Compactifying All Spatial Dimensions of the Universe

Home , Compact dimension, Conformal gravity

FACULTY OF SCIENCE

Compactifying all spatial dimensions of the universe

Quentin Decant

Thesis supervisor: Thesis submitted for the degree of prof. dr. Thomas Van Riet Master of Science of Physics, option Research

Academic year 2017 – 2018 © Copyright KU Leuven

Without written permission of the thesis supervisor and the author it is forbidden to reproduce or adapt in any form or by any means any part of this publication. Requests for obtaining the right to reproduce or utilize parts of this publication should be addressed to Faculteit Wetenschappen, Geel Huis, Kasteelpark Arenberg 11 bus 2100, B-3001 Leuven (Heverlee), +32-16-321401. A written permission of the thesis supervisor is also required to use the methods, products, schematics and programs described in this work for industrial or commercial use, and for submitting this publication in scientiﬁc contests. Preface

This thesis would not have been what it is now without the direct or indirect help of other people. I use this text to thank them. To start I would like to thank prof. dr. Thomas Van Riet for giving me the opportunity to write a thesis on such an exciting and interesting subject. I learned a lot from you this year, and the fact that you had the patience for allowing me to walk into your office anytime to ask questions was very much appreciated. I would like to express my gratitude to some of my fellow master students: David, Arno, Sam, Kristof, Kesha, for the good times at the office. My special thanks go to Vincent and Rob, with whom I had a lot of interesting and helpful discussion about physics. All the other people at the institute are also thanked, I really felt at home at the institute. The coffee breaks at four o’clock were really fun and a good relaxation moment. I also would like to thank my friends for all the good times we had together, not only this year, but the past five years. You made my student time very fun and memorable. I would like to express my deep gratitude to Anne, not only for the correcting of typos in my thesis. This thesis would simply not have been possible without all your support this past year. To conclude I am the most thankful to my parents. You always made sure that I had everything I needed to succeed. You continued to believe in me no matter what, and the freedom that you gave me due to this trust made me the person who I am now.

Quentin Decant

i Scientiﬁc summary

This thesis introduces a new approach towards solving the cosmological constant problem. This is a very subtle issue that plagues the high energy physics community for decades now. The problem is a naturalness problem which we formulated as follows: the enormous difference between the Hubble scale and the length scale of possible extra dimensions, is something very unnatural. These hypothetical extra dimensions are a fundamental concept in modern theoretical physics and are an essential ingredient in superstring theory, the most studied theory of quantum gravity. Understanding the reason that the dimensions we live are so much bigger compared to the extra dimensions is therefore an important and fundamental question. We try to understand this unnatural situation by starting from a more natural situation and evolve via a dynamical mechanism to the unnatural one. More concretely we want to start from a situation were all the spatial dimensions of our universe are treated equally. We take our natural situation to be that all dimensions are compact because we know from observations that the size of the universe has grown in time. To obtain such an initial condition for our universe we try to find one dimensional vacua of string theory, such that the scalar potential obtained from compactification admits only a meta-stable vacuum. This can be obtained by having a scalar potential which has only a local minimum for the volume modulus, no global one. In this case, classically the system will stay in this local minimum. However, quantum mechanically, the volume modulus will tunnel out of this meta-stable state, resulting in a spontaneous decompactification of some of the dimensions. This then gives rise to a universe which has some very large dimensions, and where the others remain compact. Even if there are no extra dimensions this mechanism could maybe provide initial conditions for inflation. The mistake of associating this mechanism to old inflation could easily be made, however they are not the same thing. In old inflation a four dimensional scalar field tunnels out of false vacuum state, we consider a one-dimensional field in a one-dimensional vacuum, which is a different thing. This means that the issues of old inflation do not automatically apply to our model. Under the approximation of smeared orientifold planes we derive that one- dimensional vacua do exist in string theory, and it is very likely that performing the calculation with localised planes results in the same conclusion. A simple compactification of these solutions yields an unfixed modulus, and is therefore not suited as a realisation of our mechanism. We did not investigate this further, but changed our approach. String theory and other theories with extra dimensions need to have phenomenological acceptable vacua, otherwise they can not describe the world we live in. On this fact the following reasoning is based. If we can compactify a four dimensional theory to one dimension, and this new theory admits a meta-stable vacuum, then the decompactification can lead to the theory we started from. Therefore we took four dimensional gravity combined with electromagnetism, axions and a cosmological constant, and investigated if we can compactify this to a theory admitting a one

ii SCIENTIFIC SUMMARY dimensional vacuum. This led us to a theorem excluding stable one-dimensional vacua obtained by compactifications using the above mentioned ingredients,. This turned out not to be a bad thing. We found that meta-stable pockets can appear in the scalar potential. Dependent on the quantum corrections in our model, they are classicaly stable and yield compact periodically blowing-up and deflating universes. However these pockets are only local minima, so these models can decompactify. The most interesting thing about these pockets is that they are only allowed under certain conditions. Namely the cosmological constant and the curvature have to be strictly positive, and there must exist background electromagnetic fluxes. Using these conditions we tried some compactifications, but none were successful. However we investigated only a tiny region in the space of all possible compactifications, therefore the mechanism proposed in this thesis remains very promising and should be investigated further.

iii Summary in layman’s terms

The Standard Model of particle physics is our best tested scientific theory, however it does not incorporate gravity. Therefore a lot of physicist are searching for new theories extending the Standard Model, and the best candidate at the time of writing is superstring theory. For this theory to be mathematically consistent extra dimensions, besides the three spatial dimensions we are used to, need to be introduced. Because we did not observe any extra dimensions yet, those have to be extremely small such that they are hidden from us. This idea of extra super tiny dimensions seems very strange and unnatural. Our observable universe is a ball with a radius of about thirteen billion light years, the size of the extra dimensions has to be much smaller than a billionth of the size of an atom otherwise we would have detected them. If these extra dimensions are really there, why is there such a difference between these and the dimensions we live in? This is the question which drives the research preformed in this thesis. The most natural situation for the universe to be in is the situation were all dimensions are treated on an equal footing, hence we expect that all dimensions are either large or small. We know that our universe grew in size after the big bang, and thus we expect that around that time all dimensions were small. For this reason we take the natural situation to be that all dimensions are small. Therefore we want to find solutions to our equations which describe a universe were all dimensions are compact, but were some can spontaneously grow large. If this can be found, we discovered a potential mechanism which takes us from a natural situation, to the unnatural situation we find ourselves in. This could potentially explain why there is such a difference between the dimensions we live in, and those that are hidden from us. How do we achieve this? We reversed the above reasoning and started from a universe like ours, one which has three large dimensions and were the others are compact. We then try to fold up the three remaining dimensions, to obtain the situation were all the dimensions are of comparable size. If these three dimensions can unfold, then they will probably unfold to the universe we used to construct this model, and by this demonstrate that you can evolve from a natural to an unnatural situation. We discovered that under certain assumptions the mechanism we propose is possible. What is interesting is that for it to be possible, the universes we fold up have to meet certain conditions. These are that the three large dimensions have to be positively curved, that we need to be able to fill the universe with electromagnetic fields, and that they have to posses a positive cosmological constant. These are all things which are possible in our universe, hence this is encouraging. With these conditions at hand we then tried to build some models which could be used for our proposed mechanism, but this did not succeed yet. However we only investigated a tiny fraction of all the possible universes useful to this mechanism, hence we remain positive and the search continues.

iv Contents

Preface i Scientific summary ii Summary in layman’s terms iv List of symbols viii 1 Introduction 1 2 Theoretical preliminaries 4 2.1 Gravity ...... 4 2.1.1 Basic formalism ...... 4 2.1.2 The Cosmological Constant ...... 6 2.1.3 Gravity in lower spacetime dimensions ...... 7 2.2 Scalar Lagrangian Densities ...... 9 2.3 p-Form electrodynamics & Form notation ...... 11 3 The physics of extra dimensions 14 3.1 Kaluza-Klein Dimensional Reduction ...... 14 3.1.1 Reduction of the metric ...... 15 3.1.2 Reduction of the other fields ...... 16 3.2 Modern compactifications ...... 18 3.2.1 String compactifications ...... 20 3.3 Scale separation and the CC-problem ...... 22 3.4 One dimensional compactifications ...... 23 4 Simple gravitational reductions 25 4.1 5D gravity on a circle ...... 25 4.2 Compactification over an n-torus ...... 27 2 4.2.1 M6 = M4 × T compactification with flux ...... 29 2 4.2.2 A special case: M4 = M2 × T ...... 31 4.3 Compactification over a maximally symmetric space ...... 34 4.3.1 Compactification without any flux ...... 34 4.3.2 Compactification with flux ...... 35 4.3.3 Consistency check ...... 37 4.4 One-dimensional compactification ...... 38 4.4.1 Compactification over S3 and H3 ...... 38 4.4.2 Adding flux ...... 39

v CONTENTS

4.4.3 Equations of motion from the action ...... 40 4.4.4 Reduction of the Field equation ...... 41 4.4.5 Cosmological constant ...... 43 5 Compactifications to 1D 45 5.1 Type II Supergravity ...... 45 5.1.1 Localised energy sources ...... 47 5.1.2 The equations of motion ...... 49 5.2 Vacua of type II supergravity ...... 50 5.2.1 Solution with p + 1 ≥ 2 non-compact dimensions ...... 50 5.2.2 One-dimensional vacua...... 54 5.3 Flux compactification ...... 56 5.3.1 Gravitational part ...... 56 5.3.2 Potential for the fields ...... 57 5.4 Four dimensions compactified to 1D ...... 58 5.4.1 Proof no stable 1D vacua ...... 59 5.4.2 Extrema of the potential ...... 64 5.4.3 Analysis of the scalar potential ...... 65 5.4.4 Compactification M = R × S2 × S1 with axion flux ...... 67 5.4.5 Group manifold reductions ...... 69 6 Conclusion 76 6.1 Results ...... 76 6.2 Further research ...... 78 A Conventies and theoretical background 79 A.1 Metric using vielbeins ...... 79 A.2 Derivation Cosmological constant ...... 79 A.3 Consistency substitution of vacua into action ...... 80 A.4 Conventions differential form calculus ...... 81 B Details of reductions in chapter 4 83 1 B.1 Reduction: M5 = M4 × S ...... 83 n B.2 Appendix: MD+n = MD × T ,...... 86 B.3 Reduction of a maximally symmetric space ...... 89 B.3.1 Reduction of the Ricci scalar ...... 89 B.3.2 Derivation Einstein equations with flux ...... 90 B.4 One-dimensional compactifications ...... 91 B.4.1 Equations of motion from the action ...... 91 B.4.2 Reduction of the field equations ...... 93 C Details of reductions chapter 5 96 C.1 Vacua of type II supergravity ...... 96 C.1.1 Solutions with p + 1 ≥ 2 ...... 96 C.1.2 Complete calculation one dimensional vacuum ...... 98 C.1.3 Compactification of Rb10 ...... 101 C.2 Extrema of the potential ...... 102 2 C.3 F2 contribution group manifold ...... 103

vi CONTENTS

Bibliography 105

vii List of symbols

Symbolen g Higher-dimensional metric bµbνb g Conformally rescaled higher-dimensional metric eµbνb gµν Lower dimensional metric Rb Ricci scalar of the higher-dimensional metric Re Ricci scalar of the conformally rescaled higher-dimensional metric R Ricci scalar of the lower dimensional metric λ Γeµν Christoffel symbol higher-dimensional metric λ Γµν Christoffel symbol lower-dimensional metric H Three-form ﬂux Fi i-form ﬂux φ Volume modulus / dilaton Λ Cosmological constant ∂ Partial derivative Mij Moduli matrix Laplace-Beltrami operator lower-dimensional metric e Laplace-Beltrami operator higher-dimensional metric ? Hodge-dual ∧ wedge product d Exterior derivative 2 dsb Higher-dimensional line element 2 dse Conformally rescaled higher-dimensional line elelement ds2 Lower dimensional line element

viii Chapter 1

Introduction

The domain of research of this thesis is theoretical high-energy physics. We want to understand a question which is not asked a lot, but is a very fundamental one. Why do we observe three-spatial dimensions? To make progress towards this we try to compactify all spatial dimensions of the universe. Compactification of dimensions heuristically means that you are going to make them so small that you need very high energies to see them. The more technical details will be explained later. The question you could ask is why in the universe would you want to do that? We obviously observe four dimensions, where three of them are spatial, and so making these three unobservable is a rather strange idea. It is here that the cosmological constant problem (CC-problem) appears. As is well-known, string theory is a theory were extra dimensions play a significant role. However all theories with extra dimensions have some kind of CC-problem. In this thesis we argue that extra dimensions have some very nice properties, and explain the reasons to investigate such theories. However we did not yet observe any extra dimensions than the four that we are used to, so the extra’s have to be hidden. We can then ask the question, why is there such a division? This seems very arbitrary and unnatural, and can be more concretely explained in terms of the naturalness problem of the cosmological constant. In a theory with extra dimensions the most natural situation is that all the dimensions are either decompactified and thus easily observable, or they are all compactified and unobservable. It is this unnatural situation that we try to resolve. In 2003, S.Kachru et. al. argued in their paper [1] that you can find a four- dimensional meta-stable vacuum of string theory with a positive cosmological constant. To achieve this they had to compactify six dimensions. They obtained a potential for the physical field controlling the volume of these compact dimensions, which is show in Figure 1.1 at the top of next page. We say meta-stable because classically this fields sits in a local minimum of the potential, hence the size of the compact dimensions does not change. Their construction is therefore clasically stable. However the field can quantum tunnel passed the potential barrier, out of the vacuum. The consequence of this tunneling is that the extra dimensions will decompactify and the spacetime becomes ten-dimensional Minkowski space. There is an important side remark to make, the time scale of this process can be made much larger than the time our universe exists and so this model remains viable. One of the things we learn from this paper is that dimensions can tunnel and decompactify. This is a very interesting tool to address the unnatural situation of some dimensions being compact and some not. As argued above we could expect that all spatial dimensions are treated on an equal footing, thus that all dimensions

1 Figure 1.1: This figure has been taken from [1]. The sigma denotes the field controlling the volume of the extra dimensions, V denotes the potential for this field. The values on the axis are not important to our discussion and therefore are not commented. can be expected to be compact. If some of these tunnel and decompactify, then we found a dynamical mechanism to go from a natural to an unnatural situation. This may solve the CC-problem. The goal of this thesis is to take the first step towards this idea by compactifying all spatial dimensions and checking whether perhaps three of them can tunnel out of a meta-stable state and become large, explaining why we observe three spatial dimensions. We do this using an effective field theoretical approach. This means that we start with a theory describing the classical dynamics of a higher dimensional theory, and investigate how an observer confined to some, but not all, of the dimensions experiences this theory. For instance we take five dimensional gravity, and investigate how an observer which has no access to this fifth dimension experience this. The theory describing this is obtained by integrating out the effects of the fifth dimension. Therefore the obtained theory is an approximation because using high enough energies an observer will feel the effects we integrated out, and discover that there is an extra dimension. This is why it is called an effective field theory. In this thesis we are interested in compactifications in the context of string theory. Therefore we will first investigate if string theory admits one-dimensional vacuum solutions. Once this is done, we try to construct a compactification which can tunnel to four observable dimensions. We therefore assume that in string theory we can compactify six of the dimensions to yield a four-dimensional phenomenologically acceptable vacuum. If this assumption is wrong, the calculation performed in this thesis can not by used in a string theoretical context. However if our assumption is wrong, string theory has more severe problems than the fact that we cannot relate this thesis to it. We investigate if starting from this assumption we can compactify further the three spatial dimensions such that they can decompactify via a tunneling scenario. If this is the case, then we know that these one dimensional compactifications do have the possibilty to tunnel to the four dimensional model we used to construct them. This will result in a theorem about general one dimensional

2 compactifications, which applies outside the context of string theory. This thesis is written as follows. In the next chapter we introduce the necessary computational machinery to be able to follow the calculations. We will introduce our conventions for the formulas of general relativity, explain some facts about non-linear sigma models and introduce differential forms and p-form electrodynamics. In the third chapter we introduce and motivate the concept of extra dimensions and explain at a technical level the physics which accompanies them. Once this is done we have introduced all the necessary theory needed to understand the compactifications performed in this thesis. Therefore in the fourth chapter we start with some basic models in the context of general relativity, this to become acquainted with the calculations and the physics behind them. The fifth chapter is the most important one, it is there were we obtain our results previously mentioned above. We conclude this thesis with a final chapter to summarise everything we did.

3 Chapter 2

Theoretical preliminaries

2.1 Gravity

This section is a short introduction to the most important formulas from the theory of general relativity used in this thesis. This section is inspired by the book of Sean Carroll [2] and is not at all meant to be a thorough introduction to this beautiful subject. In the theory of general relativity we are working in a D-dimensional spacetime which can have a non-trivial geometry. This means that our notion of distances is not the same as we are used to from special relativity. Hence we need to replace the Minkowski metric ηµν = diag(−1, 1,..., 1) with a more general two-index symmetric tensor gµν, called the metric, but which has the same signature as ηµν. Doing physics on a curved background, like for example calculating the motion of a particle on a sphere, is not the same as doing general relativity. This is because the background in G.R., also know as spacetime, is dynamical. Therefore the metric will have its own dynamics, but more importantly, matter can, and will, change spacetime and by this inﬂuence the metric. We can summarise this with the well-known phrase by John Archibald Wheeler: “Spacetime tells matter how to move; matter tells spacetime how to curve”.

2.1.1 Basic formalism

The ﬁrst thing we mentioned above is that the notion of distances in a curved 2 spacetime change. This means that the line element will not be given by dsD = −dt2 + d~x2, but will be replaced by the more general formula,

2 µ ν dsD = gµν(x)dx ⊗ dx , (2.1) where gµν is called the metric on spacetime and D is the dimensionality. Once we have a notion of how to measure distances we can start to describe the curvature of spacetime. To achieve this in the context of general relativity we use (pseudo-)Riemannian geometry. In this framework curvature is described by making use of a connection. These connections are used to transport tensors from one point to another on a manifold, which is an n-dimensional space which looks locally like Rn. We can deﬁne a lot of different connections on one manifold, each time resulting in another notion of curvature. The connection which is used in general relativity is called the Levi-Civita connection. This connection is characterised by a

4 2.1. Gravity number of symbols called the Christoffel symbols, and which are deﬁned by 1 Γσ := gσλ (∂ g − ∂ g + ∂ g ) . (2.2) µν 2 µ νλ λ µν ν λµ σ Using the Christoffels we can deﬁne a (1,3) tensor, the Riemann tensor R µρν . This is the object which in G.R. fully describes how spacetime is curved. In our framework this tensor is given by the following equation

σ σ σ σ λ σ λ R µρν := ∂ρΓµν − ∂νΓµρ + ΓρλΓµν − ΓνλΓµρ. (2.3)

From this tensor we can deﬁne two new important tensors, the Ricci tensor and the Ricci scalar. These are obtained by taking traces and contraction of indices of the Riemann tensor using the metric. For the Ricci tensor we have

λ Rµν := R µλν , (2.4) and for the scalar we have µν R := g Rµν. (2.5) These are all the formulas we need to describe the dynamics of the metric. The action for the metric is called the Einstein-Hilbert action, which is deﬁned as

1 p S = dDx |g|R, (2.6) EH 2κ2 ˆ with κ a constant and p|g| is the square root of the absolute value of the determinant 2 of the metric. In four dimensions we have the relation κ4 = 8πG, with G Newton’s constant. We mentioned above that matter inﬂuences the curvature of spacetime, so we need to couple this action to the matter action. To be able to do this we need the covariant derivative:

ν1···νp ν1···νp ν1 λ···νp νp ν1···λ ∇µTσ1···σq = ∂µTσ1···σq + ΓµλTσ1···σq + ... + ΓµλTσ1···σq − Γλ T ν1···νp − ... − Γλ T ν1···νp , (2.7) µσ1 λ···σq µσq σ1···λ which maps (p, q) tensors to (p, q + 1) tensors. The coupling of bosonic matter with gravity is obtained by the so-called minimal coupling, which amounts to replace partial derivatives into covariant ones and the Minkowski metric by gµν. The total action then becomes p 1 S = dDx |g| R − L . (2.8) ˆ 2κ2 matter

Performing a variation of this action with respect to the metric gives rise to the Einstein equations 1 R − g R = κ2T , (2.9) µν 2 µν µν

5 2.1. Gravity where the energy momentum tensor is deﬁned as:

2 δSmatter Tµν := . (2.10) p|g| gµν

In the appendix of this chapter we show our conventions in the tetrad formalism.

2.1.2 The Cosmological Constant

There is one quantity which plays a central role in this thesis, the the cosmological constant (CC). From classical mechanics we know that we can add a constant to the potential energy with no effect on the physics. This is because in non-gravitational theories it is not the value of the energy which is important, but differences in energy dictate the dynamics. In general relativity this is different. To illustrate this we add a constant to the Einstein-Hilbert action, which we call the cosmological constant, and derive the equations of motion from the new action 1 √ S = dDx −g R − M 2Λ , (2.11) 2κ2 ˆ p q 8π~c with Mp = κ2 the Planck mass. The Einstein equations obtained from this action are M 2Λ R = p g , (2.12) µν (D − 2) µν where the derivation is shown in the appendix A.2. If we set Λ = 0, we find the vacuum Einstein equations as it should. However if the CC is not equal to zero we see that the Ricci tensor does not vanish. This means that the manifold is no longer Ricci-flat, but becomes curved. From this we see that adding a constant to the action has a physical consequence, because it influences the curvature of spacetime. If we add a positive CC we find a positively curved universe, which is called de Sitter space (dSD), and a negative constant yields a negatively curved universe, known as Anti de Sitter space (AdSD). The subscript D refers to the dimensionality of the space. The conclusion of this discussion is that the overall value of the energy is a meaningful quantity in general relativity. In this we thesis often encounter spaces with a cosmological constant. To these spaces we can associate a length scale, which is defined by

(D − 1)(D − 2) Λ = ± , (2.13) L2 where L is the overall length scale, and the + sign is for dSD, while the − sign is for AdSD . This deﬁnition was copied from [3], but slightly modiﬁed.

6 2.1. Gravity

Using this length scale we can rewrite the Einstein equations as, M 2(D − 1) R = ± p g . (2.14) µν L2 µν From here onwards we set the Planck mass equal to 1 throughout, except in one section were it is reintroduced.

2.1.3 Gravity in lower spacetime dimensions

We want to construct one-dimensional ﬁeld theories and couple these to gravity. Note that with one-dimensional we mean no spatial dimensions, only time. To this end we introduce how gravity behaves in these lower dimensions. In three dimensions there is one essential difference with four-dimensional gravity. In four dimensions the vacuum Einstein equations

Rµν = 0, (2.15) allow non-trivial solutions for the metric. This means that in 4D, empty space does not have to be flat. For example gravitational waves can propagate in a four dimensional empty spacetime. In 3D the Riemann tensor can be written as Rκλµν = 2 gκ[µRνλ] − gλ[µRνρ] + Rgκ[µgν]λ − Cκλµν, (2.16) which is a rewriting of equation (3.147) found in [2], and where the meaning of square brackets around indices is given by (A.15). The tensor Cκλµν is called the Weyl-tensor, which in 3D is identically zero [2]. Therefore if we have a Ricci-flat surface in 3D, the Riemann tensor vanishes, meaning that the Einstein equations only have trivial solutions. This means that Ricci-flatness implies locally flat space. In two dimensions things are even stranger. In this case the Riemann tensor for an arbitrary metric can be written as R R = (g g − g g ) , (2.17) µλνσ 2 µν λσ µσ λν which was taken from exercise 7.13 in [4]. From this formula it is easy to see 1 Gµν := Rµν − 2 gµνR, which is called the Einstein tensor, vanishes identically. This has as a consequence that the energy-momentum tensor is always identically zero! This could have been derived in an other way. For a two dimensional compact boundaryless orientable manifold there is the Gauss-Bonnet Theorem which relates geometry and topology. This theorem can be stated as

p 4πχ(M) = d2x |g|R, (2.18) ˆ which is a rewriting of equation (3.199) of [2], and where

χ(M) = 2(1 − g), (2.19)

7 2.1. Gravity where g can be seen as the number of holes in a manifold. This means that integrating the Einstein-Hilbert action over a compact space yields a results which is independent of any physical field. Therefore we can eliminate this term from the total action, to obtain p S = − d2x |g|L . (2.20) ˆ mat This discussion holds for any type of 2D manifold. For non-compact manifolds the p combination |g2|R is a total derivative, and in the compact case but with boundary, equation (2.19) is modified with another constant that also only depends on the topology of the manifold. From the above discussion we see that the 2D metric is not a dynamical object anymore, thus it acts as a Lagrange multiplier in our system. This means that there is a constrain equation associated to it. The variation of the total action with respect to the inverse metric has to vanish, which is the condition we use to derive the Einstein equations. From equation (2.20) we see that a variation of the total action amounts to a variation of the matter action. The energy-momentum tensor is defined by this variation (2.10), hence it needs to vanish. This restricts the kind of matter Lagrangian we can couple to two dimensional gravity, which can be seen in the following way

2 δS δLmat = gµνLmat − 2 = 0, (2.21) p|g| δgµν δgµν

µν µν which after contraction with an inverse metric g and using g gµν = D, yields δL2D L2D = gµν mat . (2.22) mat δgµν Remark that this is an on-shell condition, so this only needs to hold for solutions of the equations of motion. A generic scalar Lagrangian coupled to two dimensional gravity is √ 1 S = d2x −g − G (φ)∂ φi(x)∂µφj(x) − V (φ(x)) . (2.23) ˆ 2 ij µ

Here Gij(φ) is a 2 × 2 symmetric matrix with signature (p, q) and V (φ(x)) is the 1 ρσ i j potential. It is clear that the kinetic term − 2 Gij(φ)g ∂ρφ (x)∂σφ (x) satisﬁes δL2D L2D = gµν kin , (2.24) kin δgµν off-shell. We can therefore eliminate this part of the action from the above condition, to obtain δV 2D V 2D = gµν mat , (2.25) mat δgµν where V is the potential. If V does not have any metric dependence, the derivative of this quantity with respect to the metric vanishes. This means that the potential

8 2.2. Scalar Lagrangian Densities always has to vanish on-shell. In the two-dimensional models we construct we do not obtain 2D gravity coupled to scalars, but we obtain Jackiw-Teitelboim gravity. This theory admits a potential, which is important because the scalar potential plays a central role in this thesis, and the absence of it would lead to problems. This is further explained in the third chapter. In one dimension we can not deﬁne something as intrinsic curvature. This can be seen using the anti-symmetric properties of the Riemann tensor, R0000 = −R0000, where we interchanged the ﬁrst and the third index. This equation implies that the Riemann tensor always vanishes, and hence we do not have an Einstein-Hilbert term in our action. However we do have the determinant of the metric in the action, which again acts as a Lagrange multiplier. This leads us to the same conclusion as in two dimension, except for an extra factor of two multiplying the right hand side of equation (2.22)

δL1D L1D = 2gµν mat . (2.26) mat δgµν This factor two is crucial. A one-dimensional scalar action coupled to gravity is given by √ 1 S = dx −g − G (φ)∂ φi(x)∂µφj(x) − V (φ(x)) , (2.27) ˆ 2 ij µ from which we see that the kinetic term does not satisfy the above constraint, equation (2.26), off-shell. Therefore this theory admits a scalar potential.

2.2 Scalar Lagrangian Densities

In this section we highlight some of the properties of scalar fields coupled to gravity. In this thesis we are interested in the vacuum structure of our obtained models, which is something that is explained in chapter three. The models we construct will have scalar fields in the action, these fields control the vacuum structure of our models. To illustrate how the scalar fields control the vacuum structure, we take a general interacting scalar theory coupled to gravity:

√ 1 1 S = dDx −g R − G (φ)∂ φi(x)∂µφj(x) − V (φ(x)) . (2.28) ˆ 2κ2 2 ij µ

Here Gij(φ) is a n×n symmetric matrix with signature (p, q). For notational simplicity the dependence of Gij(φ) on the fields is not write anymore . We investigate when this theory has one, or more vacua. These vacua are solutions of the theory which are local minima of the energy. We take Gij to be i µ j positive definite. Then due to the positive definiteness Gij∂µφ (x)∂ φ (x) ≥ 0 and identically equal to zero if and only if ∂µφ (x) = 0 for all x and µ. This means that in a

9 2.2. Scalar Lagrangian Densities local minimum, the kinetic energy terms have to vanish, hence a vacuum is a steady state. We also need to minimise the potential V (φ(x)). This is done by looking at a ∗ ∗ ∂ ∗ solution φ (x) of the equations ∂i(V )(φ (x)) = 0, where ∂i = ∂φi and ∂µφ (x) = 0. This condition gives the extrema of the potential and not the minima. Hence we need ∗ to check that the eigenvalues of the matrix ∂i∂j(V )(φ (x)) are all positive. If this is the case, φ∗ is a vacuum of our theory, and in a quantum treatment it is around these solutions that perturbation theory is applied. The lowest energy solution is called the true-vacuum. In this following short calculation we prove that vacua really solve the equations of motion. To this end we perform a variation with respect to φm of the action (2.28), and we change the partial derivatives on the scalars in covariant ones. This is always allowed because on scalars, covariant- and partial derivatives are the same thing. Performing this variation results in 1 G φi − ∂ G ∇ φi(x)∇µφi(x) = −∂ V (φ(x)), (2.29) im 2 m ij µ m where we defined φ = ∇µ∇ φ, which is called the Laplace-Beltrami operator or a µ √ √1 µν box. Because a box is also equal to φ = −g ∂µ ( −gg ∂νφ(x)), we easily see that a vacuum solves the equations of motion. The vacuum solution will act as a cosmological constant, because the Einstein equations are 2 ∗ Gµν = −κ gµνV (φ ). (2.30) In appendix A.3 we show that we can substitute the vacuum solution into the action and derive the equations with that reduced action. This is something which can not be done in general, because doing so can yield incorrect equations of motion. Another interesting fact is that if Gij is positive definite, the scalar fields form a i Riemannian manifold Ms, called the target manifold. This is because the φ can be seen as coordinates in Rn, thus the possible values of the fields give a set in Rn described by the coordinate functions φi. The target manifold is Riemannian because the matrix Gij can be used as a metric on the manifold. The conditions an object has to satisfy to be a metric are that the object is a symmetric bilinear positive definite map and transforms as a (0,2) tensor. The matrix Gij is by definition a symmetric bilinear positive definite map. The fact that it transforms as a tensor is proven as follows. First the combi- i µ j nation (Gij(φ)∂µφ ∂ φ )(x) is a scalar and therefore it is invariant under coordinate transformations, which we will denote by φe(φ). Therefore Gij(φ) transforms as

k l ! k µ l ∂φe i ∂φe µ j Gekl(φe)∂µφe ∂ φe (x) = Gekl(φe) ∂µφ ∂ φ (x), ∂φi ∂φj

k l ! ! ∂φe ∂φe i µ j = Gekl(φe) ∂µφ ∂ φ (x), ∂φi ∂φj i µ j = (Gij(φ)∂µφ ∂ φ )(x). (2.31)

10 2.3. p-Form electrodynamics & Form notation

∂φek ∂φel Here the last equality holds if and only if Gij(φ) = Gekl(φe) ∂φi ∂φj . This is exactly the transformation formula for a (0,2) tensor, which shows that Gij is a metric. The above is useful because we want to calculate the masses of the fields. For scalars these are defined by the second order terms in the fields, which can be seen by inspecting the Klein-Gordon equation. It is clear that the action given by (2.28), will not give rise to n-decoupled Klein-Gordon equations. To obtain the masses of the different scalar fields we need to rewrite it. Because Gij(φ) is a Riemannian metric, we can find a field redefinition such that ∗ ∗ at one point Gij(φ ) becomes the unit matrix δij(φe ). We denote the fields in the new coordinate system as φe. In these new coordinates it is not guaranteed that the ∗ Hessian matrix of the potential, ∂i∂j(V )(φ (x)), is diagonal, meaning that we can not just read the masses off. ∗ We investigate what the transformation properties of ∂i∂j(V )(φ (x)) are. Due to the second partial derivative this matrix is not a (0,2) tensor. For vacuum solutions we have ∗ ∗ k ∗ ∗ ∇i∂jV (φe ) = ∂i∂jV (φe ) − Γij∂kV (φe ) = ∂i∂jV (φe ), (2.32) ∗ were we used that ∂i(V )(φ (x)) = 0. This means that evaluated in the vacuum ∗ solutions ∂i∂jV (φe ) is a (0,2) tensor. Because this tensor is a symmetric bilinear form, we can diagonalise it using an orthogonal transformation. This will change the ∗ ∗ ∗ ∗ scalar fields φei in φbi and we obtain a diagonal ∂i∂jV (φbi ). Because the metric δij(φe ) is invariant under orthogonal transformations it does not transform. This means that we obtained n-decoupled Klein-Gordon equations. From these the masses of the ∗ 2 scalars φbi can be read off, these are mi /2 = di, with the di the eigenvalues of the Hessian. The one-dimensional models found in this thesis have a metric Gij which has signature (n − 1, 1). This means that there is one scalar field which has a wrong sign in its kinetic term. This implies that it can loose energy by augmenting the quantity ∂µφ, which renders the theory unstable. In 1D this is not a problem because we have that the total energy has to be equal to zero. Therefore the derivative of the scalar field is bounded by the value of the potential. There is even an advantage to it. If there is no potential, or it is equal to zero, dynamics are still allowed. In this case were the metric has signature (n−1, 1), we can use a field redefinition to transform Gij into ηij. To keep this invariant we need to diagonalise the Hessian using an SO(1, n − 1) transformation. However this is not always possible.

2.3 p-Form electrodynamics & Form notation

The classical theory of electromagnetism is given by Maxwell’s equations. These equations unify the various classical electric and magnetic phenomena observed in nature. Using the notation of special relativity we can rewrite the Maxwell equations

11 2.3. p-Form electrodynamics & Form notation as:

Fµν = ∂µAν − ∂νAµ, (2.33)

∂σFµν = 0, (2.34) µ ∂ Fµν = Jν. (2.35)

ν Here Aµ is the electromagnetic potential, Fµν the ﬁeld strength and J the current. Remark that the object Fµν is an antisymmetric tensor with all its indices down. These objects have a special name, they are called differential forms. Because Fµν has two indices it is called a two-form. The conventions used to manipulate differential forms and a short introduction of what these are, can be found in A.4 Using these differential forms we can rewrite the Maxwell equations as

F = dA, (2.36) dF = 0, (2.37) d ?F = ?J, (2.38) where A is the electromagnetic potential, J the current, both one-forms, and F the ﬁeld strength, which is a two-form. These equations can be generalised to forms with more indices as

F (p) = dB(p−1), (2.39) dF (p) = 0, (2.40) d ?F (p) = ?J (D−p+1), (2.41) where B is the higher-dimensional electromagnetic potential and F and J are just the same as in the previous case but again with more indices. Remark that the gauge transformation B(p−1) → B(p−1) + dC(p−2) leaves the field strength invariant [3]. This is a generalisation of the gauge symmetry of electrodynamics. We investigate which kind of objects could interact with these form-fields. This done by looking at standard electromagnetism. The following explanation is highly based on section 1.3 of the lecture notes on Supergravity solitons by Matthias Blau [5]. The fact that particles are charged “electrically” is reflected in their coupling to the Maxwell gauge field A, in the following way

µ 4 µ dx (τ) d xJE(x)Aµ(x) = q dτ Aµ(x(τ)) ≡ A. (2.42) ˆ ˆ dτ ˆγ

This formula was taken from [5] but slightly modified. In this formula q denotes the charge of the particle and γ is the one-dimensional world-line of the particle. The world-line is one-dimensional because the particle is zero-dimensional, it has no spatial extension. This means that an n-dimensional object will have an n + 1- dimensional world-surface. Therefore if we want to define a coupling we need to integrate over that n + 1-dimensional surface, meaning we need a vector potential with n + 1 indices. This potential will have a field strength with n + 2 indices. From

12 2.3. p-Form electrodynamics & Form notation this we conclude that p-dimensional objects couple “electrically” to an F p+2 ﬁeld strength. From now on we do not follow [5] anymore. Field strengths have their own dynamics, which can be obtained from an action principle. This action is given by: p 1 1 µ1...µp S := − dxD |g | F (p) F , (2.43) F ˆ D 2 p! µ1...µp (p) which is a generalisation of the action for ﬁeld strength of electromagnetism. Using differential forms we can rewrite it as

1 (p) (p) S (p) = − ? F ∧ F . (2.44) F 2 ˆ D

(p) p(p−1) (D−p) Because ?DF = (−1) 2 F , the action can be transformed to

1 (D−p) (D−p) S (p) = − F ∧ ? F . (2.45) F 2 ˆ D

Using (A.19) to change the differential forms of place we obtain

(D−p)p 1 (D−p) (D−p) (D−p)p S (p) = −(−1) ? F ∧ F = (−1) S (D−p) . (2.46) F 2 ˆ D F

From this we see that the action of F (p) equals the action of F (D−p), up to a possible sign difference. However the overall sign of the action does not matter for the physics, therefore we say that there is a duality between both type of ﬁeld strengths.

13 Chapter 3

The physics of extra dimensions

In this chapter we give an overview of the role that extra dimensions play in physics. We start this chapter with an introduction to Kaluza-Klein theories, then go over to more involved compactifications, such as Scherk-Schwarz-, GKP- and KKLT- compactifications. Afterwards the concept of scale separation is introduced and we relate this to the cosmological problem. To finish this chapter we motivate one- dimensional vacua, and explain how they need to be interpreted. Before introducing different models using extra dimensions, we first argue that using extra dimensions in a theory is not some obscure theoretical construction. When Einstein formulated his theory of general relativity, he de facto added an extra dimension to the worldview which governed the minds at that time. Before this revolution, the Gallilean view dominated. This consisted of three spatial dimensions, in which every object moves, and time, which is something which runs at the same pace everywhere and for everybody. Einstein combined these two in the concept of spacetime. He showed that one can move in time at different speeds, and in special cases even go backwards in time using closed timelike curves. Einstein discovered that we do not live in a three-dimensional, but in a four-dimensional world. A world where one of the dimensions, time, behaves differently from the three others.

3.1 Kaluza-Klein Dimensional Reduction

In this section we introduce the concept of Kaluza-Klein dimensional reduction. This is one of the basic tools used in this thesis. In 1918 Hermann Weyl discovered that gauging the rescaling symmetry of the metric has as consequence that the metric is not enough to describe the geometry of spacetime. An extra one-form is needed, but this one-form is only defined up to a gradient of an arbitrary function. Hence the real invariant quantity needed, in addition to the metric, is a two-form, which can be interpreted as the field strength tensor of electromagnetism [6]. This modification of G.R. is called conformal gravity and will not be treated in this thesis. Note that this was the first true appearance of the gauging of a symmetry, which is the main concept on which the Standard Model of particle physics is build. The appearance of that one-form led Kaluza in 1921 to propose another less technical approach to unify the gravitational and electric phenomena, namely a fifth dimension [7]. In 1926, Klein analysed this fifth dimension from a quantum-mechanical point of view in [8]. The unification of gravity and other forces is achieved by starting with a higher dimensional theory and stating that observers only observe an effective theory in a

14 3.1. Kaluza-Klein Dimensional Reduction lower-dimensional spacetime. Because we do not observe extra-dimensions this is a critical assumption. There are different methods to hide these extra dimensions to lower-dimensional observers, which are explained below.

3.1.1 Reduction of the metric

We explain the way unification is achieved by using an example. Take gravity defined 1 on a five dimensional space: M5 = M4 ×S , which is the product of four dimensional Minkowski space with a circle. The metric on this spacetime is given by: √ √ 2 φ(x)/ 3 2 −2φ(x)/ 3 µ µ dsb5 = e ds4 + e (dθ ⊗ Aµdx ) (dθ ⊗ Aµdx ) , (3.1) 2 where ds4 is the metric on Minkowski space and θ is the coordinate on the sphere. We can derive how five-dimensional gravity would be percieved for an observe which can not see the fifth dimension. This is done by substituting this metric into the five dimensional Einstein-Hilbert action, and perform some calculations showed in the next chapter. The result of this calculation yields 1 1 √ L = ?R − ? dφ ∧ φ − e− 3φ ? dA ∧ dA. (3.2) 4 2 4 √ φ(x)/ 3 2 What happened here? The four-dimensional part of the metric, e ds4 gave rise to the term√ R4, which describes the gravitational√ interaction. The mixed components, −2φ(x)/ 3 µ 1 − 3φ e Aµdx ⊗ dθ, gave rise to 2 e ? dA ∧ dA. This is a vector potential with a U(1) gauge symmetry, coupled to φ. The metric part e2βφdθ ⊗ dθ gave the term 1 2 ? dφ ∧ φ, which is the kinetic term of a massless scalar field. From this we see that five-dimensional gravity with one compact dimension looks to a four dimensional observer as gravity, but with the addition of electromagnetism and an extra long-range scalar force. The following is based on the references [9, 10, 11]. We saw what the effect of one extra dimension was, but now we investigate the effect of adding multiple dimensions. To explain this we start with a spacetime manifold which topologically is n n 1 1 1 a product manifold MD+n = MD × T , where T = S × S ... × S is the n-torus. In this spacetime, MD is the external space describing the non-compact dimensions observable by the lower-dimensional observers, and T n is the internal space which are the compact, extra dimensions. The metric on this manifold will be equal to, gµν gµy gµν = . (3.3) bbb gxν gxy

We use the convention that hatted indices run over all the D + n dimensions, the Greek indices run over the D dimensions of the non-compact space, and the Latin indices describe extra dimensions. Because g is a metric, it is symmetric, which implies that g andg are bµbνb µν xy

15 3.1. Kaluza-Klein Dimensional Reduction symmetric too. We also have that g = g . Therefore the metric g can be µx xµ bµbνb split into three different parts. The first part is when the indices µ,b νb = µ, ν, the second when µ,b νb = µ, y and at last µ,b νb = x, y. These three parts are in the case 1 2 µ of M5 = M4 × S respectively: ds4, Aµdx ⊗ dθ , and dθ ⊗ dθ. Transforming the external coordinates has the effect that the different parts of the metric transform respectively as: a (0,2) tensor, a 1-form and a scalar field. This means that for a D dimensional observer these different parts are: a metric (gravity), a vector- potential (electromagnetism), and a scalar field (extra force) [9]. Hence, a D + n dimensional gravitational theory where the extra dimension are form n-torus, will look to D-dimensional observers as gravity combined with n additional vector fields, and n(n + 1)/2 scalar fields. These scalar fields are called moduli. The moduli have a physical interpretation. The metric depends on the shape and the size of the manifold. The scalar fields obtained in the lower-dimensional theory come from the part of the metric on the internal manifold, therefore they describe the shape and size of it. To illustrate this look again at the example where the internal√ manifold was a circle. In that case the component g55 was equal to −2φ(x)/ 3 2 e . Because the distance on√ a circle in polar coordinates is given by rdθ , with r the radius, we see that e−2φ(x)/ 3 describes the radius of the circle. Remark that the radius can vary from place to place on the external manifold, because φ(x) is a function over it. Because of this interpretation we will make the distinction between two type of moduli: volume- and shape moduli. The volume moduli describe evolutions which change the volume of the internal space, an example of which is the blowing-up of the S1. Shape moduli describe the evolution of the internal space which do not change the volume, for example the twisting of a torus. In the reduction of five dimensional gravity there appeared a massless scalar in the reduced theory. In the case of the torus reduction, there were even n(n + 1)/2 massless scalar fields. There are several issues to this, one of which is that for every massless boson there is an associated long-range force. The force associated to the scalars will alter the gravitational force, which is something that would have been observed. This is a generic feature of these theories and this is something which needs to be resolved. This problem is cured by adding extra ingredients, such as background fluxes, to construct a potential for these moduli and stabilise them. This makes them massive, which solves the problem stated above.

3.1.2 Reduction of the other ﬁelds

We discussed what happened if you reduce the metric, but what about other fields in the higher dimensional theory? We illustrate what happens to these by taking a 1 massless scalar field in the five dimensional spacetime: M4 × S . This scalar field has as equation of motion the 5-dimensional Klein-Gordon equation,

µ − ∂µ∂ φ(x, z) + ∂z∂zφ(x, z) = 0, (3.4)

16 3.1. Kaluza-Klein Dimensional Reduction where the z-coordinate is the coordinate on the circle. Because S1 is a compact P k 2πikz/R space, φ(x, z) can be Fourier decomposed as φ(x, z) = k φe (x)e , where R is the radius of the circle. This means that the Fourier modes with k > 0 are associated to ﬂuctuations in the compact dimension. Substituting this into the Klein-Gordon equation results in

2 2 X µ k 4π k k 2πikz/R −∂µ∂ φe (x) − φe (x) e = 0. (3.5) R2 k This sum is only equal to zero if all the terms vanish independently, which means that for every Fourier mode φek(x) we have

2 2 µ k 4π k k − ∂µ∂ φe (x) = φe (x). (3.6) R2 We see that from a 4D point of view, the five-dimensional scalar field is seen as an infinite amount of scalar fields φek(x), each with mass going like ∼ k/R. In the beginning of this section we mentioned that the extra dimensions only need to be observable at very high energies not probed yet. This means that the radius R needs to be very small, and as a result the fields for which k 6= 0 become very heavy. Therefore it takes a lot of energy to excite these fields, meaning that in the lower-dimensional effective field theory only the massless term is important. Therefore we only keep the massless term of the Fourier sum, which has the effect of eliminating any dependence on the internal coordinates. This approximation is called truncation. From this we learn the first method to make the extra dimensions practically unobservable, is by taking them very small. This Fourier decomposition can be applied to any field in the higher dimensional theory. Therefore in the lower-dimensional case we expect to have higher modes of the metric carrying a dependence on the internal coordinates. In our example this was not the case because we performed a truncation, silently assuming that the extra dimensions were small enough, which is an assumption we will always make. There is an important remark about truncation which needs to made. If we want to elevate the solutions of the effective theory to higher dimensions, we need to check that after truncation they still solve the higher dimensional equations of motion. If this is the case, we call the truncation consistent. The text that follows is highly based on [11]. Consistency splits into two parts. The first part is consistency of the different Ansatze¨ we take for the massless modes of the fields. The second is consistency of the truncation of the massive modes of the higher-dimensional fields. To illustrate the first part, we look again at the case√ were the internal manifold is just a circle. Compactifying this theory with e−2φ(x)/ 3 yields as equation of motion √ µ − 3φ ∂µ∂ φ(x) = Ce ? dA ∧ dA, (3.7) where C is a constant. We see that fluctuations in φ(x) are sourced by the vector potential, therefore we can not set φ to a constant, because it would not solve the

17 3.2. Modern compactifications equation of motion. Would we have taken φ as constant in the metric Ansatz (3.1) and perform the reduction, we would not have noticed this problem [9]. Because the Ansatz with φ(x) a function is more general and thus correct, taking φ as a constant in the Ansatz is not consistent. Consistency of the truncation of the massive states is different. Consider the λ λi λi eigenfunctions Pi (y) of the Laplace operator, P = ΛiP , of a general compact manifold. Using this operator we can decompose a general scalar field on that manifold as X λi φb(x, y) = φλi (x)P (y). (3.8) i As in the case for the five dimensional scalar, there is a massless sector for which λ0 = 0. Substituting the above expression in the higher dimensional equation of motion we obtain

jφλi = Fλi , (3.9) where j refers to the Laplace operator in the lower j-dimensional theory and Fλi is a function which can depend on both the massive and massless sector. If the function Fλi vanishes when all massive φλi are constant, the truncation is consistent because the above equation is solved. Hence, we can truncate the massive sector if the massless fields do not source a massive field. This reduces the amount of compact manifolds on which we can consistently perform a dimensional reduction. Note however that it is not because we can not perform a consistent truncation, that the model is useless. The massive modes are so massive that their interaction with the massless modes is expected to be small, hence leaving out their interaction does not have to be a bad approximation. Because we are interested in low-energy effective actions, not in elevating our solutions to higher dimensions we can still use the truncated models, if the internal space is small enough [11]. How do we perform a dimensional reduction? We take all the higher-dimensional fields, decompose them via such a Fourier expansion and use this expansion into the higher dimensional equations of motion. If the internal manifold is compact the Fourier expansion will be a discrete sum. This has as a consequence that there is a mass-gap between the massless states and the massive states. If we perform a truncation to the massless sector we are performing a compactification. The truncated equations of motion are then reduced to obtain a lower-dimensional effective theory. We will mostly take a short-cut to this approach, but the different methods to find this effective theory are clearly illustrated in the next chapter.

3.2 Modern compactiﬁcations

In this section we discuss more involved compactifications. We introduce compactifications which generate non-Abelian vectors, and then introduce warped compactifications. We end this section with a discussion of the GKP- and KKLT- compactifications.

18 3.2. Modern compactiﬁcations

Figure 3.1: Schematic representation of the splitting of the higher dimensional symmetry group.

Reductions on a torus only generate vectors with a U(1) symmetry, but we know that the Standard Model of particle physics consists of the gauge groups SU(3) × SU(2) × U(1). If we want to unify the different interactions using compactifications, we need to investigate compactifications of other internal spaces besides the torus. An example of an alternative to the torus is a Lie-group G, which is an example of group manifold. Performing a compactification of gravity on a manifold Md × G can generate non-Abelain vectors. Because the higher dimensional theory is coupled to gravity, it is invariant under coordinate transformations depending on all the coordinates. If we reduce this theory on a group manifold, this diffeomorphic invariance will reduce to invariance under coordinate transformation involving only the external coordinates. However due to the reduction the lower-dimensional theory receives an extra symmetry, the symmetry of the group G. This is schematically shown in Figure 3.1. The vectors generated by this reduction will transform in the adjoint representation of the group G, thus we get a gauge theory of the group G in the lower dimensional space. Now we are still left with the problem that the scalars are massless. In these models this can be cured taking as metric Ansatz

2 2αφ 2 2βφ i i j j dsD+n = e dsD + e Mij(x)(e + A ) ⊗ (e + A ), (3.10)

i i a where e = ea(y)dy and they carry a dependence on the internal coordinates. The matrix Mij is a symmetric bilineair form having dependence on the coordinates of the external space. The ei will need to satisfy certain conditions, such that the dependence on the internal coordinates drops out of the reduced action. This Ansatz will yield a potential for the scalar fields, which we can use to stabilise these. This was first done by Scherk-Schwarz in [12]. These compactifications are referred to as group-manifold reductions, in the fifth chapter we are going to calculate some of these models and explain this in more detail. The next type of models we introduce are the RS-models [13, 14]. In these models the gauge-forces are confined to a four-dimensional membrane, but gravity acts in the fifth dimension. Due to the confined of the gauge-forces we are not able to see the presence of a fifth dimension. In their first paper they use two four-dimensional membranes which specify the boundary of a finite fifth dimension [13], in the second they send one of these to

19 3.2. Modern compactifications infinity, which makes the size of the fifth dimension infinite. The Standard Model is then confined to one of these membranes. These membranes, which are topological defects, break translational invariance in the extra dimension. Due to this we obtain the following type of metric

2A(φ) µ ν 2 dsb5 = e ηµνdx dx + rcdφ , (3.11) upon solving the Einstein equations [13]. Here the x coordinates denote the external space, φ is the coordinate on the fifth dimension and rc is just a constant describing the size of that dimension. The term e2A(φ) in front of the metric on the external space, is a function depending on the internal coordinate and is called the warp factor. Would we be able to move in the internal space, we would each time see a different external space, because the warp factor changes the normalisation of the metric on the external space. Compactifications with this type of metric are called warped compactifications.

3.2.1 String compactiﬁcations

In string and M-theory there exist extended objects called D-branes, on which there can exist a gauge field which is invariant under U(1)-transformations, and which is confined to this object. It turns out that stacking N different branes into the same place, will result in a gauge field which is invariant under the action of the group U(N) [15]. These objects are what we are going to use in the following compactifications. Cosmological models from superstring theory have been obtained by performing compactifications of ten-dimensional type II supergravity. Type II supergravity is the low-energy limit of type II superstring theory, but with low-energy we mean energies which are much larger than those probed at the LHC. Some of these models have been obtained by compactifying six of the dimensions on a compact Calabi-Yau (CY) space. This is a six dimensional Ricci-flat complex manifold, and is used because it conserves some of the supersymmetry of the higher dimensional theory. We are interested in the GKP-, [16], and the KKLT-compactification [1]. These are warped compactification, which use the above mentioned ingredients. Constructing warped compactifications in 10D supergravity is not a trivial thing to do. To cite the authors of [16]: “In pure supergravity, the integrated field equations rule out warped compactifications under broad conditions”, which has been proven in [17, 18]. Their no-go theorems can be evaded using localised negative energy sources called orientifold planes. Using these, in combination with branes and fluxes on the CY, GKP succeeded in finding a warped compactification and to stabilise the so-called complex structure moduli. These are the moduli which govern deformations of the CY keeping the volume constant. However what they do not achieve is fixing the Kaehler¨ moduli, which can roughly be seen as metric deformations changing the volume. This means that they arrive at a model where the overall volume of the CY is not fixed nor specified, leading to the same problems as those of Kaluza and

20 3.2. Modern compactiﬁcations

Klein. This problem is addressed in the KKLT-paper [1]. The scalar potential of GKP was obtained using only classical supergravity, extended objects and fluxes. In the KKLT-paper they add a quantum-non perturbative effect, yielding the potential shown in Figure 3.2 below. It is clear that this potential stabilises the Kaehler¨ modulus, and the resulting vacuum is a N = 1 supersymmetric AdS4 vacuum. It is an AdS vacuum because the field is stabilised at negative values of the potential. After obtaining this vacuum extra D-branes are added to the model. This would normally cause the overall volume to run to infinity, however it is argued that the combination of sufficient warping and the quantum corrections used, prevents this [1]. It is argued that the additional branes break the last supersymmetry and uplift the potential to obtain a dS-vacuum. This is the potential shown in the introduction, Figure 1.1. This last part of the argument is not universally accepted, and has its disbelievers. [19] The value of the cosmological constant obtained by KKLT depends on different parameters, such as the strength of the flux in the compact dimensions. This means that in this model the cosmological constant can be discretely tuned by changing these parameters [1]. It is this freedom that inspired the idea of the multiverse of string theory. Because there are different equivalent configurations that create stable vacua, they could all be realised in nature. This would mean that there does not exist one universe, but that there is a whole zoo of them. A very rough counting argument results in ∼ 10500 [20] possible universes. This allows anthropological reasoning to solve the CC-problem. The argument can be found in [21] and goes roughly as follows. Would the cosmological constant be bigger than it is, there would be no structure formation such as galaxies. Would it be negative, the universe would collapse in a finite amount of time, therefore it can not be to negative. Hence, the cosmological constant has the value we observe, because would it be much different we would not exist. If there really are ∼ 10500 universes, there is probably at least one with a cosmological constant suited for the

Figure 3.2: Figure taken from [1]. This is the KKLT-potential before the uplifting. The sigma is the Kaehler¨ modulus and V the potential. The values on the axis are not of interest to the discussion, except that the minimum is negative.

21 3.3. Scale separation and the CC-problem appearance of life, therefore this anthropic selection is possible.

3.3 Scale separation and the CC-problem

This section is based on [22]. In any compactification we want that the extra dimensions are almost unobservable. In this thesis we are studying flux compactifications, which in this context means that the masses of the KK-modes have to be out of reach from current experiments. In the second chapter we defined the length scale of an (A)dS-universe as

r 1 L ∼ , (3.12) Λ which means that a large length scale implies a small value for the cosmological constant. Thus a large value for the CC means a small length scale. Therefore if the size of all the dimensions is comparable and small, we expect that there is a large CC. 1 To every length scale we can associate a “mass”, which are related as L ∼ M . This mass reflects the energy needed to probe the associated length scale. Using this we can associate a mass to the cosmological constant, and define separation of scale in the following way: M Λ 1. (3.13) MKK From this we see that a Minkowski vacuum (Λ = 0), is always scale separated. We use this definition because if the masses of the KK-modes and the CC are comparable, their length scales are too. This definition is related to the following version of the cosmological constant problem: MΛ MP lanck. This way of stating the CC-problem comes from quantum field theory. In this framework one can calculate Feynman diagrams with no external legs. These bubble diagrams are interpreted as vacuum energy, and are therefore related to MΛ. The value of MΛ is related to the cut-off used in this calculation, which is usually taken as the Planck mass. The fact that the cut-off is around the Planck-mass means MΛ is also expected to be around that scale. However it is much much lower. The cut-off is taken at the Planck scale, because it is believed that there quantum gravity effects become important. Hence we expect new physics at those energies. This means that beyond that point, our computation can not be trusted anymore. Therefore we can weaken the above inequality to MΛ MNewP hysics. If we perform a compactification, we know that when we arrive at energies comparable to MKK , new physics enter. This means that we can replace the above inequality by M Λ 1, (3.14) MKK

22 3.4. One dimensional compactiﬁcations from which we infer that every string theory vacuum, or more generally any theory with extra dimensions has a cosmological constant problem. This is rather easy to see in our version of the cosmological constant problem. We want to ﬁnd a reason why certain dimensions are large in extend and others unobservable. cal Another problem with the CC is that MΛ is highly depended on the UV-behaviour of the theory. Small changes in the UV-regime will result in completely different values for the cosmological constant. This is an issue, because to calculate this quantity, we need the full UV-behaviour. This is a situation which normally does not appear in nature. We are able to do chemistry without the need of the Standard Model. However for a quantity which is only important at cosmological scales, we need Planckian physics to understand it.

3.4 One dimensional compactiﬁcations

The motivation of this thesis is the following naturalness argument. It is an unnatural situation that in a theory with extra dimensions, some are visible and some are not. Why is there such a separation of scale? Why are not all spatial dimensions treated on equal footing? This statement is not exactly true, but superstring theory needs ten dimensions to be consistent. Why then are we not living in the most simple solution, ten-dimensional Minkowksi space? Or if we want to compactify dimensions, the most natural thing to do would be to compactify them all. Why then are we living in a world with three non-compact spatial dimensions and six compact? To answer this question we want to find a compactification to a natural situation, namely all spatial dimensions are compact, but such that some of moduli can tunnel, as in the case in the KKLT-scenario. How do we interpret these one dimensional compactifications? The compactifications to one-dimension yield an action of the form p 1 S = dt |g | G (φ)∂ φi(t)∂tφj(t) − V (φi(t)) , (3.15) ˆ 1 2 ij t but with the extra constraint that the total energy has to vanish. Moreover the scalar fields only are time dependent. We can interpret this action in two different ways. The first way of interpreting this is the same as we did with the other compactifications. We say that the lower dimensional action is what an observer, confined to these lower dimensions, experiences. This means that an observer sits at a fixed point in space, measuring the evolution in time of the different fields, and that is it. A one-dimensional vacuum then means that the extra dimensions are stabilised and that a one-dimensional observer will classically only see constant fields. Remark that we could add fermionic matter, in the form of Grassmann numbers, to this system, however gauge fields or gravity can not be added because they have too many degrees of freedom. The second interpretation is the following. The moduli describe the size and shape of the compact space, the location of the branes, etc. Their evolution is

23 3.4. One dimensional compactifications dictated by their equations of motion, which are derived from the action given above. We interpret their dynamics as the evolution of the compact dimensions and what they contain. This evolution can be seen as a one-dimensional line into the space of all possible configurations of the system. The fact that the scalar fields only dependent on time is an approximation. When the dimensions are very small, the KK-masses are very high. We interpret this as spacetime becoming so stiff at these scales, that creating inhomogeneities not proper to the system costs an enormous amount of energy, as a result they do not appear. To make this statement mathematically rigorous, consider a small change to the configuration of a compact space with fluxes and branes will result in a change in the action, S → S + δS. If the integration S+δS volume is very large, the normalised change S will be negligible. If the space is very small this ratio will be much bigger and it is therefore less likely to happen. The same approximation in large-scale cosmology is made, but for different reasons. The reason there is because they investigate the phenomena at such a large scale that local perturbations are unimportant. This has as effect that from our compactifications, we obtain the same equations as in cosmology. The tunneling effect in this interpretation is a discontinuous jump of the line through configuration space. For this phenomenon we do not have an analogue in classical cosmology.

24 Chapter 4

Simple gravitational reductions

In this chapter we introduce the different ingredients used in the flux compactifications performed in this thesis. This is done by studying some simple models. Starting with the reduction of five-dimensional gravity on a circle. Afterwards we look at higher dimensional compactifications, and introduce fluxes. To end this chapter we compactify on curved manifolds and make contact with the FLRW-model in cosmology.

4.1 5D gravity on a circle

We start from pure gravity on a 5D manifold M5 = M4 × S1, and compactify this to a 4D effective theory. We therefore take the following Ansatz for the 5D metric:

2 2αφ(x) 2 2βφ(x) dsb5 = e ds4 + e (dθ ⊗ A) (dθ ⊗ A) . (4.1) µ Here we deﬁned A = Aµdx , which is the Kaluza-Klein vector introduced in section 3.1, and α & β are constants. φ is a function such that e2βφ(x) represents the square of the radius of the S1 circle in each 4D spacetime point. From now on, if there is a function denoted as f(x), it means that the function depends on the four dimensional spacetime coordinates: xµ with µ = 0, .., 3. Notice that the fact that φ only depends on the 4D spacetime coordinates is an approximation. This is justiﬁed as long as we are not working with energies comparable to the inverse size of the compact dimension, as was explained in section 3.1. To simplify the calculation we truncate the KK-vector (A = 0) which we is allowed because it is consistent with its equation of motion after reduction. The equation can be found in [9]. We perform this truncation is because we are interested in vacua. The presence of these depends on the effective potential we obtain after reduction. The vectors do not contribute to the potential, we are not interested in them. After performing the trucation we rewrite our Ansatz in a more useful way

2 2βφ(x) 2 2 dsb5 = e dse4 + dθ , (4.2)

2 2(α−β)φ(x) 2 where in the last line we deﬁned dse4 = e ds4. The rewriting of this metric into a conformal frame is a trick used throughout this thesis. The action of our model is:

4 µ p S = d x dθ |g5|Rc5, (4.3) ˆ b

25 4.1. 5D gravity on a circle where gb5 and Rc5 are the determinant of the 5D metric and the 5D Ricci scalar defined by (4.1). We want to get a four dimensional effective theory. Therefore we rewrite ev- 2 erything in terms of the metric g, defined by the line element ds4, and integrate out the θ-coordinate. The Ricci scalar Rb5 is defined in terms of gb, but we need an expression in terms of g. The rewriting of this term is a straightforward calculation shown in the appendix B.1. The calculation consists of using twice the formula for a Ricci scalar defined by a conformally rescaled metric, which can be found in [11]. Another important aspect of the calculation is proving that the components in the θ-direction of the Ricci tensor defined with respect to the metric ge are all vanishing. In the appendix we derive the following two formulas:

−2βφ(x) −2(α−β)φ(x) 2 2 2 Rb5 = e Re5 + e 4β (∂φ) − 8βφ(x) − 16αβ (∂φ(x)) , (4.4) and

−2(α−β)φ(x) 2 2 Re5 = e R4 − 6 (α − β) (∂φ) − 6 (α − β) φ . (4.5)

Using these two equations in the ﬁve-dimensional action results in

S = d4xµdθp|g |e(2α+β)φ(x) R + −6α2 − 4αβ − 2β2 (∂φ)2 − (6α + 2β) φ . ˆ 4 4 (4.6) Remark that we not only used the formulas for the different Ricci scalars, but we p also rewrote the factor |gb5| using that the metric is block-diagonal. Choosing the following values for our parameters in the metric 1 β = −2α and α2 = , (4.7) 12 we obtain as ﬁnal result p 1 2 S = 2π d4xµ |g | R − (∂φ) . (4.8) ˆ 4 4 2

By our choice for β we obtained the usual Einstein-Hilbert term in four dimensions. Because of this the term φ became a total derivative ,which we omitted from the action. The choice of α resulted in a canonical normalisation of the scalar, and the factor 2π came from the integral over the θ-direction. The compactification of five-dimensional gravity over a circle resulted in four- dimensional gravity but with the addition of a massless scalar field. Keeping the KK-vector in the Ansatz, results in a vector potential in the action. This potential is coupled to the scalar field. If the scalar field is taken as a constant this vector potential gives rise to the electromagnetic force described by Maxwell’s equations. This was the idea of Kaluza.

26 4.2. Compactiﬁcation over an n-torus

4.2 Compactiﬁcation over an n-torus

In this model we generalise our compactiﬁcation to the more general setting MD+n = n n 1 1 1 MD × T , where T = S × S · · · × S . This product consist of n-circles. This compactiﬁcation is valid as long as D > 2. To begin we take the following Ansatz:

2 2αφ(x) 2 2βφ(x) n m dsbD+n = e dsD + e Mmndy dy , (4.9) which was copied from [11], and where with the KK-vectors are truncated away. Here the y-coordinates are the coordinates on the torus. The matrix Mmn is a n-dimensional symmetric matrix with determinant equal to one. Note also that g is the D + n-dimensional metric, the external part of the metric is g , and bµbνb bµν −2αφ(x) gµν = e gbµν. The Ricci scalar RbD+n is obtained after a straightforward calculation highlighted in this section, but shown in detail in appendix B.2. In this calculation we reuse the trick to go in conformal frame and obtain an expression for the Ricci scalar RbD+n

−2βφ(x) 2 2 −2(α−β)φ(x) RbD+n = e ReD+n − (D + n − 1) (D + n − 2) β (∂φ) e

−2αφ(x) 2 − βe (2(D + n) − 2) φ(x) + (D − 2) (α − β)(∂φ(x)) . (4.10)

After this we need to calculate the Christoffels with respect to the metric g . These eµbνb are used to obtain an expression for the different components of R . eµbνb During this calculation we use the following trick, which is used throughout this thesis. For any square matrix A it holds that Tr(ln(A)) = ln(det(A)), which implies ∂Tr(ln(A)) = (det(A))−1 ∂(det(A)). Because by deﬁnition det (M) = 1, this formula implies that ij ∂µTr(ln(M)) = M ∂µMji = 0. (4.11)

At a certain moment in the calculation we encounter a term eMij. The box is deﬁned with respect to ge, and the indices run over the D + n-directions. In the discussion around equation (B.28) in appendix B.2 we proved that

µ µ ν eMij = ∂µ∂e Mij + Γeµν∂e Mij, (4.12) where we see that the indices only run over the external directions and not over all the dimensions. The components of the tilted Ricci tensor are given by

D+n 1 1 kl µ lk µ Re = − eMij + M ∂µMjl∂e Mki + M ∂µMik∂e Mjl , (4.13) ij 2 4 D+n D 1 ik jl Re = Re − M ∂νMkjM ∂σMli. (4.14) µν σν 4 With the Ricci tensor components at our disposal, we calculate the Ricci scalar

ij µν ReD+n = ge Reji + ge Reνµ. (4.15)

27 4.2. Compactiﬁcation over an n-torus

iλ In the above we used that ge = 0. The ﬁrst term in this sum is ij ge Reji = 0, (4.16) which was obtained after the contraction of equation (4.13) with the inverse hatted metric, and some rewriting shown in the appendix, between (B.29) and (B.33). We could have expected this result, the torus on itself is a Ricci-ﬂat surface, meaning its Ricci scalar vanishes. The second term in the sum for the Ricci scalar is

µν 1 µ il g Reνµ = ReD + ∂ M ∂eµMli. (4.17) e 4 From this we obtain our ﬁnal result

1 µ ij ReD+n = ReD + ∂ M ∂eµMji. (4.18) 4 Substituting this expression into the action functional we obtain,

1 D µ n ip S = d x d y |gD+n|RbD+n, 2κ2 ˆ b D µ n ip (n−2)βφ(x)h 1 µ ij (2β−2α)φ(x) i = d x d y |gD|e ReD + ∂ M ∂eµMji − e C , (4.19) ˆ e 4 1 where we dropped the constant 2 because it is an overall constant factor. We p 2κ rewrote the factor |gbD+n| and we used (4.10), (4.18). We also deﬁned for notational convenience C = (D + n − 1) (D + n − 2) β2 (∂φ)2 2 + 2β ((D + n) − 1) φ(x) + (D − 2) (α − β)(∂φ(x)) . (4.20)

The Ricci scalar ReD is deﬁned with respect of the D-dimensional rescaled metric 2(α−β)φ(x) geµν = e gµν. In the previous section we calculated this already, where the result was

−2(α−β)φ(x) 2 2 ReD = e RD − (D − 1) (D − 2) (α − β) (∂φ) − (2D − 2) (α − β) φ , −2(α−β)φ(x) 0 = e RD − C (4.21) Substituting this into the action we obtain p 1 i S = dDxµdnyi |g |e(nβ−2α)φ(x) R + ∂µMij∂ M − C + C0 . (4.22) ˆ D D 4 µ ji p To obtain this we rewrote |g˜D| and raised the indices on the partials with the 0 rescaled metric ge. In the appendix B.2 we show this equation but with C and C written out (B.36). We still have the freedom to choose values for our two parameters α and β in the Ansatz. Taking D − 2 n β = − α and α2 = , (4.23) n 2(D + n − 2)(D − 2)

28 4.2. Compactiﬁcation over an n-torus results after some algebra in our ﬁnal expression for the action D µp 1 2 1 µ ij S = Ve d x |gD| RD − (∂φ) + ∂ M ∂µMji . (4.24) ˆ 2 4

To obtain this we integrated over the coordinates in the internal space, resulting in a volume factor Ve of the torus. This action represents a theory of D-dimensional gravity together with n(n + 1)/2 massless scalars, where (n − 1)(n + 2)/2 scalars are packaged into the matrix Mij. The choice for our two parameters, equation (4.23), enabeling us to have canonically normalised ﬁelds, depends only on the dimensionality of the internal and external space.However these expressions are only useful as long as D > 2. In the Ansatz for the metric the vectors were truncated away, which made our metric block-diagonal. The full Ansatz is

2 2αφ 2 2βφ n n m m dsD+n = e dsD + e Mmn (dy + A ) ⊗ (dy + A ) , (4.25)

m m µ with A = Aµ dx , which represents n-different vectors. Performing the compact- iﬁcation with this Ansatz results with the same choice for α and β in the following Lagrangian 1 1 1 L = ?R − ? dφ ∧ φ + ? dM ∧ dMmn − e2(β−α)φM ? dAn ∧ dAm. (4.26) D 2 4 mn 2 mn We did not include the vectors because we do not obtain a potential for the vectors.

2 4.2.1 M6 = M4 × T compactiﬁcation with ﬂux

In this subsection we add fluxes to our model. This is the first of the three main ingredients we are going to use. We look at gravity, but coupled to a background magnetic flux running on the two torus. Note that for this field we do not use sources. 2 We take the manifold M6 = M4 × T and the action 4 2 p 1 1 ij S = d xd y |g6| Rb6 − FijF . (4.27) ˆ b 2κ2 4

This action can be split in two parts, the gravitational part and the flux part. The compactification of the gravitational part is not changed due to the occurrence of a flux. This part of the compactification only depends on the higher dimensional metric, which is not influenced by the presence of a flux. This means that we can copy the result obtained for T n. The flux part of the action is given by

p 1 S = − d4xd2y |g | F F ij. (4.28) flux ˆ b6 4 ij

29 4.2. Compactiﬁcation over an n-torus

Because Fij is a two-form living completely in the compact dimensions, we can split the integral over the internal and external space, resulting in 1 p p S = − d4x |g | d2y |g |F F ij. (4.29) flux 4 ˆ e4 ˆ e2 ij

Because F is a top-form, it is proportional to the volume-form ij. Therefore Fij = ij ik jl −4βφ(x) ik jl qij, with q ∈ R. This means that F = qg g kl = e g g kl, where p b b ij = |ge2|εij is a (0,2) tensor. From a quantum treatement of the theory it follows that q is quantised, but this is not something which we are going to explain in this thesis. Substituting our expressions obtained above into the action results in q2 S = − d4xp|g | d2yp|g | gikgjl flux ˆ e4 ˆ e2 4 ijb b kl 2 q 4 p (4α−2β)φ(x) = −Ve d x |g4|e . (4.30) 2 ˆ Here we used the formula

i1...iqlq+1...lD t [lq+1 lD] i ...i k ...k = (−1) q!(D − q)!δ . . . δ , (4.31) 1 q q+1 D [kq+1 kD] where t denotes the amount of timelike directions. This formula can be found in [11]. 2 1 Using that for a (4,2) torus compactiﬁcation we have that β = −α and α = 8 , our full action becomes

3 Ve 4 µp 1 2 1 µ ij 2 2 √ φ(x) S = d x |g | R − (∂φ) + ∂ M ∂ M − q κ e 2 . (4.32) 2κ2 ˆ 4 4 2 4 µ ji Remark that when we substituted the values of α and β, we made an explicit choice, namely α > 0. Physically this makes no difference, which is explained below. Also remark that the exponential term arises because we raise indices with the inverse hatted metric. This means that even if would we have taken the more general metric Ansatz with the vectors, we would still have the same potential. Because the flux ij only runs in the compact dimensions, and thus only the components gb play a role. √3 φ(x) The Lagrangian density in the action is clearly of the form (2.28), where q2κ2e 2 = V (φ). Because the scalar field φ(x) wants to reduce its energy, it will try to minimise its potential energy, therefore φ(x) → −∞. From our choice of α > 0 it follows that β < 0, and as a consequence the part e2βφ(x) of the metric will evolve to e2βφ(x) → +∞. The physical interpretation of this is that the volume of T 2 blows up to infinity, hence we have no vacuum. If we would have made the choice of α < 0, − √3 φ(x) then q2e 2 = V (φ), thus φ(x) → +∞ and β > 0. Therefore e2βφ(x) → +∞, which leads to the same conclusion. This means that the physics does not depend on our choice of the sign for α, which is as expected because it is a gauge choice on our metric which has no physical meaning. A last remark is that we only found a potential for the so-called volume moduli and not for the shape moduli. These are the scalar fields sitting in Mij. Therefore they remain massless, hence it does not require any input of energy to change the shape of the torus while keeping the volume fixed.

30 4.2. Compactiﬁcation over an n-torus

2 4.2.2 A special case: M4 = M2 × T

The compactiﬁcations we performed in the previous sections were valid for D > 2. Here we investigate what happens when D = 2. Therefore we perform the following 2 2 1 1 compactiﬁcation M4 = M2 × T , with T = S × S . We take the usual Ansatz, 2 2αφ(x) 2 2βφ(x) n m dsb4 = e ds2 + e Mmndy dy . (4.33) 2βφ(x) Again e Mnm is a 2-dimensional symmetric matrix where M has determinant equal to one. We reuse our convention of the previous section and call g the bµbνb −2αφ(x) four-dimensional metric , gbµν the D part metric , and gµν = e gbµν. Rewriting the metric in conformal frame and reusing equation (4.10) contracted with the inverse metric yields

−2βφ(x) 2 2 Rb4 = e Re4 − 6β (∂φe ) − 6βeφ . (4.34) The box is again deﬁned with respect to g , therefore we rewrite the box in terms of eµbνb g . From the calculation leading equation (B.20), we see that the result can just be µbνb copied while setting D = 2. This results in −2(α−β)φ eφ = e φ. (4.35) Using this in the previous equation we obtain

−2βφ 2 2 −2(α−β)φ Rb4 = e Re4 − 6β (∂φe ) − 6βe φ . (4.36) We arrived at the stage where the Christoffel symbols need to be calculated. How- ever looking back at the general case, it can be seen that nowhere the dimensionality of the metric was used. The results obtained in the general case can thus be copied up to and including the expression for ReD+n, (4.18). This yields

1 µ ij Re4 = Re2 + ∂ M ∂eµMji. (4.37) 4 Before looking at the action of our theory, we use the formula for the Ricci tensor of a rescaled metric, (4.10), a last time. We have that D = 2, n = 0, β → α − β, the hatted objects become tilted objects, and tilted objects normal objects. This results in −2(α−β)φ Re2 = e (R2 − 2(α − β)φ) . (4.38) We have all the ingredients to calculate the effective action. We start by using (4.36) to rewrite the four dimensional hatted Ricci scalar. Then we use the formula beneath it to rewrite the Ricci scalar with the tilde. Finally we rewrite the integration measure p |gb4| in terms of g2. This gives us 1 2 µ 2 ip S = d x d y |g4|Rb4 2κ2 ˆ b 1 1 = d2xµd2yi p|g |e2βφ R + ∂µMij∂ M − 6β2(∂φ)2 − (4β − 2α) φ 2κ2 ˆ 2 2 4 µ ji √ Ve2 p 1 1 = d2xµ |g |eφ/ 3 R + ∂µMij∂ M − (∂φ)2 , (4.39) 2κ2 ˆ 2 2 4 µ ji 2

31 4.2. Compactification over an n-torus where the box was cancelled, and the canonical normalisation were obtained by choosing 1 α = 2β and β = √ . (4.40) 2 3 This action is not the normal gravitational action, due to the occurrence of the exponential in front. This theory falls under the class of Jackiw-Teitelboim gravity theories. From this it seams that the kinetic terms do satisfy the condition (2.22) off-shell. Thus we would except that we cannot couple a potential to it. However this is not true due to the fact that the formula p δ |g|R = Gµν, (4.41) which is the variational identity we used to derive (2.22), is not valid anymore. To prove this we perform a variation with respect to the metric of the action p √ S = d2xµ |g |eφ/ 3R . (4.42) ˆ 2 2 The variation with respect to the metric gives us the following three terms √ √ √ δS = d2xµ eφ/ 3R δp|g | + eφ/ 3p|g |R2 δgµν + eφ/ 3p|g |gµνδR2 . ˆ 2 2 2 µν 2 µν (4.43) The first two terms combine to give the Einstein tensor multiplied with an exponential factor. This means that in our current analysis, it is the last term that will be important. The variation of the Ricci-tensor with respect to the metric is a standard formula and it is given by equation (4.65) in [2], p √ δS = d2xµ |g |eφ/ 3∇ g ∇σδgµν − ∇ δgσλ . (4.44) 3 ˆ 2 σ µν λ This term looks like a total derivative, but it is not due to the presence of the exponential in front. If it were a total derivative, we would have that this term vanishes in the action and that (4.41) holds.Before continuing remark that due to p λ µν metric compatibility, ∇σ |g| = 0 and ∇ g = 0. Using this we can integrate by parts twice to obtain p √ √ δS = d2xµ |g | g eφ/ 3 − ∇ ∇ eφ/ 3 δgµν. (4.45) 3 ˆ 2 µν µ ν From the fact that the Einstein-tensor in two dimensions vanishes, we get as final answer for the variation of the action (4.42) p √ p √ √ δ dx2xµ |g |eφ/ 3R = dx2 |g | g eφ/ 3 − ∇ ∇ eφ/ 3 δgµν. (4.46) ˆ 2 2 ˆ 2 µν µ ν From this we see that the equation (2.22) does not hold for this model. Let us investigate what happens if we couple a flux to this action. The same Ansatz for the metric is taken, but the action becomes 2 2 p 1 1 ij S = d xd y |g4| Rb4 − FijF . (4.47) ˆ b 2κ2 4

32 4.2. Compactiﬁcation over an n-torus

The compactification procedure of the gravitational part of this action is not influenced by the presences of a flux into the compact dimensions, therefore let us look at the flux part of the action. As in the previous case we take a flux which has no dependence on the x-coordinates. This means that the two-form we found in the previous section, Fij = qij, can be used. The compactification results in,

q2 S = − d2xp|g | d2yp|g | gikgjl flux ˆ e4 ˆ e2 4 ijb b kl 2 q 2 p (2α−2β)φ(x) = −Ve2 d x |g2|e . (4.48) 2 ˆ

Using this result the full action of our compactiﬁed theory is obtained,

√ Ve2 p 1 1 S = d2xµ |g |eφ/ 3 R + ∂µMij∂ M − (∂φ)2 − q2κ2 . (4.49) 2κ2 ˆ 2 2 4 µ ji 2

We derive the equations of motion of the metric. This equation of motion will act as a constrain on the system, because the metric does not have any dynamical terms in the action and is thus a Lagrange multiplier. The variation with the inverse metric yields

√ 1 δS 1 1 φ/ 3p 1 2 1 2 2 2 p µν = p µν e |g2| R2 + (∂M) − (∂φ) − q κ |g2| δg |g2| δg 4 2 √ √ h 1 1 1 = (g − ∇ ∇ ) eφ/ 3 + eφ/ 3 − g (∂M)2 − (∂φ)2 − q2κ2 µν µ ν 2 µν 4 2 1 1 i + δρδσ∂ Mij∂ M − δρδσ∂ φ∂ φ . (4.50) 4 µ ν ρ σ ji 2 µ ν ρ σ

The variation should be equal to zero, which can be interpreted as the fact that the energy momentum tensor has to vanish. This implies the following equation: √ √ eφ/ 3 1 1 0 = (g − ∇ ∇ ) eφ/ 3 − g (∂M)2 − (∂φ)2 − q2κ2 µν µ ν 2 µν 4 2 √ √ eφ/ 3 eφ/ 3 + ∂ Mij∂ M − ∂ φ∂ φ. (4.51) 4 µ ν ji 2 µ ν This equation can be contracted with the inverse two-dimensional metric and results in √ √ φ/ 3 2 2 φ/ 3 e + q κ e = 0. (4.52) This shows that the potential does not have to vanish on-shell, thus the coupling of the ﬂux is consistent. However because in a vacuum solution φ is a constant, the above equation implies that q has to vanish. This means that our model does not have a vacuum.

33 4.3. Compactiﬁcation over a maximally symmetric space

4.3 Compactiﬁcation over a maximally symmetric space

In this section we first compactify gravity over a maximally symmetric space without any flux in the compact dimensions. This is done to understand the effect that curvature has, afterwards we add a flux.

4.3.1 Compactiﬁcation without any ﬂux

The fact that the compact manifold is a maximally symmetric space implies that there aren’t any shape moduli but only one volume modulus. Would there be shape moduli, they would introduce an homogeneity, which breaks the symmetry. This simpliﬁcation is used because we are interested in the evolution of the volume modulus. The Ansatz for the metric becomes,

2 2αφ(x) 2 2βφ(x) n m ds6 = e ds4 + e Nmndy dy , (4.53) with N a symmetric matrix with dependence on the internal coordinates. We again go to conformal frame and use equation (4.10) with D + n = 6 to obtain:

−2βφ(x) −2(α−β)φ(x) 2 2 2 Rb6 = e Re6 + e −20β (∂φ) − 10β φ(x) + 2 (α − β)(φ(x)) . (4.54) To continue making progress, we use that a maximally symmetric surface is an example of an Einstein manifold. These are manifolds with the property

Rij = cgij, (4.55) with c a constant. In this calculation we look at an S2 or an H2, which is the hyperbolic plane, and which is not a compact manifold. However using a topological identiﬁca- tion, just as we did with the torus, we can make it compact. This identiﬁcation does not change the metric. For the normalised metric on this surfaces, the following holds Rn = (n − 1)nk, (4.56) where n is the dimensionality of the surface, and k = +1 for the sphere and −1 for the hyperboloid [11]. This has as a consequence that

6 µν 6 Re = gb Reµν + 2k, (4.57) µi where we used the action above and the fact that gb = 0. The metric gij does not have any dependence on the coordinates of the non-compact dimensions, geµν has not any of the compact dimensions and gbµi = 0. From this it is easily seen that the only non-vanishing Christoffel symbols are,

i λ Γejk 6= 0 and Γeµν 6= 0. (4.58)

34 4.3. Compactiﬁcation over a maximally symmetric space

µb µb µb µb λb µb λb From this and the fact that Re σµν = ∂µΓeνσ − ∂σΓeνµ + Γe Γeσν − Γe Γeµν, it is found b b b µbλb σλb b 6 4 that Reµν has no terms containing the elements of Nij. This means that Reµν = Reµν, where the second Ricci tensor is the expression obtained by only summing over the external coordinates. This means that we have the following

Re6 = Re4 + 2k. (4.59)

Using again the formula for a rescaled Ricci scalar (4.21), we ﬁnd an expression for Re4, see (B.43). Substituting all this into the action and choosing β = −α and 2 1 α = 8 , we obtain after some algebra shown in the appendix our ﬁnal result

2 4 µp 1 2 √ φ(x) S = Ve d x |g4| R4 − (∂φ) + 2ke 2 . (4.60) ˆ 2

The potential for this model is:

√2 φ(x) V (φ(x)) = −2ke 2 . (4.61)

For the sphere we have k = 1 and because the conﬁguration wants to reduce its energy φ(x) → +∞. Since β < 0 this results in e2βφ(x) → 0, which we interpret as the radius of S2 going to zero. The curvature of the sphere is inversely proportional to the radius. From this we infer that the sphere wants to increase its curvature. We can conclude that positive curvature contributes negatively to the energy. For the hyperbolic plane k = −1, so φ(x) → −∞ hence the length scale of H2 goes to inﬁnity. Hence H2 wants to reduce the absolute value of its curvature. We learn that negative curvature contributes positively to the energy. To conclude: this model does not admit a vacuum.

4.3.2 Compactiﬁcation with ﬂux

In this subsection the same spacetimes are analysed, but with ﬂux running through the compact dimensions. The action of this model is given by: 4 2 p 1 1 ij S = d xd y |g6| Rb6 − FijF . (4.62) ˆ b 16π 4

We can copy our results for the gravitational part (4.60). Looking back at the torus compactification with flux done in subsection 4.2.1, it is seen that the flux parts of the actions are the same. The resemblance goes even further, the flux is again a top form and the parameters α and β are the same as in the torus case. Therefore we can just copy the result we obtained there, equation (4.30), and plug it into our action, resulting in

2 3 Ve 4 µp 1 2 √ φ(x) 2 2 √ φ(x) S = d x |g | R − (∂φ) + 2ke 2 − q κ e 2 . (4.63) 2κ2 ˆ 4 4 2

35 4.3. Compactiﬁcation over a maximally symmetric space

The potential for these models is

2 3 √ φ(x) 2 2 √ φ(x) V (φ(x)) = −2ke 2 + q κ e 2 . (4.64)

This is an interesting result because it has a positive, and possibly a negative contribution. For H2, it is clear that because k = −1 the derivative of the potential with respect to φ can never be zero. This means that we have no vacuum. For S2 the derivative of the potential is given by

2 1 0 1 √ φ(x) 2 2 √ φ(x) V (φ(x)) = −√ e 2 4 − 3q κ e 2 . (4.65) 2 √ ∗ 4 The equation above is zero when φ (x) = 2 ln 3q2κ2 . This is a minimum of the potential because substituting φ∗ into the second derivative results in a positive number. Remark that a vacuum solution exist as long as q > 0. Quantum mechanically we need to expand our fields around vacua. Therefore we define φe(x) = φ(x)−φ∗(x) ∗ 32 −4 and rewrite the action in terms of this field. Because V (φ (x)) = − 27κ4 q we arrive at

2 3 Ve 4 µp 32 1 2 √ φe(x) 2 2 √ φe(x) S = d x |g4| R4 + − (∂φe) + e 2 − q κ e 2 . (4.66) 2κ2 ˆ 27κ4q4 2 Comparing our result with the action given by equation (A.6), it can be seen that this is a gravitational theory with a negative cosmological constant and a massive scalar ﬁeld. If we look at pure classical solutions, the scalar ﬁeld drops out of this. This means classically we obtained a gravitational theory in an AdS4 background. 2 The vacuum solutions of our theory is M6 = AdS4 × S spacetime, where the radius R of S2 is given by,

∗ 3 R2 = e2βφ (x) = q2κ2 ⇒ R ∼ qκ. (4.67) 4

Around the vacuum there is a massive scalar ﬁeld φe(x) living in that AdS4 spacetime. 00 ∗ 32 −4 −4 m2 8 −2 −2 Because V (φ (x)) = 9 κ q = 2 , the mass of φe(x) is equal to m = 3 κ q . We see that the mass of the ﬁelds reduces when the radius of the sphere increases, which is what we expect. The last thing we need to check is if there is scale separation. Reinstating the Planck mass into the action, we see that 1 Λ ∼ , (4.68) κ2q4 from equation (2.13), we see that the length scale associated to that AdS4-spacetime behaves like L ∼ κq2. (4.69) From this and (4.67) we see that R 1 ∼ , (4.70) L q

36 4.3. Compactiﬁcation over a maximally symmetric space which means that the sizes of the internal and external dimensions are comparable. To conclude we see that for H2 we have no vacuum even with a ﬂux, for S2 we 2 get an AdS4 × S vacuum but we do not have a clear separation of scale. Hence these models are not viable.

4.3.3 Consistency check

In this section we check if the reduction of the action was consistent. This is done by checking if the equations of motion obtained from the compactiﬁed action, solve the equations of motions of the higher dimensional theory. The equation of motion for the metric of the following theory µ ip 1 1 ij S = dx dy |gD+n| RbD+n − FijF , (4.71) ˆ b 2κ2 4 is derived in the appendix and is given by (B.52). Using it with D + n = 6, we obtain

2 κ ij 2 σ Rbµν = − gµνFijF + κ F b Fσν. (4.72) bb 8 bbb µb bb

The Einstein equations for a AdSD-spacetime were derived previously and are given by (A.9). For a four-dimensional spacetime these are given by 3 RAdS4 = − g . (4.73) µν L2 µν Looking at the action we obtained from our compactiﬁcation (4.66), we see that 32 6 2 4 4 Λ = − 27κ4q4 = − L2 , yielding L = 81/16κ q . Therefore our metric on the AdS4- spacetime solves the equation, 16 RAdS4 = − g . (4.74) µν 27κ4q4 µν

Returning to our 6-D Einstein equations (4.72), we see that the 4-D part can be rewritten as

2 6 κ 2αφ∗(x) 2 −4βφ∗(x) 16 Rb = − e gµν 2q e = − gµν. (4.75) µν 8 27κ2q4

2αφ∗(x) ij 2 −4βφ∗(x) Here we used that gµν = e gµν, FijF = 2q e , which was obtained in b √ section 4.2.1, α = −β = √1 , and φ∗(x) = 2 ln 4 . 2 2 3κ2q2 The part of the Einstein equations describing the compact dimensions are given by

2 κ 2βφ∗(x) 2 −4βφ∗(x) 2 σ 1 2 σ Rbij = − e gij 2q e + κ F b Fσj = − gij + κ F b Fσj. (4.76) 8 i b 3 i b

37 4.4. One-dimensional compactiﬁcation

The term F σb F is equal to i σjb 4 F σb F = g F σkb F = q2e2βφg e−4βφσkb = q2e−2βφg δk = g , (4.77) i σjb bik σjb ik σjb ik j 3κ2 ij where to go from the left to the right of the third equality we used (4.31). Substituting this into equation (4.76), we obtain

Rbij = gij. (4.78)

2 The Einstein equation of an S is the same as the equation above, and Rbµi = 0. Therefore the metric which satisfies the Einstein equations for the AdS4-spacetime and the S2-spacetime obtained by compactification, satifies the full six dimensional Einstein equations. The compactification is thus consistent.

4.4 One-dimensional compactiﬁcation

In this section we perform a compactiﬁcation to a one-dimensional manifold over a three-dimensional Einstein space. Afterwards ﬂux is added, a consistency check is performed and contact is made with the FLRW-metric used in cosmology. To end this section we add a cosmological constant and search for a vacuum.

4.4.1 Compactiﬁcation over S3 and H3

The Ansatz we take is the following metric:

2 2αφ(x) 2 2βφ(x) n m dsb4 = e ds1 + e Nmndy dy . (4.79) 3 3 Here the matrix Nmn is the usual metric on S , respectively H . We again go to conformal frame and use equation (4.10), resulting in

−2βφ(x) −2(α−β)φ(x) 2 2 2 Rb4 = e Re4 + e −6β (∂φ) − 6β φ(x) − (α − β)(φ(x)) . (4.80) For the normalised three-dimensional sphere we know that the Ricci scalar is given by (4.56), and because the metric is block-diagonal we obtain

µν 4 Re4 = g Reµν + 6k. (4.81)

We can use the same arguments as those around (4.58) to state that

i λ Γejk 6= 0 and Γeµν 6= 0, (4.82) thus Reµν does not contain terms with matrix elements Nij. Therefore we again conclude that Re4 is the sum of the Ricci scalars of the compact and non-compact

38 4.4. One-dimensional compactiﬁcation dimensions. Because the non-compact part is one-dimensional, its Ricci scalar is always equal to zero. Hence we drop this term and use the above into the action, resulting in

1 µ 3 ip S = dx d y |g4|Rb4 2κ2 ˆ b 1 = dxµd3yip|g |e(−α+3β)φ(x) − 6β2(∂φ)2 − 6β φ(x) 2κ2 ˆ 1 + 6β (α − β)(∂φ(x))2 + e(α+β)φ(x)6k. (4.83) p This expression was obtained by rewriting the term |gb4| and using our results for Rb4. Choosing √ 1 3 β = √ and α = , (4.84) 2 3 2 and omitting the box term results in

2 Ve µp 1 2 √ φ(x) S = dx |g | (∂φ) + 6ke 3 . (4.85) 2κ2 ˆ 1 2 We see that the scalar ﬁeld has the wrong sign in the kinetic term. This is due to the 2 µ fact that (∂φ) = ∂µφ∂ φ = −∂tφ∂tφ, where t represents the time coordinate. The fact that it has the wrong sign in the kinetic term means that it can loose energy by evolving. We cure this by taking the minus sign in front to obtain

2 Ve µp 1 √ φ(x) S = − dx |g | ∂ φ∂ φ − 6ke 3 . (4.86) 2κ2 ˆ 1 2 t t From this we see that the potential is

√2 φ(x) V (φ) = 6ke 3 . (4.87)

In this case positive curvature means positive energy. This is the opposite of what 2 we obtained from the compactiﬁcation of M6 = M4 × S . This is generic for a compactiﬁcation to one dimension, all the contributions to the potential switch sign. Therefore they have the opposite effect then we are used to.

4.4.2 Adding ﬂux

In this subsection a flux running through the compact dimensions is added to the model. This results in an action of the form 1 3 p 1 1 ijk S = d xd y |g4| Rb4 − FijkF . (4.88) ˆ b 16π 6 Remark that this flux is characterised by a three-form, not the usual Maxwell two-form. This means that this field does not couple to a point particle but to a one-dimensional

39 4.4. One-dimensional compactification object, as was explained in section 2.3. Because the Fijk is a top-form it is proportional to the volume-form, meaning ijk il jm kn Fijk = qijk and F = qgb gb gb lmn. Inserting these expressions into the action we obtain q2 S = − d1xp|g | d3yp|g | gilgjmgkn flux ˆ b1 ˆ b3 6 ijkb b b lmn 2 1 p (α−3β)φ(x) = −Ve q d x |g1|e , (4.89) ˆ where (4.31) was used. Substituting this into the action of the higher dimensional theory (4.88), and using the results for the gravitational part given by equation (4.85), we obtain 2 Ve µp 1 2 √ φ(x) 2 2 S = dx |g | (∂φ) + 6ke 3 − 2q κ . (4.90) 2κ2 ˆ 1 2 Remark that the exponential function which came from the flux part of the action 1 has vanished due to the fact that β = 3 α. This potential does not admit a vacuum, because taking the derivative with respect to φ shows that the flux contribution vanishes and we are left only with one exponential term. Therefore the potential has no extrema.

4.4.3 Equations of motion from the action

We reduced the above two models by substituting on-shell information, the Ansatz for the metric and the three-form, directly into the action. This is a dangerous thing to do because it can lead to errors. Because to our knowledge a one-dimensional reduction of these model has not been performed yet, we need to check for consistency. We do not do this by checking if the lower dimensional equations of motion solve the higher dimensional ones. We reduce the higher dimensional equations of motion, and we check if these are the same as the ones obtained from the reduced action. If this is the case our reduction is consistent, since reducing the equations of motion is always consistent. In the action there are two independent fields, the metric and the scalar field. The metric is a special field in one-dimension, because it is a Lagrange multiplier. Its equation of motion is a constraint equation. This equation is calculated using a variation of (4.90) with respect to the inverse metric resulting in

2 1 1 2 √ φ(x) 2 2 1 − g (∂φ) + 6ke 3 − 2κ q = − ∂ φ∂ φ. (4.91) 2 00 2 2 0 0 We perform a coordinate transformation on this equation and go to the FLRW-metric

02 2 2 n m dsb4 = −dτ + a (τ)Nmndy dy . (4.92) The equation of motion for the metric (4.91), is not a covariant expression. Hence to perfrom this coordinate transformation we rewrite it in terms of the hatted metric

40 4.4. One-dimensional compactiﬁcation such that it becomes covariant. Then we use the tensor transformation rules, to obtain as ﬁnal result a˙ 2 k 8πq2 + = . (4.93) a a2 3a6 The steps which lead to this result are shown in appendix B.4.1 between (B.55) and (B.60). The second equation of motion is obtained by varying (4.90) with respect to φ and equals √ 2 00 √ φ g ∂0∂0φ = 4 3ke 3 . (4.94) If we want to rewrite this in terms of the FLRW-metric we apply the same manipula- tions as explained above and use (4.93) to obtain

a¨ 2κ2q2 = − . (4.95) a 3a6 The details of this derivation are again shown in appendix B.4.1 between equations (B.61) and (B.66).

4.4.4 Reduction of the Field equation

In this section the full four dimensional equations of motion are derived and reduced. We then translate everything in terms of the FLRW metric and check for consistency. The four dimensional action is given by equation (4.88), 1 3 p 1 1 ijk S = d xd y |g4| Rb4 − FijkF . (4.96) ˆ b 2κ2 6

The trace reversed Einstein equations are derived in appendix B.4.2 in the discussion above (B.70) and are

κ2 σbλbρb 2 σbλb Rbµν = − gµνF F + κ F F ν . (4.97) bb 3 bbb σbλbρb σbλbµb b For the components of the compact dimensions this equation becomes

Rbij = 0, (4.98) where the details of this are shown under equation (B.71). The equation of motion for the non-compact dimension is given by

2 2 2αφ −6βφ 2 2 Rb00 = −2κ q e e g00 = −2κ q g00. (4.99)

To ﬁnd the reduced equations of motion the left part of the two equations above has to be expanded in terms of the metric g . To achieve this we need the Christoffels, µbνb

41 4.4. One-dimensional compactiﬁcation which were calculated in appendix B.4.1 of the previous section, and are given by (B.62). Using these and after a straightforward calculation, we arrive at √ 3 1 Rb00 = − ∇0∂0φ − ∂0φ∂0φ. (4.100) 2 4 If we deﬁne the same coordinate transformations as in the previous section, and introduce a(τ) we arrive at the following equation

√ 0 a¨ e− 3φ Rb0 = f −2a−6 −3 , (4.101) 00 a where the accents refer to the objects with respect to√ the new coordinates. The f comes from the coordinate transformation dτ = f(t)e 3φ/2dt. All this is explained in more detail in appendix B.4.2 between equations (B.73) and (B.77). If we use equation (4.99) but then rewritten covariantly, and transform with the new coordinates, we get

√ 2 2 2 2 −2αφ 0 2 2 −2 3φ0 −2 0 −2 2κ q Rb00 = −2κ q e g00 ⇒ Rb = −2κ q e f g = f . (4.102) 00 00 a12 Equating this with the previous equation we get

a¨ 2κ2q2 = − . (4.103) a 3a6 Remark that this is the same equation as we obtained by varying our reduced action, (4.95). The other components are 3 −4βφ 1 0 1 0 Rb0i = 0 and Rbij = R + Nije − ∂0φ∂ φ − √ ∇0∂ φ , (4.104) ij 4 2 3 Rewriting this equation in terms of the hatted metric and transforming it, substituting the value of a and equating it with equation (4.98), results in 2 1 0 0 0 0 1 0 0 0 Nij 2k + a − ∂ φ ∂ φ + √ ∂ ∂ φ = 0. (4.105) 4 0 0 2 3 0 0 All steps to obtain this are shown in the appendix. From this we see that what is in the brackets has to vanish, meaning we can continue by looking only at these terms. 0 0 The term ∂0∂0φ will yield a a¨ term, which can be replaced by using the ﬁrst equation we derived (4.103). Using this our ﬁnal result becomes

a˙ 2 k κ2q2 + = (4.106) a a2 3a6

All the steps from (4.104) to the above results are shown between (B.78) and (B.82) Remark that the two equations we obtained by a reduction of the equations of motion are the same as those we obtained by varying the reduced action. A reduction of the equations of motion is always consistent and both results we obtained are the same.

42 4.4. One-dimensional compactiﬁcation

Therefore the reduction of the action of this model to one dimension is consistent. From this we are going to assume the following: If the reduction of the action of a model to dimensions bigger than two is consistent, it will also be consistent to reduce it to one dimension. At last remark that the equations (4.103) and (4.106), are the two Friedmann equations from cosmology. These describe the evolution of the scale-factor a, which describes if the universe is expanding or contracting. These equations are obtained by assuming the homogeneity and isotropy of the universe, which is a good approximation when working at large scales. The fact that our a only depends on the time coordinate is because we assumed that the higher KK-modes are to massive to be excited. This results for this model in a homogeneous and isotropic universe. This approximation is valid as long as we are working with small length scales, which is the complete opposite of the FLRW-approximation.

4.4.5 Cosmological constant

In this model we add a cosmological constant to the previous one and investigate what the effects are on the physics. We start from the action: 2 1 1 3 p κ ijk S = d xd y |g4| Rb4 − Λ − FijkF . (4.107) 2κ2 ˆ b 3

The kinetic and ﬂux term stay the same, but we get extra terms from the cosmological constant. These are obtain from a reduction of the following action 1 S = − d1xd3yp|g |Λ Λ 2κ2 ˆ b4 Ve p √ = − d1x |g |e 3φΛ. (4.108) 2κ2 ˆ 1 Recombining everything using equation (4.90), we obtain

2 √ Ve µp 1 2 √ φ(x) 3φ 2 2 S = dx |g | (∂φ) + 6ke 3 − e Λ − 2q κ . (4.109) 2κ2 ˆ 1 2

The kinetic term of the scalar has a minus sign in the contraction because g00 is negative, which means that the kinetic term has the wrong sign. Therefore If we take this minus sign and put it in front of the action we obtain

2 √ 0 Ve µp 1 √ φ(x) 3φ 2 2 S = − dx |g | ∂ φ∂ φ − 6ke 3 + Λe + 2q κ . (4.110) 2κ2 ˆ 1 2 0 0 From this we see that the potential is equal to

2 √ √ φ(x) 3φ 2 2 V (φ) = 6ke 3 − Λe − 2q κ . (4.111)

43 4.4. One-dimensional compactiﬁcation

With the potential at hand we investigate if there are stable vacua, therefore we need the derivatives of it with respect to φ. These are given by

√ 2 √ 0 √ φ(x) 3φ 3V (φ) = 12ke 3 − 3Λe , (4.112) 2 √ 00 √ φ(x) 3φ V (φ) = 8ke 3 − 3Λe . (4.113)

It is easily seen that the ﬁrst derivative of the potential has a zero for the two following conditions k √ 4k > 0 and φ∗ = 3 ln . (4.114) Λ Λ

Therefore we found that the potential has an extremum, however for the extremum to be a vacuum it needs to solve the equation V (φ∗) = 0. After two lines of algebra we see that it does solve this equation, if the flux is tuned to the value 16 q2 = (3k − 2k) . (4.115) κ2Λ2 Hence only the solution k > 0 is permitted, which the S3 and which is exactly Einstein’s solution to the Friedmann equations. Let us check if this configuration is stable. Using our solution φ∗ in the equation for the second derivative, we obtain 128 192 V 00(φ∗) = − < 0, (4.116) Λ2 Λ2 thus we see that the configuration is unstable and we have no vacuum. This is something which could have been expected. In the previous sections we found that the compactification of the volume modulus to one dimension yields the Friedmann equations. Einstein found its static universe using a cosmological constant, positive curvature and a homogeneous matter density. In our compactification we have curvature and a cosmological constant, but also a three-form flux. This flux is homogeneously distributed, because it is proportional to the volume form of a maximally symmetric space. What is its equation of state? The two Friedmann equations are

a˙ 2 κ2ρ k = − , (4.117) a 3 a2 a¨ κ2 = − (ρ + 3p) , (4.118) a 6 where ρ is the density and p is the pressure [2]. From these equations and (4.93), and (4.95), we see that ρ = p. This means that the cosmological equivalent of the three-form flux is a scalar field with a purely kinetic term. Eventough this not the same as the ”dust” used to find Einsteins static universe, we can find a same type of solution to the Friedmann equations.

44 Chapter 5

Compactiﬁcations to 1D

In this chapter we construct a serie of compactifications to one dimension. We are interested in flux compactifications of type II superstring theory. These compactifications are best understood in the framework of type II ten-dimensional supergravity, which is the massless tree-level limit of superstring theory [23]. Therefore we begin the chapter with an introduction to type II supergravity. When the basic ingredients are mastered, we use them to obtain a compactification to one dimension and calculate the masses of some of the scalar fields in the effective theory. This is done because we want to check if the compactification admits a meta-stable vacuum. Afterwards we change our approach and leave the ten-dimensional regime. If string theory wants to describe the real world it needs a compactification to four dimensions. Therefore we assume that such a compactification exists and we try to compactify this further to one dimension. This yields some interesting constraints on the type of models which admit 1D compactifications. These constraints are used in the end of the chapter to check some possible compactifications .

5.1 Type II Supergravity

For the compactifications performed in this chapter we use two distinct ten-dimensional supergravity theories, called type IIa and type IIb. These names refer to the fact that these theories can be obtained by taking the low-energy limit of type IIa and IIb superstring theories, explaining how this limit is taken would take us too far. Note however that with the low-energy limit, we refer to energies that are beyond those probed at the LHC. In the performed compactifications we work in a regime where the size of the compact dimensions is far beyond our reach, but it is not so small that effects typical to string theory become important. This allows us to ignore them. This regime is widely used and it is were things are the best understood by the community. Due to our interest in classical vacua, we are only looking at the bosonic part of the action. The fermionic degrees describe quantum matter and are therefore omitted. From the compactification point of view, we are only interested in the parts of the action which have some metric dependence. The Chern-Simons terms in the action are topologically in nature, meaning that they are not influenced, nor do they interact with the metric. Even tough they are important for the tadpole condition, which is explained later in this section, we do not show them because they do not contribute to the compactification. To us, the useful part of the type II supergravity

45 5.1. Type II Supergravity action is given by: p+1 9−p p −2φ 2 1 2 S = d xd y |g10|e Rb10 + (4∂φb ) − |Hb| ˆ b 2

p+1 9−p p 1 X 2 − d xd y |g10| |Fbq| , (5.1) ˆ b 2 q

2 1 µ λb2···λbn which was copied from [24], with convention |An| ≡ A Ab . n! µb λb2···λbn Before we continue there are some things about this action that have to be said. This way of representing the action is called string frame, because it is obtained by taking the low-energy limit of type II superstring. Note that there is an exponential in front of the Ricci scalar and the kinetic term of the scalar has the wrong sign. If we want to interpret these fields as physical objects, we should take care to use the definitions of the fields which make them canonically normalised. By this we mean that the definition of the physical object called “the metric”, is the definition of gµν which give rise to the usual Einstein-Hilbert action. In two dimensions this is not true, but we refer to section 4.2.2 for more explanation. Because we are not in √ two-dimensions the combination g10Rb10 is not conformal invariant. This means that using a clever field redefinition of the metric we can make that exponential vanish . Therefore we redefine the metric as:

φ/2 0 gµν = e g , (5.2) bbb bµbνb where the accents refer to the new metric, and this was found in [25]. This type of redeﬁnition can always be done, it is just a mathematical renaming of the ﬁelds. This renaming has the effect that the determinant and the Ricci scalar transform to become q p 5φ/2 0 |gb10| =e |gb10|, (5.3) −φ/2 0 9 0 2 9 0 Rb10 =e Rb − (∂b φ) − b φ . (5.4) 10 2 2

The square of the p-form terms in the action carry p inverse metrics, hence they also transform. Combining everything we obtain the action in so-called Einstein frame p+1 9−p p 1 2 1 −φ 2 S = d xd y |g10| Rb10 − (∂φb ) − e |Hb| ˆ b 2 2

p+1 9−p p 1 X (5−q)φ 2 − d xd y |g10| e 2 |Fbq| , (5.5) ˆ b 2 q where we dropped the accents for notational convenience. Note that by cancelling the exponential in front of the Ricci scalar, the scalar obtained a canonical kinetic term. The other important thing about this action is that we have one action, but two supergravity theories. How can this be? The answer to this question lies in the sum

46 5.1. Type II Supergravity

of the Fcq terms. In type IIa we only keep the forms with q even and in IIb we keep the forms with q uneven. For type IIb there is an important remark. Because we are working in a ten-dimensional spacetime, F5 needs to be a self-dual form. This is something that needs to be inforced on the solutions of the equations of motion, because if this is inforced in the action it leads to incorrect equations of motion [23].

5.1.1 Localised energy sources

In the compactifications performed in this chapter we use additional ingredients besides fluxes and curvature, namely localised sources. These are objects coming from string theory. The first object is called a Dp-brane, which is a p-dimensional surface which can fluctuate in time. These branes can carry a U(1) gauge field, and their action in Einstein frame is r S = − dp+1ξ|µ |e(p−3)φ/4 | det g∗ + B + 2πα0 F |, (5.6) Dp ˆ p bp+1 ab ab which has been copied from [26], but where we added the gauge fields, and where |µp| is called the charge density or tension. The second part under the square root, 0 Bab + 2πα Fab, is associated to the gauge field and does not couple to the metric, hence we omit it from our discussion. Doing these we obtain the action used for compactifications q S = − dp+1ξ|µ |e(p−3)φ/4e−φ ||g∗ |. (5.7) Dp ˆ p bp+1

The branes sit in a certain position in spacetime, of which the geometry is described by gb10. Therefore the metric on this brane is defined by the projection of the ∗ spacetime metric onto the brane, which we denote by gbp+1. The same holds for the coordinates ξi. This action is interpreted as the volume of the brane multiplied with its tension. The second localised object is more complicated and is called an orientifold- plane (Op). Say that spacetime is a manifold M, but which possesses a discrete symmetry group G1. Then we can define a theory, not on M, but on the coset-space M/G1. If the manifold has no points which are invariant under the action of G1, the resulting coset-space is again a manifold. However, if there are invariant points these will become singularities in the coset space. When this is the case we call M/G1 an orbifold [27]. Mathematically these singularities are a problem and should be analysed with great care. However these singularities can be smoothed out by “blowing-up” these fixed points [28]. In string theory we expected that this “blowing- up” happens when strings move in the vicinity of these singularities, resulting in the fact that strings do not feel them. Therefore these singularities becomes less of a 1 problem. An example of such an orbifold is S /Z2, meaning we take the circle and make the identification x = −x. This transforms the circle into the interval [0, π], but note that 0 and π are singularities because they are invariant under the above

47 5.1. Type II Supergravity indentification [29]. When a particle moves in spacetime, we describe its evolution by a worldline. For higher-dimensional objects we can define the same, expect that we need higher dimensional surfaces called world-sheets. We can define an orientation- reversal operation on that world-sheet, denoted by Ω, and a reflection of the internal coordinates, denoted by G2. The combination of this ΩG2 is called the orientifold group. Given a theory A with the following symmetry

G = G1 ∪ ΩG2, (5.8)

0 where G1 is a discrete symmetry group, one can deﬁne a new theory A = A/G, which is called an orientifold of A [27]. If there are p + 1-dimensional surfaces which are invariant under the action of G, then these are called orientifold planes O(p). The action associated to these objects is q S = + dp+1ξ|µ |e(p−3)φ/4 ||g∗ |. (5.9) O(p) ˆ p bp+1

This action is the same as that of a Dp-brane, except that there is no gauge field and that the tension is negative. An orientifold plane consists of fixed-points and is therefore non-dynamical. This is a good thing because if they were this would render the theory unstable. Orientifold planes and D-branes act as sources for the form-fields as follows:

form SO(p) = ±µp Cp+1 Σ, (5.10) ˆΣ where Σ is the cycle that the orientifold is wrapping, µp is the charge density, and

Cp+1 Σ is the potential of Fp+2 but evaluated on Σ [30]. This action does not contain a metric, so does not contribute to the potential of the moduli of the metric. However due to this coupling, the form ﬁelds have as Bianchi-identities

dFp+2 = H ∧ Fp − µ6−pδp+3(O(6 − p)), (5.11) where the ﬁrst term comes from the Chern-Simons terms and the second from the orientifold planes. The term δp+3(O(6 − p)) is a shorthand for the normalised (n + 1) volume form transverse to the O(6−p) orientifold plane multiplied by δ(O(6−p)) [31]. The left side of the Bianchi-identity is a total derivative, therefore upon integrating this equation this term vanishes and we obtain

H ∧ F = µ δ (O(6 − p)), (5.12) ˆ p ˆ 6−p p+3 which is called the tadpole-cancellation condition. This is an identity which will turn out to be very useful in the following compactiﬁcations.

48 5.1. Type II Supergravity

5.1.2 The equations of motion

With the actions: (5.5), (5.9), (5.10) and with the addition of the Chern-Simons term we can calculate the equations of motion and Bianchi-identities of the different ﬁelds. We do not do this in this thesis, we copy them form [31]. We stress that from now till the end of this subsection, everything is almost literally taken over from this paper. We start with the trace reversed Einstein equations 1 −φ 1 2 1 2 Rµν = ∂µφ∂νφ + e |H| − gµν|H| bb 2 b b 2 µbνb 8 bb X 5−n φ 1 2 n − 1 2 1 loc 1 loc + e 2 |Fn|µν − gµν|Fn| + Tµν − T . 2 (1 + δn5) bb 16 (1 + δn5) bb 2 bb 8 n≤5 (5.13) In this equation we introduced the following notation 1 1 2 λb2···λbn 2 µb λb2···λbn |An|µν ≡ A A , |An| ≡ A A . (5.14) bb (n − 1)! µb λb2···λbn νb n! µb λb2···λbn The local stress-energy tensor contains the contribution of the localised sources, which in our case are the orientifold planes. Therefore it is given by, loc p−3 φ Tµν = e 4 µpgµνδ(Op), µ, ν = 0, 1, . . . , p. (5.15)

The µp is the magnetic charge density, and the delta-function has support on the Op worldvolume and includes a sum over parallel orientifolds planes. This delta function shows that the sources are localised in spacetime. The dilaton equation of motion is

2 −φ 1 2 X 5−n φ 5 − n 2 p − 3 p−3 φ ∇ φ = −e |H| + e 2 |F | − e 4 µ δ(Op), (5.16) 2 4 n 4 p n≤5 where ∇2 is the Laplace operator acting on the internal coordinates. The Bianchi identities for the ﬁeld strength are dH = 0, (5.17)

dFn = H ∧ Fn−2 − µ8−nδn+1(O(8 − n)), (5.18) where δn+1(O(8−n)) is shorthand for the normalized (n+1) volume form transverse to the O(8 − n) orientifold plane multiplied by δ(O(8 − n)). The equations for the RR ﬁeld strengths,

5−n φ 3−n φ n(n−1) d e 2 ?Fn = −e 2 H ∧ ?Fn+2 − (−1) 2 µn−2δ11−n(O(n − 2)), (5.19) can be obtained from the RR Bianchi identities for n > 5 upon employing the rule n−5 φ (n−1)(n−2)) e 2 Fn = (−1) 2 ?F10−n. Finally the equation of motion for the H ﬁeld strength is given by

−φ 1 X 5−n φ d(e ?H) = − e 2 ?F ∧ F , (5.20) 2 n n−2 n where the sum running over n includes all even/odd numbers up to 10 for IIA/IIB.

49 5.2. Vacua of type II supergravity

5.2 Vacua of type II supergravity

In this section we check the calculations performed in section 3.1 of [31] and while doing so we present some more details of it. We do this because in this calculation they try to find vacua of the supergravity theories exposed above, but with p ≥ 2 non-compact dimensions. Once we have these calculations under control, we expand these to check if there are vacua when there is only one non-compact dimension. If this is the case we can check a consistent subsector of the effective field theory, fluctuations in the volume and string-coupling, to see if these scalar fields are stabilised. The calculations are performed in the smeared limit, which is an approximation that makes the calculation less technical. If we are successful in finding a one- dimensional vacuum in this approximation, it is very probable that the calculation of the localised case performed in [31] can also be expanded to one dimension.

5.2.1 Solution with p + 1 ≥ 2 non-compact dimensions

To find solutions to the above equations of motion we take spacetime filling orientifold planes, where spacetime refers to the external manifold. This means that the orientifold planes are zero-dimensional objects in the internal space, hence they act as point charges for the form fields running into the interal space. To simplify the calculation we do not regard these planes as point charges, but we smear their charge all over the internal space. Due to this smearing, the source term for the dilaton, (5.16), is everywhere the same. Therefore we expect that the dilaton does not vary over the compact dimension, so we take it to be a constant. Because of the smearing out of the sources, and thus the absence of topological defects in the internal space, we do not expect warping. Therefore we take as metric Ansatz the product metric 2 2 2 ds10 = dsp+1 + ds9−p, (5.21) where we see that we have p + 1 external, and 9 − p internal dimensions. Note that we are looking for solutions of the equations of motion and thus not compactifying, which is the reason why there are no moduli in the Ansatz. Because the orientifold planes are taken to be spacetime filling, we use Op planes. These appear as sources in the Bianchi-identity for F8−p (5.18), therefore we need H and F6−p to satisfy the tadpole condition. The other forms are not sources nor needed so we take them to be equal to zero. The more technical details of the smearing of the Op-planes are the following. Smearing means that the delta functions representing the position of the orientifolds in the compact dimensions are replaced by a smooth function. This function has its support in the internal space, but it integrates to the same value as the delta function [31]. In our case smearing amounts to replacing the delta functions in the Bianchi identity for the F fields by 9−p, which is the normalized internal volume form.

50 5.2. Vacua of type II supergravity

This enables us to exactly cancel the term H ∧ F6−p, which has the effect that F8−p can be set equal to zero. This cancellation is stronger than the tadpole condition were we only required that the integrals over all space of these two quantities needed to vanish (5.12). If we would perform the calculation in the localised case this cancellation can not happen, therefore we need to take F8−p into account. In the Einstein and dilaton equations the delta functions are set to one, because we take the overall volume of the compact space to be equal to one. To find a solution, we need to solve: the Einstein equations (5.13), the Bianchi- identities for the forms (5.17) and (5.18), The equations of motion for the dilaton (5.16), and of the form fields (5.19) and (5.20). We start this computation with the trace-reversed Einstein equations in the non- compact dimensions. Because we are after vacuum solution we take the external part to be Minkowski space, and this implies that Rµν = 0. Hence using this in the trace-reversed Einstein equation (5.13) we get 1 1 1 0 = ∂ φ∂ φ + e−φ |H|2 − g |H|2 2 µ ν 2 µν 8 µν X 5−n φ 1 2 n − 1 2 1 S 1 S + e 2 |Fn|µν − gµν|Fn| + Tµν − gµνT , 2 (1 + δn5) 16 (1 + δn5) 2 8 n≤5 (5.22) where T S refers to the stress-energy tensor in the smeared approximation. In our Ansatz the dilaton is constant and the form fields live only in the internal dimensions. This means that the kinetic term for the dilaton and terms containing form fields with free indices vanish in the above equation. The trace of the energy momentum tensor (5.15), is S p−3 φ T = e 4 µp(p + 1)9−p. (5.23) Tracing the above Einstein equation, using (5.23) and that the only non-vanishing form is F6−p, we obtain

−φ 1 2 (5 − p) p−1 φ 2 7 − p p−3 φ 0 = −e |H| − e 2 |F6−p| + e 4 µp9−p. (5.24) 8 16 (1 + δp1) 16

Using |A|2 = ?A∧A, we translate this equation into differential form notation, yielding

−φ 1 (5 − p) p−1 φ 7 − p p−3 φ 0 = −e ?9−p H ∧ H − e 2 ?9−p F6−p ∧ F6−p + e 4 µp9−p. 8 16 (1 + δp1) 16 (5.25) The Bianchi identity for F8−p, which vanishes in the smeared approximation, is

0 = H ∧ F6−p − µp9−p. (5.26)

This can be used to replace the last term in the above equation, and rewrite it as

− p+1 φ 2 p+1 φ H ∧ F = e 4 ? H ∧ H + Ce 4 ? F ∧ F , (5.27) 6−p 7 − p 9−p 9−p 6−p 6−p

51 5.2. Vacua of type II supergravity with C = (5−p) . To solve this equation, we take the following Ansatz: (7−p)(1+δp1)

p+1 p − φ0 F6−p = (−1) e 4 κ ?9−p H, (5.28) and use it into the above. This results in

p − p+1 φ − p+1 φ 2 (−1) e 4 0 κH ∧ ? H = e 4 ? H ∧ H 9−p 7 − p 9−p p+1 p+1 2 φ p − φ0 + Ce 4 (−1) e 4 κ ?9−p ?9−pH ∧ ?9−pH. (5.29) This equation is a second order equation for κ and can be rewritten into (5 − p) 2 κ2 − κ + = 0. (5.30) (7 − p)(1 + δp1) (7 − p) The details are shown in the appendix between (C.1) and (C.6). We continue for the case p 6= 1, so we drop the factor (1 + δp1). We do this because otherwise we need to care about the self-duality conditions of F5, however in the ﬁnal steps we can adapt our obtained solutions to also include the case p = 1. Because at the end we are interested in the case p = 0, the p = 1 is not very important to us and so we do not do this explicitly. The two solutions of the above equation are 7 − p ± |(3 − p)| κ = , (5.31) 2(5 − p) which for p 6= 5 becomes, 2 κ = 1 and κ = . (5.32) 5 − p For p = 5, we get only κ = 1. The only solution we are interested in is κ = 1, will become clear later on. For p 6= 1 we thus get the following result

p+1 p − φ0 F6−p = (−1) e 4 ?9−p H. (5.33) We refer to this condition as the BPS condition, which for the case p = 3 this is the ISD condition on the G-flux in [16], but note that just as in GKP, the BPS condition does not necessarily imply supersymmetry [31]. For fluxes satisfying the BPS condition, the equation of motion of one flux gets mapped onto the Bianchi-identity of the other, and vice versa. This is shown in the appendix between (C.7) and (C.13). This means that if the Bianchi-identities are satisfied, the equations of motion are also automatically satisfied. We can substitute the BPS condition into equation (5.26) to relate the fluxes to the sourcing strength of the orientifold,

0 = H ∧ F6−p − µp9−p p+1 p − φ0 = H ∧ (−1) e 4 ?9−p H − µp ?9−p 1 p+1 2p − φ0 2 = (−1) e 4 |H| ?9−p 1 − µp ?9−p 1. (5.34)

52 5.2. Vacua of type II supergravity

p To obtain the last line we used H ∧ ?9−pH = (−1) ?9−p H ∧ H, and ?9−pH ∧ H = 2 |H| ?9−p 1. From this we obtain the following relation

p+1 p+1 − φ0 2 φ0 2 µp = e 4 |H| = e 4 |F6−p| . (5.35)

For the dilaton equation of motion (5.16), we have

2 −φ 1 2 p−1 φ p − 1 2 p − 3 p−3 φ 0 = ∇ φ = −e 0 |H| + e 2 0 |F | − e 4 0 µ . (5.36) 0 2 4 6−p 4 p

Using our Ansatz for F6−p (5.28), so not the BPS condition, we obtain

−φ 1 2 p−1 φ p − 1 2 − p+1 φ 2 p − 3 p−3 φ 0 = −e 0 |H| + e 2 0 κ e 2 0 |H| − e 4 0 µ 2 4 4 p −φ (p − 1) 2 2 2 p − 3 p−3 φ = −e 0 κ − |H| − e 4 0 µ . (5.37) 4 4 4 p

2 We can rewrite this by using the equation which relates µp to |H| (5.35), but then slighty modiﬁed in such a way that we did not ﬁx κ yet. This yields (p − 1) 2 2 p−3 φ p − 3 p−3 φ 0 = κ − κe 4 0 µ − e 4 0 µ . (5.38) 4 4 p 4 p

It is clear that κ = 1 is a solution of this equation, and any other solution found above for κ (5.32), is not a solution of this equation. This means that for our Ansatz for the F6−p, the only valid value is κ = 1, which is why we excluded the other solutions of the second order equation in κ. The only thing left to do is to investigate the interior part of the Einstein equations

1 1 1 R = ∂ φ ∂ φ + e−φ0 |H|2 − g |H|2 ij 2 i 0 j 0 2 ij 8 ij

5−n 1 n − 1 1 1 X φ0 2 2 S S + e 2 |Fn|ij − gij|Fn| + Tij − gijT . 2 (1 + δn5) 16 (1 + δn5) 2 8 n≤5 (5.39)

This equation simpliﬁes because the orientifold plane, and thus T S, lives only in the external dimensions, and the kinetic terms of the dilaton vanishes. This means that the above reduces to

1 1 p−1 1 5 − p −φ0 2 2 φ0 2 2 Rij = e |H|ij − gij|H| + e 2 |F6−p|ij − gij|F6−p| 2 8 2 (1 + δp1) 16 (1 + δn5) p + 1 p−3 φ − e 4 0 µ g . (5.40) 16 p 9−p ij

p+1 2 − φ0 2 2 This can be rewritten in a more insightful way using |F6−p|ij = e 2 |H| gij − |H|ij , which is derived in appendix C.1.1, but it does not hold for p = 6 because F0 has no

53 5.2. Vacua of type II supergravity index [31],

1 5 − p p−1 p + 1 p−3 −φ0 2 φ0 2 φ0 Rij = − e |H| + e 2 |F6−p| + e 4 µp9−p gij 8 16 (1 + δn5) 16 1 1 −φ0 2 2 −φ0 2 + e |H| gij − |H|ij + e |H|ij. (5.41) 2 (1 + δp1) 2 Forgetting for a moment that p = 1 is special, we use our relations found in (5.35) to obtain −φ 1 p+1 φ 5 − p p−1 φ − p+1 φ p + 1 p−3 φ R = − e 0 e 4 0 µ + e 2 0 e 4 0 µ + e 4 0 µ g ij 8 p 16 p 16 p 9−p ij

1 −φ p+1 φ + e 0 e 4 0 µ g , 2 p ij = 0. (5.42)

This means that our internal space is Ricci ﬂat. We solved all the equations of our system, thus we found a p + 1-dimensional Minkowski vacuum if and only if the following conditions are satisiﬁed:

p+1 p − φ0 F6−p = (−1) e 4 ?9−p H, (5.43)

dH = 0 and dF6−p = 0 and Fi = 0 if i 6= 6 − p (5.44) p+1 p+1 − φ0 2 φ0 2 µp = e 4 |H| = e 4 |F6−p| (5.45)

Rij = 0. (5.46)

For p = 1 the Hodge dual of F5 needs to be added. What are the physics of this solution? We used as input that we wanted Minkowski space as external manifold, and that the sourcing of the orientifold plane was smeared into the compact dimensions. This smearing enabled us to take the dilaton constant, and to work with a non-warped metric. The only-non vanishing form fields were H and F6−p, this to make sure to cancel the tadpole condition of the Op-plane. The smearing enabled us to cancel the magnetic sourcing of the term H ∧ F6−p of the F8 field, permitting us to set it to zero.The fact that F8−p was set to zero, the requirement of Minkowksi space, and the constantness of the dilaton, led us to the BPS condition. From this condition and the fact that the F8−p field vanishes, we obtained a relation between the charge of the orientifold plane and the strength of the fluxes (5.45). Using this into the trace-reversed Einstein equations for the compact dimensions, we saw that the internal manifold has to be flat. This is a consequence of the exact cancellation of the energy contributions from the fluxes and the orientifold plane, which is due to (5.45).

5.2.2 One-dimensional vacua.

In this subsection the calculation above is expanded to the case where p = 0, meaning there is only one non-compact dimension, time. This calculation is rather

54 5.2. Vacua of type II supergravity analogous to the previous one therefore it will not be shown in full details. However because it is one of our main results, the full calculation is shown in appendix C.1.2. Smearing the sources implies the following Ansatz

2 2 2 ds10 = ds1 + ds9, (5.47) and the non-vanishing fields are H, F6, and φ = φ0. Note that because we only have one external dimension, the fact that Rµν = 0, is a generic feature. Therefore the fact that we obtain flat-space after compactification is not because we want a vacuum, it is something which always holds. Looking at the trace-reversed Einstein equations in the external space, and using

1 − φ0 F6 = e 4 κ ?9 H (5.48) as Ansatz, a second order equation for κ is obtained. This is given by 1 5 7 0 = −e−φ0 ? H ∧ H − e−φ0 κ2 ? H ∧ H + κe−φ0 ? H ∧ H. (5.49) 8 9 16 9 16 9 This equation has 2 κ = 1 and κ = , (5.50) 5 as solution. In the appendix we check that if the ﬂuxes satisfy the BPS conditions, the Bianchi identities are mapped to the equations of motion, and vice versa. The Bianchi identity for F8 implies the following relation between the charge density of the orientifold plane and the H-ﬂux,

1 − φ0 2 µp = e 4 κ|H| . (5.51)

Using this in the dilaton equation of motion results in 1 1 3 0 = ∇2φ = −e−φ0 |H|2 − e−φ0 κ2|H|2 + κe−φ0 |H|2 = 0. (5.52) 0 2 4 4

2 From this we see that κ = 5 is not a solution and therefore can not be used. The internal Einstein equation is −φ 1 2 1 2 − 1 φ 1 2 5 2 1 − 3 φ R = e 0 |H| − g |H| + e 2 0 |F | − g |F | − g e 4 0 µ , ij 2 ij 8 ij 2 6 ij 16 ij 6 16 ij 0 9 (5.53) where the energy-momentum tensor was set to zero because it lives in the external dimension. Using the equation we derived in the appendix for |F6|ij (C.14), the Bianchi-identity for F8 and the BPS condition for F6, we arrive after some calculations at 1 1 5 1 R = − e−φ0 g |H|2 + e−φ0 |H|2g − g e−φ0 |H|2 − g e−φ0 |H|2. (5.54) ij 8 ij 2 ij 16 ij 16 ij

55 5.3. Flux compactiﬁcation

Hence the internal space is again Ricci-ﬂat. We obtained the following solution:

1 − φ0 F6 = e 4 ?9 H, (5.55)

dH = 0 & dF6 = 0 & Fi = 0 if i 6= 6, (5.56) 1 1 − φ0 2 φ0 2 µ9 = e 4 |H| = e 4 |F6| , (5.57)

Rij = 0. (5.58) This solution has the same physics as the one obtained in [31], except that we have p = 0.

5.3 Flux compactiﬁcation

In the above section we searched for a vacuum solution of the equations of motion of supergravity, with the input that we wanted Minkowski space as non-compact manifold. From this we found a vacuum, together with some consistency conditions on the ﬂuxes and the internal space. We now look at the p + 1-dimensional effective theory and investigate if we can stabilise the dilaton and the volume modulus in a simple way.

5.3.1 Gravitational part

We start the compactiﬁcation with the reduction of the ten-dimensional Ricci scalar in Einstein frame. We take a new Ansatz for the metric 2 2αv(x) 2 2βv(x) 2 dsb10 = e dsp+1 + e ds9−p. (5.59) We need to reduce the ten-dimensional Ricci scalar. The derivation of this is shown in the appendix C.1.3. The result of this derivation is p+1 9−p p 1 2 S = d xd y |g10| Rb10 − (∂φ) ˆ b 2 p+1 q 1 2 1 2 =Ve d x |gp+1| Rp+1 ± (∂v) − (∂φ) , (5.60) ˆ 2 2 where the sign in front of the kinetic term is negative p > 1, and positive for p = 0. The choice of the parameters in the metric is given by: −(9 − p) α = β, (5.61) (p − 1) (9 − p) α2 = , (5.62) 18(p − 1) for p > 1, and for p = 0 we have α = 9β and β = √1 . Note that for p = 0 the 12 19 equation for α coincides with the formula for general p.

56 5.3. Flux compactiﬁcation

5.3.2 Potential for the ﬁelds

Now that we have a way to obtain canonically normalised fields we investigate the potential. The action in Einstein frame for the fluxes is: p+1 9−p p −φ 1 2 1 p−1 φ 2 Sflux = − d xd y |g10| e |Hb| + e 2 |Fb6−p| . (5.63) ˆ b 2 2 From compactification of the determinant we get q β 2 2 q p ((p+1)α+(9−p)β)v (p−9) [−(9−p) +(p −1)]v |gb10| = |gp+1|e = e |gp+1|, (5.64) where we used (5.61). The H3-term receives three inverse metric contributions and the F6−p-term receives 6 − p contributions. This means we get as potential

β 1h 2 [−(9−p)2−6(p−9)+(p2−1)]v−φ Vflux(φ, v) = − |Hb| e (p−9) (5.65) 2 β p−1 2 [−(9−p)2−2(6−p)(p−9)+(p2−1)]v+ φi + |Fb6−p| e (p−9) 2 . (5.66) The orientifold is an n + 1 dimensional object, where we take n ≥ p. The action of the orientifold contains the determinant of the pullback of the metric on its world volume. This means that the determinant of the metric gives a contribution that we are not used to 2 p −1 +((n+1)−(p+1)) βv q ∗ p ((p+1)α+((n+1)−(p+1))β))v p p−9 |gbn+1| = |gn+1|e = |gn+1|e . (5.67) Therefore the contribution of the orientifold plane is given by p2−1 p−9 +((n+1)−(p+1)) βv−(p−3)φ/4 VOp (φ, v) = |µp|e . (5.68) In our compactifications we use a space filling orientifold plane, therefore n = p, so the contribution becomes p2−1 p−9 βv−(p−3)φ/4 VOp (φ, v) = |µp|e . (5.69) The total potential is therefore equal to 2 1 (p −1)βv h 2 V (φ, v) = − e (p−9) |Hb|2e−[(9−p) +6(p−9)]v−φ (5.70) 2 2 −[(9−p)2+2(6−p)(p−9)]v+ p−1 φ −(p−3)φ/4i + |Fb6−p| e 2 − 2|µp|e . (5.71) The potential is defined by an integration over the compact space, which we omitted. To continue to make progress we calculate the following term

− 1 [(9−p)2+6(p−9)]v−φ − 1 [(9−p)2+2(6−p)(p−9)]v+ p−1 φ −2 He 2 ∧ Fb6−pe 2 2 ˆ = −2 H ∧ F e−(p−3)φ/4 ˆ 6−p = −2 |µ |e−(p−3)φ/4, (5.72) ˆ p

57 5.4. Four dimensions compactiﬁed to 1D where in the last line we used the tadpole cancellation condition (5.12). With this result we can make the total potential to be a complete square:

(p2−1)βv 2 1 h − 1 Av− 1 φ − 1 Bv+ p−1 φi V (φ, v) = − e (p−9) Heb 2 2 − Fb6−pe 2 4 , (5.73) ˆ 2 where we have that A = (9 − p)2 + 6(p − 9) and B = (9 − p)2 + 2(6 − p)(p − 9). We see that if we perform a ﬁeld redeﬁnition and call

−2 q |H|2 ! (p+1) (A − B) ln 2 ϕ = v(x) + |F6−p| , (5.74) 1 0 then this field makes what is inside the square vanish. Therefore this field solves the equation ∂ϕV = 0, for any value we assign to it. Hence this field is massless, thus we proved that we have a massless direction. In this derivation we only used (5.61), which also holds for the case p = 0. This means that the compactification to one-dimension has one massless direction.

Conclusion: We found that a compactiﬁcation down to (p+1)-dimensional Minkowski space using H, F6−p, and Op planes does not stabilise all the moduli. A linear combination of the volume modulus and the dilaton ﬁeld is massless. This means that this model has no preferred scale for the compact dimensions and it costs no energy to change their size. This is a problem which could maybe be cured using non-perturbative effect. However this is not done in this thesis.

We leave the ten dimensional regime behind us and try a different method to ﬁnd a meta-stable one-dimensional vacuum.

5.4 Four dimensions compactiﬁed to 1D

We are interested in finding one-dimensional vacuum which can spontaneously decompactify to a four dimensional one. The calculations in the smeared-limit in the previous sections showed that there exist one-dimensional compactifications of type IIb, but those that were preformed had not all the moduli stabilised. The above can be seen as a proof of concept, but to connect these to four dimensional vacua we change our approach. In the literature there are some four-dimensional vacua of type IIb supergravity where the shape-moduli and sometimes the Kaehler¨ moduli of the extra dimensions are stabilised. Some notorious ones can be found in the so called GKP-, [16], and KKLT-, [1], papers. If our mechanism is real and there exist a one-dimensional vacuum which decompactifies to one of those vacua, then we could in principle compactify the remaining three dimensions of the KKLT-, or other vacua, to one- dimension. The result of this has then a chance to decompactify to four dimensional theory we compactified. This is what we are going to try to do in this section.

58 5.4. Four dimensions compactiﬁed to 1D

Note that the above assumption is not a strong one. Any theory using extra dimensions needs four dimensional phenomenologically acceptable vacua. If this is not the case this theory does not describe the world we live in. Therefore looking at one dimensional vacua of this theory, by compactifying the remaining three dimensions does seems to be a sensible thing. We begin our compactiﬁcations from the following action

1 0 3 ip 2 2 2 2 S4D = dx d x |g4| Rb4 − Λ − 2κ |H| + |F2| − |F1| . (5.75) 2κ2 ˆ b

This action is connected to string vacua via the assumption that the ingredients we use in these models have a higher dimensional origin. To be more precise, we take a manifold M10 = R × M3 × M6, meaning we have the product of a spatial three-dimensional, a spatial six-dimensional manifold, and the time. The moduli-stabilisation of six dimensions yields a cosmological constant Λ, in the four dimension remaining, but note that the CC could be zero. The CC term arises ∗ ∗ from V6D(ϕ ), where V6D is the moduli potential for six of the dimensions, and ϕ represents the critical values of the moduli associated to them. The fluxes we use wrap cycles on the M3, so play no role in the compactification of M6. We can also have axionic fields in the four dimensional theory. These come from the reduction of the higher form fields running in the compact six dimensions [32]. These scalar fields are potentially massless, however this pose less severe phenomenological problems then massless moduli. Axions are even considered as candidates for driving inflation, for dark matter, etc [32]. The contribution of these in the action is packed in the F1 and F3. We stress that even tough the calculations performed in this section can be related to string theory compactifications, they stand-alone. If it turns out that there are no extra dimensions, this calculation and its conclusion are still correct and applicable. The fact that there are fluxes wrapping cycles on the 3D space, is an assumption. This assumption can be made with or without the use of extra dimensions, however if the manifold admits background fluxes due to its non-trivial topology it would be surprising that their are none. In this subsection we proof that there are no stable one-dimensional vacua obtained by compactification over a three-dimensional compact orientable manifold using the above mentioned ingredients.

5.4.1 Proof no stable 1D vacua

The proof consists of two dinstinct parts. In the ﬁrst part it is proven that: curvature, ﬂuxes and the cosmological constant, have a universal dependence on the volume modulus. This dependence is set by the dimensionality of the manifold we are compactifying on, and nothing else. The second part uses this, and thus the universality of the equations ∂φV = 0, V = 0, to prove the theorem.

59 5.4. Four dimensions compactiﬁed to 1D

Part I: Universal dependence on the volume modulus

The proof starts by defining a metric Ansatz, we take the most general metric possible without factors dt × dei, namely: 2 2αφ 2 2βφ i j dsb4 = e ds1 + e Mij e ⊗ e , (5.76) where Mij is a positive definite symmetric matrix with dependence on the external coordinates, and the ei’s are veilbein which can have dependence on the internal coordinates. The components which contain a time-like direction and a space-like direction,dt × dei, are left out of the metric because they have no influence on the potential. The scalar potential which arises from the ingredients we use does only depend on the block diagonal components, not on the “mixed” compontents. Hence taking the Ansatz as above, encompasses also the cases where we can add “mixed” metric components. This is the most general metric for a closed orientable three-dimensional manifold. This is because for these type of manifolds there exist a proof, given in [33], which says such types of manifolds are always parallelizable. However because every compact subspace of a Hausdorff space is closed [34], 3D compact orientable manifolds are therefore always parallelizable. This means that we can globally define vielbeins, and using them we can write any metric in this form. The numbers for α and β are fixed by the fact that we want our fields to be canonically normalised. These numbers come from the gravitational part and depend on the dimensionality. They have been calculated in the reduction performed in toymodel 4 and are given by equation (4.84). The first ingredient we investigate is the cosmological constant. This part of the action is given by p S = − dxd3y |g |Λ, (5.77) Λ ˆ b4 thus its contribution to the scalar potential is completely determined by the determinant of the four-dimensional metric. Compactifying this term we arrive at √ 3φ VΛ = −Λe . (5.78) The next ingredient is the three-form flux, and from here onwards we absorbe the factor 2κ2 in equation (5.75) in the definition of the fluxes . If the manifold on which we compactify is compact and orientable we can always define a volume form. Because the magnetic three-form flux is a top-form, it is related to this form by a number. Therefore 3 Hijk = qijk, (5.79) from which we get abc ai bj ck −6βφ ai bj ck H3 = qgb gb gb ijk = qe M M M ijk. (5.80) 2 Therefore we find that |H3| is equal to 1 q2 √ |H |2 = H Hijk = e−6βφ MiaMjbMkc = q2e−6βφ det (M) = q2e− 3φ. 3 3! ijk 3! ijk abc (5.81)

60 5.4. Four dimensions compactiﬁed to 1D

√ Substituting this into the action, and using that we obtain a factor e 3φ from the determinant of the metric, we arrive at

2 VH3 = −q . (5.82)

For the two-from contribution we have,

F2 = dA1 + Σ, (5.83) where i A1 = χi(t)e + χ(t)dt (5.84) describes ﬂuctuations of the electromagnetic ﬁeld, and

1 2 0 2 0 1 Σ = α0 e ∧ e + α1 e ∧ e + α2 e ∧ e , (5.85) describes background fluxes wrapping cycles of our internal manifold. The α’s are constant, and if there are no cycles in a certain direction we put the corresponding α’s to zero. Note that the χ’s also generate scalar fields in the compactified theory. From equation (5.84) we see that when we apply the exterior derivative on A, the term χ(t)dt vanishes. This means that we can write F2 as

i i 1 2 0 2 0 1 F2 = ∂tχi(t)dt ∧ e + χi(t)de + α0 e ∧ e + α1 e ∧ e + α2 e ∧ e . (5.86)

The ﬁrst term in this equation yields kinetic terms upon compactiﬁcation, so does not contribute to the potential. The second term can be rewritten as

χ (t)dei = χ (t)d ei (y)dxµ = χ (t)∂ ei (y)dxνb ∧ dxk = χ (t)∂ ei (y)dxj ∧ dxk, i i µ i νb k i j k (5.87) where in the last step we used that the vielbein is time-independent, which is because it is deﬁned as depending only on the internal coordinates. The contribution of this term to the potential is calculated by applying the Hodge dual on this, which yields two inverse metrics with components in the compact dimensions. Combining these with the square root of the determinant of the metric, the contribution to the potential becomes √ √ √ 2 3φ −2φ/ 3 2 φ/ 3 VF2 = F2 e e = F2 e , (5.88) where we absorbed the dependence on the other moduli in the prefactor. For the one-form contribution we have that

0 1 2 F1 = ∂tf(t)dt + β0e + β1e + β2e , (5.89) where we used that A0 = f(t), so dA0 equals the ﬁrst term in the above equation. The ﬁrst term is again a kinetic term which will not contribute to the potential.

Applying the star on the other terms means that we will have√ one contribution of the −φ/ 3 inverse metric on the internal space,√ yielding a factor e . This combined with the contribution of the determinant, e 3φ, results in √ 2 2φ/ 3 VF1 = −F1 e , (5.90)

61 5.4. Four dimensions compactiﬁed to 1D

2 where the other moduli are absorbed into the F1 . The last contribution we need to calculate is from the curvature, which is obtained by reducing the higher dimensional Ricci scalar. We deﬁne as usual a tilted metric g = e2βφ e2(α−β)φds2 + M ei ⊗ ej = e2βφg , (5.91) bµbνb 1 ij eµbνb and use the formula for a rescaled Ricci tensor √ −2βφ −φ/ 3 Rb4 = e Re4 − ... = e Re4 − ... , (5.92) where the dots refer to kinetic terms. The contribution to the scalar potential from the curvature comes from the term Re3, which is the Ricci scalar of the compact manifold. However in the metric ge we see that the metric on the compact manifold does not have any terms containing the volume modulus. Therefore if we continue R our reduction,√ the dependence of e3 on the volume modulus will no longer change√ and stays e−φ/ 3. Taking the terms coming from the determinant of the metric, e 3φ, into account we can therefore state that the contribution from the curvature is √ 2φ/ 3 VR = Re . (5.93)

As usual we absorbed all the dependence on the other moduli in the factor R, which is positive for positive curvature and negative for negative curvature. We proved that the dependence of the scalar potential on the volume modulus is universal. Collecting all the different ingredients together, we have that the total potential is √ √ √ √ 2φ/ 3 2 3φ 2 φ/ 3 2 2φ/ 3 V = Re − q − Λe − F2 e − F1 e . (5.94) Note that we anticipated that the scalar modulus has the wrong sign for its kinetic term, hence all the energy contributions we are used to from the compactiﬁcations to D > 2, ﬂip sign.

Part II: Necessary conditions for vacua

With the potential at hand we can start to derive necessary conditions for the appearance of vacua. Therefore we need to solve the two equations: ∂φV = 0 and V = 0, which are given by √ √ √ √ 2φ/ 3 3φ 2 φ/ 3 2 2φ/ 3 0 =2Re − 3Λe − F2 e − 2F1 e , (5.95) √ √ √ 2 2φ/ 3 2 3φ 2 φ/ 3 0 = R − F1 e − q − Λe − F2 e . (5.96)

Let us assume that the we can fix the volume modulus. We can then make the following field redefinition: φ → φ − φc, (5.97) where φc is the critical value. This means that we can set the exponentials equal to one and check if the above equations are consistent. If they are this means that

62 5.4. Four dimensions compactified to 1D we can find a critical value for the field and the assumption was valid. If not, then the assumption that we can fix φ is false and thus we have no vacuum. Note that if we perform this redefinition the coefficients of our different contributions change by some positive constant factors containing φc, which we absorb in our definitions 2 of the parameters R, Λ, etc . Introducing new notation A = (R − F1 ), we find the following equations after the field redefinition:

2 0 =2A − 3Λ − F2 , (5.98) 0 =A − q2 − Λ − F 2. (5.99)

From the ﬁrst equation we obtain

2 F2 = 2A − 3Λ. (5.100)

2 Because F2 is positive, we obtain constraints on the parameters in the form of inequalities. These are shown below and need to be read as ”if ... holds, then we have ....”: 3Λ A > 0, Λ > 0 : A ≥ , (5.101) 2 A < 0, Λ > 0 : No solution, (5.102) 3|Λ| A < 0, Λ < 0 : |A| ≤ , (5.103) 2 A > 0, Λ < 0 : No extra conditions. (5.104)

Using equation (5.100), in the second equation we obtain

0 = A − q2 − Λ − 2A + 3Λ, (5.105) q2 = −A + 2Λ, (5.106) from which we get the inequalities:

A > 0, Λ > 0 : A ≤ 2Λ, (5.107) A < 0, Λ < 0 : |A| ≥ 2|Λ|, (5.108) A > 0, Λ < 0 : No solutions. (5.109)

The inequality obtained for the second case (5.108), is in contradiction with inequality (5.103). Therefore in the presence of a two-form ﬂux, the only choice of parameters yielding a vacuum is A > 0, Λ > 0, if they obey the following 3Λ ≤ A ≤ 2Λ. (5.110) 2

2 Taking Λ = 0, yields that F2 = 2A, and substituting this into the second equation (5.99) results in 0 = −A − q2, (5.111)

63 5.4. Four dimensions compactiﬁed to 1D

2 which has no solution. Taking A = 0 means that 3Λ = F2 , which comes from (5.100). Using this in (5.99), we obtain

0 = −q2 − 4Λ, (5.112) which has no solution. We found a critical point of the potential which solves the constraint V = 0, but we do not know if it is stable. Therefore we calculate the second derivative with respect to the potential yielding

2 2 3∂φV = 4A − 9Λ − F2 , (5.113)

2 where we multiplied this equation by a factor 3. Substituting our value for F2 we obtain 2 ∂φV = 2A − 6Λ. (5.114) From the inequality (5.110), we see that for the critical value this is always smaller than zero. This means that in the critical point, there is always a direction in ﬁeld space where the potential exhibits a runaway behaviour. Therefore, no matter what the amount of mixing between the different moduli is, this critical point is unstable in one direction.

Conclusion: If we compactify the theory given by equation (5.75) to one dimension on a compact orientable three-manifold, we see that necessary conditions to obtain 2 a solution to ∂φ = 0 and V = 0, are F2 6= 0, A > 0 and Λ > 0, in addition with some bounds on the value of A (5.110). Note that these are necessary but not sufﬁcient conditions, we worked under the assumption that the other moduli can be stabilised. However we proved that even tough we can solve these equations, the solutions will never yield a stable vacuum.

5.4.2 Extrema of the potential

In the previous section we proved that we can not ﬁnd stable vacua, however we do not abandon the idea of a compactiﬁcation from 4D to 1D. In our construction we want to tunnel from a meta-stable region, hence we need that a potential has at least two extrema, one local mimimum and one local maximum. If the scalar potential has only one extremum it means that it is either global or a saddle point, which means that we can not use them for our mechanism. In the theorem above we derived the following two equations √ √ √ √ 2φ/ 3 3φ 2 φ/ 3 2 2φ/ 3 0 =2Re − 3Λe − F2 e − 2F1 e , (5.115) √ √ √ 2 2φ/ 3 2 3φ 2 φ/ 3 0 = R − F1 e − q − Λe − F2 e . (5.116)

Instead of looking for vacua, this time we just look for extrema of the√ potential. Hence we only want to solve the ﬁrst equation. To do this we call x = eφ/ 3, and we impose

64 5.4. Four dimensions compactiﬁed to 1D the constraint x > 0. This transforms equation (5.115) into the following equation:

2 2 2 0 = x −3Λx + 2 R − F1 x − F2 . (5.117)

2 If we set one of the parameters equal to zero, except F1 , this equation has at most one solution, hence only one extremum. This means that we cannot use it for our mechanism. If we do not set a parameter equal to zero and we continue with the above second order equation, we obtain as solutions

(R − F 2) ∓ p(R − F 2)2 − 3ΛF 2 x∗ = 1 1 2 , (5.118) 3Λ where we have the constraint

2 2 2 (R − F1 ) > 3ΛF2 . (5.119)

The constraint comes from the fact that we want real solutions for x. Note that we also excluded the solution x = 0, and that all the parameters are in fact functions of the other moduli. Using the inequality x > 0 for different choice of parameters, we ﬁnd some additional constraints on the system

|(R − F 2)| + p(R − F 2)2 + 3|Λ|F 2 R − F 2 < 0 ⇒ Λ < 0 ⇒ x∗ = 1 1 2 , (5.120) 1 3|Λ| |(R − F 2)| ± p(R − F 2)2 − 3|Λ|F 2 R − F 2 > 0 & Λ > 0, ⇒ x∗ = 1 1 2 , (5.121) 1 3|Λ| −|(R − F 2)| + p(R − F 2)2 + 3|Λ|F 2 R − F 2 > 0 & Λ < 0, ⇒ x∗ = 1 1 2 . (5.122) 1 3|Λ|

The statements on the left are the different cases we can investigate, the things on the right are the extra constrains and the solutions for x. Remark that this is in accordance we what we found in the previous section, equations (5.101)-(5.104). From the above we see that the only case were the potential has two extrema is 2 described by equation (5.121), as long as F2 6= 0. In appendix C.2 we check that these correspond to a local minimum and maximum without any further restrictions on the parameters in this model.

5.4.3 Analysis of the scalar potential

We found that there is a possibility to have a local minimum of the potential. This is interesting because we proved that there are no stable vacua obtained from a four dimensional compactification satisfying our assumptions of the theorem in the previous section. This means that if we want to continue with our approach of constructing the decompactification mechanism using a four-dimensional compactification, we need to look for other methods than tunneling out of a vacuum.

65 5.4. Four dimensions compactiﬁed to 1D

In Figure 5.1√ at the√ bottom of√ this page we plotted the following function: V (φ) = 4.2e2φ/ 3 − e 3φ − 3.8eφ/ 3. In this plot we see that there is a pocket in the potential around the local minimum. If the system is in that local minimum it will not stay there, due to the fact that the total energy has to be equal to zero. Therefore because the system has negative potential energy, it will be forced to oscillate around that local minimum. If we look at Figure 5.1, we see that the volume modulus can not go over the maximum. This is due to the fact that it would require negative energy to satisfy 2 the condition Tµν = 0. Turning on H3-flux has the effect of contributing −q to the energy. The effect of this is that the potential does not change in shape, but it is just shifted downwards. Therefore adding too much H3-flux has as consequence that the local maximum becomes negative and so the volume modulus can, and will, role over it, and as a result the system decompactifies. Therefore we are not in a meta-stable state so this effect of the H3-flux needs to be taken into account. Looking at the left side of Figure (5.1), we see that the function asymptotes to the value −q2. This will always occur, for any of our choices of parameters. Therefore we see that φ can evolve to −∞, which means that the compact manifold shrinks to zero size. This would destroy the system and so it seems that we cannot use a compactification of a theory given by (5.75) to find a meta-stable one-dimensional system. However there is a way out of this situation. The ingredients we used to derive this potential are all classical. If the volume evolves to very small scale, we need to take quantum corrections to the potential into account, where an example of 2 2 which could be terms like |F2| |F2| . It is not unprobable that the potential receives positive corrections which scales like ∼ e−aφ, where a is a positive constant. If this is the case, V (φ) will evolve to infinity when φ → −∞, preventing the system to collapse. This has as consequence that we can have a universe where three spatial dimensions are compact and such that the volume of these stably oscillates. This

V( )

√ √ √ Figure 5.1: Plot of V (φ) = 4.2e2φ/ 3 − e 3φ − 3.8eφ/ 3. One dimensional slice of the multi-dimensional scalar potential. We do not show values on the axis because the only non coordinate system dependent quantity is the sign of the numbers. The actual values of the numbers, except zero, have no physical meaning.

66 5.4. Four dimensions compactified to 1D means that the dimensions blow-up and then deflate again eternally, until the volume modulus tunnels and the system decompactifies. If the corrections give negative energy contributions, V (φ) → −∞, the system becomes highly unstable and its collapse is accelerated. Because these corrections are only important when φ goes to −∞, they will probably not affect the behaviour of the potential near the local extrema too much, hence our qualitative picture of that region of the potential still holds. We stress that these quantum corrections are not something we investigated, and even tough there have to be corrections, we do not know what they are going to be. To conclude this section we stress that Figure 5.1 can be misleading. In re- alistic compactifications we do not have one modulus, but a whole collection of them. Therefore we should analyse a multidimensional potential, which is highly model dependent. The potential (5.94) describes one dimensional slices of such multidimensional potentials, therefore Figure 5.1 should be interpreted as such a slice. Hence if the other moduli are not stabilised, our analysis is not valid. Even if the other moduli are stabilised we need to take mixing between the different moduli into account. If there is too much mixing between the moduli, the oscillation of the volume modulus can destabilse the other moduli. However if for example the other moduli are much more massive than the volume modulus, there is not too much mixing and we can use the qualitative one-dimensional picture drawn above.

5.4.4 Compactiﬁcation M = R × S2 × S1 with axion ﬂux

We know that a necessary condition to find such pockets is that we have positive curvature. Therefore a compactification on a torus will not yield what we want. We also know that we need a two-form flux, so compactifications on a maximally symmetric space are also not good enough. 2 1 We try a compactification on M4 = R × S × S . For this we take the following metric Ansatz 2 2αφ 2 2γφh 2aϕ 2 2bϕ 2i ds4 = e ds1 + e e dsS2 + e dθ , (5.123)

2 where we have that dsS2 is the metric on the two-sphere. The constants a and b satisfy the relation 2a + b = 0, because we want that the overall volume of the compact dimensions is represented by φ. Another way to see this, is that we have 2aϕ 2 2bϕ 2 e dsE2 + e dθ = Mij and we want that it has determinant equal to one. For the compactiﬁcation of the gravitational part we follow what we have done for the torus and the sphere. The Rb4 term yields the following expression

−2γφ −2(α−γ)φh 2 2 i Rb4 = e Re4 − e 6γ + 12γ(α − γ) (∂φ) + 6γφ , (5.124) where we again used the trick to deﬁne a metric with a tilde and then use equation (4.10). We can split the Re4 in a one-dimensional and three-dimensional piece, but

67 5.4. Four dimensions compactiﬁed to 1D

because the one-dimensional does not contribute, we get Re4 = Re3. We can reuse the trick on this second term resulting in

−2bϕ 0 −2(a−b)ϕh 2 2 i Re3 = e R3 − e 2b + 4(a − b) (∂ϕ) + 4bϕ , (5.125)

02 2(a−b)ϕ 2 2 where the accents refer to the metric ds3 = e dsE2 + dθ . At last we have that 0 0 R3 = RE2 , because the part on the circle, being a one-dimensional manifold, does not contribute anymore. Using a last time (4.10), and remembering that R2 is an Einstein manifold we obtain

0 −2(a−b)ϕ R3 = e (2 − 2(a − b)ϕ) . (5.126) If we want to be in Einstein-frame and have a canonical normalisation for φ, we need to pick the same values for α and γ as in the case of Toymodel 4 (4.84). This means that we arrive at the following action

√ Ve3 h1 i S = dxp|g | (∂φ)2 + e2φ/ 3e−2aϕ 2 − 8a2 + 12a (∂ϕ)2 − 2a ϕ , grav 2κ2 ˆ 1 2 (5.127) where we used that 2a + b = 0. Now we fill our manifold with fluxes. Because S2 × S1 is orientable and compact 1 we can define a three-form flux H3, a one-form flux running into the S , and a two-form flux running onto S2. The three-form is as usual

1 2 H3 = βdx ∧ dx ∧ dθ, (5.128) and the one form ﬂux is C = qdθ. (5.129)

For the H3-ﬂux we have our usual contribution, for the C-ﬂux we have that

3 33 −2γφ−2bϕ C3 = q ⇒ C = gb C3 = qe , (5.130) where in the above equation we wrote explicitly the indices. The two-form ﬂux is given by 1 2 F2 = pdx ∧ dx , (5.131) where p is a constant. From this we see that we need to raise two indices with the inverse metric. Substituting these above expression into the action and using our values for α, γ and the relation between a and b we obtain √ √ p 2 2 2φ/ 3+4aϕ 2 φ/ 3−4aϕ Sflux = −Ve dx |g1| β + q e + p e . (5.132) ˆ At last we add a cosmological constant to obtain as potential √ √ √ √ V (φ, ϕ) = 2ke2φ/ 3e−2aϕ − 2β2κ2 − 2q2κ2e2φ/ 3+4aϕ − 2κ2p2eφ/ 3−4aϕ − Λe 3φ. (5.133)

68 5.4. Four dimensions compactiﬁed to 1D

2 1 Figure√ 5.2: Potential of√ the compactiﬁcation√ R × S √× S , with two moduli: V (φ, ϕ) = 0.5e2φ/ 3−2ϕ − 0.1e2φ/ 3+4ϕ − 0.05e 3φ − 0.001eφ/ 3−4ϕ − 2.5.

Finding the extrema of this potential is hard, therefore we do not do this. The figure below, Figure 5.2, shows a small part of the potential with fine-tuned values for the parameters. From this figure it is not exactly clear that we have the same behaviour of the potential as described above. We can add a quantum effect to this, which will have the effect of creating an exponential increase at the right side of the figure, which is shown in Figure 5.3. This shows that in the φ direction we have a stable oscillation, however ϕ is not stabilised. This model is therefore not viable.

5.4.5 Group manifold reductions

We know that a torus compactification or a compactification of a maximally symmetric space does not work, nor did our previous model. Therefore we move to more complicated models: group manifolds. We begin this subsection with the definition of a group manifold, afterwards we compactify (5.75) on some of them.

Deﬁnition of group manifolds:

Group manifolds are manifolds which are homogenous but not isotropic, for this reason these are often used in cosmology. From here the following is based on [11, 35, 36]. We want to introduce a dependence on the internal coordinates in the metric Ansatz, because it creates a potential for the moduli, which was mentioned in 3.2. However we want to do this in a way that the dependence on the internal coordinates does not appear in the reduced action. This is needed because

69 5.4. Four dimensions compactiﬁed to 1D

√ √ √ 2φ/ 3−2ϕ 2φ/ 3+4ϕ 3φ Figure√ 5.3: Potential V (φ, ϕ) = 0.5e − 0.1e − 0.05e − 0.001eφ/ 3−4ϕ − 2.5 + e−φ. This is the previous potential, but with the addition of a hypothetical quantum contribution e−φ. would there be any dependence on the internal coordinates, it would appear in the energy-momentum tensor in lower-dimensional Einstein equations. If this is the case the lower-dimensional metric will have a dependence on the internal coordinates sourced by the lower-dimensional energy momentum tensor. This would make the truncation of the higher modes of the lower-dimensional metric, which contain this dependence on the internal coordinates, not consistent [35]. It turns out that group manifolds have all the necessary properties we just mentioned [12], which is the reason why we compactify on them. The following is highly based on [36]. Group manifolds are special objects because in addition of the manifold structure, they also posses a group structure. To elevate a group to a manifold, we want that the group multiplication law is a diffeo- i morphism. This allows us to deﬁne a coordinate system: {y }i=1,...,dim(G), which enables us to parametrise the group elements g(y) ∈ G, where G is the group, and y denote the internal coordinates. Because we are on a manifold, we can deﬁne the following two maps

(Lh)(g) = h ∗ g and (Rh)(g) = g ∗ h, (5.134) which are called the left- and right translation, and where ∗ is the group multiplication. We can deﬁne a set of one-forms {ei} as follows i −1 e Ti = g dg, (5.135) where the Ti are the generators of the algebra associated to the group. These one-forms are invariant under left multiplication with a constant group element h, (h ∗ g)−1dh ∗ g = g−1 ∗ h−1 ∗ h ∗ dg = g−1dg. (5.136)

70 5.4. Four dimensions compactiﬁed to 1D

The one-forms defined in the above way can be used as vielbein’s. It turns out that they satisfy the following relation p r s p fmn = −2emen∂[res], (5.137) p where fmn are the structure constants of the group. From here on we do not follow [36] anymore. With these vielbeins we define the following metric 2 2αφ 2 2βφ i i j j dsD+n = e dsD + e Mij(x)(e + A ) ⊗ (e + A ), (5.138) i i a where the vielbeins are e = ea(y)dy , and they carry the dependence on the internal coordinates. The matrix Mij is a symmetric bilineair form. If we perform a KK-truncation, it means that we set all the lower-dimensional fields with dependence on the internal coordinates to zero. A field with an internal coordinate dependence transforms under a left-translation, fields with no-dependence on the internal coordinates are invariant under such a translation. This means that when we perform the truncation, we only keep the left-invariant fields, the so-called singlets. An inconsistency of the truncation would mean that we have fields invariant under left-translation in the equations of motion of fields with internal dependence [36]. This because the singlets would then source the truncated fields, so it is not consistent to put them equal to zero. However this is not possible because the metric is left-invariant by construction. Therefore the fields obtained from the reduction of the metric will only appear in the equations of motion for left-invariant fields, because you cannot construct something which is not left-invariant out of invariant objects [36]. Therefore these do not source the truncated fields, thus our reduction is consistent.

Compactiﬁcation:

We take as metric Ansatz √ √ 2 3φ 2 φ/ 3 i j ds4 = e ds1 + e Mij(x)e ⊗ e , (5.139) where it is clear that we truncated the vectors. Because we do a reduction of a 3D manifold we expect that there are six scalar fields, one of which is the volume modulus φ and the five others are in M. Two of them are so-called dilatons and the last three are axions. In [37] it is stated that the three axions can be consistently truncated away when −ϕ ϕ a(q2e − q3e ) = 0. This is a statement in the context of a compactification from 7D to 4D. Because the different terms appearing in the equations of motion do not depend, up to constant coefficients, on the dimensionality we started our compactification from, we claim that we can use their result. The compactifications we perform are on unimodular groups, for which a = 0 [37], hence we can consistently truncated the axions away. The following two formulas are taken from [37]. After some fields redefinitions we can parametrise the matrix M as √ √ √ −2σ/ 3 −ϕ+σ/ 3 ϕ+σ/ 3 Mij = diag e , e , e . (5.140)

71 5.4. Four dimensions compactiﬁed to 1D

The structure constant will satisfy the relation

i li i fjk = jklQ + 2δ[jak], (5.141)

li 1 and by choosing an appropriate basis one can always set, Q = 2 diag (q1, q2, q3), with qi = 0, ±1 and ak = (a, 0, 0). For unimodular group manifolds a = 0, thus we a see that fab = 0. We perform the reduction on Lie-groups, therefore we can use the following formula 1 decT = − f c ea ∧ ebT , (5.142) c 2 ab c c where Tc are generators of the Lie algebra associated to the Lie group, fab are the structure constants of the algebra, we copied this formula from [11]. This is the last formula we need to perform our calculation. We start by calculating the different contributions to the potential. We know that the H3 and Λ contributions are generic. 2 Form the above we know that we need F2 contributions, therefore let us calculate these. The F2 ﬂux is given by F2 = dA1 + Σ, (5.143) i where the most general Ansatz for A1 is A1 = χi(t)e + χ(t)dt, where we sum over the i-index. The χ’s are scalars and Σ is a closed but, non-exact two-form. This means that if we take

1 2 0 2 0 1 Σ = α0 e ∧ e + α1 e ∧ e + α2 e ∧ e , (5.144) with the α’s constant, we need to check for closeness and non-exactness. No matter what our choice of α’s is, this form is always closed, this for the i following reason. From equation (5.142) and the fact that fik = 0, we see that applying the derivative on Σ will result in the appearance of twice the same vielbeins in every term, hence due to the antisymmetric properties of forms it vanishes. 0 1 2 From this formula we also see that if for example f12 =6 0, then e ∧ e is an exact form. Therefore we need to put α0 = 0 to have a valid Ansatz. This reasoning can be generalized to ﬁnd i fjk 6= 0 ⇒ αi = 0. (5.145) The contribution of the two-form ﬂux to the potential is calculated in the appendix belonging to this section C.3, and it is given by √ √ √ √ φ/ 3h 2 −2σ/ 3 2 −ϕ+σ/ 3 2 ϕ+σ/ 3i V = −e A0e + A1e + A2e , (5.146)

2 i where we introduced the notation Ai = χifjk + αi . The last ingredient we need is the curvature. For group manifolds, the potential which arise from this curvature has been calculated in [37], we copy their results. Then we combine these with all the other ingredients to investigate if there is a way to obtain a pocket universe from a group manifold reduction. The potential, which

72 5.4. Four dimensions compactiﬁed to 1D

√ q 5 is taken from [37], but where we replaced 3 with 2/ 3 and we added an overall minus sign, gives 1 √ V = e2φ/ 3[Tr (MQ)2 − 2Tr (MQMQ)], (5.147) curv 2

il 1 with Q = 2 diag (q1, q2, q3). The minus sign we added is to take into account that the scalar ﬁeld φ has the wrong sign in its kinetic term. We can write the traces out, resulting in 1 √ h1 V = e2φ/ 3 (M q + M q + M q )2 curv 2 4 11 1 22 2 33 3 1 i − (M q )2 + (M q )2 + (M q )2 , (5.148) 2 11 1 22 2 33 3 with Mij the matrix elements of M. This can be rewritten as 1 √ h1 V = e2φ/ 3 (M M q q + M M q q + M M q q ) curv 2 2 11 22 1 2 22 33 2 3 33 11 3 1 1 i − (M q )2 + (M q )2 + (M q )2 , (5.149) 4 11 1 22 2 33 3 and substituting our expression for M (5.140) into this equation results in 1 √ h √ √ √ V = e2φ/ 3 e−ϕ−σ/ 3q q + e2σ/ 3q q + eϕ−σ/ 3q q curv 4 1 2 2 3 3 1 1 √ √ √ i − e−4σ/ 3q2 + e−2ϕ+2σ/ 3q2 + e2ϕ+2σ/ 3q2 . (5.150) 2 1 2 3 Before continuing we check if adding the minus sign was consistent. If we set all the q’s equal to one, we have SO(3) as group manifold. Truncating σ and ϕ away in this case, results in a reduction of S3 [37]. Doing this results in a potential with positive sign, as was the case in the S3 reduction, hence our sign convention is correct. The total potential becomes 1 √ h √ √ √ V = e2φ/ 3 e−ϕ−σ/ 3q q + e2σ/ 3q q + eϕ−σ/ 3q q 4 1 2 2 3 3 1 1 √ √ √ i − e−4σ/ 3q2 + e−2ϕ+2σ/ 3q2 + e2ϕ+2σ/ 3q2 2 1 2 3 √ √ √ √ √ 2 3φ φ/ 3h −2σ/ 3 −ϕ+σ/ 3 ϕ+σ/ 3i − q − Λe − e A0e + A1e + A2e , (5.151)

With the potential for a general unimodular group manifold, we start the analysis of the different existing ones, which we found in [37] but where classiﬁed by Bianchi.

The Heisenberg group: This manifold has q1 = q2 = 0, and q3 = 1. This means that we cannot put a two-ﬂux into the α3 direction, and we see that the terms χ1 and χ2 are equal to zero. The modulus χ3 appears quadratically into the potential, so

73 5.4. Four dimensions compactiﬁed to 1D

the equation ∂χ3 V = 0 has as solution χ3 = 0, hence we truncate this ﬁeld away. All this results in the following potential: 1 √ 1 √ √ √ h √ √ i V = − e2φ/ 3 − e2ϕ+2σ/ 3 − q2 − Λe 3φ − eφ/ 3 α2e−2σ/ 3 + α2e−ϕ+σ/ 3 . Heis 8 2 0 1 (5.152)

We see that the curvature is negative, this is not what we need.

ISO(1, 1): This group has q1 = 0, q2 = −1 and q3 = 1. We have the same argument for the χi moduli as above, so we obtain as potential 1 √ h √ 1 √ √ i V = e2φ/ 3 − e2σ/ 3 − e−2ϕ+2σ/ 3 + e2ϕ+2σ/ 3 4 2 √ √ √ 2 3φ φ/ 3h 2 −2σ/ 3i − q − Λe − e α0e . (5.153)

We see that the curvature potential is always negative, hence this is also not what we are looking for.

ISO(2): The group manifold has q1 = 0, q2 = 1 and q3 = 1, which means that our potential is 1 √ h √ √ √ i V = − e2φ/ 3 − 2e2σ/ 3 + e−2ϕ+2σ/ 3 + e2ϕ+2σ/ 3 8 √ √ √ 2 3φ φ/ 3h 2 −2σ/ 3i − q − Λe − e α0e , (5.154) which can be rewritten as 1 √ h i √ √ h √ i V = − e2(φ+σ)/ 3 sinh2(ϕ) − q2 − Λe 3φ − eφ/ 3 α2e−2σ/ 3 . (5.155) 2 0 We see that the stabilisation of the ϕ ﬁeld has as a consequence that the curvature vanishes, which is again not what we are interested in. To try to cure this we add an F1-ﬂux. This is given by 0 1 2 F1 = β0e + β1e + β2e . (5.156) Equations (5.141) and (5.142), implies that if we want closeness of this one-form, we have that qi 6= 0 ⇒ βi = 0. (5.157) Without checking for non-exactness, this means that at best

0 F1 = β0e . (5.158) √ To calculate the contribution of this ﬂux we raise this index with e−φ/ 3M00, which does not contain ϕ, and the determinant of the metric does neither. Therefore the one-form ﬂux will not change the critical value of ϕ, so ISO(2) can not be used for our purposes. The last three unimodular group manifolds are SO(2, 1), SO(3), and U(1)3. We

74 5.4. Four dimensions compactiﬁed to 1D have that U(1)3 =∼ T 3, so we do not check it, because it has no curvature. The other two have all their q’s non-zero, therefore they can not harbour a two-form ﬂux. Hence they can also not be used to construct what we want.

Conclusion: Group manifolds can not be used to construct a model of a meta-stable oscillating compact universe.

75 Chapter 6

Conclusion

In this thesis we introduced a new approach towards solving the cosmological constant problem. Before introducing this approach, we discussed the physics associated to possible extra dimensions. These can be used to solve different important problems in contemporary physics for example the gauge-hierachy in an elegant way. Extra dimensions are also a vital ingredient for superstring theory, which is in our community the most explored road towards a theory of quantum gravity, but in addition it has the ambition to unify all known interactions. However there is a price to pay when postulating extra dimensions. Besides the problem of light-moduli extra dimensions introduce a new hierarchy problem, which we argued is related to the CC-problem. The length-scale which characterises the three spatials dimensions we observe is the Hubble-scale. If there are extra compact dimensions, then they have to be smaller than any length-scale yet probed at the LHC. Therefore if those extra-dimensions are real, why is there such a difference between the three spatial dimensions we observe and the extra’s we do not observe? If there are extra dimension, why do we effectively observe a four-dimensional world? These are the questions which motivated this thesis. These questions are not asked a lot, but are very fundamental and interesting ones. Making progress in answering these questions means that we make progess towards solving the CC-problem. Even tough it has been discovered more than thirty years ago, this problem is still one of the great challenges theoretical high-energy physics faces today. Our approach to provide answers to these questions was the following. We tried to explain the unnatural situation we observe today by starting from the natural situation that all spatial dimensions are compact, and investigated if we can ﬁnd a dynamical mechanism which takes us from there to the situation we observe today. To achieve this we searched for one-dimensional classically stable vacua, but where the scalar potential is such that the volume modulus can quantum tunnel out of the vacuum, leading to a decompactiﬁcation of some of the dimensions.

6.1 Results

After introducing the physics associated to extra dimensions we performed some simple flux compactifications to become acquinted with the techniques involved in these calculations. We compactified to two dimensions to find Jackiw-Teiltelboim gravity, and to one dimension retrieving the Friedmann equations of cosmology. We remarked that in one dimension the volume modulus has the wrong sign in front of

76 6.1. Results its kinetic term, this is an important fact which we used later on. Once the different techniques were under control we moved away from the simple models. In [31], the vacuum ten dimensional supergravity equations for a product spacetime in the smeared orientifold limit are solved for an p + 1 external Minkowski spacetime, with p ≥ 1. We checked those calculations and expanded them in the case p = 0. By doing this we found a one-dimensional vacuum of type II supergravity, with a Ricci flat internal space and with H- and F6-flux, satisfying the following condition 1 − φ0 F6 = e 4 ?9 H, (6.1) where φ0 is the dilaton. After this we investigated a consistent subsector of the one- dimensional effective field theory obtained by compactification, namely we looked at fluctuations in the volume of the internal manifold and in the string coupling. This was done because we wanted to investigated if the volume modulus and the dilaton could be stabilised. We obtained a quadratic scalar potential, which had as consequence that one scalar field remained massless. This means that the ingredients mentioned above are not enough to find a compactification to one-dimension. After this result we left the ten dimensional regime behind us. A theory with extra dimension needs to have a compactification to four dimensions, therefore we investigated if we could compactify the three remaining dimensions to obtain a one-dimensional meta-stable vacuum. We took a four dimensional theory of gravity combined with electromagnetism, axions, a cosmological constant coming from the stabilising of the moduli of the extra dimensions, and investigated if such a theory can be compactified to one-dimension. Due to the universal dependency of the potential with respect to the volume modulus, we proved that stable one dimensional vacua, using the ingredients mentioned above, do not exist under our assumptions. This did not discourage us because we found that for certain values of the different parameters in our four dimensional theory there can potentially exist pockets of negative energy in the scalar potential. Depending on the total scalar potential and the quantum corrections to our model these pockets are possibly stable regions of the potential. In these pockets the universe is not static, but the volume is constantly fluctuating. This is due to the constraint that one-dimensional gravity imposes to us, namely that the total energy has to be equal to zero. If the universe is trapped in that region, the volume modulus eventually tunnels out of that pocket, leading to a spontaneous decompactification. This is not what we first had in mind, but it is exactly what we wanted. It turns out that these pockets only appear in certain circumstances. We found that the universe needs to be positively curved, and that there has to be a positive, non-vanishing, cosmological constant in combination with a background electromagnetic field. With these necessary conditions at hand we excluded possible models, for example maximally symmetric space and unimodular group manifolds. We also investigated a model where the spatial dimensions are the product of a two-sphere and a circle, but this did not work either because, one of the moduli was not stabilised.

77 6.2. Further research

6.2 Further research

There is a lot of work which needs to be done to further develop the mechanism proposed in this thesis. First of all we did not manage to construct a model with such a meta-stable pocket. Therefore other three dimensional manifolds need to be investigated to find a model which can yield the desired results. The mechanism which is proposed in this thesis can only occur if there are quantum corrections preventing the collapse of the system. It is therefore natural that we should investigate which type of corrections we obtain in our model. If we find a model which admits pockets, and which has the desired quantum corrections we are not done yet. We should try to understand what happens during the tunneling, and more importantly what happens after. This mechanism should be related to the real world, therefore it is interesting to see if it passes all the tests other cosmological models are subject to. Hence we should for example investigate what type of CMB-spectra the mechanism predicts? The last thing we mention is continuing with the ten-dimensional approach. We used a minimal amount of ingredients to construct a scalar potential, and we only investigated if we could stabilise the volume modulus in combination with the dilaton. This is a highly simplified setting, we should add extra ingredients such as for example non-perturbative effects and investigate how these affect the scalar potential.

It is clear that this thesis is only one small, but interesting step, in this whole enterprise. The CC-problem is something which was discovered more than thirty years ago, and to this day its solutions remains a mystery. Therefore it is no surprise that it has not been solved in this thesis. However I hope that the small step we took was in the right direction.

78 Appendix A

Conventies and theoretical background

A.1 Metric using vielbeins

In this thesis the tetrad formalism is not extensible used, but a metric written in this formalism does appear. The following text is almost literally copied from the book Supergravity [3]. We can diagonalise the metric in one point, but generally not on the whole manifold. This yields a b gµν = O µDabO ν , (A.1) a 0 1 D−1 with positive eigenvalues λ and such that Dab = diag(−λ , λ , . . . , λ ). Knowing this we can deﬁne locally the following a p a a eµ(x) := λ (x)O µ(x), (A.2) such that a b gµν(x) = e (x)µηabeν(x), (A.3) a a µ where ηab is just the Minkowski metric. The e (x) = eµ(x)dx are called vielbeins and are one-forms, which is something explained later in this chapter. The above equation (A.3) is the deﬁning equation for the veilbeins, and admits more solutions than those used for diagonalising the metric. This because the Minkowski metric is invariant under Lorentz transformation, so any vielbein satisfying 0a −1 a b eµ (x) = Λ b(x)eµ(x), (A.4) −1 a with Λ b(x) an inverse Lorentz transformation, is seen as equivalent to the one which diagonalises the metric. From here on we stop using [3]. With these new ingredients we can rewrite the line element as 2 µ ν a b µ ν a b dsD = gµν(x)dx ⊗ dx = e (x)µηabeν(x)dx ⊗ dx = ηabe (x) ⊗ e (x). (A.5)

A.2 Derivation Cosmological constant

We start this derivation from the Einstein-Hilbert action, but with the addition of a cosmological constant 1 √ S = dDx −g R − M 2Λ . (A.6) 2κ2 ˆ p

79 A.3. Consistency substitution of vacua into action

In the rest of the derivation we set the Planck mass equal to one. Because the only dynamical ﬁeld in this action is the metric, we perform the following variation to obtain its equation of motion, 1 δS 1 δ √ 1 δ √ √ = dDx√ −gR − √ −gΛ. (A.7) −g δgµν ˆ −g δgµν −g δgµν

√ √ µν √ 1 √ µν To continue, we use δ −gR = −gGµνδg & δ −g = − 2 −ggµνδg . These can be found in the appendix of [11]. This results in the following equations, 1 δS 1 √ = R − g (R − Λ) = 0. (A.8) −g δgµν µν 2 µν The last equality are the Einstein equations in vacuum with a cosmological constant. By contracting these equations with an inverse metric, we ﬁnd an equation for the Ricci scalar. Using this into the above, we can rewrite the Einstein equations in a more useful way. Λ R = g (A.9) µν (D − 2) µν

A.3 Consistency substitution of vacua into action

We check that is consistent to subsitute a vacuum solution into the action given by √ 1 1 S = dDx −g R − G (φ)∂ φi(x)∂µφj(x) − V (φ(x)) . (A.10) ˆ 2κ2 2 ij µ Varying this with respect to the metric, we easily see that we obtain the following Einstein equations, h1 1 1 i G = −2κ2 G (φ)∇ φi(x)∇ φj(x)− g G (φ)∇ φi(x)∇ φj(x) − V (φ) , µν 2 ij µ ν 2 µν 2 ij µ ν (A.11) where Gµν is the Einstein tensor. Substituting our vacuum solution into this we arrive at 2 ∗ Gµν = −κ gµνV (φ ), (A.12) If you substitute the vacuum solution directly into the action, (2.28), you obtain D √ 1 ∗ S ∗ = d x −g R − V (φ (x)) . (A.13) φ ˆ 2κ2 Performing a variation with respect to the inverse metric yields

2 ∗ Gµν = −κ gµνV (φ ), (A.14) which are the same equations as the one obtained above. This proves that you can substitute the vacuum solution into the action, and calculate the equations of motion with respect to this reduced action.

80 A.4. Conventions differential form calculus

A.4 Conventions differential form calculus

Before introducing differential forms we introduce the following convention 1 T = (T + even permutations − uneven permutation) , (A.15) [µ1...µn] n! µ1...µn where the permutation refers to permutation of indices.

Differential forms are antisymmetric tensors with all their indices down. Adding two antisymmetric tensors or multiplying an antisymmetric tensor with a number does not alter its antisymmetric properties. Due to this, differential forms of the D! same dimension span a vectorspace. This vectorspace has dimension p!(D−p)! , where D refers to the dimensionality of the manifold on which the forms are defined, and p indicates the number of indices of the forms. A non-vanishing form with more indices than the dimensionality of the manifold does not exist. This because there will be repeating indices, hence due to antisymmetry it vanishes. Forms with as much indices as the dimensionality of the manifold are called top-forms. The vectorspace of these top-forms is one-dimensional, and thus they are all related by a real constant. To integrate we define the volume-form, which in components reads p µ1...µn = |g|εµ1...µn , (A.16) with, ε01...n = 1, (A.17) where we followed the conventions of [11]. This is a top-form, implying that any other top-form is related by a constant to this one. From the formula of the dimensionality of the vectorspace of p-forms we see that the vectorspaces p and (D − p) have the same dimensionality. This means that we can define an isomorphism between p-forms and (D − p)-forms. This map is called the Hodge dual and it is defined as

1 µ ...µ ? (dxµ1 ∧ ... ∧ dxµp ) := 1 p dxν1 ∧ ... ∧ dxνD−p , (A.18) (D − p)! ν1...νD−p where the indices are raised using the inverse metric and we followed the conventions of [11]. We can introduce an operation to multiply a form to obtain a new forms with more indices. This is given by the wedge product, which satisﬁes the following relation

A(p) ∧ B(q) = (−1)pqB(q) ∧ A(p), (A.19) where the letters p & q denote the amount of indices the forms have, and the factor (−1)pq appears due to the antisymmetry property of forms. We deﬁne a map which maps n-forms on (n + 1)-forms in the following way

µ ν1 νn+1 dP := ∂[µPν1...νn]dx ∧ dx ... ∧ dx , (A.20)

81 A.4. Conventions differential form calculus

ν1 νn where P = Pν1...νn dx ∧ ... ∧ dx . Remark that due to the antisymmetry this operation is nilpotent meaning

d(dP ) = 0 for any P. (A.21)

This operation also satisﬁes the following property d A(p) ∧ B(q) = dA(p) ∧ B(q) + (−1)pA(p) ∧ dB(q). (A.22)

Differential forms are used in mathematics because it allows to integrate over a surface in a coordinate-independent way. This means that an integral of p-form an a p-dimensional surface, S, equals

B(p), (A.23) ˆS where there is no explicit reference to the metric.

82 Appendix B

Details of reductions in chapter 4

1 B.1 Reduction: M5 = M4 × S

In this section we perform the calculation of the reduction of ﬁve dimensional gravity over a circle in more detail. However we omit all the physics behind it, these are explained in the section 2.1. We start with a metric Ansatz given by

2 2αφ(x) 2 2βφ(x) 2 dsb5 = e ds4 + e dθ . (B.1) Here α & β are constant and φ is a scalar ﬁeld. The Ansatz can be rewritten in a more useful way as follows

2 2βφ(x) 2(α−β)φ(x) 2 2 dsb5 = e e ds4 + dθ , 2 2βφ(x) 2 2 dsb5 = e dse4 + dθ . (B.2)

2 2(α−β)φ(x) 2 In the last line we deﬁned dse4 = e ds4. The action functional is:

4 µ p S = d x dθ |g5|Rb, (B.3) ˆ b where gb5 and Rb are the determinant of the 5D metric, and the 5-D Ricci scalar deﬁned by (4.1). We want to rewrite everything in terms of gµν. To obtain this we use the following formula, which can be found in [11], to ﬁrst obtain the Ricci scalar in terms of the rescaled metric g = e−2βφ(x)g , eµbνb bµbνb R = R − (D − 2)β2(∂φ)2g bµbνb eµbνb e eµbνb +(D − 2)β2∂ φ∂ φ − (D − 2)β∇ ∂ φ − βg φ . (B.4) µb νb µb νb eµbνbe

Remark that we introduced the notation (∂φ)2, which means gµbνb∂ φ(x)∂ φ(x). Con- e e µb νb µν tracting the above with the inverse metric gbbb, we arrive at −2βφ(x) 2 2 2 2 Rb = e (Re − D(D − 2)β (∂φe ) + (D − 2)β (∂φe ) − (D − 2)βeφ − Dβeφ), −2βφ(x) 2 2 = e Re − (D − 1)(D − 2)β (∂φe ) − (2D − 2)βeφ . (B.5)

83 1 B.1. Reduction: M5 = M4 × S

The terms (∂φ)2 & φ need to be rewritten in terms of g . Therefore we use the e e √ µν following formula, φ = √1 ∂ −ggµbνb∂ φ(x), which can be found in [11]. This −g µb νb yields 1 p φ = ∂ −ggµbνb∂ φ(x) , e p µb ee νb −ge q 1 8(α−β)φ(x) −2(α−β)φ(x) µν =p ∂µ −ge e g ∂νφ(x) , −ge8(α−β)φ(x) 1 √ =√ ∂ −ge2(α−β)φ(x)gµν∂ φ(x) , −ge4(α−β)φ(x) µ ν e−4(α−β)φ(x) h √ = √ ∂ −ggµν∂ φ(x) e2(α−β)φ(x) −g µ ν √ i + 2 −g (α − β)(∂φ(x))2 e2(α−β)φ(x) ,

−2(α−β)φ(x) 2 =e φ(x) + 2 (α − β)(∂φ(x)) , (B.6) where to go from the ﬁrst to the second line we used that ∂θφ = 0. Using this in the previous equation we obtain,

h 2 Rb =e−2βφ(x) Re − 12β2 (∂φ) e−2(α−β)φ(x)

−2(α−β)φ(x) 2 i − 8βe φ(x) + 2 (α − β)(∂φ(x)) , −2βφ(x) −2(α−β)φ(x) 2 2 2 =e Re + e 4β (∂φ) − 8βφ(x) − 16αβ (∂φ(x)) (B.7)

Here the factors e−2(α−β)φ(x) appear from converting (∂φe )2 into (∂φ)2. 2 2(α−β)φ(x) 2 2 Re is deﬁned with respect to the metric dse5 = e ds4 + dθ . We calculate an interesting Christoffel symbol,

λb 1 λbσ Γe = g b ∂θgσµ + ∂µgσθ − ∂σgθµ , µθb 2e ebb beb be b 1 1 1 = gλbσb (0) + gλbσb ∂ g − gλbσb ∂ g , 2e 2e µbeσθb 2e σbeθµb 1 1 = gλθb ∂ g − gλbσb ∂ 1δ , 2e µbeθθ 2e σb θµb = 0. (B.8)

Here we used that g is blockdiagonal, therefore g = δ . Also due to the eµbνb eλθb λθb blockdiagonality one has that the inverse is also block diagonal, thus the blocks are the inverses of the blocks of the original metric. This gives us the following symbols:

θ θ µ µ Γeθθ = 0, Γeiθ = 0, Γeθθ = 0, Γeνθ = 0. (B.9) As a result the following Ricci tensor component equals,

µb µb µb µb λ µb λ Re = ∂µΓe − ∂θΓe + Γe Γeθθ − Γe Γe = 0. (B.10) θµθb b θθ θµb µλb θλ µθb

84 1 B.1. Reduction: M5 = M4 × S

The Ricci Scalar Re is obtained by contracting with the inverse metric. This gives us,

R = gµbνbR , e e eνbµb θθ θµ νθ νµ = ge Reθθ + ge Reµθ + ge Reθν + ge Reµν, νµ = ge Reµν. (B.11)

Now again Reµν is deﬁned with respect to geµν, but we need to have it with respect to −2(α−β)φ(x) gµν = e geµν. Therefore the formula found in [11] is again used, giving,

2 2 Reµν = Rµν − (D − 2) (α − β) (∂φ) geµν 2 +(D − 2) (α − β) ∂µφ∂νφ − (D − 2) (α − β) ∇µ∂νφ − (α − β) geµνφ . (B.12)

Now contracting this with the metric gµν and re-ordering some terms results in,

−2(α−β)φ(x) 2 2 Re = e R4 − (D − 1)(D − 2) (α − β) (∂φ) − (2D − 2) (α − β) φ . (B.13)

Remembering that D = 4 yields

−2(α−β)φ(x) 2 2 Re = e R4 − 6 (α − β) (∂φ) − 6 (α − β) φ . (B.14)

To be able to rewrite the action, we also need to rewrite the determinant. Therefore q p 2αφ 2βφ(x) (4α+β)φp |gb5| = |e g4 × e | = e |g4| (B.15) We arrived at the point were we can rewrite the action. We do this by ﬁrst using (B.7) and replacing the determinant, then in the next step we use (B.14). and we will use the trick of a block diagonal determinant. This all results in

4 µ p S = d x dθ |g5|Rb5, ˆ b 4 µ p (4α−β)φ −2(α−β)φ(x) 2 2 2 = d x dθ |g4|e Re4 − e 4β (∂φ) − 8β φ(x) − 16αβ (∂φ(x)) , ˆ h = d4xµdθp|g |e(2α+β)φ R − 6 (α − β)2 (∂φ)2 − 6 (α − β) φ ˆ 4 4 2 2 2 i + 4β (∂φ) − 8βφ(x) − 16αβ (∂φ(x)) ,

= d4xµdθp|g |e(2α+β)φ(x) R − 6α2 + 4αβ + 2β2 (∂φ)2 − (6α + 2β) φ . ˆ 4 4 (B.16)

Choosing β = −2α results in

S = d4xµdθp|g | R − 6α2(∂φ)2 − 2α φ ˆ 4 4

85 n B.2. Appendix: MD+n = MD × T ,

µ Note that φ = ∇µ(∂ φ), which is clearly a total derivative thus it can be dropped 1 out of the action. At last choosing α = 12 and integrating over θ our ﬁnal result becomes p 1 2 S = 2π d4xµ |g | R − (∂φ) . (B.17) ˆ 4 4 2 This is the usual Einstein-Hilbert action combined with that of a massless scalar ﬁeld.

n B.2 Appendix: MD+n = MD × T ,

n In the second model we perform the following compactification MD+n = MD × T , where T n = S1 × S1 · · · × S1. We take the following Ansatz for the metric. 2 2αφ(x) 2 2βφ(x) n m dsbD+n = e dsD + e Mmndy dy . (B.18) 2βφ(x) The y-coordinates are the coordinates on the torus and e Mnm is a n-dimensional symmetric matrix where M has determinant equal to one. We call the D + n- dimensional metric g , the metric on the external part is g , and g = e−2αφ(x)g . bµbνb bµν µν bµν During the reduction we use the same tricks as in the above reduction. We define two new metrics, g and g , which satisfy g = e2βφ(x)g = e2βφ(x) e2(α−β)φ(x)g , eµν µν bµbνb eµbνb µbνb such that we can use twice the formula for a rescaled Ricci scalar. Using the formula a first time yields

−2βφ(x) 2 2 RbD+n = e ReD+n − (D + n − 1)(D + n − 2)β (∂φe ) − (2(D + n) − 2)βeφ . (B.19)

In this equation there is a box which is deﬁned with respect to ge, but this object needs 2(α−β)φ(x) to be rewritten in terms of g. We have that geµν = e gµν, and gemn = gmn. Because nothing in y-dependent we obtain

1 p µν eφ = p ∂µ −gege ∂νφ(x) , −ge q 1 2D(α−β)φ(x) −2(α−β)φ(x) µν = p ∂µ −g e g ∂νφ(x) , −ge2D(α−β)φ(x) 1 √ = √ ∂ −ge(D−2)(α−β)φ(x)gµν∂ φ(x) , −geD(α−β)φ(x) µ ν e−2(α−β)φ(x) √ √ = √ ∂ −ggµν∂ φ(x) + (D − 2) −g (α − β)(∂φ(x))2 , −g µ ν −2(α−β)φ(x) 2 = e φ(x) + (D − 2) (α − β)(∂φ(x)) . (B.20) Substituting this into the above equation yields

−2βφ(x) 2 2 −2(α−β)φ(x) RbD+n = e ReD+n − (D + n − 1) (D + n − 2) β (∂φ) e

−2αφ(x) 2 − βe (2(D + n) − 2) φ(x) + (D − 2) (α − β)(∂φ(x)) , (B.21)

86 n B.2. Appendix: MD+n = MD × T ,

which is equation (4.10). To rewrite ReD+n we need the Christoffels. We start by calculating the following

λb 1 λbσ Γe = g b ∂µgiσ − ∂σgµi + ∂igσµ , µib 2e be b beb ebb 1 1 1 = gλbσb ∂ g − gλbσb ∂ g + gλbσb (0) . 2e µbeiσb 2e σbeµib 2e If we call the ﬁrst term of the last line A and the other B, we arrive at 1 A = gλbσb ∂ g , 2e µbeiσb 1 = gλjb ∂ g , 2e µbeji 1 = (M)kj ∂ (M ) δλ,kb , (B.22) 2 µb ij and for the B term, 1 B = − gλbσb ∂ g , 2e σbeµib 1 = − gλjb ∂ (M ) δ , 2e j ki µ,k 1 λb = − ∂e (Mki) δµ,k. (B.23) 2 Here the tilde on the partial denotes that we have raised the index using ge. This results in the following Christoffel symbols:

µ µ 1 k i j 1 jk Γe = 0, Γe = − ∂e Mij, Γe = 0, & Γe = (M) ∂µ (Mik) . νi ij 2 jk µi 2 (B.24) The last Christoffel needed complete the calculation is the following,

i 1 iσ Γe = g b ∂µgνσ − ∂σgµν + ∂νgσµ , µν 2e e b be eb 1 1 1 = giσb∂ g − giσb∂ g + giσb∂ g , 2e µeνσb 2e σbeµν 2e ν eσµb 1 1 1 = gij∂ g − gij∂ g + gij∂ g , 2e µeνj 2e jeµν 2e ν ejµ = 0. (B.25) With the Chirstoffels at our disposal we continue and calculate the Ricci tensor, and scalar. We begin with

µb µb µb µb λb µb λb Re iµyj = ∂µΓeij − ∂iΓejµ + Γe Γeij − Γe Γeµj, b b b µbλb iλb b µ µ ν k ν k µ ν l = ∂µΓeij + ΓeµνΓeij + ΓekνΓeij − ΓejµΓeki − ΓejlΓeνi, 1 µ 1 µ ν 1 kl µ = − ∂µ∂e Mij − Γe ∂e Mij − M ∂µMlk∂e Mij 2 2 µν 4 1 kl µ lk µ + M ∂µMjl∂e Mki + M ∂µMik∂e Mjl . (B.26) 4

87 n B.2. Appendix: MD+n = MD × T ,

To go from the first to the second line we expanded the first line and used that a lot of the Christoffels vanish in combination with the fact that any derivative with respect to a y-coordinate also vanishes. Going from the second to the third line was done by just filling in our results for the Christoffels, (B.24). During the reduction we want to rewrite the above equation in a more compact µ µ ν way, but to do this we need the equation eMij = ∂µ∂e Mij + Γeµν∂e Mij. This µ µ µb ν equation is true because ∇µ∂ebMij = ∂µ∂ebMij + Γe ∂ebMij, due to the fact that M b b µbνb is matrix of scalar functions and thus a scalar. This means that when we apply a partial derivative on M, the result is a vector. Now simplifying this expression we arrive at the claim

µ µb ν µ µb ν ∂µ∂ebMij + Γe ∂ebMij = ∂µ∂e Mij + Γe ∂e Mij, b µbνb µνb µ µ ν i ν = ∂µ∂e Mij + Γeµν∂e + Γeiν ∂e Mij, µ µ ν = ∂µ∂e Mij + Γeµν∂e Mij, (B.27) where the second Christoffel on the second line vanishes because it is multiplied ν ij ν with a factor ∂e Mij, resulting in M ∂e Mji which is zero due to (4.11). We thus proved that µ µ ν eMij = ∂µ∂e Mij + Γeµν∂e Mij (B.28) Doing the same for the other part of the Ricci tensor results in

µ µ µ µ b µb b b λb b λb Re σµν = ∂µΓeνσ − ∂σΓeνµ + Γe Γeσν − Γe Γeµν, b b b µbλb σλb b µ i µ i µ λ µ = ∂µΓeσν − ∂νΓeσi + ∂νΓeσµ + ΓeiµΓeσν + ΓeλµΓeσν

µ λ i j − ΓeσλΓeµν + ΓeσjΓeiν , i i λ i j = Reσν − ∂νΓeσi + ΓeiλΓeσν − ΓejνΓeiσ, 1 ik 1 ik λ i j = Reσν − ∂ν M ∂σMki + M ∂λMkiΓe − Γe Γe , 2 2 σν jν iσ D 1 ik jl = Re − M ∂νMkjM ∂σMli (B.29) σν 4 Here the same reasoning as for the ﬁrst part of the Ricci tensor was used. With the Ricci tensor components we can start to calculate the Ricci scalar

R = gµbνbR , eD+n e eνbµb ij µν = ge Reji + ge Reνµ, (B.30) iλ because ge = 0. To continue we start by rewriting the ﬁrst term in this sum ij ij 1 µ 1 kl µ lk µ g Reji =M − ∇e µ∂e Mji + M ∂µMjl∂e Mki + M ∂µMik∂e Mjl , e 2 4

1 ij 1 kl ji µ lk ij µ = − M eMji + M ∂µMljM ∂e Mik + M ∂µMkiM ∂e Mjl , 2 4 1 ij 1 µ ij = − M eMji − ∂e M ∂µMji. (B.31) 2 2

88 B.3. Reduction of a maximally symmetric space

Going from the first to the second line we inserted the factor Mij in the third position in the second and third term. The third line was obtained using that for −1 any invertible matrix A, ∂µ(A) = −A∂µ(A)A, which in components becomes, kl ij kj M ∂µMliM = −∂µM . This can further be simplified by noting the following ij µ ij µ ij µ 0 = ∇µ(M ∂e Mij) = M ∇µ∂e Mji + ∇µM ∂e Mij. (B.32) Therefore the first term in (B.30) is

ji 1 ij 1 µ ij g Reij = − M eMji − ∂e M ∂µMji = 0. (B.33) e 2 2 The second term in the sum for the Ricci scalar is

µν 1 ik jl µ g Reνµ = ReD − M ∂µMkjM ∂e Mli, e 4 1 µ il = ReD + ∂ M ∂eµMli. (B.34) 4 This results in a Ricci scalar equal to,

1 µ ij ReD+n = ReD + ∂ M ∂eµMji. (B.35) 4 We have the necessary equations to begin reducting the action. In section 4.2 these are substituted in the D + n-gravitational action. Afterwards ReD is rewritten using the formula for a conformaly rescaled Ricci scalar. After some more algebra we arrive at the following reduced action 1 S = dDxµdnyip|g | R + ∂µMij∂ M − (D + n − 1) (D + n − 2) β2 (∂φ)2 ˆ D D 4 µ ji 2 − (2 (D + n) − 2) β φ(x) + (D − 2) (α − β)(∂φ(x)) 2 2 i − (D − 1) (D − 2) (α − β) (∂φ) − (2D − 2) (α − β) φ . (B.36) Using D − 2 n β = − α & α2 = , (B.37) n 2(D + n − 2)(D − 2) we obtain our ﬁnal expression for the action D µp 1 2 1 µ ij S = Ve d x |gD| RD − (∂φ) + ∂ M ∂µMji . (B.38) ˆ 2 4

B.3 Reduction of a maximally symmetric space

B.3.1 Reduction of the Ricci scalar

In the calculation of the model of this section we use as metric 2 2αφ(x) 2 2βφ(x) n m ds6 = e ds4 + e Nmndy dy . (B.39)

89 B.3. Reduction of a maximally symmetric space

We go to conformal frame to ﬁnd an expression for Rb6, (4.54) −2βφ(x) −2(α−β)φ(x) 2 2 2 Rb6 = e Re6 + e −20β (∂φ) − 10β φ(x) + 2 (α − β)(φ(x)) . (B.40) We used the property of the Ricci tensor of an Einstein manifold to ﬁnd the Re2 of the compact dimensions and with this we found

µν 6 Re6 = ge Reµν + 2k. (B.41) Afterwards we argued that there are terms no terms which have any dependence 6 6 4 on Nij terms appearing in Reµν, which means that Reµν = Reµν and we thus get that

Re6 = Re4 + 2k. (B.42) This means that we can just reuse the expression for a conformal rescaled Ricci scalar, (4.21), to obtain

−2(α−β)φ 2 2 Re4 = e R4 − 6(α − β) (∂φ) − 6(α − β)φ . (B.43) Using all this we have that

−2βφ(x) −2(α−β)φ(x) 2 2 2 Rb6 = e Re6 + e −20β (∂φ) − 10β φ(x) + 2 (α − β)(φ(x)) , −2βφ(x) −2(α−β)φ 2 2 = e e R4 − 6(α − β) (∂φ) − 6(α − β)φ + 2k −2(α−β)φ(x) 2 2 2 + e −20β (∂φ) − 10β φ(x) + 2 (α − β)(φ(x)) . (B.44) The determinant of metric upon reduction becomes

p (4α+2β)φp |gb6| = e |g4|, (B.45) from which we see that the prefactor in front of R4 is equal to exp(2(α + β)φ). Substituting this into the action and integrating out the coordinates of the compact dimensions we obtain

Ve S = d4xµp|g |e2(α+β)φR − 6(α − β)2(∂φ)2 − 6(α − β) φ 2κ2 ˆ 4 4 2 2 2 4αφ − 20β (∂φ) − 10β φ(x) + 2 (α − β)(φ(x)) + 2ke . (B.46) Choosing now β = −α and α2 = 1/8 we arrive at the ﬁnal result

2 4 µp 1 2 √ φ(x) S = Ve d x |g4| R4 − (∂φ) + 2ke 2 (B.47) ˆ 2

B.3.2 Derivation Einstein equations with ﬂux

We start this section with the following action µ ip 1 1 ij S = dx dy |gD+n| RbD+n − FijF . (B.48) ˆ b 2κ2 4

90 B.4. One-dimensional compactiﬁcations

We are interested in the equation of the motion of the metric. For this purpose we perform the following variation 1 δS 1 δ 1 p = |gD+n|RbD p δgµbνb p δgµbνb 2κ2 b |gbD+n| b |gbD+n| b 1 δ p 1 iσ jλb − |gD+n| g bg FijF p δgµbνb b 4b b σbλb |gbD+n| b 1 1 h1 1 = G + p|g |g F F ij 2κ2 µbνb p 2 bD+n bµbνb 4 ij |gbD+n| p 1 iσ jλb i − 2 |gD+n| g bδ FijF b 4b µbνb σbλb 1 1 1 = G + g F F ij − giσbF F 2κ2 µbνb 8bµbνb ij 2b iµb σbνb 2 κ ij 2 σ = Gµν + gµνFijF − κ F b Fσν = 0. (B.49) bb 4 bbb µb bb p p To go from the ﬁrst to the second equality we used δ |g |R = |g |G δgµbνb, bD+n bD+n µbνb b p p µν iσ jλb iσ jλb δ |gD+n| = −1/2 |gD+n|gµνδgbb, and δ g bg = 2g bδ . b b bbb b b b b µbνb The D + n-dimensional Einstein equation for this theory becomes, 2 1 κ ij 2 σ Rbµν − gµν Rb − FijF = κ F b Fσν. (B.50) bb 2bbb 2 µb bb Contracting this equation with the inverse metric yields an expression for the Ricci scalar 2 D + n κ ij 2 σν Rb − Rb − FijF = κ F bbFσν, 2 2 bb

4 − (D + n) 2 ij Rb = κ FijF (B.51) 4 − 2(D + n) Substituting this into the above equation we can rewrite it as.

2 κ ij 2 σ Rbµν = gµνFijF + κ F b Fσν (B.52) bb 4 − 2(D + n)bbb µb bb

B.4 One-dimensional compactiﬁcations

B.4.1 Equations of motion from the action

In this section we give in more detail the rewriting of the equations of motions obtained by reducing the action. The metric is given by

2 2αφ(t) 2 2βφ(t) n m dsb4 = e ds1 + e Nmndy dy , (B.53)

91 B.4. One-dimensional compactiﬁcations

2 2 with ds1 = −f (t)dt. The action is equal to

2 Ve µp 1 2 √ φ(x) 2 2 S = dx |g | (∂φ) + 6ke 3 − 2q κ , (B.54) 2κ2 ˆ 1 2 from which we obtain the ﬁrst equation of motion

2 1 1 2 √ φ(x) 2 2 1 − g (∂φ) + 6ke 3 − 2κ q = − ∂ φ∂ φ, (B.55) 2 00 2 2 0 0 which was obtained by a variation of the metric. This equation written out yields

2 2 f 1 −2 2 √ φ(x) 2 2 1 − g¯ f (∂φ¯ ) + 6ke 3 − 2κ q = − ∂ φ∂ φ, (B.56) 2 00 2 2 0 0

2 where the bar refers to the fact that g00 = f g¯00 and g¯00 = −1. Performing the coordinate transformation dτ = exp (αφ(τ)) f(t)dt results in the metric

02 2 2 n m dsb4 = −dτ + a (τ)Nmndy dy , (B.57) where we deﬁned a(τ) = exp (βφ(τ)), which is the FLRW-metric used in cosmology. Remark that the equation of motion for the metric, (B.56), is not a covariant expression, hence we rewrite it in terms of the hatted metric and then we can use tensor transformation rules. This results in

−2αφ 2 e 1 2 2αφ √ φ(x) 2 2 1 − g00 (∂φb ) e + 6ke 3 − 2κ q = − ∂0φ∂0φ, (B.58) 2 b 2 2

2 where we used that gb00 = exp (2αφ(τ)) f g¯00. Using now the transformation laws we obtain,

0 −4αφ −2 2 e f 0 1 0 0 2 2αφ0 √ φ0(x) 2 2 1 0 0 −2 −2α0φ0 − g (∂b φ ) e + 6ke 3 − 2κ q = − ∂ φ∂ φf e , 2 b00 2 2 0 0 2 −2αφ0 1 0 0 2 2αφ0 √ φ0(x) 2 2 0 0 −e (∂b φ ) e + 6ke 3 − 2κ q = ∂ φ∂ φ, (B.59) 2 0 0

−1 −αφ(t) 0 where we used that dt/dτ = f e dt, gb00 = −1 and that the accents refer to the metric with the new time coordinate. Replacing a(τ) = exp (βφ(τ)) we get,

1 a˙ 2 a˙ 2 6 a6 − 6ka4 + 2κ2q2] = 12 , a6 a a a˙ 2 k κ2q2 + = . (B.60) a a2 3a6

To obtain the ﬁrst line we used that ∂0φ = a/βa˙ . The second equation of motion is obtained by varying the action (4.90) with respect to φ, and is

√ 2 −2 00 √ φ f g¯ ∂0∂0φ = 4 3ke 3 . (B.61)

92 B.4. One-dimensional compactiﬁcations

The term on the left has two derivates acting on the scalar ﬁeld, meaning that to write this expression covariantly we need the Christoffel symbols. Because the matrix Nij has no time dependence they are easily calculated and are equal to

0 0 i 0 2(β−α)φ 0 i i Γ00 = α∂0φ, Γ0i = Γ00 = 0, Γij = −βNije ∂ φ, Γ0j = β∂0φδj. (B.62)

With these (B.61) written covariantly becomes

√ √2 φ 2αφ 00 2αφ 00 0 3 e gb ∇0∂0φ + e gb Γb00∂0φ = 4 3ke , √ 2 2αφ 00 2αφ 00 √ φ e gb ∇0∂0φ + e gb α∂0φ∂0φ = 4 3ke 3 . (B.63) Now transforming the coordinates we obtain

√ √2 φ0 2αφ 000 0 0 0 2αφ 000 0 0 0 0 3 e gb ∇0∂0φ + e gb α∂0φ ∂0φ = 4 3ke . (B.64)

0 00 0i To calculate ∇0 we need Γ00 and Γ00. It is easily seen that both vanish and thus the covariant derivative becomes an ordinary partial derivative. We can now introduce a to obtain ! √ a¨ a˙ 2 √ a˙ 2 √ −2 3a6 − − 6 3a6 = 4k 3a4 a a a a¨ a˙ 2 2k + 2 = − . (B.65) a a a2

Using our ﬁrst equation, (B.60), the ﬁnal result becomes

a¨ 2κ2q2 = − . (B.66) a 3a6

B.4.2 Reduction of the ﬁeld equations

The section starts by giving the four dimensional action of our model (4.88) 1 3 p 1 1 ijk S = d xd y |g4| Rb4 − FijkF , (B.67) ˆ b 2κ2 6 and the trace-reversed Einstein equations need to be derived. To obtain these we perform a variation with respect to the inverse metric 1 δS 1 1 1 1 1 kρ p ijk p iσb jλb b = Gµν + |g4|gµν FijkF − 3 |g4| g g δµνFijkF p δgµbνb 2κ2 bb p 2 b bbb 6 b 6b b bb σbλbρb |gb4| b |gb4| κ2 ij 2 σbλb = Gµν + gµνFijF − κ F µ F = 0. (B.68) bb 6 bbb b σbλbνb

93 B.4. One-dimensional compactiﬁcations

From this the Einstein equations of this theory are 1 1 2 σbλbρb 2 σbλb Rbµν − gµν Rb − κ F F = κ F F ν . (B.69) bb 2bbb 3 σbλbρb σbλbµb b The trace reversed equations are, κ2 σbλbρb 2 σbλb Rbµν = − gµνF F + κ F F ν . (B.70) bb 3 bbb σbλbρb σbλbµb b For the components of the compact dimensions this equation becomes,

2 κ 2βφ 2 −6βφ 2 2 −4βφ Rbij = − e gij 6q e + 2κ q e gij = 0. (B.71) 3 Where we used the following identity.

σλb 2 −4βφ σλb µν F F b j = q e gb gbbσµi , σbλib bb λbνjb ρ 2 −4βφ b σλb µbνb = q e δj g g σµi , bb λbνbρb 2 −4βφ τρ σλb µν = q e gτjgbbgb gbbσµi , b bb λbνbρb = q2e−4βφg σbµbτb τjb σbµib = 2q2e−4βφg δτb, τjb i 2 −4βφ = 2q e gij, (B.72) where we used (4.31) to go from the fourth to the ﬁfth line. The Christoffel symbols for this model are √ 0 3 0 i 0 1 2(β−α)φ 0 i 1 i Γ = ∂0φ, Γ = Γ = 0, Γ = − √ Nije ∂ φ, Γ = √ ∂0φδ . 00 2 0i 00 ij 2 3 0j 2 3 j (B.73)

Using this we can expand the components of the Ricci tensor to obtain

1 1 2 µb µb µb λb µb λb Rb00 =∂µΓ00 − ∂0Γ0µ + Γ Γ00 − Γ Γµ0 = −3 √ ∇0∂0φ − 3 √ ∂0φ∂0φ b b µbλb 0λb b 2 3 2 3 √ 3 1 = − ∇ ∂ φ − ∂ φ∂ φ (B.74) 2 0 0 4 0 0 We can deﬁne the following,

√ √ 1 √ 3 φ(t) dt −1 − 3 φ(t) √ φ(τ) a˙ dτ = f(t)e 2 dt ⇒ = f e 2 a(τ) = e 2 3 ⇒ φ˙ = 2 3 , (B.75) dτ a where the dot refers to a derivative with respect to τ. Now because equation (B.74) is a covariant tensorial equation it transforms under a coordinate transformation as √ √ √ ! 0 3 1 e− 3φ f −2Rb0 = e− 3φf −2 − ∇0 ∂0 φ0 − ∂0 φ0∂0 φ0 . (B.76) 00 2 0 0 4 0 0

94 B.4. One-dimensional compactiﬁcations

Here the accents mean that it is written in the τ coordinate. Now using our deﬁnitions 00 for a and that Γ00 = 0 we can rewrite this as. √ √ 2! 0 3 √ a˙ 1 √ a˙ e− 3φ f −2Rb0 = f −2a−6 − ∂0 2 3 − 2 3 00 2 0 a 4 a a¨ a˙ 2 a˙ 2 = f −2a−6 −3 + 3 − 3 a a2 a2 a¨ = f −2a−6 −3 . (B.77) a

We reintroduce α and β for notational convenience and calculate the other components of the Ricci scalar, µb µb µb λb µb λb k k l Rb0i = ∂µΓi0 − ∂iΓ0µ + Γ Γi0 − Γ Γµ0 = 2β∂0φ Γki − δl Γki = 0, (B.78) b b µbλb iλb b and

µb µb µb λb µb λb Rbij = ∂µΓji − ∂jΓiµ + Γ Γji − Γ Γµi, b b µbλb jλb b 3 2(β−α)φ 0 2(β−α)φ 0 = Rij − 2β (β − α) Nije ∂0φ∂ φ − βNije ∇0∂ φ 2 2(β−α)φ 0 2 0 2(β−α)φ − 9β Nije ∂0φ∂ φ + 2β Nij∂0φ∂ φe , 3 −4βφ 1 0 1 0 = R + Nije − ∂0φ∂ φ − √ ∇0∂ φ. (B.79) ij 4 2 3 We write this in terms of the hat metric to have a covariant expression −2βφ −6βφ 1 0 2αφ 1 0 2αφ Rbij = 2ke gij + gije − ∂b0φ∂b φe − √ ∇b 0∂b φe . (B.80) b b 4 2 3 Transforming this expression we obtain,

1 1 Rb0 = 2ke−2βφg0 + g0 − ∂b0 φ0∂b00φ0 − √ ∇b 0 ∂b00φ0 . (B.81) ij bij bij 4 0 2 3 0 If we now write everything in terms of a we get, 0 2 1 0 0 0 0 1 0 0 0 Rb = 2kNij + a Nij − ∂ φ ∂ φ + √ ∂ ∂ φ ij 4 0 0 2 3 0 0 ! a˙ 2 a¨ a˙ 2 = 2kN − a2N −3 − + (B.82) ij ij a a a

Equating this to zero for every component, and using (4.103), we obtain (4.106), which is what we need.

95 Appendix C

Details of reductions chapter 5

C.1 Vacua of type II supergravity

C.1.1 Solutions with p + 1 ≥ 2

We start this appendix by rewriting the following formula, which is equation (5.29)

p − p+1 φ − p+1 φ 2 (−1) e 4 0 κH ∧ ? H = e 4 ? H ∧ H 9−p 7 − p 9−p p+1 p+1 2 φ p − φ0 + Ce 4 (−1) e 4 κ ?9−p ?9−pH ∧ ?9−pH. (C.1)

n(D−n)+t This equation can be rewritten using the equations ?D ?D An = (−1) An, kn and An ∧ Bk = (−1) Bk ∧ An which are taken from the appendix of [11]. For the term on the left side we obtain

p(9−p−3) p H ∧ ?9−pH = (−1) ?9−p H ∧ H = (−1) ?9−p H ∧ H. (C.2)

For the last term on the right side of the above equation we get

3(9−p−3) 2p ?9−p ?9−pH ∧ ?9−pH = (−1) H ∧ ?9−pH = (−1) ?9−p H ∧ H, (C.3) where we used that H is a three-form and the compact internal dimensions are all spacelike. Filling this into (C.1) we obtain

− p+1 φ − p+1 φ 2 e 4 0 κ ? H ∧ H = e 4 ? H ∧ H (C.4) 9−p 7 − p 9−p p+1 p+1 2 φ p − φ0 + Ce 4 (−1) e 4 κ ?9−p H ∧ H. (C.5)

From which we obtain the second order equation for κ

(5 − p) 2 κ2 − κ + = 0. (C.6) (7 − p)(1 + δp1) (7 − p)

We now need to show that using the BPS condition the equation of motion for H ﬁeld is equated to the Bianchi-identity of the F6−p ﬁeld and vice versa. The BPS condition is p+1 p − φ0 F6−p = (−1) e 4 ?9−p H, (C.7)

96 C.1. Vacua of type II supergravity and the equation of motion for H is

−φ 1 X 5−n φ d(e ?H) = − e 2 ?F ∧ F . (C.8) 2 n n−2 n Remark that a star without a subscript refers to a ten-dimensional hodge dual. Using the BPS condition, the fact that the dilation is constant and that the ﬁelds live in the compact dimensions, and thus we can split the hodge dual, we can rewrite the left part as

p+1 −φ −φ0 φ0 d(e ?H) = e d ?p+1 1 ∧ ?9−pH − e 4 ?p+1 1 ∧ dF6−p (C.9)

Because the exterior derivative on the volume form is equal to zero the ﬁrst term on the right hand side vanishes. Taking into account that the only non vanishing ﬁeld is F6−p The right side of the equation of motion

X 5−n φ 1−p φ −1−p φ −3−p φ e 2 ?Fn ∧ Fn−2 =e 2 ?F4+p ∧ F2+p + e 2 ?F6−p ∧ F4−p + e 2 ?F8−p ∧ F6−p n 1−p φ −3−p φ = e 2 F6−p ∧ F2+p + e 2 F2+p ∧ F6−p, (C.10)

1 where we dropped the overal factor of − 2 and the second term vanishes clearly. Remark that the F2+p is non-zero only when 2 + p = 6 − p, but this means that we have a wedge product of twice the same forms, which is always zero. This means that if F6−p is closed the equation of motion of H is satisﬁed. We do the same for the other equation of motion. This is given by

5−n φ 3−n φ n(n−1) d e 2 ?Fn = −e 2 H ∧ ?Fn+2 − (−1) 2 µn−2δ11−n(O(n − 2)). (C.11)

We start with the left side of this equation.

5−(6−p)φ0 (p−1) φ0 d e 2 ?F6−p =e 2 (d (?p+11 ∧ ?9−pF6−p)) ,

(p−1) −(p+1) (9−p) p φ0 φ0 =(−1) (−1) e 2 e 4 ?p+1 1 ∧ (d ?9−p ?9−pH, ) p−3 φ = − e 4 0 dH. (C.12)

To go from the ﬁrst to the second line we used that the exterior derivative on the volume form vanishes and the BPS-condition, to go to the third we used ?D ?D An = n(D−n)+t (−1) An. The right side of this equation becomes

3−(6−p) (6−p)(5−p) φ0 R.H. = − e 2 H ∧ ?F8−p − (−1) 2 µ4−pδ5+p(O(4 − p)), =O (C.13)

Here we used that the F8−p = 0, and that only µ9−p 6= O, because we have no other type of orientifold planes. From this we see that if dH = O is satisﬁed, the equation of motion for F6−p is also satisﬁed.

97 C.1. Vacua of type II supergravity

2 We now derive the expression for |F6−p|ij using the BPS-condition.

2 1 λb2···λb6−p |F6−p|ij = F F (5 − p)! i λb2···λb6−p j 1 ν λb2···λb6−p = gjνF F b (5 − p)! b i λb2···λb6−p 1 p+1 − φ0 ν λb2···λb6−p = e 2 gjν (?9−pH) (?9−pH)b (5 − p)! b i λb2···λb6−p

p+1 1 1 k k k 1 νλb2···λb6−p − 2 φ0 1 2 3 b l1l2l3 = e gjν Hk1k2k3 l l l H (5 − p)! b 3! iλb2···λb6−p 3! 1 2 3 1 p+1 − 2 φ0 νbλb2···λb6−pl1l2l3 k1k2k3 = e gjν Hl l l H (5 − p)!3!2 b iλb2···λb6−pk1k2k3 1 2 3 4! (5 − p)! p+1 − φ0 [νb l1 l2 l3] k1k2k3 = e 2 gjνδ δ δ δ H H (5 − p)!3!2 b i k1 k2 k3 l1l2l3 4! p+1 1 − φ0 µb [l1 l2 l3] l1 [νb l2 l3] k1k2k3 = e 2 gjν δ δ δ δ − 3δ δ δ δ H H 3!2 b 4 i k1 k2 k3 i k1 k2 k3 l1l2l3 3! − p+1 φ µ k k k νl l = e 2 0 g δbH H 1 2 3 − 3H H b 2 3 3!2 jνb i k1k2k3 il2l3 3! − p+1 φ k k k l l = e 2 0 g H H 1 2 3 − 3H H 2 3 3!2 ij k1k2k3 il2l3 j p+1 − φ0 2 2 = e 2 |H| gij − |H|ij (C.14) Remark that we used the following formulas, 1 F = F dxi1 ∧ ... ∧ dxi6−p (C.15) 6−p (6 − p)! i1...i6−p 1 1 = k1k2k3 H dxi1 ∧ ... ∧ dxi6−p , (C.16) (6 − p)! i1...i6−p 3! k1k2k3 which is derived using the self-duality condition, and 1 [νb l1 ··· ln] µ [l1 ··· ln] l1 [νb l2 ··· ln] δ δ δ δ Al ···l = δbδ δ δ − nδ δ δ δ δ Al ···l , (C.17) i k1 ··· kn 1 n n + 1 i k1 ··· kn i k1 k2 ··· kn 1 n where Al1···ln is an arbitrary antisymmetric tensor.

C.1.2 Complete calculation one dimensional vacuum

Take the following Ansatz for the metric 2 2 2 ds10 = ds1 + ds9, (C.18) and the non-vanishing ﬁelds are H and F6 and φ = φ0. Looking again at the trace-reversed Einstein equations in the external dimension, the time dimension gives, −φ 1 2 − 1 φ 5 2 1 S 1 S 0 = −e 0 g |H| − e 2 0 g |F | + T − g T . (C.19) 8 00 16 00 6 2 00 8 00

98 C.1. Vacua of type II supergravity

Remark that what we have done here is to replace the sum in n in the previous section by taking n = 6. Tracing this equation and ﬁlling our expression for T S, obtained by setting p = 0 in equation (5.23), and transforming everything into form notation we obtain

−φ 1 − 1 φ 5 7 − 3 φ 0 = −e 0 ? H ∧ H − e 2 0 ? F ∧ F + e 4 0 µ . (C.20) 8 9 16 9 6 6 16 0 9

Using the Bianchi identity for F8 = 0, which relates H to F6 we can rewrite the above equation as

−φ 1 − 1 φ 5 7 − 3 φ 0 = −e 0 ? H ∧ H − e 2 0 ? F ∧ F + e 4 0 H ∧ F . (C.21) 8 9 16 9 6 6 16 6 If we then use the same Ansatz for F as in the previous section, namely

1 − φ0 F6 = e 4 κ ?9 H, (C.22) and we substitute this into the above equation the result is 1 5 7 0 = −e−φ0 ? H ∧ H − e−φ0 κ2 ? ? H ∧ ? H + κe−φ0 H ∧ ? H. (C.23) 8 9 16 9 9 9 16 9 This can be rewritten as 1 5 7 0 = −e−φ0 ? H ∧ H − e−φ0 κ2 ? H ∧ H + κe−φ0 ? H ∧ H, (C.24) 8 9 16 9 16 9 which is a second order polynomial in κ, with solutions

2 κ = 1 & κ = . (C.25) 5

We now need to show that using the BPS condition the equation of motion for H ﬁeld is equated to the Bianchi-identity of the F6 ﬁeld and vice versa. The BPS condition is 1 − φ0 F6 = e 4 ?9 H, (C.26) and the equation of motion for H is

−φ 1 X 5−n φ d(e ?H) = − e 2 ?F ∧ F . (C.27) 2 n n−2 n

Using the BPS condition, the fact that the dilation is constant and that the ﬁelds live in the compact dimensions, and thus we can split the hodge dual, we can rewrite the left part as

1 −φ −φ0 φ0 d(e ?H) = e d ?1 1 ∧ ?9H − e 4 ?1 1 ∧ dF6 (C.28)

99 C.1. Vacua of type II supergravity

Because the exterior derivative on the volume form is equal to zero the ﬁrst term on the right hand side vanishes. Taking into account that the only non vanishing ﬁeld is F6, the right side of the equation of motion is

X 5−n φ 1 φ −1 φ −3 φ e 2 ?Fn ∧ Fn−2 =e 2 ?F4 ∧ F2 + e 2 ?F6 ∧ F4 + e 2 ?F8 ∧ F6 n = 0, (C.29) which is clear because in every term there is at least one form-ﬁeld that vanishes. This means that if F6 is closed the equation of motion of H is satisﬁed. We do the same for the other equation of motion. This is given by

5−n φ 3−n φ n(n−1) d e 2 ?Fn = −e 2 H ∧ ?Fn+2 − (−1) 2 µn−2δ11−n(O(n − 2)). (C.30)

We start with the left side of this equation.

5−(6)φ0 −1 φ0 d e 2 ?F6 =e 2 (d (?p1 ∧ ?9F6)) ,

−1 −1 9 φ0 φ0 =(−1) e 2 e 4 ?1 1 ∧ (d ?9 ?9H, ) −3 φ = − e 4 0 dH. (C.31)

3−6 (6)(5) φ0 R.H. = − e 2 H ∧ ?F8 − (−1) 2 µ4δ5(O(4)), =0. (C.32)

Where we used that F8 is zero and that only µ9 is not zero. We thus see that if the fields satisfy the BPS-condition, The Bianchi-identities are mapped upon the equations of motion, and vice versa. Therefore if the fields satisfy the Bianchi- identitites, they solve the equations of motion. From the BPS-condition and the Bianchi-identity of F8, we again obtain the following relation 1 − φ0 2 µp = e 4 κ|H| , (C.33) which filled in into the dilaton equation of motion gives, 1 1 3 0 = ∇2φ = −e−φ0 |H|2 − e−φ0 κ2|H|2 + κe−φ0 |H|2. (C.34) 0 2 4 4 From this we see that the second solution for κ can not be used. To end we check the internal Einstein equation −φ 1 2 1 2 − 1 φ 1 2 5 2 1 − 3 φ R = e 0 |H| − g |H| + e 2 0 |F | − g |F | − g e 4 0 µ ij 2 ij 8 ij 2 6 ij 16 ij 6 16 ij 0 9 (C.35)

100 C.1. Vacua of type II supergravity where the energy-momentum tensor was set to zero because it lives in the external dimension. Using the equation we derived in the appendix for |F6|ij, (C.14), we can rewrite the equation as −φ 1 2 1 2 − 1 φ 1 − 1 φ 2 2 5 2 R = e 0 |H| − g |H| + e 2 0 e 2 0 |H| g − |H| − g |F | ij 2 ij 8 ij 2 ij ij 16 ij 6

1 − 3 φ − g e 4 0 µ . (C.36) 16 ij 0 9 Using the Bianchi-identity we get −φ 1 2 1 2 1 −φ 2 1 −φ 2 5 − 1 φ 2 R = e 0 |H| − g |H| + e 0 |H| g − e 0 |H| − g e 2 0 |F | ij 2 ij 8 ij 2 ij 2 ij 16 ij 6

1 − 3 φ − g e 4 0 H ∧ F . (C.37) 16 ij 6

It is clear that the terms |H|ij vanish against each other. Using the Ansatz for F6 with κ = 1 1 1 5 1 R = − e−φ0 g |H|2 + e−φ0 |H|2g − g e−φ0 |H|2 − g e−φ0 |H|2. (C.38) ij 8 ij 2 ij 16 ij 16 ij The internal manifold has again no curvature. We are let to the following solution

1 − φ0 F6 = e 4 ?9 H, (C.39)

dH = 0 & dF6 = 0 & Fi = 0 if i 6= 6 − p (C.40) 1 1 − φ0 2 φ0 2 µ9 = e 4 |H| = e 4 |F6| (C.41)

Rij = 0. (C.42) This solution has the same physics as the one obtained in [31], except that we have p = 0.

C.1.3 Compactiﬁcation of Rb10

We start with the Ansatz

2 2αv 2 2βv 2 dsb10 = e dsp+1 + e ds9−p, (C.43) and perform our usual tricks to go to conformal frame. Doing this using (4.10) yields

−2βv 2 2 −2(α−β)v −2(α−β)v 2 Rb10 = e Re10 − 72β (∂v) e − 18βe v + 8 (α − β)(∂v) . (C.44)

We compactify over a Ricci ﬂat space, therefore Re10 = Rep+1. Applying a second time the trick to go to conformal frame and using (4.10) yields −2(α−β)v 2 2 Rep+1 = e Rp+1 − p (p − 1) (α − β) (∂v) − p(α − β)v . (C.45)

101 C.2. Extrema of the potential

From the determinant we get as contribution

p q α((p+1)−β(9−p))v |gb10| = |gp+1|e . (C.46)

We see that if we want to end up in Einstein frame, we need to satisfy the equation

α(p − 1) = −(9 − p)β, (C.47) therefore we deﬁne −(9 − p) α = β. (C.48) (p − 1) This is in accordance with (4.23), where we did the compactiﬁcation of an n-torus. This is as expected. Therefore we have that for p > 1 we take

(9 − p) α2 = . (C.49) 18(p − 1)

We check for the case p = 0. Doing this we see that we do not obtain kinetic terms from the second rescaling, and omitting the box term yields the kinetic term

K.T. = −72β2 − 144αβ + 144β2 (∂v)2 = 1368β2 (∂v)2 , (C.50)

Therefore if we take 1 β = √ (C.51) 12 19 we get a canonically normalised term. This choices yields the following action p+1 9−p p 1 2 S = d xd y |g10| Rb10 − (∂φ) ˆ b 2 p+1 q 1 2 1 2 =Ve d x |gp+1| Rp+1 ± (∂v) − (∂φ) (C.52) ˆ 2 2

C.2 Extrema of the potential

We calculate the second derivatives for the case which has two solutions. The equation for the second derivative is

2 2 2 3 2 3∂φV = 4 R − F x − 9Λx − F2 x. (C.53)

The two extrema were given by

|(R − F 2)| ± p(R − F 2)2 − 3|Λ|F 2 x∗ = 1 1 2 (C.54) 3|Λ|

102 2 C.3. F2 contribution group manifold

2 Substituting the solution with a plus sign, and introducing A = (R − F1 ), into this yields 4A q 2 −9|Λ| q 3 3∂2V = A + 9 A2 − 3|Λ|F 2 + A + A2 − 3|Λ|F 2 φ 9|Λ|2 2 27|Λ|3 2 1 0 q − F 2 A + A2 − 3|Λ|F 2 3|Λ| 2 q q 2 ! 2 2 2 2 02 =C 12A A + A − 3|Λ|F2 − 9 A + A − 3|Λ|F2 − 9|Λ|F2

q q 2 2 2 2 2 2 02 =C 12A + 12A A − 3|Λ|F2 − 18A − 18A A − 3|Λ|F2 + 18|Λ|F2 <0 (C.55) √ A+ A2+3|Λ|F 2 2 2 2 because A > 3|Λ|F2 , and C = 27|Λ|2 . This is something we knew, one solutions is always unstable. For the other solution the calculation is almost identical except for some signs that change. Taking this into account we obtain q q 2 h 2 2 2 2 2 2 ∂φV =C 12A − 12A A − 3|Λ|F2 − 18A + 18A A − 3|Λ|F2

02 02i + 27|Λ|F2 − 9|Λ|F2 (C.56) q 2 2 2 02 =C −6A + 6A A − 3|Λ|F2 + 18|Λ|F2 . (C.57)

This can be rewritten using the following: q q 2 2 2 2 2 2 2 (A − A − 3|Λ|F2 ) − 2A + 2A A − 3|Λ|F2 = −3|Λ|F2 , (C.58) which is obtained by expanding the ﬁrst term and rewriting. Call now D = A − p 2 2 A − 3|Λ|F2 , we arrive at the equation 2 2 − 3|Λ|F2 = D − 2AD. (C.59) Using this into equation for the second partial derivative we get 2 2 3∂φV = C(−6AD − 6D + 12AD) = −6CD(D − A), (C.60) But because D < A we ﬁnd that this is has a positive sign. We thus see that there is a minimum and a maximum of the potential. Note that the minimum is at a smaller value for φ, and thus a smaller value for the volume, then the maximum.

2 C.3 F2 contribution group manifold

For the ﬂux contributions we have

F2 = dA1 + Σ, (C.61)

103 2 C.3. F2 contribution group manifold

i 1 2 0 2 0 1 with A1 = χi(t)e + χ(t)dt and Σ = α0 e ∧ e + α1 e ∧ e + α2 e ∧ e , hence

0 1 2 1 0 2 2 0 1 F2 = χ0f12 + α0 e ∧ e + χ1f02 + α1 e ∧ e + χ2f01 + α2 e ∧ e . (C.62)

This means that ?F2 equals

0 1 2 1 0 2 2 0 1 ?F2 = χ0f12 + α0 ? ( e ∧ e ) + χ1f02 + α1 ? ( e ∧ e ) + χ2f01 + α2 ? ( e ∧ e ), −γh 0 11 22 0 1 00 22 1 = e χ0f12 + α0 M M e + χ1f02 + α1 M M e 2 00 11 2i + χ2f01 + α2 M M e (C.63)

With our choice of parametrisation of M (5.140) the contribution becomes √ √ −γh 0 2 −2σ/ 3 1 2 −ϕ+σ/ 3 ?F2 ∧ F2 = e χ0f12 + α0 e + χ1f02 + α1 e √ 2 2 ϕ+σ/ 3i + χ2f01 + α2 e . (C.64)

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