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Universit`

a degli Studi di Pisa

DIPARTIMENTO DI FISICA

Corso di Laurea Magistrale in Fisica

Holographic Quantum Chromo Dynamics and the

Generalized Skyrme Model

Relatore:

Prof Bolognesi Stefano

Candidato:

Proto Andrea

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Contents

1 Introduction 7

1.1 From Yang Mills Theory to Holography . . . 7

1.2 The Thesis . . . 10

1.2.1 The Structure . . . 10

1.2.2 The Methodology . . . 11

1.2.3 How to read this work . . . 11

2 Gauge Gravity Duality 13 2.1 Introduction . . . 13

2.2 Relativistic Point Particle . . . 13

2.3 p-brane . . . 14

2.4 The Nambu Goto Action . . . 16

2.5 The Polyakov Action . . . 16

2.6 Polyakov Action Symmetries . . . 16

2.6.1 Stress Energy Tensor and Weyl Rescaling . . . 17

2.7 Neuman and Dirichlet Boundary Conditions . . . 18

2.8 String Interactions . . . 19

2.8.1 Low Energy eective Action For Oriented Strings . . . 21

2.9 Mass Spectra for the bosonic string . . . 22

2.10 Supersymmetric String . . . 25

2.11 Supergravity . . . 28

2.12 Dp-branes Action . . . 30

2.13 T-duality and S-duality . . . 31

2.13.1 T-duality . . . 31

2.13.2 S-duality . . . 32

2.14 RR-p branes . . . 32

2.15 Anti De Sitter space . . . 34

2.15.1 Lorentzian signature . . . 34

2.15.2 Euclidean signature . . . 36

2.16 Large N Limit . . . 36

2.17 Conformal Field thery . . . 37

2.17.1 The Conformal Group . . . 37

2.17.2 The Scale Invariance . . . 41

2.17.3 Relationship between Conformal Invariance and Scale invariance . . . 42

2.17.4 The Relationship between Conformal Invariance and Beta Function . . . 43

2.17.5 The Quantum Conformal Invariance . . . 43

2.17.6 Correlation Functions Part I . . . 43

2.17.7 Correlation Functions Part II . . . 44

2.18 Killing Vectors for Ads Space . . . 45

2.19 The Relationship between RR-p brane and D-p brane . . . 45

2.20 Ads/Cft Correspondence . . . 46

2.20.1 The Correspondence . . . 46

2.20.2 A Concrete realization for the Correspondence . . . 47

2.20.3 Wilson Loop . . . 50

2.21 Ads/Cft at nite temperature . . . 53

2.21.1 The Gravity Side . . . 53

2.21.2 The Gauge Side . . . 53

2.21.3 Kaluza Klein Compactication . . . 54

2.21.4 A First Example . . . 54

2.21.5 The Entropy in the gauge side . . . 55

2.21.6 The Entropy in the Black Hole side . . . 56 3

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2.22 D4-brane background . . . 56

2.22.1 The Model . . . 56

2.22.2 The Gauge Action . . . 58

2.22.3 The Quark Anti-Quark Static Potential in D4 Background . . . 59

2.23 The Sakai Sugimoto Model . . . 59

2.23.1 Probe D8 conguration . . . 60

2.23.2 The Gauge Action . . . 62

2.23.3 The Spontaneous Symmetry Breaking . . . 63

2.23.4 The Supergravity Action . . . 63

2.23.5 The First Hodge Duality Braking in D4 Background . . . 65

2.23.6 The Second Hodge Duality Braking in D4 Background . . . 65

2.23.7 The D4 Chern Simons terms. . . 67

2.23.8 DBI term for D8 . . . 70

2.23.9 The Complete Action . . . 71

2.23.10 Implemeting a stack of N0 D6 brane . . . 71

2.24 The Sakai Sugimoto Action . . . 72

2.24.1 The pion expansion . . . 72

2.24.2 The Sakai Sugimoto Action in a New Fashion . . . 76

3 Topological Solitons 81 3.1 Homotopy group . . . 81

3.2 Topological Solitons . . . 84

3.3 Homotopy groups for coset spaces . . . 86

3.4 Topological degree . . . 87

3.5 Chern numbers . . . 87

3.5.1 The First abelian Chern number on R2 . . . 87

3.5.2 The First abelian Chern number on a compact space with and without boundary . 88 3.5.3 The First abelian Chern number on a bundle . . . 88

3.5.4 The Second abelian Chern number . . . 89

3.5.5 The First non abelian Chern number . . . 89

3.5.6 The Second non abelian Chern number . . . 90

3.6 Derrick's Theorem . . . 91

3.7 The Moduli Space . . . 93

3.8 Kinks and Sine Gordon Kinks . . . 93

3.8.1 Kinks . . . 93

3.8.2 Sine Gordon Kinks . . . 96

3.9 Vortices . . . 97

3.9.1 How Vortices arise . . . 97

3.9.2 Vortices at critical coupling . . . 99

3.9.3 The Vortices Moduli Space . . . 99

3.10 Monopoles . . . 99

3.10.1 The Dirac Monopole . . . 99

3.10.2 Wu-Yang Monopole . . . 105

3.10.3 't Hooft-Polyakov Monopole . . . 107

3.11 Instantons . . . 111

3.11.1 The Idea . . . 111

3.11.2 The Topological Sector and the BPST solution . . . 113

3.11.3 The t'Hooft Ansatz . . . 115

3.11.4 The N Instanton solution . . . 116

3.11.5 Theta Vacua . . . 118

3.11.6 Gribov Copies . . . 120

3.11.7 The Instanton Moduli Space . . . 122

3.12 Sigma Model Lumps . . . 122

3.12.1 The Model . . . 122

3.12.2 The Topological Sector and Static Energy . . . 123

3.12.3 From O(3) Sigma Model to CP(1) Model . . . 124

3.12.4 The Lumps Moduli Space . . . 125

3.13 The Baby Skyrme Model . . . 125

3.13.1 The Model . . . 125

3.13.2 Bogomolny Bounds . . . 125

3.13.3 Minimum Energy Solutions . . . 127

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CONTENTS 5

3.14 The Skyrme Model . . . 131

3.14.1 The Model . . . 131

3.14.2 The Topological Sector . . . 132

3.14.3 A Geometrical Point of View Pt 2 . . . 133

3.14.4 The HedgeHog Ansatz . . . 134

3.14.5 Holonomy . . . 136

3.14.6 BPS Skyrme Model . . . 136

3.14.7 Skyrme Moduli Space . . . 140

3.14.8 Skyrme Model and Nuclear Physics . . . 140

4 The Sextic Term Extraction from the SS Action 143 4.1 Approaching The Target . . . 143

4.2 Recovery of The Sextic Term . . . 143

4.2.1 The Set Up . . . 143

4.2.2 The Limit for Small Lambda . . . 144

4.2.3 Approximation in the E.O.M. . . 145

4.2.4 Solving the E.O.M. . . 145

4.2.5 The SS Action On-Shell . . . 147

4.2.6 The Final On-Shell Action . . . 149

4.2.7 Evaluating The Static Energy . . . 150

4.3 Conistency of the Solution . . . 152

4.3.1 Checking the Goodness of the Behaviour . . . 152

4.3.2 Checking the Goodness of the Approximation . . . 154

5 Conclusions 157 6 Appendix 161 6.1 General Relativity in the Newtonian Limit . . . 161

6.1.1 Linearized Gravity . . . 161

6.1.2 Newtonian Limit . . . 162

6.1.3 Gravitational red shift . . . 163

6.2 Dierential Forms . . . 163

6.2.1 Preliminaries . . . 163

6.2.2 Denition and proprieties . . . 164

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Chapter 1

Introduction

1.1 From Yang Mills Theory to Holography

The early 1950 was the start of new period for physicsts. The rst accelerators were on the point to surpass energies of 1 Mev and physicsts were discovering a "particle zoo". So people was asking how to organize the particles that were discovered continuously. In those days the strong interactions were known to exist but nobody had idea about how to realize them from the theoretical point of view. In 1954 Yang and Mills pubblished one of the most famous and important paper in the history of physics, where they proposed to extend the QED Lagrangian to non abelian elds [84].

In their original paper they proposed to extend the isospin symmetry to a local one. In this way they discovered a new Lagrangian, which was called Yang Mills Lagrangian in their honour, where the isospin symmetry was gauged under a non abelian group. At the beginning this group was SU(2), due to the fact that they were thinking in the isospin context.

In 1961 Gell Mann published an article in which the particles discovered were organized inside SU(3) avour multiplets. In this picture the whole zoo particle known was organized into SU(3) avour mul-tiplets. Since the particles corresponding to the fundamental representation of SU(3) (avour), i.e. the quarks, hadn't been observed, people started to think that they could have been elementary fermions that, for some unknown reason, hadn't still be observed. However the original idea of Y.M. had a terrible issue inside, apparently. In February 1954 Robert Oppenheimer invited Yang to present the upcoming idea at a seminar at the Institute for Advanced Study (IAS) in Princeton, New Jersey.

Wolfgang Pauli was present. We knew that, one year before, Pauli developed the rst consistent generalization of the 5D Kaluza-Klein Theory to a higher dimensional internal space. Since he saw no way to give masses to gauge bosons, he refrained from publishing his results formally. Giving mass to gauge bosons was important because, for the Heisenberg uncertainty principle, physicists thought that a gauge particles, carrying a short-range force (as the strong interaction), should have been massive.

In this spirit Pauli interrupted the Yang presentation twice demanding for the mass of the B eld without receiving a satisfactory answer. After the third interruption, Yang responded that him and Mills had no denite conclusions. Then Pauli snapped back "It is not a sucient excuse" in a very hostile way. At this point Yang sat down and the whole room fell in an awkward silence.

Finally Oppenheimer, who was the chairman of the seminar, said "we should let Frank proceed". Then Yang resumed and Pauli didn't ask any more questions during the seminar. Despite of what happened, Yang Mills nevertheless decided to publish their paper one later, even if it was clear to every body that Pauli wasn't wrong and that the issue had to be redeemed. Indeed

• Apparently if the gauge bosons were massless, they carry only a long range force • If the gauge bosons were massive, the Yang Mills theory was non-renormalizable and

spoiled of gauge invariance =⇒ eective and not unitary theory (1.1) So Yang Mills theory was at an impasse and was abandoned. It seemed to be an useless tool for about 20years. However new scenarios were appearing on the horizon.

In 1965 Nambu proposed to associate the SU(N) gauge symmetry of the Yang Mills Lagrangian to SU(Nc), where Ncstands for color number. In this context the particles carrying the strong interaction,

called gluons, transform under the adjoint representation of SU(N) and carry a color charge, while the fermions, which we now call quarks, transform under the fundamental representation and carry a color charge as the gluon. This necessary step was provided from the discovery of the ∆++ resonance, a

baryon that required three up quarks with parallel spins and vanishing orbital angular momentum. Since the ∆++ resonance wasn't described by an antisymmetric wave function, as it should have been, a new

quantum number was needed. This hidden quantum number was exactly the color number. 7

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Due to the postulate of the color connement, today we know that only "color singlets" can be observed in nature. This postulate is needed and explains the reason for which no free quark has never been observed. Indeed baryons (three quark composite particles) and mesons (a pair quark anti-quark composite particles) are color singlets.

In spite of the developments, a full comprehension of the strong interaction was missing. This lack was partially lled in 1974 when Wilczek and Gross published their work intitled "Ultraviolet behaviour of non abelian gauge freedom" [32], in which they showed the Asymptotic freedom for non abelian gauge theories. This has been a fundamental step for the comprehension of strong force because they showed that, at high energies, i.e. small distances, the coupling running constant goes to zero (asymptotic freedom), while at low energies, i.e. great distances, the coupling running constant becomes ∼ 1 (infrared slavery).

This was related to the fact that strong force is mediated by massless gauge bosons, i.e. the gluons, which possess a color charge themselves and interact one with each other, causing the short range inte-raction, contrarily to photons who carry no charge of any type. From the equations it was also clear that there was a lower bound energy of the QCD validity, in the weak perturbative regime, of some hundred of Mev's.

However still new problems arose. Indeed the whole physicists community was interested in under-standing the nuclear force mechanism. As we know and as it was clear in those early times, the nuclear force in nuclei is a residual of the original strong force involving quarks and gluons and it becomes real-ly strong when the energies are low and the running coupling constant is ∼ 1. In these conditions no perturbative expansion may be achieved and physicists started demanding a new way to overtake these limits. In literature this became known as connement problem and lies in nding a mathematical proof of the infrared slavery of QCD.

As time went on it became clear that it was not an easy solvable problem. In 1974 't Hooft proposed the 1/N expansion in which a number of innite colors was taken, keeping the coupling constant squared times the color number large but xed. 't Hooft showed that the large N limit of QCD provided a theory involving only glueballs and mesons [65]. Glueballs are bound states of two or more gluons and they are thought to be massive. In literature this is indicated as the mass gap problem and is one of the conjectures that wait to be proved axiomatically, together with the connement problem [25]. So the large N limit of QCD provided a theory involving mesons. This was very interesting because, as Nambu showed in 1961 in his pioneering paper, a theory involving mesons could be achieved involving a concept that was at the windows in those years and that became crucial very soon: The Spontaneous Symmetry breaking.

Consider a theory at certain energies. This theory will have a vacuum. If the system approaches the vacuum and it is not invariant under the initial symmetry of the system, we say that the symmetry is spontaneously broken. A theorem establishes that, in the situation just described, new objects make the appearance, necessary to maintain the correct number of degrees of freedom. This objects are called Goldstone bosons. If the symmetry spontaneously broken is global the Goldstone bosons are massless. If, on the other hand,the symmetry spontaneously broken is local, the Goldstone bosons "become" massive. Furthermore, we underline that in S.S.B. the symmetry is broken only at the level of vacuum and that, expanding the Lagrangian under a small uctuation around the vacuum, one sees that the symmetry is still existing, although it's hidden. Nambu explored the rst case and showed that the spontaneous symmetry breaking of the avour symmetry group of QCD caused the appearance of pions as (pseudo)-Goldstone bosons.

On the other side the spontaneously symmetry breaking of the local symmetry will be at the basis of the Higgs Mechanism. Turning on the global case, physicists saw that, imposing the Goldstone bosons to respect the initial symmetry, it's possible to build by hand a new Lagrangian, working at energies below the QCD ones, at which the only involved degrees of freedom are pions and baryons. This was done very soon and it took the name of "eective pion eld theory" or "chiral perturbation theory" . As the Glashow-Weinberg-Salam model is the UV completition of the four-fermions Fermi theory, QCD is the UV completition of the pion eective Theory. Still nowadays a direct derivation of the eective pion Lagrangian from the Yang Mills one is missing. Furthermore the eective pion Lagrangian is not renormalizable (since it is eective), although contains very interesting object inside called Skyrmions.

In his pioneering works [60], [61] Skyrme showed that baryons could be seen as solitonic solutions of the eective pion Lagrangian truncated at the two rst orders. We remind that solitons are static solutions with nite energy for a given system. They may be topological or not, depending on the system considered. The Skyrme solitons are topological and in literature they are known as Skyrmions. Later, Finklestein and Rubinstein showed that the Skyrmions, made of pions, could obey the Fermi statistics [29]. With the advent of QCD a lot of progress were made, especially thanks to Witten works [11] [81]. More in detail Witten showed that the baryons interpretation of Skyrmions could be reliable in the large N limit. This interpretation seems to be supported from data experiments, especially for lower nuclei, although a lot of progresses await to be done.

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1.1. FROM YANG MILLS THEORY TO HOLOGRAPHY 9 Going forward, the situation involving the comprehension of strong interactions was in a theoretical stall and the only way in which the QCD could be approached, in the non-perturbative regime, with quite satisfactory results, was the lattice QCD. This situation remained unchanged till when Maldacena, in his famous paper, published the AdS/CFT conjecture [13]. This conjecture, which is still a conjecture without a mathematical proof, even if several reasons to believe it's true exist, stated that a precise type of string theory in the bulk is dynamically equivalent to a conformal theory on the boundary. Dynami-cally equivalent means that the observables are the same. Conformal means the theory is invariant under a group called Conformal group, which is an extension of the Poincarè one. In the original paper [13], the boundary theory was a N = 4 Super Yang Mills theory in four dimensions, where N is the number of supercharges. However this theory is Supersymmetric and we have no direct-indirect conrm of the Supersymmetry at the experimental level. Furthermore this duality stands only in the limit of innite color number.1

As regards the String theory we will only make few remarks. The String Theory is a theoretical fra-mework in which the point-like particles are "replaced" by one dimensional objects called strings. The idea comes from pioneering paper of Nambu, who studied a model constituted of a quark anti-quark pair at xed ends rotating around z axis from the relativistic point of view. So the idea of string arises from the necessity of explaining the potential in a quark anti-quark pair. From the historical point of view, the birth of string theory is intrinsically related to QCD.

As time went on, string theory took and independent path and, the more it procedeed, the more the physicist community became drastically divided, becouse became soon clear that string theory doesn't obey the falsiability criterion. This criterion, due to the philosopher Karl Popper, states that a theory is genuinely scientic only if it is possible, in principle, to enstamblish that is false, throught experiments for examples. Since the string lives at the Planck Scale, it's impossible for our current technology to observe them (LHC arrives at 15 Tev, while the Planck Mass is set at 1015Tev). However this is not the

only issue. Indeed there is also a problem arising from the dimensional compactication, which causes the presence of something like 10500 metastable vacua, i.e. states at local minima of the energy which

may decade into lower energy vacua. According to string theory, our universe should live in one of these metastable vacua and so on. Since all these metastable vacua seems share the same probability, there is no "selection rule" and there is no reason for which we should live in one among these 10500. This is

known as String Landscape problem. Even if dierent solutions has been proposed2, no one of these seem

good enough and the idea of string theory as the Ultimate Theory seems resized, also because, as we have said before, we have no direct-indirect conrm of Supersymmetry and string theory makes a great use of Supersymmetry in his most important developments.

Turning to the original problem, the issue became how to build a duality between a non-Supersymmetric Yang Mills theory and a consistent string theory. This issue was partially solved by Witten one year later then the Maldacena paper, in 1998, considering a particular background, in which the string theory is dual to a Pure Yang Mills theory in the large N limit, where Pure stands for no quark involved. To complete the picture, the quark needed to be added to the holographic system. This was achieved by Sakai and Sugimoto in 2004, when they proposed an holographic dual of Yang Mills theory with massless quarks. As we will see later, the Sakai Sugimoto Model may be reduced to the Skyrme Model (the pion eective Lagrangian truncated at the rst two terms which are the quadratic and the quartic ones), using a particular ansatz.

The Sakai Sugimoto Model also provides the baryon-instantons duality, a feature already known between the Skyrmions and SU(2) instantons [15], but promoted at an holographic level. Later, using an extension of the Sakai Sugimoto Model, it has been possible to introduce massive quark in the holographic picture [34].

These developments have allowed to predict, using the holographic dual correlation functions, the scattering cross sections involving the low energy states of strong interactions such as glueballs, mesons and so on. This has been achieved putting N = 3 in the end, without considering the large N limit

1We remark that,in principle, one may have a string/gauge duality without holography, i.e. the gauge theory may not

live on the boundary of the string one. In general other forms or dualities have been achieved to which is often referred in literature as non AdS/non CFT dualities.

2At the beginning of 90's some of the most important developer of string theory, as Susskind, advanced the idea tha the

anthropic principle could solve the question. In synthesis the anthropic principle states that, the only reason for which an arbitrary phyisical state or a parameter is the same that we obseve lies in the fact that those quantities are the only ones that allow the existence of intelligent life able to address the problem and to indentify the physical state or the parameter. In the physical state case the anthropic principle is referred to the landscape problem while, in the parameter case, the anthropic principle is referred to the cosmological constant problem. Indeed the cosmological constant is too little and no derivation is known. In 1989, using the anthropic principle, Weinberg predicted a value of hundred times the current value [76].

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assumption in the beginning. In this work we will not consider the quark mass implementation and we will deal with the original Sakai Sugimoto Model. Since the Sakai Sugimoto Model may be reduced to the Skyrme Model using a particular eld conguration, we will nd a method to recover the sextic term and we will study the energy behaviour of the Skyrmions solutions for various (small) 't Hooft coupling values.

The Sakai Sugimoto Model is the closest holographic top-down approach to QCD at non perturbative regime. Top-down approach means that we start with a string setting in order to arrive at a particular gauge Lagrangian in QFT. It's rather dierent from the the bottom up one, where one usually starts from a ve-dimensional gauge-theory dened on an AdS background. The most important references about the Sakai Sugimoto Model are [35], [56], [57], [58].

Holography and lattice QCD are not the only non perturbative attempts to QCD existing nowadays. A huge variety of attempts have been done, others are still in working, and the hope is that, in the future, the strong interaction phenomenology can be better understood.

1.2 The Thesis

The aim of this work is to recover the sextic term in the Skyrme Model appearing on the pion expansion in the Sakai Sugimoto Action. Once that this goal will be achieved, we will study the instantons solutions involving the sextic term in the small lambda limit.

1.2.1 The Structure

The thesis is divided into three chapters. Chapter II

Here we will study the string and the superstring theory basics and we will build the rst quantized states. In sections (2.2),(2.3),(2.4) we will study the classical bosonic string and we will introduce the fundamentals such as the Nambu Goto Action and the Polyakov Action, whose symmetries will be studied in section (2.6).

Then we will study which boundary conditions may be imposed on a string and which structures arise. This will be provided in section (2.7). In section (2.8) we will see the string interactions and we will write the low energy eective action for an open and closed string.

In section (2.9) we will approach the quantization procedure of the bosonic string and we will provide the Mass Spectra. Especially, we will focus on the ground state and the rst excited states.

After this, we will introduce the Supersymmetric string providing the Mass Spectra focusing on the rst states as before. This will be provided in section (2.10).

In section (2.11) we will write the Supergravity Action, which will be the base point for our goals. Then we will study Dp-branes and will see how a gauge theory may arise from the Dp-brane Action. This will be accomplished in section (2.12). A very short excursus about T-Duality and S-Duality will be provided in section (2.13)

At this point we will introduce the Ads/CFT basics. We will start with the RR-p branes in section (2.14), the AdS space in (2.15), the large N limit in section (2.16) and we will give some basics about the conformal eld theory in section (2.16).

Then, fully armed, we will enter into the core of the Maldacena Conjecture. We will start with the duality between RR-p branes and D-p branes in section (2.19). Then we will build the original conjecture and we will show the original duality between the Super Yang Mills Theory in 4D and the Type II B string in AdS5× S5. All this will be provided in sections (2.19) and (2.20).

In section (2.21) we will show how to extend the Maldacena's Conjecture to nite temperature systems. This will be necessary to introduce the D4 Witten Background which is the rst holographic system dual to a Non Supersymmetric Pure Yang Mills Theory.

Finally we will study the Sakai Sugimoto Model in most of his crucial aspects with all his implications. We will derive all the quantities and we will put the basis for the last part, which is the original one. This will be provided in sections (2.23) and (2.24).

Chapter III

Here we will study the topological solitons. In sections (3.1), (3.2) and (3.3) we will review some fundamental aspects such as homotopy groups, nite energy conditions and coset spaces.

Then we will introduce some mathematical tools such as Chern numbers and Derrick's Theorem. This will be provided in sections (3.5) and (3.6).

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1.2. THE THESIS 11 Then, after having introduced the Moduli Space idea in section (3.7), we will study the most important theoretical models involving topological solitons. We will start examining the kinks in section (3.8), while in sections (3.10) and (3.11) we will study monopoles and instantons.

In (3.12) we will study sigma model lumps while in (3.13) and (3.14) we will study Baby Skyrmions and Skyrmions.

We remark that Instantons and Skyrmions will play a crucial role in this work. Chapter IV

This is the original part. Here we will consider the small t'Hooft coupling limit. The conjecture we are going to verify is that the Sakai Sugimoto soliton (instanton) approaches to at space for xed but small 't Hooft coupling. We will lay the foundation of this conjecture in section (4.1).

After having veried that this limit may be taken, compatibly to our assumptions, we will provide a mechanism to extrapolate the sextic term of the Skyrme Model from the Sakai Sugimoto Model, valid in the original lambda limit and adaptable to the small lambda one. This will be done combining dierent ansatz's for the gauge eld, already presented in Chapter II, and then putting the system on-shell using one of the motion equations of the Sakai Sugimoto Action in the static Ansatz. This procedure requires an initial approximation which will be validated a posteriori

Then we will verify the initial conjecture using the Derrick's theorem and we will nd the solutions for various values of the 't Hooft coupling. We will see that, in the limit of small 't Hooft coupling, the soliton solutions, in addition to approach the at space, have the interesting property that the energy corresponding to the quartic term of the Skyrme Model is suppressed. All this machinery will be provided in section (4.2).

In the section (4.3) we will verify the assumptions made in (4.2). Firstly we will solve the system examined in the previous section without the quartic term, getting the same amount in energy for the same lambda values. Then we will show a posteriori that the soliton solutions satises the approximation taken in (4.2).

1.2.2 The Methodology

In this work we will take for granted the basics of Quantum Chromo Dynamics and the Supersymmetry. In particular every calculation involving the last one will be skipped. We have made this choice due to the lack of space in whis work. However we want give some references where Supersymmetry may be examined in depth: [20], [62], [69].

1.2.3 How to read this work

For the reader who approaches the Guauge/Gravity duality for the rst time we strongly recommend to start from the references and to make all the evaluations as will be made in next chapters.

If instead the reader has familiarity with the whole ensemble of Gauge / Gravity tools and its concepts, we let him to jump directly to the last, original part.

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Chapter 2

Gauge Gravity Duality

2.1 Introduction

A string is a special case of a p-brane1. The string evolution in time is described by a two dimensional

surface in spacetime, which is called the string world sheet of a string. The interactions in string theory are derived from the world sheet topology. There are two mainly dierences between string theory and QFT. Firstly, in string theory, the couplings are made with its own quantization states. Secondly there is no need to renormalize the theory because there are not UV divergences. This is due to the fact that in string interactions there are not short-distances singularities.

In string theory 1/ls 2 acts as a sort of cuto. We can say that, how we will see later, when we

make the string perturbation theory we are keeping the energy low so that the string length ls becomes

bigger then the string length scale lp. Due to the fact that the string physics is inaccessible to the

experiments, and will be probably so in the next future, we underline that the considerations made above are methodological and not epistemological ones.

Now the simplest string theory is the bosonic string theory. This theory is not complete and it has a lot of problems that we will not discuss here but it's the most natural starting point for our aims. For further discussions about string and superstring theory we refer to [18], [21], [36], [47], [50], [51], [52], [70], [75], [86], [53], [37].

2.2 Relativistic Point Particle

Let's start form the following Action3

S0= −α

Z

ds . (2.1)

This is the well known Relativistic Point Particle Action.

Here the Action is the length of the time-like world line γ times a constant (α). Making the non relativistic limit of (2.1) one can easly see that α = m. Now we know that4

ds2= −GM NdXMdXN , (2.2)

where M = N = 0, ..., D − 1. Now we make a reparametrization introducing a real parameter that we will call λ. So we will write XM → XM(λ). Now XM(λ) are the target space coordinates which are

elds on the world line parametrized by λ and provide the location of the particle in the spacetime. 5

Applying what has been said we get, from (2.2) S0= −m Z ds= −m Z p −dMdXNG M N = −m Z dλ√−gλλ , (2.3)

1In this context p is referred to the number of spatial dimensions on which the object is extended. A 0-brane is the

point particle, a 1-brane is the string, a 2-brane is simply a 2-brane. In literature is common to nd this objects referred to as D-branes instead of p-brane, where now D is the number of the total dimension of the object (D = p + 1). We will not use this notation because make create ambiguity, since D is the total dimension of space-time and p + 1 is the total dimension of the world-volume,as we will see.

2lsis the string length and is dierent from lp. The last one is the string length scale estimated by dimensional analysis:

lp=



~G

c2. The Planck mass is given by Mp=

q

~c G. 3From now on we will use the convention ~ = c = 1. 4We will use the mostly plus signature where η

M N= diag(−, +, +, +).

5Alternately one could say that XM(λ)are the embedding coordinates of the world line into the bulk.

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where gλλ(λ) = dXM dλ dXN dλ GM N(X) . (2.4)

We call gλλ(λ) the induced metric on the world line. We can see that the Action (2.3) is invariant under

the change λ → λ0(λ)with boundary conditions λ0

0) = λ0 and λ0(λ1) = λ1. Now this Action has two

problems.

The Square Root

There are two types of diculties which arise from the presence of the square root. The rst one is the ambiguity due to the sign of the expression. The second one arises from the fact that, if we want quantize the theory, the description of energy as a combination of creation and destruction operators applied to a vacuum state becomes dicult due to the presence of a non linear object as a square root.

• Zero Mass Limit

The Action (2.3) in the zero mass limit vanishes so that the massless relativistic point particle dynamics cannot be extrapolated.

In order to avoid these issues one can introduce an auxiliary eld, that we will call6 e(λ), and write a

new Action which takes the form ˜ S0= 1 2 Z dλ  e−1dX M dλ dXN dλ GM N− m 2e  . (2.5)

The equation of motion for e(λ) is given by m2e2+ dX M dλ dXN dλ GM N 2 = 0 . (2.6)

If we solve (2.6) for e(λ) and insert this into (2.5) we get back (2.3). So we can say that (2.3) is the on shell version of (2.5) . 7

2.3 p-brane

Now let's generalize the world line description to the one for a p-dimensional extended object. As before we have the target space coordinates XM but, instead of λ, we have σm with m = (0, 1, ..., p).8 This

means that

xM → XM(σ) . (2.7)

Obviously now we will have a world volume9instead of a world line and the Action must be proportional

to the world volume swept out by the p-brane. Furthermore we must impose p < D for the consistency of the theory. Analogously to what has been said in the previous section, if we now consider p-branes, we can choose the world volume to be p + 1 dimensional. In this way we have

Sp= −Tp× (volume) = −Tp

Z

dp+1σp−gmn(σ) , (2.8)

where the induced metric on the world volume is given by

gmn(X(σ)) = ∂mXM∂nXNGM N . (2.9)

Here Tp has the role of a tension. Indeed its dimension in mass is p + 1. Now consider a (p + 1)-form

gauge eld potential Ap+1(x)such as

Ap+1=

1

(p + 1)!AM1....Mp+1dx

M1∧ ... ∧ dxMp+1 , (2.10)

6"e" stands for einbein.

7May be interesting to see that (2.5) is invariant under reparametrization of the world line if e(λ) transforms in a special

way.

8σmrepresents the world volume coordinates.

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2.3. P-BRANE 15 where AM1....Mp+1 is a totally antisymmetric tensor eld. If we take the gauge transformation

δAp+1= dθp(x) , (2.11)

with θp(x) a p-form gauge parameter, we have that the (p + 2)-form eld strenght FM1...Mp+2, dened

by Fp+2= dAp+1 10, is invariant. So we can dene thee kinetic term for the (p + 1)-form eld as follows

S(p+1)kin= − 1 2 Z M F(p+2)∧ ∗F(p+2)= − 1 2 Z M dDx√−G | Fp+2|2 . (2.13)

Now we are ready to introduce the minimal coupling between the (p+1) form gauge eld and the p-brane. Sint= −qp Z Σ Ap+1 = −qp Z Σ 1 (p + 1)!AM1....Mp+1dx M1∧ ... ∧ dxMp+1 = − qp (p + 1)! Z Σ AM1....Mp+1(X)(∂m1X M1)...(∂ mp+1X Mp+1)dσm1∧ ... ∧ dσmp+1 . (2.14)

If we want Sintto be gauge invariant, we must impose the p-brane not to have any boundary. The factor

in front of (2.14) is the charge of the p-from eld and the integration is made over the (p+1) dimensional world volume. To see this we can start form the equations of motion of the gauge eld

∇MFM N1....Np+1 = JN1....Np+1 , (2.15)

which can be written equivalently as

d ∗ Fp+2− ∗Jp+1= 0 , (2.16) where JM1...Mp+1 =qp −G Z Σ (∂m1X M1)..(∂ mp+1X Mp+1D(x − X(σ)) dΣ , (2.17)

with dΣ = dσm1∧ .... ∧ dσmp+1. Now take (2.15) and make the following steps

∇MFM AN1...Np= JAN1...Np=⇒ ∇A∇M∇MFM AN1...Np= ∇AJAN1...Np= 0 , (2.18)

using to the antisymmetry of FM AN1...Np. In this way we obtain the continuity equation

∇MJM N1...Np= 0 , (2.19)

which can be also written as

d ∗ Jp+1= 0 . (2.20)

This implies we have11

qp= Z BD−p−1 ∗Jp+1= Z BD−p−1 d ∗ Fp+2= Z SD−p−2 ∗Fp+2 . (2.21)

We also remember that

∗Fp+2≡ ˜FD−p−2= d ˜AD−p−3 . (2.22)

10The Hodge Dual of F

p+2 is given by ∗Fp+2 = d ˜AD−p−3. This dual couples with a (D − p − 3) dimensional world

volume, which correspends to a (D − p − 4)-brane. In this way the p-brane and the (D − p − 4)-brane are dual to each other. It's also possible to show that from these considerations follows the Dirac quantization condition

qpqD−p−4= 2πn (2.12)

11May be useful recalling something about the denition of an n-ball Bnand of an n sphere Sn. A 0-ball is a point, a

1-ball is a line between the points −1 and +1 on R, a 2-ball is the area enclosed by a circle of unitary radius, a 3 ball is the space surrounded by the sphere and so on. Now the n-sphere is dened as the boundary of an (n + 1) − ball. In this way we have that:

S0is the boundary of B1 and is given by the extreme points of the line (−1 and +1).

S1is the boundary of B2 so it's a circle.

S2is the boundary of B3 so it's a sphere.

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2.4 The Nambu Goto Action

Let's take (2.8) for p = 1

S1= −T1

Z

d2σp−det(gmn(σ)) . (2.23)

Here the string sweeps out a two dimensional surface as it moves through space-time. This surface is known in literature as world sheet. If p > 1 the surface will become an hypersurface of dimension p + 1, which is known as world-volume.

Now we are going to approach the p = 1 case. Here we can set σ0= τ and σ1= σ. Now the dimension

in mass for T1 is 2 in mass units, so we can rewrite (2.23) as

S1= − 1 2πα0 Z dτ dσp−det(gmn(σ)) , (2.24) with α0= l2

s. The (2.25) is called Nambu Goto Action. As we said before in the case p = 0 this Action is

not a good starting point for quantization. So we have to write the Nambu Goto Action in a new fashion.

2.5 The Polyakov Action

In order to accomplish our goal we can introduce an auxiliary eld as we did for the relativistic point particle. This time we must introduce a new object which is the auxiliary world sheet metric γmn and

that plays the role of the world sheet metric. It's important to underline that it's conceptually dierent from the induced metric gmn.

Now we will start from the end writing the Polyakov Action and then we will see how this provides an on shell equivalent description of the Nambu Goto Action. The Polyakov Action is

˜ S1≡ − 1 4πα0 Z Σ √ −γγmn mXM∂nXNGM N(X) | {z } same gmnas (2.9) = − 1 4πα0 Z Σ √ −γγmng mn . (2.25)

This Action has a large amount of symmetries that we will discuss later. For the moment let's write the equation of motion for a 2D world sheet. We will get

√ −g ≡p−det(gmn) = 1 2 √ −γγmng mn . (2.26) So we nd ((2.26) (2.25) =⇒ ˜S1|on-shell= − 1 2πα0 Z Σ p−det(gmn) =⇒(2.8) for p = 1 , (2.27)

where we have used the denition for the string tension T1, encountered in the previous section.

2.6 Polyakov Action Symmetries

The Symmetries are the following • Poincarè Transformations

The Poincarè transformation may be interpreted as a global internal symmetry 12 from the perspective

of the 2D world sheet theory

XM(σ) → ΛMNX N

(σ) + VM, Λ ∈ SO(1, D − 1) . (2.28)

12An internal symmetry is a transformation acting only in the elds, therefore not transforming spacetime point, and

leaving the Lagrangian or the physical results invariant. Example of internal symmetries are global and local (gauge) symmetries. The local symmetries are spacetime dependent transformations in the sense that they are dierent for each spacetime point. The external symmetries are the symmetries of spacetime. The Poincarè group is the isometry group of the Minkowski space. However, if we consider the embedding coordinates XM as elds of the 2D world sheet theory,

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2.6. POLYAKOV ACTION SYMMETRIES 17 • Reparametrization of the world sheet coordinates

The invariance under Reparametrization means that the Action is invariant under a reparametrization of τ and σ. For example if we consider

σm→ σ0m= fm(σ) , γmn= ∂fα ∂σm ∂fβ ∂σnγ 0 αβ(σ0), (2.29)

and setting J = det∂f0α

∂σm, we have consequently

d2f = d2σ0= dτ00= Jd2σ= Jdτ dσ ,

p−det(γ0) =p−det(γ) , (2.30)

so that

d2σ0p−detγ0= d2σp−detγ0 . (2.31)

In this way the change of the world sheet coordinates leaves the Action invariant. Reparametrization of the world sheet coordinates may be seen as a local dieomorphism.

• Weyl Transformations

A Weyl Transformation or Weyl Rescaling is a conformal transformation of the world sheet metric. It acts as follows

γmn→ eφ(σ,τ )γmn , δXM = 0 . (2.32)

From the expression above we have√−γ → eφ√−γ and γmn→ e−φγ

mn. This transformation leaves our

Action invariant again. It's good remember that Poincarè transformations are global symmetries in the string context, while weyl transformations and reparametrization are local ones.

2.6.1 Stress Energy Tensor and Weyl Rescaling

From G.R. we know that the Belinfante-Rosenfeld stress energy momentum tensor in the 2D world-sheet is given by Tmn≡ 2 √ −γ δ ˜S1 δγmn =⇒(2.25) =⇒ Tmn= − 1 2πα0  −1 2γmnγ αβg αβ+ gmn  . (2.33) From the relation above it's easy to see that

γmnTmn= 0 . (2.34)

where we have used the basic property γmnγ

mn= 2. In this way we have seen that the classical invariance

under Weyl rescaling for the Polyakov Action has as consequence the stress energy momentum tensor to be traceless.13 This may be seen in the following way.

Now consider a generic world-sheet metric dependent Action and his variation. We will have something like

δS= Z δS

δγmn

δγmn. (2.35)

We know that for a Weyl rescaling

δγmn= Ω2(τ, σ)γmn , (2.36)

holds, where Ω stands for any function whose variables are the world-sheet coordinates τ and σ. So, using the denition for Tmn we nd

(2.35) together with (2.36) =⇒ δS =Z Ω2(τ, σ) δS

δγmn

γmn,using the denition for Tmn from (2.33)

=⇒ δS ∼ Z

Tmnγmn=⇒ Weyl invariance requieres δS = 0 so Tmnγmn= 0 . (2.37)

13We underline that the stress energy tensor Tmnis vanishing if evaluated on shell respect to the equations of motion

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So we have that, if the theory is Weyl invariant, then the stress energy momentum tensor on the 2D world sheet is traceless.

2.7 Neuman and Dirichlet Boundary Conditions

Consider the variation of the Polyakov Action (2.25) respect to XM. We get

δ ˜S1= 1 2πα0 Z Σ √ −γ(∇2XM+ γ mn∂nXA∂mXBΓMBA)GM NδXN − 1 2πα0 Z ∂Σ (dΣ)m mXMGM NδXN . (2.38)

In order to have δ ˜S1 = 0 we have to set the rst and the second term to zero. The rst one gives the

equation of motion for XM(σ). The second one must be treated with care. Indeed we must distinguish

between two dierent cases: • Closed String

If the string is closed, the boundary term vanishes. 14

• Open String

If the string is open we have to ways to send the second term in (2.38) to zero. • Neumann Boundary Condition =⇒ nm

mXM |∂Σ= 0.

Dirichlet Boundary Condition =⇒ XM |

∂Σ= cM where nmis the unit normal vector to the

boundary while cM is a constant which xes the position at the endpoints of the string. 15

Figure 2.1: On the left a string stretching between two D-p branes. On the right a string starting and ending on the same D-p brane

Both the boundary conditions preserve the Weyl symmetry. Now consider the Neumann one. If the choice is made for all M's, this condition respects D-dimensional Poincarè invariance. This is due to the fact that, if we impose the Neumann Boundary Condition for all M's, there is no way to introduce the Dirichlet Boundary Condition, since there are not remaining M's. The Dirichlet Boundary Condition breaks the Poincarè invariance because the endpoints of the strings are obliged to have a specic position in the target space-time, so the translational invariance is broken. 16

14The classical string motion equations extremize the world sheet area. If we take a at background geometry we will

have a string sweeping out the world sheet described by the surface of a cylinder innitely extended. It's easy to see that this object doesn't have any boundary.

15To provide a better comprehension of the Neumann Boundary condition consider a at world sheet endlessly extended

on τ with two boundaries sited at σ = 0 and σ = l. From this it's clear that the unit normal vector to the boundary must be n = (0, 1) so that the Neumann Boundary Condition reduces to ∂σXM|σ=0and ∂σXM|σ=l.

16Poincarè invariance means invariance also under spacetime translations. By the Noether Theorem one sees that the

conserved current is given by the momentum. So at the endpoints of an object, where Dirichlet Boundary Conditions are imposed, the momentum conservation is broken.

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2.8. STRING INTERACTIONS 19 D-p Branes

Now let's generalize what has been said considering the following situation for a generic p + 1 worldsheet embedded in a D dimensional space time

M = (0, ..., p

| {z }

Neumann Boundary Conditions

, p+ 1, ..., D − 1)

| {z }

Dirichlet Boundary Conditions

. (2.39)

Here we can see that the Neumann Boundary Conditions are imposed on world sheet coordinates whose index goes from 0 to p. Now, as explained before, we know that the momentum conservation is broken under Dirichlet Boundary Conditions. This means intuitively that an extended dynamical object must be attached to our system, so that the momentum can scroll through. The hypersurface made by the string endpoints in the target space-time may be seen as the volume of the object we are referring to.

Now, since hypersurface has dimension p + 1, the object will have dimension p. In literature this dynamical extended object is called Dirichlet Brane or, in short, Dp-Brane . 17 In 1995 Polchinski showed

the equivalence between D branes and D-p branes [49], giving the go-ahead to the second superstring revolution.18

2.8 String Interactions

Consider the following Action SΦ0 = Φ0  1 4π Z Σ d2σ√−γR(γ) + 1 2π Z ∂Σ dsK  . (2.40)

This is the equivalent of an Einstein-Hilbert Action in 2D spacetime with boundary. Here R(γ) is the scalar curvature of γmnσ)and K is the extrinsic curvature of the boundary. Now, due to the

Riemann-Roch theorem, we know that (2.40) gives us the Euler number χ which is a topological invariant of the world sheet . SΦ0= Φ0  1 4π Z Σ d2σ√−γ + 1 2π Z ∂Σ dsK  = χ = 2 − 2g − b − c , (2.41) where g is the number of handles, b is the number of boundaries, c is the number of crosscaps of the surface (orientable or not) we consider. 19 20

It goes without saying that, if we add (2.40) to the Polyakov Action, we don't change the equations of motion. It's possible generalize the Action above writing

SΦ=  1 4π Z Σ d2σ√−γR(γ)Φ(X) + 1 2π Z ∂Σ dsKΦ(X)  , (2.42)

where now Φ0= Φ(X → ∞). Φ(X) is known in literature as the dilaton eld and will arise automatically

when we will quantize the bosonic string in the next chapter. We must also say that the insertion of the dilaton eld breaks the classical Weyl invariance. Since we want the classical theory to be Weyl invariant, we must view Sφ as a term that will enter at higher loops respect to the term of the complete Action

which we will write afterwards (see (2.61) and (2.62)).

The Action just seen is not the only one which preserves the Weyl symmetry. Another one can be SB = 1 4πα0 Z Σ B2= 1 4πα0 Z Σ d2σ√−γmn∂mXM∂nXNBM N(X) , (2.43)

17There is a radical dierence between a p-brane and a Dp-brane. Roughly speaking a Dp-brane is a p-brane where the

ends of open strings are localized on the target space-time (often improperly called brane).

18The rst superstring revolution was the realization of the description of all elementary particles, as well as their

interactions, in terms of strings.

19A crosscap is the self-intersection of a one-sided surface. The crosscap can be thought as the object produced by

puncturing a surface a single time, attaching two zips around the puncture in the same direction, distorting the hole so that the zips line up, requiring that the surface intersects itself, and then zipping up.

20The Euler Number (also called Euler Characteristic) is a topological invariant in the sense that describes a topological

space's shape regardless the way it is bent. For a sphere we have χ = 2 (g = 0, b = 0, c = 0), while for a disk we have χ = 1 (g = 0, b = 0, c = 0). There are four well known cases in which we can have χ = 0.

1)Torus =⇒ g = 1, b = 0, c = 0 2)Cylinder =⇒ g = 0, b = 2, c = 0 3)Moebius Strip =⇒ g = 0, b = 1, c = 1 4)Klein Bottle =⇒ g = 0, b = 0, c = 2

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where B2(X)is a two form external gauge potential and mn(σ)is the Levi Civita tensor in 2D world

sheet.21 BM N is also called Kalb-Ramond eld and arises, as for dilaton, from the quantization of the

bosonic string. This Action is invariant under Weyl symmetry; indeed it doesn't depend explicitly from the world sheet metric. As regards the gauge invariance of (2.43) the question is dierent. A Kalb-Ramond eld BM N is dened as an antisymmetric Lorentz tensor with the following gauge symmetry

transformation

δBM N = ∂MΛN− ∂NΛM . (2.44)

This gauge transformation has the following redundancy

Λ0M = ΛM + ∂MΥ , (2.45)

under which BM N remains invariant. Of course we can dene a eld strength and an Action for the eld

BM N HM N P = ∂MBN P + ∂NBP M+ ∂PBM N , SKR= − 1 12 Z dDxHM N P HM N P . (2.46)

The Action (2.58) is present only for oriented strings. This is explained by the fact that, if we make an orientifold projection22, we are able to eliminate the Kalb-Ramond eld. 23. If we make the gauge

variation (2.44) for SB we get

δSB = 1 4πα0 Z Σ δB2= 1 4πα0 Z Σ d2σ√−γmn mXM∂nXNδBM N(X), using (2.44) =⇒ δSB = 1 2πα0 Z Σ d2σ√−γmn mXM∂nXN∂MΛN =⇒ ∂mΛN ≡ ∂ΛN ∂σm = ∂ΛN ∂XM ∂XM ∂σm =⇒ δSB = 1 2πα0 Z Σ d2σmn∂nXN∂mΛN = − 1 2πα0 Z dΣmmnΛN∂mXM , (2.47)

where dΣ1= dτ and dΣ2= dσ. Now, since we are using the 2D world-sheet coordinates, we are free to

write (2.47) =⇒ δSN = − 1 2πα0 Z dτmnΛ N∂τXM σ=l σ=0− Z dσmnΛ N∂σXM τ =+∞ τ =−∞  . (2.48) Here we will assume Λ to vanish at the endpoints of time. We remain so with

δSN = − 1 2πα0 Z dτmnΛ N∂τXM σ=l σ=0 , (2.49)

which is a boundary term. Let's see the closed and open string cases. • Closed String

In this case the world sheet has not any boundary. So (2.49) goes to zero. In this way (2.43) is automatically gauge invariant.

• Open String

In the open string case the (2.49) doesn't vanish identically since we have a boundary. So we must compensate the gauge variation (2.49) with a new term to add to the original SB. A possible guess is

SBoundary= 1 2πα0

Z dΣm

mXMAM(X) . (2.50)

Let's make the gauge variation of (2.50) under the gauge transformation δAM(X) ≡ ∂Mθ ≡ΛM where

Λis constricted to be the same as before δSBoundary= 1 2πα0 Z dΣm mXMδAM(X) = 1 2πα0 Z dΣm mXMΛM , (2.51)

21The Levi Civita tensor has been introduced due to the antisymmetry of BM N. It's fundamental recall that mn(σ) = ˜

mn

−γ where ˜mnis the original Levi Civita tensor.

22We will not discuss the orientifold projection here. For insights we refer to rifstring1.

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2.8. STRING INTERACTIONS 21 the variation is exactly the same as the one in (2.47) with the opposite sign. Now we have the whole ingredients to build the Bosonic Oriented Closed and Open Strings Actions 24

• Closed Oriented String Action Using (2.25), (2.42), (2.43) we arrive at S= 1 4πα0 Z Σ d2σ[γmn∂mXM∂nXNGM N(X) + imn∂mXM∂nXNBM N(X) + α0R(γ)Φ .(X)] (2.52)

• Open Oriented String Action

Using (2.25), (2.42), (2.43) and (2.50) we arrive at S= 1 4πα0 Z Σ d2σ[γmn mXM∂nXNGM N(X) + imn∂mXM∂nXNBM N(X) + α0R(γ)Φ(X)] + 1 2πα0 Z ∂Σ dΣm∂mXMAM(X) . (2.53)

This is also known as Dirac-Born-Ineld Action.

2.8.1 Low Energy eective Action For Oriented Strings

Now we will consider world sheet interactions as perturbations. One of the most common ways to achieve this goal is called Background Field Method. We can start writing

XM(σ) = ˜XM +√α0YM(σ) , (2.54)

where XM are supposed to be Eulero Lagrange equations solutions. If we expand (2.52) or (2.53) using

(2.54) we have to evaluate, rst of all

GM N(X(σ)) = GM N |X˜M +α

0YM∂GM N

∂XM |YM=0 . (2.55)

Knowing that the metric tensor of the target space-time GM N is dimensionless we can quietly say that

∂GM N

∂XM =

1 rS

, (2.56)

where now rS is the characteristic length scale of the target space-time. If we want make perturbation

theory we must set

√ α0

rS

<<1 . (2.57)

Now, being √α0 = l

s, where lsis the string length for our system, the condition (2.57) means that the

string length must be smaller then the characteristic scale of the target space time. This means that we have to deal with almost point like strings that don't see the curvature. Equivalently we could say that the necessary condition to make perturbation theory is that the String length must be very much smaller then the curvature so that the theory, at low (gravity) energies, doesn't feel string physics.

Now we can see that the Action (2.52) is not invariant, classically speaking, under Weyl rescaling due to the dilaton eld. As we outlined previously when we introduced the Dilaton Action, we can make the dilaton term enter at higher loops, so that the three level contribution arising from the Weyl transformation of the dilaton term cancels againts one loop contributions coming from the Weyl anomalies of the remaining terms in (2.52). In this way classical invariance under Weyl rescaling may be restored.25

If we evaluate the quantum corrections to the stress energy momentum tensor we can restore classical invariance at one loop. In order to make this, we can evaluate the trace of energy momentum tensor at

24We will adopt the Euclidean signature.

25When a symmetry of a classical theory is broken by radiative corrections, so that we can't add local counterterms

to the Action to recover the symmetry, we have an Anomaly. If the Anomaly is referred to a global symmetry, there are not inconsistencies. Take the Axial Anomaly in QCD as example. If, instead, the Anomaly is referred to a local one, this implies that we are not able to produce a consistent theory, due to the fact that we can't eliminate unphysical degrees of freedom at the quantum level. So the the theory is not unitary.

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one loop for the sigma model and at the tree level of the string perturbation, getting hTm mi = − 1 2πα0β G M Nγ mn mXM∂nXN − i 2α0β B M N mn mXM∂nXN− 1 2βΦR , (2.58) where βM NG = α0  RM N + 2∇M∇NΦ − 1 4HM KLH KL n  + O(α02) βM NB = α0  −1 2∇ KH KM N+ (∂KΦ)HKM N  + O(α02) βΦ=D − Dc 6 + α 0  −1 2∇ 2Φ + (∂ MΦ)2− 1 24HKM NH KM N  + O(α02) , (2.59) where HKM N is the eld strength of the two form gauge eld BM N and Dc= 26for the bosonic string.

If we want (2.58) to to 0, we must set

βM NG = βBM N = β Φ

= 0 . (2.60)

This condition may be seen as a set of Euler Lagrange equations derived from the following Action • Low Energy eective Action For Oriented Closed String

Seclosed= Z M dDx √ −G 2κ2 0 e−2Φ  R+ 4(∂MΦ)2− HM N LHM N L 12 − 2(D − Dc) 3α0  , (2.61) where we have discarded the O(α02)contribution.

• Low Energy eective Action For Oriented Open String The Action is the following

Seopen= −TD−1

Z

M

e−ΦdDxp−det(GM N+ BM N+ 2πα0FM N) . (2.62)

2.9 Mass Spectra for the bosonic string

Before quantizing the bosonic string is preferable to choose a at background and x a gauge for the 2D world sheet metric. We know that the world sheet metric looks like

γαβ=

γ00 γ01

γ10 γ11



, (2.63)

where γ01 = γ10 so that we have three independent parameters. Counting the degrees of freedom we

see that, using the invariance under reparametrization and under Weyl rescaling, we can locally gauge away all the parameters (i.e. the components of hαβ. Indeed we know that the world sheet metric hαβ

has p(p+1)

2 degrees of freedom, where p is the dimension of the generic world sheet. The Weyl rescaling

has 1 d.o.f. while reparametrization has p d.o.f. So we have that the number of remaining independent parameters for a p-dimensional world sheet metric tensor is given by

number of degrees of freedom =p(p + 1)

2 − (p + 1) =

(p − 2)(p + 1)

2 . (2.64)

We can see that for p = 2 all the parameters may be gauged away locally. Proceeding analytically one can see that under a Weyl rescaling we have

−hR −→√−h R − 2∇2φ(τ, σ)

. (2.65)

If we solve the equation R = 2∇2φ(τ, φ), we get R = 0 locally. This implies that also Riemann tensor

vanishes. 26 The vanishing of the Riemann tensor implies that we have a at metric tensor, at least

26In general we can say that, if the metric is at, we have Riemann Tensor = 0 =⇒ Ricci Tensor = 0 =⇒

Scalar Curvature = 0. The converse is true, at least locally. Indeed, if the Riemann Tensor vanishes identically, the metric is, at least locally, at. So we can look for weaker constraints, i.e. we can search for the conditions which cause a at metric without setting directly the Riemann Tensor to zero. We want get the Riemann Tensor vanishing as a consequence of vanishing tensors or scalars built from the same Riemann Tensor. This result may be achieved but depends on the dimension

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2.9. MASS SPECTRA FOR THE BOSONIC STRING 23 locally. In the end, using Reparametrization, we can write the world sheet metric tensor in the following way γαβ= −1 0 0 1  . (2.67)

In literature (2.67) is called at gauge. 27Now (2.67) holds locally. If we want (2.67) to hold globally we

must avoid topological obstructions, i.e. we need vanishing Euler characteristic. This means that we can treat torus-like and cylinder-like world sheets (for oriented strings). Now consider (2.25) and let's apply the variation XM → XM + δXM. So we have

δ ˜S1= 1 2πα0 Z dτ dσ(∂τXM)(∂τδXM) − (∂σXM)(∂σδXM) = 1 2πα0 Z dσ[(∂τXM)(δXM)] |τ =+∞τ =−∞− 1 2πα0 Z dτ[(∂σXM)(δXM)] |σ=lσ=0 + 1 2πα0 Z dτ dσ[∂2 σXM− ∂τ2XM]δXM , (2.68)

where we have supposed the world sheet to have two boundaries at σ = 0 and σ = l. Now we must examine the three terms from (2.68). First of all we will make the assumption that (δXM) |τ =+∞τ =−∞= 0.

In this way the string endpoints variation goes to 0 as the "world sheet proper time" goes to innity. This choice is coherent with the Principle of Least Action. Then we set ∂2

∂σ2 −

∂2

∂τ2



XM = 0getting the Euler Lagrange equations for (2.68). In the end we remain with the second term. We must distinguish between the closed string and open string case. To make this we will use the same boundary conditions explained in section (2.7). So, using what has been said previously, we write the second term in (2.68) as follows δ ˜S1= − 1 2πα0 Z dτ[(∂σXM)(δXM)] |σ=lσ=0 , (2.69)

• Dirichlet Boundary Conditions

Consider a closed string. By denition a closed string has no boundary. So, from the point of view of Dirichlet Boundary Conditions, it's like setting the periodicity condition XM(τ, 0) = XM(τ, l) in the

whole space-time except M = 0. In this way (2.69) vanishes identically.

Now let's consider the open string case. We look for the most general condition on δXM needed to

make (2.69) vanish. This goal may be achieved imposing • Dirichlet Boundary Conditions =⇒

δXM(τ, σ) |σ=0= δXM(τ, σ) |σ=l= 0

∂XM

∂τ (τ, σ = σ

) = 0 =⇒ XM(τ, σ = σ) = cM for every M 6= 0 , (2.70)

where σ∗ is the space world-sheet coordinate living on the boundary. So the Dirichlet Boundary

Condi-tions oblige the string endpoints to be xed as time ows. We remember the in the open string case the boundary may be not unique and that cM may take dierent values on the same coordinate. In every

case we will name (2.70) as DD boundary conditions. • Neumann Boundary Conditions

Now we look for the most general conguration on the sigma derivatives by which (2.69) vanishes. This may be achieved imposing the Neumann Boundary Condition already encountered in the previous

considered. More specically, the lower is dimension, the weaker is the constraint. In one dimension the Riemann and Ricci tensor are identically zero,as well as the Scalar Curvature, so that every metric is at. In two dimensions the Riemann Tensor is proportional to the Scalar Curvature, so that, if the latter is zero, the metric is at. In three dimensions the Riemann tensor depends on borh the Ricci Tensor and the Scalar Curvature. So if the Ricci Tensor vanishes, the metric is at. If, on the other side, the Scalar Curvature vanishes, the metric is not at in general. In four dimensions the Riemann Tensor depends on the Ricci Tensor and the Weyl Tensor, where the latter is the traceless part of the Riemann Tensor. In one, two and three dimensions the Weyl tensor vanishes identically. In four dimensions there are plenty of examples of non at metrics with both Ricci Tensor and Scalar Curvature equal to zero. So in four dimensions we need the full Riemann Tensor to vanish in order to have a at metric. Furthermore we underline that a general property involving the Weyl Tensor holds

Weyl Tensor = 0 ⇐⇒ The metric is conformally at for D ≥ 4 . (2.66) As regards the meaning of the word conformal, we refer to section (2.17).

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sections.

• Neumann Boundary Conditions =⇒

∂σXM(τ, σ) |σ=0= ∂σXM(τ, σ) |σ=l= 0 . (2.71)

Here the two string endpoints are treated as independent. Consequently string endpoints has to vanish separately. We will refer to (2.71) as NN boundary conditions.

• Mixed Boundary Conditions

Now we look for a mixed conditions. This may be obtained imposing • DN Boundary Condition =⇒ XM(τ, σ) |

σ=0= 0 ∧ ∂σXM(τ, σ) |σ=l= 0

• ND Boundary Condition =⇒ ∂σXM(τ, σ) |σ=0= 0 ∧ XM(τ, σ) |σ=l= 0 (2.72)

The conditions above, as the ones shown previously, will be essential in the quantum approach. • D-p Branes again

Now, instead of considering the previous conditions, imagine to have something like

•DD from 0 to p •NN from p + 1 to D − 1 (2.73) where the NN conditions are taken on the unique boundary so that cM is a unique, constant spacetime

vector. This is the same "ansatz" encountered in (2.39), only in a dierent notation. More and more extended Dp-Brane congurations may be obtained mixing the whole ensemble of conditions shown above. For example, considering the NN conditions on two boundaries, it's possible to describe a string stretching between 2 D-p Branes. This conguration will be studied later in the last part of (2.23.2).

Entering in the Quantum Level

So we have found four dierent ways to make the second term vanish: NN, DD, DN, ND. Now, rewriting the Euler Lagrange equations in light-cone coordinates 28, we get ∂+XM = 0. The most general

solution for this equation is given by a sum of right and left movers. Using the boundary conditions (2.70), (2.71) and (2.72) we get the well known modes for our strings which we will not write here for brevity. It's possible to approach the quantum description dening the following commutation rules

[XM, pN] = iηM N , αM m,α˜ N n] = [α M m, α N n] = mδm+n , [αMm,α˜ N n] = 0 , (2.74)

and the Fock space setting αM

m|0; ki = ˜αMm = 0. Now we realize we have an issue. Indeed dening

(αM m, αm†M) = αM m √ m, αM−m

m, setting n = −m and N = M = 0, one arrives at [a 0

m, a†0m] = −1, i.e. we have

negative norm states. A such theory would violate causality and unitarity. We will discuss this in a moment. Firstly dene the quantum Virasoro generators

Lm= 1 2 +∞ X n=−∞ : αM m−nαnM : , L˜m= 1 2 +∞ X n=−∞ :αM˜ m−nαnM˜ : , (2.75)

where the double dots stands for normal ordering.29. The commutation rules for the generators are

[Lm, Ln] = (m − n)Lm+n+

c 12m(m

2− 1)δ

m+n . (2.76)

where c is called "central charge" and arises only if we consider a quantum theory as in this case. Furthermore may be shown that the normal ordering aects only L0 (and ˜L0).

28In light-cone coordinates we have σ±= τ ± σ. Consequently ∂σ

±=

(∂τ±∂σ)

2 .

29This step may be tricky. QFT may help us. When we quantize the electromagnetic eld we get an Hamiltonian which

diverges because we have something like h0| aa†|0i. At his point one writes the previous expression in the normal order way

getting h0| a†a |0i + [h0| a, a|0i]. Now the second term also diverges but if we consider the dierence among energy levels

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2.10. SUPERSYMMETRIC STRING 25 Indeed we have L0= 12α20+12

P+∞

n=1α M

−nαnM− awhere a is a constant which we must evaluate. To

achieve this goal we must write the stress energy momentum tensor in light-cone coordinates. It reads T++= ∂+XM∂+XM

T−−= ∂−XM∂−Xm

T+−= T−+= 0 . (2.77)

To have classical Weyl invariance we should set T+++ T−− = 0. Now, due to the fact that we have

chosen the at gauge, we must impose the vanishing of the stress energy momentum tensor as an external constraint. We will make this at the quantum level. Rewriting the Virasoro generators in terms as Fourier Transforms of T++ and T−−we set

(Lm− aδm) |φi , ( ˜Lm− aδm) |φi , m ≥0 , (2.78)

which is the equivalent of (2.77) at the quantum level. Now, using (2.78) for m = 0 we get L0|φi = a |φi

and (L0− ˜L0) |φi = 0. Using the rst relation on the left and the rst of (2.75) we are able to write

aclosed= (D−2)

24 30. For an open string, after taking (D − n) NN plus DD directions and n ND directions,

one nds aopen = (D−2)2416n. Now using the expressions for L0 and ˜L0 and that p2= −M2 in mostly

plus signature and that we can write the mass spectra • Closed String Mass Spectrum

Mclosed2 = 2 α02 (+∞ X n=1 α−nαn+ +∞ X n=1 ˜ α−nα˜n− D −2 12 ) . (2.79)

Where we used the relation αM

0 = ˜αM0 =

q

α0

2p

M valid for the closed string.31

• Open String Mass Spectrum

Mopen2 = 1 α0    +∞ X n=1 α−ni αni + X n∈Z+1 2>0 ˜ αa−nα˜an− aopen    +(x0− x1) 2 (2πα0)2 . (2.80)

Where we used the relation αM 0 =

2α0pM valid for the open string. The previous relations may be

also rewritten dening Nclosed ≡P +∞ n=1α−nαn = ˜Nclosed ≡P +∞ n=1α˜−nα˜n and Nopen ≡P +∞ n=1α i −nαin+ P+∞

n=1α˜a−nα˜an. Now, the closed string aclosedmay be seen as the number of transverse modes of a closed

string in D dimensions. Due to the fact that in light-cone gauge all the excitations are generated by acting with the transverse modes we conclude that excited states must belong to a (D − 2) component vector representation of the rotation group SO = (D −2).32 This means that, to have a Lorentz invariant

theory, the vector states generated by the rst excitations must be massless. In this way we have found that the critical dimension for the bosonic theory is D = 26. So aclosed = 1 and aopen = 1 − 16n. The

related Hilbert Space is displayed on the table above.

2.10 Supersymmetric String

Now we want extend world sheet symmetry to 2D local Supersymmetry. This will take us to the Neveu-Schwarz-Ramond superstring. If, on the contrary, we choose to enlarge the invariance of the Polyakov

30The expressions have been regularized. We have provided the analytic continuation for Riemann Zeta function ζ(s, q) =

P+∞ n=1(n + q) −swriting ζ(−1, q) = P+∞ n=1(n + q) = − 6q2−6q+1 12 . 31For closed string also P+∞

n=1α−nαn=P+∞n=1α˜−nα˜nholds.

32In a Lorentz invariant theory in D dimensions a state forms an irreducible representation under the subgroup of

SO(1, D − 1)that leaves its momentum invariant. This is called Little Group or Stabilizer Group. We must distinguish between three cases.

1)p2= 0 , p = (ω, 0

D−2, ω) =⇒Little Group = SO(D − 2).

2)p2< 0 , p = (m, 0

D−1) =⇒Little Group = SO(D − 1).

3)p2> 0 , p = (0, ρ, 0

D−2) =⇒Little Group = SO(1, D − 2).

If, conversely, we consider invariance under Poincarè group we may repeat the same argument as before and arrive at the same point above except for the case p2= 0. Here the Little group is ISO(D − 2) which is the semi-direct product between

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