University of Heidelberg - Max Planck Society...University of Heidelberg Department of Physics and Astronomy Master Thesis The Axion-Instanton Weak Gravity Conjecture and Scalar Fields

University of Heidelberg

Department of Physics and Astronomy

Master Thesis

The Axion-Instanton Weak Gravity Conjectureand Scalar Fields

Submitted by:Clemens Vittmann

Supervised by:PD Dr. Eran Palti

Referees:Prof. Dr. Arthur Hebecker

Prof. Dr. Timo Weigand

This thesis has been carried out

at the

Institute for Theoretical Physics

Saturday 4th August, 2018

http://www.uni-heidelberg.de

https://www.thphys.uni-heidelberg.de/~palti/teaching.html

https://www.thphys.uni-heidelberg.de/~hebecker/

https://www.thphys.uni-heidelberg.de/~weigand/

https://www.thphys.uni-heidelberg.de

Acknowledgements

Foremost, I thank Eran Palti for giving me the opportunity to write this thesis,for guiding me through the process and for the hours of discussing and explainingeven after he left Heidelberg University for a position in Munich.

Second, I thank Arthur Hebecker, whose lectures introduced me to quantum fieldtheory, and Timo Weigand, who kept me busy with string theory at the same time,for being willing to be the referees of my thesis.

I am grateful to Marc Merstorf and Clemens Fruböse as well as Sascha Leon-hardt and Thomas Mikhail for tons of helpful and entertaining discussions (notonly) about physics and for providing a very pleasant work atmosphere at thePhilosophenweg 19.

My last thanks goes to Mikio Nakahara for writing his excellent textbook on differ-ential geometry, topology and physics which taught me most of the mathematicsnecessary for dealing with complex manifolds.

iii

Abstract

We study the Weak Gravity Conjecture in the presence of scalar fields. The WeakGravity Conjecture is a consistency condition for a theory of quantum gravityasserting that for a U(1) gauge field, there is a particle charged under this fieldwhose mass is bounded by its charge. It was extended to a statement about anycanonical pair of (p − 1)-dimensional object and p-form coupling to it, in par-ticular to axion-instanton pairs. The gauge-scalar Weak Gravity Conjecture is amodification of this bound that includes scalar interactions. We propose a similarextension to cases where scalar fields are present for the axion-instanton WeakGravity Conjecture and provide evidence from Type IIA supergravity.

Zusammenfassung

Wir untersuchen die Weak Gravity Conjecture in Anwesenheit skalarer Felder. DieWeak Gravity Conjecture ist eine Konsistenzbedingung für Theorien der Quan-tengravitation und behauptet, dass es zu einem U(1)-Eichfeld ein Teilchen gibt,welches Ladung unter diesem Feld trägt und dessen Masse von der Ladung be-schränkt ist. Sie wurde zu einer Aussage über jedes kanonische Paar bestehend aus(p− 1)-dimensionalem Objekt und daran koppelnder p-Form verallgemeinert, ins-besondere zu einer Bedingung für Axion-Instanton-Paare. Die Gauge-Scalar WeakGravity Conjecture ist eine Erweiterung dieser Bedingung, die Wechselwirkungenmit skalaren Feldern beinhaltet. Wir schlagen eine ähnliche Verallgemeinerung zuFällen, in denen skalare Felder vorhanden sind, für die Axion-Instanton-Varianteder Weak Gravity Conjecture vor und untermauern diese mit Indizien von Typ-IIA-Supergravitation.

v

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Weak Gravity Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Absence of global symmetries in quantum gravity . . . . . . . . . . . 72.2 Weak Gravity Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Generalized Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Calabi-Yau compactification . . . . . . . . . . . . . . . . . . . . . . . 213.1 Kaluza-Klein reduction . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Calabi-Yau requirement . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Reduction on the Calabi-Yau . . . . . . . . . . . . . . . . . . . . . . 263.4 Moduli spaces of Calabi-Yau threefolds . . . . . . . . . . . . . . . . . 273.5 Complex structure moduli space . . . . . . . . . . . . . . . . . . . . 303.6 Kähler moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.7 Mirror symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 A Scalar WGC for Type IIB Particles . . . . . . . . . . . . . . . . . 374.1 Supersymmetric black holes . . . . . . . . . . . . . . . . . . . . . . . 374.2 Gauge fields from reduction of Type IIB supergravity . . . . . . . . . 414.3 Particles from D3-branes . . . . . . . . . . . . . . . . . . . . . . . . 464.4 A scalar WGC for the particle . . . . . . . . . . . . . . . . . . . . . 48

5 A Scalar WGC for Type IIA Instantons . . . . . . . . . . . . . . . . 535.1 Axions from reduction of Type IIA supergravity . . . . . . . . . . . 535.2 Instantons from E2-Branes . . . . . . . . . . . . . . . . . . . . . . . 565.3 A scalar WGC for instantons and axions . . . . . . . . . . . . . . . . 57

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 63A.1 Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A.2 Kähler geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.3 Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.4 Some integrals on Calabi-Yau threefolds . . . . . . . . . . . . . . . . 71

B Type II Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 73B.1 Type IIA SUGRA from compactification of 11-dim. SUGRA . . . . 73B.2 Type IIB SUGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.3 N = 2 supergravity in d = 4 and special geometry . . . . . . . . . . 76

vii

C Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81C.1 Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81C.2 Supersymmetric black holes . . . . . . . . . . . . . . . . . . . . . . . 83C.3 Gauge-coupling matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 83

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

viii

List of Figures

1 Feynman diagram and string worldsheet . . . . . . . . . . . . . . . . 2

2 Swampland surrounding string landscape . . . . . . . . . . . . . . . . 3

3 Particles satisfying gauge-salar WGC . . . . . . . . . . . . . . . . . . 4

4 D3-brane wrapping three-cycle . . . . . . . . . . . . . . . . . . . . . . 5

5 Increasing black hole charge . . . . . . . . . . . . . . . . . . . . . . . 7

6 Energy conservation for decaying black hole . . . . . . . . . . . . . . 8

7 Extremal black hole decaying . . . . . . . . . . . . . . . . . . . . . . 11

8 Convex hull condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

9 Moduli space of a quantum theory of gravity . . . . . . . . . . . . . . 18

10 Compactification of two-dimensional surface . . . . . . . . . . . . . . 21

11 Running couplings in minimal supersymmetric extension of standardmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

12 Kähler cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

13 E2-brane wrapping three-cycle . . . . . . . . . . . . . . . . . . . . . . 56

14 Hodge Diamond of Calabi-Yau threefold . . . . . . . . . . . . . . . . 70

All figures were created by the author.

List of Tables

1 Bases for cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Metrics and Kähler potentials . . . . . . . . . . . . . . . . . . . . . . . 36

ix

3 Type IIB multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Type IIA multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 N = 2 supergravity multiplets . . . . . . . . . . . . . . . . . . . . . . 77

x

1 Introduction

When attempts to find a description of the strong interaction were made fifty yearsago - the force binding protons and neutrons together - young Gabriele Venezianofound that the Euler beta function had certain features one would expect fromthe scattering amplitude of strongly interacting particles. No theory was knownat that time, though, that would produce such a scattering amplitude. It wasabout two years later that Yoichiro Nambu, Holger Bech Nielsen and LeonardSusskind independently discovered that it was not a theory of point particles butone of vibrating strings that would give rise to Veneziano’s amplitude. Soon, thoseworking on the physics of such strings realized two things: First, this model wasmaking predictions about the strong force which were not in accordance with theexperimental findings and the theory of quarks and gluons developed at that time -quantum chromodynamics - turned out to be superior as a description of the stronginteraction. Second - and quite surprisingly - this theory of strings seemed to offera solution to a very different but fundamental problem of theoretical physics:Bringing together Einstein’s general relativity and the Standard Model in a singleunifying theory.After the two revolutions of theoretical physics in the first half of the twenti-

eth century - the discoveries of general relativity and quantum mechanics - thesetwo fields developed essentially separate from each other and appeared to be dras-tically different: For example, in quantum field theory - the relativistic quantumframework now underlying our description of the electromagnetic, weak and stronginteractions in what is known as the Standard Model - fields at two points in space-time whose separation is space-like should (anti-)commute. Meanwhile, in generalrelativity - the theory describing the remaining fundamental interaction, namelygravity - the metric is dynamical and one does a priori not even know whethera distance is space-like [1]. Above all, general relativity is classical and one runsinto the following difficulties when attempting to perform the same procedure ofperturbative quantization which proved so successful for the other fundamentalinteractions: Unlike the Standard Model, gravity is non-renormalizable meaningthat the infinities one encounters in the quantization process cannot be removedby a finite number of counter-terms. Although the resulting theory is predictiveat low energies - i.e. below the Planck mass - it breaks down at shorter distances[2]. While this does not play a role for most practical purposes in physics, thingsdo become problematic in the quantum description of small black holes and earlyuniverse cosmology.Without elaborating on the idea of unification as a driving force in the history of

physics, we want to briefly review why it is plausible to assume that the StandardModel and gravity themselves are - in spite of their success in describing basicallyall experimentally observed phenomena - not fundamental descriptions but rather

1 Introduction

limits of such a complete theory. We already mentioned the problem of infinitiesin quantum field theories. While it is well-known that the Standard Model is arenormalizable QFT, which means that these infinities can be absorbed in a finitenumber of counter terms, their very appearance suggests that this description ismerely an effective one, a low-energy limit of a more fundamental theory. Thesame is true with regard to the spacetime singularities in general relativity whichare contained in the centers of black holes. The circumstance that the StandardModel has about twenty free parameters - like electron mass or mixing angleswhich have to be taken from experiment - renders the theory arbitrary. Evenworse, some of these parameters appear to be fine-tuned. We will not elaborate onthis any further but go back to the UV infinities of quantum field theory. It was

point ofinteraction

interactionsmeared out

Figure 1: Feynman diagram of point particles (l.h.s.) and closedstrings scattering diagram (r.h.s.).

realized that if the fundamental entities of nature were not point-like but ratherof finite length, then these UV divergences would not occur. This was provedfor the one- and two-loop diagrams analogous to the Feynman diagrams of pointparticles and there is no reason to expect anything different at higher orders. Evenwithout performing the actual calculation, this behavior is intuitively accessiblewhen looking at such stringy diagrams like the one in fig. 1 on the right-handside: Unlike the case of point particles, there is no single point in spacetime wherescattering strings interact. Instead, the worldsheet in fig. 1 always looks locallylike that of a single freely propagating string and it is only the topology of thisworldsheet which encodes any interaction.Even more surprisingly, the theory included a particle in its spectrum that had

precisely the numbers of freedom one would expect from a particle that carriesthe gravitational force, the graviton. At the same time, some problems like theambiguity arising from free parameters would not appear: String theory has nofree dimensionless parameters, all couplings are expectation values of fields and

the theory seems to be an up to dualities unique and consistent theory of quantumgravity.While string theory is unique in ten spacetime dimensions, there are many con-

sistent ways to obtain a four -dimensional effective theory corresponding to thechoice of compactification manifold. The set of these solutions to string theory isknown as the string landscape and one is led to the question if, conversely, anyconsistent effective field theory can be coupled to gravity1). About thirteen years

StringLandscape

Swampland

particular low-energyeffective theory

Figure 2: Swampland of theories thatlack quantum consistencysurrounding string land-scape.

ago, Vafa argued [3] that this was notthe case and termed the set of those the-ories which lack quantum consistency theswampland.Several criteria were proposed [3, 4] to

distinguish effective theories that can beUV-completed to a consistent theory ofquantum gravity from those that merely liein the “surrounding” swampland. The ar-guably best known criterion to exclude atheory from the landscape is the “folk the-orem” that quantum gravity does not allowfor global symmetries. This is turned intoa quantitative statement by the conjectureproposed by Arkani-Hamed et al. in [5] that“gravity must be the weakest force”. Moreprecisely: For a consistent quantum theorycontaining gravity and a U(1) gauge fieldthere must exist a particle charged underthis gauge field with a mass bounded fromabove by its charge. This Weak GravityConjecture (WGC) has attracted much interest in the past decade and foundapplication e.g. in cosmology, where it is used to constrain models of large fieldinflation [6, 7, 8]. It was generalized to settings with several U(1)s [9], p-forms [10]and quite recently to situations where scalar fields are present [11, 12, 13].This last generalization is particularly important for us. It proposes bounds

(henceforth called Scalar and Gauge-Scalar WGC )

m2 < µ2M2p , g2M2

p ≥ m2 + µ2M2p (1.1)

1)If that were the case, one would not try to construct four-dimensional theories as compactifi-cations of ten-dimensional string theory but rather take some effective theory that fits bestto experiment [3].

– 3 –

1 Introduction

gauge scalar

gravitational

Figure 3: Two particles satisfying the Gauge-Scalar WGC.

for a particle m coupled to a scalar field with coupling µ. The second inequalityis saturated by BPS states and for two such particles, the combined scalar, vectorand gravitational forces cancel as sketched in fig. 3. We will elaborate on this insection 2.3.4. In the main part of this thesis, we will eventually suggest a similarcondition for axion-instanton pairs in the presence of scalar fields. Before doing so,we need to lay the basis. Section 2 begins with a review of the no-global-symmetriesconjecture and the black hole based argument supporting it. The remaining partof the section contains a review of the Weak Gravity Conjecture, a discussion ofmotivation as well as evidence for it and we will present some of the extensions tomore general settings.Evidence for our conjecture about axions, instantons and scalar fields will come

from compactification of Type IIA string theory where the scalars will be themoduli of the compactification manifold. In order to lay the ground for sections4 and 5 and make the thesis self-contained - we review the idea and procedure ofCalabi-Yau compactification in section 3, focusing on the moduli spaces of Calabi-Yau manifolds. In particular, we discuss the special Kähler structure, which thecomplex structure and Kähler moduli spaces carry. While a basic knowledge of dif-ferential geometry is of course indispensable, appendix A gives a recap of complexmanifolds while B.3.2 introduces the notion of special geometry.In section 4 which marks the beginning of the main part of this thesis, we will

see that compactification of Type IIB supergravity on a Calabi-Yau three-foldgives rise to several vector fields in spacetime while D3-branes wrapping super-symmetric three-cycles look like particles charged under these gauge fields fromthe perspective of the four-dimensional theory. The latter is due to the fact thatwe can perform the spatial integrations in the brane action which results in a one-dimensional path in spacetime as illustrated in fig. 4. We will show that theseparticles satisfy the condition

qAGABqB = 8V

(m2 + 4GAB∇Am∇Bm

). (1.2)

– 4 –

spacetimeD3-brane

three-cycle one-dimensonalworld-line

Figure 4: D3-brane wrapping three-cycle looks like particle fromlow-energy perspective. Surface symbolizing brane iscut open to expose cycle underneath.

Here, qA are the charges corresponding to the vector fields, GAB is the metric onthe moduli space, V the compactification manifold’s volume, the particle mass mis a function of the moduli and units are chosen such that Mp = 1.The same can be done in Type IIA supergravity, where Euclidean E2-branes

couple to the axions ξ and ξ arising from one of the forms upon compactifica-tion to four dimensions. In section 5, we will find that wrapping supersymmetricthree-cycles, these E2-branes look like points in space-time and can therefore beinterpreted as instantons. We will show that these satisfy a condition which is sim-ilar to the Gauge-Scalar WGC and relates instanton action, axion decay constantsand the scalars:

Q2 = S2 +Gab∇aS∇bS (1.3)

with S the instanton action and Q2 defined as

Q2 =1

2

(p q

)(−(I +RI−1R) RI−1

I−1R −I−1

)(pq

). (1.4)

In this expression, the charges q and p are the couplings to ξ and ξ respectivelywhile the matrix entries come from the kinetic terms for the axions in the four-dimensional action. The matrices I and R are the imaginary and real parts ofthe gauge-coupling matrixM which is defined in terms of the prepotential for the

– 5 –

1 Introduction

Calabi-Yau moduli. We will establish (1.3) for the axions with kinetic term∫ [(I)−1

]abdξa ∧ ∗dξb (1.5)

which corresponds to having purely p-charges but expect it to hold also for chargesq or both, q and p.The Gauge-Scalar WGC can be phrased as the statement that for two particles,

the gauge repulsion exceeds the combined gravitational and scalar attraction whileall forces cancel in case of two BPS states. It is therefore natural to propose thatanalogously, the equality (1.3) becomes an inequality

Q2 ≥ S2 +Gab∇aS∇bS (1.6)

in the absence of supersymmetry. For a single axion with decay constant f andonly one scalar field φ, this translates to

S2 + ∂φS∂φS ≤ 1/f 2 (1.7)

Further, we conjecture that this relation is a general extension of the axion-instanton WGC to situations with scalar fields.Finally, some (especially lengthy) calculations were put in appendix C and are

cited when needed in order to make the thesis clear and readable.

– 6 –

2 Weak Gravity Conjecture

2.1 Absence of global symmetries in quantum gravity

Since in the limit g → 0 a gauge symmetry becomes a global one, the WGC isstrongly motivated by the statement that quantum gravity does not allow for globalsymmetries. Before giving a sharper formulation of the conjecture, we review the

Q,M=const.

Figure 5: Throwing particles into the black hole to increase itscharge to Q while radiating the excess mass away.

general black-hole based argument [14, 15] for the absence of global symmetries.While a black hole charged under a gauge theory needs to satisfy the extremalitybound2)

M ≥ QMp, (2.1)

for a global symmetry we can construct a black hole at a fixed mass with arbitrarilyhigh charge Q: We can always increase the charge by throwing enough chargedparticles into the black hole while keeping the black-hole mass constant. The latteris achieved by waiting for the excess mass to be radiated away by emission ofuncharged particles (e.g. photons) via Hawking radiation before throwing anotherparticle into the black hole. This is depicted in fig. 5.Having established that we can consider a black hole with any global charge Q,

we now see what happens if we let such a black hole evaporate its mass. At somepoint, its Hawking temperature TH = M2

p/M will eventually exceed the mass mof the lightest charged particle. In order to get rid of its charge, the black holeneeds to have a mass that is equal to at least Q times m at this point,

M ≥ Qm. (2.2)

2)See equation (2.14) below.


This situation is sketched in fig. 6 on the right-hand side. From the two

Q,M

Figure 6: Bound demanded by energy conser-vation for decaying BH.

bounds (2.1) and (2.2) we conclude

Q ≤(Mp

m

)2

, (2.3)

which is violated if we only take Qbig enough.While a similar argument can

also be applied for gauge theorieswith tiny couplings [5], in the caseof a global symmetry, Hawking ra-diation produces equal numbers ofparticles with charge +q and −qsince there is no electric field out-side the black hole to produce achemical potential term that favorsdischarge [14]. Thus, the black hole cannot decay completely and what remains isa stable black hole remnant, an object with size and mass of Planck order. Sincethere is no upper bound on the global charge, this leads to an infinite number ofsuch remnants, which are argued [16] to let the entropy per unit area go to infinityand Newton’s constant to zero. This strongly suggests that global symmetriesshould be absent from a quantum theory of gravity which is further supported bystring theory, where in fact all symmetries are gauged [17].

2.2 Weak Gravity Conjecture

In view of the above claim, something should prevent us from taking the limitg → 0 for the coupling of a local symmetry where it becomes indistinguishablefrom a global one. Naturally, the question arises if small but non-zero couplingsare also problematic and if so, what the lower bound for allowed couplings is. Intheir paper [5], Arkani-Hamed et al. proposed the following:

– 8 –

2.2 Weak Gravity Conjecture

Weak Gravity Conjecture

In a theory containing a U(1) gauge field with coupling g and gravity,

i) there must be a particle carrying charge Q under the gauge field withits mass satisfying the bound

m ≤ qMp, (2.4)

where we defined q =√

2Qg,

ii) and the effective theory breaks down at a scale Λ . gMp.

These statements are called electric and magnetic WGC. The latter actually fol-lows from the former by considering a magnetic monopole: The Dirac-quantizationcondition demands gmag ∼ 1/gel. Since the monopole mass m needs to account forthe energy stored in its magnetic field, we have

m ≥ g2magΛ, (2.5)

where Λ is the cutoff of the effective theory. Applying the electric WGC,

m . gmagMp, (2.6)

it follows thatgmagΛ .Mp, (2.7)

which is the magnetic WGC for gmag ∼ 1/gel. Phrased differently, the WGC thussuggests that there is a lower bound on the strengths of interactions associatedwith the gauge boson and hence turns the merely qualitative argument that globalsymmetries should be absent from a quantum theory of gravity into a quantitativecriterion. Before discussing the conjecture in more detail, we want to point outthat while generally thought to be true, the WGC still is somewhat speculativeand the best evidence so far is that all models obtained from string theory seem tosatisfy it [18]. We will test the conjecture (or rather an extension of it) explicitlyin section 4.The argument we made to exclude global symmetries relied on the fact that

the charge of the black hole was not observable from outside, which is no longertrue if we consider non-zero gauge couplings. Nevertheless, we can adapt theabove argument as follows: Consider a black hole in four dimensions with massM electrically charged under a U(1) field with coupling g. Such a solution of the

– 9 –


Einstein action

S =1

2κ2

∫R ∗ 1− 1

2g2

∫F ∧ ∗F (2.8)

is called a Reissner-Nordström black hole. We chose convention such that

F = dA, A =g2Q

4πr, (2.9)

i.e. the charges are defined as

Q =1

g2

∫S2

∗F. (2.10)

Its metric is [19]

ds2 = −∆dt2 + ∆−1dr2 + r2dΩ22, (2.11)

where

∆ = 1− 2MG

r+

(gQ)2G

4πr2

= 1− Mκ2

4πr+

(gQ)2κ2

4π(8π)r2(2.12)

with κ = M−1p . Looking for the roots of ∆(r),

r = MG±

√(Mκ2

8π

)2

− g2Q2κ2

(4π)(8π), (2.13)

we can see that the r = 0 singularity is only shielded by an event horizon if theextremality bound

M ≥√

2gQMp (2.14)

is satisfied. According to the cosmic censorship conjecture [19, 20], naked sin-gularities should not appear in physical situations and therefore, (2.14) needs tobe fulfilled. We demand that an extremal black hole is still able to decay. If welabel the final states by an index i and write mi, qi for their masses and chargesrespectively as depicted in fig. 7, energy conservation demands that the black hole

– 10 –

2.3 Generalized Conjectures

Q,M charge qimass mi

Figure 7: Extremal black hole decaying to collection of states ofmasses mi and charges qi.

mass M =√

2gQMp amounts to at least the sum of the masses of these states,

M ≥∑i

mi. (2.15)

At the same time, we have

Q =∑i

qi (2.16)

due to charge conservation. Until now, we did not give any specification regardingthe WGC particle. We will now argue that the weak gravity bound is satisfied bythe particle whose charge-to-mass ratio is maximal. Let zi := qi/mi. Then

Q =∑i

zimi ≤ zmax

∑i

M = zmaxM, (2.17)

where zmax = q/m denotes the particle with maximal charge-to-mass ratio. UsingM =

√2gQMp, we conclude

1 ≤√

2q

mgMp, (2.18)

which is precisely the electric Weak Gravity Conjecture (2.4).


If we take a look at the weak gravity bound (2.4) again, we find that there aredifferent possible ways in which it could be modified. First, one can ask if ananalogous bound holds also for higher-dimensional objects charged under somep-form. Indeed, the argument we just gave is not restricted to zero-dimensional

– 11 –


objects, i.e. particles. Second, we can stick to particles but consider a setup wherethese are charged under several gauge fields and see how this restricts the particlemass. At last, we can take additional - namely scalar - forces into account. Wewill now discuss these modifications in turn.

2.3.1 The WGC for p-form gauge fields

As mentioned, it seems only natural to extend the Weak Gravity Conjecture togeneral p-form gauge fields, as was argued in [5]. We assume that p-forms appearas

1

2g

∫Fp+1 ∧ ∗Fp+1. (2.19)

The statement as given in [10] is:

p-Form Weak Gravity Conjecture

In d dimensions, for each Abelian p-form gauge field with coupling g, theremust be a (p− 1)-dimensional object (a (p− 1)-brane) with tension Tp thatcarries integer charge Q under this gauge field and satisfies

p(d− p− 2)

d− 2T 2p ≤ g2Q2Md−2

d , (2.20)

where Md is the d-dimensional Planck mass.

We will not discuss the precise nature of the prefactor and are content withseeing that (2.20) reduces to the precise form of the electric WGC given in (2.4)if we set p = 1 and d = 4, which corresponds to the case of a point-particle withmass m = T1 in four-dimensional spacetime that couples to a U(1)-gauge field. Inthat case, the inequality (2.20) reads

1

2m2 ≤ g2Q2M2

4 . (2.21)

With Mp = M4, we recover (2.4).The conjecture is supported by the same argument as the one for the Weak

Gravity Conjecture for a single U(1): We demand that extremal black branesshould be able do decay. There are some values of p where this fails though,namely p = 0 and p ≥ d− 2, and we need to address them separately:

i) A d− 1 form is non-dynamical and a d-form couples to a (d− 1)-brane whichis space-time filling.

– 12 –


ii) While the p = 0 case is particularly interesting since it corresponds to ax-ions and could yield a falsifiable prediction for the QCD axion. The obviouscandidates for (−1)-dimensional objects are instantons that couple to theseaxions with the inverse decay constant playing the role of the gauge coupling.Evidence for this relation comes from string theory and we will come back tothis issue in section 5.

2.3.2 The WGC for axions

It was suggested in [5] that the WGC can be extended to axions and instantonsthat couple to them in the following way:

Weak Gravity Conjecture for Axions

For any axion with decay constant f there must be an instanton with actionSE coupling to the axion such that

SE .1

fMp (2.22)

is satisfied.

We see that the euclidean instanton action SE is analogue to the mass “m”while the inverse of the axion decay constant f plays the role of the coupling “g”.That this statement is a consequence of the standard WGC can be derived fromT-dualities in string theory, where particles are mapped to instantons and viceversa [8].One needs SE > 1 in order to have e−SE < 1 for an instanton and hence, we

conclude that the axion decay constant is at most of order of the Planck mass,

f .Mp. (2.23)

This is very interesting inasmuch as it provides a potentially falsifiable predictionfor the QCD axion.

2.3.3 The WGC for multiple U(1)s

In [9], the Weak Gravity Conjecture was extended from U(1) to a product gaugegroup of several U(1)s. Before we give the statement, note that the WGC can berephrased in a slightly different way: Consider a U(1) gauge theory with coupling

– 13 –


g and particles of mass mi and charge qi3). We define the (dimensionless) ratios

zi := QiMp

mi

, where Qi :=√

2gqi. (2.24)

Then, the electric WGC (2.4) is the statement that there exists a species k suchthat

1 ≤ zk. (2.25)

Figure 8: Example with n = 2 that satis-fies the convex hull condition.

Now, let us extend the gauge group toa product

∏na=1 U(1)a with couplings

ga such that each particle mi carriescharges qai . Similar to before, we definethe ratios

zai := Qai

Mp

mi

, where Qai :=√

2qai ga.

(2.26)

For i fixed, qai and zai are vectors ofSO(n) and we write ~qi and ~zi respec-tively.It turns out that the “most obvi-

ous” generalization of the WGC to thepresent case, namely to conjecture thatthere is a species k such that |~zk| > 1,is not sufficient. However, demandingthat all vectors ~zi have |~zi| > 1 is toostrict. To determine the proper condition, we consider a black hole with charge ~Qand mass M and - following the same line of reasoning as in 2.2 - demand that itbe able to decay into a state with ni particles of species i for i = 1, ..., n. Due tocharge conservation,

~Q =∑i

~Qi (2.27)

3)That is, the index i labels the species, each consisting of particles and antiparticles of massmi and charge qi and −qi respectively.

– 14 –


and with ~Z = ~QMp/M and the vectors ~zi as defined above, it follows that

~Z =∑i

~QiMp

M=

1

M

∑i

nimi~zi. (2.28)

Thus, we can interpret ~Z as weighted average of the charge vectors ~zi. From energyconservation it follows that the mass M has to account for at least the masses ofthe decay particles, i.e.

1 >1

M

∑i

nimi. (2.29)

Hence, the black hole charge vector ~Z lies in the convex hull spanned by the vectors±~zi. Since for an extremal black hole |~Z| = 1, this translates to the requirementthat the unit ball must be enclosed by the complex hull. This motivates thefollowing conjecture:

Weak Gravity Conjecture for Several U(1)s

Let mi be a number of particles carrying charge ~qi = (qai ) under a productgauge group

∏Na=1 U(1)a with couplings ga and vectors ~Qi be defined via

Qai =√

2qai ga. Then, the convex hull spanned by the (dimensionless) charge-

to-mass ratios ~zi = ~QiMp/mi encloses the unit ball.

Fig. 8 illustrates a situation with two U(1)s where the conjecture is satisfied.Note that although |~z3| < 1, the unit ball is enclosed in the convex hull. Likewise,one can easily think of a situation where all charge vectors ~zi have |~zi| > 1 thatwould still violate the conjecture: Take for example n = 2 and assume there aretwo species with ~z1,2 orthogonal to each other. Then, we can take their lengthsslightly bigger than one but such that their convex hull intersects the unit disk.Since the boundary of the unit disk consists of extremal black hole solutions, thiswould correspond to a situation where stable black hole remnants exist that renderthe theory unphysical.

2.3.4 Gauge-Scalar Weak Gravity Conjecture

Finally, we come to an extension of the weak gravity conjecture that takes not onlygauge and gravitational but also scalar forces into consideration. This is going toplay a major role in the main part of the thesis where it is elaborated in moredetail. Here - for the sake of simplicity - we will restrict to a situation where the

– 15 –


WGC particle couples to a single scalar field. In that case, it was proposed in [11]that the WGC bound (2.4) needs to be altered in the following way:

Gauge-Scalar Weak Gravity Conjecture

If the WGC particle is coupled to a scalar field with coupling µ, the WGCbound is modified to

m2 + µ2M2p ≤ g2M2

p (2.30)

where absorbed the charge q in the definition of g.

It is easy to give a physical interpretation to (2.30). We consider a fermion ψas WGC particle whose mass m = m(φ) is parameterized by a scalar φ as wellas a single gauge-field Aµ under which the WGC particle is charged. Expandingthe scalar field about its VEV, φ = 〈φ〉 + δφ introduces a coupling to the WGCparticle: We have m(φ) = m(〈φ〉) + ∂φm(〈φ〉)δφ and the Lagrangian contains aterm

L ⊃ ∂φmδφψψ. (2.31)

Similarly, if the WGC particle is itself a (complex) scalar ϕ, the term m2(φ)ϕϕ∗

gives rise to a coupling

L ⊃ 2m∂φmδφϕϕ∗ (2.32)

in the Lagrangian. We stick with the fermion case where

L ⊃ mψψ +QAµγµψψ + ∂φmδφψψ. (2.33)

– 16 –


Clearly, this gives rise to the following tree-level interactions

between two WGC particles via gravity, the gauge field and the scalar. Associ-ated with these are forces of the form

F ∼ g2

r2, (2.34)

where g is the respective coupling. These are attractive for gravity and the scalarand repulsive for the gauge field. Thus, (2.30) can be read as the statement that thegauge repulsion exceeds the combined forces of gravitational and scalar attraction.This was already sketched in figure 3.We will establish in section 4.3 that equality holds for BPS states. Due to thedifferent signs, in this case the sum over all forces vanishes: Two similar BPSparticles put next to each other do not feel any force. For gravity to truly bethe weakest force, we need that in addition to (2.30), the scalar interaction mustexceed the gravitational force:

Scalar Weak Gravity Conjecture

If the WGC particle is coupled to a scalar field with coupling µ, then

|µ|Mp > m. (2.35)

Note that with µ = ∂φm, this is a differential equation that is easily integrated:

m ∼ e− φMp . (2.36)

Interestingly, this connects the WGC with another quantum-consistency conditionproposed in [4] which we discuss as last conjecture about quantum gravity beforeturning to Calabi-Yau compactification.

– 17 –


different superselection sectors

Figure 9: Moduli space of a quantum theory of gravity. Pointscorrespond to distinct low-energy effective theories.For a fixed point p0, it is conjectured that for any T >0, there is a point p at a distance d(p, p0) ≥ T .

2.3.5 Swampland Conjecture

It is well-known that the moduli space of a consistent quantum theory of gravity(this is true in string theory but assumed to be a general feature of quantumgravity) is parameterized by the expectation values of massless scalar fields. Hence,we can talk about the geometry of the moduli space by defining a metric via thekinetic terms of these scalars. A single point (as depicted in fig. 9) correspondsto a certain low-energy effective action and Ooguri and Vafa conjectured thatdisplacements from such a point in the moduli space of a quantum theory ofgravity lead to an infinite tower of states that become light exponentially fast.Distances in the moduli space can be defined as shortest geodesics with respect tothe metric mentioned above.

– 18 –


Swampland Conjecture

LetM denote the moduli space of a quantum theory of gravity and let thedistance d(p, q) between two points of M be defined as shortest geodesicbetween them. Then, for any p0 ∈M, the interval

d(p, p0)|p ∈M (2.37)

is not bounded from above and the theory at p has an infinite tower of stateswith mass of order

m ∼ e−αd(p,p0) (2.38)

with some α > 0.

The conjecture implies that as the distance diverges, the low-energy effectivetheory breaks down due to the appearance of an infinite tower of light states.Hence, the theory corresponding to a particular point p0 only makes sense in somefinite region around this point.

– 19 –

3 Calabi-Yau compactification

In the thesis, we will mostly be dealing with the low-energy actions of Type IIAand IIB string theory. These are formulated in ten spacetime dimensions. Sincethis is in contradiction with our every-day experience, six of these dimensions mustbe such that they are not detectable in experiment. A possible way to resolve thisissue is requiring the extra-dimensions to be small and compact such that theyare invisible above a certain length scale. This is illustrated in fig. 10: Identifyingtwo opposite sides of a rectangle yields a cylinder. Doing the same with the tworemaining sides, one arrives at a torus which looks like a single point if we take itsradii small enough. Before turning to the Type II theories, we will briefly reviewKaluza-Klein reduction and discuss the moduli spaces of Calabi-Yau manifolds.

point-likeshrinking shrinking

Figure 10: Compactified two-dimensional surface looks like asingle point from low-dimensional perspective.

3.1 Kaluza-Klein reduction

In the Kaluza-Klein ansatz, one assumes that spacetimeM has product structure4)

M =M1,3 × Y, (3.1)

whereM1,3 is some maximally symmetric space with four non-compact dimensionsthat represents our observed world - e.g. Minkowski space - and Y is a compact4)More generally, one can consider a so-called warped product but we will assume M to be a

direct product.

– 21 –


manifold called internal or compactification manifold. We denote the coordinatesonM1,3 by x and those on the internal manifold by y. The ansatz (3.1) correspondsto having a metric of the form

GMN(x, y) =

(g

(4)µν (x) 0

0 g(6)mn(y)

). (3.2)

The field content of IIA/IIB supergravity consists - apart from the metric - ofp-forms. The equation of motion of a p-form Cp is

d ∗ dCp = 0 (3.3)

and together with the gauge condition

d ∗ Cp = 0, (3.4)

this can be written as

∆(10)Cp = 0, (3.5)

where the Laplacian is defined as ∆(10) = (d + d†)2. In order to obtain a four-dimensional effective theory, one expands such a p-form Cp into a sum

Cp(x, y) = Ck(x)ϕk(y), (3.6)

where the Ck and ϕk are fields onM(1,3) and Y respectively. With the compacti-fication ansatz (3.1), the ten-dimensional Laplacian decomposes as

∆(10) = ∆(4) + ∆(6). (3.7)

In order for the four dimensional field to remain massless, we need

∆(4)Cp = 0 (3.8)

and consequently

∆(6)ϕk = 0. (3.9)

This tells us that the expansion is one in terms of harmonic forms on the internalmanifold.

– 22 –

3.2 Calabi-Yau requirement

Figure 11: Running couplings in minimal supersymmetric exten-sion of standard model.


So far, it is not clear what to chose as a compactification manifold, since weposed no further restrictions. But surely, the obtained four-dimensional theorywould depend on this choice in a crucial way. The easiest ansatz Y = T6, i.e.taking the six-torus as compactification manifold, does not yield an appealingtheory from a phenomenological point of view, as it leaves all (N = 4 or N = 8)supersymmetry unbroken in four dimensions. Still - despite the fact that so far, notraces of supersymmetry have been observed in experiment - it seems reasonable[21] to expect that at least some supersymmetry survives compactification to afour-dimensional theory. In particular, one can look for a model that possessesN = 1 supersymmetry in four dimensions at high energies which are low comparedto the compactification scale. A striking piece of evidence for this assumption isthe resulting unification of the three gauge-couplings at about 1016 GeV in such asupersymmetric extension of the standard model as illustrated in fig. 11. Whatexactly does the requirement of unbroken supersymmetry imply for the structureof the compactification manifold Y ? Spontaneous symmetry breaking is associatedwith non-vanishing vacuum expectation values: Consider a SUSY charge Q withSUSY parameter ε. Having unbroken supersymmetry means (see e.g. [22] on this)that

εQ|0〉 = 0. (3.10)

– 23 –


In order to have SUSY preserved by the vacuum, we therefore need all supersym-metry variations to vanish in the vacuum,

〈δεΦ〉 = 〈0|[εQ,Φ]|0〉 = 0 for all fields Φ. (3.11)

Note that the variation δb of a boson is fermionic and thus does not have a non-zerovacuum expectation value. Hence, we need only demand 〈δf〉 = 0 for fermions.In Type II string theory, the SUSY charges form two independent superalgebras.

Associated with each of these is a ten-dimensional Majorana-Weyl spinor εi withi = 1, 2. The gravitino which is present in both Type IIA and IIB transforms as[23]

δψiA = ∇Aεi + · · · (3.12)

where 〈· · · 〉 = 0 in the absence of fluxes which we do not consider. Thus, thecondition for unbroken SUSY is

〈∇Aεi〉 = 0, (3.13)

i.e. εi must be covariantly constant with respect to the background metric. Clearly,this restricts the allowed compactification manifolds5). We will examine what thisimplies. In Type IIA the two spinors transform in the 16 and 16’ of SO(1, 9),while in IIB both transform in the chiral representation 16. With the ansatz (3.1),SO(1, 9) decomposes as

SO(1, 9)→ SO(1, 3)× SO(6) (3.14)

and we see that on Y , the spinors have two pieces that transform6) as 4 and 4of SU(4) ∼= SO(6). If the spinors are covariantly constant on Y , they are leftinvariant upon parallel transport around any closed loop. But this is just anotherway to say that they transform as a singlet under the holonomy group7)

Hol(Y) ⊂ SO(6) ∼= SU(4). (3.15)

There are different possibilities for Hol(Y). Let us consider Hol(Y) = SU(3) where4 decomposes into 3 + 1 and - in a consequence - we arrive at one covariantly

5)Consider e.g. the two-sphere S2. It is well-known that one cannot define a vector field thatvanishes nowhere on S2. But a covariantly constant vector field that vanishes at one pointvanishes everywhere.

6)See e.g. [24].7)We assume Y to be orientable. Otherwise, Hol(Y) ⊂ O(6).

– 24 –


constant spinor of each chirality. To make this clear, we write (for Type IIA)

ε1 = ε1+ ⊗ η+ + ε1

− ⊗ η−,ε2 = ε2

+ ⊗ η− + ε2− ⊗ η−. (3.16)

The spinors ε1+ and ε2

+ are two independent Weyl-spinors in four dimensions andfor each covariantly constant η+, we get 4 + 4 supercharges in four dimensions -one for each component of εi+. Since we found one η+, this corresponds to N = 2SUSY in four dimensions.We saw that the existence of exactly one covariantly constant spinor means

reducing the holonomy of our compactification manifold to SU(3). For Kählermanifolds, this is the requirement to be Calabi-Yau, which we discuss in detailin appendix A. See in particular definition (A.13) from where we quote that acompact Kähler n-fold is Calabi-Yau iff it has holonomy Hol ⊂ SU(n). We didnot yet establish that the internal manifold is Kähler, though. As discussed in theappendix, we need to show that the manifold is complex and has a closed Kählerform. For the former, it suffices to construct an almost complex structure withvanishing Nijenhuis tensor field. By means of the covariantly constant spinor, webuild a bilinear

Jmn := iη†+γmpgpnη+ = −iη†−γmpgpnη−, (3.17)

where γmn = 12[γm, γn] and γm are the internal gamma matrices. With help of the

Fierz transformation formula (see e.g. [1]), one finds

JmpJpn = −δnm, (3.18)

that is, the compactification manifold is almost complex with

J = Jmndxm ⊗ ∂

∂xn(3.19)

the almost complex structure. Since η+, η− and the metric are covariantly constant,J is also covariantly constant and consequently, the Nijenhuis tensor vanishes:

Nmnp = Jnl∂[lJp]m − Jpl∂[lJn]

m = 0. (3.20)

This implies that Y is complex8). Finally, we can - according to (A.15) - definecomplex coordinates zi, z j such that the metric is Hermitian. Bearing in mind that

8)See (A.13) and below.

– 25 –


the metric is also covariantly constant, one sees that the Kähler form

J := igijdzi ∧ dz j (3.21)

is closed. We briefly recapitulate: In order to maintain a minimal amount ofsupersymmetry, the six-dimensional compactification space is required to be aKähler manifold with closed Kähler form and SU(3) holonomy - in short: A Calabi-Yau threefold.

3.3 Reduction on the Calabi-Yau

We found that in Kaluza-Klein reduction, ten-dimensional fields are expanded inharmonic forms on the internal manifold. As discussed in the appendix, these arein one-to-one correspondence with elements of the cohomology groups (see (A.34))and therefore counted by the Hodge numbers which - for a Calabi-Yau - take theform shown in fig. 14.To make clear how this works in practice, consider as an example a three-form

C3 with components CLMN in d = 10. Then, Cµνσ does not carry any Calabi-Yauindices and thus is a scalar from the perspective of the compactification manifold.Likewise, Cµνi is a (1, 0)-form (which does not exist on the internal space), Cijk a(2, 1)-form and so on. This leads us to the correspondence

Cµνσ ↔ H0,0(Y ), Cijk ↔ H3,0(Y ), Cµij ↔ H1,1(Y ),

Cijk ↔ H2,1(Y ), Cijk ↔ H1,2(Y ) (3.22)

and

Cµνi ↔ H1,0(Y ) = ∅, Cµij ↔ H2,0(Y ) = ∅ (3.23)

since h1,0 = 0 = h1,1. We will follow this scheme when performing the compactifi-cation of Type IIA and IIB.In the appendix, a complex basis for H2,1(Y ) is defined consisting of the (2, 1)-

forms ηa with a = 1, ..., h2,1. Another choice [25] is to consider a basis for thewhole space

H3(Y ) = H3,0 ⊕H2,1 ⊕H1,2 ⊕H0,3. (3.24)

Note that h3,0 = 1 = h0,3 (the corresponding cohomology groups only consist ofΩ and Ω respectively) and hence, we can chose a real basis αa, βa with a =

– 26 –

3.4 Moduli spaces of Calabi-Yau threefolds

Cohomology Basis Defined

H1,1(Y ) ωA in (A.35)

H2,2(Y ) ωA in (A.35)

H2,1(Y ) ηa below (A.35)

H1,2(Y ) ηa below (A.35)

H3(Y ) (αa, βb) in (3.25)

Table 1: Bases for cohomology groups of Y .

0, 1, ..., h2,1 satisfying∫Y

αa ∧ αb = 0 =

∫Y

βa ∧ β b, 1

V0

∫Y

αa ∧ β b = δba. (3.25)

The various basis forms of the different cohomology groups are listed in table 1.The relations (3.25) are preserved under the symplectic group Sp(2h2,1 + 2), i.e.

under transformations (β

α

)7→(A BC D

)(β

α

)(3.26)

with

ATD − CTB = 1 = (ATD − CTB)T (3.27)

and

ATC = (ATC)T , BTD = (BTD)T . (3.28)

We will make use of this structure in order to define a symplectic section thatserves as projective coordinates on the Calabi-Yau moduli space.


A Calabi-Yau three-fold Y with given Hodge numbers is not uniquely determined.Instead, we can consider perturbations

g → g + δg (3.29)

– 27 –


of its Kähler metric that leave it Ricci-flat. In the following discussion of this, wefollow [26] and [27]. In order to still have Rmn(g + δg) = 0 and thus maintain theCalabi-Yau property, the variations δg need to satisfy the Lichnerowicz equation

∇k∇kδgmn + 2Rmpnqδgpq = 0. (3.30)

In complex coordinates, this splits into two independent equations for perturba-tions δgij with mixed indices and such with pure indices, δgij and δgij, respectively.1. The mixed variations correspond to a real (1, 1)-form

iδgijdyi ∧ dyj ∈ H1,1(Y ) (3.31)

which allows us to expand

(gij + δgij)(x, y) = −ivA(x)(ωA)ij(y), A = 1, ..., h1,1, (3.32)

where ωA is the basis of H1,1(Y ) ∼= Harm1,1(Y ). Since the metric is directly re-lated to the Kähler form, (A.24), we have J = vAωA and call the real

Figure 12: Kähler cone

scalars vA Kähler moduli. Note that since J ∧J ∧ J is a volume form9), that is∫

Mk

k∧J > 0 (3.33)

for k = 1, 2, 3 and any complex k-dimensionalsubmanifold of Y , these determine the volumeof the internal manifold. In general, the met-ric deformations which preserve (3.33) form acone as illustrated in the figure on the right:If the equation holds for J , then it holds forany rJ with r > 0. Together with the h1,1

real scalars bA arising from the expansion ofthe two-form B2 which appears together withthe metric in Type II string theory, B2(x, y) =B2(x) + bA(x)ωA(y), we define h1,1 complexscalar fields

tA := bA + ivA (3.34)

forming the so-called complexified Kähler cone Mks.

9)See (A.25)

– 28 –


2. Since there are no (2, 0)-forms on a Calabi-Yau, the pure variations cannotbe expanded directly. Instead, they correspond to a complex (2, 1)-form

Ωijkgkmδgmldy

i ∧ dyj ∧ dy l ∈ H2,1(Y ) (3.35)

where Ω is the holomorphic (3, 0)-form. We expand

Ωijkgkmδgml = za(ηa)ijl, a = 1, ..., h1,2 (3.36)

or

δgij =i

‖Ω‖2za(ηa)iklΩ

klj, ‖Ω‖2 :=

1

3!Ωijkg

ilgjmgknΩlmn (3.37)

in terms of h2,1 complex scalar fields za and the basis ηa for H1,2(Y ). There aretwo things that need to be noted at this point. First, we have

dz1 ∧ dz2 ∧ dz3 ∧ dz1 ∧ dz2 ∧ dz3 = i33!dx1 ∧ dx2 ∧ dx3 ∧ dy1 ∧ dy2 ∧ dy3

(3.38)

in coordinates zi = xi+iyi10) Second, since the components Ωijk are antisymmetric,we need to have

Ωijk(z) = f(z)εijk (3.39)

for some holomorphic (and nowhere vanishing) function f and εijk the epsilontensor. Hence,

‖Ω‖2 = |f |2(√g)−1 (3.40)

and with this∫Ω ∧ Ω =

∫1

3!|f |2εijkεlmndzi ∧ dzj ∧ dzk ∧ dzl ∧ dzm ∧ dzn

=i33!

3!

∫‖Ω‖2d6x

√q

= −iV‖Ω‖2 (3.41)

or

‖Ω‖2 =i

V

∫Y

Ω ∧ Ω. (3.42)

10)We write d6x for dx1dx2dx3dy1dy2dy3.

– 29 –


Deformations (3.37) usually yield a metric which does no longer satisfy (A.19)and thus fails to be Hermitian. By a suitable coordinate transformation, themetric can be put in a form where the mixed-index components again vanish.But such a transformation is not holomorphic and this metric is thus Hermitianwhich respect to new complex coordinates, i.e. another complex structure [26].The corresponding scalars za are therefore called complex structure moduli and wedenote the space they span byMcs.Together, these scalars span the geometric moduli space of the Calabi-Yau,

which locally is a product

M =Mcs ×Mks (3.43)

of complex structure and Kähler structure moduli space respectively.The most general metric [26] one can write forM is

ds2 = − 1

2V

∫d6x√ggikgjl

(δgijδgkl + δgilδgjk − δBilδBjk

)(3.44)

with V the volume of the Calabi-Yau.

3.5 Complex structure moduli space

The part of the metric (3.44) corresponding to the complex structure moduli is

2Gabzazb := − 1

2V

∫Y

d6x√ggikgjlδgijδgkl. (3.45)

and using the expansion (3.37) as well as (3.38) again, we have

2Gabzazb = −2i‖Ω‖2

V‖Ω‖4

∫ηa ∧ ηbzazb, (3.46)

that is,

Gab =−iV‖Ω‖2

∫Y

ηa ∧ ηb, (3.47)

We will work in the real basis defined in (3.25) and denote the three-cycles dualto αa, βa by Aa,Ba, i.e. one has

Aa ∩ Bb = δab

= −Bb ∩ Aa (3.48)

– 30 –

3.5 Complex structure moduli space

and ∫Aaαb =

√V0δ

ab

= −∫Bbβa. (3.49)

In terms of this basis, Ω can be expanded as

Ω = Z aαa −Fbβb (3.50)

with the periods

Z a =1√V0

∫Aa

Ω, Fa =1√V0

∫Ba

Ω. (3.51)

The coordinates Z a are actually projective because Ω is homogeneous of degreeone,

(Z0, Z1, ..., ) ∼= (λZ0, λZ1, ...), (3.52)

which allows us to chose

za =Za

Z0. (3.53)

This is discussed in great detail in B.3.2. The expansion

∂zaΩ = kaΩ + iηa (3.54)

which is derived in C.1.1 can be used to define a Kähler potential Kcs for themetric (3.47) via

e−Kcs

:=i

V0

∫Y

Ω ∧ Ω =VV0

‖Ω‖2 (3.55)

– 31 –


since

∂za∂zbKcs = − ∂za

(1∫

YΩ ∧ Ω

∫Y

Ω ∧ (kbΩ− iηb))

=1

(−iV‖Ω‖2)2

∫Y

(kaΩ + iηa) ∧ Ω

∫Y

Ω ∧ (kbΩ− iηb)

− 1

−iV‖Ω‖2

∫Y

(kaΩ + iηa) ∧ (kbΩ− iηb)

= kakb − kakb −i

V‖Ω‖2

∫Y

ηa ∧ ηb

= Gab. (3.56)

Plugging in the expansion (3.50) for Ω,

i

V0

∫Y

Ω ∧ Ω =i

V0

∫Y

(Z aαa −Fbβb) ∧ (Z cαc − Fdβ

d)

= i(Z aFa − Z aFa), (3.57)

we find that this expression is equal to the symplectic product introduced in B.3.2,

e−Kcs

= −i〈v, v〉 where v =

(Z a

Fa

). (3.58)

Hence, the Kähler metric can be written as the Kähler and symplectic covariantexpression (B.37):

Gab = i〈∇aV, ∇bV 〉 (3.59)

Here, V = eKcsv and the Kähler covariant derivatives are introduced in (B.34).

The ka appearing in the expansion (3.54) can be determined explicitly from

∂a

∫Y

Ω ∧ Ω = −iV0e−Kcs

∂aKcs (3.60)

and

∂a

∫Y

Ω ∧ Ω =

∫Y

(kaΩ + iηa) ∧ Ω = iV0e−Kcs

ka. (3.61)

– 32 –

3.6 Kähler moduli space

From this we conclude

ka = −∂aKcs,

∂aΩ = −(∂aKcs) + iηa (3.62)

and can therefore use the Kähler covariant derivative to write

iηa = ∇aΩ (3.63)

and thus the metric on the complex structure moduli space as

Gab =−iV‖Ω‖2

∫∇aΩ ∧ ∇bΩ = −

∫∇aΩ ∧ ∇bΩ∫

Ω ∧ Ω(3.64)

3.6 Kähler moduli space

The metric on the Kähler moduli space is

GAB =1

4V

∫Y

ωA ∧ ∗ωB (3.65)

which we derive in C.1.2. Using the volume form (A.25), we define

K :=

∫Y

J ∧ J ∧ J = 6V (3.66)

and

KABC :=

∫Y

ωA ∧ ωB ∧ ωC , KAB := KABCvC , KA := KABvB. (3.67)

Note that with this notation, K = KAvA. Using the fact [28] that

∗ωA = −J ∧ ωA +3KA2K

J ∧ J, (3.68)

the metric GAB onMks can then be written as

GAB =3

2

(KABK− 3

2

KAKBK2

). (3.69)

– 33 –


It has a Kähler potential11)

Kks := − ln4

3K. (3.70)

To see this, note that we have ∂tA = −∂tA = − 12i∂vA on a function f(tA) = f(vA).

Thus,

∂tBK =i

2∂vBKCDEvCvDvE =

3i

2KB, ∂tAKB =

1

2i∂tAKBCDvCvD =

1

iKAB.

(3.71)

Using this, we have

∂tA∂tB

(− ln

4

3K)

= −∂tA(

3i

2KKB)

= −3i

2

(−1

K2

3

2iKAKB +

1

K1

iKAB

)= −3

2

(KABK− 3

2KKAKB

), (3.72)

which shows that Kks indeed is a Kähler potential for the metric. We define theinverse KAB via

KABKBC = δCA (3.73)

which implies

KBKBC = KBDvDKBC = vDδCD = vC . (3.74)

The inverse metric GAB can be written as

GAB = −2K3

(KAB − 3

vAvB

K

)(3.75)

11)We included the prefactor of 43 for later convenience.

– 34 –

3.7 Mirror symmetry

as can be seen by direct computation:

GABGBC =

(KAB −

3

2

KAKBK

)(KBC − 3

vBvC

K

)= δCA +

KAvC

K

(−3

2− 3 +

9

2

)= δCA . (3.76)

The Kähler manifold actually is of a certain kind type which is discussed in detailin appendix B.3.2. Such Kks can be expressed in terms of a prepotential F ,

e−Kks

= i(XAFA −XAFA), (3.77)

which is defined as

F := − 1

3!KABC

XAXBXC

X0, FA := ∂XAF , (3.78)

with coordinates XA := (1, tA). For explanations on the projective nature of thecoordinates and a more general discussion of special Kähler manifolds we againrefer to the appendix on special geometry. In C.1.3, we show that (3.77) indeed isa prepotential, i.e.

i(XAFA −XAFA) =

4

3K = 8V = e−K

ks

. (3.79)

We sum up metric and Kähler potential for the two moduli spaces in table 2for later reference. There is an interesting connection between the moduli spaceswhich we will exploit later. Hence, there is one topic to be discussed before turningto the main part.

3.7 Mirror symmetry

For a Calabi-Yau manifold Y (we only consider threefolds), mirror symmetry statesthe following:

- there is a so-called mirror Calabi-Yau Y with even and odd cohomologies iden-tified. That is, their Hodge numbers are related via

h1,1(Y ) = h2,1(Y ), h2,1(Y ) = h1,1(Y ) (3.80)

which amounts to reflecting the Hodge numbers in the Hodge diamond (see fig.14) along the diagonal.

– 35 –


Complex Structure Moduli Kähler Moduli

Metric

KählerPotential

Table 2: Metrics and Kähler potentials for moduli spaces.

- The complex structure and Kähler moduli spaces of the mirror Calabi-Yau Yare identified with the Kähler and complex structure moduli spaces of Y ,

Mks(Y ) =Mcs(Y ), Mcs(Y ) =Mks(Y ), (3.81)

and the action resulting from compactification of Type IIA supergravity on someCalabi-Yau Y is identical to the action of Type IIB supergravity compactified onthe mirror manifold Y . In particular, one can also assume a cubic prepotential

F := − 1

3!KABC

XAXBXC

X0, FA := ∂XAF , (3.82)

for the complex structure moduli space by switching to the mirror Calabi-Yau. Wewill not go into any more detail about mirror symmetry but refer to [29] for moreinformation.Having acquired the tools necessary for dealing with the moduli spaces of Calabi-

Yaus, we can now begin the main part of the thesis.

– 36 –

4 A Scalar WGC for Type IIB Particles

4.1 Supersymmetric black holes

The argument we gave in support of the Weak Gravity Conjecture relied on therequirement that charged extremal black holes should be able to decay. Sincecompactification of Type II supergravity gives rise to N = 2 SUGRA in d = 4 andin SUGRA, gravity is coupled to various scalars φa, we will consider extremal blackhole solutions of N = 2 supergravity in presence of scalar fields [11]. The latterenter the action of a theory containing U(1) gauge fields V A via symmetric realfunctions IAB(φ) and RAB(φ). The bosonic part of the action takes the genericform [30]

S =

∫∗R −Gabdφ

a ∧ ∗dφb +1

2IABFA ∧ ∗FB +

1

2RABF

A ∧ FB (4.1)

where FA = dV A and the metric g - entering implicitly via the Hodge-∗ - is afunction of the fields φ.

4.1.1 Electromagnetic duality

The theory enjoys a symmetry that is analogous to the well-known12) duality ofMaxwell’s electromagnetism, i.e. the interchange of F ↔ ∗F . Obviously, thispreserves the set of vacuum Maxwell equations

dF = 0, dG = 0 (4.2)

where the dual field strength G is related to F via

G =δ

δF

∫1

2F ∧ ∗F = ∗F. (4.3)

In the theory described in (4.1), the dual field strengths GA are defined similarlybut mix the forms FA and their Hodge duals:

GA :=δS

δFA= IAB ∗ FA +RABF

A (4.4)

with Hodge-dual13)

∗GA = IABFA −RAB ∗ FB. (4.5)

12)See e.g. [31] for an extensive treatment.13)Note that ∗2 = −1 when acting on even forms on the Calabi-Yau-threefold.

– 37 –


With this definition, we can write the Bianchi identities and equation of motionsimilar to (4.2) as

dFA = 0, dGA = 0 (4.6)

and look for a symmetry that preserves this set of equations as well as the definitionGA = δS/δFA. The latter requirement restricts [23] the allowed transformationsto elements of the symplectic group,(

FA

GA

)→ S

(FA

GA

), (4.7)

with S ∈ Sp(2h,R). Here, h denotes the number of vector fields V A, i.e. A =1, ..., h. Since we are considering charged, spherically symmetric and asymptoti-cally flat black-hole solutions, we define the magnetic and electric charges (pA, qA)by the integrals

pA :=1

4π

∫S2∞

FA, qA :=1

4π

∫S2∞

GA (4.8)

performed over a sphere at infinity.14) By definition,

Γ :=

(pA

qA

)(4.9)

transforms as symplectic vector.A discussion of the structure underlyingN = 2 supergravity is given in appendix

B.3. From there, we quote that the scalars parameterizing IAB and RAB are the hcomplex scalars zA from the gauge multiplets to which the one-forms V A belong.As discussed in the appendix, the manifold spanned by these scalars is endowedwith special Kähler geometry. Hence, there is a symplectic section (ZA,FA) withthe h+ 1 fields ZA(zA) serving as projective coordinates on the manifold and FAare functions of these coordinates. The Kähler potential is given by the symplecticinvariant expression

K = i〈v, v〉 (4.10)

with the symplectic vector v defined in (B.22). The metric can be written as theKähler and symplectic invariant expression

GAB = i〈∇AV, ∇BV 〉 (4.11)

14)Note that the convention differs slightly from the one used in (2.4).

– 38 –

4.1 Supersymmetric black holes

in terms of the Kähler covariant derivative (B.34) and

V = e12Kv = e

12K

(ZA(zA)

FA(zA)

)(4.12)

as shown in (B.38).

4.1.2 Superalgebras with central charge and BPS bound

Remember that for massive representations of the N = 1, d = 4 SUSY-algebra,the anti-commutator of the supercharges in the rest frame takes the form [22]

Qα, Q†β = 2mδαβ (4.13)

where m is the mass of the states and α, β = 1, ..., 4 label the Majorana spinorcomponents. For extended supersymmetry, i.e. N > 1, there are additional termsthat are allowed by Lorentz invariance and which we thus include:

QIα, Q

†Jβ = 2mδIJδαβ + 2iZIJΓ0

αβ (4.14)

where I, J = 1, ...,N with conserved quantities ZIJ . These are called centralcharges since they commute with all other generators in the superalgebra. Notethat the matrix ZIJ is necessarily antisymmetric which is why central charges canonly appear for extended supersymmetries. We will only consider N = 2 wherethe central charge matrix can be brought15) to the form

ZIJ =

(0 Z−Z 0

). (4.15)

Since the left-hand side of (4.14) is non-negative, the eigenvalues of the right-handside, namely 2m + 2Z and 2m − 2Z, must also be non-negative. The resultinginequality

m ≥ |Z| (4.16)

is called BPS bound. The central charges are electric and magnetic charges couplingto the gauge fields [19]. States that saturate (4.16) are called BPS states.

15)To do so, perform unitary transformations Z 7→ UTZU .

– 39 –


4.1.3 Relation for the central charge

We will now derive a relation for the central charge of the superalgebra that isgoing to play an important role in the next section.First, note that the name graviphoton of the vector in the N = 2 supergravity

multiplet (see table 5) stems from the fact that its field strength appears in thetransformation of the gravitino [23]. It is given by the (scalar field dependent)combination of field strengths

e12K(ZAGA −FAF

A) (4.17)

and therefore, we have the relation

Z =

∫e

12K(ZAGA −FAF

A)

= e12KZAqA −FAp

A (4.18)

which identifies the central charge with the graviphoton charge at infinity. Wewrite this in the concise form

Z = 〈V,Γ〉 with Γ =

(pA

qA

), V = e

12K

(XA

FA

). (4.19)

The sections FA and XB are related (see eq. (C.32)) via the gauge-coupling matrixM as

FA =MABXB (4.20)

and one finds

Z = XA(qA −MABp

B), ∇aZ = ∇aX

A(qA −MABpB) (4.21)

where we used thatM is symmetric. Hence,

|Z|2 +∇aZ∇bZGab = (qA −MACpC)(qB − MBDp

D)(XAXB +∇aX

A∇bXBGab

)(4.22)

and we identify the third factor on the right-hand side as −12I−1. To simplify this

equation further, we use the matrix introduced in [32]:

M =

(− (I +RI−1R)AB (RI−1)A

B

(I−1R)AB −(I−1)

AB

)(4.23)

– 40 –

4.2 Gauge fields from reduction of Type IIB supergravity

and we define

Q2 :=1

2ΓTM Γ. (4.24)

In terms of these, (4.22) takes the form

Q2 = |Z|2 +∇aZ∇bZGab (4.25)

which we derive in C.2.1. Note that (4.25) is a statement about BPS states whichare extremal with respect to (4.16), m = |Z|. For a complex function f , one has∂|f | = 1

2|f |(∂f)f and therefore, ∂|f |∂|f | = 14∂f∂f . Hence, we can write this as

Q2 = m2 + 4Gab∇am∇bm. (4.26)


Compactification of Type IIB supergravity on a Calabi-Yau three-fold gives riseto a number of U(1)s in the four-dimensional theory. As we will see, D3-braneswrapping three-cycles can be viewed as particles carrying charge under these gaugefields and thus should satisfy the weak gravity conjecture. The compactificationof Type II supergravity follows mainly the discussions in [25, 33, 34, 35]. Foran overview of compactification in the presence of background fluxes see [36]. Asummary of results on Calabi-Yau manifolds and their moduli spaces and a shortdiscussion of Type II SUGRA is given in the appendices A and B.

4.2.1 Action for Type IIB SUGRA

In appendix B, we discuss the ten-dimensional supergravities Type IIA and IIBwhich are the low-energy limits of the Type II string theories. From there we citethe action (B.14) where we now put hats on the ten-dimensional quantities:

S(10)IIB =

1

2κ2

∫e−2φ

(∗R+ 4dφ ∧ ∗dφ− 1

2H3 ∧ ∗H3

)− 1

4κ2

∫ (F1 ∧ ∗F1 + F3 ∧ ∗F3 + F5 ∧ ∗F5

)− 1

4κ2

∫C4 ∧ H3 ∧ F3

(4.27)

– 41 –


with the Kalb-Ramond 2-form B2, the dilaton φ and metric g from the NS-NSsector as well as the axion C0, 2-form C2 and 4-form C4 from the R-R sector and

H3 = dB2,

F1 = dC0,

F3 = dC2 − C0H3, F5 = dC4 − C2 ∧ H3, (4.28)

4.2.2 Expanding the fields

We make the Kaluza-Klein reduction ansatz discussed in section 3.1 and expandthe fields (4.28) in terms of harmonic forms on the Calabi-Yau, i.e. in terms of thebasis forms listed in table 1. We will discuss these fields in turn.

i) The axion C0 has no internal index and corresponds to H0,0(Y ).

ii) The two-form B2 (and likewise C2) decomposes in Bµν corresponding toH0,0(Y )

and Bµi corresponding to H1,1(Y ). There are no further terms, see our dis-cussion in section 3.3.

iii) The Cµν part of C4 is a four-form in d = 4 and therefore closed. Hence, it doesnot contribute to the action. The pieces Cijk, Cijk, Cijk and Cijk correspondto H3(Y ) = H3,0(Y )⊕H2,1(Y )⊕H1,2(Y )⊕H0,3(Y ) and the remaining Cµνijand Cijkl to H1,1(Y ) and H2,2(Y ) respectively.

Accordingly, we expand

B2(x, y) = B2(x) + bA(x)ωA(y),

C2(x, y) = C2(x) + cA(x)ωA(y),

C4(x, y) = DA2 (x) ∧ ωA(y) + %A(x)ωA(y) + V a(x) ∧ αa(y)− Ua(x) ∧ βa(y) (4.29)

with

a = 0, ..., h2,1, a = 1, ..., h2,1,

A = 0, ..., h1,1, A = 1, ..., h1,1. (4.30)

From now on, we will not write the coordinates explicitly. The vectors V a and Ua(as well as DA

2 and %A) are actually not independent but related by the self-dualityF5 = ∗F5 of the field strength F5. We will keep the one-forms V a and scalars %A.

– 42 –


Multiplet (Massless) Field Content Number

Gravity multiplet (gµν , V0) 1

Gauge multiplets (V a, za) h2,1

Hypermultiplets (bA, cA, vA, %A) h1,1

(hB, hC , φ, C0) 1

Table 3: Type IIB multiplets in d = 4.

Defining H3 := dB2, F a := dV a and Ga := dUa we have

H3 = H3 + dbA ∧ ωA,dC2 = H3 + dcA ∧ ωA,dC4 = dDA

2 ∧ ωA + F a ∧ αa −Ga ∧ βa + d%A ∧ ωA. (4.31)

We are only interested in the action for the gauge fields V a and Ga which comesfrom the F5 ∧ ∗F5-term in the action (4.27). Plugging in the above-mentionedexpansion for the expression of the field strength F5 in (4.28), we obtain

F5 = dDA2 ∧ ωA + F a ∧ αa −Ga ∧ βa + d%A ∧ ωA − (C2 + cAωA) ∧ (H3 + dbAωA)

= F a ∧ αa −Ga ∧ βa + d%A ∧ ωA + (dDA2 − C2 ∧ dbA − cA ∧H3) ∧ ωA

− cAdbB ∧ ωA ∧ ωB. (4.32)

4.2.3 Field content of Type IIB in four dimensions

As discussed in section 3, the four-dimensional theory possesses N = 2 SUSY.Hence, the massless fields form three kinds of irreducible representations, namelya gravity multiplet, gauge multiplets and hypermultiplets [23]. Again, we willdiscuss these in turn.

i) The gravity multiplet contains the graviton gµν and a one-form V 0.

ii) A gauge multiplet in N = 2, d = 4 contains a one-form and a complex scalar.The only (remaining) one-forms are V a which combine with the complex struc-ture moduli za.

iii) Hypermultiplets contain four real scalars. Hence, bA and the Kähler modulivA combine with cA and %A to h1,1 multiplets.

– 43 –


iv) The two-forms B2 and C2 are actually Poincaré dual to scalars hB, hc [25]which form another hypermultiplet with the dilaton φ and C0.

The various multiplets are collected in table 3.

4.2.4 Integrating on the Calabi-Yau

The non-vanishing terms in the integral over F5∧∗F5 containing the field strengthsF a and Ga are

I1 = − 1

4κ2

∫Y

(F a ∧ αa) ∧ ∗(F b ∧ αb −Gb ∧ βb) (4.33)

and

I2 = − 1

4κ2

∫Y

(−Ga ∧ βa) ∧ ∗(F b ∧ αb −Gb ∧ βb). (4.34)

We use the integrals from A.4 to get

4κ2I1 =− F a ∧ ∗F b[ImM+ (ReM)(ImM)−1(ReM)]ab

+ F a ∧ ∗Gb[(ReM)(ImM)−1]ba,

4κ2I2 = Ga ∧ ∗F b[(ReM)(ImM)−1]ba −Ga ∧ ∗Gb[(ImM)−1]ab (4.35)

where we defined the four-dimensional coupling κ2 = κ2/V0. This can be writtenin a more compact way. For example,

I := ImM−1(MF ) ∧ ∗(MF ) (4.36)

is the F ∧ F -term form above,

I = [(ImM)−1]ab(ReM+ i ImM)acFc(ReM− i ImM)bd ∧ ∗F

d

= [(ImM)−1]ab(ReM)ac(ReM)bdFc ∧ ∗F d

+ [(ImM)−1]ab(ImM)ac(ImM)bdFc ∧ ∗F d

= (ReM)ac[(ImM)−1]cd(ReM)dbFa ∧ ∗F b + (ImM)abF

a ∧ ∗F b. (4.37)

One can see that together with the remaining terms, this takes the form

ImM−1(G−MF ) ∧ ∗(G− MF ). (4.38)

There are, of course, other non-vanishing terms arising from the F5 ∧ ∗F5 integralbut we are not interested in these.

– 44 –


4.2.5 Action for the gauge fields

We have not yet imposed the self-duality condition F5 = ∗F5. First, note thatF5 = ∗F5 implies

∗F a ∧ ∗αa − ∗Ga ∧ ∗βa = ∗ F a ∧(

[(ReM)(ImM)−1]baαb+

[− ImM− (ReM)(ImM)−1(ReM)]abβb)−

∗Ga ∧(

[(ImM)−1]abαb

)− [(ReM)(ImM)−1]a

bβ b

= F a ∧ αa −Ga ∧ βa (4.39)

and by equating coefficients16)

∗G = ReM∗ F − ImMF

G = ReMF + ImM∗ F. (4.40)

The self-duality can then be imposed via the equation of motion for Ga by addingthe term

1

2F a ∧Ga (4.41)

as a Lagrange multiplier to the Lagrangian for the fields strengths F a and Ga:

LF a =1

4(ImM)−1(G−MF ) ∧ ∗(G− MF ) +

1

2F a ∧Ga. (4.42)

Variation with respect to Ga shows that (4.40) is now implemented:

δGLF a =1

4(ImM)−1(∗(G− MF ) + ∗(G−MF )) +

1

2F

=1

2(ImM)−1(∗G− ReM∗ F ) +

1

2F = 0, (4.43)

i.e.

∗G = ReM∗ F − ImMF. (4.44)

16)See footnote 13) to eq. (4.5).

– 45 –


Eliminating Ga in LF a via its equation of motion yields

LF a =1

4(ImM)−1(ReMF + ImM ∗ F −MF ) ∧ (ReM∗ F − ImMF − M ∗ F )

+1

2F ∧ (ReMF + ImM∗ F )

=1

4(ImM)−1(ImM∗ F − i ImMF ) ∧ (− ImMF + i Im ∗F )

+1

2F ∧ (ReMF + ImM ∗ F )

=1

2ImMabF

a ∧ ∗F b +1

2ReMabF

a ∧ F b (4.45)

which is a kinetic and a topological term for the gauge fields V a. We note:

Sgauge =

∫1

2ImMabF

a ∧ ∗F b +1

2ReMabF

a ∧ F b. (4.46)

This is precisely what we had expected for the action of the gauge fields in N = 2supergravity - see section 4.1. By comparison with the action (4.1), we see thatthe roles of I and R are played by the imaginary and real part of the matrixM.

4.3 Particles from D3-branes

As already mentioned, D3-branes wrapping three-cycles in Type IIB look likeparticles in the four-dimensional theory and it will turn out that these satisfy amodified version of the WGC.We begin with the action [22, 37]

S(10)D3 = −µ3

∫D3

d4ξe−φ√− det(G) + µ3

∫D3

C4 (4.47)

of a D3-brane coupled to the four-form C4 and wrapping a three-cycle C. The branetension µ3 is related17) to the ten-dimensional gravitational coupling as µ3 =

√π/κ

and G denotes the pullback of the spacetime metric onto the brane’s world-volume.Introducing the integer charges qa and pa by expanding C in terms of the three-cycles Aa and Ba dual to the three-forms αa and βa (see 3.49),

C = qaAa + paBa, (4.48)

17)For a Dp-brane we have µp = (√π/κ)(2π

√α′)3−p, see [22] eq. (13.3.7).

– 46 –

4.3 Particles from D3-branes

and using the expansion (4.29), the second integral in (4.47) reads

µ3

∫D3

C4 = µ3

√V0

(qa

∫V a + pa

∫Ua

). (4.49)

To compute the first integral, we perform a Weyl rescaling (B.7) with e−φ2 to get

rid of the dilaton factor and with (V0/V)−1/4 such that

−µ3

√V0

V

∫D3

d4ξ√− det(G). (4.50)

The volume of the cycle has a lower bound

Vol(C) ≥ 1√‖Ω‖

∣∣∣∣∫C

Ω

∣∣∣∣ =

√VV0

e12Kcs

∣∣∣∣∫C

Ω

∣∣∣∣ (4.51)

where we used (3.42) and the definition of the Kähler potential (3.55). Note thatwe did only choose a homology class in (4.48) and thus cannot give more than thisbound for the volume. We can specify the cycle as to minimize this volume18) inwhich case an equal sign holds. This is true for a BPS state. We will come backto this issue later and assume for now that the volume is minimized. Using theexpansion (3.50) of the holomorphic three-form, we can perform the integrationover C, ∫

CΩ =

∫C(Z aαa −Fbβ

b) =√V0

∫dτ(paZ

a − qbFb), (4.52)

to arrive at

−µ3

√V0

VVol(C) = −µ3

√V0e

12Kcs∣∣∣qaZ a − pbFb

∣∣∣ ∫ dτ. (4.53)

We have

µ3 =

√π

κ=

√π

κ√V0

(4.54)

18)Such a cycle is called supersymmetric special Lagrangian.

– 47 –


and are working in units where√π/κ = 1. Thus, the four-dimensional action is

S(4)D3 = −e

12Kcs∣∣∣qaZ a − pbFb

∣∣∣ ∫ dτ + qa

∫V a + pa

∫Ua. (4.55)

Clearly, this is the action of a particle with mass

m = e12Kcs∣∣∣qaZ a − pbFb

∣∣∣ (4.56)

charged under the gauge fields V a and Ua. One recognizes m as the central charge(4.19). Hence, a brane wrapping a supersymmetric cycle gives rise to a particlethat is extremal with respect to the BPS bound (4.16),

m = |Z| . (4.57)

4.4 A scalar WGC for the particle

4.4.1 Applying the central charge relation

We found out that the central charge is related to the symplectic charges and gaugecouplings via (4.25) and that a D3-brane wrapping a supersymmetric Lagrangianthree-cycle looks like a particle that is extremal with respect to the BPS boundupon compactification to four dimensional spacetime. Hence, we can apply thecentral charge relation derived in the last section,

Q2 = m2 + 4Gab∇am∇bm. (4.58)

If we consider only electric charges and set

Γ =

(0

qa

), (4.59)

the quantity Q defined in (4.24) becomes

Q2 = −1

2qa(I−1)abqb (4.60)

where I = ImM. In that case, Q is the analogue to the electric charge Q ap-pearing in the electric weak gravity conjecture (2.4) and we will make this clearby explicit calculation from the prepotential. Before doing so, we should point outthe following: Recall that we assumed that the brane is wrapping the three-cycle

– 48 –


such that it minimizes its volume and that only in this case the mass is extremalwith respect to the BPS bound. Dropping this assumption would lead to m ≥ |Z|but it is not obvious how the second term with derivatives of m would be affected.We keep the volume minimized.The equality (4.58) is in accordance with a generalization of the Gauge-Scalar

WGC as we presented it in (2.30) to situations with several gauge and scalar fields.In particular - since the second term on the right-hand side is positive definite - itimplies the Weak Gravity Conjecture

m2 ≤ Q2. (4.61)

4.4.2 Explicit calculation from prepotential

Now, to get better understanding of (4.58), we explicitly carry out the calculationin the prepotential formalism. To do so, we need an explicit expression of thegauge-coupling matrix M. Recall19) that mirror symmetry allows us to assumea cubic prepotential for the moduli in the following way: Let Y be the mirrorCalabi-Yau of Y . Since by mirror symmetry dimCH

2,1(Y ) = dimCH1,1(Y ) and

dimCH1,1(Y ) = dimCH

2,1(Y ), we interpret Kks := Kcs as Kähler potential forthe Kähler moduli on Y and likewise, XA := Z a, FA := Fa as belonging to itsprepotential. We will drop the tilde in the following.We only consider charges under the vectors in the gauge multiplet and hence

set qA = (0, qA), pA = 0, such that

m2 = eKks ∣∣qAtA∣∣2 . (4.62)

We can now make use of an explicit expression for the gauge-coupling matrixM. This can be derived directly from the prepotential but takes some time andeffort while at the same time being not very illuminating. Hence, we perform thecalculation in the appendix and from there cite eq. (C.59):

(ImM−1)AB = − 6

K

(1 bA

bA 14GAB + bAbB

). (4.63)

Therefore,

Q2 = −1

2qA(ImM−1)ABqB

= eKks (

qAGABqB + 4(qAb

A)2). (4.64)

19)See section 3.7.

– 49 –


We can choose tA = bA + ivA to be purely imaginary20) such that

Q2 = eKks

qAGABqB,

m2 = eKks ∣∣qAivA∣∣2 . (4.65)

Next, we compute the Kähler covariant derivatives

∇Am = ∂tAm+1

2

(∂tAK

ks)m (4.66)

that appear on the right-hand side of (4.58):

∇Am =1

2i

(∂vA +

1

2∂vAK

ks

)m

=1

2ie

12Kks (

∂vA + ∂vAKks)

(qAvA). (4.67)

With

∂vAKks = − 1

K∂vAKBCDvBvCvD = −3KA

K, (4.68)

we find

∇Am =1

2ie

12Kks

(qA −

3

2

KAKqBv

B

). (4.69)

We need the following expressions:

GABKA = −2K3

(KAB − 3vAvB

K

)KA

= −2K3

(vB − 3vB)

= e−Kks

vB (4.70)

and

GABKAKB =4

3KvAKB =

4

3K2. (4.71)

20)This is because the metric G does not depend on the axions.

– 50 –


With these, the right-hand side of (4.58) takes the form

m2 + 4GAB∇Am∇Bm =1

4eK

ks

(qAvA)2+

GABeKks

(qA −

3

2

KAKqCv

C

)(qB −

3

2

KBKqDv

D

)= eK

ks

[(qAv

A)2 + qAGABqB − 2

3

2

4

3(qAv

A)2 +9

4

4

3(qAv

A)2

]= eK

ks

qAGABqB. (4.72)

This is precisely what we found in (4.65) and thus we have shown that

qAGABqB = e−K

ks (m2 + 4GAB∇Am∇Bm

), (4.73)

confirming eq. (4.58). In the next section, we will perform a similar calculation inorder to establish a weak-gravity bound for an axion-instanton-pair.

– 51 –

5 A Scalar WGC for Type IIA Instantons

5.1 Axions from reduction of Type IIA supergravity

We now turn to Type IIA supergravity where compactification gives rise to axionsξa and ξa in the four dimensional theory. Unlike before, there is no self-dualitythat relates these two sets of fields.Our starting point is the ten-dimensional low-energy action (B.11) that is ob-

tained from eleven-dimensional supergravity as discussed in appendix B.1:

S(10)IIA = − 1

4κ2

∫e−2φH3 ∧ ∗H3 −

1

4κ2

∫F2 ∧ ∗F2 −

1

4κ2

∫F4 ∧ ∗F4

+1

4κ2

∫H3 ∧ C3 ∧ dC3 +

1

2κ2

∫e−2φ

(∗R+ 4dφ ∧ ∗dφ

)(5.1)

We recall the definitions

H3 = dB2, F2 = dA1, F4 = dC3 − A1 ∧ H3 (5.2)

and proceed similar to the discussion of Type IIB in the last section.

5.1.1 Expanding the fields

By now, it should be clear how the four-dimensional fields arise. The forms areexpanded as

A1 = A0

C3 = C3 + AA ∧ ωA + ξaαa − ξaβa

B2 = B2 + bAωA, (5.3)

where

a = 1, ..., h2,1, a = 0, 1, ..., h2,1,

A = 1, ..., h1,1, A = 0, 1, ..., h1,1. (5.4)

The corresponding field strengths are

dA1 = dA0,

dC3 = dC3 + dAA ∧ ωA + dξa ∧ αa − dξa ∧ βa,H3 = H3 + dbA ∧ ωA (5.5)

– 53 –


Multiplet (Massless) Field Content Number

Gravity multiplet (gµν , V0) 1

Gauge multiplets (V A, tA) h1,1

Hypermultiplets (za, ξa, ξa) h2,1

(hB, φ, ξ0, ξ0) 1

Table 4: Type IIA multiplets in d = 4.

with H3 := dB2. Since we are eventually interested in the axions ξa, ξa, we take acloser look at the expansion of F4. Plugging in the above expressions this becomes

F4 = dC3 + dAA ∧ ωA + dξa ∧ αa − dξa ∧ βa − A0 ∧H3 − A0 ∧ dbA ∧ ωA. (5.6)

5.1.2 Field content of Type IIA in four dimensions

Note that the three-form C3 is dual to a constant in d = 4 and thus does not carryany degree of freedom. The remaining fields form the following multiplets

i) The gravity multiplet consists of the graviton gµν and the one-form V 0.

ii) The one-forms V A combine with the complex tA = bA + ivA to h1,1 gaugemultiplets.

iii) Complex structure moduli za form h2,1 hypermultiplets with the scalars ξaand ξa.

iv) The two-form B2 is dual to a scalar hB which combines in an additionalhypermultiplet with the dilaton φ, ξ0 and ξ0.

For later reference, these multiplets are collected in table 4.Now, we will derive the four-dimensional actions for the axions ξ and ξ.

– 54 –

5.1 Axions from reduction of Type IIA supergravity

5.1.3 Integrating on the Calabi-Yau

The only non-vanishing terms in the integral of F4 ∧ ∗F4 are

I1 :=

∫Y

(dC3 − A0 ∧H3) ∧ ∗(dC3 − A0 ∧H3),

I2 :=

∫Y

(dAA − A0 ∧ dbA) ∧ ωA ∧ ∗[(dAB − A0 ∧ dbB) ∧ ωB],

I3 :=

∫Y

(dξa ∧ αa − dξa ∧ βa) ∧ ∗(dξ b ∧ αb − dξb ∧ βb) (5.7)

and of these, only I3 contributes a kinetic term for the axions:

I3 = − dξa ∧ ∗dξ b[ImM+ (ReM)(ImM)−1(ReM)]ab

+ 2dξa ∧ ∗dξ b[(ReM)(ImM)−1]ab− dξa ∧ ∗dξb[(ImM)−1]ab

= (ImM)−1ab[dξa +Macdξc] ∧ ∗[dξb + Mbddξ

d]. (5.8)

The kinetic part in the action for the axions ξa is therefore described by the inverseI−1 = (ImM)−1,

Sξ =

∫(I−1)abdξa ∧ ∗dξb. (5.9)

The one for the axions ξa is

Sξ =

∫(ImM−1)abMacMbddξ

c ∧ ∗dξd

=

∫(ImM−1)ab(ReM+ i ImM)ac(ReM− i ImM)bddξ

c ∧ ∗dξd

=

∫ [(ImM−1)ab(ReMac ReMbd) + ImMcd

]dξ c ∧ ∗dξd (5.10)

which we write as

Sξ =

∫ (RI−1R+ I

)ab

dξa ∧ ∗dξ b. (5.11)

The next section will focus on the axions ξ in order to make the calculation easier.

– 55 –


zero-dimensionalworld-volume

spacetime

E2-brane

three-cycle

Figure 13: E2-brane wrapping three-cycle looks like instantonfrom low-energy perspective. Surface symbolizingbrane is cut open to expose cycle underneath.

5.2 Instantons from E2-Branes

We saw in section 4.3 that a D3-brane wrapping a three-cycle in Type IIB lookslike a zero-dimensional object - a point particle - in four dimensions that carriescharge under the one-forms which arise from dimensional reduction of the four-form coupling to the D3-brane. Further, it was shown that this particle satisfies arefined form of the Weak Gravity Conjecture.In the present section, we consider a setting with a Euclidean E2-brane21) that

couples to the three-form C3. Wrapping such E2-branes around appropriate cyclesgives rise to (−1)-dimensional objects upon compactification to four dimensions.This is symbolized by the sphere in fig. 13 which from afar looks like a single pointin spacetime. We will now derive the four-dimensional action for such a brane.Our starting point is

S(10)E2 = −µ2

∫E2

d3Eξe

−φ√− det(G) +

√2µ2

∫E2

C3, (5.12)

the action of an E2-brane coupled to the three-form C3 which (like the D3-branein the last section) wraps a supersymmetric three-cycle

C = qaAa − pbFb. (5.13)

21)An Ep-brane is a p-brane whose world-volume time is euclideanized. See e.g. [1].

– 56 –

5.3 A scalar WGC for instantons and axions

This time, the integral is Euclidean and therefore yields the brane volume. Wefind:

S(4)E2 = −e

12Kks∣∣∣qaZ a − pbFb

∣∣∣+ qaξa + paξa. (5.14)

This can be interpreted as an object localized (see the illustration in fig. 13) inspacetime - an instanton - with action

S = e12Kks∣∣∣qaZ a − pbFb

∣∣∣ (5.15)

coupled to the axions ξa and ξa.


We will set qa = 0, i.e. consider charges

Γ =

(pa

0

), pa = (0, pa), (5.16)

implying that the instanton couples to ξ only. Similar to the last section, wecompute the expression

S2 +∇aS∇bSGab. (5.17)

5.3.1 Calculation

The prepotential and coordinates are related via Fa = MabZb (see eq. (C.32))

and therefore,

S = e12K |paMab| . (5.18)

Here, the axions ba = 0 were set to zero again such that

Mab = −i12e−KGab. (5.19)

The calculation goes through as before. We have

∇aS = ∂aS +1

2(∂aK)S, where ∂aS =

1

2(∂aK))S + e

12K∂a|paFa|, (5.20)

– 57 –


and thus

∇aS∇bS =1

4eK(−3Ka

2KpcMcdv

d +Macpc

)(−3Kb

2KpcMcdv

d + Mbdpd

). (5.21)

Hence,

4eKGab∇aS∇bS = Gab

(Gac −

3Ka2K

Gcdvc

)(Gbe −

3KbK

Gefvf

)pcpe

= Gabpapb − 2 · 3Ka

2KGabGbeGcdv

dpcpe +9

4

4

3(Gabp

avb)2

= Gabpapb − 3 · 4

3(Gabp

avb)2 + 3(Gabpavb)2, (5.22)

that is

Gab∇aS∇bS =1

4e−K

(Gabp

apb − (Gabpavb)2

). (5.23)

In the last steps, we used

GabKa =4

3Kvb, GabKaKb =

4

3K2, (5.24)

several times. The second term on the right-hand side of (5.23) is equal to minusS2:

S2 = eK∣∣∣∣pa(− i2e−KGabiv

b

)∣∣∣∣2= e−K

1

4(paGabv

b)2. (5.25)

Putting all terms together,

S2 +Gab∇aS∇bS =1

4e−KpaGabp

b. (5.26)

Remember that we found a kinetic term22)∫ [(ImM)−1

]abdξa ∧ ∗dξb (5.27)

22)Note the appearance of the inverse (ImM)−1 in contrast to what we had in the last section.

– 58 –


for the axions ξ that couple to the instanton with charges p. Hence, the corre-sponding charge Q2 is

Q2 = −1

2pa(ImM)abp

b

=1

4e−KpaGabp

b (5.28)

This is equal to the right-hand side of (5.26) such that we finally arrive at theidentity

Q2 = S2 +Gab∇aS∇bS. (5.29)

Note that we established (5.29) as a statement about the axions ξ by turning offthe charges q23). However, it is sensible to assume that it holds also for the axionsξ. For these we found (see eq. (5.11)) a kinetic term(

RI−1R+ I)ab

dξa ∧ ∗dξ b (5.30)

which makes the calculation leading to (5.29) pretty awful. We will not carry itout but instead propose that (5.29) also holds if the instanton couples to ξ or bothtypes of axions, corresponding to having q or both, q and p charges.

5.3.2 Result

Equation (5.29) is an extension of the Weak Gravity Conjecture (2.22) for axionsto a situation with scalar fields present. For simplicity, we will now consider thecase of a single scalar field φ and a single axion with decay constant f coupled tothe instanton. With Q = 1/f and still working in units where Mp = 1, we thenhave

1 = f 2(S2 + ∂φS∂φS). (5.31)

In particular - since the second term is positive definite - we recover the axionWGC

1 ≥ fS, (5.32)

that is, f < 1 if we assume S > 1.

23)This corresponds to the lower right part of M in (4.23).

– 59 –


We have established (5.29) for branes wrapping supersymmetric cycles. Thisis similar to the situation discussed in the last section where we saw that a D3-brane wrapping a supersymmetric three-cycle gives rise to a BPS particle. Weargued that for two such particles, gauge, gravitational and scalar interactionwould cancel, while for non-BPS states, the former would exceed the combinedgravitational and scalar forces. Hence, it seems natural to propose that withoutsupersymmetry, equation(5.29) becomes a bound

Q2 ≥ S2 +Gab∇aS∇bS. (5.33)

Motivated by this evidence, we propose that there is a general extension of theaxion-instanton Weak Gravity Conjecture to situations with scalar fields whichis analogous to the gauge-scalar Weak Gravity Conjecture and (reinstalling thePlanck mass) takes the form

M2p/f

2 ≥ S2 + |∂φS|2M2p . (5.34)

Since the second term on the left-hand side of this relation is positive-definite,this lowers the bound f < Mp for the axion decay constant which follows fromdemanding e−S < 1 for an instanton and we have the new bound

f < Mp/√

1 + |∂φS|2M2p ≤Mp. (5.35)

– 60 –

6 Conclusion

We have discussed the Weak Gravity Conjecture in the presence of scalar fields andpresented evidence from IIB supergravity where three-branes wrapping supersym-metric Lagrangian three-cycles look like (BPS) particles after compactification tofour-dimensional spacetime. Such a particle is charged under the gauge fields aris-ing from the expansion of the Type IIB four-form and saturates the Gauge-ScalarWeak Gravity Conjecture. If the particle carries charge under a single gauge fieldand with only one scalar field present, the bound takes the form

m2 + |∂φm|2M2p ≤ g2M2

p .

For two BPS particles, this translates to the statement that the combined gravi-tational, gauge and scalar forces between them vanishes.Motivated by this, we studied compactification of Type IIA string theory, where

a Euclidean E2-brane wrapping a supersymmetric Lagrangian three-cycle looks likean instanton in the four-dimensional theory. Expansion of the Type IIA three-formgave rise to certain axions and we found a relation for these which is similar to theGauge-Scalar Weak Gravity Conjecture:

Q2 = S2 +Gab∇aS∇bS

where S is the instanton action and Q2 describes the coupling of these axions tothe instanton. More precisely,

Q2 =1

2

(p q

)(−(I +RI−1R) RI−1

I−1R −I−1

)(pq

)where the matrices I and R are defined in terms of the prepotential for the Calabi-Yau moduli.We therefore argued that the axion-instanton Weak Gravity Conjecture might be

extended to situations where scalar fields are present. More precisely, we proposeda bound of the form

S2 + |∂φS|2M2p ≤M2

p/f2

relating instanton action S, axion decay constant f and scalar field φ. Whilewe gave a physical interpretation of the Gauge-Scalar Weak Gravity Conjecturein terms of forces, it is not clear to us how the additional ∂φS∂φS should beunderstood in the case of axions and instantons. Since this term is positive definite,we argued that scalar fields actually lower the bound f < Mp that follows fromthe axion-instanton Weak Gravity Conjecture when demanding e−S < 1 for theinstanton action.

– 61 –

A Mathematical Preliminaries

While the reader is expected to be familiar with basic differential geometry [31],we review some of the mathematical definitions and results on complex manifoldsthat are used in the thesis. More detailed discussions on this can be found in [38,39, 40] while a brief treatment of Kähler and Calabi-Yau manifolds is given in [27].

A.1 Complex manifolds

Simply put, a complex manifold is a manifold M that allows us to define holomor-phic functions f : M → C. To do so in a patch independent way, one needs thetransition functions to be holomorphic. We define

Definition A.1 (Complex manifold)A complex manifold is a differentiable manifold with a holomorphic atlas.

Note that a complex manifold necessarily has even real dimension since the co-ordinate functions are maps onto Cm. Instead of pondering on complex manifolds,we will begin with differential manifolds that are not necessarily complex but ad-mit a so-called almost complex structure which resembles the one that complexmanifolds carry: While they might not even locally look like Cm, their tangentspaces are complex. Eventually, we will think of complex manifolds as differen-tiable manifolds with almost complex structure and vanishing Nijenhuis tensorfield. The reason we are choosing this approach is that it is the one that comesnaturally when we discussing string compactification.

Definition A.2 (Almost complex structure)A (1, 1)-tensor field J on a differentiable manifold M satisfying

J 2 = − id (A.1)

is called almost complex structure. In that case, (M,J ) is called almost complexmanifold.

Note that since TM ⊗ T ∗M = End(TM), the field J defines an endomorphismJp of every fiber TpM . Writing Jp in local coordinates,

Jp = Jµν(p)dxµ ⊗∂

∂xν, (A.2)

– 63 –


condition (A.1) becomes

Jµσ(p)Jσν(p) = −δνµ. (A.3)

Taking the determinant of (A.3), we find that almost complex manifolds needto have even real dimension. Not every 2m-dimensional differentiable manifoldsadmits an almost complex structure, though. For example, the only spheres thatdo are S2 and S6 [39].The almost complex structure now allows us to define (anti-)holomorphic vector

fields as follows: We extend Jp to the complexification24) TpMC. Since J 2

p =− idTpM , the almost complex structure Jp has eigenvalues ±i and we can thereforedecompose

TpMC = TpM

+ ⊕ TpM− (A.4)

where the two disjoint spaces TpM± are spanned by the eigenvectors with eigen-value ±i. This can be extended to the whole tangent bundle and we write

TMC = TM+ ⊕ TM−. (A.5)

The complexification Γ(TM)C consists of the vector fields Z = X + iY withX, Y ∈ Γ(TM) and we define the complex conjugate Z := X − iY . With aprojection

P± :=1

2(id∓iJ ) , (A.6)

we can then decompose any vector field W ∈ Γ(TM)C as

W = Z1 + Z2 (A.7)

with Z1 := P+W and Z2 := P−W .

Definition A.3 ((Anti-)holomorphic vector field)A vector field Z = P+Z is called holomorphic and a vector field Z = P−Z anti-holomorphic.

24)By the complexification V C of a real vector space V we mean the complex space spanned bythe linear combinations a + ib with a, b ∈ V and all vector space operations defined in theobvious way. It has complex dimension dimC V

C = dimR V .

– 64 –

A.1 Complex manifolds

The decomposition of the complexified tangent bundle (A.5) also defines thecomplexified cotangent bundle

T ∗MC := T ∗M+ ⊕ T ∗M−. (A.8)

This gives us a way to define the space Ωr,s(M) of (r, s)-forms on M : The exteriorproduct ΛkT ∗MC decomposes as

ΛkT ∗MC = Λ0,kM ⊕ Λ1,k−1M ⊕ · · · ⊕ Λk,0M (A.9)

where

Λr,sM := ΛrT ∗M+ ⊗ ΛsT ∗M−. (A.10)

Definition A.4 ((r, s)-forms)A section of Λr,sM is called a (r, s)-form on M . We denote the set of (r, s)-formswith Ωr,s(M).

For a complex manifold, the exterior derivative of a (r, s) form can be decom-posed [40] as

dωr,a = αr+1,s + βr,s+1 (A.11)

which amounts to a decomposition d = ∂ + ∂. We define:

Definition A.5 (Dolbeault operators)The operators

∂ : Ωr,s → Ωr+1,s, ∂ : Ωr,s → Ωr,s+1 (A.12)

from the decomposition d = ∂+∂ are called Dolbeault operators. A p-form ω ∈ Ωp,0

for which ∂ω = 0 is called holomorphic.

We are now ready to give a criterion to determine whether an almost complexstructure actually is a complex structure.

Theorem A.6An almost complex manifold (M,J ) admits a complex structure if and only if theNijenhuis tensor field N : Γ(TM)× Γ(TM)→ Γ(TM) defined by

N(X, Y ) := [X, Y ] + J [JX, Y ] + J [X,J Y ]− [JX,J Y ] (A.13)

– 65 –


vanishes [38].

In local coordinates (A.2), the components of N are

Nµνδ = Jνσ∂[σJδ]µ − Jδσ∂[σJν]

µ. (A.14)

If this condition is fulfilled, one can define complex coordinates in every patch interms of which the almost complex structure J can be written as

J = idzj ⊗ ∂

∂zj− idz j ⊗ ∂

∂z j. (A.15)

Definition A.7 (Hermitian metric)A Riemannian metric g on a complex manifold (M,J ) is called Hermitian, if itsatisfies

g(X, Y ) = g(JX,J Y ) (A.16)

for all vector fields X, Y on M . In that case, (M, g) is called Hermitian manifold.

It is easy to show that every complex manifold admits a Hermitian metric [39].In local coordinates, g = gµνdx

µ ⊗ dxν and condition (A.16) translates as

gµν = JµαJνβgαβ. (A.17)

We note the following property [38]: The components of a Hermitian metric withrespect to a complex basis,

gij(p) = gp

(∂

∂zi,∂

∂zj

), gij(p) = gp

(∂

∂zi,∂

∂zj

), ..., (A.18)

satisfy

gij = 0 = gij. (A.19)

Thus, one can write

g = gij

(dzi ⊗ dz j + dz j ⊗ dzi

). (A.20)

– 66 –

A.2 Kähler geometry

A.2 Kähler geometry

Definition A.8 (Kähler form)On a Hermitian manifold (M, g,J ), the tensor field J defined by

J(X, Y ) := g(JX, Y ) for all X, Y ∈ Γ(TM) (A.21)

is antisymmetric25) and thus a two-form. Extending J from Γ(TM) to Γ(TM)C,it becomes a two-form of type (1, 1) and is called Kähler form of the Hermitianmetric.

It’s easy top see that the Kähler form is invariant under J , i.e. J(JX,J Y ) =J(X, Y ) for all vector fields X, Y . If we write J in a complex basis,

J = Jijdzi ⊗ dzj + Jijdz

i ⊗ dz j + · · · (A.22)

with

Jij = J

(∂

∂zi,

∂

∂zj

)= igij, Jij = igij, ···, (A.23)

we find that due to (A.19) the pure index components vanish and we can write

J = igijdzi ∧ dz j. (A.24)

Note that the Kähler form is real, J = J . The Kähler form of a Hermitian manifoldM with complex dimension m allows us to define a 2m-form

J ∧ · · · ∧ J. (A.25)

Writing zi = xi + iyi and using √g = 2m det gij, we find

1

m!

∫Y

J ∧ · · · ∧ J =

∫Y

√gdx1 ∧ · · · ∧ dxm ∧ dy1 ∧ · · · ∧ dym = Vol(Y ) (A.26)

Hence, (A.25) serves as a volume form on the manifold.

Definition A.9 (Kähler manifold)A Hermitian manifold (M, g) with closed Kähler form dJ = 0 is called Kählermanifold and its metric Kähler metric.

25)We have g(JX,Y ) = g(J 2X,J Y ) = −g(X,J Y ) = −g(J Y,X).

– 67 –


In local complex coordinates,

dJ = (∂ + ∂)igijdzi ∧ dz j = 0

from which we conclude

∂gij∂zl

=∂glj∂zi

,∂gij

∂z l=∂gik∂z j

.

Thus, we can write J in terms of a so-called Kähler potential K:

gij =∂2K

∂zi∂z j, J = ∂∂K. (A.27)

Note that the metric can be expressed this way in a given patch Ui. Given twocharts (Ui, ϕi) and (Uj, ϕj) with Ui∩Uj 6= ∅ and coordinates z = ϕi(p), w = ϕj(p),the Kähler potentials Ki and Kj do in general not coincide but are related by aKähler transformation

Kj(w, w) = Ki(z, z) + fij(z) + gij(z) (A.28)

with holomorphic (antiholomorphic) functions f and g [38].With help of the Dolbeault operator ∂, we can generalize de-Rham cohomology

and define

Definition A.10 (Dolbeault cohomology group)For a complex manifold M , we call the quotient space of ∂-closed modulo ∂-exact(r, s)-forms,

Hr,s(M) :=ker (∂ : Ωr,s(M)→ Ωr,s+1(M))

im(∂ : Ωr,s−1(M)→ Ωr,s(M))(A.29)

the X(r, s)-th Dolbeault cohomology group. Its complex dimension

hr,s := dimCHr,s(M) (A.30)

is called Hodge number.

Note that since the Kähler form J is closed, J ∈ Hr,s(M). The correspondingequivalence class [J ] is called Kähler class.We further define the adjoint operators ∂† := − ∗ ∂∗ and ∂† := − ∗ ∂∗ with the

Hodge-∗ operator and the Laplacians ∆∂ := (∂ + ∂†)2 and ∆∂ := (∂ + ∂†)2.

– 68 –

A.3 Calabi-Yau manifolds

We quote the following properties of a Kähler manifold with complex dimensionm [38]:

i) The Hodge numbers hr,s are related to the Betti numbers bp via

bp =∑r+s=p

hr,s, (A.31)

ii) they are symmetric, hr,s = hs,r and hr,s = hm−r,s−r and

iii) the Laplacians defined above coincide with ∆ = (d + d†)2,

∆ = 2∆∂ = 2∆∂. (A.32)

Definition A.11 (Harmonic form)A form satisfying ∆∂ = 0 is called harmonic and we denote the set of harmonic(r, s)-forms by Harmr,s(M).

The Kähler form is harmonic. Forms of this type are of great importance to ussince massless forms in the type II supergravity actions are expanded in terms ofharmonic forms on the compactification manifold. This is possible due to Hodge’stheorem: For Ωr,s(M) there is a unique decomposition

Ωr,s(M) = ∂Ωr,s−1(M)⊕ ∂Ωr,s+1(M)⊕ Harmr,s(M). (A.33)

From this follows in particular that

Harmr,s(M) ∼= Hr,s(M). (A.34)

A.3 Calabi-Yau manifolds

There are many different ways to define what a Calabi-Yau is. We cite from [40]:

Theorem A.12Let (Y, J, g) be a compact Kähler manifold with complex dimension dimC(Y ) = n.The following statements are equivalent:

i) Y is Ricci-flat, that is it has vanishing Ricci-form R = 0.

ii) It admits a globally defined and nowhere vanishing holomorphic n-form.

iii) It has holonomy Hol(g) ⊂ SU(n).

iv) Its first Chern class vanishes.

– 69 –


Definition A.13 (Calabi-Yau n-fold)A compact Kähler n-fold with one of the properties listed in A.12 is called Calabi-Yau manifold.

From now on, we denote the coordinates on a Calabi-Yau Y by yi, yj and onlyconsider those of complex dimension dimC Y = 3. For a Calabi-Yau threefold, theHodge numbers are depicted in the following figure:In particular, there is only one (3, 0)-form which we denote by Ω (with Ω the

h0,0

h1,0 h0,1

h2,0 h1,1 h0,2

h3,0 h2,1 h1,2 h0,3

h3,1 h2,2 h1,2

h3,2 h2,3

h3,3

=

1

0 0

0 h1,1 0

1 h2,1 h1,2 1

0 h1,1 0

0 0

1

Figure 14: Hodge Diamond of a Calabi-Yau threefold.

conjugate (0, 3)-form), while there are no one- or five-forms. It is worth noticingthat since h3,3 = 1, the (3, 3)-form Ω ∧ Ω must be proportional to J ∧ J ∧ J andthus to the volume form.As presented in the main text, for a Calabi-Yau Y the cohomology groups

H1,1(Y ) and H2,1(Y ) are associated with Kähler and complex structure modulirespectively. Thus, we introduce

i) a basis ωA for H1,1(Y ) with A = 1, ..., h1,1 as well as a dual basis ωA forH2,2(Y ) normalized such that

1

V0

∫Y

ωA ∧ ωB = δAB, (A.35)

where we introduced a six-dimensional fixed Calabi-Yau references volume V0.

ii) a basis ηa for H2,1(Y ) and ηa for H1,2(Y ) with a = 1, ..., h2,1.

– 70 –

A.4 Some integrals on Calabi-Yau threefolds

A.4 Some integrals on Calabi-Yau threefolds

We collect a couple of integrals used in the thesis for the compactification of TypeII supergravity. First, we need to know how deal with αa ∧ ∗αb and similar terms.The real three-forms αa and βa were defined to be orthogonal as stated in (3.25).Since their Hodge duals are again three-forms, we can expand

∗αa = Aabαb +Babβ

b, ∗βa = C abαb +Dabβ

b (A.36)

in terms of some coefficient matrices A,B,C and D. Thus,

1

V0

∫Y

αa ∧ ∗αb =1

V0

∫Y

αa ∧ (Bbcβc) = Bacδ

ca = Bba = Bab (A.37)

and analogously

1

V0

∫Y

βa ∧ ∗β b = −C ba = −C ab,1

V0

∫Y

αa ∧ ∗β b = Dba = −Aab. (A.38)

These matrices can be written in terms of a matrix M via

A = (ReM)(ImM)−1,

B = −(ImM)− (ReM)(ImM)−1(ReM),

C = (ImM)−1, (A.39)

which again is determined by the moduli. This is derived in C.3 and we note atthis point:

1

V0

∫Y

αa ∧ ∗αb = −[ImM+ (ReM)(ImM)−1(ReM)]ab,

1

V0

∫Y

βa ∧ ∗β b = −[(ImM)−1]ab,

1

V0

∫Y

αa ∧ ∗β b = −[(ReM)(ImM)−1]ba,

1

V0

∫Y

αa ∧ β b = δba (A.40)

If we integrate one of the four-dimensional forms Λ over the internal Calabi-YauY , we just get the form times the volume,∫

Y

Λ ∧ ∗Λ = Λ ∧ ∗Λ∫Y

∗1 = Λ ∧ ∗ΛV =1

6KΛ ∧ ∗Λ. (A.41)

– 71 –

B Type II Supergravity

In the thesis, we are dealing with the low energy effective actions of Type IIA andIIB string theory, namely Type IIA and Type IIB supergravity (SUGRA). Thisappendix reviews some facts about Type II supergravity that are important forthe thesis.In supersymmetry (SUSY ), the Poincaré and internal symmetries of a quantum

theory are extended to include N charges which are spinors. The Poincaré andSUSY charges form a so-called superalgebra that necessarily includes bosonic andfermionic elements. Supergravity is supersymmetry where the SUSY parametersε = ε(x) are local. The name hints at the fact that local supersymmetry necessarilyincludes gravity and conversely, a consistent supersymmetric theory of gravitymust be local. This is standard textbook material and a thorough treatment canbe found in the literature. See [21, 22], [19] or [41] for general string theory and[42], [23] or the appendix of [22] for supersymmetry and supergravity. Both hereand in the rest of the thesis, a basic knowledge of these topics is presumed.Supergravity restricts the number of spacetime dimensions in which it can be

consistently formulated to eleven as a higher-dimensional theory leads to morethan eight gravitinos upon toroidal compactification to four dimensions. Thesecan only be embedded in a representation that also has spins ≥ 5/2 for which - asis well-known - no consistent interactions exist [23].

B.1 Type IIA SUGRA from compactification of 11-dim. SUGRA

We begin with SUGRA in the largest allowed dimension d = 11. This is notonly the low-energy effective action of M-theory but we will construct Type IIAsupergravity by compactifying d = 11-SUGRA on a circle. It contains:

• The graviton gMN transforming in the symmetric traceless of SO(d − 2)which has dimension d(d− 3)/2 = 44.

• Its superpartner, the gravitino ψM : A Majorana spinor in the vector-spinorrepresentation of SO(d− 2) which has dimension (d− 3)2(d−2)/2 = 128.

• A three-form C3 that accounts for the remaining 128 − 44 = 84 =(d−2

3

)bosonic states.

The unique action keeping only the bosonic fields is

S(11) =1

2κ211

∫∗R − 1

2κ211

∫F4 ∧ ∗F4 −

1

6κ211

∫F4 ∧ F4 ∧ C3 (B.1)

– 73 –


with F4 = dC3 and the gravitational coupling κ11. In d dimensions, the latterrelates to the d-dimensional Newton’s constant Gd via

κ2d = 8πGd. (B.2)

As mentioned before, Type IIA SUGRA is obtained upon dimensional reductionof S(11), i.e. compactification on S1 keeping only the massless modes26). In thefollowing, we will sketch the usual Kaluza-Klein procedure as presented for instancein [21].We take one of the spatial dimensions and call it y, demanding that y ∼= y+2πR,

where R is the radius of the compactification circle. This corresponds to R1,10 →R1,9×S1 and gives rise to a scalar σ and a gauge field A1 from the lower-dimensionalperspective. To see this, consider the general ansatz

GMN(x, y)dxMdxN = Gµν(x) + e2σ(x)(dy + Aµ(x)dxν)2, µ, ν = 0, ..., 9, (B.3)

or

GMN =

(Gµν + e2σAµAν e2σAµ

e2σAν e2σ

), (B.4)

where the hats denote 11-dim. quantities. Likewise, the three-form C3 splits intoa three-form C3 and a two-form B2. The precise procedure is discussed in section 3and we won’t carry out the explicit compactification but rather quote the resultingten-dimensional action from [22]:

S(10) =1

2κ210

∫ (eσ ∗ R − 1

2e3σF2 ∧ ∗F2

)− 1

4κ210

∫ (e−σH3 ∧ ∗H3 + eσF4 ∧ ∗F4

)− 1

4κ210

∫B2 ∧ F4 ∧ F4, (B.5)

where we defined the ten-dimensional coupling κ2 = 2πRκ210 and

H3 = dB2, F4 = dC3 − A1 ∧H3. (B.6)

To make contact with string theory, we perform a Weyl-rescaling [35, 37]

Gµν → Ω−2Gµν (B.7)

26)This is due to the fact that M-theory compactified on a circle with radius R is correspondingto Type IIA string theory with string coupling gs = R/

√α′

– 74 –

B.2 Type IIB SUGRA

under which

Gµν → Ω2Gµν ,√−G→ Ω−d

√−G,∫

dxd√−GR →

∫dxd√−GΩ2−d (R+ (d− 1)(d− 2)Ω−2∂µΩ∂µΩ

). (B.8)

Note that wedge products of the form Cp ∧ ∗Cp contain√−G and p times the

metric, such that

Cp ∧ ∗Cp → Ω2p−dCp ∧ ∗Cp. (B.9)

With Ω = eσ2 and d = 10, the action is

S(10) =1

2κ210

∫ [e−3σ (∗R+ 9 · 2dσ ∧ ∗dσ)− 1

2F2 ∧ ∗F2

]− 1

4κ210

∫ (e−3σH3 ∧ ∗H3 + F4 ∧ ∗F4

)− 1

4κ210

∫B2 ∧ F4 ∧ F4. (B.10)

Defining σ = 23φ and rearranging the terms, we finally get

S(10)IIA =

1

2κ210

∫e−2φ

(∗R+ 4dφ ∧ ∗dφ− 1

2H3 ∧ ∗H3

)− 1

4κ210

∫(F2 ∧ ∗F2 + F4 ∧ ∗F4)− 1

4κ210

∫B2 ∧ F4 ∧ F4 (B.11)

which has the structure

S(10)IIA = SNS + S IIA

R + SIIACS (B.12)

with fields in the NS-NS and R-R sector of Type IIA string theory and a Chern-Simons term.

B.2 Type IIB SUGRA

Since we only consider massless fields, the difference between the field content ofType IIA and Type IIB string theory lies in the R-R sector where the formercontains the odd p-form gauge fields A1 and C3 while the latter has the even fields

– 75 –


C0, C2 and C4. The form of S(10)IIA thus suggests to define similarly

S(10)IIB = SNS + SIB

R + SIIBCS , (B.13)

explicitly,

S(10)IIB =

1

2κ210

∫e−2φ

(∗R+ 4dφ ∧ ∗dφ− 1

2H3 ∧ ∗H3

)− 1

4κ210

∫(F1 ∧ ∗F1 + F3 ∧ ∗F3 + F5 ∧ ∗F5)− 1

4κ210

∫C4 ∧H3 ∧ F3

(B.14)

with

H3 = dB2,

F1 = dC0,

F3 = dC2 − C0H3,

F5 = dC4 +B2 ∧ dC2. (B.15)

There is an obstacle though, since self-duality F5 = ∗F5 is not implied by (B.14)and thus must be imposed by an additional constraint on the solution. As aconsequence, the above action does not have the same number of bosonic andfermionic degrees of freedom in thus is not supersymmetric. The equations ofmotion, though, are supersymmetric if the constraint is imposed.

B.3 N = 2 supergravity in d = 4 and special geometry

As discussed in section 3, the four-dimensional theory resulting from compactifica-tion of Type II string theory on a Calabi-Yau manifold has N = 2 supersymmetry.Hence, the actions obtained from the Type II supergravities are N = 2 supergrav-ities in four dimensions. We briefly review some of their features needed in thethesis - a comprehensive treatment can be found e.g. in [23]

B.3.1 Multiplets of N = 2 supergravity

An extensive treatment on this can be found in [23]. The (massless) field contentof N = 2 SUGRA in d = 4 is given by nV one-forms V A and complex scalars zA,the metric g and a one-form V 0 called the graviphoton as well as 2nH additionalcomplex scalars ta, ξa. These form three multiplets that are listed in table 5. Theσ-model describing their self-interactions factorizes in a product of a special Kählermanifold for the scalars zA in the gauge multiplets and a quaterionic manifold for

– 76 –


Multiplet (Massless) field content Number

Gravity multiplet (gµν , A0) 1

Gauge multiplets (V A, zA) nV

Hypermultiplets (ta, ξa) nH

Table 5: N = 2 supergravity multiplets in d = 4.

the scalars in the hypermultiplets [30]. It is the former scalars that parameterizethe matrices I and R in 4.1. Since there are nV + 1 vectors, the special Kählermanifold is projective and we can use nv + 1 holomorphic sections ZA(zA) asprojective coordinates on the scalar manifold as well as nV +1 dual vector field FAsuch that the sections (ZA,FA) form symplectic vectors. We will now elaborate onthis in more detail. First, we will discuss special geometry for global SUSY sinceit is a little easier before turning to SUGRA. For further discussion on this topicsee [37, 43] and especially [23].

B.3.2 Rigid special Kähler manifolds

Definition B.1 (Rigid special Kähler geometry)Let (M, g) be a Kähler manifold with dimCM = n on which complex coordinatesza and 2n fields XA(z), FA(z) are defined where the latter transform as a vectorof the symplecic group Sp(2n,R). Writing V = (XA,FA), we can define an innerproduct

〈V,W 〉 = V T

(0 1

−1 0

)W (B.16)

which obviously is invariant under symplectic transformations. We call M a rigidspecial Kähler manifold if

gij = i〈∂iV, ∂jV 〉, 〈∂iV, ∂jV 〉 = 0. (B.17)

Note that i〈∂iV, ∂jV 〉 = ∂i∂ji〈V, V 〉, i.e. the symplectic invariant i〈V, V 〉 servesas a Kähler potential for the metric. Since both indices i and I range from 1, ..., n,the expression

∂iXI =

∂XI

∂zi(B.18)

– 77 –


is an n×n matrix. If it is invertible - which is the case if g is positive definite [23]- then we can choose the XI as new coordinates and multiply the second equationin (B.17) with the inverse ∂zi/∂XK to get

∂FK∂zj

=∂FI∂XK

∂XI

∂zj. (B.19)

This is the condition for FI to possess a prepotential F which is a holomorphicfunction of the coordinates X with

FI(X) =∂F(X)

∂XI. (B.20)

Since we are eventually interested in N = 2, d = 4 SUGRA obtained from Type IIstring theory, we now turn to so-called projective special Kähler geometry. Hence-forth, we will drop the word “projective”.

B.3.3 Special Kähler manifolds

Special geometry in its symplectic formulation is suited to describe the modulispaces of Calabi-Yau threefolds. In (3.50), the fields Z a and Fa, which are definedas periods of the holomorphic three-form27),

Z a =

∫Aa

Ω, Fa =

∫Ba

Ω, (B.21)

form a symplectic vector

v :=

(Z a(z)

Fa(z)

)(B.22)

with a = 0, 1, ..., h2,1. Note that unlike the case of rigid special geometry, this isone degree of freedom more than the number of coordinates za. However, sincea rescaling of Ω amounts to a rescaling of the periods Z a while the holomorphicthree-form is defined only up to a complex rescaling, the Z a are projective,

(Z0, Z1, ..., ) ∼= (λZ0, λZ1, ...). (B.23)

27)Note that unlike the main text, we are not careful about a Calabi-Yau reference volume factorV0 in this appendix. It can always be reinstalled by dimensional analysis, though.

– 78 –


Defining h2,1 inhomogeneous coordinates28) Z a/Z0 - provided the matrix

∂a

(Zb

Z0

)(B.24)

is invertible - we have the right number to chose

za =Za

Z0(B.25)

or Z a = (1, za). Hence, we can write the periods Fa as functions29) of the co-ordinates Z. We arrive at a condition similar to the second equation of (B.17)from ∫

∂aΩ ∧ Ω =

∫(αa − ∂aFbβ

b) ∧ (Z aαa −Faβa)

= −Fa + (∂aFb)Zb, (B.26)

where ∂a = ∂/∂Z a, since this integral vanishes. This follows directly from theexpansion

∂aΩ = kaΩ + iηa (B.27)

derived in (C.1.1) and we conclude

Fa = (∂aFb)Zb =

1

2∂a

(FbZ

b). (B.28)

Thus, similar to (B.20), the periods FI can be written in terms of a prepotentialvia

Fa =∂F(Z)

∂Z a, where F(Z) =

1

2FaZ a. (B.29)

Unlike the rigid case, the symplectic product i〈v, v〉 does not serve as a Kählerpotential. Rather, we define

e−K := −i〈v, v〉 (B.30)

and we will see in a minute, that K is a Kähler potential for the metric on thecomplex structure moduli space. As mentioned several times, the holomorphic

28)In a chart where Z0 = 0, we can chose a Zb 6= 0, since the Z a do not simultaneously vanish,and rename the indices.

29)By a small abuse of notation, we use the same name for Fa(z) and Fa(Z) again.

– 79 –


three-form is not uniquely defined but be can consider redefinitions

Ω→ ef(Z)Ω (B.31)

that should not change the physics. We see that under (B.31), the quantity Ktransform as

K → K − f(Z)− f(Z) (B.32)

which is a Kähler transformation. Now, we are almost in place to write down anexpression for the Kähler metric similar to the rigid case (B.17). First, we definethe vectors

V := e12Kv (B.33)

as well as Kähler covariant derivatives

∇aZb := ∂aZ

b + (∂aK)Z b,

∇aZb := ∂aZ

b + (∂aK)Z b. (B.34)

Note that these transform as

∇aZb → ∇aZ

be−f , ∇aZb → ∇aZ

be−f (B.35)

under a combined Kähler transformation

Z a → Z ae−f , Z a → Z ae−f , K → K + f + f . (B.36)

Then, the expression

i〈∇aV, ∇bV 〉 (B.37)

where ∇aV = eK∇v is Kähler and symplectic covariant. Explicitly,

i〈∇aV, ∇bV 〉 = ieK (∂a∂b〈v, v〉+ ∂a〈v, v〉∂bK + ∂aK∂b〈v, v〉+ ∂aK∂bK〈v, v〉)= (∂a∂bK + ∂aK∂bK) + (−∂aK∂bK) + (−∂aK∂bK) + (∂aK∂bK)

= ∂a∂bK, (B.38)

which means that it is the Kähler metric corresponding to the Kähler potentialK. In the discussion of the complex structure moduli space in section 3.5 we seethat this is the Kähler metric Gab.

– 80 –

C Calculations

C.1 Compactification

C.1.1 Expansion of ∂zaΩ

We derive eq. (3.54). First, we show that ∂zaΩ ∈ H3,0 +H2,1:

∂zaΩ =1

3!(∂zaΩijk)dy

i ∧ dyj ∧ dyk +1

2Ωijk∂za(dy

i) ∧ dyj ∧ dyk. (C.1)

The first part is a (3, 0)-form, while the second term is the wedge product of thederivative ∂zadyv and a (2, 0)-form. We will take a closer look at this derivative.Expanding

yi(za + δza) = yi(za) + Λiaδz

a, (C.2)

we find

∂za(dyi) = dΛi

a =∂Λi

a

∂yjdyj +

∂Λia

dyjdyj. (C.3)

Thus, ∂zadyv is composed of a (1, 0)-form and a (0, 1)-form, i.e. ∂zaΩ ∈ H3,0+H2,1.We calculate the (2, 1)-part

1

2Ωijk

∂Λia

∂y ldy l ∧ dyj ∧ dyk (C.4)

by differentiating the Calabi-Yau metric:

0 = ∂za(2gijdyidyj) = 2

∂gij∂za

dyidyj + 4gkj∂Λk

a

∂yidyidyj (C.5)

and find

gikgki∂gij∂za

=∂gij∂za

= −2gkj∂iΛka. (C.6)

From (3.37), we have

∂gij∂za

= − i

‖Ω‖2(ηa)iklΩ

klj (C.7)

– 81 –

C Calculations

with ∂i := ∂/∂yi, that is,

∂iΛka =

i

2‖Ω‖2(ηa)ilmΩlmk. (C.8)

Thus, we confirm that (C.4) is given by ηa, i.e.

∂

∂zaΩ = kaΩ + iηa. (C.9)

C.1.2 Kähler moduli metric

We derive the metric (3.69) from the Kähler moduli part of (3.44):

2GABtAtB = − 1

2V

∫d6x√ggikgjl

(δgilδgjk − δBilδBjk

)= − 1

2V

∫d6x√ggikgjl

[(−i)2vA(ωA)ilv

B(ωB)jk − bA(ωA)ilbB(ωB)jk

]=

1

2V

∫d6x√ggikgjl(vAvB + bAbB)(ωA)il(ωB)jk

=1

2V

∫ωA ∧ ∗ωBtAtB, (C.10)

that is,

GAB =1

4V

∫ωA ∧ ∗ωB. (C.11)

C.1.3 Kähler moduli space prepotential

It is shown that the cubic prepotential indeed is a prepotential for the Kählermoduli:

i(XAFA −XAFA) =

−i3!KABC

(−X0X

AXBXC

X02 + 3XAXBXC

X0− c.c.

)=

i

3!KABC

(tAtBtC + 3tAtBtC − tAtB tC − 3tAtB tC

)=

i

3!KABC(tA − tA)(tB − tB)(tC − tC)

=i

3!KABC(2i)3vAvBvC

=4

3K (C.12)

– 82 –

C.2 Supersymmetric black holes

C.2 Supersymmetric black holes

C.2.1 MatrixM in relation for central charge

We show that

1

2

(p q

)(− (I +RI−1R) RI−1

I−1R −I−1

)(p

q

)= −1

2(q −N p)TI−1(q − Np)

(C.13)

in order to prove (4.25). After multiplying both sides by 12, the left-hand side reads

−pA(I +RI−1R)ABpB + pA(RI−1)A

BqB + qA(I−1R)ABp

B − qA(I−1)ABqB.(C.14)

We find the same qTI−1q-term on both sides and the remaining terms on theright-hand side are

−NACpC(I−1)ABqB = −pC(R+ iI)AC(I−1)ABqB

= −pA(RI−1)ABqB − ipAqA,

−qA(I−1)ABNBCpC = −qA(I−1)AB(R− iI)BCpC

= −qA(I−1R)ABpB + ipAqA,

NACpC(I−1)ABNBDpD = pC(R+ iI)AC(I−1)AB(R− iI)BDpD

= −pA(I +RI−1R)ABpB, (C.15)

which adds up to precisely what we found for the left-hand side.

C.3 Gauge-coupling matrix

The matrices defined by the expansions

∗αa = Aabαb +Babβ

b, ∗βa = C abαb +Dabβ

b (C.16)

can be expressed in terms of the moduli. To do so, notice that the Hodge star actson a (3, 0)-form as ∗Ω = −iΩ which lets us write (3.50) as

∗Ω = Z a∗αa −Fb∗βb

= Z a(Aabαb +Babβ

b)−Fb(Cbcαc − Acbβ c)

= −iΩ

= −i(Z aαa −Fbβb), (C.17)

– 83 –

C Calculations

and by equating coefficients

Z aAab −FaC ab = −iZ b,

Z aBab + Aba = iFb. (C.18)

We found the expression (3.62) for ka in the expansion of Ω, which we now calculateexplicitly with help of the prepotential:

ka = ∂za ln(Z bFb − Z

bFb)

=Z bFab − Fa

i(Z bFb − Z bFb

) , Fa = FabZb

=Z bFab − FabZ b

Z bZ cFbc − Z bZ cFbc

=2 ImFabZ b

2 ImFbcZ bZ c. (C.19)

Thus,

∂zaΩ =1

ImFbcZ bZ cImFabZ

bΩ + iηa. (C.20)

Since also ∂z0Ω ∈ H3,0 +H2,1, one can define η0 via

∂Z0Ω =1

ImFbcZ bZ cImF0bZ

bΩ + iη0. (C.21)

Using equation (C.18), it follows that

αa −Fabβb = ∂ZaΩ =

1

ImFbcZ bZ cImFabZ

bΩ + iηa

=1

ImFbcZ bZ cImFabZ

b(Z cαc −Fcβ c) + iηa, (C.22)

– 84 –


i.e.

iηa =

(δca −

ImFabZ bZ c

ImFbdZ bZ d

)αc −

(Fac −

ImFabZ bFcImFbdZ bZ d

)β c

= ∗ ηa

=− i

(δca −

ImFabZ bZ c

ImFbdZ bZ d

)(Aecαe +Bceβ

e)

+ i

(Fac −


)(C ceαe − Aceβ e) (C.23)

from which we find (δca −

ImFabZ bZ c

ImFbdZ bZ d

)(δec + iAec) = i

(Fac −


)C ce

i(−AecZ c + C ceFc)ImFabZ b

ImFbdZ bZ d+ δea −

ImFabZ bZ ee

ImFbdZ bZ d= −i(Aea −FacC ce). (C.24)

Using (C.18) again, this is

−Z e ImFabZ b

ImFbdZ bZ d+ δea −

ImFabZ bZ e

ImFbdZ bZ d= −i(Aea −FacC ce), (C.25)

i.e.

Aba −FacC cb = iδba −2i

ImFcdZ cZ dImFacZ cZ b (C.26)

and separating real and imaginary part

Aba = ReFacC cb + δba + i ImFac

(C cb − 2

ImFcdZ cZ dZ cZ b

). (C.27)

Likewise, one finds

Bab + FacAcb = −iFab +2i

ImFcdZ cZ dImFacZ cFb (C.28)

– 85 –

C Calculations

Introducing a matrix

Mab := Fab +2i

Z a ImFabZ bImFacZ c ImFbdZ

d, (C.29)

we can write

A = (ReM)(ImM)−1,

B = −(ImM)− (ReM)(ImM)−1(ReM),

C = (ImM)−1. (C.30)

We derive another identity which we use in the thesis:

MabZc = FabZ

b + 2i ImFacZ c

= FabZb, (C.31)

i.e.

Fa =MabZb. (C.32)

For a prepotential of the form

F := − 1

3!Kabc

ZaZbZc

Z0, Fa := ∂ZaF , Z = (1, tA = bA + ivA), (C.33)

like the one for the Kähler moduli space in section 3.6, the matrix M can bebrought to an explicit form which we will now to. We have

F00 =− 1

3!∂0Kabc

ZaZbZc

−(Z0)2

=− 1

3KabcZaZbZc,

Fa0 =1

3!Kbcd∂a

ZbZcZd

(Z0)2

=1

2KabcZbZc,

Fab =−KabcZc (C.34)

– 86 –


and

ImFab =−Kabcvc = −Kab,

ImF00 =− 1

3Im[(ba + iva)(bb + ivb)(bc + ivc)

]=− 1

3Kabc(vabbbc + vbbabc + vcbabb − vavbvc)

=−Kabbabb +1

3K with K = Kava = Kabvavb = Kabcvavbvc,

ImFa0 =1

2Kabc Im

[(bb + ivb)(bc + ivc)

]=

1

2Kabc(bbvc + vbbc)

=Kabbb. (C.35)

Thus,

Z a ImFabZb =

(−Kabbabb +

1

3K)

+ 2 (Kabba)Zb +(−KabZaZb

). (C.36)

With

KabZaZb =Kab(ba + iva)(bb + ivb)

=Kabbabb + 2iKaba −K,KabbaZb =Kabbabb + iKaba, (C.37)

we have

Z a ImFabZb =Kabbabb +

1

3K + 2iKaba −Kabbabb − 2iKaba +K

=4

3K. (C.38)

Thus,

M00 =− 1

3Kabc(ba − iva)(bb − ivb)(bc − ivc) +

2i43K

(ImF0bZb)2 (C.39)

– 87 –

C Calculations

where

i(ImF0bZb)2 =i(−Kabbabb +

1

3K +KabbaZb)2

=i

(−Kabbabb +Kabbabb +

1

3K + iKabbavb

)2

=i

(1

3K + iKaba

)2

=

(1

3K + iKaba

)(i1

3K −Kaba

). (C.40)

The imaginary part is

ImM00 =− 1

3Kabc(−3vabbbc + vavbvc) +

3

2K

[(1

3K)2

−KabaKbbb]

=Kabbabb −1

3K +

K6− 3

2KKabaKbbb

=− K6

(1− 6

KKab +

9

K2KaKb

)babb

=− K6

(1 + 4Gabb

abb). (C.41)

The 0a-term is

M0a = F0a +3i

2K(ImFabZ

b)(ImF0cZc) (C.42)

with

(ImFabZb) = ImFa0 + ImFabZb

=Kabbb −Kab(bb + ivb)

=− iKa, (C.43)

– 88 –


i.e.

ImM0a = −Kabbb + Im

[3i

2K(−iKa)

(1

3K + iKbbb

)]= −Kabbb +

3

2KKaKbbb

= −K6

(−4−3

2K

)(Kab −

3

2KKaKb

)bb

= −K6

(−4Gabbb). (C.44)

The ab-term is

Mab = Fab +3i

2KImFacZ c ImFbdZ

d

= −Kabc(bc − ivc) +3i

2K(−iKa)(−iKb) (C.45)

and has imaginary part

ImMab = Kab −3

2KKaKb

= −K6

4

(− 3

2K

)(Kab −

3

2KKaKb

)= −K

64Gab. (C.46)

Gathering the terms (C.49), (C.42) and (C.46), this can be written in matrixnotation as

ImMab = −K6

(1 + 4Gabb

abb −4Gabbb

−4Gabbb 4Gab

). (C.47)

Now, we’ll compute

ReMab = Re

(Fab +

2i

Z a ImFabZ bImFacZ c ImFbdZ

d

). (C.48)

To do so, recall that we already found

M00 =− 1

3Kabc(ba − iva)(bb − ivb)(bc − ivc) +

2i43K

(ImF0bZb)2 (C.49)

– 89 –

C Calculations

where

i(ImF0bZb)2 =

(1

3K + iKaba

)(i1

3K −Kaba

). (C.50)

Thus,

ReM00 = −1

3Kabcbabbbc +Kaba +

3

2K

(−1

3KKaba −

1

3KabaK

)= −1

3Kabcbabbbc +Kaba −Kaba

= −1

3Kabcbabbbc. (C.51)

In order to compute the real part of

M0a = F0a +3i

2K(ImFabZ

b)(ImF0cZc), (C.52)

we note that

ReFa0 =1

2Re[Kabc(bb + ivb)(bc + ivc)

]=

1

2Kabc(bbbc − vbvc)

=1

2Kabcbbbc −

1

2Ka (C.53)

and recall

(ImFabZb) = −iKa. (C.54)

Thus,

ReMa0 =1

2Kabcbbbc −

1

2Ka +

3

2KRe

[i(−iKa)

(1

3K + iKbbb

)]=

1

2Kabcbbbc −

1

2Ka +

3

2KKa

1

3K

=1

2Kabcbbbc. (C.55)

– 90 –


The ab-term is

Mab = Fab +3i

2KImFacZ c ImFbdZ

d

= −Kabc(bc − ivc) +3i

2K(−iKa)(−iKb) (C.56)

and has real part

ReMab = −Kabcbc. (C.57)

Gathering all terms, we have

ReMab =

(−1

3Kcdebcbdbe 1

2Kacdbcbd

12Kacdbcbd −Kabcbc

). (C.58)

After this lengthy calculations, we finally arrive at the explicit expression

M =

(−1

3Kcdebcbdbe 1

2Kacdbcbd

12Kacdbcbd −Kabcbc

)+ i−K6

(1 + 4Gabb

abb −4Gabbb

−4Gabbb 4Gab

).

(C.59)

– 91 –

References

[1] Katrin Becker, Melanie Becker, and Andrew Strominger. “Five-branes, mem-branes and nonperturbative string theory”. In: Nucl. Phys. B456 (1995),pp. 130–152. doi: 10.1016/0550- 3213(95)00487- 1. arXiv: hep- th/9507158 [hep-th].

[2] Matthew D. Schwartz. Quantum Field Theory and the Standard Model. Cam-bridge Univ. Press, 2015.

[3] Cumrun Vafa. “The String landscape and the swampland”. In: (2005). arXiv:hep-th/0509212 [hep-th].

[4] Hirosi Ooguri and Cumrun Vafa. “On the Geometry of the String Landscapeand the Swampland”. In: Nucl. Phys. B766 (2007), pp. 21–33. doi: 10.1016/j.nuclphysb.2006.10.033. arXiv: hep-th/0605264 [hep-th].

[5] Nima Arkani-Hamed et al. “The String landscape, black holes and gravityas the weakest force”. In: JHEP 06 (2007), p. 060. doi: 10.1088/1126-6708/2007/06/060. arXiv: hep-th/0601001 [hep-th].

[6] Tom Rudelius. “Constraints on Axion Inflation from the Weak Gravity Con-jecture”. In: JCAP 1509.09 (2015), p. 020. doi: 10.1088/1475-7516/2015/09/020,10.1088/1475-7516/2015/9/020. arXiv: 1503.00795 [hep-th].

[7] Daniel Junghans. “Large-Field Inflation with Multiple Axions and the WeakGravity Conjecture”. In: JHEP 02 (2016), p. 128. doi: 10 . 1007 / JHEP02(2016)128. arXiv: 1504.03566 [hep-th].

[8] Jon Brown et al. “Fencing in the Swampland: Quantum Gravity Constraintson Large Field Inflation”. In: JHEP 10 (2015), p. 023. doi: 10.1007/JHEP10(2015)023. arXiv: 1503.04783 [hep-th].

[9] Clifford Cheung and Grant N. Remmen. “Naturalness and the Weak GravityConjecture”. In: Phys. Rev. Lett. 113 (2014), p. 051601. doi: 10 . 1103 /PhysRevLett.113.051601. arXiv: 1402.2287 [hep-ph].

[10] Ben Heidenreich, Matthew Reece, and Tom Rudelius. “Sharpening the WeakGravity Conjecture with Dimensional Reduction”. In: JHEP 02 (2016), p. 140.doi: 10.1007/JHEP02(2016)140. arXiv: 1509.06374 [hep-th].

[11] Eran Palti. “The Weak Gravity Conjecture and Scalar Fields”. In: JHEP08 (2017), p. 034. doi: 10.1007/JHEP08(2017)034. arXiv: 1705.04328[hep-th].

[12] Thomas W. Grimm, Eran Palti, and Irene Valenzuela. “Infinite Distances inField Space and Massless Towers of States”. In: (2018). arXiv: 1802.08264[hep-th].

– 93 –

https://doi.org/10.1016/0550-3213(95)00487-1

http://arxiv.org/abs/hep-th/9507158



https://doi.org/10.1016/j.nuclphysb.2006.10.033



https://doi.org/10.1088/1126-6708/2007/06/060

https://doi.org/10.1088/1126-6708/2007/06/060


https://doi.org/10.1088/1475-7516/2015/09/020, 10.1088/1475-7516/2015/9/020

https://doi.org/10.1088/1475-7516/2015/09/020, 10.1088/1475-7516/2015/9/020

http://arxiv.org/abs/1503.00795

https://doi.org/10.1007/JHEP02(2016)128






https://doi.org/10.1103/PhysRevLett.113.051601










References

[13] Dieter Lüst and Eran Palti. “Scalar Fields, Hierarchical UV/IR Mixing andThe Weak Gravity Conjecture”. In: JHEP 02 (2018), p. 040. doi: 10.1007/JHEP02(2018)040. arXiv: 1709.01790 [hep-th].

[14] Tom Banks, Matt Johnson, and Assaf Shomer. “A Note on Gauge TheoriesCoupled to Gravity”. In: JHEP 09 (2006), p. 049. doi: 10.1088/1126-6708/2006/09/049. arXiv: hep-th/0606277 [hep-th].

[15] Tom Banks and Nathan Seiberg. “Symmetries and Strings in Field Theoryand Gravity”. In: Phys. Rev. D83 (2011), p. 084019. doi: 10.1103/PhysRevD.83.084019. arXiv: 1011.5120 [hep-th].

[16] Leonard Susskind. “Trouble for remnants”. In: (1995). arXiv: hep-th/9501106[hep-th].

[17] Tom Banks and Lance J. Dixon. “Constraints on String Vacua with Space-Time Supersymmetry”. In: Nucl. Phys. B307 (1988), pp. 93–108. doi: 10.1016/0550-3213(88)90523-8.

[18] Gary Shiu, Pablo Soler, and William Cottrell. “Weak Gravity Conjectureand Extremal Black Holes”. In: (2016). arXiv: 1611.06270 [hep-th].

[19] K. Becker, M. Becker, and J. H. Schwarz. String theory and M-theory: A mod-ern introduction. Cambridge University Press, 2006. isbn: 9780511254864,9780521860697.

[20] Renata Kallosh et al. “Supersymmetry as a cosmic censor”. In: Phys. Rev.D46 (1992), pp. 5278–5302. doi: 10.1103/PhysRevD.46.5278. arXiv: hep-th/9205027 [hep-th].

[21] J. Polchinski. String theory. Vol. 1: An introduction to the bosonic string.Cambridge University Press, 2007. isbn: 9780521633031.

[22] J. Polchinski. String theory. Vol. 2: Superstring theory and beyond. Cam-bridge University Press, 2007. isbn: 9780521672283.

[23] Daniel Z. Freedman and Antoine Van Proeyen. Supergravity. Cambridge,UK: Cambridge Univ. Press, 2012. isbn: 9781139368063, 9780521194013.

[24] A. Zee. Group Theory in a Nutshell for Physicists. In a nutshell. PrincetonUniversity Press, 2016. isbn: 9780691162690.

[25] Jan Louis and Andrei Micu. “Type 2 theories compactified on Calabi-Yauthreefolds in the presence of background fluxes”. In: Nucl. Phys. B635 (2002),pp. 395–431. doi: 10.1016/S0550- 3213(02)00338- 3. arXiv: hep- th/0202168 [hep-th].

[26] Philip Candelas and Xenia de la Ossa. “Moduli Space of Calabi-Yau Man-ifolds”. In: Nucl. Phys. B355 (1991), pp. 455–481. doi: 10.1016/0550-3213(91)90122-E.

– 94 –




https://doi.org/10.1088/1126-6708/2006/09/049

https://doi.org/10.1088/1126-6708/2006/09/049


https://doi.org/10.1103/PhysRevD.83.084019





https://doi.org/10.1016/0550-3213(88)90523-8

https://doi.org/10.1016/0550-3213(88)90523-8





https://doi.org/10.1016/S0550-3213(02)00338-3



https://doi.org/10.1016/0550-3213(91)90122-E

https://doi.org/10.1016/0550-3213(91)90122-E

References

[27] Brian R. Greene. “String theory on Calabi-Yau manifolds”. In: Fields, stringsand duality. Proceedings, Summer School, Theoretical Advanced Study Insti-tute in Elementary Particle Physics, TASI’96, Boulder, USA, June 2-28,1996. 1996, pp. 543–726. arXiv: hep-th/9702155 [hep-th].

[28] Andrew Strominger. “Yukawa Couplings in Superstring Compactification”.In: Physical Review Letters 55 (23 1985), pp. 2547–2550. doi: 10.1103/PhysRevLett.55.2547. url: https://link.aps.org/doi/10.1103/PhysRevLett.55.2547.

[29] S. Hosono, A. Klemm, and S. Theisen. “Lectures on mirror symmetry”. In:Lect. Notes Phys. 436 (1994), pp. 235–280. doi: 10.1007/3-540-58453-6_13. arXiv: hep-th/9403096 [hep-th].

[30] Gianguido Dall’Agata. “Black holes in supergravity: flow equations and du-ality”. In: Springer Proc. Phys. 142 (2013), pp. 1–45. doi: 10.1007/978-3-642-31380-6_1. arXiv: 1106.2611 [hep-th].

[31] J. Baez and J. P. Muniain. Gauge fields, knots and gravity. 1995.

[32] Anna Ceresole, R. D’Auria, and S. Ferrara. “The Symplectic structure ofN=2 supergravity and its central extension”. In: Nucl. Phys. Proc. Suppl.46 (1996), pp. 67–74. doi: 10.1016/0920-5632(96)00008-4. arXiv: hep-th/9509160 [hep-th].

[33] Thomas W. Grimm. “The Effective action of type II Calabi-Yau orientifolds”.In: Fortsch. Phys. 53 (2005), pp. 1179–1271. doi: 10.1002/prop.200510253.arXiv: hep-th/0507153 [hep-th].

[34] Sebastien Gurrieri. “N=2 and N=4 supergravities as compactifications fromstring theories in 10 dimensions”. PhD thesis. Marseille, CPT, 2003. arXiv:hep-th/0408044 [hep-th].

[35] Thomas W. Grimm and Jan Louis. “The Effective action of N = 1 Calabi-Yau orientifolds”. In: Nucl. Phys. B699 (2004), pp. 387–426. doi: 10.1016/j.nuclphysb.2004.08.005. arXiv: hep-th/0403067 [hep-th].

[36] Mariana Grana. “Flux compactifications in string theory: A Comprehensivereview”. In: Phys. Rept. 423 (2006), pp. 91–158. doi: 10.1016/j.physrep.2005.10.008. arXiv: hep-th/0509003 [hep-th].

[37] Eran Palti. “Aspects of moduli stabilisation in string and M-theory”. PhDthesis. Sussex U., 2006. arXiv: hep-th/0608033 [hep-th].

[38] M. Nakahara. Geometry, topology and physics. 2003.

– 95 –




https://link.aps.org/doi/10.1103/PhysRevLett.55.2547

https://link.aps.org/doi/10.1103/PhysRevLett.55.2547

https://doi.org/10.1007/3-540-58453-6_13

https://doi.org/10.1007/3-540-58453-6_13


https://doi.org/10.1007/978-3-642-31380-6_1

https://doi.org/10.1007/978-3-642-31380-6_1


https://doi.org/10.1016/0920-5632(96)00008-4



https://doi.org/10.1002/prop.200510253






https://doi.org/10.1016/j.physrep.2005.10.008

https://doi.org/10.1016/j.physrep.2005.10.008



References

[39] Stefan Vandoren. “Lectures on Riemannian Geometry, Part II: ComplexManifolds”. In: 2009. url: https://www.semanticscholar.org/paper/Lectures-on-Riemannian-Geometry%2C-Part-II%3A-Complex-Vandoren/9eef42d482293387ffc37ca54d8979b43e651258.

[40] Vincent Bouchard. “Lectures on complex geometry, Calabi-Yau manifoldsand toric geometry”. In: (2007). arXiv: hep-th/0702063 [HEP-TH].

[41] Luis E. Ibanez and Angel M. Uranga. String theory and particle physics:An introduction to string phenomenology. Cambridge University Press, 2012.isbn: 9781139227421.

[42] Adel Bilal. “Introduction to supersymmetry”. In: (2001). arXiv: hep- th/0101055 [hep-th].

[43] Marijn Davidse. “Membrane and fivebrane instantons and quaternionic ge-ometry”. PhD thesis. Utrecht U., 2006. arXiv: hep-th/0603073 [hep-th].url: https://inspirehep.net/record/712090/files/arXiv:hep-th_0603073.pdf.

– 96 –

https://www.semanticscholar.org/paper/Lectures-on-Riemannian-Geometry%2C-Part-II%3A-Complex-Vandoren/9eef42d482293387ffc37ca54d8979b43e651258







https://inspirehep.net/record/712090/files/arXiv:hep-th_0603073.pdf

https://inspirehep.net/record/712090/files/arXiv:hep-th_0603073.pdf

Declaration of Authorship

Erklärung:

Ich versichere, dass ich diese Arbeit selbstständig verfasst und keine anderen alsdie angegebenen Quellen und Hilfsmittel benutzt habe.

Heidelberg, den 4. August, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Documents

University of Heidelberg - Max Planck Society...University of Heidelberg Department of Physics and Astronomy Master Thesis The Axion-Instanton Weak Gravity Conjecture and Scalar Fields