The Existence of Four-Dimensional Non-Supersymmetric Meta-Stable String Vacua

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The Existence of Four-Dimensional

Non-Supersymmetric Meta-Stable String Vacua

Zhi-Zhen Wang

Amsterdam, North-Holland, Netherlands

University of Amsterdam (2020)

A Thesis Presented to the Faculty of Science

of the University of Amsterdam in Candidacy for the Degree of

Master of Science in Theoretical Physics

Supervisor: Jan Pieter van der Schaar, Associate Professor

Examiner: Jácome Armas, Assistant Professor

Institute of Physics (IoP), Faculty of Science

University of Amsterdam

Aug, 2020

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⃝Copyright by Zhi-Zhen Wang 2020

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Abstract

String theory is one of the most promising candidates of quantum gravity. Nonetheless, finding a com-pactification of supersymmetric ten-dimensional string theory consistent with our four-dimensional reality (e.g. standard model, cosmology) remains an unsolved problem. In the first part of this the-sis, we will explain how string theory can realise a consistent four-dimensional non-supersymmetric inflationary cosmology and review a typical approach, which is known as KPV/KKLT, to construct it via compactification. Moreover, the stability of KPV/KKLT is highly debated because of the presence of potential singularities. Based on recent works, we will then argue that these singularities can be cloaked by a black hole horizon in blackfold texhniques, and refine earlier arguments present in the literature. This thesis provides a new perspective on KPV/KKLT approach and suggests that one can indeed achieve a four-dimensional realistic theory starting from superstring theory when a certain parameter bound on warp factors is satisfied.

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Acknowledgments

This thesis is dedicated to my parents.

Spring in Amstelveen, summer in Zandvoort, autumn in Zuid-kennemerland, and winter in Valkenburg. The four seasons in Netherlands, the four semesters in the University of Amsterdam (UvA), and the four quarters in capsuling the memory of them.

I still remembered the date when I first met t’Hooft in the Kobayashi-Masukawa Institute. On the banquet, I asked him a question and he praised me for that good question. Actually, I can’t recall which question I have asked, but his recommendation is still echoing in my brain now: why not consider the universities in Netherlands?

I have never imagined to have my post-graduate life spent in Europe before that. In my senior year, I applied only one master program, and that has been proved to be the best choice I have made during the application period because of the wonderful experience in Netherlands and Amsterdam I have had these two years. The magic coincidence always makes me feel that it’s not me but the fate chose the UvA. And the kind people I met here always convince me that I am right.

Outside the Horizon: the visible ones

Firstly, I should present my second best thanks to my supervisor, Dr. Jan Pieter van der Schaar, who is so nice that gives me the highest degree of freedom to conduct researches which I am interested in and always support me if I have either academic or non-academic questions. Jan Pieter led me to a field which I have never touched, but his enthusiasm and profound insights help me to build a recognition and successively improve my taste of the string cosmology. Now I am proud that he can be my supervisor, and I hope that I can become stronger in this field and let him to be proud of me some day.

I should also say a lot of thanks to my examiner Dr. Jácome Armas. Jay has a long curly hair like me and always let me feel that we are familiar with each other. He is also a workaholic, sometimes I sent email to ask a question at twelve or one a.m. and he usually replies in minutes. Specifically, for academic questions, he is more attacking (not derogatory) than Jan Pieter and can accurately point out the defect of my notes and the problems of my understanding.

As for this thesis, I must appreciate Jan Pieter and Jay, without their help, this thesis can never be completed and I may still hover outside the gate of string theory.

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discussed the coadjoint orbit and entanglement wedge with me, and gave me many suggestions on my note of p-adic AdS/CFT. Thanks to Guilherme Pimentel for communicating the new developments of de Sitter ampltiudes with me and encourage me to discuss the problems about scattering amplitudes with him. Thanks to Ioana Coman for dating an appointment for a discussion on the topological string theory with me, though this appointment was canceled because of some of her emergencies. Thanks to Guglielmo Lockhart for recommending me the great materials to learn the chiral algebra related to topological string. Thanks to Jasper van Wezel for his exhaustive and patient explanation for every aﬀairs of theoretical track, also a special thank to him for encouraging me to apply to transfer the track to mathematical physics. Thanks to Jasper Stokman and Hessel Posthuma for their kind explanation for the regulation of transferring to mathematical physics track and reviewed my application profile. Thanks to Sergey Shadrin for giving me a brief lecture on cohomological field theory and tolerating my naive undergraduate-level questions in mathematics. Oh, a special “thank” to Diego Hofman, for his insane homework and the take-home exam of advanced quantum field theory lecture, but I have to say it really improves me a lot.

I also enjoy the time spending with my friends in Amsterdam. Thanks to Stathis, this Greece guy could be the most clever master student I met in UvA, I really enjoy the time listening to your joke, discussing the physics and complaining the too-easy QFT lecture by Daniel. Thanks to Keivan, although a little introverted, is a nice guy who always be there to hear your voice. On the train to Leiden holography meeting, we discussed a lot about Iran’s history and politics. Thanks to Archishna, who is the first friend I have made in UvA, we usually attend the string journal club and eat the free empanada together at the beginning of the first year. Thanks to Vao Hoang, who is a positive and enthusiastic guy. He is always glad to hear the physics of diﬀerent areas and positively raise questions. We attended the diﬀerential geometry lecture in Utrecht together. It is nearly impossible to list all of you here, but I would like to thank every mate of theoretical physics track who spent the two years with me. Besides, thanks to Jun-Meng Duan and Jing Meng who spent the bad time in presence of coronavirus with me. Thanks to Lei Zhang for having leisure time in Netherlands together. Special thank to Rikako Aida who always brings me happy, confidence and the power to love the world.

Lastly, I need to present my love and the best thanks to my parents. They deserve all the sublime praises. Whatever the problem I am facing on, they are always in my back and support me. I love you forever.

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Inside the Horizon: the invisible ones

During the hard time caused by the disease, many people are getting depressed at home. But we should say many thanks to those doctors and nurses who are working in the hospital and facing the infectious disease every day. They are the true warrior protect us from coronavirus. I would like to say thank you as a common people protected by you.

I also thank the staﬀ who are responsible for the non-academic aﬀairs in UvA. They are around us but we may usually ignore them. Either the worker repairing electric equipment, or the cleaner making our workplace as comfortable as enough, they all deserve a thank you. Although the break time may cause the unemployment for many of them, I hope they can still happily face the life.

At the last, let me end this lengthy acknowledgement with the thank to myself. Thank you for choosing UvA and spent such a lovely two years here. In academic, you have raised to a person with your own taste and interested fields. You will never forget the days spending on reading Weinberg’s QFT, Polchinski’s string theory, and Wess & Begger’s SUSY. You can now peacefully discuss academic problems with others and properly raise your questions. Well done! In life, you have understood that physics is not all of your world. You have the people who love you and you love. You also spend more leisure time on enjoying the life. You made videos, played rocks, learned cooking and baking, etc. Two years seem too short to enjoy all of these. And I hope you will still keep your love in the life.

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List of Figures

2-1 A stack of branes with endpoints of open strings located. . . 13 3-1 The comparison between the resolved conifold and the deformed conifold. This figure

is cited from [38]. . . 40 3-2 The illustration of KS geometry with a long warped throat. This figure is cited from

[39]. . . 42 3-3 The eﬀective potential with p/M = 0.03 in the upper panel, and p/M = 0.08 in

the lower panel, respectively. When p/M ≲ 0.08, the meta-stable de Sitter vacua is admitted. This figure is cited from [7]. . . 48 3-4 The sketch of brane/flux annihilation for p/M = 0.03 and p/M = 0.08, respectively.

This figure is cited from [7]. . . 49 3-5 The uplifted eﬀective potential in presence of anti-branes. Here we set W0 =−10−4,

B = 1, D = 3× 10−9. This figure is cited from [8]. . . 52 4-1 The sketch of the full eﬀective potential in range of ψ ∈ (0, π). It globally has the

same trend as KPV approach in extremal limit. This figure is cited from [9]. . . 86 4-2 The zoom of sketch of the full eﬀective potential in the range of ψ ∈ (0, 0.3π). With

the global entropy varied to a critical value, the new uncovered unstable point (black dots) will merge with the meta-stable point (blue dots). Note that in both Fig. 4-1 and Fig. 4-2, the p/M value is fixed as 0.03. This figure is cited from [9]. . . . 87 4-3 The fatness curve in terms of horizon parameter α. The blue curve corresponds to

the brane transition at the unstable point while the orange one denotes the NS5-brane transition at the meta-stable point. At the merger point, the bulk radius r0 will

be in the same order with the genus radius and the intersection of blue and orange curves appears at a place very close to 1. This figure is cited from [9]. . . 88

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4-4 A geometric sketch of the transition of NS5-branes. As an NS5 moving to the North pole, its bulk radius become smaller and smaller and can eﬀectively polarize to an D3-brane at the tip of the throat. . . 89 5-1 The integration in dependence of a. This is a linear line with positive slope and

intercept so that one can arbitrarily choose a. This requirement also constrains the parameter space to b1 < 0 and b1 > b3. The setting of parameters in this illustration

is b1 =−1, b2 = 1, b3 =−1.1, rh = 10. . . 108

5-2 The integration in dependence of rh with limit b2 = 0. The setting of parameters in

the left panel is b2 = 0, b3 = 0.5 and that in the right panel is b2 = 0, b3 = −0.5.

Because of the vanishing of b2, b1 terms do not contribute. Only in the case of b3 < 0

can we obtain a positive integration. . . 108 5-3 The integration in dependence of rh with limit b1 = 0. For the bounds b2, b3 < 0,

the integration is positive. Besides, if we tune a little for b1, the integration exhibits

double extrema, as the right panel shows. The setting of parameters in the left panel is b1 = 0, b2 = −0.5, b3 = −1 and that in the right panel is b1 = −0.03, b2 = −0.5,

b3 =−1. . . 109

5-4 The integration in dependence of rh with limit b3 = 0. For the bounds b2, b1 < 0,

the integration is positive. The setting of parameters in the left panel is b1 =−0.04,

b2 =−0.5, b3 = 0, that in the middle panel is b1 =−0.04, b2 = 0.5, b3 = 0 and that

in the right panel is b1 = −1, b2 = −0.5, b3 = 0. This improves the constraints on

parameter space to b1 ≲ 0, b2 < 0 and b1 > b3. . . 110

5-5 The integration in dependence of rh with relaxed constraints. It suggests that there

is another bound set with b1 < b3, b1+ b2 < 0 and b2 + b3 < 0 which also guarantees

a positive integration. The setting of parameters in this illustration is b1 = −1,

b2 = 0.466, b3 =−0.5. It also has double extrema and will drastically disappear when

b2 moves. . . 112

5-6 The integration in dependence of rh and b2. The surface is exponentially increasing on

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5-7 The integration in dependence of b2 with non-vanishing a1 and a3. Because b2 is

proportional to p/M , the non-vanishing horizontal intercept of this illustration reveals that there is a little range of instability with negative integration, and the positive integration corresponds to the meta-stable vacua exist for a large range of positive

b2. Besides, the value of horizontal intercept—0.00443, which corresponds to p/M ∼

0.0796—implies that the KPV bound p/M ∼ 0.08 has been restored by tuning a1 and

a3. The setting of parameters in the left panel is a1 = 1, a2 = 0, a3 =−3.2, b1 =−0.2,

b3 = 0.1, rh = 4, a = 1. . . 115

5-8 The integration in dependence of rh in perspective of supersymmetric KS solution.

Here x denotes rh while y denotes the integration value. In the region rh > 0, the

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Chapter 1 Prologue

Many years later, as she faced the distortion engine, Shiho Aida was to remember that distant afternoon when Witten took her to discover string theory.

It was on a lecture, Shiho fell asleep because of the periodic and hypnotic tick-tack from the clock. The professor was still writing the formula on the blackboard, the lecture room was so quiet that one can even hear the voice of light led by the movement of foliage.

“Shiho, wake up, wake up.” A twilight comes into mind with an old voice.

Shiho opened her eyes and noticed a person bowed looking at her. Small brown glasses, Jewish face, straight nose, square chin and slightly bald brain, all of these are showing the brightness of this man.

“Who are you?” Asked by Shiho.

“Edward Witten. A man. A dad of three children. A mathematician. A string theorist. And the most intelligent man on the earth.” The man answered with a mysterious smile.

“Wouldn’t that be quite weird to introduce yourself as the most intelligent man? But what does string theorist mean?”

“Well, it seems you have missed what your professor was teaching,” Witten get a little frowned, “well, must be, otherwise you won’t meet me here.”

“Actually I don’t WANT to meet you,” not sure if Shiho misheard or just wants to make a lame joke, “so another question, where is here?”

“Look at your back,” Shiho turned back at once and saw the shocking hexagonal pattern, while Witten’s voice comes again, “it’s my realm, the string realm of Physionia.”

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and ability on controlling them.”

Ignored the complex emotion on Shiho’s face, Witten went on:“Look at this pattern. They consist of five types of consistent superstring and an eleven-dimensional supergravity (SUGRA).”

“Wait, wait, so many notions, I think it is too quick for me.” Shiho waves her hands and want to pause Witten’s eloquent speech. But it seems useless.

“In the land of Physionia, there is a devil dragon called Gravity. It governs the motion of celestial and terrestrial objects and deprived our right to observe or understand how to use that power. It was until 1687 that our first prophet Issac Newton broke the ban and predicted that Gravity’s power is universal and always moves the orbit elliptically. However, that is not the true strength of Gravity. In 1915, the Great Warrior Albert Einstein constructed the magic of general relativity. This magic finally helped us to understand the power of Gravity and find its den. We once defeated it, but...”

“Hold on. I think I know this story... Isn’t this simply the history of physics in gravity? Why there are so many game-style adjectives and nouns?”

“Hmm... I think describing the history in such a way sounds cooler and more exciting. After all, no one will read the details of this thesis except your supervisor. We have to try to make the introduction and conclusion parts as funny as enough.”

“You know what? Japanese call your symptom as chu-ni-byou (middle school disease1).”

“Don’t be so serious. Let me move on,” Witten ignored Shiho’s suggestion and went on his epic. “What we didn’t know is, Gravity dragon has two lives. The one we have defeated and killed is the Classical life which governs the motion in macro-scope, while the second one is the Quantum life which can dominate the energy scale far beyond we have reached. In this resurrection, Gravity obtains the power of quantum field theory and evolved to the Quantum Gravity (QG). No one can defeat it and we do not even know where its den is now. Besides, this time the Gravity becomes more cunning, it instigates internal disorder among humankind. There are two sects against QG dragon. One is seeking the harmony and wants to use the same way as Warrior Einstein’s magic to beat QG dragon, which known as loop quantum gravity (LQG) sect, founded by Ashtekar, Rovelli and Smolin. And here, the string realm, of course named by our string theory sect, is founded by Scherk, Schwarz and Venezieno, and then handover to Green, Maldacena, Seiberg, Vafa and me. Instead of holding stereotypes, we are rather radical and introduced some new notions to fight against QG dragon. As I have noted, one is ‘super’, and one is ‘string’.”

“Strings are the virtual objects which play the role as one-dimensional generalization of

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sionless particles. The interesting feature is that it preserves more symmetries and can produce many possible configurations because of its plentiful topological structures. While the ‘super’ typi-cally stands for the supersymmetry (SUSY) which allow us to admit a symmetry between bosonic and fermionic degrees of freedom on the worldsheet swept by the string.”

“Then I have to ask, why I should believe you but not the LQG sect. I don’t mean that I can’t believe the string theory, just feel you are suspicious.” Shiho performs her mistrust to Witten.

“Well, listen, my dear, note the top corner of that hexagonal figure? Why do you think I mentioned a SUGRA there?” Witten pointed the pattern and asked me.

“Although SUSY could be unnatural to the reality, it is a strong tool for us to reveal the mask of complex four-dimensional quantum gravity. In presence of it, string theory can be naturally degenerate to a gravity with SUSY. This is a surprising signal, SUGRA is exactly a low-energy eﬀective version of string theory. If we can govern this power, we will have stronger power than QG dragon and it can no longer be our rival. LQG sect can never do that, they focus more on the spacetime itself rather than the excited states inside it. But I concede that we all hope that our humans can assemble together, with both these two doctrines unified, to fight against the QG dragon. We don’t know if it still has a third life, we have to be prudent this time.”

“Furthermore, the six corners of this pattern can be further unified to a giant magic—M-theory, they all play the role as the low-energy eﬀective versions of M-theory. And now it is the time to pass the baton to you.”

“You seems so adept in your speech words. I really suspect that you said that to every greenhorn like me.” Shiho questions again.

“Absolutely. As I told the media, we have to nurse hundreds of pre-warriors, because we don’t know which one of them is not the cannon fodder.” Witten answered without hesitation.

“Well, then I have my last question. Now that either SUSY or string is unobservable in the experiments nowadays, how can you convince me that string theory deserves to be discovered as a PHYSICAL theory, oh magic in your language?”

“And that’s why we need you. Look at this arrow,” Witten waved his hand and an arrow with diﬀerence scales appears, “there are various hierarchies in physics. But we are only familiar with the scale lower than the QFT scale with fundamental particle interactions. Luckily, the cosmological observation suggests that our cosmology is inflationary and could have experienced a big-bang process at the early age of evolution. As an intermediate approach, if we can first examine that the string vacuum in the inflation scale can have large-scale cosmological interactions, it will be a giant leap

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for us to finally connect QFT and string and defeat the QG dragon.” “Great. So where should I get start?”

“The string realm consists of four pieces. North Seiberg, focusing on the string-described QFT; South Maldacena, focusing on the quantum information and entanglement entropy; East Nima, focusing on the scattering amplitudes and; West Vafa, focusing on the string-described cosmology. And you need to pass through eighty one suﬀerings to obtain and learn the doctrine of Vafa. This is not a plagiarism to Journey to the West since you only have yourself one person without animals.” Witten exposes his mysterious smile again.

“To be honest, I hate you. But I know if I do not follow your advice, this thesis cannot be completed and I can’t say sayonara (farewell) to you, right?”

“Exactly. But a good new is that I can provide you a tour guide. Young brave, go and seek your own dragon.” After these words, Witten threw a thin guidebook to Shiho’s hands and disappeared.

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1.1 Guidebook

Your journey has been arranged as follow:

First, you should be familiar with the rudimentary magic principle about strings and branes. They are the elements of string magic. Their properties and actions will be manifest in Flux Com-pactification Village[1, 2, 3], where their low-energy eﬀective actions in terms of fluxes will also be given. The notion of fluxes are significant for you to make the magic you cast be stable. Further-more, you will learn the first technique known as flux compactification, with the assistance of fluxes or supersymmetries[4, 5].

Second, you should go to String Vacua Valley. Here you will learn how to stabilize the scalar fields with certain vacuum expectation value and obtain the visualizable eﬀective potential minima. There are eight hermits living in string vacua valley. Klebanov and Strassler[6] will tell you how to obtain string vacua for a warped geometry with non-compact direction. Kachru, Pearson and Verlinde[7] will teach you how to correctly introduce anti-branes to polarize branes and break SUSY, so that a meta-stable state constrained by fluxes is produced. Lastly, Kachru, Kallosh, Linde and Trivedi[8] will let you know how to raise the negative AdS4 vacuum to a positive dS4 vacuum. To this end, it

will be easy for you to produce a four-dimensional non-supersymmetric meta-stable state, known as the D3-NS5 bound state.

The third destination is the Blackfold Formalism Hot Spring. Based on the eﬀective worldvolume theory[46], you are able to extract the non-extremal solution with finite temperature and horizon radius of D3-NS5 bound state. With the assistance of the work by Armas et al.[9], it is pointed out that the thermal singularity previously proposed can be cloaked by the horizon geometry.

Then you will reach the Misconception Clarified Forest. This is a place for you to improve yourself and seek and slay your first dragon. With the smeared and localized sources, and the Maxwell and Page charges clarified, you will be able to conjecture a parameter bound which can determine whether a Klebanov–Strassler solution can be described by blackfold analysis. This can helps us to capture more dragons and check if they are evil. We will make a refined statement for [10, 11] based on the numerically constructed Klebanov–Strassler black hole solution[12].

At the final destination, you should meet Vafa and conclude what you have learned during this journey and the prospects what can be further considered to suit ten-dimensional string theory to four-dimensional realistic gravity and conform your conclusions with the Swampland conjecture[13,

14].

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interest you. Appendix A reviews the basic knowledge about the mathematics of complex geometry; while appendix B introduces the holographic description of Klebanov–Strassler geometry.

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Chapter 2 Flux Compactification

As the starting point, it is necessary for the readers to keep basic knowledge about string theory and related D-branes. Here we will first give a minimal preliminary to ensure that everybody can understand the contents discussed in this thesis, and then review the physics of string compactifica-tion.

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2.1 Physical Preliminary

2.1.1 String Theory in a Nutshell

String theory is a quantum theory described by (0 + 1)-dimensional objects, strings, rather than (0+0)-dimensional particles as that in QFT. And its action in the spacetime will allow a worldsheet on which the string-like objects (can generate bosonic and fermionic fields in the spectrum) embedded. The toy model of string theory is the one with only bosonic fields encoded, which is also called

bosonic string theory. Suppose there is a bosonic string propagating in the D-dimensional Minkowski

spacetime, the bosonic string action can be given by[1,2, 3]

SP =−

1 4πα′

∫

d2σ√−hhab∂aXµ(σ)∂bXν(σ)ηµν, (2.1)

where Xµ_{, with µ = 0,}_{· · · , D −1, are the coordinates in the target spacetime; σ}a _{are the coordinates}

on the string worldsheet, with a = 0, 1 denotes two diﬀerent local parameters used to parametrizing the two-dimensional worldsheet; hab_{is an induced metric; and 2πα}′ _{is the inverse of the string tension.}

This action describes a two-dimensional field theory with D scalar fields.

To make the worldsheet theory more realistic, one expects to also contain fermionic string into the action. Supersymmetry was believed to be the most promising internal symmetry beyond Poincaré invariance which naturally pairs one boson to its fermionic super partner, and vice versa. While recent experiments on the Large Hadron Collider (LHC) strongly suggest that the nature dislikes SUSY. Nevertheless, supersymmetry is still one of the most significant baseline in this thesis, we will see how this condition can be relaxed by some physical mechanism in the last sections of this chapter. The simplest total action admitting the N = (1, 1) worldsheet supersymmetry takes the form S = SP+ SF =− 1 4πα′ ∫ d2σ(∂aXµ∂aXµ− i ¯ψµρa∂aψµ ) , (2.2)

where ρa are two-dimensional Dirac matrices obeying the Cliﬀord algebra {ρa, ρb} = −2ηab; ψµ is a Dirac spinor on the worldsheet that transforms as a vector under Lorentz transformations in the target space. One usually express ψµ in terms of its two components as

ψµ ≡   ψ−µ ψµ   , (2.3)

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and the fermionic part action can be rewritten as SF = i 2πα′ ∫ d2σ(ψM₋∂+ψ₋N + ψM+∂−ψ N + ) ηM N, (2.4)

where ∂_± ≡ 1₂(∂0 ± ∂1). This implies that the worldsheet fermions naturally carry left-moving and

right-moving modes, like that of closed strings. These fermionic degrees of freedom satisfy anti-commutation relation of superconformal field theory (SCFT) and contribute a central charge. This central charge should be vanishing to keep a worldsheet theory non-anomalous. Such condition, combined with that obtained from bosonic part action, require that the worldsheet has a critical dimension D = 10.

Furthermore, action (2.4) does not completely determine the spacetime spectrum of the theory: it admits an arbitrary choice for specifying the periodicity of the fermions under transport around the closed string worldsheet. For common closed strings, the periodic condition is Xµ(σ + π) = Xµ(σ), similarly, periodic fermions obeying ψµ_±(σ + π) = ψµ_±(σ). The fermions with periodic condition are said to be in the Ramond (R) sector, while for the fermions obey anti-periodic condition: ψ_±µ(σ +π) =

−ψµ

±(σ) are said to be in the Neveu-Schwarz (NS) sector. This choice can be made separately for

the left-moving and right-moving fermions, so that there are four possible sectors: NS-NS, NS-R, R-NS, R-R. In the ten-dimensional eﬀective theory, bosonic fields arise from string states in the NS-NS and R-R sectors and the NS-R and R-NS sectors give rise to spacetime fermions. In addition to the periodicity constraint, it turns out to be necessary to impose a particular projection, GSO

projection, on the spectrum to obtain a consistent closed string theory coupling fermions. This entails

one further choice: If NS-R and R-NS sectors admit identical GSO projections, chirality will emerge (in particular, the two gravitinos have the same chirality); while if these two sectors have opposite GSO projections, a non-chiral spectrum will be performed. The chiral closed string theory is called

type IIB superstring theory and the non-chiral one is called type IIA superstring theory.

Moreover, three other consistent superstring theories are known. Type I string theory, SO(32)

heterotic string and E8×E8 heterotic string. Type I string theory allows the classification of oriented

strings and unoriented strings, GSO projection removes one of the two gravitinos from the spectrum since both R-NS and NS-R sectors have the same spectra. The two heterotic theories are constructed by left-moving bosons and right-moving fermions or left-moving fermions and right-moving bosons, respectively. All of these three string theories haveN = 1 supersymmetry.

The five superstring theories described above are interrelated by S-duality and T-duality and correspond to diﬀerent limits of an underlying theory which is called M-theory. A higher version of

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unifying string theory is called F-theory, which is a twelve-dimensional theory and supposed to be consistent if and only if compactifying it on a two-torus.

2.1.2 Low-energy Eﬀective String Theory

In this thesis, our goal is to understand how a string theory can be consistent to realistic theories (e.g. standard model, inflationary cosmology) at low energy scale (much less than the Planck scale, in which we assume that only massless string states contribute). Thus we will focus on the low-energy effective versions of string theory. As in QFT, the idea to construct an effective action is to perform the path integral. First integrating out massive excitations of the string, and then leaving an effective action for the massless modes. Still start from the bosonic action, on the worldsheet we have,

SB = 1 2κ2 D ∫ dDX√−Ge−2Φ ( R + 4(∂Φ)2−1 2|H3| 2 +O (α′) ) , (2.5)

where G = GµνdXµdXν with Gµν the symmetric traceless graviton, H3 = dBµν with Bµν the

anti-symmetric two-form field, Φ the scalar known as the dilaton; and R ≡ R(h) is the Ricci scalar constructed from hab.

For the consistent superstring theories, each of their low-energy eﬀective theories is a ten-dimensional supergravity. Here we will focus on type IIA and type IIB which play important roles in this thesis. The actions for the other three theories can be found in [1,2, 3].

The NS-NS sector of 10D type II supergravity have exactly the same spectrum as (2.5) has, just with D=10 and all higher excited states omitted. In this case, 2κ2₁₀ = (2π)7(α′)4. And H3 = dB2 is

now defined as NS-NS fields since they are characteristic p-form fields of NS-NS massless spectrum. For type II theories, there also exist R-R fields which diﬀer by the chirality. In type IIA SUGRA, there are one-form C1and three-form C3, which induce F2 = dC1, F4 = dC3. Combining a topological

term required by the gauge invariance, the full IIA SUGRA action takes the form

SIIA= SNS-NS+ SR-RIIA + S IIA CS = 1 2κ2 ∫ d10X{√−G [ e−2Φ ( R + 4(∂Φ)2 −1 2|H3| 2 )] − 1 2 ( |F2| 2 + ˜F4 2)} − 1 4κ2 ∫ B2∧ F4∧ F4, (2.6)

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field strength. The full action is SIIB = SNS-NS+ SR-RIIB + S IIB CS = 1 2κ2 ∫ d10X{√−G [ e−2Φ ( R + 4(∂Φ)2−1 2|H3| 2 )] − 1 2 ( |F1| 2 + ˜F3 2 + ˜F5 2)} − 1 4κ2 ∫ C4∧ H3∧ F3, (2.7) where ˜F5 = F5− 1₂C2 ∧ H3+ 1₂B2∧ F3 is self-dual: ˜F5 = ⋆ ˜F5.

2.1.3 D-branes

In addition to the strings, there are higher-dimensional objects which admit solitonic degrees of freedom and higher-form fields. These objects are called branes whose sweeping contributes to a worldvolume embedded in the spacetime. Since there is no periodic boundary conditions to restrict open strings, it is suggested that open strings are stretching among branes or having their two end-points on the same brane, as Fig.2-1showed. Then the endpoints of open string will obey Neumann boundary condition for which the spatial derivative of the string coordinate in a specific direction vanishes but can move freely; or Dirichlet boundary condition for which the string coordinate is fixed at a certain position. Now we again keep our eyes on the open string sector in type II theory with Neumann and Dirichlet boundary condition. Imagine an open string moving in (9 + 1)-dimensional flat spacetime with p + 1 Neumann boundary condition and 9− p Dirichlet condition. This means the string endpoints are fixed on a (p + 1)-dimensional hypersurface called Dp-brane. Polchinski pointed out[15] that these D-branes are dynamic objects of equal importance in string theory as the string itself. For example, a single D-brane naturally carries U (1) symmetry and strings can have one endpoint located on the brane to produce gauge coupling; a stack of N branes will enhance U (1) to U (N ) and induce Higgs mechanism.

All D-branes can be present in type IIA/IIB theory, while their stabilities have dependence on the dimension of the worldvolume p. In type IIA, only the D-branes with even p are stable; in type IIB, only the D-branes with odd p are stable. The stable D-branes of type I and type II string theories are BPS objects which preserve half of the spacetime supersymmetry. A BPS D-brane corresponds to a higher-dimensional generalization of an extremal black hole, with tension equal to its charge.

D-branes have tension Tp and interact gravitationally with the closed string. As the examples

showed, D-branes have a gauge-theory-like structure as a common electromagnetic theory, with possible non-abelian structure group. Thus it is natural to guess that the brane action can be

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Figure 2-1: A stack of branes with endpoints of open strings located.

described by a non-linear electromagnetic action, which is also called Dirac–Born–Infeld action (DBI action): SDBI =− ∑ p Tp ∫ dp+1σe−Φ√− det P(G − 2πα′F)ab, (2.8)

where 2πα′Fab = Bab− 2πα′Fab, with dB a closed NS-NS field (known as Kalb–Ramond field) and

F the field strength associated with the worldvolume gauge boson A. This action exhibits the

non-perturbative nature of D-branes since it is independent of gs used in perturbation theory. The symbol

P denotes the pull-back action which satisfies: PGab =

∂Xµ

∂σa

∂Xν

∂σb Gµν, (2.9)

with X ≡ X(σ) the embedding of brane coordinates σ in spacetime X(σ).

By expanding the determinant to the second order derivative under fluctuation Xa_{= x}a_{+ 2πα}′_ξa

(xa _{denotes the position of D-brane), one finds}

SDBI =− ∑ p Tp ∫ √ −Ge−Φ(_{1 + (πα}′₎2_F2_{+ 2(πα}′₎2_(∂σ)2_{· · ·})_. _(2.10)

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The first term is just the Casimir energy, the second gives the kinetic term for the gauge field and scalars. All matter fields in the gauge multiplet transform in the bifundamental, for strings starting and ending on the same brane this becomes the adjoint representation of the gauge group.Furthermore, the above identifies the gauge coupling ggauge as

g_gauge2 = gs

Tp(2πα′)2

= g_s2(2π)p−2α′p−32 . (2.11)

Thus the tension of a D-brane can be written as

Tp ≡

1 (2π)p_g

s(α′)(p+1)/2

, (2.12)

the coupling to the R-R sector is described in the Chern–Simons (CS) action, we often make use of polyforms and the democratic formulation of type II theory[16] to extract CS action. The idea is to double the number of gauge potentials and field strength such that we can derive both the equations of motion and Bianchi identities from the action in order to describe only the physical degrees of freedom with correct duality relation, e.g. we relate F7 to its Hodge dual F3. And the CS action is

SCS =− ∑ p (−1)pµp ∫ Σp+1 Pξ(e−2πα′F ∧ C ) , (2.13)

where Σp+1 is the Dp-brane worldvolume. For D-branes, Tp = gs−1|µ|p, µp is defined as brane charge.

After momentarily erasing theF term, we obtain a topological charge induced in the worldvolume,

SCS = µp

∫

Σp+1

Cp+1, (2.14)

This implies that we have extended objects (D-branes) that are charged under R-R form potentials

Cp+1 and we can identify µp with the charge of the Dp-brane.

The coupling of D-branes to the background fields has important consequences. A D-brane can contribute to the massless spectrum of closed string theory. In particular, it holds a localized source of energy-momentum tensor and R-R charge, which will source curvature and R-R fluxes in proportion to the tension and charge of this D-brane. This fact is known as backreaction in gravity-like theory. And these backreaction eﬀects are the cores introducing the methodologies and the problems this thesis most concerns.

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spatially extended extremal black hole, named as extremal p-brane. For N coincident Dp-branes, the characteristic radius of curvature r+ is given by

r7₊−p = dpgsN (α′) 7−p 2 , (2.15) where dp ≡ (4π) 5−p 2 Γ(7−p

2 ). The dilaton in dependence of radial coordinate can be written as

eΦ = gs ( 1 + (_r + r )7−p) 3−p 4 (2.16)

A classical supergravity description is valid if the curvatures are small and gs ≪ 1. For p < 7,

Note that from (2.15) we can see the curvature is small at large gsN . For p = 3 case which we

will be most interested in, the dilaton is constant and can be small everywhere, which implies that D3-branes are decoupled from the scalar dilaton. Therefore, if p = 3, the α′ and gs corrections to

leading order classical supergravity can be neglected everywhere if

1≪ gsN ≪ N. (2.17)

We will have more discussions in Chap. 4, where the non-extremal branes will also contribute.

2.1.4 Charges and Fluxes of D-branes

Let us have a closer look at the eﬀective D-brane actions. To understand R-R fields and potentials, we assume that both Kalb–Ramond potential B and gauge potential A are vanishing. Then we have

S =− 1 2κ2 10 ∫ 1 2⋆ Fp+2∧ Fp+2− (−1) p_µ p ∫ ξ (Cp+1)∧ δ9−p, (2.18)

where δ9−p is a δ-function form on the worldvolume. This gives the equation of motion:

1 2κ2

10

d ⋆ Fp+2 =−µpξδ9−p, (2.19)

and the Bianchi identity:

1 2κ2

10

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The Ramond charge can then be obtained by integrating F8−p over appropriate sphere Qp = ∫ S8−p F8−p = 2κ210µp = (2π √ α′)7−p. (2.21)

For M coinciding branes, this charge can keep track of the number of branes as QM = M Qp ∈

(2π√α′)7−p_Z.

The above charge is an electric analogy, one can also compute the dual magnetic charge for F8−p

with the electric charge Qe ∼

∫

⋆Fp+2. Each pair of dual charges can be related by Dirac charge

quantization.

Fluxes can be thought of as vacuum expectation values (vev) of the R-R forms (NS-NS forms can also have fluxes in the same sense). We first detect the case with NS-NS fields vanishing and there are no local sources for R-R forms such that their equations of motion and Bianchi identity just reveal a topologically trivial background, where the p-form fields are closed and exact, namely:

dF = 0, d ⋆ F = 0. (2.22)

This implies that only those topologically non-trivial manifolds can admit non-vanishing fluxes. This is the motivation to introduce compactification in the presence of fluxes, known as flux

compactifi-cation. Fluxes provide background values for the Ramond forms in the absence of local sources and

quantized by string theory. The same goes for NS-NS flux, with the charges written as

QK =

∫

H ∈ 4π2α′Z. (2.23)

We have seen that D-branes are being charged under the corresponding R-R field strengths, and there is a corresponding five-brane for the NS-NS H field. This brane is called the NS5-brane and is magnetically charged under H as:

dH = µ5δN S5. (2.24)

The fundamental string (known as F1) couples electrically to H. At perturbative level, we have a good understanding of D-branes, while that is not so obvious for NS5-branes. The only insights we have are through S-duality, which can be understood as strong-weak coupling duality, of string theory. We will discuss this in detail in Sec.3.1. D3 and NS5 branes are the most important blocks building our thesis, one of the aims in this thesis is to better understand these exotic branes in

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non-extremal case.

2.2 Kaluza-Klein Reduction

Now that we have understood what flux means, it is the time to go to the compactification farm. As the introduction has noted, to coincide four-dimensional QFT to ten-dimensional spacetime, the compactification used to separate Minkowski spacetime and compact manifold is necessary. We consider the geometry decomposes as

M10 =M4× X6, (2.25)

where X6 is a six-dimensional compact manifold. And the metric can be decomposed as

ds2₁₀= e2A(ym)gµνdxµdxν + gmndymdyn. (2.26)

Here gmn is the internal metric with m, n ∈ 4, 5, · · · , 9, and the warping factor A is a function in

terms of internal coordinates ym_{. Before explicitly discuss various grotesque compactifications, let}

us treat a simplest case—Kaluza–Klein reduction.

Kaluza–Klein (KK) reduction is human’s first trial to understand the physics on the compact manifold. Kaluza and Klein unified gravity and electromagnetism in four dimensions by deriving both interactions from pure gravity in five dimensions.

2.2.1 Original Formalism

Kaluza suggested that all matter forces are simply a manifestation of pure geometry and the four-dimensional physical degrees of freedom are obtained from the wrapping of higher-dimensional geometry. Specifically, for five-dimensional gravity, we should have Tab = 0, and correspondingly

Gab= 0 or Rab = 0. (2.27)

And the five-dimensional gravitational action can be written as the Einstein–Hilbert action:

S5 =

1 16πG5

∫

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The five-dimensional metric gab can be block-diagonalized to gab =   gµν + k2ϕ2AµAν kϕ2Aµ kϕ2_A ν ϕ2   , (2.29)

where gµν still the four-dimensional metric, Aµ is the electromagnetic four-potential, and ϕ denote

some scalar fields; k is a constant scaling the electromagnetic potential, k = 4√πG4. One can then

variate the action and derive equations of motion of the diagonal elements and oﬀ-diagonal elements, respectively. It is easy to note that the oﬀ-diagonal elements give the equations of motion of electro-magnetic tensor, while two diagonal elements correspond to metric tensor and scalar, respectively. Namely, Gµν = k2_ϕ2 2 Tµν− 1 ϕ[∇µ(∂νϕ)− gµν□ϕ] , (2.30) ∇µ_F µν =−3 ∂µϕ ϕ Fµν, (2.31) □ϕ = k2ϕ3 4 FµνF µν_, _(2.32)

where Tµν = 1₄gµνFρσFρσ − FρµFρν is the electromagnetic energy-momentum tensor. Choosing the

scalar field ϕ to be constant throughout the spacetime, then the third equation vanishes and the first two degenerate to the Einstein gravity coupling Maxwell equations. As Kaluza did, we choose ϕ = 1, the equations become

Gµν = 8πG4Tµν, ∇µFµν = 0. (2.33)

And it is clearly that the scalars will determine the equations of motion after compactification.

2.2.2 Dimensional Reduction

To understand how the eﬀective four-dimensional theory is obtained from the higher-dimensional gravity, we usually use a method called dimensional reduction. For KK reduction, it is equivalent to state that there is a reduction leads to M4× S1, with the one coordinate flat and compact.

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Scalar Fields

First consider a real massless scalar in D = 5 with action

S =−1

2 ∫

d4xdy∂µφ∂µφ. (2.34)

Suppose that S1 is a circle of radius R, for a complete set of coordinates xµ= (xa, y), y in S1 should

satisfy periodic condition y = y + 2πR. The field φ satisfies the equation of motion

□φ = 0 ⇒ ∂µ

∂µφ + ∂y2φ = 0. (2.35)

Since y admits periodic condition, the fields should satisfy φ(x, y) = φ(x, y + 2πR), we can write the Fourier expansion as φ(x, y) = √1 2πR ∞ ∑ n=−∞ φn(x)einy/R. (2.36)

Notice that Yn(y)≡ √_2πR1 einy/R are the orthonormalized eigenfunctions of ∂y2 on S1. We can insert

(2.36) into (2.35) so that

∂_µ2φn−

n2

R2φn = 0. (2.37)

This implies that φn(x) are four-dimensional scalar fields with mass _Rn. By observing such fact in

the action, we have

S =− ∞ ∑ n=−∞ 1 2 ∫ d4x [ ∂µφn∂µφ∗n+ n2 R2φ ∗ nφn ] . (2.38)

In addition to the fact that the mass of scalar fields is n/R, the action also reveals that in four dimensions there is one massless scalar φ0 plus an infinite tower of massive scalars φn, known as KK

towers. In the limit of R→ 0, only φ0 is still light and all heavier excited states can be neglected in

the path integral. φ0 is also called the zero mode of a compactified theory, which is the only degrees

of freedom to be kept in dimensional reduction. Generally speaking, dimensional reduction in this restricted sense is compactification on a torus Sp_{× S}d _{with all massive modes integrated out. Note}

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solution of the lower-dimensional equations of motion is still a solution of the full higher-dimensional theory.

Gauge Fields

Following the same logic, one can generalize to the action with gauge fields. The action is

S = ∫ dx4dy ( −1 4FabF ab₊1 2(∂aA5− ∂ 5_A a)(∂aA5− ∂5Aa) ) . (2.39)

The Fourier expansion gives

Aµ(xa, y) = √1

2πR ∑

n

Aµ_n(x) (xµ) eiRny, (2.40)

then the action can be rewritten as

S = ∫ d4x∑ n [ F(n)abFnab+ 1 2 ( ∂aA(−n)5+ i n RA(n)a ) ( ∂aA(n)5− i n RA a n )] . (2.41)

The mixed term can be removed by gauge transformation

A(n)a → A(n)a− i

R

n∂aA(n)5, (2.42)

A(n)5→ 0 (n ̸= 0). (2.43)

After gauge fixing, the action becomes

S = ∫ dx4 ( −1 4F(0)abF ab 0 + 1 2∂aA(0)5 1 2∂ a A(0)5 ) + 2∑ n≥1 ( −1 4F(−n)abF ab n + 1 2 n2 R2A(−n)aA(n)a ) . (2.44)

The five-dimensional Maxwell action thus describes a four-dimensional gauge field and a real scalar field in the zero mode. The nonzero modes correspond to the maasive vector fields.

In general, the fields in D-dimensional spacetime conform to various representations of the Lorentz group SO(1, D− 1). In particular, for the compactification MD → Md× XD−d, we would like to

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gauge field Aµ transforming in the fundamental representation D, we have the decomposition

D = (d, 1) + (1, D− d), (2.45)

which suggests that Aµ splits into gauge fields Aa, a = 0,· · · , d − 1 under SO(1, d − 1) and singlet

scalars AM, M = d,· · · , D − 1 under SO(D − d). The same conclusion applies to two-form fields:

Bµν → Bab(antisymmetric tensor)⊕ BaM(vector)⊕ BM N(scalar). (2.46)

2.3 Compactification

Now let us come to the compact manifold and dive into the sea of compactification. In this section, we take the convention that Greek indices label the four-dimensional coordinates while Roman indices label the local coordinates of six-dimensional compact manifold, and the capital Roman letters stand for the indices labeling ten-dimensional spacetime. Recall (2.26), we can generalize it to

ds2 = e2A(y)gµνdxµdxν + gabdyadyb, (2.47)

where A(y) is the warp factor, gµν is a Minkowski, dS4 or AdS4 metric. Furthermore, we will assume

that the four-dimensional spacetime admits maximal spacetime symmetry, this requires the vev of the fermionic fields to be vanishing. Thus the background should be purely bosonic. Moreover, only the fluxes whose frame is zero-trad or tetrad can be permitted to turn on. E.g. NS-NS flux H3

must be internal; in type II theory, IIA admits F4 and IIB admits F5, respectively, in the external

spacetime.

2.3.1 Killing Spinor

Again, to add fermions into the compactified theory, supersymmetry is necessary. We require that the vacuum satisfies ¯ϵQ|0⟩ = 0, where ϵ(xM_{) parametrizes the supersymmetry transformation}

which is generated by Q, they are spinors of SO(1, D− 1). Since fermionic fields are spinors which non-trivially transforming under SO(1, d− 1), ⟨δϵψ⟩ ∼ ⟨ψ⟩ = 0, and we only need to worry about

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dilatinos λA. In the supergravity approximation, they transform as

δϵψM =∇Mϵ +· · · , (2.48)

δϵλ =· · · , (2.49)

where ψM denotes a combination vector of the two Majorana–Weyl spinors ψ1M and ψM2 (same/opposite

chirality in IIB/IIA), and the ellipses stand for terms which contain other bosonic fields which will contribute to vanishing vevs. Then, the vanishing of δϵψM implies

⟨∇Mϵ⟩ = 0 ⇒ ∇a,µϵ = 0. (2.50)

The spinor fields ϵ satisfy (2.50) is known as Killing spinors, they play an important role in performing the geometric properties of compact manifold, e.g. holonomy, cohomology group, etc.

In particular, in the absense of flux, (2.50) can be written in components as

∇µϵ +

1

2(γµγ5⊗ /∇A)ϵ = 0. (2.51)

This yields the integrability condition

[∇µ,∇ν] ϵ =− 1 2(∇mA) (∇ m A) γµνϵ = 1 4Rµνλργ λρ ϵ = k 2γµνϵ, (2.52)

where we have used the fact that Rµµλρ = k(gµλgνρ−gµρgνλ), which is correct for maximally symmetric

spacetime, in the last two equalities above. And this gives

k +∇m∇mA = 0. (2.53)

The only possible constant value of (∇A)2 _{on a compact manifold is zero, which implies that the}

warp factor is constant and the four-dimensional manifold can only be Minkowski spacetime. The existence of covariantly constant spinors is a necessary condition for a supersymmetric com-pactification. This is also a very strong restriction from the diﬀerential geometric perspective. A Killing spinor in the internal manifold demands that this manifold must have reduced holonomy, e.g. The CY manifold which we are familiar with, has only one Killing spinor and maximal SU(3) holonomy, and preserving the maximal N = 4 supersymmetry; If there are two Killing spinors on the CY manifold, it will degenerates to four-dimensional K3 surface which holds SU(2) holonomy

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and N = 2 SUSY. If one is allowed to turn on fluxes, this condition will be relaxed.

However, the case most interests us is that with fluxes. These newly released piggies are the bravest fighter against supersymmetric conditions. That is also the motivation for us to consider flux compactification to obtain a suﬃciently consistent four-dimensional theory from superstring. Nevertheless, in order to have some supercharges preserved (e.g. 4d N = 1 field theory), or even in the case where all of them are spontaneously broken by the fluxes, we need to have globally well-defined supercurrents. This requires that the structure group of internal manifold must be reduced. For a review of G-structures in the context of compactifications with fluxes, readers can see [17]. For some basic diﬀerential and complex geometry knowledge, one can read App.A. Here we will briefly show the main results from principal complex bundle perspective.

2.3.2 Supersymmetric Compactification with Fluxes

First consider the definition of G-structure.

Definition 2.3.1. A G-structure is a principal sub-bundle of the frame bundle with fiber G ⊂

GL(d,R).

Physically speaking, this means we allow transition functions which are elements of a subgroup

G ⊂ GL(d, R) when we transform fields from one patch to another on the base manifold. The

logic behind the use of G-structures to describe physical situations is to interpret the field content of a theory as a topological data for the structure and to relate the equations they obey to some integrability condition. Once the correspondence is stated, one can exploit representation theory to look for solutions, a problem which is usually easier than to directly address the field equations.

An easiest example is a type II supergravity which gives the CY geometry. For a configuration without fluxes, one can set dilaton and warp factor to be zero, which yields a reduced structure group O(6). Moreover, the spinor ϵ in supersymmetry transformations has to be globally defined and non-vanishing everywhere, and we can further reduce the structure group to the stabilizer of ϵ in SO(6), which is just SU(3). This can be translated into the existence of globally defined higher-forms together with some compactibility condition on X6. They define an almost complex structure. To

understand this argument more clearly, let us come back to non-flux situation for a while.

As we stated above, supersymmetry requires the existence of Killing spinor on the internal mani-fold, this fact is two-fold. First, the existence of a non-vanishing globally defined spinor, and second, it is covariantly constant. The first condition implies the existence of two four-dimensional

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super-on the manifolds with reduced structure group. All vector, tensor and spinor representatisuper-ons should be decomposed in representations of d-dimensional reduced structure group SO(d)/{stabilizer}.

Specifically, for six-dimensional compact manifold, one naturally has 4 of SO(6) as the spinor representation, which can be decomposed in SU(3) as

4→ 3 ⊕ 1. (2.54)

Therefore, there is a SU(3) triplet of 3 and a SU(3) singlet of 1. And the existence of the singlet implies that there is a spinor trivially depends on the tangent bundle of the manifold, therefore, the spinor is well-defined and non-vanishing everywhere. That is why we have SU(3) as reduced structure group in a six-dimensional manifold in absence of fluxes.

Now let us further decompose other SO(6) representations, in SO(6), its fundamental represen-tation gives

42→ 1 ⊕ 6 ⊕ 15 ⊕ 20, (2.55)

where 6 stands for vectors, 15 stands for two-form tensor fields and 20 stands for three-form ones. In the representations of SU(3), this further decompose as

6→ 3 ⊕ ¯3, (2.56)

15→ 8 ⊕ 3 ⊕ ¯3 ⊕ 1 (2.57)

20→ 6 ⊕ ¯6 ⊕ 3 ⊕ ¯3 ⊕ 1 ⊕ 1. (2.58)

And there are also singlets in the decomposition of two-forms and three-forms. Thus the higher-forms we analyzed that should also be globally defined in six-dimensional manifold are just real two-form and complex three-form which usually denoted as J and Ω. These two can determine an internal metric. Let us again treat SUGRA as an example to see how these higher-forms work.

Four-dimensional N = 1 supergravity

For a general 4DN = 1 SUGRA, the bosonic degrees of freedom are clear, the metric field gµν,

gauge field F = dA, and complex scalar field ϕi_{. In the absence of gauge coupling, the low-energy}

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be clearly discussed later on. The Lagrangian is

Lint. =−Ki¯j∂µϕi∂µϕ¯j− VF, (2.59)

where Ki¯j = ∂i∂¯_jK is the Kähler metric1 of the Kähler potential K(ϕi, ¯ϕi), VF can be expressed as

VF(ϕi, ¯ϕi) = e K M 2 pl [ Ki¯jDiW DjW − 3 M2 pl |W |2 ] , (2.60) where DiW ≡ ∂iW + _M12 pl

(∂iK)W , and W ≡ W (ϕi) stands for the so called superpotential. Kähler

potential is a real analytic function of the fields while superpotential is a holomorphic one, in de-pendence of ϕi. The Kähler potential is a natural emergence of complex geometry background we are discussing. Without dive into mathematical details, we will simply show you some important conclusions lead to Kähler potential and few features of it. Note that we will set Mpl = 1 in this

chapter unless specified.

Calabi–Yau compactifications with N = 2 supersymmetry

As we pointed out, an non-vanishing Killing spinor usually corresponds to two supersymmetric parameters and leads to N = 2 SUSY. Let us focusing on the CY compactifications with N = 2 SUSY now. Recall the ten-dimensional metric (2.47) with warp factor set to be 1 for convenience2_.

For X6 a CY manifold, the definition is

Definition 2.3.2. A Calabi–Yau manifold is a compact Kähler manifold with a vanishing c1 and a

Ricci-flat metric, where c1 stands for the first Chern class of a manifold.

Definition 2.3.3. A Kähler manifold is a manifold with a complex structure, a Riemannian

struc-ture, and a symplectic structure. Those three structures are compatible to each other.

Example 2.3.1. Every smooth complex projective plane is a Kähler manifold. Thus gab is a

Ricci-flat metric (Rab = 0 as we have noted). And the compactification over X6 will obtain a 4d N = 2

spacetime3_.

1_{This is just a definition.}

2_{In fact, supersymmetric compactification is subtly diﬀerent to the warped compactification, we will discuss the}

latter in the following chapter.

3_{This statement is correct only for type II string theory. For compactification of type I and heterotic ones, one}

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Definition 2.3.4. A Hermitian metric g is a covariant tensor field of the form

n

∑

i,j=1

g_i¯_jdzi⊗ d¯z, (2.61)

where g_i¯_j = g_i¯_j(z) such that g_j¯_i(z) = g_i¯_j(z) and g_i¯_j(z) is a positive definite matrix.

Remark 2.3.1. To any Hermitian metric, we can associate a (1, 1)− form

ω = i

n

∑

i,j=1

g_i¯_jdzi∧ d¯zj. (2.62)

ω is known as the fundamental form associated with the Hermitian metric g.

Definition 2.3.5. Given a Hermitian metric K_i¯_j, if the associated form is closed, namely dω = 0, we call Ki¯j a Kähler metric. A complex manifold endowed with a Kähler metric is also called as

Kähler manifold. And ω is the Kähler form.

Remark 2.3.2. dω = 0⇒ ∂ω = ¯∂ω = 0 can be reinterpreted as

∂iKj¯k = ∂jKi¯k, ∂¯iKj¯k= ¯∂kKj¯i. (2.63)

These are the explicit Kähler condition.

There are some scalar fields, like dilatons, which can stretch or compress the compactified volume, known as moduli. Rigorously speaking, moduli correspond to the deformation parameter of a compact manifold and behave as degeneracy of vacuum. For example, S1 _{has one modulus: R the radius.}

The fact that any value of R is allowed manifests itself in the spacetime theory as a massless scalar field with vanishing potential. And T2 _{has one Kähler modulus and one complex structure modulus.}

Here, the geometric moduli are required to deform gab while preserving the CY conditions.

Ac-cording to Def. 2.3.2, a CY manifold can be understood as a three-layer folded T2 _{whose moduli can}

be parametrized by Kähler modulus and complex structure modulus. Thus CY manifold also leads to Kähler moduli and complex structure moduli, respectively. The Kähler moduli are deformations of the Kähler form

J ≡ ig_i¯_jdzi∧ d¯z¯j, (2.64)

where zi_{, ¯}_z¯j_{, i, ¯}_{j = 1, 2, 3 are complex coordinates on X}

6. And the complex structure moduli are

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the Kähler moduli by harmonic (1, 1)-forms ω which form a basis for the Dolbeault cohomology group

H1,1_{. This also implies that there exists index contraction between ω and the h}1,1 _{scalar fields, with}

the index rounds over 1,· · · , h1,1. Here h1,1 is the Hodge number characterizes H1,1. The Kähler form is

J = tI(x)ωI, I = 1,· · · , h1,1, (2.65)

where tI_{(x) denotes the h}1,1 _{four-dimensional scalar fields, namely the Kähler moduli.}

Definition 2.3.6. The Dolbeault cohomology groups are defined as

Hp,q(M) = Z

p,q₍M)

¯

∂(Ωp,q−1₍M)). (2.66)

This definition is nearly the same as de Rham cohomology. While the cochain is defined over complex operator ¯∂, Z(M) is the cycle of that cochain, and the term in denominator is just the boundary after acting ¯∂ on a complex form Ωp,q−1_.

Definition 2.3.7. hp,q₍M) ≡ dim

C(Hp,q(M)) called the Hodge number. hp,q= h3−p,3−q.

Now let us define the complex structure moduli in base of Kähler moduli. One expects that the Ricci-flatness of gab is preserved, which implies

Rab(g + δg) = 0⇔ δgi¯j = 0, δgij = 0. (2.67)

Since holomorphic coordinate transformations do not change the type of index, it is clear that δgij can

only be removed by a non-holomorphic transformation. But this means that the new metric is Kähler with respect to a diﬀerent complex structure compared to the original metric (the non-holomorphic change of coordinates generates a g_i¯_j).

Theorem 2.3.1. There is an unique holomorphic (3,0)-form of Dolbeault cohomology. Corollary 2.3.1. H0,1₍_{M, T}1,0₎_{→ H}∼= 2,1₍_{M) by defining the complex (2,1)-forms}

Ωijkδg¯_lkdzi∧ dzj ∧ ¯z ¯

l

. (2.68)

Thus the complex structure deformations can be expanded in a basis of harmonic (2, 1)-forms as:

δgij = i ||Ω||2ζ J_(x)(χ J)ij¯iΩ j¯i j = 0, (2.69)

The Existence of Four-Dimensional Non-Supersymmetric Meta-Stable String Vacua