|) = (0,∞). It clearly shows that two vortices feel repulsive forces, inboth the real and internal space, for every distance and relative orientation.The minima of the potential has a flat direction along the d-axis where theorientations are anti-parallel (η = 0) and along the η axis at infinite distance(d =∞). Therefore the anti-parallel vortices do not interact.

0.45

0.50

V

0 1 2 3 4 5d

0

2.5

57.5

10|!|

Figure 6.13: Numerical plot of the reduced effective potential V(η, d).

In the Type I case (λ < 1) the effective potential is upside-down of thatof the Type II case. There is unique minimum of the potential at (d, |η|) =(0,∞). This means that attractive force works not only for the distancein real space but also among the internal orientations. Configurations withanti-parallel orientations do not interact, but these configurations representunstable points of equilibrium. Type I vortices always stick together.

6.2.4 Vortices with generic couplings

In this Section we will shift from the fine-tuned U(N) model (6.2.1) to themore general model defined in Eqs. (6.2.6) and (6.2.7). This will lead tomore complicated algebra, but we will also clarify interactions with differentqualitative behaviors. There are 5 parameters (g, e, λg, λe, v) in the model,but we can reduce their number with the following rescaling

H → vH, Wµ → evWµ, xµ →xµev. (6.2.44)

117

Then the Lagrangian (6.2.6) is expressed as follows

L = Tr

[− 1

2γ2FµνF

µν − 1

2fµνf

µν +DµH(DµH)†]−

− γ2λ2g

4Tr X2 − λ2

e

4Tr(X0t0 − 1N

)2, (6.2.45)

where L = L/e2v4 and we have introduced the ratio of the two gauge cou-plings

γ ≡ g

e. (6.2.46)

We have three effective couplings γ, λe, λg and the rescaled masses of particlesare

MU(1) = 1, MSU(N) = γ, Ms = λe, Mad = γλg. (6.2.47)

The Lagrangian can be thought of as the bosonic part of a supersymmetricmodel only when both the parameters λe and λg are unity. In the BPS casewe can easily find the BPS equations

DH = 0, F12 =γ2

2X, F 0

12t0 =

1

2

(X0t0 − 1N

). (6.2.48)

The last two equations can be rewritten in a compact way

F12 =1

2(X − 1N) +

γ2 − 1

2X. (6.2.49)

The first BPS equation in Eq. (6.2.48) can be solved using the moduli matrixin the usual way

H = S−1(z, z)H0(z), W = −iS−1∂S, (6.2.50)

where S is a GL(N,C) matrix. We would like to stress that this solutiondoes not depend on γ so that the moduli space of the BPS vortices is thesame as that of the well investigated vortices in the equal gauge couplingtheory g = e. The Eq. (6.2.49) can be rewritten in a gauge invariant fashionas

∂(Ω∂Ω−1

)=

1

4

(Ω0Ω−1 − 1N

)+γ2 − 1

4

(Ω0Ω−1 − Tr (Ω0Ω−1)

N1N

)where Ω = SS† is same as before and Ω0 ≡ H0H

†0.

118

Now we are ready to investigate interactions between two almost BPSvortices by using same strategy as we used in Sec. 4. An effective action ofthe moduli dynamics for appropriately small |1 − λ2

e,g| 1 is obtained byplugging BPS solutions into the action. Then we get

V (η, d; γ, λe, λg)

2πv2=

∫dx2

(γ2(λ2

g − 1)

8πTr X2 +

λ2e − 1

8πTr(X0t0 − 1N

)2)

=1

2π

∫dx2 Tr

[λ2g − 1

γ2(F12)2 + (λ2

e − 1)(F 012t

0)2

],

(6.2.51)

where we have used the BPS equations in the second line. Let us define theAbelian and the non-Abelian potentials as

Ve(η, d; γ) =

∫dx2 Tr (F 0

12t0)2, Vg(η, d; γ) =

∫dx2 Tr (F12)2. (6.2.52)

The true potential is a linear combination of them

V (η, d; γ, λe, λg) = (λ2e − 1)Ve(η, d; γ) +

λ2g − 1

γ2Vg(η, d; γ). (6.2.53)

Notice that Ve,g is determined by the BPS solutions, so it does not dependon λe,g.

Equal gauge coupling γ = 1 revisited

In Sec. 4 we have discussed the effective potential for two vortices for anyseparation and with any relative orientation in a model with γ = 1 andλ = λg = λe. The potential is shown in Fig. (6.13). The effective potentialcomes from two pieces: the Abelian Ve and the non-Abelian Vg potentialgiven in Eq. (7.2). In Figs. (6.14) and (6.15) we show Ve and Vg takingseveral slices of Fig. (6.13).

Let us consider the case with λ2e − 1 > 0 and λ2

g − 1 > 0. In this case theeffective potential will have the same qualitative behaviors like the reducedpotentials in the Figs. (6.14), (6.15) and (6.16). The figures shows how Veand Vg behaves very differently. In particular, the Abelian potential is alwaysrepulsive, both in the real and internal space8 (see the red dots in Figs. (6.14)and (6.15)). The non-Abelian potential is on the contrary sensitive on theorientations. In particular, Fig. (6.14) shows that it is repulsive for parallel

8Attraction in the internal space here just means that orientations tend tobecome the same.

119

1 2 3 4 5d

0.1

0.2

0.3

0.4

1 2 3 4 5d

0.1

0.2

0.3

0.4

1 2 3 4 5d

0.1

0.2

0.3

0.4

anti-parallel intermediate parallel

Figure 6.14: Effective potential vs. distance of vortices d. Red dots denote Vewhile blue denote Vg. All plots are for γ = 1. The left figure shows the twoanti-parallel vortices η = 0, the middle shows η = 4.1 and the right shows the twoparallel vortices η =∞.

2 4 6 8 10!

0.1

0.2

0.3

0.4

2 4 6 8 10!

0.1

0.2

0.3

0.4

2 4 6 8 10!

0.1

0.2

0.3

0.4

d = 0 d = 1 d = 4

Figure 6.15: Effective potential vs. η. Red dots denote Ve while blue denote Vg.

vortices while it is attractive for anti-parallel ones. Furthermore, the non-Abelian potential becomes almost flat (along the spatial coordinate d) withthe orientation η ∼ 4, see the middle of Fig. (6.14). The blue dots in Fig.(6.15) reveal that the non-Abelian potential always gives attractive forces inthe internal space. When the two scalar couplings are equal, λ2

e = λ2g, the

left picture in Fig. (6.14) clearly shows how the two potentials exactly cancelfor anti-parallel vortices, recovering the result of the previous Section.

Of course, the true effective potential depends on λe and λg through thecombination in Eq. (6.2.53). This indicates the interaction between non-Abelian vortices is quite rich in comparison with that of the ANO vortices.Let us make a further example. The rightmost panel in Fig. (6.14) showsthat Ve and Vg for parallel vortices (η = ∞) are identical9. Thus if we takeλ2e+λ2

g = 2, the interactions between parallel vortices vanishes while betweenanti-parallel vortices they do not canceled out. This is just opposite to whatwe found in the case λ2

e = λ2g. We summarize the possible behaviors of the

Abelian and non-Abelian forces, when the gauge couplings are equal e = g

9This statement can be proved analytically.

120

0.20.250.30.350.4

02.5

57.5

10

0 2 4

Ve

d

|!|

0

0.1

0.2

02.5

5 7.510

0 2 4

Vg

d

|!|

Figure 6.16: The Abelian potential Ve (left) and the non-Abelian potential Vg(right) for γ = 1.

λ2g > 1 λ2

g = 1 λ2g < 1

λ2e > 1

N=(−,+)A=+

N=0A=+

N=(+,−)A=+

λ2e = 1 N=(−,+)

A=0

N=0A=0

N=(+,−)A=0

λ2e < 1

N=(−,+)A=−

N=0A=−

N=(+,−)A=−

Table 6.3: The forces between two vortices in the γ = 1 case. N and A standfor non-Abelian force and Abelian force, respectively. 0 means no force, + meansrepulsive and − means attractive. N=(−,+) means that there is an attractiveforce for anti-parallel vortices and a repulsive force for parallel vortices.

(γ = 1), in Table (6.3).

Different gauge coupling γ 6= 1

We now consider interactions between non-Abelian vortices with differentgauge coupling e 6= g (γ 6= 1). Several numerical solutions are given in theleft panel of Fig. (6.12). In Figs. (6.17) and (6.18) we show two numericalexamples for the reduced effective potentials Ve, Vg given in Eq. (7.2).

The plots show that the qualitative features of Ve and Vg are basicallythe same as what is discussed in the equal gauge coupling case (γ = 1).Therefore, the qualitative classification of the forces given in Table (6.3) isstill valid for γ 6= 1. We observe that the Abelian potential tends, at largedistances, to a value smaller than the non-Abelian one for γ < 1, whileopposite happens for γ > 1. The only things that have non dependence

121

1 2 3 4 5d

0.1

0.2

0.3

0.4

1 2 3 4 5d

0.1

0.2

0.3

0.4

1 2 3 4 5d

0.1

0.2

0.3

0.4

anti-parallel (η = 0) intermediate (η = 4) parallel (η =∞)

Figure 6.17: Effective potential with γ = 1/2 vs. separation. (red,blue)=(Ve,Vg).

1 2 3 4 5d

0.1

0.2

0.3

0.4

1 2 3 4 5d

0.1

0.2

0.3

0.4

1 2 3 4 5d

0.1

0.2

0.3

0.4

anti-parallel (η = 0) intermediate (η = 4) parallel (η =∞)

Figure 6.18: Effective potential with γ = 1.3 vs. separation. (red,blue)=(Ve,Vg).

on γ are the values of the potentials at (1, 1)-vortices with (d, η) = (0, 0).Regardless of the gauge couplings Vg(η = 0, d = 0) = 0 while Ve(η = 0, d =0) ; 0.41. This is because the corresponding solution is proportional to theunit matrix, so that there are no contributions from the non-Abelian part.

The effective potential is obtained from the linear combination in Eq.(6.2.53) and also depends on three parameters γ, λe and λg. With this bigfreedom we can obtain a lot of interesting interactions. For example, wecan have potentials which develop a global minimum at some finite non zerodistance. In such cases two vortices may be bounded at that distance. Wecan show a concrete potential in Fig. (6.19). The figure shows the presenceof a minimum around d ∼ 2.10 This kind of behavior have not been foundfor the ANO Type I/II vortices and the possibility of bounded vortices reallyresults from the non-Abelian symmetry11.

10These kind of potentials are really generic. In fact, the presence of this minimadoes not require strong constraints on the couplings.

11Similar behaviors in the static intervortex potential were also observed for Z3

vortices in [129].

122

1 2 3 4 5

0.16

0.17

0.18

0.19

0.21

0.22

Figure 6.19: γ = 1/2, λe = 1.2, λg = 1.06: From η = 0 (green) to η = 7 (blue)with d = 0 ∼ 5 for each η.

6.2.5 Interaction at large vortex separation

Vortices in fine-tuned models e = g and λe = λg

In this subsection we will obtain an analytic formula for the asymptotic forcesbetween vortices at large separation. We follow the technique developed inRefs. [115, 17]. First of all, we need to find asymptotic behaviors of thescalar and the gauge fields. We again consider the (1, 0)-vortex

H0(z)(1,0) =

(z 00 1

), ~φ

(1,0)1 =

(10

). (6.2.54)

When we are sufficiently far from the core of the vortex, Y (r) and f(r) inEqs. (6.2.31) can be written as

Y = 2 log r + δY, f = 1 + δf (6.2.55)

where δY and δf are small quantities. Plugging these into Eqs. (6.2.31) andtaking only linear terms in δY and δf , we obtain the following linearizedequations

δY ′′ +1

rδY ′ − λ2δY − 2

(δf ′′ +

1

rδf ′ − λ2δf

)= 0, (6.2.56)

δY ′′′ +1

rδY ′′ − 1

r2δY ′ − δY ′ = 0. (6.2.57)

Solutions to these equations are analytically obtained to be

δY − 2δf = − qπK0(λr), δY =

m

πK0(r)− C, (6.2.58)

123

where K0(r) is the modified Bessel’s function of zeroth order and q,m andC are integration constants. C must be 0 because δY → 0 as r → ∞ whileq,m should be determined by the original equation of motion. In the BPScase, δf ≡ 0 (f ≡ 1), thus q = −m. Eq. (6.2.58) leads to the well knownasymptotic behavior of the ANO vortex

H[1,1] = fe−12Y z '

(1 + δf − 1

2δY

)eiθ =

(1 +

q

2πK0(λr)

)eiθ, (6.2.59)

W[1,1] = − i2∂Y ' − i

4eiθ

d

dr(2 log r + δY ) = − i

2

(1

r− m

2πK1(r)

)eiθ,(6.2.60)

where K1 ≡ −K ′0 and we have defined H[1,1] and W[1,1] as [1, 1] elements ofH and W in Eq. (6.2.15) with the k = 1 ansatz (6.2.30).

Next we treat the vortices as point particles in a linear field theory coupledwith a scalar source ρ and a vector current jµ. To linearize the Yang-Mills-Higgs Lagrangian, we choose a gauge such that the Higgs fields is given bythe following hermitian matrix

H =

(1 00 1

)+

1

2

(h0 + h3 h1 − ih2

h1 + ih2 h0 − h3

), Wµ =

1

2

(w0µ + w3

µ w1µ − iw2

µ

w1µ + iw2

µ w0µ − w3

µ

).

(6.2.61)

Here all the fields ha, waµ are real. Then the quadratic part of the Lagrangian(6.2.11) is of the form

L(2)free =

3∑a=0

[−1

4faµνf

aµν +1

2waµw

aµ +1

2∂µh

a∂µha − λ2

2(ha)2

](6.2.62)

where we have defined the Abelian field strength faµν ≡ ∂µwaν − ∂νw

aµ. We

also take into account the external source terms to realize the point vortex

Lsource =3∑

a=0

[ρaha − jaµwaµ

]. (6.2.63)

The scalar and the vector sources should be determined so that the asymp-totic behavior of the fields in Eqs. (6.2.59) and (6.2.60) are replicated. Equa-tions of motions are of the form(

+ λ2)ha = ρa, (+ 1)waµ = jaµ. (6.2.64)

In order to replicate the (1, 0)-vortex corresponding to the k = 1 ansatz(6.2.30), we just need to mimic the result of Refs. [115, 17] because the single

124

non-Abelian vortex is a mere embedding of the ANO vortex as mentionedearlier. In fact, only (h0, w0

µ, ρ0, j0

µ) = (h3, w3µ, ρ

3, j3µ) are relevant and all the

others are zero:

h0 = h3 =q

2πK0(λr), ρ0 = ρ3 = qδ(r),

w0 = w3 = −m2π

k×∇K0(r), j0 = j3 = −mk×∇δ(r) , (6.2.65)

where k is a spatial fictitious unit vector along the vortex world-volume. Thevortex configuration with general orientation is also treated easily, since theorigin of the orientation is the Nambu-Goldstone mode associated with thebroken SU(2) color-flavor symmetry

H0 → H0(z)(1,0)UF, ~φ2 = U †F~φ

(1,0)1 =

(α∗

β∗

), UF ≡

(α β−β∗ α∗

),

(6.2.66)

where |α|2 + |β|2 = 1. The fields H and Wµ receive the following transfor-mations, keeping the hermitian form of (6.2.61).(

X 00 0

)→ U †F

(X 00 0

)UF =

(|α|2 α∗βαβ∗ |β|2

)X. (6.2.67)

The scalar interaction between a vortex at x = x1 with the orientation ~φ1

and another vortex at x = x2 with the orientation ~φ1 can be obtained by

Lh =

∫dx2 Tr

[(h0(x− x1) 0

0 0

)(|α|2 α∗βαβ∗ |β|2

)ρ0(x− x2)

]= |α|2 q

2

2πK0(λ|x1 − x2|). (6.2.68)

The gauge interaction is also obtained by similar way

Lw = −∫dx2 Tr

[(w0(x− x1) 0

0 0

)·(|α|2 α∗βαβ∗ |β|2

)j0(x− x2)

]= −|α|2m

2

2πK0(|x1 − x2|). (6.2.69)

Then total potential is Vint = −Lh − Lw

Vint = −

∣∣∣~φ†1~φ2

∣∣∣2∣∣∣~φ1

∣∣∣2 ∣∣∣~φ2

∣∣∣2(q2

2πK0(λ|x1 − x2|)−

m2

2πK0(|x1 − x2|)

), (6.2.70)

125

where |α|2 =|~φ†1~φ2|2|~φ1|2|~φ2|2 is invariant under the global color-flavor rotation.

When two vortices have parallel orientations, this potential becomes that oftwo ANO vortices [115]. On the other hand, the potential vanishes when theirorientations are anti-parallel. This agrees with the numerical result found inthe previous Sections. In the BPS limit λ = 1 (q = m), the interactionbecomes precisely zero.

Since K0(λr) ∼√π/2λre−λr, the potential asymptotically reduces to

Vint '

− |

~φ†1~φ2|2

|~φ1|2|~φ2|2q2

2π

√π

2λre−λr for λ < 1 Type I

|~φ†1~φ2|2|~φ1|2|~φ2|2

m2

2π

√π

2re−r for λ > 1 Type II

(6.2.71)

where r ≡ |x1 − x2| 1. If we fix the relative orientation being some finitevalue, the force Fr = −∂rVint between two vortices is attractive for λ < 1and repulsive for λ > 1 similar to the force between ANO vortices. The forcevanishes when the relative orientation becomes anti-parallel. If we fix thedistance by hand, the orientations tend to be anti-parallel for the Type IIwhile the parallel configuration are preferred for the Type I case.

Vortices with general couplings

It is quite straightforward to generalize the results of the previous Sectionto the case of generic couplings. We can of course use the same gauge as inEq. (6.2.61). The quadratic Lagrangian (6.2.6) is of the form

L(2)free =

3∑a=1

[− 1

4γ2faµνf

aµν +1

2waµw

aµ +1

2∂µh

a∂µha − λ2gγ

2

2(ha)2

]+

[−1

4f 0µνf

0µν +1

2w0µw

0µ +1

2∂µh

0∂µh0 − λ2e

2(h0)2

]. (6.2.72)

The external sources can be still reproduced by source terms as in Eqs. (6.2.63),(6.2.64). The linearized equations following from the above Lagrangian areof the form (

1

γ2+ 1

)waµ = jaµ,

(+ λ2

gγ2)ha = ρa,

(+ 1)w0µ = j0

µ,(+ λ2

e

)h0 = ρ0. (6.2.73)

For the (1, 0)-vortex the only non-zero profile functions are (h0, w0µ, ρ

0, j0µ)

and (h3, w3µ, ρ

3, j3µ). But these profiles are no longer equal and we need to

126

deal with them independently. The only difference from the similar equations(6.2.64) is for the masses of the particles. The masses are directly related toasymptotic tails of vector and scalar fields. We can easily find the solutionsby doubling Eqs. (6.2.65)

h0 =q0

2πK0(λer), w0 = −m

0

2πk×∇K0(r),

ρ0 = q0 δ(r), j0 = −m0 k×∇δ(r),

h3 =q3

2πK0(λgγr), w3 = −m

3

2πk×∇K0(γr),

ρ3 = q3 δ(r), j3 = −m3 k×∇δ(r). (6.2.74)

The vortex with the orientation ~φ1 in Eq. (6.2.66) can be obtained by per-forming an SU(2)C+F rotation like Eq. (6.2.67)

(X0+X3

20

0 X0−X3

2

)→(

X0

2+ (|α|2 − |β|2) X3

2(α∗β − βα∗) X0

2+ (α∗β + βα∗) X3

2

(αβ∗ − β∗α) X0

2+ (α∗β + βα∗) X3

2X0

2− (|α|2 − |β|2) X3

2

).

Similar to Eqs. (6.2.68) and (6.2.69), we find the total potential Vint

Vint =1

2

(−(q0)2

2πK0(λe|x1 − x2|) +

(m0)2

2πK0(|x1 − x2|)

)+

+

∣∣∣~φ†1~φ2

∣∣∣2∣∣∣~φ1

∣∣∣2 ∣∣∣~φ2

∣∣∣2 −1

2

(−(q3)2

2πK0(λgγ|x1 − x2|) +

(m3)2

2πK0(γ|x1 − x2|)

).

When we tune the parameters to γ = 1 and λg = λe (q0 = q3, m0 = m3),this effective potential is exactly identical to that of Eq. (6.2.70). In the BPSlimit λe = λg = 1, the interaction becomes precisely zero because q0 = m0

and q3 = m3.

At large distance, the interactions between vortices are dominated by the

127

particles with the lowest mass Mlow. There are four possible regimes

Vint =

−(q0)2

4π

√π

2λere−λer for Mlow = Ms, Type I

−(|~φ†1~φ2|2|~φ1|2|~φ2|2 −

12

)(q3)2

2π

√π

2λgγre−λgγr for Mlow = Mad, Type I∗

(m0)2

4π

√π

2re−r for Mlow = MU(1), Type II(

|~φ†1~φ2|2|~φ1|2|~φ2|2 −

12

)(m3)2

2π

√π

2γre−γr for Mlow = MSU(2), Type II∗

(6.2.75)

which exhaust all the possible kinds of asymptotic potentials of this system.This generalizes the Type I/II classification of Abelian superconductors. Wehave found two new categories, called Type I∗ and Type II∗, in which theforce can be attractive or repulsive depending on the relative orientation.The Type I force is always attractive and the Type II force is repulsiveregardless of the relative orientation. On the other hand, Type I∗ and TypeII∗ depend on the relative orientation. In the Type I∗ case the forces betweenparallel vortices are attractive while anti-parallel vortices repel each other.The Type II∗ vortices feel opposite forces to the type I∗12. Note that wehave used the same terms Type I/II for the fine-tuned model in sect. (6.2.5).In the perspective of this Section, they should be called Type I+I∗/II+II∗

because of the degeneracy of some masses. It is interesting to compare theseresults with the recently studied asymptotic interactions between non-BPSnon-Abelian global vortices [130, 124, 125]. A very different feature of globalvortices is that the interactions are always repulsive. This is because theyare mediated by Nambu-Goldstone zero modes, whereas in our model theseparticles are all eaten by the gauge bosons thanks to the Higgs mechanism.

We find a nice matching of qualitative features between the numericalresults of the previous Sections and the semi-analytical results in this Section.Let us look at Figs. (6.14), (6.17) and (6.18). In all the cases, we found thatthe Abelian potentials are attractive regardless of the orientations while thenon-Abelian potentials are sensitive to those. These properties are well shownalso in the semi-analytical results in (6.2.75). The Type I/II interactionsoriginated by the U(1) part are independent of the orientations whereas theType I∗/II∗ which are coming from the SU(N) part do depend on them.

12In the previous Section we have found a similar result in a supersymmetrictheory. The relation between the two notations is explained in the Appendix. Thesupersymmetric theory shows only type I/I∗ behaviors.

128

The result in Eq. (6.2.75) is easily extended to the general case of U(1)×SU(N). This can be done by just thinking of the orientation vectors ~φ astaking values in CPN−1.

6.2.6 Summary

In this Section we have studied static interactions between non-BPS vor-tices in SU(N) × U(1) gauge theories with Higgs fields in the fundamentalrepresentation. We have discussed models with arbitrary gauge and scalarcouplings. We have numerically computed the effective potential for almostBPS configurations for arbitrary separations and any internal orientations.We have also obtained analytic expressions for the static forces between wellseparated non-BPS vortices. This expression is valid also for models far fromBPS limit.

For the fine-tuned model we found interaction pattern similar to thatof the ANO vortices in the Abelian-Higgs model. The numerical effectivepotential is given in Fig. (6.13), it depends on both the relative distance andthe orientations. The asymptotic potential between two vortices is given byEq. (6.2.71). In this model the mass of the U(1) and of the SU(N) vectorbosons are same MSU(N) = MU(1), and also all the scalars have the samemasses Ms = Mad. We thus have only two mass scales, which corresponds totwo different asymptotic regimes. For λ < 1 (Ms < MU(1)) there is universalattraction (Type I) and for λ > 1 (Ms > MU(1)) universal repulsion (TypeII). Both the numerical and the analytical result show that the interactionsbetween two anti-parallel vortices vanish; this configuration is unstable forType I vortices and stable for Type II. So in this last case the part of themoduli space which corresponds to vortices with opposite CP 1 orientationsat arbitrary distance survives the non-BPS perturbation.

In models with arbitrary couplings, on the other hand, the pattern ofinteractions becomes richer. In this case we considered separately the Abelianand non-Abelian contributions Ve and Vg to the effective potential. The twoshow very different qualitative behavior. While the Abelian contribution isalways attractive (or repulsive) for a given choice of the parameters, the non-Abelian one can be attractive or repulsive depending on the relative internalorientation of the two vortices. These properties, combined with the fact thatthe full effective potential is the linear combination given in Eq. (6.2.53),give rise to many qualitatively different type of interactions depending onthe choice of the parameters of the theory. Such a variety also appears inthe possible asymptotic behavior of the interactions. In the theory there arefour different mass scales, the masses of the SU(N) vector bosons MSU(N),the U(1) vector boson MU(1), the adjoint scalars Mad and the singlet scalar

129

Ms under the color-flavor symmetry. This leads to four different asymptoticregimes, classified in Eq. (6.2.75). We found, in addition to Type I and TypeII, new types of interaction mediated by the non-Abelian particles, whichwe call Type I∗ and Type II∗. The Type I (Type II) force is attractive(repulsive), and occurs when the singlet scalar (U(1) vector boson) has thesmallest mass. These forces do not depend on the relative orientation. In theType I∗ case there is an attractive force for parallel orientations and repulsivefor anti-parallel ones where the asymptotic force is mediated by the lightestnon-Abelian scalar fields. On the other hand for Type II∗ there is repulsionfor parallel orientation and attraction for anti-parallel ones and the force atlarge distance is mediated by the lightest non-Abelian vector field.

The dynamics of the interactions of the non-BPS non-Abelian vortices isquite rich. Let us give comments on some possible further directions:

• Non-Abelian vortices and Abrikosov lattice: In the usual Type II su-perconductor (the Abelian-Higgs model), if a large number of vorticespenetrate a region of given area A, they will form a hexagonal lattice(Abrikosov lattice) rather than forming a square one. This is verifiedexperimentally. If we consider the same for non-Abelian vortices, weexpect that the property of the lattice can be quite different, say, latticespacing and/or form can change. An interesting possibility is the ap-pearance of phase transitions due to the change of the lattice structurewhen the density of vortices change. This eventuality is suggested bythe possible presence of interactions which change with distances. Forexample in the case of Fig. (6.19) there are repulsive forces at shortdistances while at large distances they are attractive.

• Quantum aspects: In this paper we focused on the classical aspectsof the interactions between non-Abelian vortices. In the theoreticalset-up that we have discussed, the quantum aspects of the infraredphysics of a single vortex are described by an effective bosonic CPN−1

sigma model (the theoretical setting is in this sense similar to the onediscussed Ref. [21]). Let us consider two vortices at large distance; inthe Type I∗ and in the Type II∗ regimes the quantum physics will bedescribed by two CPN−1 sigma models with an interaction potentialgiven by Eq. (6.2.75). It would be interesting to study the effect of thisterm in the sigma model physics. Another interesting problem is thenumerical determination of the effective theory for vortices at genericseparations. To do this one has to determine the Manton metric on thefull moduli space.

130

Chapter 7

Stability of non-Abeliansemi-local vortices

In this Chapter we study the stability of semi-local vortices in the presenceof the N = 1 perturbation which makes the vortices non-BPS. In the case ofa theory with a mass gap (Nf = Nc), the ’t Hooft standard classification ofpossible massive phases [131] is applicable; as a result, there is a very clearrelation between the Higgs mechanism and magnetic confinement. In thecase with several additional flavors (Nf > Nc), this is not a priori clear dueto the presence of massless degrees of freedom (see Ref. [132] for some of thesubtleties associated with the massless case).

The question is also important from the point of view that the vortexsystem (with gauge group H) being studied is actually a low-energy approx-imation of an underlying system with a larger gauge group, G (See Chapter(2). The homotopy-map argument shows that the regular monopoles aris-ing from the symmetry breaking G → H are actually unstable in the fulltheory, when the low-energy VEVs breaking completely the gauge group aretaken into account. In other words, such monopoles are actually confined bythe vortices developed in the low-energy H system: this allows us to relatethe continuous moduli and group transformation properties of the vorticesto those of the monopoles appearing at the ends, thus explaining the ori-gin of the dual gauge groups, such as the Goddard-Nuyts-Olive-Weinberg(GNOW) duals [46]. The stability problem on non-BPS non-Abelian stringshave also been studied in another system: the Seiberg-dual theory of theN = 1 supersymmetric SO(Nc) QCD [133].

In these considerations a subtle but crucial point is that when small termsarising from the symmetry breaking G→ H are taken into account, neitherthe high energy system (describing the symmetry breaking G → H andregular monopoles) nor the low-energy system (in the Higgs phase H →

131

1 describing the vortices) is BPS-saturated any longer. This on the onehand allows the monopoles and vortices to be related in a one-to-one map(each vortex ends up on a monopole); on the other hand, the system isno longer BPS and we must carefully check the fate of the moduli of theBPS vortices which do not survive the perturbation. In fact, zero modesrelated to global symmetries (orientational modes) still survive, because oftheir origin as Goldstone bosons, while other modes are no more protectedby supersymmetry. These issues are the subject for investigation in thisChapter.

7.1 Abelian Vortices

The question of stability of the semi-local vortices [73, 74, 94] for Abeliansystems has been investigated by Hindmarsh [74] and other authors [134,109, 110], some time ago. The model which has been studied is an AbelianHiggs system with more than one flavor (Nf > 1),

L = − 1

4e2FµνF

µν +Dµφ(Dµφ)† − λ

2

(φφ† − ξ

)2, (7.1.1)

whereDµ = ∂µ−iAµ is the standard covariant derivative, φ = (φ1, φ2, . . . , φNf )represents a set of complex scalar matter fields of the same charge. Thismodel is sometimes called the semi-local model since not all global symme-tries, i.e. U(Nf ) here, are gauged. As a consequence, the vacuum manifoldM is CPNf−1 = SU(Nf )/[SU(Nf − 1) × U(1)]. Since the first homotopygroup of M is trivial for Nf > 1, vortex solutions are not necessarily stable.For β ≡ λ/e2 < 1 (i.e. Type I superconductors) the vortex of ANO-type[116, 135] is found to be stable. In the interesting special (BPS) case, β = 1,there is a family of vortex solutions with the same tension, T = 2πξ. Exceptfor the special values of the moduli (in the space of solutions), which repre-sents the ANO vortex (sometimes called a “local vortex”), the vortex has apower-like tail in the profile function, hence the vortex width (thickness ofthe string) can be of an arbitrary size. In the limit of large size, the vortexessentially reduces to the CPNf−1 sigma-model lump (or two-dimensionalskyrmion), characterized by π2(CPNf−1) = Z.

For β > 1 (i.e. Type II superconductors), vortices are found to be unstableagainst fluctuations of the extra fields (flavors) which increase the size (andspread out the flux).

132

7.2 Non-Abelian Vortices

We consider the same N = 2 supersymmetric (U(Nc) gauge theory of Section(1.1. The only difference here is the we consider a generic number (Nf > Nc)of hypermultiplets Qf , Qf (f = 1, . . . , Nf ). Since all the essential featuresare already present in the minimal case, Nc = 2 and Nf = 3, we will con-centrate in the U(2) mode. The extension to more general cases, Nc > 2, isstraightforward.

If the squark multiplets are kept massless, as usual, the theory has adegenerate set of vacua in the Higgs phase where the gauge symmetry iscompletely broken, which is the cotangent bundle over the complex Grass-manian manifold [84]

Mvac = GrNf ,Nc ' SU(Nf )/[SU(Nc)× SU(Nf −Nc)× U(1)] (7.2.1)

for general Nc. It is important to notice that the first homotopy group ofthe Grassmanian manifold is trivial (for Nf > Nc). Up to gauge and flavorrotations, we can choose the following VEV for the scalar fields

Q = Q† =

√ξ

2

(1 0 0 · · · 00 1 0 · · · 0

), a0 = 0 , a = 0 , (7.2.2)

where a ≡ aiτi/2. The vacuum is invariant under the following global color-

flavor locked rotations: (Uc ∈ SU(2), Uf ∈ SU(2) ⊂ SU(Nf ))

Q→ UcQU†f , Q→ U †f QUc , a→ UcaU

†c , Fµν → UcFµνU

†c . (7.2.3)

In addition to the case with Nc = Nf , the vacuum is also invariant underpure SU(Nf − 2) flavor rotations which act on the last Nf − 2 columns.

When the bare masses for the squarks are tuned to some special val-ues, there exist quantum vacua in which the non-Abelian gauge symmetrySU(2) × U(1) is preserved (the so-called r = 2 vacua) [64]. The low energyeffective theory in these vacua is exactly the theory we are studying here. Asthe quantum vacua with an SU(2) magnetic gauge group require the pres-ence of a sufficient number of flavors, Nf ≥ 2 r = 4, we must necessarilydeal with a system with an excess number of flavors (Nf > Nc). As thesesystems in the BPS approximation (η, µ = 0) contain semi-local vortices witharbitrarily large widths, the necessity of studying the fate of these vorticesin the presence of the perturbations η, µ presents itself quite naturally.

We will start by simply embedding the well-known solution for Nf = 2

133

flavors (6.1.10) in the present model with additional flavors:

Q =

(φ0(r)eiθ 0 0 . . . 0

0 φ1(r) 0 . . . 0

), a0 = λ0(r) , a = λ1(r)

τ 3

2,

Ai = Aαiτα

2= −εijxj

r2[1− f1(r)]

τ 3

2, A0

i = −εijxjr2

[1− f0(r)] , (7.2.4)

The six functions φ0, φ1, f0, f1, λ0, λ1 are found numerically by using thesame techniques of the previous Section. A numerical result is shown in Fig.(7.1).

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4r

f0f1!0!1"0"1

Figure 7.1: Profile functions for the vortex in the radial direction r: f0 (unbro-ken/red), f1 (short-dashed/green), λ0 (long-dashed/blue), λ1 (dash-dotted/black),φ0 (dash-double-dotted/magenta), φ1 (long-dash short-dash-dotted/grey). Thenumerical values are taken as ξ = 2, e = 2, g = 1, η = 0.2, µ = 0.4. For thisconfiguration T = 0.964TBPS.

In the BPS limit (η = µ = 0), λ0(x) = λ1(x) = 0, the substitution of thisconfiguration into the Lagrangian (1.1.3) gives the BPS value, TBPS = 2πξ,even if it is no longer a solution to the vortex equations (6.1.9), for η, µ 6= 0.This means that the non-BPS vortex tension derived from Eq. (6.1.9) is nec-essarily less than the BPS value. In other words, the model we are consideringalways yields type I superconductivity [136]. Our result below – the stabilityof the ANO-like vortices – thus generalizes naturally the known correlationbetween the type of superconductor and the kinds of stable vortices, foundin the Abelian Higgs models [74, 137, 134]. Also, in so much as we study asoftly-broken N = 2 supersymmetric gauge theory and its low-energy vortexsystem, the present work can be regarded as an extension of the one in SU(2)gauge theory [11], even though in the latter the low-energy system was nat-urally Abelian (U(1)). We have verified that the same qualitative conclusion

134

holds in that case. In the next Section the stability of this solution underperturbations of the additional flavors will be studied.

7.2.1 Fluctuation Analysis

In this Section, we will generalize the fluctuation analysis of the stabilityof the Abelian vortex of Ref. [74] to the non-Abelian vortex. As we havementioned, the first homotopy group of the vacuum manifold is trivial, thusthe local vortex found in the previous Section can be unstable. To study thestability of non-BPS local vortices, we must consider the quadratic variationsof the Lagrangian due to small perturbations of the background fields. Itturns out that the quadratic variation of the Lagrangian can be written asthe sum of two pieces

δ2L = δ2L∣∣local fields

+ δ2L∣∣semi−local fields

, (7.2.5)

where the first term denotes the variation with respect to the fields describinga non-trivial background vortex configuration (i.e. the “local” fields), whilethe second term is the variation with respect to the “semi-local” fields (Qf , Qf

with Nf ≥ f > Nc = 2). The key point is that there are no mixed terms (atthe second order) between the variations of the background fields and the“semi-local” fields. The first term cannot give rise to instabilities (i.e. thelocal vortices with Nf = 2 are topologically stable). Hence, it suffices for ourpurpose to study only the second term

δ2L∣∣semi−local fields

= (DbµδQ3)†DµbδQ3 + (DbµδQ3)(DµbδQ3)† − δ2V∣∣Q3,Q3

,

(7.2.6)

where we have taken, for simplicity and without loss of generality, Nf = 3.The subscript ‘b’ denotes background fields which we fix to be given by theansatz (6.1.10) in the following. Let us work out this variation explicitly.

Even if Qb†

= Qb in the background fields we keep their variations indepen-dent.

Let us first consider the variation of the potential (1.1.6), piece by piece:

Variation of V1

Remember that we are only interested in the variations that involve the semi-local fields, i.e. the variation with respect to the third flavor. The crucialobservation is that the background value for this field is zero: Q†3b = Q3b =0. This means that the variation of terms such as Trf [Q

†τ iQ] are alreadyquadratic in the perturbation. Furthermore, the term εijkajak evaluated on

135

the background fields is zero, due to a3 being the only non-zero non-Abelianadjoint field. Thus, the variation of V1 with respect to the semi-local fields isat least cubic, coming from the cross terms involving the adjoint field. Hence,the quadratic variation vanishes

δ2V1

∣∣Q3,Q3

= 0 . (7.2.7)

Variation of V2

V2 contains no adjoint fields. The same argument used for V1 goes through:the variations of V2 are at least quartic

δ2V2

∣∣Q3,Q3

= 0 . (7.2.8)

Variation of V3

The variation of V3 gives us the first non-trivial contribution. In this casethe variation with respect to the semi-local fields is at least quadratic, andit is given by

δ2V3

∣∣Q3,Q3

=g2

2

(Trf(Q

†bτ

3Qb) + 2µa3b

)(δQ3τ

3δQ3 + c.c.)

+e2

2

(Trf(Q

†bQb)− ξ + 2 ηa0b

)(δQ3δQ3 + c.c.

). (7.2.9)

Variation of V4

The variation of V4 is also quadratic and is simply

δ2V4

∣∣Q3,Q3

=1

2δQ†3

(a0b + τ 3a3b

)2δQ3 +

1

2δQ3

(a0b + τ 3a3b

)2δQ†3 .

(7.2.10)

The variations are not diagonal, thus involve mixed terms like δQ3δQ3.We can easily diagonalize them by the following change of coordinates (keep-ing the kinetic terms canonical)

δQ3 =1√2

(q + q†) , δQ3 =1√2

(q† − q) , (7.2.11)

136

which yields the following variation of the potential to second order

δ2V∣∣Q3,Q3

=g2

2

(Trf(Q

†bτ

3Qb) + 2µa3b

) (|qI |2 − |qI |2 − |qII |2 + |qII |2

)+e2

2

(Trf(QbQb)− ξ + 2 ηa0b

) (|qI |2 − |qI |2 + |qII |2 − |qII |2

)+

1

2(a0b + a3b)2 (|qI |2 + |qI |2

)+

1

2(a0b − a3b)2 (|qII |2 + |qII |2

),

(7.2.12)

where the capital indices I and II label the color components. The problemof stability is now reduced to studying four decoupled Schrodinger equations.We expand the fluctuations as

qI,II ≡∑k

ψ(k)I,IIe

ikθ , qI,II ≡∑k

ψ(k)I,IIe

−ikθ . (7.2.13)

Using these expansions, the quadratic variations of the energy density (7.2.6)give rise to the following Schrodinger equations[

−1

r

d

dr

(rd

dr

)+ V

(k)I,II

]ψ

(k)I,II = M

(k)I,IIψ

(k)I,II ,[

−1

r

d

dr

(rd

dr

)+ V

(k)I,II

]ψ

(k)I,II = M

(k)I,IIψ

(k)I,II , (7.2.14)

which must be solved with the following boundary conditions at small r:

-1.6-1.4-1.2

-1-0.8-0.6-0.4-0.2

0 0.2

0 1 2 3 4 5 6 7 8 9 10r

VI(0)

VII(0)

(a)

-0.2 0

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

0 1 2 3 4 5 6 7 8 9 10r

V~ I(0)

V~ II(0)

(b)

0

2

4

6

8

10

0 1 2 3 4 5 6 7 8 9 10r

VI(1)

V~ I(1)

(c)

Figure 7.2: Potentials as functions of r: (a) V (0)I and V (0)

II . (b) V (0)I and V (0)

II . (c)V

(1)I and V (1)

I . The results are plotted for e = 2, g = 1, ξ = 2, η = 0.2 and µ = 0.4.

ψ(k)I,II(r), ψ

(k)I,II(r) = rk +O(rk+1). (7.2.15)

137

The effective potentials are

V(k)I,II =

1

4r2[f0 − 1± (f1 − 1) + 2k]2 +

1

2(λ0 ± λ1)2 (7.2.16)

+e2

2(φ2

0 + φ21 − ξ + 2 ηλ0)± g2

2(φ2

0 − φ21 + 2µλ1)

V(k)I,II =

1

4r2[f0 − 1± (f1 − 1) + 2k]2 +

1

2(λ0 ± λ1)2 (7.2.17)

− e2

2(φ2

0 + φ21 − ξ + 2 ηλ0)∓ g2

2(φ2

0 − φ21 + 2µλ1) . (7.2.18)

The upper signs refer to the color-index I, while the lower to index II. Sinceonly the first terms are dependent on k, it is easy to check that(

V(k)I , V

(k)II

)≥(

min[V

(0)I , V

(1)I

], V

(0)II

),(

V(k)I , V

(k)II

)≥(

min[V

(0)I , V

(1)I

], V

(0)II

), (7.2.19)

for all k. To prove the above inequalities, it is sufficient to recall that theprofile functions for the gauge fields satisfy: 0 ≤ f0,1 ≤ 1.

-0.2

0

0.2

0.4

0.6

0.8

1

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

!I(0

)

M

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35 40

!I(0

)

r

Figure 7.3: Left panel: the value of the wave function ψ(0)I at large distance

(r = 20, in the unit of the vortex size), as a function of M (see Eqs. (7.2.14)).There are no zeros at negative values of M : this implies the absence of normalizabletachyonic modes. Right panel: the wave function for the zero energy fluctuation:M = 0. The linear behavior is a common feature for zero energy wave functions.The numerical values of the parameters are chosen as in Fig. (7.2).

From a well-known theorem in one-dimensional quantum mechanics, itfollows that (valid for the lowest eigenvalues)(

M(k)I ,M

(k)II

)≥(

min[M

(0)I ,M

(1)I

],M

(0)II

),(

M(k)I , M

(k)II

)≥(

min[M

(0)I , M

(1)I

], M

(0)II

). (7.2.20)

138

-2

-1.5

-1

-0.5

0

0.5

0 1 2 3 4 5 6 7 8 9 10

VI(0

)

r

(a)

-0.2

0

0.2

0.4

0.6

0.8

1

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

!I(0

)

M

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 5 10 15 20 25 30 35 40

!I(0

)

r

(c)

Figure 7.4: Numerical results in the BPS case: e = 2, g = 1, ξ = 2, η = µ = 0.(a) The potential V (0)

I has a zero energy fluctuation. (b) The value of ψ(0)I at a

large distance (r = 20) as function of M . (c) The zero energy wave function ψ(0)I

as function of r.

Hence, in order to exclude the existence of negative eigenvalues, it suffices tostudy the six potentials: V

(0,1)I , V

(0,1)I , V

(0)II and V

(0)II .

Fig. (7.2) shows that only the potentials V(0)I and V

(0)II can be sufficiently

negative, such that one could expect negative eigenvalues. The potentialsV

(1)I and V

(1)I are even divergent at the core of the vortex. Indeed, the

results of the numerical analysis show (up to the numerical precision) thatthere are no negative eigenvalues for any of the potentials, V 0,1

I , V 0,1I , V 0

II

and V 0II . We have checked this statement in a very wide range of values of

the couplings of the theory. This strongly suggests the absence of negativeeigenvalues for all values of the couplings. The result is shown in Fig. (7.3)for a particular choice of parameters. Note that all the potentials go to zeroat large distance. This means that there will be a set of positive eigenvaluesthat represent the continuum states, which can be interpreted as interactionmodes with the massless Goldstone bosons of the bulk theory.

The status of the zero energy wave functions however is subtler. Thewave functions for the non-BPS (Fig. (7.3)) and the BPS cases (Fig. (7.4))are both non-normalizable, hence they should both be interpreted as a partof the continuum. However, there is an important difference between thetwo cases. In the BPS case, the wave function is limited and goes to zero atinfinity; its squared norm diverges only logarithmically with the transversevolume (∼ logL2). In the non-BPS case, the wave function is unlimited,and it diverges linearly (Fig. (7.3)). This implies a divergence of its squarednorm being quadratic in the transverse volume (∼ L4). Such a fluctuationchanges the vacuum expectation values of the scalars at infinity, thus doesnot correspond to a size zero mode (collective coordinate). The situation isclearer when our vortex system is put into a finite volume. In the BPS case,the wave function of the semi-local mode asymptotically approaches zero andis an (approximately) acceptable wave function. In a box, it represents a zero

139

energy bound state. On the contrary, the wave function for the non-BPScase is non-normalizable. The corresponding zero energy state is eliminatedand the semi-local excitations have a mass gap. This observation stronglysuggests that in the non-BPS case the size moduli disappears.

As a further check, we note that the potential which gives rise to the sizezero modes in the BPS limit (which we know exist in that case) is precisely

V(0)I . The zero mode is thus given by a fluctuation of the field ψ

(0)I . Using

Eq. (7.2.11) we see that δQ3I = δQ3I =√

2ψ(0)I , while all other components

of Q3 and Q3 vanish. This is exactly the fluctuation we expect for the semi-local vortex, which in the BPS limit is known to have the following form[6, 138]:

Q =

(φ0(r)eiθ 0 χ(r)

0 φ1(r) 0

). (7.2.21)

7.3 Meta-stability versus Absolute Stability

The results of the previous Section indicate that the local (ANO-like) vortexembedded in a model with additional flavors is a local minimum of the action.The problem whether or not this solution is a global minimum cannot beaddressed with a fluctuation analysis only. In principle, the existence of aninstability related to large fluctuations of the fields is still not excluded.

However, we have a reason to believe that the local vortex is truly stable,being the global minimum of the action. Note that a vortex cannot dilutein the whole space. In fact, such diluted configuration would have an energythat is equal to the BPS bound, T∞ = 2πξ = TBPS, while the tension ofthe local vortices, in the class of theories we are considering, is always less:T < TBPS, as already noted. To see this, we will first show that the semi-localvortex solution, obtained in the BPS limit, is an approximate solution to thenon-BPS equations, in the limit of an infinite size of the vortex. Then, it iseasy to check that the tension of such approximate configuration convergesto the BPS value. Let us write the equations of motion for the semi-local

140

configuration (7.2.21)

f ′′0 −f ′0r

= e2(f1(φ2

0 − φ21 + χ2) + f0(φ2

0 + φ21 + χ2)− 2χ2

),

f ′′1 −f ′1r

= g2(f1(φ2

0 + φ21 + χ2) + f0(φ2

0 − φ21 + χ2)− 2χ2

),

φ′′0 +φ′0r− (f0 + f1)2φ0

4r2=φ0

2

((λ0 + λ1)2 + e2A+ g2B

),

φ′′1 +φ′1r− (f0 − f1)2φ1

4r2=φ1

2

((λ0 − λ1)2 + e2A− g2B

),

χ′′ +χ′

r− (f0 + f1 − 2)2χ

4r2=χ

2

((λ0 + λ1)2 + e2A+ g2B

),

λ′′0 +λ′0r

= e2((λ0 + λ1)(φ2

0 + χ2) + (λ0 − λ1)φ21 + e2η A

),

λ′′1 +λ′1r

= g2((λ0 + λ1)(φ2

0 + χ2)− (λ0 − λ1)φ21 + g2µB

),

(7.3.1)

where we have defined

A ≡ φ20 + φ2

1 + χ2 − ξ + 2ηλ0 , B ≡ φ20 − φ2

1 + χ2 + 2µλ1 . (7.3.2)

For simplicity, we take the gauge couplings to be equal: e = g. Let us considerthe following set of approximate solutions corresponding to a semi-local BPSvortex in the limit of large size (lump limit): a 1/e

√ξ:

(φ0, φ1, χ) =

√ξ

2

(r√

r2 + |a|2, 1,

a√r2 + |a|2

),

f0 = f1 =|a|2

r2 + |a|2 , λ0 = λ1 = 0 . (7.3.3)

The right hand sides of Eqs. (7.3.1) calculated with this configuration vanishobviously. The left hand sides are derivative terms which disappear in thelarge size limit: the set of fields above, trivially satisfy the fourth and the lasttwo equations, while they satisfy the other equations up to terms of orderO(1/|a|2) ∣∣∣∣f ′′0,1 − f ′0,1

r

∣∣∣∣ =8|a|2r2

(r2 + |a|2)3≤ 32

27|a|2 ,∣∣∣∣φ′′0 +φ′0r− φ0(f0 + f1)2

4r2

∣∣∣∣ =

√ξ

2

2|a|2r(|a|2 + r2)5/2

≤√ξ

2

(4

5

) 52 1

|a|2 ,∣∣∣∣χ′′ + χ′

r− χ(−2 + f0 + f1)2

4r2

∣∣∣∣ =

√ξ

2

2|a|3(|a|2 + r2)5/2

≤√ξ

2

2

|a|2 .

141

Thus, we see that the configuration of Eq. (7.3.3), in the limit |a| → ∞, isa solution to the non-BPS equations of motion: it represents an infinitelydiluted vortex, having the tension equal to the BPS value.

We thus see that an infinitely wide vortex is not favored energetically. Theonly remaining possibility is the existence of a true minimum for a vortexwith some intermediate size, which involves also a non-trivial configurationfor the semi-local fields. This case is very unlikely. In fact, we could not findany such solution in our numerical surveys. The relaxation method shouldhave detected such an unexpected stable vortex.

We consider our results as a strong evidence for the fact that the localvortex is a true minimum, absolutely stable against becoming a semi-localvortex.

7.4 Low-energy Effective Theory

In this short Section we only make some considerations on the low-energyeffective theory of the vortex which follow from the results of the previousSections.

In the semi-local case, the vacuum has no mass gap, thus there are mass-less Nambu-Goldstone bosons in the bulk. This fact gives rise to subtleties(which we encountered in Sec. (7.2.1)) when defining an effective theory forthe vortex only. This is because both the vortex excitations and the masslessparticles in the bulk appear as light modes. As previously mentioned, semi-local excitations are non-normalizable and the corresponding states mustbe considered as bulk excitations. In a rigorous effective theory, we shouldtake into account only localizable and normalizable modes, which discardsthe bulk excitations1. This holds both in the BPS and non-BPS cases. Infact, effective actions which include semi-local excitations have already beenproposed in the literature. The effective actions for BPS semi-local vorticesderived in Ref. [6, 75], however, are valid only in a finite volume.

It seems to us that the following point has not been stressed clearly enoughin the literature. We consider all the excitations related to non-normalizablezero energy wave functions as part of the massless states of the bulk theory.In the BPS case, the surviving supersymmetry enables us to describe thesefluctuations as collective coordinates of the vortex. It is known, for example,that a single non-Abelian BPS vortex has non-normalizable size parameters(in the case Nf > Nc). From the point of view that we propose here, however,

1Similar issues have been already discussed for monopoles and vortices, see forexample Refs. [139, 96, 140].

142

the effective action on the local vortex is given by the orientational part only(normalizable states). The other collective coordinates (size) give rise to zeroenergy fluctuations which represent bulk excitations.

While these considerations might just be a matter of interpretation in theBPS case, they could be important if one wants to push further the study ofthe non-BPS vortices we started in this work.

7.5 Summary

We have examined the stability of the non-Abelian vortices in the contextof an N = 2, H = U(Nc) model with Nf > Nc flavors and an N = 1perturbation. The particular perturbation chosen (the adjoint scalar massterms) makes our vortices non-BPS. Local (ANO-like) vortices are found tobe stable; the vortex moduli corresponding to the semi-local vortices (presentin the BPS case) disappear, leaving intact the orientational moduli – CPNc−1

in this particular model – related to the exact symmetry of the system.This conclusion seems to be very reasonable, as the narrow, ANO-like

vortices are needed to eliminate the regular monopoles from the spectrumof the full system, in the case our H gauge theory arises as a low-energyapproximation of an underlying G theory, after the symmetry breaking G→H. If the semi-local vortex would have survived the non-BPS perturbations,the magnetic monopoles would be deconfined [6]. Happily, this is not thecase; the Higgs phase of the (electric) low-energy theory is actually a magneticconfinement phase.

143

Chapter 8

A Dual Model of Confinementfor SO(N) Gauge Theories

We will apply the results on the moduli space of composite vortices to testour ideas about duality transformations, introduced in Chapter (2), againstanother class of theories,

SO(2N + 1)〈φ1〉6=0−→ SU(r)× U(1)N−r+1 〈φ2〉6=0−→ 1. (8.0.1)

One of the reasons why this case is interesting is that the semiclassical

monopoles arising from the symmetry breaking SO(2N + 1)〈φ1〉6=0−→ U(N)

appear to belong to the second-rank symmetric tensor representation ofSU(N) [51, 37]. Another, related reason is the fact that since π1(G) =π1(SO(2N + 1)) = Z2, the homotopy map Eq. (2.2.1) is less trivial in thiscase. Thirdly, according to the detailed analysis of the softly-broken N = 2theories with SO(N) gauge group [141] the quantum mechanical behavior ofthe monopoles is different for r = N and for r < N . Non-Abelian monopolesbelonging to the fundamental representation of the dual SU(r) group appearsonly for r ≤ Nf/2, and because of the requirement of asymptotic freedom ofthe original theory (Nf < 2N − 1), this is possible only for r < N. It is veryencouraging that such a difference in the behavior of non-Abelian monopolesindeed follows, as we shall see, from the way we define the dual group thoughthe transformation properties of mixed monopole-vortex configurations andhomotopy map.

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8.1 Maximal SU factor; SO(5)→ U(2)→ 1

Let us first consider the case the SU(N) factor has the maximum rank,

SO(2N + 1)〈φ1〉6=0−→ U(N).

To be concrete, let us consider the case of an SO(5) theory, where a scalarVEV of the form

〈Φ〉 =

0 i v 0 0 0−i v 0 0 0 0

0 0 0 i v 00 0 −i v 0 00 0 0 0 0

(8.1.1)

breaking the gauge group as SO(5) → H = SU(2) × U(1)/Z2 = U(2). Weassume that at lower energies some other scalar VEVs break H completely,leaving however a color-flavor diagonal SU(2) group unbroken. This modelarises semiclassically in softly broken N = 2 supersymmetric SO(5) gaugetheory with large, equal bare quark masses, m, and with a small adjointscalar mass µ, with scalar VEVs given by v = m/

√2 in Eq. (8.1.1) and

Q = Q† =

õm

2

1 0 0 · · ·i 0 0 · · ·0 1 0 · · ·0 i 0 · · ·0 0 0 · · ·

. (8.1.2)

(See Appendix (??), also the Section 2 of [141], for more details).The SO(4) ∼ SU(2)× SU(2) subgroup living on the upper-left corner is

broken to SU(2)×U(1), giving rise to a single ’t Hooft-Polyakov monopole.On the other hand, by embedding the ’t Hooft-Polyakov monopole in thetwo SO(3) subgroups (in the (125) and (345) subspaces), one finds twomore monopoles. All three of them are degenerate. Actually, E. Weinberg[142, 129] has found a continuous set of degenerate monopole solutions in-terpolating these, and noted that the transformations among them are notsimply related to the unbroken SU(2) group.1

From the point of view of stability argument, Eq. (2.2.1), this case is verysimilar to the case considered by ’t Hooft, as π1(SO(5)) = Z2: a singularZ2 Dirac monopole can be introduced in the theory. The minimal vortex of

1 This and similar cases are sometimes referred to as “accidentally degeneratecase” in the literature.

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the low-energy theory is truly stable in this case, as a minimal non-trivialelement of π1(H) represents also a non-trivial element of π1(G). This can beseen as follows. A minimum element of π1(H) = π1(U(2)) ∼ Z correspondsto simultaneous rotations of angle π in the (12) and (34) planes (which isa half circle of U(1)), which brings the origin to the Z2 element of SU(2),diag (−1,−1,−1,−1, 1), followed by an SU(2) transformation back to theorigin, an angle −π rotation in the (12) plane and an angle π rotation around(34) plane. The net effect is a 2π rotation in the (34) plane, which is indeeda non-trivial element of π1(SO(5)) = Z2. Such a vortex would confine thesingular Dirac monopole, if introduced into the theory (See Fig. (2.1)).

On the other hand, there are classes of vortices which appear to be stablein the low-energy approximation, but are not so in the full theory. In factnon-minimal k = 2 elements of π1(H) = π1(SU(2) × U(1)/Z2) ∼ Z areactually trivial in the full theory. This means that the k = 2 vortices mustend at a regular monopole. Vice versa, as π2(SO(5)) = 1, the regular ’tHooft Polyakov monopoles of high-energy theory must be confined by thesenon-minimal vortices and disappear from the spectrum.

The transformation property of k = 2 vortices has been studied recentlyin [90, 91], and in particular, in Section (4.1.1). It turns out that the modulispace of the k = 2 vortices is a CP 2 with a conic singularity. It was shownthat the generic k = 2 vortices transform under the SU(2)c+f group as atriplet. At a particular point of the moduli - an orbifold singularity - thevortex is Abelian: it is a singlet of SU(2)c+f .

2

As the full theory has an exact, unbroken SU(2)c+f symmetry, it followsfrom the homotopy-group argument of Section (2.2) that the monopoles inthe high-energy SO(5) → U(2) theory have components transforming as atriplet and a singlet of SU(2)c+f .

Note that it is not easy to see this result – and is somewhat mislead-ing to attempt to do so – based solely on the semi-classical construction ofthe monopoles or on the zero-mode analysis around such solutions, wherethe unbroken color-flavor symmetry is not appropriately taken into account.Generically, the “unbroken” color SU(2) group suffers from the topologicalobstruction or, perturbatively, from the pathology of non-normalizable gaugezero-modes [47, 48, 49, 50, 51]), as we noted already.

Nevertheless, there are indications that the findings by E. Weinberg [142,129] are consistent with the properties of the k = 2 vortices. In the standard

2 In another complex codimension-one subspace, they appear to transform asa doublet. However quantum states of any triplet of SU(2) contains such an orbit.The state of maximum Sz, |1, 1〉, transforms under SU(2) as an SO(3) vector,staying on a subspace S2 ∼ CP 1 ⊂ CP 2.

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!

"!

#

Figure 8.1: Non-zero root vectors of SO(5)

way to embed SU(2) subgroups through the Cartan decomposition (we followhere the notation of [142, 129]),

t1(ν) =1

(2 ν2)−1/2(Eν + E−ν); t2(ν) =

−i(2 ν2)−1/2

(Eν − E−ν);

t3 = (ν2)−1 νj Tj, (8.1.3)

where ν denotes the non-vanishing root vectors of SO(5) Fig. (8.1), theunbroken SU(2) group is generated by γ. The monopole associated with theroot vector β and the (equivalent) one given by µ naturally form a doubletof the “unbroken” SU(2), while the monopole with the α charges is a singlet.The continuous set of monopoles interpolating among these monopoles foundby Weinberg are analogous to the continuous set of vortices we found, whichform the points of the CP 2, which transform as a triplet. (See the Fig. (4.1)).

An even more concrete hint of consistency comes from the structure ofthe moduli space of the monopoles. The moduli metric found in [142, 129] is

ds2 = M dx2 +16π2

Mdχ2 +

+ k

[d b2

b+ b (dα2 + sin2 α dβ2 + (d γ + cosα dβ)2)

]. (8.1.4)

By performing a simple change of coordinate, B ≡ 2√b, it becomes evident

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that the moduli space has the structure

C2/Z2, (8.1.5)

apart from the irrelevant factor R3 (the position of the monopole) and S1

(U(1) phase).3 Eq. (8.1.5) coincides with the moduli space of the k = 2co-axial vortices, seen in the central (1, 1) patch.

These considerations strengthen our conclusion that the continuous setof monopoles found in [142, 129] belongs to a singlet and a triplet represen-tations of the dual SU(2) group. Although the detailed properties of themoduli spaces for monopoles and vortices are different4, this could be relatedto the fact that one should ultimately consider a smooth monopole-vortexmixed configurations in the full theory, not each of them separately. Also, re-lated to this point, there remains the fact that the dual group which is exactand under which monopoles transform, is not the original SU(2) subgroupbut involves the flavor group essentially.

Note that our conclusion is based on the exact symmetry, and should bereliable. However, the degeneracy among all the vortices (or the monopoles)lying in the entire moduli space CP 2/Z2 found in the BPS limits, is an ar-tifact of the lowest-order approximation. Only the degeneracy among thevortices (or among the monopoles) belonging to the same multiplet is ex-pected to survive quantum mechanically. 1 and 3 vortex tensions (monopolemasses) will split. Which of the multiplets (1 or 3) will remain stable, afterquantum corrections are taken into account, is a question just lying beyondthe power of semiclassical considerations.

In the context of asymptotically-free N = 2 supersymmetric models,there are no indications that the triplet monopoles of SO(5)→ U(2) theorysurvive quantum mechanically. This result can be actually understood by asimple renormalization-group argument:

• In a SO(2N + 1) theories with N = 2, 1 supersymmetries, the condi-tion for the original theory to be asymptotic-free (Nf less than 2N −1,3(2N−2)

2, respectively)5 is not compatible with the low-energy SU(N)

theory being non-asymptotic-free (Nf ≥ 2N and Nf ≥ 3N , respec-tively.)

3 The monopole modulus due to the unbroken U(1) ⊂ U(2) is not present inthe full system, where the gauge group is completely broken.

4 The first is known to be hyper-Kahler and the second Kahler – indeed CP 2/Z2

does not admit hyper-Kahler structure.5The counting is made for the appropriate supersymmetry multiplets, Nf hy-

permultiplets for N = 2; Nf chiral multiplets for N = 1 supersymmetric SO(N)theory.

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The problem would not arise if the rank of the unbroken SU(r) weresmaller. That such a “sign-flip” of the beta function is a necessary conditionfor the emergence of low-energy non-Abelian monopoles has been pointedout some time ago by one of the authors [143], even though the validity ofsuch an argument for non-supersymmetric theories is perhaps not obvious.

If the condition of asymptotic freedom of the ultraviolet theory is dropped,then there are no such constraints, and it makes sense to consider symmetrybreaking patterns such as SO(2N + 1) → U(N). Our conclusion that themonopoles of SO(5)→ U(2) system transform as a triplet or a singlet wouldapply under such conditions. Analogously, we expect the monopoles in thesystem SO(2N + 1) → U(N) to transform as a second-rank symmetric orantisymmetric representation.

8.2 SO(2N+1)→ SU(r)×U(1)N−r−1 → 1 (r < N)

Consider now the cases in which the unbroken SU(r) factor has a smallerrank, SO(2N + 1)→ SU(r)× U(1)N−r+1 → 1, where r < N . For concrete-ness, let us discuss the case of an SO(7) theory,

SO(7)〈φ1〉6=0−→ U(2)× U(1)

〈φ2〉6=0−→ 1. (8.2.1)

As we are interested in a concrete dynamical realization of this, we considerthe softly broken N = 2 theory, with Nf = 4 quark hypermultiplets. Sucha number of flavors ensures both the original SO(7) theory being asymp-totically free and the SU(2) subgroup being non-asymptotically free. Thelow-energy gauge group U(2) × U(1) is completely broken by the squarkVEV’s similar to Eq. (8.1.2). The large VEV 〈φ1〉 has the form:

〈φ1〉 =

0 iv0 0 0 0 0 0−iv0 0 0 0 0 0 0

0 0 0 iv0 0 0 00 0 −iv0 0 0 0 00 0 0 0 0 iv1 00 0 0 0 −iv1 0 00 0 0 0 0 0 0

, v1 6= v0. (8.2.2)

The “unbroken” U(2) lies in SO(4)1234 ∼ SU(2) × SU(2) while the U(1)factor corresponds to the rotations in the 56 plane (see Appendix (??)). Thesemiclassical monopoles of high-energy theory are 6

6Within the softly broken N = 2 theory, the quantum mechanical vacua withSU(2) × U(1)2 gauge symmetry, in the limit mi = m ' Λ, appears to arise from

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(i) a triplet of degenerate monopoles of mass 2 |v0|/g (they arise as in theSO(5) theory discussed above);

(ii) a doublet of degenerate monopoles of mass |v0 − v1|/g: they arise fromthe breaking of SU+(2) ⊂ SO(4)1256 and SU+(2) ⊂ SO(4)3456 (seeAppendix (??));

(iii) a doublet of degenerate monopoles of mass |v0 + v1|/g: they also arisefrom the breaking of SU−(2) ⊂ SO(4)1256 and SU−(2) ⊂ SO(4)3456;

(iv) a singlet monopole of mass 2 |v1|/g arising from the breaking of SO(3)567.

Which of these semiclassical monopoles are the lightest and which of themare stable against decay into lighter monopole pairs, depend on the variousVEVs. It is possible that the monopoles (ii) or (iii) are the lightest of all. Ofcourse more detailed issues such as which of the degeneracies survives quan-tum effects, are questions which go beyond the semiclassical approximations.

In fact, when v0, v1 ∼ Λ the standard semi-classical reasoning fails togive any reliable answer: a fully quantum-mechanical analysis is needed.Fortunately, in the softly broken N = 2 theory such analyses have beenperformed [141] and we do know that the light monopoles in the fundamentalrepresentation (2) of SU(2) appear in an appropriate vacuum.

Knowing this, we might try to understand how such a result may followfrom our definition of the dual group. At low energies the gauge groupU(2) × U(1) is completely broken, leaving a color-flavor diagonal SU(2)c+fsymmetry unbroken. The theory possesses vortices of

π1(U(2)× U(1)) = Z× Z. (8.2.3)

The minimal vortices corresponding to π1(U(2)) = Z transform as a 2 ofSU(2)c+f .

A minimum element of π1(U(2) × U(1)) such as an angle 2π rotation inthe U(1)56 factor, or the minimal U(2) loop, corresponds to vortices stable

the semiclassical vacua of the form of Eq. (8.2.2), with v0 = m/√

2 Λ, v1 = 0,with classical symmetry SU(2) × U(1) × SO(3)567. The SO(3)567 gauge sector(pure N = 2 theory) becomes strongly-coupled at low energies and breaks itselfto U(1). Thus it would be more correct to say v1 ∼ Λ, but then the discussionabout semiclassical monopole masses ∼ v1/g, etc., should not be taken too liter-ally. If one wishes, one could consider a larger gauge group, e.g., SO(9), to do astraightforward semiclassical analysis for an unbroken SU(2) group. In general,the relation between the classical vacua and the fully quantum mechanical vacuais a rather subtle issue. See for instance the discussions in [62].

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in the full theory. They would confine Dirac monopoles associated withπ1(SO(7)) = Z2, if the latter were introduced in the theory.

The regular monopoles in which we are interested in, are instead confinedby some non-minimal (k = 2) vortices of the low-energy theory. However,in contrast to the SO(5) theory discussed in the preceding subsection, thisdoes not necessarily imply a second-rank tensor representation of SU(2)c+fof these monopoles. In fact, the monopoles of the (ii) group, for instance,carry the minimum charge of U(2) and an unit charge of U(1). Therefore, therelevant k = 2 vortex corresponds to the minimum element both of π1(U(2))and of π1(U(1)), generated by a 2π rotation in the 56 plane together with aminimal loop of π1(U(2)), analogous to the one discussed in the precedingsubsection. As a consequence the monopoles confined by such vortices, byour discussion of Section 3, transform as a doublet of the dual group SU(2) ∼SU(2)c+f .

This discussion naturally generalizes to all other cases with symmetrybreaking, SO(2N+1)→ SU(r)×U(1)N−r+1 → 1, r < N . The dual magneticSU(r) group observed in the low-energy effective theory [141], under whichthe light monopoles transform as a fundamental multiplet, thus matchesnicely with the properties of the dual SU(r) ∼ SU(r)c+f group.

The cases of SO(2N)→ SU(r)×U(1)N−r+1 → 1, r < N − 1 are similar.We expect that there is a qualitative difference between the breaking withthe maximum (or next to the maximum) rank SU factor and smaller SU(r)unbroken groups. Such a difference is indeed observed in the fully quantummechanical analysis of SO(N) theory [141].

The behavior of monopoles in asymptotic-free USp(2N) theories (Nf <2N+2) is more similar to those appearing in the SU(N) theories, because ofthe property, π1(USp(2N)) = 1. All monopoles are regular monopoles dueto the partial symmetry breaking, USp(2N)→ SU(r)×U(1)N−r+1, r ≤ N .The transformation property of these monopoles, in the theory with exactunbroken SU(r)c+f global symmetry, is deduced from the transformationproperties among the non-Abelian vortices of the low-energy system SU(r)×U(1)N−r+1 → 1 : they transform as r of SU(r)c+f . Such a result is consistentdynamically, as long as r ≤ Nf/2. It is comfortable that these are preciselywhat is found from the quantum mechanical analysis [64].

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8.3 Other symmetry breaking patterns and

GNOW duality

Before concluding this Section, let us add a few remarks on other symmetrybreaking patterns such as SO(2N+3)→ SO(2N+1)×U(1) and USp(2N+2) → USp(2N) × U(1), and the resulting GNOW monopoles. These casesmight be interesting as the GNOW dual groups are different from the originalone: the dual of SO(2N + 1) is USp(2N) and vice versa. It is possible toanalyze these systems, again setting up models so that the “unbroken group”is completely broken at a much lower mass scales by the set of squark VEVs.Indeed such a preliminary study has been made in [144], where the emergenceof the GNOW dual group has been found.

However, the quantum fate of these GNOW dual monopoles is unclear.More precisely, within the concrete N = 2 models we are working on wherethe exact quantum fate of the semiclassical monopoles is known from theanalyses made at small m,µ [141], we know that these GNOW monopolesdo not survive quantum effects. Only the monopoles carrying the quantumnumbers of the SU(r) subgroups discussed in the previous subsection appear.On the other hand, there is clearly a reason why the GNOW monopolescannot appear at low energies in these cases: the low-energy effective actionwould have a wrong global symmetry. GNOW monopoles are not alwaysrelevant quantum mechanically 7. These and other peculiar (but consistent)quantum properties of non-Abelian monopoles have been recently discussedin [62].

8.4 Summary

In this Chapter we have examined an idea about the “non-Abelian monopoles”,put forward some time ago [36], more systematically and by using some re-cent results on the non-Abelian vortices. According to this idea, the dualtransformation of non-Abelian monopoles occurring in a system with gaugesymmetry breaking G −→ H is to be defined by setting the low-energy Hsystem in Higgs phase, so that the dual system is in confinement phase. Thetransformation law of the monopoles follows from that of monopole-vortexmixed configurations in the system

Gv1−→ H

v2−→ 1, (v1 v2)

7Seiberg duals of N = 1 supersymmetric theories with various matter contents,provide us with more than enough evidence for it.

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under an unbroken, exact color-flavor diagonal symmetry Hc+f ∼ H. Thetransformation properties of the regular monopoles (classified by π2(G/H))follow from those among the non-Abelian vortices (classified by π1(H)), viathe isomorphism π1(G) ∼ π1(H)/π2(G/H). Our results, obtained in thesemiclassical approximation (reliable at v1 v2 Λ) of softly-brokenN = 2supersymmetric SU(N) and SO(N) theories, are – very non-trivially – foundto be consistent with the fully quantum-mechanical low-energy effective ac-tion description (valid at v1, v2 ∼ Λ), available in these theories.

For G = SU(N+1), H = U(N), GF = SU(Nf ), Nf ≥ 2N , this argumentproves that the monopoles induced by the G/H breaking transform as N ofH = SU(N). Analogous result holds for G = SU(N + 1), H = U(r),GF = SU(Nf ), r ≤ Nf/2, where the semi-classical monopoles transformas in the fundamental multiplets (r) (as well as some singlets) of SU(r).These results are in agreement with what was found in the fully quantummechanical treatment of the system [59, 60, 64].

For G = SO(2N + 1), H = U(r) × U(1)N−r, GF = SU(Nf ) (withr ≤ Nf/2, r < N) we find monopoles which transform in the fundamen-

tal representation of the dual SU(r) = SU(r)c+f group. This result is againconsistent with the fully quantum mechanical analysis of N = 2 supersym-metric SO(N) models [141] and in agreement with the universality of certainsuperconformal theories discovered in this context by Eguchi et. al. [145].

In the case of maximal-rank SU subgroup, such as G = SO(5), H = U(2),there is a qualitative difference both in our duality argument and in the fullquantum results. For instance the set of monopoles found earlier by E.Weinberg is shown to belong to a singlet and a triplet representations of thedual SU(2) group, but their quantum fate is not known. In supersymmetricmodels a renormalization-group argument suggests (and the explicit analysisof softly broken N = 2 theory shows) that the triplet does not survive thequantum effects, as long as the underlying SO(5) theory is asymptoticallyfree.

For G = SO(2N), H = U(r) × U(1)N−r, GF = SU(Nf ) the situation issimilar. When r < N − 1, r ≤ Nf/2 we find monopoles transforming in the

r representation of the dual SU(r) = SU(r)c+f , whereas the maximal andnext-to-maximal cases, r = N,N − 1, encounter the same renormalization-group constraint as in SO(2N + 1).

Finally for G = USp(2N), H = U(r)× U(1)N−r, GF = SU(Nf ) the pic-ture is very much like in SU(N +1). We have monopoles in the fundamental

representation of the dual SU(r) = SU(r)c+f as long as Nf ≥ 2 r.

Summarizing, in the context of softly-broken N = 2 supersymmetricgauge theories with SU , SO and USp groups, where fully quantum me-

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chanical results are available by combining the various knowledges such asthe Seiberg-Witten curves, decoupling theorem, Nambu-Goldstone theorem,non-renormalization of Higgs branches, N = 1 ADS instanton superpoten-tial, vacuum counting, universality of conformal theories, etc., our idea onnon-Abelian monopoles is in agreement with these known exact results.

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Chapter 9

Universal Reconnection ofnon-Abelian Cosmic Strings

The issue of reconnection (intercommutation, recombination) of colliding cos-mic strings attracts much interest recently (see [146, 146, 147] for reviews),owing to the fact that the reconnection probability is related to the numberdensity of the cosmic strings, which is strongly correlated with possible obser-vation of them. However, solitonic strings may appear in numerous varietiesof field theories, which certainly makes any prediction complicated. In thisChapter, we employ the moduli matrix formalism to show that, in a widevariety of field theories admitting supersymmetric generalization, inevitablereconnection of colliding solitonic strings (i.e. reconnection probability isunity) is universal. The inevitable reconnection of local strings in AbelianHiggs model [148, 149] (see also [150]) has been known for decades, and thisuniversality was found in [90] for non-Abelian local strings in NC = NF gaugetheories. Via a different logic and explicit computations, the results of thisChapter extends the universality to semi-local strings [73, 74] with NC < NF,which is consistent with recent numerical simulations [151] ([152]).

The reconnection of the vortex strings can be understood [148, 149] asright-angle scattering of vortices in head-on collisions [153, 154, 155, 156,157, 158, 159, 160] appearing in a spatial slice. We use moduli space ap-proximation where the motion of the strings is slow enough, to find universalright-angle scattering of vortices on two spatial dimensions. The moduli ma-trix formalism gives a well-defined set of moduli coordinates, and with thatthe analysis of the motion is quite simple and robust.

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9.1 Reconnection of non-Abelian local strings

We deal with the local strings (NC = NF), followed by the semi-local strings(NC < NF). We will find that essential feature can be captured in the caseNC = NF = 2. Single vortex (k = 1) moduli space is C × CP 1 with C theposition of the vortex string in z-plane and CP 1 the orientational moduliconcerning the internal color-flavor space [7, 8, 16], while the moduli space ofseparated two (k = 2) vortices is a symmetric product (C×CP 1)2/S2. Thereconnection problem is related to how they collide in the full k = 2 modulispace, parameterized by the moduli matrices (see Chapter 2):

H(0,2)0 =

(1 −az − b0 z2−αz−β

), H

(1,1)0 =

(z−φ −η−η z−φ

). (9.1.1)

The superscripts label patches covering the moduli space, but one more patch(2,0) is needed to cover the whole manifold. Since the (0,2) patch covers allthe moduli space except lower-dimensional submanifolds, this is sufficientfor computing the reconnection probability. The moduli space of the twocoincident vortices in this theory has been studied in Chapter 2 and foundto be C × WCP 2

(2,1,1) ' C × CP 2/Z2, which any collision of strings goesthrough. The locations z1 and z2 of the vortices and the orientation vectors~φ1 and ~φ2 of the internal moduli for each vortex are determined by

detH0 = (z − z1)(z − z2), H0(z = zi)~φi = 0. (9.1.2)

We parameterize the vectors as ~φi = (bi, 1)T with bi = azi + b, and therelations to the original parameters are

a =b1−b2

z1−z2

, b =b2z1−b1z2

z1−z2

, α = z1+z2, β = −z1z2. (9.1.3)

Physical meaning of the parameters (zi, bi) is clear, but they can cover onlythe subspace z1 6= z2 because the relations (9.1.3) are not defined at z1 = z2.

Let us consider slow motion of the moduli parameters, a la Manton [96],to show the universal right-angle scattering in the vortex collision. We haveto use the parameters (a, b, α, β), not (zi, bi), because, as we have shown, themoduli space metric with respect to the former parameters (which appearlinearly in the moduli matrix H0) is smooth and non-vanishing. With these“well-defined” parameters of the moduli space, at least for a certain period oftime around the collision moment, one can approximate the moduli motionas linear functions of t (since the coordinates are subject to free motion):

a = a0 + ε1t+O(t2), b = b0 + ε2t+O(t2), (9.1.4)

α = 0 +O(t2), β = ε3t+O(t2), (9.1.5)

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where εi, a0 and b0 are constant. Here α is the center of mass of the vortices(see the later discussion for identifying the decoupled center-of-mass param-eter), and thus set to be zero around t = 0. We have used a time translationso that a constant term in β(t) vanishes. This is equivalent to choose thecollision moment as t = 0.

Physical interpretation of the motion (9.1.4) and (9.1.5) can be extractedby looking at the solution in terms of zi and bi. From (9.1.3), we obtain

z1 = −z2 =√ε3t+O(t3/2), (9.1.6)

bi = b0 + (−1)i−1a0

√ε3t+O(t). (9.1.7)

The first equation shows that the vortices are scattered by the right angle;since the time dependence is

√t, when time varies from negative to positive,

the vortex moves from the imaginary axis to the real axis. As stressed before,this right-angle scattering means that the vortex strings are reconnected. So,generic collision results always in reconnection.

When a0 = 0 in (9.1.7), the orientational moduli for each vortex coincide,which corresponds to a reduction to the case of the Abelian-Higgs model.Here we have shown that even when a0 6= 0 and the non-Abelian stringshave different orientational moduli at the initial time, as they approach eachother in the real space, the internal moduli approach each other; in particular,bi experiences the right-angle scattering, too. This is the only consistentsolution to the moduli equations of motion, with generic initial conditions.Note that this understanding comes from the re-description in terms of biand zi, while the true and correct motion in the moduli space is determinedby the moduli parameters (a, b, α, β), which have linear dependence in t.

Although we have shown (by using the (0,2) patch) that the reconnectionprobability is unity, it is instructive to look at the other patches to see whathappens in the submanifold(s) of the moduli space which cannot be describedby the (0,2) patch. In fact the submanifold includes the Z2 singularity of theCP 2/Z2. This corresponds to the situation where the vortices sit in twodecoupled U(1) sub-sectors of the U(2) in the original field theory and wherestrings should pass through each other in collision in that special case. Inthe (1,1) patch, the condition for coincident vortices, namely detH0 = z2,reads

φ = −φ, φφ− ηη = 0, (9.1.8)

which can be parameterized by X and Y through XY = −φ = φ, X2 ≡η, Y 2 ≡ −η. The Z2 symmetry (X, Y ) ∼ (−X,−Y ) is manifest. Note thatthe orbifold singularity X = Y = 0 (η= φ= η= φ= 0) is present only in thesubmanifold z1 = z2, while the full moduli space is smooth. One can confirm

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this by computing the Kahler potential explicitly around the origin of the(1,1) patch, K = 2πc(|φ|2+|φ|2+|η|2+|η|2)+higher, which shows that therethe metric is smooth and non-vanishing. Going to the (X, Y ) coordinates onthe submanifold, we obtain a metric of a Z2 orbifold, K ∝ (|X|2 + |Y |2)2.

Let us study geodesic motion on the moduli space to see the reconnection.After imposing the center-of-mass condition z1 = −z2, we obtain the motionof the moduli parameters

φ = −φ = −XY + s1t+O(t2), (9.1.9)

η = X2 + s2t+O(t2), η = −Y 2 + s3t+O(t2), (9.1.10)

where X, Y and s1,2,3 are constant. We have chosen the collision momentto be t = 0, so that the constant terms in the above satisfy the constraint(9.1.8). The orientational moduli bi are obtained as bi = η/(zi − φ).

From this generic solution of the equations of motion, we compute (for|X|2 + |Y |2 6= 0)

z1 = −z2 =√φ2 + ηη =

√st+O(t3/2), (9.1.11)

bi = XY −1 + (−1)iY −2√st+O(t), (9.1.12)

where s ≡ −2s1XY + s3X2 − s2Y

2. Therefore, we confirm the generic re-connection for s 6= 0. The condition s = 0 is equivalent to ε3 = 0 in theanalysis of the (0,2) patch, because among the patches we have a relationβ = ηη − φφ = st. s = ε3 = 0 can be achieved only by finely tuned initialconditions, so we are not interested in it.

When X = Y = 0 (this point is not covered by the (0,2) patch, so theidentification s = ε3 fails), we obtain

z1 = −z2 =√s2

1 + s2s3 t+O(t3/2), (9.1.13)

bi = s1s−13 + (−1)i−1s−1

3

√s2

1 + s2s3 +O(t1/2), (9.1.14)

which shows no reconnection. Note that this finely tuned collision allowsconstant non-parallel orientations b1 6= b2 at the collision, in contrast to thegeneral case (9.1.7) (9.1.12) where b1 = b2 at t = 0. One observes thatthe reconnection is intimately related to the parallelism of the orientationvectors bi, as is along the intuition. But the significant is that parallel bi atthe collision moment follows from generic initial conditions, which is clarifiedhere in the explicit computations in the moduli matrix formalism.

ForNC =NF > 2 (the orientational moduli space is CPNC−1), the same ar-gument finds that the probability is unity. The moduli matrix of (0, · · · , 0, 2)

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patch is

H(0,··· ,0,2)0 =

(1NC−1 ~az −~b~0 T z2 − αz − β

). (9.1.15)

The center-of-mass parameter is identified with α and we put it zero. Then,we have β = z2

1 , and the solution of the equation of motion for β is thesame as (9.1.5), after the time translation. Finally we have (9.1.6), there-fore we conclude that reconnection occurs, irrespective of the other moduliparameters ~a and ~b. Because the (0,0,· · · ,2) patch covers generic points ofthe moduli space, the reconnection probability is unity. The results are com-pletely consistent with [90] which used a different logic though.

9.2 Reconnection of semi-local strings

We shall show that the reconnection probability is unity also for the semi-local strings, NC < NF. We follow the same logic and find that it appliesto rather generic theories, showing universality of reconnection. Since non-Abelian case can be examined straightforwardly, we concentrate on Abeliancase with NF = 2. The moduli matrix is [75]

H0 = (z2 − αz − β, az + b). (9.2.1)

In the following, we shall show that (i) even in this semi-local case the center-of-mass coordinate is α and thus put to be zero, and (ii) the parameter a(which is associated with the size of the vortex) is non-normalizable and putto be constant. Using these facts, the logic leading to the reconnection is thesame for the remaining normalizable parameters: z1 =

√β =√ε3t. We find

the universality in reconnection. Note that the additional moduli parametersappearing from the extra flavors, a and b, does not play any role in showingthe reconnection. This is clearly the same even for non-Abelian semi-localstrings. With the help of the moduli matrix, one can also show that thereconnected semi-local strings have the same width, which is expected froma geometrical viewpoint.

Let us identify the non-normalizable mode by studying possible infra-red divergence in the Kahler potential (5.4.1). The asymptotic boundarycondition for the master equation (3.1.15) is Ω→ (1/c)H0H

†0, and using the

expression of H0 (9.2.1), we find only the first term in (5.4.1) is relevant. Bythe Kahler transformation K → K+f +f ∗ with f ≡ −c

∫d2z log(z2−αz−β),

we obtain for large |z|

K ∼ c

∫d2z log

[1 +|a|2|z|2]∼ c

∫d2z|a|2|z|2 = 2πc|a|2 logL

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where in the last expression we introduced a cut-off radius L(→ ∞). Thisdivergence shows that the parameter a is non-normalizable. We have to fixthis mode to be constant, so that the effective Lagrangian is finite. In otherwords, motion of the parameter a is frozen because the kinetic term of adiverges and any motion costs infinite energy.

Next, we provide a method to determine the center-of-mass parameter,which is decoupled from the others. We write the moduli matrix in thefollowing form,

H0 =((z − z1)(z − z2), a(z − z3)

). (9.2.2)

in which the parameters are not the “well-defined” parameters. In this form,there is a translation symmetry z → z+δ, zi → zi+δ. Let us assume that z0,which is a linear combination of z1, z2 and z3, is the center-of-mass parameter.The other two parameters independent of z0 should be selected properly fromthe three z′i ≡ zi − z0 (i = 1, 2, 3). We compute the metric from the Kahlerpotential, for this set of independent coordinates. The complete decouplingof z0 from the remaining parameters is ensured if the metric componentgi0 ≡ δ2K/δz′iδz0 vanishes. We can compute it as

gi0 = − δ

δz′i

∫d2z

δ

δzK(z, z0, z

′j) = − δ

δz′i

∮dzK, (9.2.3)

where K is the integrand of the Kahler potential, and we used the fact thatz0 dependence in K is always through the combination z − z0. The explicitexpression (9.2.2) gives, after an appropriate Kahler transformation, for large|z|,

− δ

δz′i

∮dzc log

(1− z1+z2

z− z1+z2

z+· · ·

)= 4πc

δ

δz′i

z1+z2

2.

Vanishing of this means that z′i is orthogonal to the combination z1 + z2,which shows that the center-of-mass parameter is z0 = (z1 + z2)/2 = α/2.This result is non-trivial, because there is another dimensionful parameterz3 which might have been involved with the definition of the center-of-mass.

9.3 Summary

While we studied the critical coupling, non-critical region (which can besmoothly deformed from the critical coupling) has the same universality, sincein the moduli space it is described by introduction of potential terms alongrelative position moduli induced by attractive/repulsive force between Type

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I/II strings. Even for the repulsive case two strings must collide, becauseparts of two strings far from the collision point do not feel a force and thepotential induced around the collision point is negligible compared with thetotal string energy. Adding small mass terms breaking flavor symmetry canbe treated similarly (see for example [90]).

The universal reconnection we found is valid in a low energy regime wherethe moduli space approximation is valid classically. However, as in the caseof Abelian-Higgs model, numerical simulations [151] showed robustness ofthe reconnection even for high energy collisions. We hope that, in the futureobservation, this universality may help for distinguishing solitonic stringsfrom cosmic superstrings/D-branes which have lower reconnection probabil-ities [159, 150]. The moduli matrix formalism has opened up new paths toanalyze BPS solitons. It would be intriguing to apply it further to moreinvolved/realistic situations, such as cosmic string webs and thermal phasetransitions.

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Part III

Vortices in Arbitrary GaugeTheories

162

Chapter 10

Vortices and Lump

In Chapter 5 we discussed semi-local vortices and their relations with thesigma model lumps which arise in the theory at strong coupling. In thisChapter, we will deepen these relations. We will state the existence of aprecise correspondence between vortices and lumps in a very wide class oftheory, including and generalizing the discussions of Chapter 5. The existenceof these links, enable us to consider a bottom-up approach to the constructionof vortices. First, we will introduce a generalized construction for lumps.This construction will straightforwardly give us a construction for vortices,as soon as the gauge couplings are restored to finite values.

10.1 Model and BPS Vortex Equations

Let us start with presenting the general class of models in which we willconstruct vortices. We consider non-Abelian gauge theories with gauge groupG = G′ × U(1), whose Lagrangian is given by

L = − 1

4e2F 0µνF

0µν − 1

4g2F aµνF

aµν + (DµHA)†DµHA

−e2

2

∣∣∣∣H†At0HA −v2

√2N

∣∣∣∣2 − g2

2|H†AtaHA|2 . (10.1.1)

The matter content of the model consists of N flavors of Higgs scalar fieldsin the fundamental representation, with a common U(1) charge, written asa color-flavor mixed N ×N matrix H. This model is a direct generalizationof (3.1.2), which has G′ = SU(N). The flavor symmetry of the model isSU(N)F. It is important, as always, to remember that the model can beembedded in an N = 2 supersymmetric gauge theory, which explains the

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particular form of the potential, ensuring at the same time its stability againstradiative corrections.

An important consequence of this is that we have several useful descrip-tions of the moduli space of vacua, which will be important in the upcomingdiscussion [161]:

MG = H |D-term conditions /G (10.1.2a)

= H //GC (10.1.2b)

=I ijG , holomorphicG-invariants

/algebraic relations(10.1.2c)

Furthermore, N = 2 supersymmetry implies the existence of BPS satu-rated vortex solutions which are supported by the non-trivial first homotopygroup π1(G = G′ × U(1)) = Z. The standard Bogomol’nyi completion forstatic, x3-independent, configurations

T =

∫d2x

[1

2e2

∣∣∣∣F 012 − e2

(H†At

0HA −v2

√2N

)∣∣∣∣2 +

+ 4 |DzH|2 +1

2g2

∣∣∣F a12 − g2H†At

aHA

∣∣∣2 − v2

√2N

F 012

]≥ − v2

√2N

∫d2xF 0

12 , (10.1.3)

gives the BPS equations for vortices.We will focus our attention on the classical Lie groups SU(N), SO(2M)

and USp(2M). For G′ = SO(2M), USp(2M) their group elements are em-bedded into SU(N) (N = 2M) by constraints of the form, UTJU = J , whereJ is the rank-2 invariant tensor

J =

(0M 1Mε1M 0M

), (10.1.4)

where ε = +1 for SO(2M), while ε = −1 for USp(2M). Specializing (10.1.3)to these cases we get the BPS vortex equations

DzH = 0, (10.1.5a)√2N

e2F 0

12 −(Tr(HH†)− v2

)= 0, (10.1.5b)

4

g2F a

12ta −

(HH† − J†(HH†)TJ

)= 0, (10.1.5c)

where the complex coordinate z ≡ x1+ix2 has been introduced. Eq. (10.1.5c)reads for G′ = SU(N) instead:

2

g2F a

12ta −

[HH† − 1N

NTr(HH†)

]= 0 . (10.1.6)

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10.2 The Strong Coupling Limit

Another class of theories we are interested in, is the set of non-linear sigmamodels which can be obtained from the theories (10.1.1) by taking the strongcoupling limit: e, g → ∞. In a fully Higgsed vacuum, all gauge fields aremassive and can be integrated out, and the limit is reliable. The resultingtheory is an N = 2 NLσM in 3+1 dimensions whose target space is thevacuum manifold of the gauge theory:

LNLσM = gi,j∂µφi∂µφj, gi,j = ∂φi∂φjK(φi, φj),

MNLσM = MG. (10.2.1)

The fields φi are a conveniently chosen set of independent coordinates on thetarget space while K is the Kahler potential of the gauge theory.

This class of models also admits BPS, x3-independent, stringy solitoncalled lumps. This kind of solitons are supported by a non-trivial homotopygroup π2(MNLσM). It is indeed very easy to find the equations for lumpswithout going into the explicit computation of the Kahler potential. In fact,taking the strong coupling limit in (3.1.2) without explicitly intagrating outthe gauge fields, we obtain a description of the models (10.2.1) in terms ofthe fields of the original gauge theory (H,Wµ). Lump equations are simplygiven by the strong coupling limit of the vortex equations (10.1.5).

DzH = 0, (10.2.2a)

Tr(HH†)− v2 = 0, (10.2.2b)

HH† − J†(HH†)TJ = 0. (10.2.2c)

Notice that Eq. (10.2.2b) and Eq. (10.2.2c) are nothing but the D-termvacuum equations, whose solutions define MNLσM.

A general property of the models (10.2.1) is the classical scale invariance.Lumps develop a moduli space of degenerate solutions with arbitrary sizeand consequently there appear singular solutions with zero size, called small-lump singularities. Another way to understand these singularities is to noticethat the NLσMs are well defined only on a fully Higgsed vacuum, while thetheories (3.1.2) have a moduli space of vacua that usually includes manybranches with different physical properties: Higgs and Coulomb branches,mixed Higgs-Coulomb phases in which the gauge symmetry is only partiallybroken. Lump solutions can be understood as maps from the C-plane orthog-onal to the x3-axis, into the target spaceMNLσM and we expect singularitieswhen this map hit a point which corresponds to a phase of the original gaugetheory with an unbroken gauge symmetry. In fact, in this points, the appear-ance of massless particles gives rise to singularities in the Kahler potential

165

of the gauge theory, and thus on the target manifold of the NLσM. For ex-ample, small-lump singularities are usually related to conical singularities inthe target space1.

Before giving the recipes for the construction of lumps, we will comment,in the next section, on the correspondence between vortices in the gaugetheories (3.1.2) and lumps in the related NLσM (10.2.1). A consequence ofthis correspondence is that we get, as promised, the moduli space of vortices.

10.3 Vortex-Lump Correspondence

The deep connection between vortices and lumps was first noticed by Hind-marsh [74]. In the extended Abelian Higgs model “semilocal” vortices [73,163] appear, which are generalizations of the usual Abrikosov-Nielsen-Olesen(ANO) vortices: they possess zero modes related to an arbitrary size parame-ter. Semilocal vortices “interpolate” between ANO vortices and CP n lumps,in the sense that zero-size semilocal vortices are ANO (“local”) vortices, whilesemilocal vortices with increasing size become identical to lumps.

We consider this relationship from another, but equivalent, point of view[6]: taking the strong coupling limit e, g → ∞ we map the gauge theory(3.1.2) into the sigma model (10.2.1); at the same time, we map any vortexappearing in the first theory into a lump of the latter:

G′ × U(1) Gauge theory −→ NLσM onMG

Vortices −→ Lumps. (10.3.1)

This correspondence can also be better understood if one consider more quan-titative relations.

BPS correspondence. Vortices and lumps are BPS saturated objects.Their masses do not depend on the gauge couplings, being proportional tothe central charge of the supersymmetry algebra: Tvor = Mlum

Topological correspondence. In a generic case in which the vac-uum manifold is simply connected2, the following relations hold: π2(MG) =π2(Mvacman/G) = π1(G), which define a correspondence between the topo-logical charges of vortices and lumps.

It is important, however, to notice that some vortex configurations will bemapped into singular lumps. For example, ANO vortices, with a typical size

1MNLσM can contain also different kind of singularities, like curvature singu-larities. They give rise to different kinds of singular lumps [162].

2This property of the vacuum manifold is the necessary condition for the exis-tence of semilocal vortices [163].

166

∼ 1/gv, shrink to singular spikes of energy, and are mapped into small-lumpsingularities. Conversely, small-lump singularities are regularized into ANOvortices when finite gauge couplings are restored. The discussion above leadsto the following relation among the moduli spaces of vortices and lumps:

Mvor =Mlum ⊕ singular lumps (10.3.2)

10.4 A Generalized Rational Map Construc-

tion for Lumps

After the general discussion of the previous Section, we are now ready toexplicitly construct the moduli space of lumps arising from (10.2.1).

We choose the fully Higgsed, color-flavor locked vacuum: Hvev = v√N

1N .

The G′ × U(1) × SU(N)F invariance of the theory is broken to the globalcolor-flavor diagonal G′C+F. The first step toward the solutions of Eq. 10.2.2 isto switch from the description (10.1.2a) for the moduli space of vacua to thatof (10.1.2b). We can thus forget the D-term vacuum equations (10.2.2b) and(10.2.2c), at the prize of lifting the gauge symmetry G to its complexificationGC. Eq. 10.2.2a has now a very intuitive meaning. It can be thought as a“covariant holomorphicity” condition. This means that H is holomorphic,up to a complexified gauge transformation:

H = S−1H0(z) = S−1e S ′

−1H0(z) , (10.4.1)

where S ∈ GC, while H0(z) is a matrix whose elements are holomorphicin z. We denote H0(z) the moduli matrix [81], as it encodes all moduliparameters3. Eq. 10.4.1 does not fix completely GC. H0(z) is in fact onlydefined up to holomorphic gauge transformations V (z):

H0 ∼ V (z)H0 = Ve V′(z)H0 , V ′ ∈ G′C, Ve ∈ C∗ . (10.4.2)

The next step is to consider the holomorphic invariants I(i,j)G (H) made of

H, which are invariant under GC, with (i, j) labeling them. This is equivalentto changing our description ofMG from Eq. 10.1.2b to Eq. 10.1.2c. The keypoint here is that, thanks to Eq. 10.4.1, these invariant are also holomorphicfunctions of the coordinate z:

I(i,j)G (H) = I

(i,j)G

(S−1H0(z)

)= I

(i,j)G (H0)(z). (10.4.3)

3To reconstruct the solution in terms of the original Higgs fields H, one hasstill to solve Eq. 10.2.2 as algebraic equations for S.

167

We have thus identified a lump as a holomorphic map from the complexplane C to the set of holomorphic invariants I

(i,j)G

4.The last step is to impose the following boundary conditions to the holo-

morphic invariants:

I(i,j)G (H)

∣∣∣|z|→∞

= I(i,j)vev . (10.4.4)

It is convenient to translate the above conditions in terms of the holomorphicinvariants of G′, IjG′(H0):

I(i,j)G (H0) ≡ I iG′(H0)

IjG′(H0), (10.4.5)

where we must take ratios of invariants with the same U(1) charge: ni = nj5.

Eq. 10.4.5 defines the generalized rational map construction for lumps.It is easy to check that Eq. 10.4.4 and Eq. 10.4.5 together imply the

following:

I iG′(H0)∣∣∣|z|→∞

= I ivev zνni (10.4.6)

where ν is an integer common to all invariants. As I iG′(H0(z)) are holo-morphic, the above condition means that I iG′(H0(z)) are polynomials in z.Furthermore, ν ni must be a positive integer for all i:

ν ni ∈ Z+ → ν = k/n0 , k ∈ Z+ , (10.4.7)

where (GCD = the greatest common divisor)

n0 ≡ GCDni | I ivev 6= 0 . (10.4.8)

Note that a U(1) gauge transformation e2πi/n0 leaves I iG′(H) invariant:

I iG′(H′) = e2πini/n0I iG′(H) = I iG′(H) : (10.4.9)

the phase rotation e2πi/n0 ∈ Zn0 changes no physics, and the true gaugegroup is thus G = U(1)×G′/Zn0 , where Zn0 is the center of G′. A simplehomotopy argument tells us that 1/n0 is the U(1) winding for the minimal,

4All the points at the infinity of z are identified, so that the map is in factdefined on CP 1.

5The U(1) charge ni is simply defined as: IiG′(αH0) = αniIiG′(H0), for anycomplex number α.

168

k = 1, topologically non-trivial configuration. Finally, for a given k thefollowing important relation holds

I iG′(H0) = I ivevzkni/n0 +O(zkni/n0−1) , (10.4.10)

which implies nontrivial constraints on H0(z). The set of all inequivalentH0(z) satisfying Eq. 10.4.10 identify a lump solution, and thus defines thefull moduli space.

10.5 Moduli Space of Vortices

The lump construction explained in the last Section is only partially com-plete, because one has to identify all singular lump solutions, and this is noteasy in general. This problem disappears when we lift the construction forlump to that for vortices in the related gauge theory, according to the dis-cussion of Sec. 10.3. As explained there, singular lumps are regularized intovortices as we turn on finite gauge couplings, and Eq. (10.4.10) now give thefull and complete description of the moduli space of non-Abelian vortices.

To reconstruct the original Higgs fields H = S−1H0 we now have tosolve the BPS differential equations Eq. (10.1.5b) and Eq. (10.1.5c) (or Eq.(10.1.6)) in terms of S, while from Eq. (10.1.5a) we find the gauge fields:

W1 + iW2 = −2i S−1∂S. (10.5.1)

The tension of the BPS vortices can be written as

T = − v2

√2N

∫d2x F 0

12 = 2v2

∫d2x ∂∂ log(SeS

†e) . (10.5.2)

The asymptotic behavior Se ∼ |z|ν then determines the tension

T = 2πv2ν , (10.5.3)

thus ν has to be identified with the U(1) winding number of the vortexconfiguration.

Let us now apply the general discussion above to concrete examples. ForG′ = SU(N), with N flavors, there only exists one invariant

ISU = det(H) , (10.5.4)

with charge N . Thus the minimal winding is equal to 1/N and the conditionfor k vortices is given by:

AN−1 : detH0(z) = zk +O(zk−1), ν = k/N . (10.5.5)

169

For G′ = SO(2M), USp(2M), there are N(N ± 1)/2 invariants

(ISO,USp)rs = (HTJH)rs, 1 ≤ r ≤ s ≤ N , (10.5.6)

in addition to (10.5.4). The constraints are:

CM , DM : HT0 (z)JH0(z) = zkJ +O(zk−1) , ν = k/2 . (10.5.7)

Thus, vortices in the SO(2M) and Usp(2M) theories are quantized in halfintegers of the U(1) winding [164].

Explicitly, the minimal vortices in SU(N) and SO(2M) or USp(2M)theories are given respectively by the moduli matrices

H0 =

(z − a 0

b 1N−1

),

(z1M −A CS/A

BA/S 1M

), (10.5.8)

together with all permutations of lines and columns. The moduli parametersare all complex. For SU(N), a is just a number; b is a column vector.For SO(2M) or USp(2M), the matrix CS/A for instance is symmetric orantisymmetric, respectively. And vice versa for B. A detailed derivation ofthe matrices written above, together with the construction of moduli matricesfor SO(2M + 1) as well as those for k = 2 vortices in SU, SO,USp theories,will be given explicitly in the next Chapter (see Refs. [162, 165] for moredetails).

The complex dimension of the moduli space, given by the number ofcomplex parameters in the moduli matrices (10.5.8), is

dimC (MG′,k) =k N2

n0

. (10.5.9)

This result was firstly obtained in Ref. [7] for SU(N); The dimension of themoduli space, inferred from the moduli matrices, agrees with the one givenin Eq. (10.5.9) and is the same for any choice of the gauge group G′, whereN is the dimension of the fundamental representation of G′ for which Zn0 isthe center.

Except for the SU(N) case, our model has a non-trivial Higgs branch(flat directions). The color-flavor locked vacuum Hvev ∝ 1N is just one of thepossible (albeit the most symmetric) choices for the vacuum; our discussioncan readily be generalized to a generic vacuum on the Higgs branch. Thisfact, however, implies that our non-Abelian vortices have semi-local modulieven for Nf = N . In contrast to the Abelian or SU(N) cases, moreover, theyexhibit new, interesting phenomena such as “fractional” vortices [165].

The generalization to exceptional groups can be done if the invarianttensors of each group are known. Indeed, this is the case, and Tab. 10.1 liststhem all.

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G′ AN−1 BM CM , DM E6 E7 E8 F4 G2

R N 2M + 1 2M 27 56 248 26 7rank inv − 2 2 3 2, 4 2, 3, 8 2, 3 2, 3n0 N 1 2 3 2 1 1 1

Table 10.1: The dimension of the fundamental representation (R), the rankof the other invariants [166] and the minimal tension ν = 1/n0 i.e. the centerZn0 of G′. The determinant of the R × R matrix gives one invariant withcharge, dim R.

10.6 Local (ANO-like) Vortices

For various considerations, we are interested in knowing which of the moduliparameters describe the local vortices, the ANO-type vortices with expo-nential tails. The moduli space of fundamental local vortices, for example,is completely generated by the global symmetries of the vacuum (see below)and it can eventually survive non-BPS deformations that preserve these sym-metries. Furthermore, we found in Section 5.4 that local vortices correspondto the subset of solutions with the maximum number of normalizable moduli.

Local vortices correspond, as we mentioned, to small lump singularities inthe lump construction. These configurations are obtained when the rationalmap construction is degenerate, e.g., when all the invariants I iG′ share acommon zero:

I iG′ = (z − z0)ni/n0 I ′iG′ . (10.6.1)

The rational map in Eq. (10.4.5), does not feel any of these common zeros,which can be interpreted as insertions of singular spikes of energy, or localvortices. Configurations with only local vortices are obtained imposing acomplete degeneration of the rational map:

I iG′(H0,local) =

[k∏`=1

(z − z0`)

]ni/n0

I ivev . (10.6.2)

For G′ = SO(2M), USp(2M) with ISO,USp of Eq. (10.5.6) we find that thecondition for vortices to be of local type is

HT0,local(z)JH0,local(z) =

k∏`=1

(z − z0`) J . (10.6.3)

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Let us now discuss a few concrete examples. The general solution for theminimal vortex (10.5.8) for G′ = SU(N), SO(2M), USp(2M) is reducedto a local vortex if we restrict it to be of the form:

H0,local =

(z − a 0

b 1N−1

),

((z − a)1M 0

BA/S 1M

). (10.6.4)

The vortex position is given by a, while b for SU(N) and BA/S for SO(2M) orUSp(2M) encode the Nambu-Goldstone modes associated with the breakingof the color-flavor symmetry by the vortex G′C+F → HG′ . The moduli spacesare direct products of a complex number and the Hermitian symmetric spaces

MlocalG′,k=1 ' C×G′C+F/HG′ , (10.6.5)

HSU(N) = U(N − 1) while HSO(2M),USp(2M) = U(M). The SU(N) andSO(2M) cases have already been studied in Refs. [7, 16, 164]. In the nextChapter we will going to discuss in more details all these cases. Furthermore,we will generalize these results to USp and SO(2M + 1) theories.

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Chapter 11

Orthogonal and SymplecticVortices

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Chapter 12

Semi-classical hints of GNOWDuality from vortex side

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Conclusions and Outlook

The developments in our knowledge about non-Abelian vortices, like those Idescribed in this Thesis are a new, big, step in our capacity to handle solitonsin supersymmetric gauge theories. In particular, the potential extension ofthe moduli matrix formalism to all gauge theories supporting BPS solitons,will give us the possibility to significantly improve our understanding aboutthe physics of solitons.

Now we have the technologies to start a broad and systematic study ofsolitons, for which the following points are only possible examples of futureline of research.

Constructing new types of solitons

It will be interesting to explicitly apply our construction to models differentfrom those studied until now. The first step has been already partially ad-dressed, constructing non-Abelian vortices in theories with SO(N) × U(1)or USp(2M) × U(1) gauge groups. Other interesting possibilities would beto study how the properties of solitons change with the choice of the mattercontent and in particular with respect to the choice of the representations ofthe fields. Exceptional groups can also be considered. Groups like F4 andG2 have attracted much attention, especially in lattice simulations. Theyhave a trivial center, avoiding the possibility that confinement is given bycondensation of center vortices (instead of being due to the condensation ofmonopoles).

The generalized moduli matrix construction should also be applied, in thenear future, to explore different kinds of solitons in this new set of theories.We think of domain-walls and monopoles, together with composite config-uration of solitons, like domain-wall-vortex junctions, monopole as confinedkinks on vortices and so on. These configurations have already been studiedin U(N) gauge theories with fields in the fundamental representation. Theycan now be constructed in a more general set-up, giving rise to the possi-bility of making a real complete and systematic survey of BPS solitons insupersymmetric gauge theories.

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This program could be useful in the “reverse engineering” of string theoryfrom field theory. As is well-known, string theory has been used to construct,with a geometrical approach, field theories and their solitons. These con-structions have been very useful in the understanding of many properties ofgauge theories. More recently, it has become clear that the inverse processis also possible. For example, domain-walls have been used to “reproduce”the physics on D-branes. The biggest difficulty in this project is to show theactual localization of a non-Abelian gauge field on the domain-wall. Froma general point of view, a wider understanding of solitons is crucial to fullyappreciate the relations between string theory and gauge theories.

Non-Abelian vortices and 4-d gauge dynamics

There are many interesting developments which could be realized along thisline of research. Until now, the matching between the vortex theory and thebulk theory has been demonstrated only in the case of U(N) gauge theories.The first step forward would be the study of the quantum physics of thevortex theory in the generic case G′×U(1). This would give us a new set of2d/4-d correspondences.

A more recent progress is regarding to the study of the same issue intheories with N = 1 supersymmetry. The quantum physics of the vortex(now dubbed heterotic) has been shown to capture the physics of the bulktheory. In particular, the quantum deformations of the moduli space of vacuaare reproduced from the vortex side, together with supersymmetry breakingand a successive restoration mechanism, as soon as a meson field expecta-tion value is turned on. It is very interesting to consider the possibility thatthe vortex theory could also mimic the physics of Seiberg duality. To pushforward this program, it is crucial to understand how the meson fields ofthe Seiberg-dual theory will deform the vortex theory. Semi-local vorticeswill also play a crucial role in this context. These considerations may in-dicate the existence of dual (vortex) theories. Indeed, there are hints thatthe moduli space of vortices transforms in an interesting fashion under thetransformation Nf → Nf −Nc.

Non-Abelian monopoles from the vortex side

Non-Abelian vortices may also help to push forward the study of non-Abelianmonopoles, using similar strategies to those I have used in my past research.The main step forward will be the generalization of previous works on thehierarchical system G → G′ × U(1) → 1, where G′ × U(1) is a subgroupof a generic simple group G. We have now the technology to construct

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non-Abelian vortices arising from the low-energy breaking G′ × U(1) → 1.These vortices will confine a very general class of non-Abelian monopoleswhich arise from the G → G′ × U(1) breaking. We believe that the modulispace of the confining vortex captures the important properties of that of theconfined monopole, including the quantum ones. This strategy seems to beconsistent in the G = U(N) case. From this point of view, the vortex shouldprovide a regularization of the non-normalizable modes of the monopoles(these modes are responsible for the well-known difficulties in quantizingthem). The proposed generalization is mostly important in the context ofverifying GNOW duality. While the U(N) group is self-dual, other gaugegroups have really different duals, for example: SO ↔ USp. This is crucialto really have strong hint of the appearance of new symmetries in theoriessupporting non-Abelian monopoles.

Further research lines

It is tempting to speculate that non-Abelian vortices may play a role in thephysics of a newly proposed mechanism of dynamical supersymmetry break-ing [167]. These new result have been important for a renewed hope to con-struct supersymmetric models with a true phenomenological relevance. Thisnew framework is intriguing. First it gives a solution to the long-standingproblem of supersymmetry breaking. Moreover, it puts together the phe-nomenology of supersymmetry with the (strong coupling) physics of Seibergduality. Seiberg duality may be the link between dynamical supersymmetrybreaking and the vortex physics. In fact, a study of vortices in metastablevacua has already been started in some depth.

Another interesting and recently developed field is the gauge/gravity cor-respondence (ADS/CFT). The correspondence gives us the possibility tostudy the strong coupling regime, and confinement, from a different perspec-tive. Gravity duals of vortices in confining, N = 1∗, theories have alreadybeen studied, and a Casimir scaling law for the tension has been checked.It would be interesting to construct gravity duals of non-Abelian vortices intheories like those we have previously considered.

Non-Abelian vortices may also be relevant in future experiments of con-densed matter. They appear in many low-energy phenomena, like supercon-ductivity and superfluidity. Their importance is also recognized in the phe-nomenology of the fractional quantum Hall effect. In the last case, vorticesappear as solitons in a (2+1)d Chern-Simons effective theory. Non-abelianvortices can be be detected in systems with non-Abelian statistic, which canbe realized, for example, in multi-layered materials. One interesting ques-tion is the formation of lattices. In a type II superconductor, for example,

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vortices form the hexagonal Abrikosov lattice. Recently found 1.5 supercon-ductors admit vortices which have interaction similar to those we found inthe non-Abelian case, with static inter-vortex forces, which change sign withrespect to the relative distance. Our study about interactions is a first stepto understand the shape and the structure of lattices of non-Abelian vortices.

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Acknowledgements

I am very grateful to my supervisor, Prof. Kenichi Konishi and to all mycollaborators that shared with me the pleasure of new understandings: Mi-noru Eto, Jarah Evslin, Koji Hashimoto, Giacomo Marmorini, Muneto Nitta,Keisuke Ohashi, Naoto Yokoi.

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