Katholieke Universiteit Leuven

Faculteit der Wetenschapp en

Instituut vo or Theoretische Fysica

Low energy physics from

typ e IIB theory

Frederik Denef

Promotor Prof Dr W Tro ost Pro efschrift ingediend voor

het b ehalen van de graad van

Do ctor in de Wetenschapp en

Deze thesis kwam tot stand met de nanciele steun van

het Fonds voor Wetenschapp elijk Onderzo ek

Een geest een en al logica is als een mes

dat een en al lemmet is

Het doet de hand die het gebruikt bloeden

Tagore

De wijze waarop het eten wordt opgediend is

minstens even belangrijk als de wijze waarop het

wordt toebereid

Ons Kookboek KVLV

Vo orwo ord

Een thesis schrijf je niet alleen Ik b en daarom iedereen die de vo orbije zes

entwintig jaar heeft bijgedragen tot het tot stand komen van dit werk bijzonder

dankbaar

In de eerste plaats denk ik daarbij natuurlijk aan mijn promotor Walter

Tro ost voor het op gang brengen en aanwakkeren van mijn interesse in string

theorie en theoretische fysica in het algemeen vo or zijn advies hulp en aanmo edi

ging en om me te laten meegenieten van zijn scherp e fysische inzichten humor en

relativeringsvermogen Ook Toine Van Pro eyen zou ik sp eciaal willen bedanken

omwille van wat hij mij op wetenschapp elijk en menselijk vlak heeft bijgebracht

voor de internationale contacten en vo or zijn no oit tanende gedrevenheid waaraan

het instituut wellicht een gro ot deel van haar huidige internationale uitstraling te

danken heeft Heel wat collegafysici ben ik bovendien dankbaar voor de soms

verduisterende maar meestal verhelderende uiteenzettingen en discussies die ik

van en met hen mo cht hebb en Joris Raeymaekers vo or de talloze metafysische

conversaties al vanaf de eerste kandidatuur Fred Ro ose vo or de al dan niet met

Ouzo overgoten brainstormsessies tot er geen krijt of b ord meer over was Ben

Craps vo or zijn consequentscherp e en verhelderende vragen en antwo orden Jan

Tro ost voor zijn diep e inzichten en intellectuele provo caties ook ver buiten de

fysica Marco Billo als b ondgeno ot in vele samenwerkingen Piet Claus en Jeanne

Dejaegher voor de uitwisseling van fysicop olitieke ideeen Daniela Zanon en Pietro

erkingen en Italiaanse gastvrijheid Alex Sevrin vo or zijn enthou Fre vo or samenw

siasme en aanmo ediging Martijn Derix vo or zijn ecientie en droge humor Mieke

De Co ck vo or een verdubb eling van de aangenaamheid van het leven op het zesde

Lies Van Moaert voor haar attentheid Magda Marganska om mee te help en

het instituut o ok s nachts b emand te houden Kor Van Ho of vo or zijn gevatheid

Chris Van Den Bro ecken PieterJan Desmet voor hun losoe Christine voor

de gezellige babb els en AndreVerb eure Christ Maes Mark Fannes Desire Bolle

en Raymond Gastmans vo or geanimeerde koepauzediscussies en wat ik van hen

mo cht leren niet alleen over olie en ko elvlo eistof Een fysicus die ik tenslotte

zeer sp eciaal zou willen b edanken is Henk Schets onder andere voor de talloze

met zin en onzin gevulde avonden die er jaren lang vo or gezorgd hebb en dat de

fysica mij niet in een so ciale coma dreef

Twee mensen ben ik verder onmetelijk dankbaar voor de impact die ze ge

had hebb en op mijn wetenschapp elijke ontplo oiing De eerste is mijn nonkel

wiskundige Jan Denef Iedereen die o oit met hem in contact is gekomen weet

ho e heerlijk aanstekelijk zijn passie voor de wiskunde werkt Onze regelmatige

discussienamiddagen die wegens onze genetische tijdsb esefhandicap vaak uit

liep en tot diep in de nacht en waarin ik de ene keer mo cht pro even van zijn

wonderlijkegave zijn diep inzicht in de wiskunde over te brengen aan de jargon

analfab eet en de andere keer een dankbaar oprecht enthousiast oor vond voor

mijn verhalen over strings waren een p ermanente bron van energie en motiva

tie De andere p erso on die ik in dat opzicht zou willen b edanken is Herman Van

Bauwel mijn leraar wiskunde en fysica in de middelbare scho ol Ik denk niet dat

iemand meer invlo ed heeft gehad op mijn manier van denken Zijn consequente

logica zijn inzicht in het feit dat de vo ornaamste taak v an een lesgever niet het

bijbrengen van kennis maar het bijbrengen van een denkkader is en zijn enthou

siaste b eleving van de natuurkunde heb ik gedurende heel mijn opleiding zelden

of no oit do or iemand evenaard gezien

Maar de wereld b estaat niet uit fysica alleen De meest intense herinneringen

van de vo orbije vier jaar heb ik to chovergehouden aan vrienden en familie Mijn

bro ertje Vincent en zusjes Natalie en Barbara zou ik heel hard willen b edanken

vo or de plezierige momenten samen en voor hun onvoorwaardelijkehulp en steun

op elk moment zoals je die alleen van bro ers en zussen kan krijgen Mijn nonkel

loso of Philipp e Van Parijs b en ik zeer erkentelijk vo or zijn interesse en wijze

wo orden op mo eilijkere momenten Als een tweede familie waren de vrienden

waarmee ik heb samengewo ond gedurende de laatste drie jaar van mijn doktoraat

waarvan ik sommigen al kende van vo or ik b esefte dat de aarde rond de zon draait

met wie ik zoveel heb meegemaakt en aan wie ik zo veel dank versch uldigd b en

dat een opsomming daarvan onmogelijk is Rigo Belpaire Joachim De Weerdt Jo

Seldeslachts Rudy Mecano Bjorn Van Camp enhout en Katleen Van Den Bro eck

Bjorn en zijn sprankelende zielsgeno otje Els Lecoutere zou ik hier b ovendien nog

eens heel heel sp eciaal willen b edanken voor de ontelbare prachtige dingen die we

samen b eleefd hebb en de vo orbije twee jaar voor hun b egrip steun en opvang

wanneer ik er weer in slaagde de trein of iets anders te missen vo or de grappige

situaties waarin we samen gerold zijn en hop elijk nog zullen rollen en voor nog

zoveel meer Voor een p erso on vind ik het bijna onmogelijk de go ede wo orden

te vinden om haar gepast te b edanken en dat is Katleen Zonder haar zou deze

thesis er no oit geweest zijn Zij heeft ervo or gezorgd dat ik niet van ondervoeding

en eenzaamheid omkwam gedurende de drie maanden afzondering die ik mijzelf

vo or het schrijven van dit werk had opgelegd Zij was het die me een jaar geleden

w eer de kracht gaf de fysica in te duiken Zij is het vo orbije jaar mijn reddingsb o ei

muze en zon geweest Aan haar in drie b ep erkte wo orden thanks for living

Tenslotte zou ik mijn ouders willen b edanken voor een kwart eeuw liefde

aanmo ediging tro ost wijze raad b ezorgdheid en steun voor mijn eerste fysica

lessen met b ehulp van twee app elsienen en een lamp vo or mijn eerste schrijessen

met b ehulp van Fisher Price letters en een ko elkast en vo or al het andere Aan

hen draag ik deze thesis op

Contents

Intro duction

Why not strings

Why

Why not

Beyond p erturbation theory

Strings and low energy physics

Outline and summary of results

Low energy eective eld theories

PI Wilson and Seib ergWitten

PI eective action

Wilsonian eective action

Seib ergWitten eective action

Relation with b etafunction

From lo cal to rigid sp ecial geometry

Lo cal sp ecial geometry

Rigid sp ecial geometry

The rigid limit of lo cal sp ecial geometry

Low energy eective action of N YangMills theory

Contents

CalabiYau compactications of I IB

Four dimensional low energy eective action

Typ e I IB string theory

CalabiYau manifolds and sp ecial geometry of the complex

structure mo duli space

Four dimensional massless sp ectrum

Four dimensional action

BPS states from wrapp ed

States in string theory

BPS branes

Mass and charge of wrapp ed branes

Spin

BPS states from the D eective action attractors

Static spherically symmetric congurations

BPS solutions the attractor ow equations

Existence of the BPS state

Multicenter case

Attractor technology

Validity of the D and eld theory pictures

Quantum YangMills from I IB strings

Near ALE brations and the light BPS sp ectrum

Vector branes

Hyp er branes

Reduction to Seib ergWitten p erio ds

The weak gravity limit

Unication scales and the heterotic picture

Dynamics of the dynamical dynamically generated scale

An explicit example

Contents

Fibration structure

Construction of cycles and p erio ds

The SU rigid limit

Attractors at weak gravity

Attractor ow equations in the weak gravity limit

Spherically symmetric congurations at weak gravity

Rigid limit

Solutions in the rigid limit

Multi monop ole dynamics

A Samenvatting

Chapter

Intro duction

Two thousand ve hundred years ago Plato had the brilliant insight that

space time and matter as we see it might very well be only a pale shadowofa

much more complex underlying reality inaccessible by direct observation In a

b eautiful pap er he describ ed the following gedanken exp eriment Imagine a

bunch of p eople sitting next to each other staring at the wall of a cave They

are heavily chained and only able to see this wall and have b een sitting there all

their conscious life Behind their back guards have made huge res such that

the shadows of the chained p eople as well as of other ob jects passing in front

of the re are visible on the wall of the cave The entire observable reality of

the chained is atwo dimensional world of shadows Direct observation provides

anumb er of rules for the shadow dynamics though they are intricate require a

priori sp ecication of a large numb er of undetermined quantities and break down

when pushed to o far It will b e exceedingly dicult for those p eople to arriveat

the insight that there is actually an underlying unifying three dimensional reality

and this will most certainly require brave new ideas and farreaching abstract

theoretical developments As discussed in it will moreo ver b e almost imp ossibly

hard to convince the chained of the reality and the intrinsic b eauty of this three

dimensional world even if they were dragged out of the darkness of the cave and

taken outside into the Sunshine

Twentieth century physics has shown Platos insight to b e astonishingly true

As years progressed the search for the fundamental laws of physics required a

picture of reality which got seemingly further and further removed from everyday

observations but at the same time got increasingly b eautiful and elegant At

present the top of this evolution is dominated by a remarkably rich and unifying

theoretical construct the theory of strings

Chapter Intro duction

From the p oint of view of us the chained the central question in any candidate

underlying theory is of course what it implies for our observable world In this

thesis we will address asp ects of this problem in the light of the quite drastic new

developments in string theory the past few years

Why not strings

Why

The fundamental laws of physics needed to understand and predict the results

of exp eriments in present day particle accelerators are to an impressively high

degree of accuracy given by the so called StandardModel of elementary particles

This describ es all known elementary particles together with the three fundamental

interactions b etween them which are relevant at the energies which can now be

reached in accelerators the electromagnetic weak and strong interaction

The framework of the Standard Mo del is quantum eld theory In a quantum

eld theory elementary particles are represented as p ointlikeobjectsinteracting

with each other by emitting and absorbing other particlesfor example the electro

magnetic force b etween electrons is due to exchange of photons Each absorption

or emission has a certain probability prop ortional to a certain constant g which

only dep ends on the typ e of interaction and the energy scale of the pro cess This

constant has to be determined exp erimentally and is called the eective cou

pling constant of the interaction For electromagnetic interactions at low energies

this is g Clearly the smaller the coupling constant the smaller the emis

sionabsorption probability and the weaker the interaction force If as for the

electromagnetic interaction at presently accessible energies g is suciently small

one can hop e to calculate with go o d accuracy the outcome of eg particle collision

Such calculations are called pertur exp eriments as a truncated p ower series in g

bative and are in most quantum eld theories the only p ossible way to extract

precise quantitative predictions There exists an attractive diagrammatic repre

sentation of such series due to Feynman where one has to sum over all p ossible

intermediate emissionabsorption pro cesses ordered according to the number of

interactions

For electromagnetic and weak interactions p erturbation theory is useful at all

accessible energies For the strong interaction this is not the case at low energies

but at high energies MeV it is All in all it turns out that the standard

Wehave in mind here the center of mass energy of a particle collision exp eriment say



At least for one value of the energy scale Given all the particles sp ecies which exist in

nature the theory then provides the coupling constant at all other energies

Why not strings

mo del givesavery go o d description of presently accessible particle physics with

strong predictivepower provided one plugs in a set of ab out twenty parameters

coupling constants and particle masses

Despite its exp erimental successes the Standard Mo del is very likely not

the end of the story The mo del shows some striking structures such as the

app earance of three fermion generations b egging for an explanation which the

theory itself cannot provide It also contains ab out twenty a priori undetermined

dimensionless adjustable external parameters of which many have values which

are eyebrowraisingly unnatural in the context of the Standard Mo del

The main reason to doubt ab out the Standard Mo del as the ultimate theory

however is that it fails to describ e the gravitational interaction adequately For

gravity the quantum eld theory framework fails miserably at least in p erturba

tion theory Roughly one can understand this as follows The gravitational force

between two particles is prop ortional to their mass and hence their energy In

other words the eective coupling constant increases with increasing energy scale

This implies that gravity is unimp ortantatlow energies but on the other hand

also that it rapidly grows strong at high energies The characteristic energy scale

determining the meaning of low and high here is the so called four dimensional

Planck mass M given in terms of the Newton constant G as

P N

M G GeV

P

N

we use units with c h As quantum mechanics allows arbitrarily large

energy uctuations provided these are lo calized in a suciently small region of

spacetime and the p oint particle picture of quantum eld theory on the other

hand indeed allows two particles to b e lo calized in an arbitrarily small region of

spacetime there is no limit on the eective gravitational interaction strength in

intermediate emissionabsorption pro cesses during particle collisions As a result

p erturbation theory breaks down unless a high energy or short distance cuto is

articially intro duced But this in turn intro duces a large degree of arbitrariness

in the theory greatly reducing its predictivepower and annihilating its credibility

as a truly fundamental theory

There are two p ossible resolutions The rst one is that the problem with

gravity is an artifact of p erturbation theory and that it disapp ears when the theory

is solved exactly Evidently the latter is not an easy task and this program has

b een unsuccessful thus far The second p ossibility is that quantum eld theory

is simply not the right description of physics all the waydown to distance scales

of order M where gravity b ecomes imp ortant Somehow interactions should

P

b e smeared out in a natural wayavoiding the short distance divergence of the

gravitational interaction and allowing a consistent p erturbative description of

gravity

Chapter Intro duction

There is at present only one known way to achieve this and that is string

theory The idea b ehind string theory is simply to replace the dierent particle

sp ecies p erturbatively interacting with each other via emission and absorption

of other particles by dierent vibrational mo des of a single kind of string in

teracting simply by splitting and joining of strings Instead of one dimensional

particle worldlines connected to each other in interaction vertices we now get

smo oth swept out by the strings in spacetime The theory is thus

extremely simple in its basic ingredients but nevertheless when one tries to set

up such a consistent theory of strings which is not manifestly incompatible with

observation one necessarily nds the following rich set of features

Gravity Every consistent string theory contains a state with the prop erties

of a the particle mediating the gravitational force whose interac

tions reduce at low energies to

Finite perturbation theory Due to the eective smearing out of interactions

string theory gives a p erturbation theory which is nite order by order In

particular it provides a nite unitary p erturbative description of quantum

gravity in sharp contrast with eld theory

Unication Apart from the graviton and gravity string theory also leads to

other particles and forces including those of the Standard Mo del There are

also particles and interactions not present in the Standard Mo del but those

do not necessarily contradict observations All forces are thus describ ed

on the same fo oting

Extra dimensions String theory requires a denite number of spacetime

dimensions For all theories of which consistency has b een established this

numb er is To b e consistent with observations six of those havetobeso

small that they cannot b e resolv ed by present day exp eriments This is com

patible with the consistency constraints of the theory The extra dimensions

give rise to additional particles and interactions in the four visible dimen

sions dep ending on the geometry of the compact internal space Some of

the allowed geometries pro duce the particle sp ectrum of the Standard Mo del

Recall Platos cave

All established consistent string theories are sup ersymmet

ric meaning that there is a fundamental symmetry between b osonic and



We use the word string theory here in a broad sense including candidate nonp erturbative

extensions such as the matrix mo del



They could b e discovered in the next generation of accelerator exp eriments however eg at

LHC the app earance of TeV scale eects of extra dimensions are not excluded by presentday

observations

Why not strings

fermionic degrees of freedom in the theory This symmetry is clearly broken

in the world as weknow it but there is quite some evidence in favor of its

presence at the level of the fundamental theory from theoretical as well as

exp erimental considerations There is hop e that the LHC collider at CERN

under construction at the time of writing will settle this issue

No freeparameters String theory has no adjustable external dimensionless

constants It only has one fundamental energy scale at least in p erturba

tion theory identied with the tension of the string and called the string

scale M Sometimes the string scale is identied with the Planck scale M

S P

but this is wrong in general see b elow Now though there are no external

adjustable constants there are in a certain sense internal adjustable con

stants string theory seems to allowa huge continuous family of dierent

consistentvacua with dierent exp ectation values of certain massless elds

which can b e considered as condensates of the corresp onding massless par

ticles Particular examples are the metric g and the eld The

vacuum exp ectation value of the exp onential of the latter g he i app ears

s

as a sort of coupling constant in string p erturbation theory a string world

sheet with don ut holes and n b oundary comp onents gives a contribution

g n

prop ortional to g String p erturbation theory is therefore accurate

s

when g is small Incidentally the ratio of the ten dimensional Planck scale

s

and the string scale turns out to b e M M g implying that the

s

P S

string mass scale is always smaller than the D Planck scale at weak string

coupling

Uniqueness Though there are several distinct p erturbative string theories

those are now b elieved to be just dierent p erturbative expansions ab out

dierentvacua of a single underlying theory of which the fundamental for

mulation is not yet known

Why not

Despite its theoretical successes also string theory is very likely not the end of

the story at least not in a formulation where strings are the truly fundamental

degrees of freedom There are some ob jections against string theory as describ ed

ab ove

The theory is dened in an intrinsically p erturbativeway scattering proba

bilites of particles are given by an asymptotic series in p owers of the string

coupling constant g The k th order term in this series corresp onds to a sum

s



The number g n is called the Euler characteristic of the surface

Chapter Intro duction

over dierent p ossible string splitting and joining pro cesses during the col

lision where each p ossibilitysweeps out a in spacetime of xed

Euler characteristic g n k As usual for a p erturbative series

this only makes sense for suciently small coupling constant g More

s

over p erturbation theory will clearly fail to give an adequate description

of the physics as so on as there are energies involved which are suciently

high to pro duce states which are not in the elementary p erturbative sp ec

trum of the theory The prototyp e examples of such states are the magnetic

monop ole in ordinary quantum eld theory and the in general

relativity On top of that any p erturbativepower series in a coupling con

stant g will miss contributions to the observ ables which do not haveaTaylor

expansion ab out g the most notable example b eing the so called in

S

n

cl

g

with n stanton contributions which are typically of the form e

S As the string coupling constant g can take arbitrary values de

cl s

p ending on the chosen vacuum there denitely exists a sector of the theory

where the p erturbative formulation breaks down completely In absence of

a nonp erturbative formulation this is evidently a serious problem

Unlike particles which can be given a nonp erturbative background in

dep endent and at the same time very convenient description by going to

quantum eld theory strings apparently do not have such a second quan

tized formulation at least not of comparable elegance and simplicity There

has b een substantial work on this sub ject but the yield has b een rather

disapp ointing and at present there is a widespread b elief that second quan

tized strings are not the fundamental degrees of freedom needed to go b eyond

p erturbative string theory

In its p erturbative formulation string theory has a plethora of energetically

degenerate vacua which strongly reduce its predictive power One hop es

that sup ersymmetry breaking and nonp erturbative eects will lift most or

even all of this degeneracy though this is far from sure

Despite recent successes in explaining the thermo dynamics of some near

sup ersymmetric black holes including a microscopic derivation of the Beckenstein

Hawking entropy a direct understanding of the quantum mechanics of arbi

trary black holes is still lacking

The extreme richness and equally extreme tightness of the theory itself hints

to the existence of underlying organizing principles which are not known yet

String theory fails as miserably as its predecessors in accounting for the

ridiculously low exp erimental value of the cosmological constant given the

fact that sup ersymmetry should b e suciently strongly broken to pro duce

Why not strings

particle masses of the order of those app earing in the Standard Mo del This

seems hard to avoid even b eyond p erturbation theory and maywell b e the

most outstanding puzzle of presentday theoretical high energy physics

Beyond p erturbation theory

So there are enough reasons to go on and try to get some insight in asp ects of

nonp erturbative string theory whatever the fundamental underlying theory will

turn out to b e There has really b een a lot of progress here during the past ve

years This is largely due to the discovery of string dualities and Dbranes making

it p ossible to prob e many asp ects of nonp erturbativephysics within the framework

of string p erturbation theory

Let us start by explaining the concept of a Dirichlet pbrane Dbrane in

short Instead of intro ducing its denition ad hocwe will try to argue for its exis

tence and usefulness in string theory from physical arguments Consider therefore

rst a black pbrane in an asymptotically at noncompact ten dimensional space

time with p This is an ob ject innitely extended in p dimensions

time and p spatial dimensions with the prop erty that the gravitational force is

so strong in its vicinity that anything coming to o close can never escap e again

More precisely there exists a certain p dimensional surface surrounding the

ob ject called the horizon beyond which return tickets are no longer available

A pulse signal sent out by someb o dy falling in will arriveto a distant observer

with ever decreasing frequency as escaping b ecomes increasingly dicult for the

signal eventually coming to a standstill when the p erson falling in reaches the

horizon so from the p oint of view of the external observer the p erson falling in

actually never passes through the horizon One says there is an innite grav

itational redshift at the horizon Since frequency is tantamountto energy this

implies that any ob ject with nite energy with resp ect to a freely falling observer

at the horizon will have zero energy with resp ect to a distant observer Suchblack

pbranes are well known solutions of classical gravity theoriesthey are the higher

dimensional generalizations of the p case the black hole

One can also consider multiple black pbrane congurations but these will in

general b e unstable Only when the gravitational attraction is exactly canceled

by another repulsive force b etween the branes such a system can remain in equi

librium In sup ersymmetric theories such congurations can easily b e realized by

taking a numb er of socalled BPS pbranes aligned parallel to each other but oth

erwise at arbitrary relative p ositions Mass and charge densities of a BP S pbrane

satisfyavery sp ecic relation saturating a lower b ound on the mass densit y for

the given charge density



The sux brane stems from the word membrane

Chapter Intro duction

Imagine nowtwo parallel BPS pbranes and a numb er of closed strings oating

around Some of the closed strings will b e captured by the rst pbrane some by

the second one and some will oat around in spacetime forever Also some will

b e captured by both pbranes at the same time that is one part of the string will

be stuck to the horizon of the rst brane while the other part will be stuck to

the horizon of the second brane As argued ab ove the parts of the string at the

horizons do not contribute to the total energy due to the innite redshift at least

if near horizon interactions meet some niteness conditions All energy resides

in the part of the string stretched b etween the two branes Such a closed string

will thus eectively b ehavevery much liketwo op en strings with their endp oints

conned to the brane horizons Moving one of those two comp onents o to innity

along one of the p noncompact dimensions of the brane we are left with one

eectively op en string stretched b etween the pbranes

In the following we always cho ose our energy units such that the string mass

scale M equals Such units are called string units Now supp ose we send

S

the string coupling constant g to zero while keeping the pbranes of xed charge

s

density and at xed distance from each other As the D Planck length M

P

g which determines the strength of gravity go es to zero in string units when

s

g spacetime at any xed nonzero distance from the brane b ecomes at at

s

least if the BPS mass density do es not grow to o fast when g If this is

s

the case then when g we are simply left with at spacetime containing two

S

rigid p dimensional ob jects b etween which op en strings can b e stretched Apart

from the stretched op en strings there can also b e closed strings oating around

in spacetime as well as op en strings with b oth endp oints on one brane Thus to

lowest order in string p erturbation theory such pbranes merely provide Dirichlet

b oundary conditions for string worldsheets propagating in trivial at spacetime

hence their name Dirichlet pbranes or simply Dbranes

There are other ways to arrive at Dbranes in string theory for example via

Tduality see b elow or the sup ersymmetry algebra The idea of a Dbrane b eing

established one can try to use the Dbrane prescription in string p erturbation

theory in other circumstances such as curved backgrounds Dbranes wrapp ed

around compact dimensions coincident Dbranes and so on Quantizing the strings

ending on Dbranes yields a sp ectrum of excitations which are interpreted as the

degrees of freedom of the brane just as closed strings in bulk spacetime yield the

degrees of freedom of the bulk including uctuations of its geometry As usual

in string theory quantization gives also consistency conditions on the allowed D



This is of course not p ossible for p or for branes with nite spatial extent Those require

a separate discussion but we will not go into this here



This dep ends on the details of the theory For example in typ e IIB string theory this

requirement excludes the NSbrane which indeed cannot b e represented by Dbranes in string

p erturbation theory

Why not strings

brane emb eddings whichhave the form of equations of motion for the Dbranes

As could b e exp ected intuitively isolated Dbranes have ripples in their excitation

sp ectrum propagating as waves over the brane with the obvious interpretation

of emb edding uctuations For N coincident Dbranes the massless degrees

of freedom are more exotic instead of simple co ordinate uctuations one nds

N N hermitean matrix uctuations suggesting the emergence of noncommutative

geometry

Thus it b ecomes p ossible to study various asp ects of the dynamics of branes

ob jects which are clearly not in the p erturbative string Fock sp ectrum within

the realms of string p erturbation theory Furthermore many asp ects of their low

energy dynamics can be extrap olated to strong string coupling and their mere

existence provides hints for string dualities see below and the nonp erturbative

structure of the theory Dbranes thus provetobeavery p owerful to ol in uncov

ering the mysteries of nonp erturbative string theory

The other cornerstone of the progress made in understanding nonp erturba

tive string theory is duality In short duality is the physical equivalence of two

seemingly dierent theories Massive interest in dualities was sparked with the

seminal work of Seib erg and Witten in where an exact expression was de

rived for the quantum low energy eective action of N sup ersymmetric SU

YangMills theory Of central imp ortance in their work was a duality between

N SU YangMills and an N theory of magnetic monop oles coupled to

a U ab elian The former description is weakly coupled when the

latter is strongly coupled and vice versa making p ossible accurate p erturbative

calculations in one picture when one is far into the nonp erturbative region of the

other one Convincing evidence accumulated that similar and even more p ower

ful dualities are presentin string theory Equivalences emerged b etween weakly

coupled string theories on dierent backgrounds this is called Tduality and in

cludes mirror symmetry and b etween weak and extrap olated strong coupling

regimes of the same or even dierent string theories this is called Sduality

strongweak coupling duality or string dep ending on context and

author Gradually a picture develop ed in which all known p erturbative string

theories can b e understo o d as convenient p erturbative expansions ab out dierent

p erturbative vacua or p erhaps b etter in dierent regimes of the same under

lying theory With some luck scattering amplitudes in a vacuum far outside the

validity region of one p erturbative string picture can simply b e calculated p ertur

batively in another picture This do es not mean that for anyvalue of the vacuum

parameters wehave a suitable p erturbative picture available as there are vacua of

intermediate coupling where none of the p erturbative string theories is adequate

but at least it limits how exotic the theory can get at strong coupling Perhaps

the situation here is morally sp eaking comparable to the description of water

Chapter Intro duction

in dierentphases Dep ending on whether we have the solid the liquid or the

gas phase totally dierent approximate descriptions are appropriate though the

fundamental laws are of course always the same The fact that we have here a

duality b etween ice water and steam do es however not mean that we can all of a

sudden calculate all wewant ab out eg water turbulence But for someb o dy who

would have learned the theory of ice the theory of water and the theory of steam

as three distinct items in a dicult green textb o ok the revelation that they are

merely dierent descriptions of the same fundamental thing would denitely b e

quite exciting

Tdualities relating two p erturbative descriptions in dierentbackground ge

ometries are prettywell under control and in many cases rigorously established

order by order in p erturbation theory Sdualities on the other hand are equally

dicult to tackle as it would be to demonstrate theoretically the presence of a

single set of fundamental laws describing b oth ice and steam in the absence of

a theory of atoms Consequently Sdualities are still largely conjectural though

indirect tests and their aesthetic attractiveness have placed their existence at

least in a rough form b eyond reasonable doubt

Another kind of duality has emerged ab out two years ago the Maldacena

corresp ondence This relates a string theory on a certain curved background to

a certain lower dimensional eld theory without gravitythus realizing in a certain

sense the of t Ho oft and Susskind This principle

states in its original form that the fundamental degrees of freedom of a certain

region of space should actually live on the boundary of this region with ab out one

binary degree of freedom p er unit Planck area

Given all these dualities b etween string theories the prize question is of course

what the underlying molecular theory is This as yet elusive theory has tenta

tively b een given the name Mtheory

Strings and low energy physics

One of the main results of renormalization theory is that physics at low energies

is apart from a few parameters indep endent of the details of physics at high

energies This is fortunate since it allows us to predict the outcome of present

day particle collision exp eriments without having to knowanything ab out Planck

scale physics but on the other hand it also implies that these exp eriments will

This simple analogy of course fails on many p oints we are not necessarily sp eaking ab out

dierent phases of string theory there can even b e regimes in which several adequate descrip

tions overlap and the richness and p ower of string dualities is of course substantially bigger

Outline and summary of results

teach us virtually nothing ab out string theory or whatever it is that governs the

Planck scale

This do es not mean string theory is useless at low energy scales There are of

course the twenty undetermined parameters of the Standard Mo del which string

theory might provide but we are still far from achieving this However string

theory has turned out to b e useful for low energy physics in a completely dierent

way namely as a p owerful geometrical to ol in analyzing nonp erturbative asp ects

of quantum eld theoriesrather surprisingly string theory leads to quantum eld

theory results which would be virtually imp ossible to obtain within the conven

tional framework of quantum eld theory itself There are two lines of attack

here One is to exploit the app earance of nonab elian YangMills theories in the

description of the dynamics of coincident noncompact branes and to make use

of various string theory results and dualities to derive quantum asp ects of those

Typical results thus obtained are exact low energy eective actions repro

ducing and extending the Seib ergWitten solution of low energy quantum N

SU YangMills the BPS sp ectrum of the theory and the phase structures of

gauge theories including connement phases The results are mostly restricted

to sup ersymmetric theories and probably related to this fact very elegant

and geometric in nature The restriction to low energies and to a certain extent

also the restriction to sup ersymmetric theories is avoided in approaches based on

the Maldacena corresp ondence The other line of attack

the one we will follow in this thesis is to make use of the representation

of massivecharged gauge particles as branes wrapp ed around nontrivial cycles

of the six compact dimensions of spacetime and of various nonrenormalization

theorems together with string dualities and geometrical constructions of the four

dimensional low energy eective action This approach is usually a bit more in

volved but has the advantage that a larger class of eld theories can b e studied

and that gravity can also b e included Here as well the results are very elegant

but mostly restricted to sup ersymmetric theories

Outline and summary of results

In this thesis we will study some asp ects of low energy physics extracted from

string theory We will fo cus on two sub jects the derivation of low energy eective

actions including gravity and low energy prop erties of charged BPS particle

states We will work almost exclusively in the context of type IIB string theory in

a spacetime with six compact dimensions forming a CalabiYau manifold Typ e

I IB theory is a p erturbativ e string theory which has a thirtytwo sup ersymmetry

generators when spacetime is at A CalabiYau manifold of real dimension

six is a space with the prop erty that the group of transformations on vectors

Chapter Intro duction

induced by parallel transp ort along closed lo ops is isomorphic to SU instead

of the usual SO Such a manifold has a metric satisfying the vacuum Einstein

equations The direct pro duct of four dimensional Minkowsky space with a Calabi

Yau manifold is an exact solution of the string theory equations of motion In such

a background typ e I IB string theory remains invariant under eight sup ersymmetry

generators yielding at low energies an N sup ersymmetric theory in the four

noncompact dimensions We will furthermore restrict most of the time to the

socalled vectormultiplet sector of the theory The reason why we make these

particular choices is that they provide a setting in which exact low energy results

can b e obtained which are at the same time very nontrivial

The outline of this thesis is as follows

In chapter the concept of an eective action is intro duced b oth in eld and in

string theory The dierence and relation b etween the one particle irreducible and

the Wilsonian eective action are outlined and the meaning of the Seib ergWitten

eective action is explained in this context Rigid and lo cal sp ecial geometry

are dened and their central role in four dimensional N YangMills resp

sup ergravity actions is discussed Following our work in we show in

general how rigid sp ecial geometry arises as a certain limit of lo cal sp ecial geometry

We conclude the chapter with a comp endium of Seib ergWitten theory and its

generalizations This chapter contains mainly known material but some eort is

done to tie up some lo ose ends in the usual Seib ergWitten review literature

In chapter we study in great detail CalabiYau compactications of typ e

IIB string theory The b osonic four dimensional massless sp ectrum and its low

energy eective action is derived The result is well known but we use a formal

ism whichismoreintrinsically geometric than usual This is esp ecially useful to

discuss BPS states obtained from wrapping Dbranes around nontrivial cycles

in the CalabiYau manifold Before we get to this we elab orate on how one can

pragmatically describ e a state in string theory and in particular how this re

lates to classical backgrounds Next we fo cus on BPS states in four dimensional

N theories and the sup ersymmetry multiplets in which they are organized It

is explained how they are realized as sp ecial Lagrangian emb eddings of branes in

the CalabiYau manifold Following our work in we derive the precise reduc

tion of the b osonic part of the wrapp ed Dbrane action to a BPS particle action

in four dimensions Here in addition we also discuss the curvature couplings As

an application we calculate the electromagnetic force b etween twomoving BPS

test particles with arbitrary charges Finally we discuss the problem of the de

termination of the typ e of sup ersymmetry multiplet to which a certain wrapp ed

Dbrane gives rise We present an easy shortcut for this which can b e applied in

some cases

Most of the remaining part of the chapter is devoted to the eective eld

Outline and summary of results

theory picture of the BPS states and in particular the attractor mechanism We

derive the spherically symmetric equations of motion in an invariant geometrical

framework slightly relaxing the usual ansatz made in the literature and in par

ticular wework out in detail the analogy with the dynamics of a nonrelativistic

particle moving in mo duli space We use this to giveanintuitive discussion of the

general solution including nonBPS black holes We p oint out that the attractor

mechanism do es not arise as a result of damp ed motion as is often claimed in

the literature but rather as a result of the nite energy condition together with

the unstability of the system as usual in soliton physics Next we turn to the

discussion of some BPS solutions with sp ecial emphasis on the nongeneric ones

and we obtain a novel solution which rst app eared in our work in and which

will play a prominent role in chapter We end this part on attractors with a dis

cussion of the multicenter case and the presentation of some p owerful techniques

for solving the attractor ow equations providing an intrinsic geometrical Kahler

gauge invariant formulation of the metho ds dev elop ed in

We conclude the chapter by presenting a discussion of the validity of Dbrane

and eld theory pictures in this sp ecic setting Wesketch the Maldacena corre

sp ondence is this context When the same reasoning is applied to nonblack hole

BPS states in four dimensions we seem to nd a new kind of corresp ondence

which is reminiscent of the well known mathematical Nahm duality b etween N

monop oles and a certain N N hermitean matrix system We leave this as an

intriguing op en issue

In chapter we study in detail how the low energy eective action of N

quantum YangMills theoryweakly coupled to gravity is obtained from typ e I IB

string theory compactied on a CalabiYau manifold This is based on the well

The main novel fea known geometrical engineering techniques

tures here are the explicit coupling to gravity and to the dynamical dynamically

generated scale and the fact that we deriveeverything directly and completely

within the typ e I IB theory without restricting to lo cal considerations andor in

voking S or T dualities Also the reduction of lo cal to rigid sp ecial geometry is

demonstrated explicitly Our approachisdowntoearth but fairly general

We start by presenting a detailed derivation of the Dbrane and CalabiYau

geometries needed to get the correct lightYangMills sp ectra in four dimensions

This involves some singularity theory Next we prove the reduction of lo cal to

rigid sp ecial geometry and we obtain the low energy eective action including

the weak coupling to gravity and the dynamical dynamically generated scale

This repro duces and extends the Seib ergWitten solution Wegoonby analyzing

the various unication scales which emerge and we make the connection with the

Sdual heterotic string picture To get some insight in the physical content of

our results we plug in some exp erimental data though the N mo dels we

Chapter Intro duction

are studying of course do not really match phenomenological observations We

end the chapter with a detailed case study of an explicit example based on our

work in in which cycles p erio ds mono dromies and sp ecial geometry data are

constructed explicitly with results supp orting our general discussion

Finallyinchapter we study BPS states in the eective eld theory picture

at weak gravity Most results here are new We reconsider the eective particle

picture of the spherically symmetric equations of motion in the weak gravity limit

We p oint out an interesting O G correction to the eective action which is from

N

the microscopic quantum eld theory p oint of view purely due to the backreaction

of quantum uctuations on the dynamical scale The weak gravity attractor ow

equations are used to give a eld theoretic description of Stromingers massless

black holes moving at the sp eed of light We further sp ecialize to the exactly

rigid limit where gravity is decoupled completely with as protot yp e example

N SU YangMills theory Only for monop oles and elementary dyons a

spherically symmetric BPS solution exists Some prop erties of these solutions are

investigated and we establish in particular the existence of stable equilibrium

p oints of various charged test particles at nite distance from the monop ole or

elementary dyon core Next weinvestigate in detail what go es wrong in trying

to construct spherically symmetric solutions for the other charges A picture

of those states which include the Wb osons as b ound states of the monop ole

and the elementary dyon emerges making contact with the socalled pronged

string representation of BPS states We conclude the chapter with the start of a

discussion on the low energy dynamics of N monop oles Wegive a general formula

for the mo duli space metric and evaluate this for N pro ducing the exp ected

result

A note

Though a substantial part of the results in this thesis require rather sophisticated

geometrical techniques wehave tried to make the discussion everywhere as phys

ical and intuitive as p ossible Sometimes this has b een at the price of a certain

loss of rigor However as the p ower and imp ortance of go o d intuitive pictures in

physics can hardly b e overestimated we hop e the b enets outweigh the costs in this case

Chapter

Low energy eective eld

theories

At suciently low energies any reasonable physical theory satisfying some basic

assumptions like relativistic invariance lo cality and unitarity can be given an

eective quantum eld theory description see eg In principle for any

theory with asymptotic particle states one can even construct a quantum eective

eld theory from which one can calculate any quantum scattering amplitude of

the original theory at the classical level of the eective theory that is in the limit

h or diagrammatically at tree level no lo ops In practice however such

quantum eective theories are usually extremely nonlo cal ugly and complicated

and extremely dicult to nd since they are equivalent to solving all amplitudes

of the full quantum theory However again at suciently low energies one can

approximate the full quantum eective theory by a lo cal eld theory describ ed

by an action with at most two derivative and four fermion terms two fermions is

equivalent to one derivative One can of course rene the approximation by adding

higher derivative terms Suchtwo derivative eective eld theories are usually still

very dicult to nd since they are equivalentto solving all amplitudes at very

low energy and one has to make still further approximations usually in the

form of a p erturbative series in p owers of the coupling constant However when

sucient sup ersymmetry is present it turns out to b e p ossible to nd the exact two

derivativelow energy quantum eective action in some nontrivial cases thanks to

The meaning of low is very context dep endent In string theory it usually means lower

than the energy of any massive state of the theory Generically this is the Planck scale but as

we will see there are imp ortant exceptions



again muchlower than any relevant mass scale in the theory

Chapter Low energy eective eld theories

Figure The general idea b ehind low energy eective actions is that the eects

of the virtual heavy particles can b e summarized in eectiveinteractions b etween

the light particles obtained byintegrating out the heavy ones

the extremely strong constraints sup ersymmetry puts on the form of those eective

actions For example for N theories in four dimensions sup ersymmetry

imp oses the structure of special Kahler geometry on the two derivative part of

the action Seminal nontrivial examples of exact solutions are typ e I I string

theory compactied on a CalabiYau manifold and N YangMills theory in

four dimensions a review of b oth can be found for example in we do not

intend to give complete references here

Obviouslythelow energy eective action of a theory provides a very p owerful

to ol to extract various asp ects of the low energy ph ysics Often esp ecially in string

theory the low energy eective theory also teaches us a lot ab out the structure

and symmetries of the full theory itself

Since eective eld theories and their geometrical structure play a ma jor role

in this work it is probably worth sp ending some time on the basics of this sub ject

In this chapter we will therefore review some basic facts ab out eective actions

and sp ecial geometry Though we will not go into much detail we will attempt

to clarify some but not all conceptual issues concerning eective actions which

are usually left obscure in the literature

PI Wilson and Seib ergWitten

The idea b ehind all low energy eective descriptions is to construct an action for

the light elds which gives their quantum dynamics for low energetic excitations

One distinguishes essentially b etween two dierent kinds of eective actions the generating functional of one particle irreducible connected diagrams in short the

PI Wilson and Seib ergWitten

PI eective action and the Wilsonian eective action

PI eective action

In eld theory the exact PI eective action can b e dened as the Legendre

transform of the logarithm of the partition function W J which in p erturbation

theory is the generating functional of connected diagrams This is explained in

any eld theory textb o ok eg so we will only briey rep eat the main fea

x

tures Denoting the elds collectively by with x a generalized index spacetime

p osition and other indices wethus dene

Z

x

iS h J

x

cl

W J ln D e

x

J W J

x

where J is given byinverting the relation

W

J h i

J

J

that is J is the external current needed to induce the exp ectation value for

the eld One can show that repro duces at tree level all scattering amplitudes

of the original theory that is when formally used as if it were the classical action

S repro duces all original scattering amplitudes in the limit h

cl

W J lim W J

h

Furthermore the physical eld exp ectation values at J are stationary p oints

of Thus indeed deserves to b e called the quantum eective action

Now in p erturbation theory any connected diagram in the expansion of W J

can b e regarded as a tree whose vertices consists of one particle irreducible sub di

agrams So in order for to b e correct the p erturbative expansion in p owers

of some small coupling constant of must be the sum of all PI connected

diagrams with arbitrary numb ers of external lines each external line corresp ond

ing to a factor rather than a propagator or wave function Hence the part

of which can be written as a p erturbative series is indeed the PI generating

functional Note however that our denition of extends beyond p erturbation



g

theory for example it can contain contributions e which are

invisible in a p ower series expansion in the coupling constant g

The two derivative low energy approximation of can simply be obtained

by putting all heavy elds to zero and expanding to second order in the eld

momenta or derivatives

Chapter Low energy eective eld theories

String theory do esnt have an analogous straightforward second quantized

oshell eld theoretic formulation It has a certain innite p erturbativeFock

particle sp ectrum however to which one can asso ciate elds as usual The massless

elds can b e consistently used as background for the p erturbative string worldsheet

path integral but only if they are on shel l that is if they ob ey certain equations

of motion obtained by requiring Weyl invariance of the worldsheet theory which

is needed for its consistency

One denes the PI eective action for these elds directly in p erturbation

theory simply as the or rather a spacetime action which at tree level repro duces

the string scattering amplitudes In particular the quantum equations of motion

ie the minima of the eective action are obtained by requiring the tadp ole

amplitudes amplitudes with one external particle to vanish Of course to get an

eective action whichisthe integral over a lo cal Lagrangian density one has to

make a momentum expansion of the amplitudes whichby dimensional analysis is

equivalenttoan expansion To get something sensible one thus has to restrict

p

energies to b e muchlower than the string scale making the eective eld

theory description of string theory only useful for elds much lighter than the string

scale these are all massless in a maximally symmetric vacuum of the theory

Now in order for the ab oveto make sense the equations obtained from re

quiring worldsheet Weyl invariance up to order k in the string coupling constant

needed for setting up a consistent nite p erturbation theory up to order k should

imply the equations of motion derived from the k th order corrected eective ac

tion Perhaps this statement needs rst some clarication One might think that

since Weyl invariance is a purely lo cal prop erty and the Weyl is a pure

worldsheet UV eect the requirementofWeyl invariance should in particular b e

indep endentofworldsheet top ology and hence receive no correction b eyond string

tree level This is not so If one sums over top ologies and integrates over the

Riemann surface mo duli space as one should in string p erturbation theory one

has to include worldsheets with arbitrarily small handles attac hed These UV

handles contribute to the total Weyl anomaly thus correcting the consistency

equations of motion for the background including particle mass corrections Ig

noring this contribution and simply taking the background needed for tree level

conformal invariance results in divergent lo op tadp oles and related amplitudes

so one really has to take the corrections to the background into account to get

all amplitudes nite This is known as the FischlerSusskind mechanism see

for a review Fortunately most sup ersymmetric backgrounds do not suer from

these corrections



Considering the fact that one can only directly dene scattering amplitudes with all particles

on shell in string theory and on the other hand the fact that an action provides an arbitrary

oshell extension of these amplitudes such an action cannot b e unique



b eing the inverse of the string tension

PI Wilson and Seib ergWitten

Let us rst see howtreelevel conformal invariance indeed implies the tree level

tadp oles to vanish Here tree level means the sphere contribution for closed and

the disk contribution for op en strings The argumentistaken from p

Consider for instance a genus massless closed string tadp ole amplitude whichcan

b e represented as the exp ectation value hV i of a massless closed string vertex

op erator V inserted at z in a conformal eld theory on the complex z plane

Conformal invariance means in particular that this amplitude should b e invariant

under the rescaling z z On the other hand under such a transformation a

massless closed string vertex op erator transforms as V z V z Therefore

wehave

hV i hV i

so hV i aswewanted to show

For disk diagrams the same trick can b e used to show that tree lev el conformal

invariance implies vanishing tadp oles inserted at the boundary of the worldsheet

One can further argue that the higher order corrections to the background

needed for Weyl invariance of the full amplitudes at order k are also precisely the

corrections needed to have still vanishing tadp ole amplitudes at this order So

happily the twoways of obtaining equations of motion from string p erturbation

theory give exactly the same results

Note that the disk diagram is of order g while the sphere diagram is of

S

order g where g is the string coupling constant Therefore at tree level in

S

S

p erturbation theory the elds from the op en strings will feel the elds from the

closed strings but conversely the closed string elds will not feel the op en string

elds the eect of their presence app ears only as a correction g For theories

S

with Dbranes represented as spacetime defects on which strings can end this im

plies that to lo west order in p erturbation theory one should not takeinto account

the backreaction of the Dbranes on the bulk eldsthe bulk bac kground should

b e a solution of the sourceless eld equations This is very fortunate since it al

lows one to keep on doing at space p erturbation theory even when Dbranes are

present as long as the string coupling constant is small considering backreaction

as small p erturbations

One cautionary note to end this part from the ab ove one might get the

impression that string theory is exactly equivalent to a certain complicated non

lo cal eective eld theory at all energies In particular its fundamendal degrees

of freedom might seem to b e describable as an innite tower of elds of increasing

mass with interactions that are eectively cut o in the ultraviolet This is not

the case Of course the exact eective eld theory will b e horrendously nonlo cal

and can hardly b e thought of as a eld theory in the conventional sense But much deep er is the fact that the construction of this eld theory dep ends completely

Chapter Low energy eective eld theories

on string p erturbation theory which is only dened as an asymptotic series and

which breaks down as so on as circumstances b ecome to o extreme In particular

p erturbation theory will denitely have broken down when the circumstances are

such that states like particles can b e pro duced which are not in the p erturba

tive Fock sp ectrum eg in gravitongraviton scattering at energies which are

suciently high to pro duce a black hole p or certain wrapp ed Dbrane

states The latter p ossibility can even o ccur at arbitrarily low energies as a matter

of fact even at zero energy So we cannot drawany conclusion at this p ointabout

the true high energy fundamental degrees of freedom of the full nonp erturbative

string theory assuming such a thing exists and certainly not that it would b e

just an innite tower of elds

Actuallywe should b e rather happy ab out this equivalence with any eld the

oryeven an eectively UV cuto one satisfying some minimal lo cality principles

would probably be incompatible with the holographic principle implying

among other disastrous things that string theory would not b e able to explain the

BeckensteinHawking black hole entropy formula

This brings us to the question then what is string theory b eyond p erturbation

theory It is probably not a theory of second quantized strings It could be

Matrix theory It could b e a more general holographic theory It could b e

something completely dierent Nobody knows

Wilsonian eective action

The idea of the Wilsonian eective action S is to split the degrees of freedom of

W

our theory in heavy and light mo des and to integrate out somehow the heavy

mo des so as to obtain an eective theory for the light mo des alone Thus roughly

we dene in eld theory

Z

iS iS

W

lig ht

e D e

heav y

The scattering amplitudes for the light mo des are then obtained by quantizing

the theory given by the classical action S The two derivative low energy

W

Wilsonian eective action is obtained by making a two derivative approximation

of S As an example the quantum eld theory describing the Standard Mo del

W

can b e considered as a low energy Wilsonian eective theory derived from whatever



This is the case for example for string theory compactied on a CalabiYau manifold with a

singularity Dbranes which are nonp erturbative states wrapping the vanishing cycle

have zero mass so string p erturbation theory breaks down at zero energywhich explains why

the low energy theories of such compactications are in fact singular See chapters and for a more elab orate discussion

PI Wilson and Seib ergWitten

will turn out the fundamental theory of nature valid well b elow the energy scale

where the details of this fundamental theory b ecome imp ortant which is at most

the Planck scale since there ordinary eld theory obviously breaks down

Clearly there is some vagueness in this denition For example the split in

heavy and light degrees of freedom can be done by intro ducing a scale and

integrating out all Fourier mo des of the elds ab ove or by splitting the elds

in light and heavy elds and integrating out completely the heavy ones or in

any other convenientway dep ending on the case at hand However as we will

see for the tw o derivative approximation in favorable circumstances sucient

sup ersymmetry these ambiguities do not matter

One can also dene a Wilsonian eective action in string theory namely

as the eective action obtained as ab ove but restricting the string path integral

to the massive string states This is somewhat articial of course and indeed this

Wilsonian eective action do es not resp ect some of the symmetries of string theory

On the other hand the very low energy part two derivative for example has

sometimes useful holomorphicity prop erties which the more physical PI eective

action do es not necessarily have p

Seib ergWitten eective action

To make all this a bit clearer and to discuss the relation b etween S and in

W

eld theory let us consider the example of N sup ersymmetric SU Yang

Mills theory in four dimensions This theory was solved in the two derivativelow

energy approximation by Seib erg and Witten in We will wait to give their

solution till wehave discussed some necessary technicalities of sp ecial geometry

but we will already use some prop erties for the purp ose of illustration here For

simplicitywe will only consider the b osonic degrees of freedom but it is of course

thanks to the balancing fermionic degrees of freedom that this theory is so well

b ehaved and under control at the quantum level

The b osonic elds of N SU pure YangMills are a complex scalar

a a

triplet a andavector triplet A b oth transforming in the adjoint

a

of SU the fermionic elds are two SU adjointWeyl fermion triplets and

a

butwe will not consider those further here The b osonic part of the classical



In a sense of course the PI eective action for string theory is also Wilsonian since it

leaves out even in principle the usually massive nonp erturbative string states and breaks down

together with p erturbation theory at the energy scale of these states Also in practice even

the p erturbative massive elds are not included and the action is written down in a low energy

expansion Perhaps we should call the action dened here string p erturbative Wilsonian action

Chapter Low energy eective eld theories

action is

Z Z

a a a a a a

S D D F F F F

g g

Z

c abc b

d x

g

where

a a abc b c

D d A

a a abc b c

F dA A A

This theory has at directions that is a continuous family of classical minimal

energy congurations namely

a

const

c abc b

One can eliminate such at directions from the path integral by xing at spatial

a

innity Take a Note that by the Higgs eect for nonzero a the elds

A A and their sup erpartners are massive with mass jaj We will

dene our Wilsonian eective action here as the action obtained byintegrating

out completely those massive elds and integrating out the momentum Fourier

mo des of the massless elds ab ove a scale The range of validity of our eective

theory is then for energies lower than and the Wb oson mass scale jaj Denote

from nowonwith the low momentum mo des of and with A those of A

Wethus have

Z

a a

iS A iS A

W

e D D D e

p

It can b e shown that the sup ersymmetry prevents the generation of an eective

p otential for the massless elds That is the at directions remain at and the

derivative momentum expansion of the b osonic part of S gives

W

Z

S d d

W

g

Z

F F F F

g

where the dots indicate higher derivative terms and g are the eld

dep endent eective coupling constant resp thetaangle which also dep end on

PI Wilson and Seib ergWitten

the dynamically generated scale of the theory replacing the classical

ig as a free parameter of the theory

An exact expression for the two derivative part of S was derived by Seib erg

W

and Witten in from sup ersymmetry and a numberofphysical consistency argu

ments see section Their arguments are valid for any strictly p ositivevalue of

the massless momentum cuto scale and since those arguments determine the

two derivativepartof S uniquely we conclude that the two derivative part of

W

S is actually indep endentof Since lowering means including more quantum

W

corrections from the massless eld lo ops to S this is equivalent to the statement

W

that the two derivative part of S do es not receivequantum correction from the

W

massless elds at nonzero momentum Presumably this nonrenormalization the

orem can b e argued directly from sup ersymmetry Note for example that there are

no lo op diagrams of the microscopic theory involving the massless elds only

Since it is known that thanks to sup ersymmetry there are no p erturbative correc

tions b eyond one lo op this implies indeed that the only perturbative corrections to

the eective action of the massless elds come from single massive particle lo ops

It is not clear to us how to extend this argument to nonp erturbative corrections

so weleave this p oint op en See however for a p ossible starting p ointtothe

relevant literature

This nonrenormalisation theorem also implies that we can calculate the exact

quantum scattering amplitudes of the massless particles in the very lowenergy

limit simply at tree level from the two derivative part of the Wilsonian eective

action S Or in other words the PI quantum eective action obtained

W W

by taking S as classical action with and J only consisting of mo des with

W

momentum below and only path integrating over the remaining mo des with

momentum b elow has the same two derivative part as the tree level eective

action S This eliminates already one of the ambiguities of the interpretation

W

of the Seib ergWitten eective action

One could also wonder whether the PI eective action obtained by using

W

S as classical action is the same as the full PI eective action with the massive

W

a

elds put to zero This is indeed the case denoting all elds collectively as



is a complex number so it is not really a scale Its rather its mo dulus jj which

deserves this name Wewillhowever follow common parlance and call the scale anyway The

complex character of is needed b ecause it takes over the role of the complex free parameter



ig of the classical theory



Actuallyeven if the action is written in the form indep endent of a scalar vacuum exp ectation

value their arguments only x a family of actions diering from eachotherby the dynamically

generated scale So in the following when we talk ab out the Seib ergWitten eective action

we actually mean this complete family

There is a lo ophole here our argument do es not exclude that there are extra contributions

to from lo ops at zero momentum These and other infrared subtleties do indeed o ccur in

W

generic N sup ersymmetric theories but apparently not here

Chapter Low energy eective eld theories

with a the adjoint SU index wehave

Z Z

R

a

iS J

a a

a

cl

J ln D e

a

where J is the external current needed to induce the vacuum exp ectation

a

a a

value for Now if J J and a we always have

h i h i no matter what J is This follows physically from the fact that

excitations of the elds alone do not act as sources for the and elds

R

a

and mathematically from the invariance of S J the PI measure and

cl a

the b oundary condition a and hence of under

when J J so that indeed h i h i and h i h i So

the external current J needed to induce the vacuum exp ectation value zero for

and has simply J J that is J Therefore with

J only consisting of mo des with momentum lower than and

Z Z

R



J iS



cl

J ln D D D D e

p p

Z Z

R

 

iS J

W 

J ln D e

p

W

Thus we conclude that the two derivativelow energy parts of

and S are in fact identical We can simply refer to this ob ject as the two

W W

derivative low energy eective action without having to worry ab out p otential

ambiguities in the meaning of the word From the arguments it follows that we

will always b e able to do this as long as

the massless elds do not renormalize the two derivative terms in the low

energy eective action and

pure massless eld excitations do not act as sources for the massive elds

Relation with b etafunction

Let us further consider the example of SU N YangMills An obvious

question is what the relation is of the eective coupling constant g whichis

determined by the Seib ergWitten solution still to b e discussed in with

the usual running coupling constantg M in eld theory and in particular if we

can see the b eta function app earing Let us see what we can say ab out this

PI Wilson and Seib ergWitten

If we dene the running coupling g M ofthe SU theory eg as in p

wehavethat g M is prop ortional to the single photon exchange diagram

of the eective action between two charged heavy particles at momentum

transfer q M Therefore since at vanishing momentum transfer the photon

propagator is obtained from the two derivative low energy part of the eective

action as given in wehave

gM g a

On the other hand at weak coupling wehave for M jaj b ecause of asymptotic

freedom

d g g g

M g N N

c f

dM

hence

M

ln

g jj

where the integration constant is the dynamically generated scale intro duced

earlier Together this gives a picture for the running coupling gM which

starts atg g a for lowvalues of the energy scale M and then matches onto

the asymptotic running for large values of M

If on the other hand we dene the couplingg M in the spirit of p

we should take our energy scale M jhij jaj and simply put

gM g M

w energy region However Clearly both denitions are quite dierent in the lo

as argued in p any reasonable denition of the coupling gives the

same function up to fourth order in the coupling constant Therefore b ecause

of asymptotic freedom we exp ect the same asymptotic running for M for

b oth denitions In particular this means we should nd for large

lnjj

g

Happily this indeed turns out to b e the case Despite some claims made in

the literature much more is not known ab out the relation between exact b eta

function and the Seib ergWitten eective action

without further sp ecication at that time here we see the emergence of its meaning at least

of its mo dulus The phase of is dened by removing the mo dulus signs and replacing the left

hand side with the complexied coupling constant

Chapter Low energy eective eld theories

From lo cal to rigid sp ecial geometry

The two derivative part of the action of any four dimensional N sup ersym

metric theory describing massless ab elian vector multiplets is governed by sp ecial

Kahler geometry local sp ecial Kahler if the sup ersymmetry is realized lo cally that

is if it is a sup ergravity theory rigid sp ecial Kahler if it is realized only globally

that is if it is a quantum eld theory in at space without gravity

We giveaquick review here referring to for more details

Lo cal sp ecial geometry

A lo cal sp ecial Kahler manifold M is an n complex dimensional Kahler manifold

with Kahler p otential of the form

t

V K lniV Q

where V is a certain holomorphic lo cal section of a rank n symplectic vec

tor bundle over M and Q an invertible antisymmetric and constant matrix the

a

symplectic form The Kahler metric is given b y g K where the z



a

b

ab

z

z

a n are complex co ordinates on M Transition functions b etween dier

ent lo cal sections should b e such that the Kahler metric is globally well dened

The symplectic section furthermore has to satisfy the following integrability con

dition

t

D V Q D V

a b

where D K

a a a

Assuming a symplectic basis has b een chosen suchthat Q is of standard form

Q

I

and such that V can be split as V X F I n with the F

I I

I

lo cally expressible as functions of the X a particularly ecient device enco ding

all geometric quantities in sp ecial geometry is the prepotential F This ob ject is

dened lo cally as a degree homogeneous function of the X given by F X

I

F

I

X F We then have thanks to the integrability condition F

I I

I

X

a a

and in sp ecial co ordinates t X X and with F t X F

i

K a a

e jX j FF t t F F

a a

We will not address here the subtleties arising for the case n

From lo cal to rigid sp ecial geometry

The curvature of M satises in these co ordinates the constraint

K ef

R g g g g e C g C

ace

abcd ab cd ad cb f bd

with C F This is yet another intrinsic geometrical ob ject named

abc a b c

quite dierently dep ending on the physical or mathematical context Yukawa cou

plings in N heterotic compactications magnetic moments in typ e I I N

sup ergravity op erator pro duct co ecients or p oint functions in the context of

conformal or top ological eld theory on the worldsheet and triple intersection num

b ers from the p oint of view of CalabiYau geometry We will simply call it the

coupling

In geometrya is usually a space parametrizing a family of dif

ferent p ossible geometrical structures like complex structures Kahler structures

quaternionic structures and so on In eld theory a mo duli space is a space

parametrizing dierent p ossible scalar vacuum exp ectation values and thus a

family of degenerate vacua If these scalars are dynamical and describ ed by

an eective action the physical metric on mo duli space is given by the scalar

kinetic term at zero energy

Four dimensional N sup ergravity coupled to n vector multiplets has a

V

n complex dimensional scalar mo duli space M on which sup ersymmetry im

V V

p oses lo cal sp ecial geometry The form of the two derivative action is com

pletely determined by sp ecial geometry actually byanintegral over sup erspace

of the prep otential The b osonic part has the following form

Z Z

p

a b I

d x S GR g dz dz F G

I

ab

Here is the gravitational constant G is the spacetime metric with determi

a

nant G and Ricci scalar R the z a n are the mass dimensionless

V

I

scalar mo duli elds F I n is the form eld strength of an

V

I

ab elian form p otential A is a constant dep ending on the normalisation of the

I

A of order one with the usual conventions and

J J

G ReN F Im N F

I IJ IJ

where N is a mo duli dep endent symmetric matrix

IJ

K L

Im F X Im F X

IK JL

N F i

IJ IJ

M N

X Im F X

MN

with F F and F the prep otential In order for the vector kinetic energy to

IJ I J

b e p ositive Im N should b e negative denite The couplings also app ear but IJ

Chapter Low energy eective eld theories

only in the fermionic part of the action The fully general N twoderivative

action including fermions and hyp ermultiplets can b e found in

For typ e IIB string theory compactied on a CalabiYau manifold which

will b e analyzed in detail in the next chapter the vector multiplet scalar mo duli

space coincides with the complex structure mo duli space of the internal CalabiYau

manifold This space has lo cal sp ecial Kahler geometry with

Z

V

C



Q C C

where is the holomorphic form on the CY fC g is a basis of cycles and

the dot denotes the intersection pro duct

Rigid sp ecial geometry

A rigid sp ecial Kahler manifold M is an r complex dimensional Kahler manifold

with Kahler p otential of the form

t

K iv q v

Here v is a holomorphic section of a rank r symplectic vector bundle with sym

i

plectic form q and the Kahler metric is given b K where the u y g



i

j ij

u

u

i r are co ordinates on M Transition functions between dierent lo cal

sections should b e such that the Kahler metric is globally well dened The inte

grability condition nowis

t

v q v

i j

Assuming a symplectic basis has b een chosen suchthatq is of standard form

q

A

and such that the v can b e split as v A r with the

DA DA

A

lo cally expressible as functions of the we can again dene a prep otential F

For rigid sp ecial geometry this is no longer a homogeneous function Using the

A

integrability condition it can b e dened as a function of the by

F

DA

A



The factor is intro duced to get the standard rigid sp ecial geometry formulas in the con

ventions of eg

From lo cal to rigid sp ecial geometry

Again all geometrical ob jects can b e derived from the prep otential for example

F

A

K Im

A

A

and in the co ordinates

g Im

AB

AB

where is the mo duli dep endent symmetric matrix given by

AB

F

AB A B

r

Rigid N sup ersymmetric U YangMills eld theories havean r com

plex dimensional scalar mo duli space M on which sup ersymmetry imp oses a rigid

v

sp ecial geometry As with sup ergravity the form of the two derivative action is

completely determined by rigid sp ecial geometry again byanintegral over sup er

space of the prep otential The b osonic part has the following form

Z

A A B

g d d F G S

A

AB

A A

Here the A r are the mass dimension scalar elds F is the

A

form eld strength of an ab elian form p otential A and

B B

G Re F Im F

A AB AB

with as in Note that in order for the kinetic energy to b e p ositive

AB

Im should b e p ositive denite Our conventions are those of

AB

The reason whywe use mass dimensionful scalar elds here rather than di

mensionless ones as we did for sup ergravity is that we do not wanttointro duce

a dimensionful parameter in the classical YangMills theory with constant scalar

metric In the sup ergravity theorywe already had such a parameter the gravita

tional constant Of course there we could equally well havechosen to w ork with

dimensionful scalar elds However the presence of dimensionful scalars implies

that we will have to intro duce a dimensionful parameter as so on as the scalar

metric is no longer constant This parameter indeed app ears in quantum eective

actions it is the dynamically generated scale So in the quantum case since we

have this scale parameter anywaywe could equally well work with dimensionless

scalars Note that with these conventions the symplectic vector has dimension

mass and the Kahler p otential dimension mass squared while in sup ergravity

these quantities are dimensionless

Chapter Low energy eective eld theories

Rigid sp ecial geometry is realized on the complex structure mo duli space of

certain classes of Riemann surfaces as follows

Z

v

SW

c

q c c

where is a scale parameter the dynamically generated scale is a certain

SW

meromorphic form on the Riemann surface and fc g is a certain set of r inde

p endent cycles Not any class of Riemann surfaces endowed with a meromorphic

oneform and a set of cycles satises the axioms of rigid sp ecial geometry It is

an op en question what the criterion is for a subspace of mo duli space to have rigid

sp ecial geometry

Such geometric mo duli spaces app ear b eautifully in the exact solution of the

quantum low energy eective action of N eld theories as we will see in the

section

The rigid limit of lo cal sp ecial geometry

Quite generally a rigid limit of lo cal sp ecial geometry can b e obtained as follows

Supp ose there is a region in mo duli space where we can cho ose a subset of co ordi

nates u u and symplectic vector comp onents V V v suchthat

r r

the Kahler p otential can b e written as

t

K ln L iv q v R

where L is real and indep endent of the u q is a real invertible and antisymmetric

i

matrix and R is a remainder such that

t

v q v R

t

L v q v

when approaching a certain lo cus in this region Then close to this lo cus wecan

make the following expansions

v v

t

q v iv K ln L

L

D v v

u u

where the dots indicate subleading terms that can b e neglected in the limit under

consideration Note that is up to a uindep endent term the expression

Low energy eective action of N YangMills theory

for the Kahler potential in rigid sp ecial geometry Moreover the integrability

condition reduces to

v q v

u

which is precisely the integrability condition dening rigid sp ecial geometry Thus

we nd that the geometry of the mo duli subspace parametrized by the mo duli u

i

and endowed with the symplectic vector v is essentially rigid sp ecial Kahler

The limit describ ed ab ove will be called a rigid limit and can be thought

of as sending the Planck mass to innity eectively decoupling gravity from the

degrees of freedom asso ciated with the rigid mo duli u We will come back to this

i

in great detail in chapter

Low energy eective action of N  Yang

Mills theory

For generic exp ectation values of the scalars of a nonab elian N superYang

Mills theory the gauge group is broken at low energies by the Higgs eect to its

r

Cartan subgroup U where r is the rank of the original gauge group Therefore

from the ab ove discussion it follows that the exact solution of the two derivative

quantum low energy eective action of such theories simply amounts to nding

the correct underlying rigid sp ecial Kahler manifold

It turns out that for a very large class of such theories this can b e done in

terms of a certain mo duli space of Riemann surfaces This was originally argued

for the pure SU theory by Seib erg and Witten in and later extended to

include matter hyp ermultiplets and higher rank gauge groups by

using physical arguments based on sp ectrum considerations mono dromies and

will deduce these asymptotic behavior from the one lo op b eta function As we

low energy eective actions directly from string theory in chapter we will not

review those arguments here some go o d reviews are eg Instead we

will just give a comp endium of the relevant mo duli spaces and some asp ects of the

corresp onding physics without justication for a numb er of example theories

Pure SU

This is the case originally considered by Seib erg and Witten The rank of

SU is The rigid sp ecial Kahler manifold underlying the eective action



For the nonexp ert reader the details of the following summary are not really necessary

to understand the main line of this thesis We will make contact with Seib ergWitten theory

in chapter and use some of the formulas for the SU mo duli space metric given b elowin

chapter

Chapter Low energy eective eld theories

is the mo duli space of genus curves given by the equation

z x u

z

where z x are the ambient space co ordinates and u is the mo dulus deter

mining the vacuum

h i u

VAC

Here is the at weak coupling elementary scalar eld of the unbroken U

vector multiplet and is the dynamically generated scale Fig shows

the structure of the Seib ergWitten Riemann surface and g gives a

picture of the mo duli space metric g The set of two indep endent cycles

uu

is a basis ofH Z with The meromorphic Seib ergWitten

form is given by

dz

p

x

SW

z

When is chosen to b e equal to twice the unit circle in the z plane lifted to

the Riemann surface and to b e the cycle encircling the branchpoints given

R R

and z u in the z plane wehave by

SW SW D

z

where canbeshown to corresp ond to the scalar of a dual magnetic U

D

massless vector multiplet which b ecomes weakly coupled when the original

electric theory b ecomes strongly coupled see b elow

The theory contains massivecharged BPS particles these are particles with

minimal mass for the given charge with electric andor magnetic charges

A BPS particle with electric charge n and magnetic charge m has mass

p

equal to jn m j In the conventions we use particles in the adjoint

D

of SU haveinteger electric and magnetic charges while particles in the

fundamental have halfinteger electric and integer magnetic charges By

the BPS condition and the triangle inequality the BPS particles are abso

lutely stable except when is real then they are only marginally stable

D

Marginal stability o ccurs in the uplane on a closed ellipslikeline passing

through u Outside this line of marginal stability the theory is well

describ ed in terms of the original variables with the coupling running to

zero by asymptotic freedom for u and running to at the singu

laritues u in mo duli space Outside the curve of marginal stability

the BPS sp ectrum consists of the W b osons charge a magnetic

monop ole charge which b ecomes massless at u an elementary

dyon charge dep ending on cut conventions which b ecomes mass

less at u and dyons of charge n n Z and of course all charge

conjugates of those Note that the latter can b e obtained from the monop ole

Low energy eective action of N YangMills theory

Figure Schematic picture of the Seib ergWitten Riemann surface represented

as a sheeted cover of the z plane The z plane is mapp ed by z ln z to a

strip with opp osite sides identied The two sheets of the surface are represented

by two such strips connected via the branch cuts running to innity from z

p

u u The cycle consists of two disconnected lo ops one on each sheet

running from one side of the strips to the other The cycle runs around the

branch p oints encircling the hole in the picture More details can b e found in the text

Chapter Low energy eective eld theories

Figure The metric g on the SU Seib ergWitten mo duli space plotted as

uu

a function of u The spikes at u are strong coupling singularities pro duced

byintegrating out massless charged particles

i

or the elementary dyon by a mono dromy u e u ab out u Indeed

such a mono dromy transforms and als follows

From our general discussion on eective actions one would exp ect the ef

fective action approximation to break down at p oints where integratedout

charged particles b ecome massless This is indeed the case the mo duli space

has singularities at u see g

Inside the line of marginal stability the only stable BPS particles in the

sp ectrum are the monop ole and the elementary dyon The theory in the

neighb orho o d of the strong coupling p oint in the original variables u

is well describ ed in terms of the dual magnetic massless U eld coupled

to the light monop ole Such a theory is infrared free and b ecomes weakly

coupled for u See g Analogously close to u one nds es

sentially a dual dyonic U theory coupled to the elementary dyon There

is no p oint in mo duli space where the original SU gauge symmetry is restored

Low energy eective action of N YangMills theory

Figure The dual coupling constant squared g g plotted as a

D

D D

function of u in a neighboorhood of u At u the dual coupling is zero

Approximate expressions for some sp ecial geometry quantities near u

are

p

u

p

i

u ln u

D

juj

Kjj ln juj

ln juj

g jj

uu

juj

ln g

and near u

u lnu

i

u

D

Re u ju j ln ju j Kjj

g jj ln ju j

uu

Chapter Low energy eective eld theories

D

ln g

D D

The prep otential in the weak coupling region has the following form

k

X

i

F ln c

k

k

The rst term is the onelo op correction while the k th term in the sum arises

as a contribution from k The Seib ergWitten solution amounts

to an exact sp ecication of all co ecients c

k

The results for SU can be generalized in various directions either by

generalizing the gauge group or by adding matter Wenow consider some of

those generalizations

Pure SU N

c

The rank of the group is r N The relevant Riemann surfaces form an

c

r dimensional family of hyp erelliptic genus r curves emb edded in C given

by the equation

r r

z x u x u x u

r

z

where z x are the ambient space co ordinates and the u are the mo duli As

i

set of cycles we can just takeany basis of cycles which wil contain indeed

precisely r cycles The meromorphic form is again given by

SU with one massive fundamental hypermultiplet

Actually this generalization requires a slight extension of the rigid sp ecial

geometry framework presented in the previous section since sp ecial geometry

only governs the low energy eective action of vector multiplets However it

turns out that the low energy theory of asymptotically free or conformal

invariant N YangMills theories with additional matter multiplets is

still completely determined by the geometry of a certain family of Riemann

surfaces endowed with a certain meromorphic form Since we will mainly

restrict to the pure YangMills case in this thesis anywaywe will not go into

the details of this construction

The relevant curves for SU with one hyp ermultiplet of mass m in the

fundamental of SU are given by

z mz x u

z

Low energy eective action of N YangMills theory



where as in the pure YangMills case u is identied with h i Again



the meromorphic form is the metric on the vector multiplet mo duli

space is rigid sp ecial Kahler constructed as for the pure SU case and

BPS masses can be calculated simply as p erio ds of if we consider a

SW

residue of also as a p erio d

SW

Pure E

We include this case just to show that life isnt always that easy

For the E exceptional group without matter the meromorphic form is

as usual but the mo duli space is given by the following parameter family

of genus surfaces

x Z Q x Z Q

where

Z z u

z

Q x u x u x u x u u x

u u x u u u x u u u x

u u x u u u x u u u u x

u u x u

Q P Q P

P x u x u x u x u x u u x u x

u x u

u x u P x u x u x u x u u x u x

Out of the indep endent cycles spanning H Z there are sp ecial

ones providing the indep endent p erio ds dening rigid sp ecial geometry

The construction of those cycles can b e found in The underlying orga

nizing structure under this apparent mess is of course group theory here E

representation theory Other unifying connections are Toda integrable sys

tems and singularity theory The latter connection emerges naturally from

string theory We will come back to this in chapter

Many other generalizations have b een constructed for whichwe refer to the

literature

In summary we have argued in this chapter that we can safely talk ab out

action of a certain quantum N the two derivative low energy eective

Chapter Low energy eective eld theories

YangMills theory Its form is b eautifully governed by the rigid sp ecial geometry

of a certain geometrical mo duli space of Riemann surfaces in the sense that for

example the scalar kinetic term of the low energy eective action app ears with the

natural sp ecial Kahler metric on this mo duli space

Chapter

CalabiYau compactications

of IIB string theory

In this chapter we will investigate some low energy asp ects of typ e IIB string

theory compactied down to four dimensions on a CalabiYau manifold Wewill

start by demonstrating how four dimensional N sup ergravity coupled to a

numberofvector and hyp ermultiplets arises as the low energy theory of the four

dimensional massless elds Next we will study the massive particlelike BPS states

in the D theory originating from branes wrapp ed around nontrivial cycles of

the CalabiYau manifold These BPS states can also b e obtained as solitonic BPS

solutions of the low energy D eective eld theory As we will see these solitons

have the remarkable prop ertyof b eing attractors for the scalars in the vector

multiplets that is towards the center of the soliton the scalars always ow to

certain xed values only dep endent on the electric and magnetic charges of the

state We will study the attractor solutions in quite some detail emphasizing the

physical intuition b ehind the equations We conclude the chapter with a discussion

of the range of validity of eld theory and Dbrane pictures

Part of the material in this chapter is a review of well known facts though we

will present these in a not so conventional way and we will elab orate a bit more

than usual on some conceptual issues

and branes cannot generate D BPS states in this way since there are no nontrivial one

or ve dimensional cycles in a CalabiYau manifold at least with the denition of a CY manifold

whichwe use

Chapter CalabiYau compactications of I IB string theory

Figure Sketch of a CalabiYau compactication The mo duli of the Calabi

Yau X can vary over the noncompact curved four dimensional spacetime M

Four dimensional low energy eective action

We will immediately turn to the tree level two derivative low energy action of

the massless elds in the theory For the world sheet p ersp ective higher order

corrections and many other asp ects which we cannot cover here Polchinskys

book is highly recommended We will also use mostly his notations and

conventions

Typ e IIB string theory

Low energy typ e I IB theory has N sup ersymmetry sup ersymmetry gen

erators in dimensions with equal The sup ersymmetry algebra is

A B AB

fQ Q g P



this means we consider the eective action for these elds to leading order in the string

coupling constant g e and the gravitational constant or

s

Four dimensional low energy eective action

where

with andA B The high amount of sup er

symmetry completely xes the low energy theory at tree level which is just D

typ e I IB sup ergravity The massless b osonic elds are

A scalar a form p otential B with eld strength H and the metric g

NeveuSchwarz NeveuSchwarz sector

A scalar C a form potential C with eld strength F and a form

p otential C with selfdual eld strength F RamondRamond sector

We adopt conventions such that the mass dimensions of all comp onents of the

p oten tials are zero To write down the action it is convenient and it makes

Sduality manifest to dene

C ie

j j Re

M

ij

Re

Im

H

i

F

F

C H B F F F

We will also work always in the Einstein frame which has canonical Einstein

Hilb ert term and is related to the string frame by G e G where

E S

is the dilaton g he i is the string coupling constant Due to the presence of the

selfdual form eld strength there is actually no simple manifestly covariant

action for this theory but the following comes close

Z

p

d x GR S

IIB

Z

j

i

F F d d M F F

ij

Im

ij

j

i

C F F

At the classical level one only has to imp ose the additional selfduality constraint

F F after varying this action to get the correct equations of motion The

 p

so the dimension of the pform itself is length

Chapter CalabiYau compactications of I IB string theory

constant is related to the constant app earing in the worldsheet action of

string p erturbation theory by It can be identied as

the ten dimensional gravitational constant Note however that this scale on itself

do esnt have a physical meaning only ratios of mass scales are meaningful and

measurable Whichcombination of and the coupling constantwe call the ten

dimensional gravitational constant is therefore purely conventional Something

which does have a physical meaning for example is the ratio of the mass of the

lowest excited string oscillator state g in the Einstein frame and the

physically measured D Planck mass This ratio is g and in

dep endent of frameit tells ho w strong spacetime is deformed when a minimally

excited string is present Notice that the meaning of as used eg in in

the string frame is dierent from the meaning of in the Einstein frame in the

Einstein frame it is the physically measured gravitational constant in the string

frame it diers from the physical gravitational constant with a factor g Confusion

can always b e eliminated by calculating mass ratios

The action is invariant under the following SL R symmetry

a b

c d

d c

i i

F F

b a

F F

G G

There is strong evidence that the discrete subgroup SL Z is actually an exact

symmetry of string theory not only at tree level and low energies this is the

typ e I IB S dualitygroup

Typ e IIB string theory contains Dirichlet pbranes with odd p We are in

w particular interested in the case p which has the following tree level lo

energy action for the b osonic degrees of freedom

S S S

D D D B I D W Z

The rst term is apart from curvature corrections of DiracBornInfeld

form and indep endent of the RR elds In the Einstein frame and without the

curvature corrections this is

p

Z

q

d deth e B e F S

ab ab ab D D B I

Four dimensional low energy eective action

where h and B are the pullback of spacetime metric and NSNS B eld to the

ab ab

brane and F is the eld strength of the gauge eld A living on the brane F

ab

a b

F dx dx dA The curvature corrections are of a rather complicated form

ab

see and we will not give them here For the cases we are interested in these

terms happ en to cancel anywayin a nontrival wayhowever The second term

is of WessZumino form Apart from the curvature couplings it is indep endentof

the metric and the other NSNS elds

p

Z

S C F B C

D W Z

F B trR trR C

T N

where R R is the curvature of the tangent normal bundle of the brane

T N

There are further nonp erturbative instanton corrections to S making

D

this action invariant under S duality as it should

CalabiYau manifolds and sp ecial geometry of the com

plex structure mo duli space

We assume the reader is familiar with the basic facts ab out dierential geometry

and CalabiYau manifolds Excellent reviews can b e found in See also

We dene a CalabiYau nfold as a compact n complex dimensional Kahler

manifold with vanishing rst Chern class and no nontrivial cycles equipp ed with

a Ricciat metric whichhas SU n holonomy A theorem of Calabi and Yau says

that for any xed complex and Kahler structure of the manifold such a metric

exists and is unique

In the following we restrict to the case n The mo duli space of complex

structures of a CalabiYau fold X equipp ed with arbitrary complex co ordinates

a

z a h has lo cal sp ecial geometry We will not prove this well known

fact here but merely review the main features and intro duce some notations which

we will need in the sequel Readers not interested in the technicalities of sp ecial

geometry can probably skip this section

Denote the up to normalisation unique holomorphic form on X by The



We will adopt the following general notational convention for form comp onents F

p

F dx dx This is as in cf page

p

p

Chapter CalabiYau compactications of I IB string theory

Kahler p otential for the mo duli space metric is

Z

K ln i

X

with Kahler metric

g K

a

ab b

The couplings akaYukawa couplingsmagnetic momentsp oint functionstriple

intersection numb ers are

Z

C

abc a b c

X

I

Cho ose a symplectic basis of cycles fA B g I h and denote the

I

I

corresp onding p erio ds as X F

I

Z

I

X

I

A

Z

F

I

B

I

With this choice the prep oten tial is given by

I

F X F

I

I

and X F is of course nothing but the symplectic vector of sp ecial geometry

I

One furthermore has the prop erty

H

H H

a

H H H

a b

H H H H

a b c

Dening the Kahler covariantderivative

D K

a a a



We attribute mass dimension zero to as well as to other abstract geometric quantities

such as cohomology classes and hence p erio ds

Four dimensional low energy eective action

we see from and that D a h forms a basis

a

of H X and that

Z

K

D D g ie

a

b ab

X

Thus we can decomp ose any real harmonic form on X according to the

Ho dge decomp osition theorem

H X C H X H X H X H X

as

Z Z

K K ab

ie D cc i e D g

a

b

X X

This decomp osition is useful b ecause it diagonalizes the Ho dge star op erator

pp p pp

X i X

It can be used to nd an explicit expression in terms of central charges for the

intersection and Ho dge scalar pro ducts on X

Z

ab

ImZ Z g D Z D Z

a

b

X

Z

ab

ReZ Z g D Z D Z

a

b

X

where we dened the central charge Z of as

Z Z

K

Z e

X

and D Z K Z Here by slight abuse of notation wehave denoted

a a a

and its Poincare dual by the same symbol If no confusion is p ossible wewill

always do this that is

Z Z

B B B

X

where the dot denotes the intersection pro duct the number of p oints in the in

tersection counted with signs We will use the same notational conventions for

forms and cycles of any degree Note also the relative minus sign b etween the rst

terms in the rhs of and

 p q

In general for an ob ject f which transforms as f f under a Kahler transformation

we dene D f p Kf and D f q Kf

a a a a a a

Chapter CalabiYau compactications of I IB string theory

Four dimensional massless sp ectrum

The four dimensional massless sp ectrum of typ e IIB string theory compactied

on the CalabiYau manifold X can b e deduced as follows see eg Cho ose

A

a basis A h b b of H X Z or by Poincare duality of

H X Z and denote the corresp onding dual basis of H X ZH X Z by

A

B B I

ie As ab ove cho ose a symplectic basis fA B g I h

A I

A

of H X ZorH X Z X has h complex structure deformations and h

Kahler class deformations By Torellis theorem the complex structure deforma

tions can be parametrized lo cally on mo duli space by the p erio ds of the holo

morphic form on X

I I

X B F A

I I

Note that indeed since i and the normalisation of is irrelevant a set of

i I

h indep endentvariables eg t X X is sucient to parametrize lo cally

all complex structures The complexied Kahler class deformations on the other

hand can b e parametrized again lo cally by the p erio ds of the complexied Kahler

where B is the NSNS form p otential and J the usual Kahler form B iJ

form on X

A

t

A

Z

A

t

A

As already mentioned ab ove Yaus theorem states that there is a unique Ricci at

metric on X for anygiven complex structure and real Kahler class The relation

between metric uctuations and complex structure and Kahler class deformations

is given by

g J

mn mn

rs

g kk

mn mrs n

mnr

where kk

mnr

Now the D massless elds pro duce D massless descendants by decomp osing

M X in the harmonic form bases them according to the compactication M

constructed ab ove

I I A A

C A B B A T S

I I A A

A

C c C

A D

A

B b B

A D

Four dimensional low energy eective action

Here is an arbitrary mass scale parameter which has b een intro duced to give

the four dimensional elds their usual dimensions that is our D forms have

I

dimension zero Note that the selfduality constrainton F relates the form A

A

to the form B and the scalar S to the form T

I A

A A

Now dualize the D forms B C and T to B C and T resp ectively

D D

A A A A

so for example dB dB Then t and t c iT together with

D

t B ie and t C iC form the b osonic elds of n h D

H

I

hyp ermultiplets while the four dimensional metric G the vectors A and the

D

I I I

scalars X mo d X X form the b osonic elds of the gravitymultiplet plus

n h vector multiplets

V

This is precisely the b osonic sp ectrum of four dimensional N sup ergravity

ultiplets Apart from coupled to n h hyp ermultiplets and n h vectorm

H V

the precise numberofvector and hyp ermultiplets this could have b een deduced

purely from sup ersymmetry Indeed it is known that a compactication on

a CalabiYau fold preserves one quarter of the original sup ersymmetries ie

sup ersymmetry generators or N in four dimensions

For later use we record here the D N sup ersymmetry algebra in its

most general form including central charges of p ointlike ob jects

A B AB AB

fQ Q g P i Re Z ImZ

where A B and Ma jorana rep Z is the central charge

which commutes with all other generators of the sup erPoincare group We added

the tilde to distinguish this op erator from the geometric ob ject Z as it was intro

duced in We will later see that Z and Z are in fact simply prop ortional

to each other

Four dimensional action

Since in this thesis we will only b e interested in the physics of the four dimensional

massless vector multiplets we will take all four dimensional massless hyp ermult

plet elds to be trivial constant scalars It can be checked from the typ e IIB

action and the dimensional reduction formulas that this

is a consistent truncation of the theory the vector and gravitational elds do not

app ear as sources of the hyp ermultiplet elds



So only half of these D elds are indep endent which elds wecho ose as fundamental degrees

of freedom is a matter of choice dierentchoices are related by duality transformations



We dont b other ab out the precise grouping and normalisation of these scalars here since

we will not consider the massless hyp ermultiplets in detail in the remaining part of this thesis

Chapter CalabiYau compactications of I IB string theory

In that case the dimensional reduction of the action yields taking into

acount the selfduality constrainton F and using and sp ecial

geometry of the complex structure mo duli space

Z Z

p

a b I

S GR g dz dz d x F G

D I

ab

M M

 

where

VolX

p

VolX

I I

with dened in F dA and G dB is to be considered as a

I I

I

function of A given by the selfduality constrainton F

I I I I

F B G A F B G A

I I X I I X

Here denotes the Ho dge star op erator on the spacetime manifold M and

X

I

the one on the CalabiYau manifold X Integrating this equation over A B and

I

using gives a system of equation which can b e used to express the G in

I

I

terms of the F

J J

G ReN F Im N F

I IJ IJ

where N is a mo duli dep endent symmetric matrix We will not need its explicit

IJ

form but give it here anyway for completeness

K L

Im F X Im F X

IK JL

N F i

IJ IJ

M N

X Im F X

MN

with F F and F the prep otential as given in We will discuss

IJ I J

the range of validity of this action in section

I

While the action is only determined up to a choice of symplectic basis A B

I

the electromagnetic energy is unambiguously dened once a spacetime decomp o

sition has b een chosen Intro ducing the notation

I I

dA F F F B G A

I I

where the second equalityis just with the hyp ermultiplets taken to be

constant and denoting the spatial comp onents of F as F the electromagnetic

s

energy density is simply given by

Z

H dt F F

em s s

X

Because of the subtleties asso ciated to the selfduality constraint the action for the vectors

which come from the selfdual form is actually determined by requiring that the correct

equations of motion are repro duced and by matching the expressions for eg the energy to get

the right normalisation See also

BPS states from wrapp ed branes

This can b e given a more explicit expression using sp ecial geometry Denote

Z

K

F e

s

X

then from

ab

H dt g D D

em a

b

The low energy theory thus obtained is as exp ected N sup ergravity

coupled to n vectormultiplets with scalar manifold given by the complex struc

V

ture mo duli space of X The action is indeed identical to In

principle is just the classical low energy eective action and though its

form is xed by sup ersymmetry there could still be an innite number of very

complicated string lo op and worldsheet instanton corrections However a miracle

happ ens From considerations of sup ersymmetry and sp ecial geometry it can

be shown that there cannot b e any couplings b etween massless vector and hyp er

multiplets at the twoderivativelow energy level Since the dilaton in typ e I I

theory is in a hyp ermultiplet and since the string coupling constant is given by

e there can therefore b e no string lo op corrections to the vectormultiplet action

at this low energy level Furthermore since string worldsheet instantons always

R

come with factors exp B iJ which again contain hyp ermultiplet elds

there can also b e no worldsheet instanton corrections So we conclude that the

twoderivativelow energy eective action of D N sup ergravity coupled to

anumber of vector multiplets does not receive any quantum corrections It is ex

act The imp ortance of this result cannot b e overestimated It allows us to extract

exact quantum results from classical geometry and is at the core of such diverse

things as curve counting via mirror symmetry exact solutions of quantum

eld theories and p ossibly even the solution of Hilb erts twelfth problem

BPS states from wrapp ed branes

States in string theory

String theorists tend to b e a bit sloppy ab out the meaning of the word state

This is in part b ecause it is simply not clear what the full set of states of string

In typ e I IA theory on the other hand these are vector multiplet elds and indeed the D

low energy vector multiplet action obtained from typ e I IA string theory do es receive this kind of corrections

Chapter CalabiYau compactications of I IB string theory

theory is and in part just a manifestation of the often convenient habit of abusing

terminology in physics We will try to explain this word in the context of string

theory a bit more precisely here though we do not intend to give a full solid

denition

Stated abstractly a state is roughly just something which asso ciates ex

p ectation values to physical observables In a quantum theory usually but not

necessarily states are given a Hilb ert space representation In string theory one

usually talks ab out states with this Hilb ert space framework in mind though a

more general framework might actually b e more appropriate or even necessary

as string theory in its present formulation do es not allow a constructive denition

of its full underlying Hilb ert space This is simply b ecause unlike in quantum

eld theory the nonp erturbative fundamental degrees of freedom of string theory

are not known

Note that we are talking ab out states of the full spacetime theory here not

ab out the states of one of the sp ecic worldsheet theories arising in p erturbation

theory from the string path integrals Those two are not entirely disconnected how

ever For example the states of say the typ e I IB worldsheet theory on a cylinder

in a at ten dimensional background provide the dierent oneparticle states of

the typ e I IB spacetime theory expanded around the at vacuum These are the

quanta of the spacetime degrees of freedom Incidentally replacing worldsheets by

worldlines the same applies of course to eld theory though there one needs dif

ferenttyp es of worldline theories to get dierenttyp es of particle sp ecies When

the at spacetime in whichwewere considering our typ e I IB example also contains

a Dbrane we can analogously consider the states of the worldsheet theory on a

strip with b oundaries glued to the Dbrane cf chapter This gives a sp ectrum

of particle states living on the brane which are the quanta of the brane degrees

of freedom ripples and so on

As in eld theory one can consider scattering exp eriments with several widely

separated and eectively free particles in the far past and future interacting for a

nite time with each other in the lab oratory String theory in its original formula

tion do es nothing more than giving a prescription how to calculate such scattering

according to worldsheet top ology amplitudes as a p erturbative series ordered

and the main go o d news here is that the results are free of UV divergencies Now

we can consistently formulate all this with strings propagating in various back

ground geometries with or without Dbranes and with various other elds turned

on p ossibly including solitonic congurations at least provided all those ob jects

satisfy certain equations of motion The imp ortant thing to remember is however

There do es exist a sort of a second quantized string eld theory but there are quite some

ob jections against it and it is now widely b elieved that this is not the prop er description of

nonp erturbative string theory

BPS states from wrapp ed branes

that in string theory a priori scattering amplitudes are al l we have to probe the

physics We could equivalently say that the physical observables wehave at our

disp osal in this p erturbative framework are built out of particle annihilation and

creation op erators in the far past and future

What is the meaning of the classical backgrounds then They are denitely

not directly observable However dierent backgrounds will in general give dier

ent scattering amplitudes ie dierent exp ectation values of our observables so

they do dene dierent states Of course by the usual path integral saddle p oint

argument at suciently large time and distance scales the scattering amplitudes

will b e very muchaswewould exp ect from classical particles propagating in the

given classical background Usually in string scattering theory the background is

taken to b e time indep endent and stable and then one refers to the corresp onding

state as a p erturbative vacuum Note that though it may still seem o dd that

for example a Dparticle at rest and lo calized at a certain p oint in space could

give rise to a consistentquantum state in p erturbative string theory this actually

is not that strange when the string coupling g is sent to zero the mass of the

Dparticle diverges in string units so the quantum mechanical spreading of its

wave function is indeed suppressed with resp ect to the string scale when g

At nite g however the use of such classical particlelike solutions having nite

mass is questionable in p erturbative string scattering theory We will encounter

precisely such particlelike states However as their low energy prop erties which

we will study turn out to be indep endent of g in the Einstein frame we can

always imagine g so that the mass of the particle diverges in string units and

we dont have to face this complication

Apart from varying the background within a given p erturbative string theory

liket ypeItyp e I IA typ e I IB E E heterotic and so on we can also change

the p erturbative string theory itself The picture which emerged during the past

veyears is that after sucha change we are not lo oking at another fundamental

theory but merely at another p erturbative description of the same underlying

theory For example it can be proven that typ e IIA p erturbative string theory

on a space with one compact dimension of radius R in string units and string

coupling g has actually precisely the same physics as typ e I IB string theory on a

space with one compact dimension of radius R and string coupling gR AIIA

string winding n times around the compact circle corresp onds to a I IB string with

momentum n along the circle Scattering amplitudes are likewise mapp ed to each

other So at least to all orders in p erturbation theorywehaveanidentication

of the I IAR g and the I IBR gR state This is Tduality It shows very

clearly the fact that the background geometry do esnt have an absolute meaning

it is just part of a nonunique state lab el Similar equivalences b etween I IA and



For higher noncompact branes there is no problem of this kind

Chapter CalabiYau compactications of I IB string theory

Figure An artist view of the coninuum of states describ ed in the text including

Mtheory What the picture wants to emphasize is the connectedness of this

space the dierent p erturbative theories b eing no more than dierent suitable

p erturbative descriptions in dierent corners of the space

I IB exist when Dbranes are added even branes in I IA corresp onding to certain

o dd ones in I IB or when the geometry is made more complicated for example

CalabiYau compactications where the map is given by mirror symmetry

Note that the ab ove I IA and I IB p erturbative string theories b etween which

the Tduality map acts are actually b oth at the same time sensible p erturbative

descriptions only if g and g R So in some regions of the lab els IIA is

accurate in some IIB in some b oth and in some none Another kind of duality

which is much harder to test as it interchanges strong and weak coupling and

hence requires always a sort of extrap olation of the theories a way from p erturbation

theory is Sduality An example is the SL Zdualityoftyp e I IB theory already

mentioned in section Another example is the dualitybetween heterotic theory

on T and typ e IIA theory on K Most of the tests of Sdualities rely on low

energy extrap olations combined with sup ersymmetry constraints

One thus arrives at a picture of one fundamental theory with a coninuum of

states of which some corners have a go o d description in terms of one p erturbative

string theory and others in terms of another Some states mighthave no p ertur

bative string description whatso ever However thanks to various Sdualities the

extreme strong coupling limits are often dual to weak coupling and hence p ertur

bative string limits The most notable exception is the strong coupling limit of

typ e I IA theory This is the magical mysterious matricious Mtheorywhichat

BPS states from wrapp ed branes

low energies reduces to eleven dimensional sup ergravity It do es certainly not

have a consistent p erturbative string description How to do quantum physics in

this regime is therefore not clear The numb er one candidate for this is the matrix

mo del of Banks Fischler Shenker and Susskind The picture one thus gets for

this coninuum of states is shown in g

BPS branes

An imp ortant step towards understanding nonp erturbative asp ects of strings has

b een the discovery that often p erturbative particle states in one p erturbative

string picture are realized as Dbranes in another Sdual p erturbative string

picture This is very useful b ecause strong coupling asp ects of these states in

one picture can then b e calculated simply at weak coupling in the other picture

We will see examples of this b elow Till now quantitative results have b een

mainly obtained for BPS states whichhave a certain amount of sup ersymmetry

and therefore relatively controllable quantum corrections We will esp ecially be

interested in such BPS states arising in typ e IIB CalabiYau compactications

whichhave a particle or black hole interpretation in the D low energy eective

theory

Let us therefore briey recall what is meantby a BPS state in a D N

theory Consider a state with denite central charge Z and energymomentum P

and go to a Lorentz frame where the spatial momentum P is zero and P M

i

On such a state the algebra b ecomes

B y

A AB AB

fQ Q g M i Re Z ImZ

The left hand side considered as a matrix in A B is nonnegative denite

so the eigenvalues M jZ j of the right hand side have to b e nonnegativeaswell

implying

M jZ j

A state is called BPS when its mass is minimal for a given Z ie M jZ j From

the algebra it then follows that half of the sup ercharges annihilate the state so

we keep N sup ersymmetry The remaining four sup ercharges can be taken

together to form complex fermionic op erators say b and b furnishing

together a spin representation of the little group SO for a massivestate

and satisfying the fermionic oscillator algebra

y

f b b g

i ij

j

The smallest representation of this algebra is four dimensional so anyBPSstate

in a D N theory will be part of a multiplet of at least four states

Chapter CalabiYau compactications of I IB string theory

y

degenerate in mass since the b commute with P One can construct such a

i

multiplet by starting from one or more fermionic oscillator ground states also

called fermionic oscillator vacua ji lab eled byanindex r which aresupposed

r

to furnish again arepresentation of the little group and to satisfy b ji and

i r

y

subsequently acting on those with one or more of the creation op erators b For

i

single particle states this gives typically one of the following BPS multiplets for

M

a halfhyp ermultiplet this has one spin fermionic oscillator vacuum

states The spin content is

a full hyp ermultiplet two spin fermionic oscillator vacua of opp osite

charge states Spin contentis

a vectormultiplet spin fermionic oscillator vacuum states Spins

BPS massive gravity multiplet spin fermionic oscillator vacuum

states

Such BPS multiplets can b e seen as Goldstone multiplets of broken sup ersymmetry

Note that these BPS multiplets are shorter than generic massivemultiplets a

generic massive state breaks all sup ersymmetry and is therefore part of a multiplet

of at least states This is whyin an N sup ersymmetric vacuum a

particle which is in a short BPS multiplet at vanishing coupling is exp ected to

stay BPS M jZ j when interactions become stronger at least as long as it

exists as a stable state the opp osite would imply a discontinuous jump upwards

in the numb er of states in the multiplet This argument and analogous reasonings

underly many results in strongly coupled string theory since it allows one to

extend some quantities obtained in the Dbrane picture at weak string coupling

where the Dbrane can just b e represented as an ob ject on which strings can end

without having to take into account backreaction on the ambientspace to the

strong coupling regime

Let us turn now to the Dbrane representation of these D BPS states in

typ e I IB string theory compactied on a CalabiYau manifold Typ e I IB theory

contains only odd branes and since the only odd dimensional nontrivial cycles

in a CalabiYau have dimension the only p ossibility to get particlelike BPS

states in four dimensions from Dbranes is to wrap a brane ab out a cycle

in the CalabiYau in such a way that one half of the sup ersymmetries remains



the dierent fermionic oscillator vacua are usually also connected to each other by cre ationannihilation op erators constructed from certain fermionic zero mo des see eg

BPS states from wrapp ed branes

intact More precisely the latter condition means that it should be p ossible to

comp ensate the action of one half of the sup ersymmetries of the given Calabi

Yau background on the fermionic elds of the brane worldvolume theorybya

worldvolume transformation We will call suchanemb edding supersymmetric

This problem was studied by Becker Becker and Strominger in resulting in

the following criterion for the emb edding f of the brane in the CalabiYau to

b e sup ersymmetric

The pullbackoftheKahler form to the brane should vanish

f J

The pullback of the holomorphic form should have constant phase

i

f e d

i

where is a constant the i are co ordinates on the spatial part

of the brane and is a real p ositive function on the brane

In the mathematics literature a submanifold satisfying these conditions is known

as a spe cial Lagrangian submanifold It can b e shown that a brane satisfying

these conditions has minimal volume in a given homology class see also b elow

The analysis of implicitly assumes triviality of all background elds including

the worldsheet U eld strength except for the metric and selfdual form eld

strength since the brane itself is a source for these elds This is compatible with

the assumptions we made earlier Considering the known curvature corrections

to the Dbrane action one could worry ab out the fact that these curvature

couplings could cause the wrapp ed brane to act as a source for C and This

is not the case for sup ersymmetric branes as can b e shown directly from the ex

pressions in at least if the prop erty holds that a sup ersymmetric cycle in a

CalabiYau manifold is totally geo desic The only terms to worry ab out are those

in the DBI part of the action since those in the WZ part vanish automatically

b ecause of the direct pro duct structure of the brane Under the assumption of

b eing totally geo desic using the fact that normal and tangent bundle of a sup er

symmetric cycle in a CalabiYau manifold are isomorphic we nd that the

various terms in the DBI part as given in cancel exactly

Not every homology class has a sup ersymmetric representative The existence

of such a representative is equivalent to the existence in the D eectivetheoryofa

BPS state with charges corresp onding to the homology class under consideration

Unfortunately it is not easy to establish existence of sup ersymmetric cycles in a

generic CalabiYau



We did not sayanything ab out the worldsheet U eld strength yet but it is clear that

turning on this eld would add extra energy to the state whichwould moveitaway from the

BPS b ound and break the remaining sup ersymmetry not what wewant

Chapter CalabiYau compactications of I IB string theory

Mass and charge of wrapp ed branes

Let us now compute the mass and charges of such a BPS particle obtained from

wrapping branes ab out CalabiYau cycles This can be most easily done by

studying how the Dbrane action reduces to a particle action in four

dimensions

We consider the brane as a prob e that is we neglect backreaction on the

spacetime elds As discussed in sections and this is what we should do

if wewant to use the branes as Dirichletbranes in lowest order string p erturbation

theory In particular this means that the mo duli are taken to b e constantover the

four dimensional spacetime When one do es takeinto account backreaction the

mo duli at the worldline are for a BPS state always at their socalled attractor

values see section and the mass one thus nds by reduction of the D action

is not the actual mass of the state but only the bare mass of the minimal brane

as measured by an observer at r which is the horizon when the BPS state

is a black hole From the p oint of view of an observer at innity the complete

mass of the state is in the surrounding elds as will b e sho wn in section It is

p erhaps surprising that the bare mass in the prob e brane picture is the same as

the mass of the elds when the full backreaction is taken into account However

this is just a consequence of sup ersymmetry it is the BPS mass formula at work

We will deal with the backreaction issue in section and discuss the validity

of the Dbrane and eective eld theory pictures in section For now we simply

put the branes in a xed background

Asp ects of the relation b etween wrapp ed Dbranes and black hole solutions

are studied eg in Boundary states for wrapp ed Dbranes were studied

in

Now with our assumptions ab out the background elds the Dbrane action

b ecomes

p p

Z Z

p

S d det h C

D ab

We rst consider the DBI part Dimensional reduction gives

p

Z

S ds V s

DB I D

where s is the prop er time measured with the D metric and V is the volume of

D

the spatial wrapp ed part of the brane

Z

p

V d det h

D ij



the bare mass if any is redshifted away

BPS states from wrapp ed branes

m

Denote with y the holomorphic co ordinates on the CalabiYau manifold X Then

in general

m n

y y

h G

ij mn

i j

but if the rst BPS condition f J is satised we can drop the symmetrization

m n

y y

h G

ij mn

i j

and

m

y

det h det G det

ij mn

i

Now note that

mnk

kk det G j j

mnk mn

Using this equation and the fact that is covariantly constantover X so kk

is simply a constant we nd

Z

K

e i kk V

X

X

where V is the volume of the CalabiYau X Plugging this backinto we

X

obtain

K

det G e V j j

mn X

Using this in to compute det h we get the following expression for the

ij

volume of the brane if the rst BPS condition is satised

Z

p

p

K

V V e jf j

D X

So we nd the inequality

Z

p

p

K

V V e j f j

D X

where equality is satised if and only if the phase of f is constant whichwas

precisely the second BPS condition So recalling we conclude that for

a BPS brane wrapp ed around a cycle in X the DBI part of the

Dbrane action reduces to

p p

p

Z Z

V

X

S jZ jds jZ jds

DB I

Chapter CalabiYau compactications of I IB string theory

Therefore the mass of the D BPS particle is

p

p

jZ j M jZ j

BP S

G

N

with G the four dimensional Newton constant This also suggests the following

N

relationship between the geometric central charge Z and the central charge Z

from the sup ersymmetry algebra

p

Z Z

G

N

This can indeed b e veried from the representation of the sup ersymmetry algebra

on the typ e I IB elds but we will not go into this

The reduction of the WZ part of the Dbrane action is straightforward

Recalling the notation we get from and

p

Z

S A

WZ

I I

so if our Dbrane is wrapp ed around a cycle n A m B we nd recalling

I I

the denition given under

p p

Z Z

I I

S A n A m B

WZ I I

So if we consider the Avectors as opp osed to the B vectors as the elemen

tary massless excitations of the theorywe can identify the electric charges with

winding numb ers ab out the Acycles and magnetic charges with winding numb ers

ab out the B cycles

From the equations of motion obtained from the total action S S S

D D

with S as given in it follows that the electromagnetic eld pro duced by

D

a static brane wrapp ed around at r in a spherically symmetric background

is in the notation given by

F

X

where

p

sin d d



Note that this is purely a conventional identication there is a priori no invariant distinction

between A and B cycles and no invariant distinction b etween electric and magnetic charges This

manifests itself as the electricmagnetic dualityoflow energy theory

BPS states from wrapp ed branes

Here and are spherical co ordinates ab out r This can be expressed in

comp onents using and More interestinglywe can use the same

formulae together with to calculate the electromagnetic force on a test

particle with charges moving with velo city v in the eld of a static charge

From and we nd this is

e e

r r

F v

r r

R

Here the dot denotes as usual the intersection pro duct which

X

can b e calculated for linear combinations of integral cycles and their Ho dge duals

using In particular for two particles with colinear charges Q we

i i

thus obtain

e

r

F Q Q V z z

r

with

ab ab

V z z jZ j g D Z D Z jZ j g jZ j jZ j

a a

b b

with Z the central charge of

Spin

It is a much more dicult problem to determine the D spin of the particles

obtained from wrapp ed branes or more precisely to determine the kind of N

sup ermultiplet one obtains by wrapping a brane ab out a particular cycle In

on principle this could be done by investigating the sp ectrum of strings ending

the Dbrane but in practice this is waytooinvolved

When the homology class under consideration has a unique sup ersymmetric

representative there is usually no problem in that case one exp ects a unique

fermionic oscillator ground state which must have spin zero since higher spin

implies a degenerate ground state on which one can build a halfhyp ermultiplet

by acting with combinations of the broken sup ersymmetries as outlined ab ove

The other half of the hyp ermultiplet is obtained by CPT conjugation which simply

reverses the orientation of the brane and hence the charge

The situation b ecomes problematic when there is a continuous family or mo d

uli space of sup ersymmetric cycles in a given homology class The sup erpartners

of these mo duli are fermionic zeromo des which can b e used to build up a degen

erate fermionic oscillator vacuum on which one can then build N multiplets

in the usual way However the degeneracy and spin of these fermionic oscillator

vacua and hence the kind of multiplet is a priori undetermined and in principle

Chapter CalabiYau compactications of I IB string theory

a detailed analysis of sup ersymmetric quantum mechanics on the mo duli space is

needed to gure this out This is analogous to the determination of the spin of

eg a monop ole in eld theory which can also b e quite nontrivial see eg

Fortunately in some cases there is a shortcut We would like to present a

particularly simple and useful one here The idea is the following Supp ose wecan

increase the number of remaining sup ersymmetries eg from to generators

of the original generators of typ e I IB string theory byachange of the com

pactication geometry away from the brane For example if the compactication

manifold is a K bration over a certain base manifold B and the brane a K

cycle bration over a nontrivial circle in B we can put the brane in a T K

compactication B T to double the sup ersymmetry Often more sup ersym

metry makes it easier to deduce the multiplets pro duced by the brane b ecause

there are fewer p ossible multiplets and b ecause the compactication geometry is

usually simpler as the holonomy group b ecomes smaller Supp ose wethus man

age to nd the multiplet in the case with higher sup ersymmetry As usual this

multiplet can b e constructed with creationannihilation op erators which are com

binations of the sup ercharges broken purely by the presence of the brane Those

sup ercharges are spinors invariant under the holonomy of the compactication

manifold Now when we restore the original compactication geometry the holon

omy group increases and some of these sup ercharges will no longer be invariant

under it Consequently some of the states in the original extended susy BPS

multiplet will b ecome noninvariant under the holonomyand leavethe multiplet

of BPS states States remaining invariant will staytoformamultiplet of our

original sup ersymmetry algebra A state which always satises this is the top

spin state of the extended multiplet eg the vector in an N vector multiplet

or the graviton in an N gravitymultiplet Indeed this state is always invari

ant under the lo cal Lorentz group of the compactication manifold since there is

only one highest spin state in a sup ermultiplet and the spin value has to remain

invariant under internal symmetries this state must be in a singlet of the lo cal

Lorentz group of the compactication manifold so a fortiori under the internal

holonomy group On a more p edestrian level we could simply say that this is

b ecause the top spin eld has all its indices in the noncompact spacetime

least So in the cases where such a trick is p ossible the brane pro duces at



However by adding one ore more suitable extra fermionic zeromo des which might b e induced

by the change of compactication this noninvariance can b e canceled again leading to more

multiplets To analyze this requires a more sophisticated approach involving twists of the Lorentz

algebra see eg the third reference in



It is p ossible that the brane pro duces still more multiplets bythemechanism discussed in

the previous fo otnote

Note that nothing guarantees the original fermionic oscillator vacuum itself to stay in the

multiplet since the original annihilation op erators can get mixed up with the original creation

op erators under the enlarged holonomy group

BPS states from the D eective action attractors

the N multiplet lab eled by this highest spin state a massive BPS vector

multiplet if the extended multiplet is of vector typ e or a massive BPS gravity

multiplet if the extended multiplet is of gravity typ e In the ab ove example of

the K bration wewould get a vector multiplet We will discuss this and other

examples in much more detail in the next chapter

BPS states from the D eective action at

tractors

At least for suciently low energies that is for slowly varying eld exp ectation

values it should also b e p ossible to nd the BPS states of our theory as solutions

of the four dimensional low energy eective action Furthermore there should

also b e a nontrivial representation of the four broken sup ersymmetry generators

on the elds generating a m ultiplet of BPS solutions which up on quantiza

tion should repro duce the quantum multiplet structure discussed earlier in section

Thus the two complex Grassmann parameter family of BPS solutions gen

erated out of a single spherically symmetric spin BPS solution will corresp ond

to a halfhyp ermultiplet while the family of BPS solutions corresp onding to a vec

tormultiplet will b e generated out of a more intricate family of BPS solutions

For an example of how the generation of sup erpartner solutions works con

cretely in N sup ergravity see where a hyp ermultiplet of solutions is

constructed from a spherically symmetric black hole solution Another example

is the N SU monop ole whichisworked out in detail including nonBPS

b ound states in In some physical prop erties of the sup erpartner solu

tions are studied

In this thesis we will restrict ourselves mainly to static spherically symmetric

BPS solutions whichthus should corresp ond to BPS hyp ermultiplets of the four

dimensional N theory



It is not entirely clear to us how exactly this family should lo ok like but it will denitely

not b e a spherically symmetric solution since this corresp onds to a hyp ermultiplet We will call

them spin fermionic oscillator vacuum solutions in analogy with the spin fermionic

vacuum state used to build the quantum vector multiplet At this p oint this is just a name

Chapter CalabiYau compactications of I IB string theory

Static spherically symmetric congurations

We take a general static spherically symmetric ansatz for the metric

U r U r

ds e dt e dr r d

f r

or changing variables r c sinh c f r h cosh c

c c

U U

ds e dt e d d

h

sinh c sinh c

Here U r and f r are functions of the radial co ordinate r to b e determined by

the BPS equations of motion We require asymptotic atness of spacetime so

U and f h at spatial innityr We furthermore assume

the mo duli and electromagnetic elds to b e spherically symmetric and pro duced

by a source with charge H X Zatr that is according to

F

X

U

e

p p

where sin d d d dt Note that with this ansatz the

h

equations of motion for F are automatically satised

The metric has Ricci scalar

c cosh c sinh c

U

h h R he U hU c c

c sinh c h

sinh c

Putting all fermionic elds to zero and plugging all ansatze for the b osonic elds

into we nd for the eective action mo dulo b oundary terms

Z

U

e

b

a

d fhU g z z c G V z z S T

N D

ab

G h

N

c

h g

h

sinh c



This ansatz is more general than the one used in However it reduces to the latter by

the equations of motion as we show here

R

 I

The action S T F G is only determined up to symplectic duality

D I





I

transformations ie up to choice of symplectic basis A B However to get a manifestly

I

I I

consistent reduced action principle at xed eld strength F F dA should only app ear in

I

the action such that the action is not varied via changes of F when the other elds are varied

With the ansatz we use this is the case when wecho ose our basis suchthat B and then

I

the electromagnetic part of the action is equal to the electromagnetic energy leading to

the ab ove expression for S

D

BPS states from the D eective action attractors

R

d

where T dt is the elapsed time and the dot denotes The scalar p otential

d

V z z is derived from and

ab ab

V z jZ j g D Z D Z jZ j g jZ j jZ j

a a

b b

Note that as usual for static solutions the action p er unit time is equal to minus

the p otential energy p ossibly up to b oundary terms

Now the equation of motion for h obtained from variations of corre

sp onding to radial dieomorphisms ie f f for a function f acting

a

on the elds U z and h together with the h equation of motion actually implies

k

h tanh c with k an arbitrary parameter Solutions with dierentvalues



c

of k are related by a radial dieomorphism corresp onding to a change of the till

now arbitrary constant c Let us now gauge x this remaining dieomorphism

by putting k so h

Varying h on the other hand implies at h the constraint

b

U a

G e V z c U g z z

N

ab

However since at h by translational invariance of the reduced action

the left hand side is a conserved quantity along translations anyway this just

determines the value of c from the b oundary conditions leaving no nontrivial

constraint indep endent of the other equations of motion Bearing this in mind we

can simply put h and write as

Z

b

a U

S T d fU g z z c G e V z g

D N

ab

G

N

or by completing squares as

Z

p p

U a U ab

S T G e jZ j kz G e g d fU jZ jk c g

D N N

b

G

N

U

e

p

jZ j

G

N

BPS solutions the attractor ow equations

We will rst try to get some intuition in the general solution of the mo del and

then pro ceed to the more precise and technical study of the equations

Lo oking at the form of the action we see that our system is equivalent

to a particle moving on R M with R parametrized by U and M the complex

Chapter CalabiYau compactications of I IB string theory

U

Figure Sketch of the p otential e V in which the eective particle is mov

U

ing plotted as a function of e and the mo duli z For generic initial conditions

the particle will run away to innity pro ducing an unphysical singular antigrav

itating sup ergravity solution tra jectory a If U is suciently large

the particle keeps on moving all the wayto U at least if z isatthe

same time tuned such that z This corresp onds to a nite energy

p

nonextremal black hole solution with ADM mass M U G

AD M N

tra jectory b The critical tra jectory with minimal M for given initial z

AD M

barely reaching the top of the p otential at corresp onds to an extremal

black hole tra jectory c

U

structure mo duli space of X sub ject to the p otential e V The minus sign in

front of the p otential is crucial The parameter R plays the role of time

Contrary to what is usually claimed in the literature the motion on this space

is not damp ed as energy is conserved according to Close to lo cal maxima

U U

of e V we exp ect oscillatory solutions Close to lo cal minima of e V we exp ect

unstable b ehavior and runaway solutions except when the initial conditions are

netuned such that the particle climbs the hill with just enough energy to reach

the top ending its motion there see g Such nite particle action solutions

corresp ond to nite energy eld congurations in our original eld theory they

are the solitons we are lo oking for

U

Since critical points minima in particular of e V are thus of central im

BPS states from the D eective action attractors

p ortance to nd nite energy solutions let us study those a bit closer Clearly

critical p oints of jZ j are also critical p oints of V with V jZ j The converse

cr cr

is not necessarily true A counterexample is given eg by taking essentially

z z

Z z O z and g O z which has V but Z at z We

z z

will mainly fo cus on critical p oints of Z as they will turn out to b e the relevant

ones for BPS solutions Using some sp ecial geometry identities one can show

that at critical p oints of jZ j we furthermore have V jZ j and

a b cr a b cr

V jZ j g jZ j Therefore since the metric g is p ositive

a cr a cr cr

b b ab ab

denite such critical p oints are always minima Happily minima are precisely

what wewant

U

For nite U minima of V are only minima of e V if Z that is when

cr

the cycle under consideration has a vanishing p oint in mo duli space This case

will b e discussed in section and in chapter In the generic case however

U

Z and in order for e V to be critical we need U Therefore

cr

recalling corresp onds to r nite energy solutions must have U

at r Typically this gives a black hole with horizon at r Now U

U

is always a critical p ointofe V for any nite z Therefore for suciently large

U the motion in the U direction has no turning p oint and the particle

U

runs all the way to the U top of the e V hill at least if we tune z

such that at the same time z when if not the motion will in general

explo de due to unstability of the system and the particle will fall back Hence in

this case we always nd a nite energy solution with no preferred mo duli values at

the horizon This is not surprising since the ADM mass of the solution is given by

p

M U G and we indeed exp ect nonextremal black hole solutions for

AD M N

any suciently large value of M Now for xed initial values of the mo duli

AD M

when we start to lower U at a certain p oint U will fail to reach innity for all

initial mo duli velo cities and consequently nite energy solutions no longer exist

The critical tra jectory with minimal ADM mass for the given mo duli at

will have U at and we exp ect it to just reach a minimum of V there

U U

since minimal V means minimal pushback force e V e V in the

U

p ositive U direction leaving no ro om to lower the initial jU j even more while still

reaching the top of the mountain in the U direction Consequently for such a

critical solution wehave b oth kinetic and p otential particle energy equal to zero

at implying c according to Thus we exp ect to nd solutions

with minimal mass for the given charge and mo duli at innity which furthermore

are extremal c Again this is not surprising such solutions are simply the

extremal black holes of N sup ergravity

From the ab ove argument we thus exp ect that the mo duli at r the



This limit is of course a bit subtle but we will see b elow that this intuition is indeed conrmed

by the equations

Chapter CalabiYau compactications of I IB string theory

horizon will b e xed at a critical p ointofV invariant under continuous variations

of the mo duli at innity This is the famous attractor mechanism The

attractor prop erty has nothing to do with damping of the eective particle motion

as often wrongly stated Quite the contrary it arises from the nite energy

condition thanks to the unstability of the particle motion in a way which is actually

very common in soliton physics

Note that extremal black holes ie solutions with c are not necessarily

BPS If wehave a critical p ointofV at z z which is not a critical p ointofZ and

we takesay z z so z const straightforward solution of the equations

p

of motion derived from yields the c solution U ln G V

N

p p

p

a

with G M V jZ j jZ j jZ j jZ j The converse is true

N AD M a

however BPS black holes are always extremal

Let us see nowhowwe get the BPS solutions from the equations

From it is clear that the reduced action at xed values of c and the

b oundary mo duli has a minimum when

p

U

U G e jZ j

N

p

a U ab

z G e g jZ j

N

b

Equation then implies c so r and

U U

ds e dt e dx

which has Ricci curvature

t U

R e U

t

U

R e U U

U

R R e U

these solutions satu Assuming asymptotic atness ie U at spatial innity

rate the BPS b ound

p

jZ j M

G

N

Here wehave dropp ed the b oundary term since and im

U

ply that b oth e and jZ j are monotonously decreasing functions see also equation

p

U

b elow satisfying the estimate e jZ jminfjZ j G jZ j jZ jg

N



The nonextremal metric with c is obtained from this one simply by replacing the factor



 

by sinh c c

BPS states from the D eective action attractors

U

and hence e jZ jwhen Note that this estimate also implies that when

Z the solution is a black hole with horizon at r Indeed from the

form of the metric and direct analysis of equation in the limit

we then get an AdS S near horizon metric

ds dt G jZ j dx

N

G jZ j

N

with horizon area

A G jZ j

N

and hence with thermo dynamic BeckensteinHawking entropy

A

jZ j S

BH

G

N

Cho osing the other sign p ossibility in do es not give an acceptable

U U

solution now e and jZ j are increasing functions satisfying the estimate e jZ j

p

G jZ j for any xed hence anynontrivial solution develops jZ j

N

a singularity at nite distance from the origin and has innite action and energy

Furthermore these solutions would b e gravitationally repulsive They corresp ond

to the unstable solutions falling down the hill along BPS tra jectories in the eective

particle picture Note however that in a nite region of spacetime preventing

to run to innity such solutions might b e acceptable and p ossibly imp ortant

Equations and are called the attractor ow equations This

is b ecause their solutions converge to xed mo duli values at namely to

those values for which jZ j is minimal Indeed from we see

p

d

U ab

G e g jZ j jZ j jZ j

N a

b

d

so the mo duli will owdown the hill till a minimal value of jZ j is reached with

vanishing norm of its gradient This is intuitively clear in the brane picture since

we exp ect the brane to pull the mo duli such that its volume is as small as

p ossible It is of course also precisely what we exp ected from the intuitive particle

picture describ ed ab ove

To describ e the ow in mo duli space it is sometimes convenienttointro duce

p

U

a scale variable e G in terms of which the attractor ow equations

N



at least not for our purp oses in it is discussed in what sense such solutions could still

b e meaningful



a comparable situation is p erhaps the o ccurrence of exp onential tunneling solutions of

Maxwells equations b etween two dielectrics

Chapter CalabiYau compactications of I IB string theory

take the form

d

jZ j

d

a

dz

ab

g ln jZ j

b

d

suggesting an interpretation as a renormalisation group ow in mo duli space

The attractor ow equations can also be obtained from the requirement of

conservation of one half of the sup ersymmetries Solutions and gener

alisations have b een discussed for example in

Before we go on we would like to make the following side remark which

is further not consequential for this thesis Supp ose we restrict the region of

spacetime which we consider to r with very small The ac

tion principle tells us that for any value of the elds on and

we can nd a solution of the equations of motion Supp ose we keep z and U

xed at and deviating a nite amount from the attractor val

ues at the latter p oint Now let Because of the instability of the eec

tive particle system the solution U z of the equations of motion given these

b oundary values will be arbitrarily close to the attractor ow solution U z

attr

for not to o close to More precisely for any xed and nite we have

lim U z U z Also some closer analysis of the action

attr

shows that for black holes the action of the deviating solution approaches that

p

of the attractor solution lim S U z S U z T jZ j G

attr N

essentially b ecause the deviating solution only diers signicantly from the at

U

tractor one close to r but there the factor e is very small damping out all

dierence in action On the other hand the value of the mo duli at r

hence at the horizon of lim z will by construction dier from the attractor

values by a nite amount The limiting solution is discontinuous and therefore not

acceptable as a smo oth classical solution but it indicates the presence of degrees

of freedom at the horizon of which the excitations have zero energy as seen from

any nite radius The solutions deviating from the attractor ow innitesimally

close the horizon would be as imp ortant as the attractor ows themselves in a

Feynman path integral This can b e understo o d as an innite redshift eect The

degenerate excitations on the horizon give the black hole an entropy which by

extensivitywe exp ect to b e prop ortional to the area A of the horizon in agree

ment with the BeckensteinHawking formula Unfortunately when one actually

tries to count the excitations on the horizon in this way the result is divergent

when arbitrarily short wavelengths are allowed A cuto of the order of the Planck

scale is needed



this is not the case for nonblack hole solutions

BPS states from the D eective action attractors

String theory should eectively provide such a cuto in principle but going

into the sub ject of string theory counting of black hole states would lead us

way to far aeld so we will drop the sub ject here

Existence of the BPS state

We can trust the low energy sup ergravity approximation if the curvature stays

suciently small From and the attractor ow equations it follows that

this is the case if and only if

U

G e V z

N

Note that as long as the curvature is everywhere nite we can always make

it arbitrarily small by adding equal charges at r Indeed obviously the

a

solution U z to the attractor ow equations for multiple charge N can

b e obtained from the solution for single charge as

N

N

The curvature then scales corresp ondingly as

R R N

N

N

so we can always make the curvature if nite arbitrarily small such that it satis

es bytakingN large The same argument holds for the derivatives of the

curvature and of all the other elds So in general we can exp ect sup ergravity

to b e reliable in the large N limit which is of course no surprise

Let us see what the attractor ow equations can teach us ab out the existence

of a BP S hyp ermultiplet with given charge and given mo duli at spatial innity

There are three p ossibilities

jZ j has a nonvanishing lo cal minimum say at z z The ow gradient

eld given by the rhs of the attractor ow equations in mo duli space is

shown schematically in g In this case as outlined ab ove we get a

en by The condition black hole solution with near horizon metric giv

for our sup ergravity approximation to be valid translates then to

jZ z j which can always be obtained by taking a sucient amount

of charges Therefore in this case the low energy theory establishes the

existence of a BPS hyp ermultiplet of the given charge



A spherically symmetric solution gives a hyp ermultiplet as we argued earlier The nonex

istence of a static spherically symmetric solution do es however not exclude the existence of

nonspherically symmetric solutions and hence of multiplets built on those solutions

Chapter CalabiYau compactications of I IB string theory

Figure Typical ow gradient eld in mo duli space represented bythe z plane

close to a generic attractor p oint with nite Z The gradientvectors vanish at

cr it

the attractor p oint

We argued earlier that the existence of a BPS state is equivalent to the

existence of a sup ersymmetric representative in the given homology class

so a remarkable connection emerges here between existence of sp ecial La

grangian emb eddings in CalabiYau manifolds and existence of solutions to

the attractor equations More concretely in the case at hand this

connection predicts the existence of a sp ecial Lagrangian representativein

homology classes which have nonzero minimal jZ j From the mathe

matical point of view this is a rather nontrivial conjecture But one can

go even further the degeneracy of the black hole ground state predicted

by the BeckensteinHawking entropy formula grows for large Z according

to as exp jZ j This degeneracy supp osedly is roughly equal

attr

to the dimension of the mo duli space of the wrapp ed branes that is the

mo duli space of sp ecial Lagrangian emb eddings of the given homology class

The mathematical consequences of this statement are enormous as was

realized and worked out by Mo ore in and further in It could even

lead to a solution of Hilb erts twelfth problem We will not go deep er

into this fascinating but very dicult sub ject however

Z has a zero at a regular p oint z z in mo duli space Regularity implies

R

that the holomorphic p erio d Y then must have a zero at z z The

holomorphic equation X z in a regular neighborhood of z actually

denes a complex co dimension one set of zeros of Z in mo duli space Take

BPS states from the D eective action attractors

Figure Flow gradient eld in the transversal Y plane close to a regular lo cus

Y in mo duli space where Z The gradientvectors do not vanishatthe

attractor p oint

Y as the co ordinate transverse to the zero lo cus Close to Y the

attractor owisapproximately given by

U

U k e jY j

U

Y k e YjY j

where k and k are certain p ositive constants See g The approximate

solution for small Y of this system is

arg Y const

jY j k k

U k k jY j

where the k are again p ositive constants whose precise relation to each

i

other and to k and k is not imp ortant here

Weve got a problem here from it follows that we reach Y at

nite and that b eyond this p oint we cannot continue the solution The

ow breaks down a spherically symmetric BPS solution to the equations



Here we are assuming Y do es not have a double zero in the p oint under consideration we

classify suchapoint as singular If it do es the analysis and conclusions will change drastically See item

Chapter CalabiYau compactications of I IB string theory

of motion do es not exist in this case Continuing the ow by just keeping

Y in the interior do es not yield a solution of the equations of motion

Continuing the owbyinverting the sign of the rhs of the attractor ow

equations do es not give an acceptable solution either b ecause it is highly

singular and antigravitating as discussed earlier Note that if the metric

Y Y

comp onent g has nonvanishing gradientatY which is the generic

case the point under consideration will not even b e a stationary p ointof

V so from the eective particle picture wedo not exp ect it to be a true

attractor p ointforany solution BPS or not

On the other hand the curvature and all its derivatives stays everywhere

nite so at least for large charge N we can trust this low energy analysis

We are therefore led to conclude that there do es not exist a BPS hyp er

multiplet with the given charge in this case that is for mo duli at innity

suciently close to the Y lo cus

Fortunately this is exactly what wewould exp ect physically Indeed if such

a BPS hyp ermultiplet would exist wewould have a massless charged hyp er

multiplet in a vacuum with Y This should however give logarithmic

corrections to the dynamics of the U vector multiplet and in particular

there should b e a singular onelo op correction to the mo duli space geometry

as in pro ducing a singularity at Y which would contradict our

initial regularity assumption So this all ts nicely

note that these considerations do not exclude the existence of an Finally

extremal c but nonBPS solution to the equations of motion with the

given charge and mo duli at spatial innity actuallybylowering the ADM

mass of a nonextremal black hole one eventually exp ects to end up with an

extremal one so extremal solutions should generically exist

Z has a zero at a singular or b oundary p oint of mo duli space In this case

the analysis is much more subtle an dep ends strongly on the case at hand

Wegive three examples here following and extending

Example

Consider a one parameter family of CalabiYau manifolds near a large com

plex structure limit Denoting the mo dulus with t and taking the large

complex structure limit to b e at Im t the Kahler p otential and metric

at large Im t for this case can b e taken to b e

K

e Im t



The precise geometrical meaning of this is not imp ortantat this point The reader not

familiar with such terminology can follow the example starting from the given expressions for

Kahler p otential and p erio ds

BPS states from the D eective action attractors

U

Figure Flow gradient eld in the e Im tplane for a cycle with vanishing Z

in the large complex structure limit Im t

g Im t

tt

There are four indep endent p erio ds asymptotically prop ortional to tt t

Combinations of the rst two p erio ds havevanishing central charge in the

limit Im t

q q t

p

Z

Im t

Let us take q for simplicity The more general case yields qualitatively

the same results though some exp onents etc in the solutions are dierent

The attractor ow equations in this limit are then

p

U

U q G e Im t

N

p

U

t iq G e Im t

N

g with large asymptotics

U

e

t i

This solution has a naked singularity the curvature diverges at the origin

of spacetime At a nite radius the sup ergravityapproximation

breaks down So in principle we cannot draw conclusions ab out the existence

Chapter CalabiYau compactications of I IB string theory

in the full string theory of a BPS hyp ermultiplet with the given charge close

to the large complex structure limit However the physically undesirable

app earance of a naked singularity for all values of the parameters leads us

rather to b elieve that suchahyp ermultiplet do es not exist This would also

b e in agreement with physical exp ectations based on the brane picture and

mirror symmetry Indeed a brane wrapp ed around the cycle considered

here corresp onds in the typ e IIA theory on the mirror CalabiYau to a

brane A brane in a background with sup ercharges eg atorus

compactication is in a BPS massive gravitymultiplet states By the

trick discussed at the end of section we conclude that it should also

give rise to a BPS massive gravitymultiplet in our CalabiYau background

sup ercharges This has states with a spin fermionic oscillator

vacuum Therefore we indeed do not exp ect a valid spherically BPS solution

of the b osonic low energy eective action for this charge as this w ould give

ahyp ermultiplet It would be interesting to check whether a nonsingular

nonspherically symmetric multiplet of BPS solution can b e constructed We

have not done this

Example

For our second example we consider a vanishing cycle near a conifold lo cus

in mo duli space Since the internal CalabiYau degenerates at this point

we cannot necessarily trust the sup ergravity approximation However the

results obtained in this case are interesting so we will ignore this p otential

problem and pro ceed For simplicitywe will again consider only one mo d

ulus z which we take to be the p erio d of the vanishing cycle Then the

Kahler p otential and metric for z can b e taken to b e

K

e k jz j ln jz j k Re z

g ln jz j

z z

k

where k and k are p ositive constants This is similar to the geometry

near the singularities u in the Seib ergWitten plane discussed in the

previous chapter see also g The central charge of N times the

vanishing cycle close to z is

N

z Z

k



Actually it is not necessary to go to the mirror for this argument the brane considered here

is a torus which as a wrapp ed brane in a torus compactication would also give the N

BPS massivegravitymultiplet in four dimensions However the trick part of the argumentis

p erhaps clearer in the brane picture and furthermore it is nice to see how the mirror symmetry

conjecture works out ne again

BPS states from the D eective action attractors

Figure The p otential V z z for the conifold cycle as a function of z near its

minimum at the conifold p oint z The gradientvectors vanish at the attractor

p oint

The form of the corresp onding p otential V z znearz isshown in g

The ow gradient eld in the z plane is given in g The attractor

ow equations in this limit are

p

U

U G Ne jz j

N

k

p

z

U

z k G Ne

N

jz j ln jz j

with solution approximately for z given by

arg z const

p

jz j ln jz j jz j ln jz j k G N

N

jz j ln jz j jz j ln jz j U

k

Again the ow terminates at nite distance from the origin namely at

p

r jz j ln jz j

k G N

N

However this time the ow can b e smo othly continued b eyond this p oint



actually only once continuously dierentiable but this can b e interpreted as an artifact of

the two derivativelow energy approximation we are using

Chapter CalabiYau compactications of I IB string theory

Figure Flow gradient eld for the vanishing conifold cycle in the transversal

z plane close to the conifold lo cus z in mo duli space

just by taking z and U U for g Indeed

U z when as can b e seen immediately from the attractor ow

equations Also the curvature is easily seen to stay everywhere nite in

fact arbitrarily small for large N or small z Only the higher derivatives of

curvature and mo dulus diverge close to So strictly sp eaking the low en

ergy approximation breaks down there Nevertheless we b elieve that higher

derivative corrections will not alter qualitatively the conclusionsw e exp ect

those merely to smo oth out the solution eg by replacing the constant

inner solution by an almostconstant exp onentially decreasing one In

any case as long as we sticktothelow energy approximation this solution

is p erfectly well b ehaved and physically acceptable Therefore we are here

rather led to conclude that such a BPS hyp ermultiplet indeed exists Again

this is in agreement with physical exp ectations starting with the work of

Strominger one has argued from various p oints of view that there should

indeed b e a BPS hyp ermultiplet near the conifold with charge corresp onding

to the vanishing cycle It is this hyp ermultiplet which is held resp onsible



Morally sp eaking this is analogous to the fact that we do not exp ect the motion of a falling

piece of inelastic mud to the surface of the earth to b e inuenced muchby higher derivative

corrections though its idealized motion denitely has divergentderivative of acceleration at the

attractor p oint the surface of the earth



The issue whether the N states can b e b ound ie of single particle typ e cannot directly

b e addressed here Of course if there exist an N state there will exist an unbound N particle

state for any N see also next section but to check for bound states one should nd out more

BPS states from the D eective action attractors

Figure The space dep endence of the mo dulus z for the conifold attractor

Inside a critical radius r r the mo dulus is constant and equal to its attractor

value z

for the app earance of the conifold singularity in mo duli space hence in the

low energy dynamics in the low energy action as given earlier this hyp er

multiplet is as all other massive states integrated out However precisely

at the conifold p oint this hyp ermultiplet b ecomes massless and we should no

longer integrate it out if we still do so the one lo op correction of this hy

p ermultiplet pro duces a logarithmic singularity in the low energy eective

action of the other massless elds

Note that an alternative solution of the equations of motion is obtained by

inverting the signs in the rhs of the attractor ow equations for

However again as discussed earlier such a solution is very unphysical it is

antigravitating has singular curvature at nite radius and innite energy

in stark contrast with the smo oth nite energy solution found by stopping

the ow

Conifold transitions are not p ossible either in this case since this requires

more than one cycle vanishing at the same time

The unphysical inverted ow solution has app eared a numb er of times in

various guises in the literature eg causing quite some confusion

The nite energy solution was missed b ecause of the fact that the earlier

ab out the dynamics eg in a low energy soliton mo duli space approximation It has b een argued

from the precise form of the lo op corrections to the mo duli space geometry that N BPS b ound states do not exist in this case

Chapter CalabiYau compactications of I IB string theory

analysis was done in a formalism and with co ordinates on mo duli space which

precisely b ecome ill dened at the conifold singularity of mo duli space To

get an idea what go es wrong consider the nonholomorphic co ordinates

i

x Im e z

i

y Im e iz ln z

where argz These are roughly the natural co ordinates arising

in the formalism of as the imaginary parts of a symplectic p erio d basis

in a particular Kahler gauge The advantage of these co ordinates is that

they are harmonic functions on spacetime for BPS solutions indeed using

f f and we see that trivially x y One would

then be tempted to conclude that the solutions are either constant or r

divergent This would corresp ond to an in verted ow solution For our

solution withaow terminating at r wehave x constand

y but the latter only for r For r wehave y

r

Our solution is smo oth continuously dierentiable when expressed in go o d

co ordinates on mo duli space but not when expressed in the y co ordinate

simply b ecause y is not a go o d co ordinate around the conifold p oint y

The prop er treatment of the exterior conifold attractor ow and in par

ticular the fact that the ow terminates app eared for the rst time in

In we observed that the ow could be smo othly continued to the in

terior and the physical acceptability of this solution and its generalization

to weakly gravitating eectiveYangMills BPS states was emphasized Re

cently analogous sp ontaneous cuto solutions have emerged in the context

of DiracBornInfeld brane dynamics

Prop erties of this and similar solutions will b e discussed in detail in chapter

Example

As a third example we consider a zero of order of Z at a p oint with

Z z and g Again starting suciently nondegenerate metric say

z z

close to Z will give only small variation of the metric function U while

a short calculation analogous to the ab ove examples gives for the central

charge itself if





jZ j jZ j

and if

jZ j jZ je

The phase is again constant For wendaowwhich stops at

nite and cannot b e continued These cases also corresp ond to divergent

BPS states from the D eective action attractors

maxima of V in stead of the required minima For the ow stops

at nite but can b e continued as Z to the interior region For

the ow runs smo othly all the wayto At least for suciently small

Z none of these solutions is a black hole

It could b e interesting to compare these solutions with the p erturbative het

erotic BPS string states which should corresp ond to it by heterotic typ e I I duality

This would extend the results in

Multicenter case

The previous discussion is readily extended to the extremal multicenter case with

equal charges byintro ducing an eective radial co ordinate

N

X

i

N

i

where i runs over the N dierent centers and is dened as in the previous

i

discussion relative to the ith center Surfaces of equal can be considered as

equip otential surfaces for the multisource conguration g The ansatz

for the metric is the extremal

U U

ds e dt e dx

The electromagnetic eld is given by sup erp osition and has exactly the same form

P

U 

e

p

d dt The scalar elds are supp osed as with N

i

i

to b e functions of only

Since the complete setup is formally the same as for the spherically symmet

ric case so are the attractor ow equations Therefore everything said ab out

the extremal spherically symmetric case applies to the extremal equal charge

multicenter case as well

It is easy to see that the total force b etween equal static BPS particles indeed

vanishes The magnitude of the repulsive static electromagnetic force is from

given by

ab

jZ j g jZ j jZ j

a

b

r

 

Multicenter solutions with charges corresp onding to dierent elements of H X Z and

in particular with mutually nonlo cal charges nonzero intersection pro duct are muchmore

dicult to study and their existence is not clear

Chapter CalabiYau compactications of I IB string theory

Figure Some surfaces of equal in the center case

The rst term is comp ensated by gravity the second one by scalar exchange

One could also contemplate replacing the function with an arbitrary har

monic function However in order to get nonsingular solutions of the attractor

ow equations this function should b e b ounded from b elow Therefore all consis

tent harmonic functions are equivalent to a distribution of equal charge sources

In particular single center harmonic functions containing spherical harmonics do

not give consistent solutions

Attractor technology

Here we would like to give some formulas which can be useful for solving the

attractor ow equations The results in this section provide

among other things an intrinsic Kahler gauge indep endent formulation of the

formalism develop ed in

We will put the Newton constant G throughout this section

N

i

Using jZ j e D Z with arg Z and D K we can rewrite

i i i i i

the attractor ow equations as

U i

Z U e e

i a U ab

e z e g D Z

b

From this a straightforward but slightly tedious calculation shows that the central

charge Z of a charge satises

d

U i U ab

e e Z e Z Z g D Z D Z

a

b d

Validityofthe Dbrane and eld theory pictures

and

d

U i ab

e e Z Z Z g D Z D Z

a

b

d

Taking the real part of and the imaginary part of and using

gives the following nice geometric formulae

d

U i U

Re e e Z e

d

d

U i

Im e e Z

d

Since the intersection pro duct is a top ological invariant the second of these equa

tions can readily b e integrated

i U U i

Im e Z e f Im e e Z g

This is a very p owerful identity since by cho osing for a basis of cycles

it gives provided U is known the solution of the attractor equations in terms of

i

co ordinates Im e Z on mo duli space where some care has to be taken to

check that the ow do es not pass through a singularity where it could stop as in

some of the examples ab ove By cho osing a p osition dep endent normalisation

of such that Z is real and K U this is a choice of Kahler gauge and

K

writing Z e X b ecomes for and N

U

Im X Q N e Im X

where Q is the intersection matrix This expression can b e compared

with

Note that implies at a nonsingular attractor p oint wehave

Im ZZ Q N

This equation in principle determines the p osition of the regular attractor p oints

in mo duli space It is therefore often called the attractor equation

Validity of the Dbrane and eld theory pic

tures

To end this chapter we would like to make some comments on the domain of

validity of the eective eld theory sup ergravity and the Dbrane pictures of



Note that this gauge choice is singular at a black hole horizon where U



At a conifold singularity the gradient term in do es not necessarily vanish

Chapter CalabiYau compactications of I IB string theory

the dynamics of nonp erturbative states We will also briey touch the Maldacena

conjecture and sp eculate on a p ossible extension

The Dbrane picture in the sense of Polchinskis prescription spacetime de

fects where strings can end is in general exp ected to b e a go o d description of the

physics when the string coupling constant g is suciently small such that string

p erturbation theory is adequate In case wehave a stack of a large number N of

Dbranes wemust havethatgN is small since the sum over the dierent branes

makes gN rather than g the eective coupling in brane amplitudes as in the

large N t Ho oft expansion of eld theories

The low energy eld theory picture on the other hand is exp ected to be a

go o d description for suciently slowly varying elds which is in general exp ected

to b e the case when the system under consideration is macroso copic ie at large

N

So one would exp ect the regime where the Dbrane picture is valid and the

regime where the eld theory picture is valid to b e more or less complementary in

general

These statements are of course rather vague and imprecise In particular the

meaning of large and small is not really clear Let us therefore try to make this

more precise in the case of typ e I IB CalabiYau compactications

Massive states limiting the eld theory picture

Consider the eective action In general in order for a low energy eective

eld theory of the massless elds to makes sense all relevant energy scales haveto

be much smaller than the lightest massive states in the full theory whichcoupleto

the massless elds Higher derivative terms in the eective action are suppressed

byinverse p owers of these masses

Now which of those light massive states can we p ossibly have in typ e IIB

string theory compactied on a CalabiYau manifold Candidates are fundamen

tal F strings Dstrings wrapp ed Dbranes either completely around a cycle

or partially around a cycle pro ducing an eective string in four dimensions

wrapp ed NS and Dbranes partially around a cycle wrapp ed or

Dbranes partially around the full CalabiYau and of course also simply excita

tions of the D massless particles in the internal space X KaluzaKlein states

Actually it is not entirely clear whether we should include partially wrapp ed brane

states leading to strings in four dimensions This is in the spirit of the selfdual



One can also consider more generally p q strings and Dbranes and other b ound states

but it is sucient to consider the extremes ie and to single out the lightest ones

which of those two will b e the lightest dep ends on whether g is larger or smaller than

Validityofthe Dbrane and eld theory pictures

strings arising in I IB K compactications from partially wrapp ed branes

used in to count black hole microstates We dont knowofany reason not

to include such states here as well Mirror symmetry also supp orts this p ointof

view Indeed typ e I IB Dstrings living in four dimensional spacetime are mapp ed

to certain partially wrapp ed Dbranes in the mirror typ e I IA theory It is then

natural to consider arbitrary partially wrapp ed typ e I IA Dbranes which are in

turn mapp ed back to certain wrapp ed D D and Dbranes in typ e IIB On

the other hand it is not clear that we can simply treat those ob jects as eective

strings and deduce features of their excitation sp ectrum accordingly Nevertheless

we will do so in the following and see what comes out

Denote the IIB string coupling constant e with g The mass of the low

p

est excited Fstring states is in the string frame of the order of where

hence in the Einstein frame in whichwehave b een work

The Einstein mass of an excited D string is M ing M g

SD SF

g The mass of the lightest excited in the D eective string sense

partially wrapp ed branes is M k V where V is the volume

SD X X

of the CalabiYau X and k is a dimensionless factor dep ending on the Kahler

mo duli of X That of a completely wrapp ed Dbrane is M cV

D X

where c is the mo dulus of the central charge of the lightest cycle dep ending

on the complex structure mo duli For a partially wrapp ed Dbrane we have

M k g V where k again is a dimensionless factor dep end

SD X

ing on the Kahler structure mo duli for its NS analogon we have M

SN S

and for a partially wrapp ed Dbrane M k g V

X

SD

g V resp M g V Finally the mass of atypical

X X

SD

low KaluzaKlein excitation in X is M V

KK

X

Expressed in units where the four dimensional Planck length V

X

equals and intro ducing the dimensionless ratios

V V

X X

V

X

g

M

SF

which can b e considered as the radius of X in D Planck units resp Fstring

Chapter CalabiYau compactications of I IB string theory

units this b ecomes

M g

KK

M g g

SF

M g g

SD

M k k g

SD

M k g k g

SD

M k g k

SN S

M g g

SD

M g g

SD

M c c

D

In D Planck units all this has to be multiplied by g

Note the invariance of the sp ectrum under g g which is of course just typ e

I IB Sduality We can therefore assume g

Asawarmup example we see for instance that for xed energies in D or

D Planck units the four dimensional eective eld theory description breaks

down when or This is evident since this limit corresp onds to

decompactication A p erhaps more surprising feature is that for xed CalabiYau

mo duli ie xed k k c and or and again xed energies in Planck units

the four dimensional eective eld theory description always breaks down in the

weak string coupling limit g simply b ecause eg the fundamental string mass

b ecomes much lighter than the D Planck mass

The limit g with xed and xed X mo duli

When g with and the other compactication mo duli xed corresp onding

to vanishing string coupling with a xed internal space size compared to the string

scale the I IB p erturbative string states with xed KK and oscillator excitatition

numb er masses prop ortional M and M havevanishing mass compared to

KK SF

the nonp erturbative wrapp ed Dbrane states The bulk dynamics of the elds at

energies nite with resp ect to the string scale is in this case p erfectly well decrib ed

by I IB string p erturbation theory At energies far b elow the lowest massive string

state M andthelowest KaluzaKlein state M the two derivative low

SF KK

energy D eld theory description of the bulk elds is moreover perfectly good

as well



Actuallywehave assumed for simplicity that the RR scalar C in deriving our mass

formulas A nonzero value would make the mass formulas and S duality a bit more complicated

without altering qualitatively the conclusions however



where the action is since we are in the weak string coupling regime the D string tree

level eective action This can dier from the dimensionally reduced D action byworldsheet

Validityofthe Dbrane and eld theory pictures

The description of the dynamics of the nonp erturbative states on the other

hand is more subtle In the limit under consideration the Compton wavelengths

of the nonp erturbative ob jects b ecome vanishingly small compared to the string

scale M so they are seen as lo calized ob jects from the p erturbative string p oint

SF

of view Moreover when g with xed Dbrane charges since the string scale

b ecomes much larger than any other relevant distance scale in the D theory

interactions b etween dierent Dbranes and backreaction of Dbranes on the bulk

elds vanish at any in string units xed distance from the branes at least if this

interaction and backreaction fall o with distance Therefore one can set up

a go o d p erturbative description of the nonp erturbative states as Dbranes repre

sented as spacetime defects on which strings can end simply sup erp osed on the

original background without backreaction Interactions b etween the branes and

backreaction on the bulk elds are then taken into account as p erturbative cor

rections Subsequent terms in the p erturbative expansion with increasing number

of string b oundary comp onents on the branes b ecome smaller with p owers of gN

when and p owers of gN when where N is the order of magnitude

of the brane charges under consideration The factor N comes from summing

over the dierent branes on which the strings can end The factor can be

understo o d from the fact that when is small the Dirichlet b oundary conditions

eliminate for every string b oundary comp onent translational degrees of freedom

in X This factor is also the reason why the four dimensional gravitational back

reaction of a wrapp ed brane is indep endentof though its mass compared to

the string scale M grows as when is increased

SF

The stringy typ e I IB p erturbative description of these states is therefore

valid as long as

gN for gN for

In particular this implies that in this regime the dynamics of N coincident

branes totally wrapp ed around a certain sup ersymmetric cycle at energies well

b elow M g will be given by a certain twisted sup ersymmetric

SF

U N YangMills theory on R For generic cycles the KK mo des of the elds

on the brane will have energies of the order of the KK mo des of the bulk elds

in X ie M M so for energies b elow this scale as well automatic if

KK SF

is of order or less the low energy dynamics will actually b e this Y angMills



k



corresp onding to eucledian string world sheets wrapping non instanton corrections e

trivial cycles in X Such corrections only aect the hyp ermultiplet part of the two derivative

low energy eective action since the instanton factors involvehyp ermultiplet mo duli and those

are by sup ersymmetry forbidden to couple to the vectors at the two derivativelevel



This is not the case if there are conning interactions present



When is large and the dierent endp oints are close to each other those degrees of freedom

are eliminated automatically by the tension of the string

Chapter CalabiYau compactications of I IB string theory

theory dimensionally reduced to the worldline of the corresp onsing D particle

that is a matrix mo del The Dbrane worldvolume YangMills coupling constant

is g g so the coupling constant of the matrix mo del is using

p

g V g gM

D

SF

jZ j

where Z is the central charge of The eective p erturbative expansion parameter

for N coincident branes at energy scale E will b e g NE If we stay in the low

energy EM regime the dynamics of the branes will keep on b eing describ ed

SF

bythe U N matrix mo del even when N is increased such that is no longer

valid though the mo del is eectively strongly coupled now and p erturbation theory

can no longer b e used

What ab out the D eective eld theory description of these states Consider

again the case of N coincident totally wrapp ed BPS branes As wehave seen in

the sections on attractors the spatial gradient of the dimensionless elds and

hence the energy scale E of the corresp onding eective eld theory solution

scales as N More precisely as the total eld energy M of such a state is

D

generically of order in D Planck units we have E N As discussed

ab ove in the limit under consideration validity of the D eld theory picture

requires E to b e much less than M and M that is

SF KK

gN for gN for

Comparison with learns that for the D eld theory and string p er

turbative regimes are precisely complementary For there is an additional

gap this is the regime where D eld theory is the go o d description Note that

for the dynamics of N coincident branes at energies far b elow M and M

SF KK

alid descriptions if is satised and g D eective eld wehavetwov

theory and a strongly coupled U N matrix mo del As the relevant energy scale

of this regime of simultaneous validity go es to zero with g compared to the D

Planck scale the relevant eective eld excitations in the generic c case are

forced to vanish everywhere except innitesimally close to the horizon where the

diverging red shift factor allows nite eld excitations with arbitrarily low ener

gies As the geometry is AdS S there wethus arrive in this limit at a duality

between low energy eld theory on AdS S and a strongly coupled U N matrix

mo del Similar considerations hold in higher dimensions This is the celebrated

Maldacena conjecture



This reduction can b e nontrivial as the emb edding geometry of the brane can induce

eld twists and additional zeromo des



Actually precisely in the AdS case this duality turns out to b e much more subtle than in 

Validityofthe Dbrane and eld theory pictures

The c case

In this thesis we will be mainly interested in BPS states pro duced by totally

wrapp ed branes with mass M very much smaller than the D Planckmass

D

that is the c case We have already seen the example of a brane wrapp ed

around a vanishing conifold cycle in section In chapter we will study

such states in detail Gravity decouples in this limit and the states are rigid eld

theory states not black holes

If g and the remaining Kahler mo duli are kept xed while sending c

the lightest states in the sp ectrum are the totally wrapp ed branes of mass c

in D Planck units Typ e I IB string p erturbation theory has clearly broken down

at the energy scale of these states however via heterotictyp e I I duality heterotic

p erturbation theory can be used at energies suciently ab ove this scale as the

wrapp ed brane then corresp ond to weakly coupled p erturbative string states in

the heterotic picture this will be discussed in more detail in chapter The

eld theory picture of N coincident BPS particles on the other hand is valid when

the energy scale of the eld variations is muchlower than the mass c which

amounts simply to N when only light charges M c are considered

and to cN when generic charges M are involved

If one wants to keep typ e I IB p erturbation theory valid in a certain energy

range as well one should take g down to zero together with c in suchway that

moreover gc Then p erturbation theory is valid for energies nite with

resp ect to the string scale Note that this energy range necessarily go es to zero

with resp ect to the mass of the brane states Taking for simplicity

the condition for p erturbation theory to b e valid in the presence of N coincident

wrapp ed Dbranes is now

gN

while the condition for the eld theory picture to b e valid b ecomes

g

N

c

In their resp ective regimes the p erturbative description yields a U N matrix

mo del for the dynamics while the eld theory description can b e truncated to two

derivatives provided the energy scale E is far b elow the lightest massive states

ie

g g

E M

D

c

higher dimensions This is related to the fact that unlike their noncompact higher dimen

sional generalisations the D BPS particles have also arbitrarily low energy mo des corresp onding

to slow translational motion away from each other

Chapter CalabiYau compactications of I IB string theory

Note that and can easily be satised simultaneously At energy

scales given by the dynamics of rigid QFT states in the large N limit can

thus b e describ ed byaweakly coupled matrix mo del This seems to b e a new kind

of Maldacenalike corresp ondence giving a holographic description of a rigid quan

tum eld theory at least at low energies Interestingly the AtiyahDonaldson

DrinfeldHitchinManinNahmNaka jima construction of the mo duli space metric

of N nonab elian nonsup ersymmetric SU YangMills monop oles is based

precisely on a purely mathematical corresp ondence with a certain b osonic

U N matrix system which is of the form one would exp ect for the corresp ondence

we prop ose here We b elieve this deserves further study

Other limits

The limits considered in the previous discussion were adapted to typ e IIB F

string p erturbation theory most naturally expressed in terms of the radius of X

compared to the Fstring scale From one sees that S duality is manifest

when all masses are expressed in terms of the radius of X expressed in D

Einstein frame Planck units A natural limit to consider in these co ordinates is

g with xed Assuming partially wrapp ed Dbranes b ehaving eectively

like strings in four dimensions indeed exist shows that in this limit typ e

IIB string p erturbation theory breaks down since the lightest massive states in

the sp ectrum are now partially wrapp ed branes rather than strings Low

energy eective eld theory will be valid at energies far b elow this mass scale

g However even at zero energy the hyp ermultiplet action will have

instanton corrections which unlike in the xed case do not vanish when g

euclidean NS D D D F and D branes and their p q combinations

can wrap X a cycle a cycle or a point Their contributions are weighted

S

cl

factors roughly with the following e

g

D e

  

k k g

 

F e e

  

k g k g



D e e

 

k g k



D e e

  

g g

D e e

   

g g

NS e e

As it should in the g xed limit where string p erturbation theory is

supp osed to b e valid the nonp erturbative instanton corrections are exp onentially



As the nonrenormalization theorem for the two derivativelow energy ective action of the

vector multiplet action do es not rely on p erturbation theory the exactness of the classical vector

multiplet action continues to hold in this regime as well

Validityofthe Dbrane and eld theory pictures

small The only surviving instanton contributions are those from euclidean F

strings wrapp ed around nontrivial cycles which are included in p erturbation

theory They b ecome arbitrarily small in the large radius limit When

on the other hand is xed while g instanton corrections from eucledian

wrapp ed Fstrings Dbranes and p ossibly for small Dbranes are imp ortant

It is not clear if there exists a weakly coupled dual formulation of the theory

in this regime

Another interesting limit is the one arising from the Mtheory matrix mo del

mirrored to typ e I IB whichamounts to sending g and to zero with g

The relevant energy scale here is the energy of states with nite eleven dimensional

light cone energy in eleven dimensional Planck units E g M The

DK P S F

eld theory breaks down at this scale in the limit under consideration due to eg

partially wrapp ed Dbranes Indeed physics is supp osedly not describ ed bya

eld theory in this regime but bya U N matrix mo del describing the dynamics of

N D particles obtained by wrapping a brane around a certain sup ersymmetric

T cycle in X

Chapter CalabiYau compactications of I IB string theory

Chapter

Quantum YangMills

gravity from IIB strings

Till ab out typ e II string theories were considered to be incapable of

describing phenomenologically interesting nonab elian gauge theories since their

bulk p erturbative sp ectrum can only carry a very limitited range of gauge groups

to o limited for the standard mo del as it turns out For example as wehave

seen in detail in chapter typ e I IB compactied on a CalabiYau manifold has



h

ab elian gauge theory sector Heterotic and typ e I theories typically only a U

on the other hand supp ort big nonab elian gauge groups and therefore those were

considered to b e the only p ossibly phenomenologically relevant theories

This turned out to b e completely wrong With the advent of string dualities

and Dbranes probing nonp erturbative asp ects of string theory it b ecame clear

that actuallytyp e I I theories even do b etter they do describ e nonab elian gauge

theories and often even their exact low energy quantum dynamics The rst

indications of this remarkable fact emerged from Heterotic typ e II dual

ity evidence accumulated suggesting that nonp erturbatively completed

compactied on T K is equivalent to certain I IA and

hence IIB by mirror symmetry CalabiYau compactications Since the former

theory has nonab elian vectors so should the latter The obvious question is then

where the nonab elian vectors are hiding in typ e II theory The answer turned

out to b e b eautifully simple those are precisely the states obtained by wrapping

Dbranes ab out nontrivial cycles in the compactication manifold Once this

was realised a magical window was op ened to the derivation of quantum eld

theory results from string theory For example this insight can b e used to obtain

Chapter Quantum YangMills gravity from I IB strings

Figure Asketch of a CalabiYau manifold which at low energies in a compact

ication of typ e I I string theorygives rise to a nonab elian gauge theory weakly

coupled to gravity Branes wrapp ed around a generic cycle will havetypically

Planck scale masses However after suitable tuning of the mo duli some cycles

can b ecome muchlighter than the Planck mass

the exact quantum twoderivativelow energy eective action for a plethora of four

dimensional N eld theories a go o d review can be found

in Indeed by tuning the compactication mo duli such that the masses of

anumb er of the branes corresp onding to nonab elian vectors b ecomes very small

compared to the Planck mass g one exp ects their low energy dynamics to

be governed byanN nonab elian YangMills theory p ossibly with additional

matter

r

Nowwe only get an eective action for the massless unbroken U vectors

from string theory but at energies well b elo w the mass of the wrapp ed branes this

is sucient On the other hand as wehave seen in chapter the straightforwardly

obtained D two derivativelow energy eective action of typ e I IB string theory

compactied on a CalabiYau manifold do es not receiveany quantum corrections

So if we manage to identify the nonab elian quantum eld theory governing the

dynamics of the light branes wehave obtained the exact quantum twoderivative

low energy eective action for the massless elds in this theory

The price wehavetopay for exact quantum results is the restriction to low

energies Indeed as discussed in chapter p erturbative string theory and hence

the eective actions extracted from it breaks down at energies of states not in

the p erturbative sp ectrum Usually these states have masses of the order of the

Planck mass but precisely in the case at hand they are much lower since the

nonp erturbative wrapp ed Dbrane states havetoprovide the gauge particles So

there is really no hop e to extend these exact results in the p erturbativetyp e II

picture to energies higher than the lightest gauge particle masses not even in

principle However this is precisely where the p erturbative heterotic picture

takes over

In many cases the pro cedure outlined ab ove reduces the lo cal sp ecial geom

etry of the CalabiYau mo duli space to the rigid sp ecial geometry of the mo d

uli space of a certain class of Riemann surfaces repro ducing and extending the

Seib ergWitten solution of N quantum YangMills theory Furthermore many

features of quantum eld theory have a b eautiful geometrical interpretation in this

framew ork and this provides quite elegant solutions to problems whichwould b e

hard to tackle with ordinary eld theory techniques like for example the existence

and stability of BPS states or connement

Note that such a reduction from lo cal to rigid is by our discussion in chapter

necessary if wewant to extract rigid quantum eld theory results from string

theory Decoupling gravity is tantamount to going to a rigid limit of sp ecial

geometry This limit will therefore playacentral role in this chapter

A very large class of N d quantum eld theories and even more

exotic theories can be engineered and solved geometrically in this way The

usual pro cedure is to nd a lo cal IIA mo del which in the rigid limit

pro duces the eld theory to b e solvedto map this I IA theory to an equiv alentIIB

theory using lo cal mirror symmetry and nally to solve this IIB theory exactly

in the low energy limit using classical geometry One argues that the restriction

to lo cal mo dels and lo cal mirror symmetry where lo cal means that one only

considers a certain small region of the CalabiYau manifold is allowed roughly

b ecause the relevant light brane degrees of freedom are all lo calized well inside

that region The drawback of such strictly lo cal considerations is that one cannot

see directly how the rigid limit is obtained in the low energy string theoryand

more imp ortant that one lo oses all information ab out the coupling of the gauge

theory to gravity and the rest of the ambient string theory In the literature the

fo cus has consequently been almost entirely on the decoupled rigid eld theory

asp ects of this construction

We will directly work in the typ e I IB theory without assuming a priori that

we can restrict ourselves to lo cal considerations The reduction of lo cal to rigid

sp ecial geometry will be demonstrated explicitly This will allow us to derive

the lowest order coupling of the eective quantum eld theory to gravity and the

other remaining string theory degrees of freedom which as wewill sho w has a

universal form Some interesting physics will b e derived from this as well likethe

gravitational backreaction of quantum uctuations of the gauge theory and the dynamics of the dynamical dynamically generated scale

Chapter Quantum YangMills gravity from I IB strings

Apart from the geometric engineering strategy discussed ab ove there are

several other metho ds to extract nonp erturbative quantum eld theory results

from string theory Unlike the bulk spacetime approach of geometrical

engineering these alternatives all have as starting p oint the realization of gauge

theories on multiple Dbrane world volumes In one considers various em

b eddings of large branes into p ossibly compactied ten dimensional spacetime

where time together with three noncompact spatial directions of the brane form

the four dimensional spacetime on which the quantum eld theory under study is

dened By making use of various prop erties of M theory andor brane dynamics

one derives the low energy quantum gauge theory dynamics in a fairly straight

forward and elegantway In an exact equivalence is conjectured b etween four

dimensional N large N SU N YangMills theory and typ e I IB string theory

on AdS S with N units of F ux The string p erturbative expansion cor

resp onds to a certain N expansion of the YangMills theory This remarkable

corresp ondence makes it even p ossible to go b eyond the low energy approximation

of the quantum YangMills theory

Though these alternative approaches are often more powerful or at least

simpler than the approachwe will follow they are not very useful for our purp oses

since nothing can b e learned from them ab out the interplaybetween YangMills

theory and gravityand other stringy degrees of freedom This is b ecause brane

worldvolume theories never contain gravity From the p oint of view of physics this

is the main advantage of our approach and approaches based on bulk spacetime

compactications in general

This chapter is organized as follows We rst discuss the geometric structures

needed to get nonab elian YangMills theories in four dimensions We will argue

that CalabiYau manifolds which are lo cally of near ALE bration form give rise

at low energies to a N YangMills theory weakly coupled to gravity Next we

consider this weak gravity limit and derive its universal form Finallywe comment

on the interpretation of the low energy eective theory thus obtained including

the eects of gravity and compare with the dual heterotic picture To illustrate

and clarify the general results we conclude this chapter with the detailed analysis

of an explicit example This example inspired the general arguments and if the

latter would get to o obscure to the reader at a certain p oint it could b e helpful

to have a lo ok at the example rst

In the next chapter we will use these results to discuss BPS states and in

particular the attractor equations in the weak gravity limit

Near ALE brations and the light BPS sp ectrum

Figure The compactication T K A cycle can be constructed as a

pro duct of a K cycle and a torus cycle

Near ALE brations and the light BPS sp ec

trum

Vector branes

From the previous discussion and chapter it follows that we have to lo ok for

CalabiYau manifolds with cycles which by wrapping branes around them

give rise to light massiveBPSvector multiplets in four dimensions To nd out

what kind of cycles pro duces vector multiplets we are going to use the trick

hapter section Thus let us rst try to nd out which explained in c

cycles pro duce a vector multiplet in an N compactication To get N in

four dimensions by compactication of typ e I IB string theory in the way describ ed

in chapter wehave to take the compact manifold to b e T K g Since

K has no nontrivial odd dimensional cycles all nontrivial cycles in T K

have to be the pro duct of a torus cycle and a K cycle Denoting the

emb edding of in the K by f the conditions and to have a





A massive BPS N vector multiplet has states with spin con tent 

Chapter Quantum YangMills gravity from I IB strings

supersymmetric cycle b ecome

f J

K

arg f const

K

m n

Here and J ig dz dz are the the holomorphic form resp Kahler

K K mn

form on K By a suitable choice of the phase of we can take To

K

get b etter insight in the meaning of these conditions let us recall some basic

facts ab out K geometry for a review see A K manifold dened as the

up to dieomorphisms unique two dimensional CalabiYau manifold has actually

indep endent anticommuting covariant constant complex structures J J J

Denote the asso ciated forms by J that is J g J Achoice of complex

i i i

structure sp ecies the Kahler form J and the holomorphic form in terms

K K

of these for example

J J J iJ

K K

or any other choice corresp onding to a rotation of the J Now let us go backto

i

our original K There we already assumed a certain choice of complex structure

but we can also consider another rotated one dened by sp ecifying for

J Re Im iJ

K K K

K K

In the primed complex structure the condition for a sup ersymmetric cycle thus

b ecomes simply

f

K

If we cho ose complex co ordinates z z compatible with the primed complex

structure and describ e lo cally by writing z as a function of z and z



z



immediately gives So the condition for a sup ersymmetric cycle simply re



z

duces to holomorphicity of the emb edding of with resp ect to the primed complex

structure

Now in contrast to sp ecial Lagrangian emb eddings of cycles in CalabiYau

folds holomorphic emb eddings of cycles in K are extensively studied and

very well understo o d in the mathematics literature A mathematical result

whichisvery imp ortant for us is that a K homology class with selfintersection

g g has a g complex parameter family of holomorphic representatives

of genus g The area of these curves is minimal in the given homology class and

R



Note that for any Kahler form J we have by a short calculation J J V

K 

K 

Therefore implies for a given K volume V achoice of normalisation mo dulo phase

K 

R

of V

K  K 



This implies that the brane mo duli space has complex dimension g g from the emb edding

deformations and g from the at U brane worldvolume gauge eld deformations Such brane

ground state degeneracies can b e shown to repro duce exactly the BekensteinHawking entropy

formula for black holes

Near ALE brations and the light BPS sp ectrum

given by

Z Z

A J j j

K

K

In particular for cycles with selfintersection wehavea unique holomorphic

representative which is a minimal area sphere One can also show that the volume

of cycles with g is actually b ounded from b elow on K mo duli space with

p

lower b ound prop ortional to g Since we are lo oking for branes whichcan

be made aribtrary light this leaves us with the g cases as candidates for

branes representing gauge particles

Do es such a cycle give rise to a N vector multiplet in four dimensions

The easiest way to see this is to p erform a T dualtityonthe T in the direction

of This maps I IB to I IA and our brane to the brane given by but leaves

the physics and in particular the typeofmultiplet which our brane gives in four

dimensions invariant Let us rst consider the theory as I IA compactied to six

dimensions on K This has N sup ersymmetry in six dimensions Since

for a xed p osition in the D spacetime has a unique sup ersymmetric repre

sentative the brane has a unique zero mo de fermionic oscillator ground state

where the fermionic oscillator creation and annihilation op erators are comp osed

of the sup ersymmetries broken by the brane cf chapter Therefore this ground

state or vacuum must b e in the trivial representation of the SO SO

little group Consequently the six dimensional N sup ersymmetry multi

plet of states built on this ground state is a D massiveBPSvector

multiplet When we compactify further on T this D vector multiplet trivially

reduces to a D N massive BPS vector multiplet which is precisely what

wewere lo oking for

All in all we conclude that the light four dimensional N vector multiplets

are pro duced in typ e I IB compactied on T K by branes wrapp ed around

the pro duct of a T cycle and a K cycle with selfintersection whichisa

minimal area sphere

Now what ab out N compactications How can we get our lightvector

multiplets there We follow the reasoning outlined in chapter section

We transform our N T K compactication to an N compactication

bychanging the geometry away from a set of N vector branes This

i

can b e done for example by making the mo duli of K p osition dep endentonT

replacing the direct pro duct by a K bration We can furthermore change the

basis B of the bration away from to eg a sphere or a higher genus Riemann

surface or change the K geometry away from the to another fold K g

i



With a more general normalisation of the right hand side is replaced by

R R





j V j

K 

K 

Chapter Quantum YangMills gravity from I IB strings

Figure We break N sup ersymmetry to N by deforming the T K

direct pro duct geometry to a CalabiYau bration geometry with b er K and

base B

Of course the resulting manifold should be CalabiYau in order to have

an N compactication We furthermore take care that our cycles remain

nontrivial and that the geometry in their neighbourhood do es not change to o

muchlets sa ywe assume that the geometry in a neighb orho o d of the cycles can

b e continuously deformed to the original pro duct geometry in this neighb orho o d

and that wetake the deformation parameters suciently small suchthatwe still

have a sup ersymmetric representative of top ology S S for each of our cycles

Thus at least some of the BPS states in the N multiplet will survivethe

change of compactication geometry Not all however since the holonomy group

of the compactication manifold has b een enlarged from SU to SU and

therefore some of the states which are dened by annihilationcreation op erators

comp osed of the sup ercharges invariant under the SU but not necessarily

the SU holonomy will leave the multiplet As argued in section it is not

guaranteed that eg the original fermionic osillator ground state remains in the

multiplet but certainly the highest spin state of the multiplet do es that is the

vector So we are again left with at least an N massive BPS vector multiplet

A massiveBPS N vector multiplet has states The fermionic oscillator

ground state is a spin doublet In the context of the ab ove argument this doublet

corresp onds to two states of the original states in the N multiplet which

Near ALE brations and the light BPS sp ectrum

remain invariant in passing from SU to SU holonomy and which are anni

hilated by the annihilation op erators formed with the N sup ercharges Note

that this ground state degeneracy is not in contradiction with our earlier assertion

that a unique sup ersymmetric representative gives a spin fermionic oscillator

ground state the representativeis not unique here since a brane with top ology

S S has at least one real mo dulus corresp onding to turning on a at U brane

worldvolume gauge eld along the S comp onent and one corresp onding to em

b edding deformation of the brane The deformations can b e thoughtofasa

shift of in the base B of the bration A nontrivial brane mo duli space indeed

typically gives rise to sucha multiplet of fermionic oscillator ground states with

resp ect to the fermionic creationannihilation op erators made from the N

susy generators but a direct computation of this requires a detailed analysis of

sup ersymmetric quantum mechanics on the mo duli space analogous to monop ole

mo duli space analysis From general considerations we exp ect however that in

abackground with unbroken sup ercharges as is the case here the n umber of

fermionic oscillator ground states is equal to the numb er of holomorphic forms on

mo duli space satisfying certain b oundary conditions if there is a b oundary If

we take our base manifold B to b e a sphere the mo duli space will b e one complex

dimensional and top ologically a cilinder so indeed we exp ect no more than two

harmonic forms the constant function and a holomorphic one form Therefore

in this case we exp ect precisely one vector multiplet for every cycle of the kind

describ ed ab ove

This vector multiplet can b e made arbitrarily lightby tuning the mo duli such

that the cycles in K remain arbitrarily small in a neighborhood of in the

i

base B What do es K lo ok like in the neighborhood of these almost vanishing

spheres Consider rst the case where wehave just one sphere Then in the limit

of a zero size sphere wehaveanA singularity Any complex deformation

of an A singularity which is lo cally CalabiYau is lo cally given by the following

equation in C

w y x u

where u is the deformation parameter and x y w are co ordinates on C This

space is an A ALE Asymptotically Lo cally Euclidean space The sphere is

given bytheintersection with Im u x y w Taking the natural choices

dxdy

and J idx dx dy dy dw dw it is easy to see that this

w

representant is also the sup ersymmetric one Its area is prop ortional to juj

If wehave more spheres the limiting singularity and the lo cal geometry de



If B is a torus with K bre indep endent of the base we get an extra N hyp ermultiplet

as this is simply the N case The corresp ondence of states with holomorphic forms do es not

longer hold b ecause of the enlarged numb er of sup ersymmetries Wehave not analyzed other cases

Chapter Quantum YangMills gravity from I IB strings

Figure The deformed D SO singularity has a basis of four vanishing

spheres intersecting each other as indicated

p end on the way those spheres intersect Mathematicians discovered a b eautiful

corresp ondence b etween singularities and Lie algebras In this corresp ondence

nontrivial spheres in the deformed singularity are identied with Lie algebra ro ots

and sphere intersections with minus the inner pro duct of the ro ots Corresp ond

ing to the set of p ositive ro ots there is a set of p ostive spheres in terms of which

all other spheres can b e expressed as linear combination in homology with only

negative or only p ositive co ecients The p ossible intersections hence the p ossi

ble singularities can b e enco ded by Dynkin diagrams a dot indicates a sphere

and each line connecting two dots indicates an intersection Fig shows as an

example the D SO case In the cases whichwe are considering singulari

ties app earing by collapsing spheres in K all spheres have selfintersection

and mutual intersections or Therefore the corresp onding Lie algebra will b e

simply laced Actually as it turns out it will b e of ADE typ e such singularities

are called simple For a singularityoftyp e S the lo cal geometry is given by

complex deformations of W where

S

k

w y x W

A

k

k

W w y x x

D

k

W w y z

E



W w y yx

E



Note that as it should for this corresp ondence to make sense the set of spheres is indeed only

a subset of the total integer homology group for a homology class to have a sup ersymmetric

sphere representative its selfintersection has to b e at least in the cases relevant to us by the result stated under

Near ALE brations and the light BPS sp ectrum

Figure A sphere can b e constructed as a circle bration over an op en path

with vanishing circle bres at the endp oints

W w y x

E

It is sucient to consider deformations up to p olynomial orders strictly lower than

W z w y x since higher order deformations can lo cally b e absorb ed in a

z S

co ordinate transformation For example the A deformations are

n

n n

w y x U x U x U

n

The cycles in this space can b e constructed as follows First note that the

space dened by at xed x consists of copies of the complex plane

connected with a throat so this space has cylinder top ology The nontrivial

n n

cycle around the throat vanishes when P x x U x U x U

n

So by transp orting this nontrivial cycle along a path in the xplane running from

one zero x to another zero x of P x we obtain a cycle with S top ology

i j ij

g We can carry out this construction for all nn oriented pairs of

zeros of P x so we exp ect to nd nn dierent oriented minimal spheres

of which n are also homologically indep endent Note also that

ij kl il jk ik jl

In general the number r of linearly indep endent homology classes in the

S

space given by W equals the rank of the corresp onding Lie algebra S eg

S

r k This is equal to the numb er of deformation mo duli one can consider the

A

k

p erio ds of a basis of cycles to b e the mo duli The numb er of sup ersymmetric

spheres counting separately opp osite orientations equals the numb er of ro ots of

Chapter Quantum YangMills gravity from I IB strings

the corresp onding Lie algebra ie the dimension of the adjoint representation

minus r

S

An interesting prop erty of the nontrivial spheres of the space given by W is

S

that they are p ermuted byWeyl reections of the Lie algebra S under mono dromies

in the deformation mo duli space Indeed by the PicardLefschetz formula a

cycle transforms as follows under a mono dromy ab out a lo cus in mo duli space

where the twosphere vanishes

i

i

Consequently the brane states jii constructed from the cycles will b e only

i

physically distinguishable up to Weyl transformations Since Weyl transformations

are precisely the residual gauge transformations which one has after sp ontaneous

breaking of the gauge symmetry the nonab elian gauge theory supp osedly describ

ing the dynamics of the massive vector multiplets corresp onding to the jii must

have a gauge group with Lie algebra S The massless v ector multiplets corre

r

S

sp onding to the unbroken U gauge group which is generated by the Cartan

subalgebra are the vector multiplets asso ciated to the deformation mo duli cf

chapter Note also that the results of chapter imply that the massivevector

r

S

multiplet states jii have precisely the right charges under this unbroken U

which one would exp ect for a gauge group with Lie algebra S To see this cho ose

I

the symplectic basis A B of cycles such that a subset of the Acycles are

I

i i

given by A by this we mean that A is a bration over the nontriv

i

ial cycle in the base B with bre From the discussion around

i

it follows that the vector multiplets corresp onding to branes wrapp ed around

i i i

n have U charge n wrt the massless vector A

i

Thus we arriveat the following picture for CalabiYau compactications of

typ e IIB string theory Supp ose we have a CalabiYau X which local ly can be

continuously deformed to a pro duct of a cylinder and an S typ e ALE space Then

the brane states obtained by wrapping the brane on the nontrivial cylinder

cycle and the ALE spheres deformed to X together with the massless vector

multiplets from the p erturbative string sp ectrum ll out a nonab elian vector mul

tiplet of the Lie algebra S This vector multiplet can b e made arbitrarily lightby

shrinking the ALE spheres Therefore when the spheres are small we exp ect

the dynamics of this vector multiplet at energies much smaller than the Planck

scale to b e given byanN YangMills theory p ossibly with matter coming

from other branes in X with gauge group algebra S weakly coupled to gravity

In the limit of vanishing spheres gravity decouples completely from the vec

tor dynamics In particular since the two derivative low energy eective action



We assume the deformation suciently small such that the sup ersymmetric representative exists throughout the deformation

Near ALE brations and the light BPS sp ectrum

of the massless vector multiplets in typ e I IB compactied on a CalabiYau

do es not receive quantum corrections cf chapter we exp ect to repro duce the

Seib ergWitten solution for the low energy eective action of quantum N

YangMills theory This will indeed turn out to b e the case Moreover we will b e

b e able to extend their result by deriving the coupling of this theory to gravity

Chapter Quantum YangMills gravity from I IB strings

Figure Construction of a nontrivial sphere in X The dierent elements are

explained in the text

Figure Distribution over the base B of the p oints where a sphere in the b er

vanishes The base manifold B lo cally a cylinder is represented as the punctured

complex plane with mapp ed to the unit circle The sphere here i

i

vanishes in precisely p oints P and P one inside and one outside

i i

Near ALE brations and the light BPS sp ectrum

Hyp er branes

In general the nontrivial bration structure of our CalabiYau manifold X as

opp osed to the trivial direct pro duct structure of the N T K compacti

cation yields new nontrivial cycles and therefore new massivecharged BPS

particles in the sp ectrum Typically this go es as follows Because of the non

trivial dep endence of the bre K on the p osition in the base B some nontrivial

cycles in K will vanish at one or more p oints P in the base One can then

j

construct a cycle in X as a bration with b er over a path in B starting

and ending on elements of fP g such that the bre vanishes at the endp oints

j j

g If has the top ology of a sphere the resulting cycle will have S

top ology this is analogous to the construction of the S in g Assuming

there exists a sup ersymmetric representative for this cycle wethus nd another

N BPS multiplet Now since sup ersymmetric deformations of cycles on

CalabiYau manifolds are in one to one corresp ondence with harmonic forms on

the brane worldvolume the sup ersymmetric emb edding of such a brane with

S top ology is unique By the same reasoning as used in chapter section

and in this chapter in the N case wethus exp ect a unique N fermionic

oscillator vacuum in four dimensions which has necessarily spin since it should

supp ort a representation of the little group SO Consequentlyitisinahalf

hyp ermultiplet The conjugate halfhyp ermultiplet is obtained by reversing the

orientation of the brane

Genericallyifwe tune the mo duli of X such that the S S vectormultiplet

branes b ecome light also some S hyp ermultiplet branes will b ecome light namely

K and a nite at least those constructed with the vanishing ALE spheres in

path in B So the hyp ermultiplet content of the four dimensional eld theory

describing the light BPS states will b e determined by the dep endence of the lo cal

ALE ie the singularity deformation parameters on the base B

For concreteness supp ose wewant to obtain pure SU n N YangMills

Then we should have the following prop erties of the hyp ermultiplet sp ectrum

From semiclassical considerations it is known that at weak but nonzero cou

pling there are BPS hyp ermultiplets with nonzero magnetic charge namely

the monop ole and dyon hyp ermultiplets

By denition there are no purely electrically charged hyp ermultiplets

Translated to the brane picture of the sp ectrum this means

There should be some light S branes and so by the ab ove construction

some p oints P inB where an ALE sphere vanishes

j i i

Chapter Quantum YangMills gravity from I IB strings

For a given the P should be distributed such that necessarily the

i j i

interconnecting paths intersect the path which was the base for the

vector multiplet cycles since no intersection means no magnetic charge

In other words there should b e precisely one zero of at b oth sides of

i

g and there should be no other nontrivial spheres except those

constructed from a path interconnecting those pairs of zeros

We already know that in order to havean SU nvector multiplet the lo cal

ALE bration should haveanA bre The dep endence of the bre on the base

n

p oints is given by sp ecifying the U as analytic functions of a co ordinate on

i

the base Wecho ose the co ordinate such that the lo cal cylinder structure of B

is represented as the punctured plane and the nontrivial cycle wrapping the

cylinder is given by the unit circle j j From the discussion b elow

it follows that the p oints in the base where an ALE sphere degenerates are the

n n

zeros of the discriminantu of P x x U x U x

n

U where two zeros of P x coincide From elementary considerations ab out

p olynomial zeros one can see that the only p ossibilit y to satisfy the ab ovetwo

requirements to havea pureN YangMills sp ectrum in this case is to have

all U u indep endentof except U which should b e given by

i i

b

U u

where band u are constants The constant b can b e put to by a suitable

rescaling of co ordinates and mo duli

Indeed the symmetry guarantees a pairing of zeros on the two sides

of the unit circle j j and with the ab ove dep endence of the U there are

i

precisely n homological ly independent spheres in the ALE bre whichvanish

eachatapoint inside the unit circle g shows a typical n conguration

Higher p owers of or dep endence of the U with i would intro duce more

i

than n zeros of the discriminant on one side of the unit circle with since the

dimension of the ALE homology is n necessarily some of the corresp onding

cycles homologically dep endent on the others whichwould make it p ossible to

not intersecting the unit circle construct S cy cl es

With this choice of dep endence there are n homologically indep endent

S cycles precisely as many as there are S S cycles Together they form a

setofn indep endent cycles with nondegenerate intersection matrix which

can b e put in standard symplectic form after suitable recombination of the basis

elements and they can be taken to form part of a basis of cycles of the full

CalabiYau

Near ALE brations and the light BPS sp ectrum

This ALE bration can b e written in nonsingular p olynomial form as

n

W y w x u b

Considering this as a lo cal approximation for an algebraic CalabiYau manifold

the unique holomorphic form on this manifold is in the patch w given by

d dx dy dx dy d

W w

w

where is an arbitrary normalisation factor Note that sending b to zero in

corresp onds to the deformation to the trivial direct pro duct bration con

sidered in section The dyon hyp ermultiplets b ecome innitely massiveinthe

limit b so this should corresp ond to turning o the YangMills coupling since

in semiclassical YangMills theorydyon masses are prop ortional to g

YM

Analogous considerations can b e made for the other ADE gauge groups

Thus we conclude that in order to have a pure YangMills sp ectrum in four

dimensions the dep endence of the ALE b er on the base co ordinate must be

given by with all other U constant

i

Reduction to Seib ergWitten periods

We consider again the pure SU N case Consider a YangMills cycle

ij

constructed as a sphere bration over a path in the plane with the sphere

bre constructed as under from a path in the xplane running b etween

ij

zeros x and x of

i j

N N

P x x u x u

N

With the holomorphic form given by the p erio d of is then

ij

Z Z

d

i x x

j i

ij

H

Here we used that dy y k i for a contour encircling the cut b etween

y k in the y plane Equation describ es precisely the genus N

SU N Seib ergWitten Riemann surface as given in Moreover if we

denote by the cycle on this Riemann surface obtained as the lift of to the

ij

sheet x minus the lift of to the sheet x we nd

j i

Z Z

p

i

SW

ij ij

Chapter Quantum YangMills gravity from I IB strings

d

p

where x is nothing but the Seib ergWitten meromorphic form

SW

Thus in the lo cal ALE bration approximation of the CalabiYau X

our light p erio ds reduce precisely to the Seib ergWitten p erio ds Note also that

implies that up to a sign the intersection pro duct is conserved under the

map

ij ij

ij il jk ik jl ij

kl kl

The weak gravity limit

We will now study in more detail the pro cedure of tuning the CalabiYau mo duli

such that the branes wrapp ed around the YangMills cycles ie the small cycles

to the Planck pro ducing the YangMills sp ectrum b ecome very light compared

mass From it follows that this is equivalent to tuning the periods of

those cycles to b e much smaller than the p erio ds of the generic cycles Assuming

a sp ecic but rather general form of the CalabiYau we will deduce the b ehavior

of the p erio ds in such a limit and from this we will obtain the general form of the

Kahler p otential on the complex structure mo duli space in this limit We call this

limit the weak gravity limit since the YangMills dynamics on which we fo cus

will b e arbitrarily weakly coupled to gravity in this limit

We will restrict to the pure SU N YangMills case though generalizations

are clearly p ossible

Consider an arbitrary compact algebraic CalabiYau manifold X emb edded

in a pro jective space and given in a certain ane co ordinate patch with co ordinates

x y wby a p olynomial equation

W x y w

SU N ALE bration To get a manifold X which is lo cally given by the pure

where we take lo cal to mean small x y w and nite with the YangMills

ALE p erio ds much smaller than the others we assume W to b e of the form

W W W x y w

AF

where

N N N N

W w y x L u x L u x L u L

AF N

with L very large and the u nite and where W is a p olynomial containing only

i

N

terms of order x w y L and higher p ossibly still dep ending on mo duli

The weak gravity limit

dierent from the u which will not be imp ortant in what follows Indeed

i

N

by rescaling w L w y L y and x L x we see that for nite

tilde variables which corresp ond to small nontilde variables x y w we get the

required lo cal geometry

N N

W W L w y x u x u x u

AF N

For obvious reasons we will call the mo duli u the rigid moduli

i

The holomorphic form on the other hand is in a patchwith W

w

d dx dy d dx dy

W w W

w w

where is an arbitrary normalisation factor For nite tilde variables indeed

N

reduces to with L

d dx dy

N

L

w

We will assume that it is p ossible to construct a basis of cycles for X entirely

lo calized in the co ordinate patch parametrized bynite x y w This puts some

constraints on the global structure of X For example it excludes the p ossibility

that X is a bration over a higher genus Riemann surface of which would

then parametrize only a certain co ordinate patch Roughly this assumption thus

says that X is a bration over a sphere the plane but actually wedonthave

to be that precise since how exactly X is compactied at innity wrt the

co ordinates x y w do es not matter for our purp oses

Now cho ose a maximal set f g of N indep endent YangMills

bration ie nite in the tilde variables cycles which are compact on the ALE

Denote the nondegenerate intersection matrix by q Extend this

basis with cycles to a basis for H X Q in suchway that Since

q is nondegenerate this is always p ossible however in general it is not p ossible

to construct such a basis for H X Z Denote the intersection matrix of the

cycles by Q

Note that since there are no intersections of the cycles with the cycles

there is no obstruction for deforming the cycles away from the region of small



This can b e generalized to cases where W also dep ends on but to keep the already

quite technical arguments a bit transparent we will not consider this here We will see an

example of this in section however

for example the bration structure do es not necessarily have to b e extendable to the p oints

at innity wrt the co ordinates x y w

Chapter Quantum YangMills gravity from I IB strings

Figure If a cycle bred over a path in the plane encircles or stretches

between branch p oints moving to innity when L this cycle will

b e stretched to innity

x y w nite tilde variables where the cycles are lo calized We will further

more assume that we can keep the cycles at nite values of x y w when L

which is not a strong assumption

R

N

can be seen from to be prop ortional to L The p erio ds

As explained in section they can be reduced to Seib ergWitten p erio ds

Furthermore in that section it was also explained that the intersection form of

those cycles equals minus the corresp onding intersection form on the Seib erg

Witten Riemann surface

We now turn to the p erio ds These can be divergent when L but

since for L the cycles stay nite in x y w and away from the singularity

lo cus nite tilde variables so that W w W w x y is b ounded from

w w

below on the only p otential source of divergencies is the fact that X factorizes in

the co ordinate patch under consideration as a direct pro duct of the punctured

plane and a complex fold W b ecomes indep endent of when L

consequently some cycles can and will be stretched to innity in the

plane g Since W is p olynomial in such cycles will b e typically stretched

to values of prop ortional to certain p ositive or negative p owers

of L So we see from since we assumed w x y to stay nite and away from

the singularity lo cus on the cycles that these stretched cycles will at most b e

logarithmically divergent that is

Z

a ln L b



The weak gravity limit

where a and b could a priori still b e dep endent on all mo duli dierent from L

We will now showhowever that if the normalisation factor is uindep endent the

N

leading order udep endent term is actually at most prop ortional to L ln Lso

that a and b must b e independent of the rigid moduli u Indeed using

i

and we nd

Z Z

W W

w

w

u W W u

k w w k

Z

W

w

k kN

x L

W

w

Z

W

w

kN k

L x

w W

w

Again b ecause stays nite in x y w and away from the singularity lo cus of nite

tilde variables so that W stays b ounded from b elow the integral factor in

w

the rhs of can at most b e prop ortional to ln L and since k N the

N

full p erio d at most to L ln L This is what wewanted to show Now since

uindep endence of the a and b will turn out to b e essential to extract the rigid

limit as describ ed in general in section we will thus take the normalisation

factor to b e uindep endent

Combining all this to compute the form of the Kahler p otential we

nd

Z

K lni

X

Z Z Z Z

lniQ iq



N

lna ln jLj b jLj K u u

N

jLj

K u u lna ln jLj b

a ln jLj b

Here a and b are uindep endent real constants b ecause a and b are a

Z Z

K u u j j iq

SW SW

here we used and and the dots include uindep endent terms

N

of nonzero order in L and udep endent terms higher than second order in

N

L The reason for the absence in of terms prop ortional to ln L ln L or

Note that this argument fails for the p erio ds b ecause on these W is not b ounded from

w

N

b elow and indeed the uderivatives of those p erio ds are prop ortional to L

Chapter Quantum YangMills gravity from I IB strings

R

K

must b e invariant under the mono dromy lnLL is the fact that e i

X

iN

L e L The presence of the divergent term can also b e inferred from the

form of and the fact that X degenerates to the direct pro duct of an innite

cylinder the punctured plane and a compact manifold The integral over the

cylinder parametrized by gives a logarithmically divergent factor Note that

this divergent term would b e absent for compactications on eg the direct pro d

uct of a torus and K yielding N in four dimensions Such a cases

could b e discussed along the same lines though some features will b e qualitatively

dierent

Note that the form of the Kahler p otential we nd is precisely the one needed

to have a reduction from lo cal to rigid sp ecial geometry for the umo duli as was

explained in chapter section Clearly K u u is closely related to the

SU N Seib ergWitten Kahler p otential K u u but let us make this more

SW

precise From and it follows that

K u u j j jj K u u

SW

where is the scale of the gauge theory which app eared as an arbitrary param

eter in chapter equation it is the prop ortionalityfactorbetween the

dimensionful YangMills scalars and the dimensionless geometric p erio ds recall

that the Kahler p otential in rigid YangMills theory has dimension mass squared

that is expressed in the u variables it has an overal factor jj is xed here in

terms of L and the four dimensional Planck length by requiring the YangMills

eldstohave canonical kinetic terms as in Comparing the scalar kinetic

terms in and we see that this puts

N

j j jLj

jj

a ln jLj b

Alternatively one could compare the BPS mass formulas in rigid YangMills theory

and sup ergravity cf chapter or to x the phase of as well compare central

charges

p

Z

sugra

p

Z

susy

G

N

This gives

p

N

L

a ln jLj b

Large L therefore corresp onds to a gauge theory scale which is small compared

to the Planck scale hence sending L while keeping nite and

in the way prescrib ed by corresp onds to decoupling gravity from the

nite energy nontrivial gauge theory dynamics The relationship with and the

volume of X is given by

Unication scales and the heterotic picture

So all in all wend

K lna ln jLj b K uu

SW

Note that the terms wehave dropp ed are prop ortional to p ositivepowers of

Thus weseehow the Seib ergWitten solution for the two derivativelowenergy

eective action of the massless elds in N YangMills indeed b eautifully

emerges from purely geometric concepts in typ e I IB string theory But we have

more now wehaveapowerful geometric representation of the massive BPS states

as sup ersymmetric branes wrapp ed around CalabiYau cycles and wehavea

consistentemb edding of quantum low energy YangMills theory in string theory

including gravity

Unication scales and the heterotic picture

In order to compare our results with standard treatments of gauge theory physics

b

in heterotic string theory see eg chapter we assume ln L and

a

rewrite with

b

S ln L ln L

a

as

K lnS S K jjuu

SW

where wehave dropp ed an irrelevant term ln a and

expSN

p

k

Re S

where

p

p

k

a

p

Note that jk j is indep endent of the choice of normalisation of since a scales

as j j under rescaling of It dep ends on the details of the chosen mo del for X

however including mo duli dierent from L and the u if those are present

i

The asymptotic running of the pure N SU N YangMills coupling which

can b e extracted from the solution of the low energy eective theory wehave found

agreeing with the wellknown lo op N b eta function is then given for energy

Chapter Quantum YangMills gravity from I IB strings

scales M by

N M

ln

g M

YM

N M

SU

Re S ln

M

where

M jk j Re S

SU

The mass scale M canbeinterpreted as the string gauge unication scale if we

SU

would haveseveral SU N gauge groups induced by the CalabiYau compactica

tion thiswould b e the value of M where the dierent gauge groups have equal

coupling equal to

g M

SU

YM

Re S

to b e compared with eq Note that from a phenomenological p oint

of view the string gauge unication scale is not necessarily equal to the scale

M where the standard mo del group SU SU U unies to a simple

GU T

group like SU though this would b e quite pleasing of course

The string gauge unication scale is closely related but not equal to the

string scale M which can be extracted from the low energy gauge physics as

S

eq

g M

YM SU

p p

M M

S SU

jk j

Re S

Expressions similar to the ab ove formulas typically app ear in the context of

four dimensional gauge theory physics obtained from heterotic string theory

Compare for example with a and with in In



heterotic string theory S is the dilatonaxion scalar S e ia where

is the four dimensional heterotic dilaton and a is the axion eq

This is a manifestation of the dualitybetween heterotic and typ e I I string theory

Let us have a short closer lo ok at this corresp ondence Consider for example

Though we did not consider this case it can b e obtained straightforwardly by replacing the

p olynomial in x in by a p olynomial which has two or more nitely separated bunches of

closely spaced zeros We will see an example of this in the next section The unication scale is

then the energy scale of the gauge theory branes when the u mo duli are taken so large that all

zeros are more or less equally spaced Generically this is indeed close to the string scale but if

it happ ens that twobunches of zeros are already quite near each other at order u values the

corresp onding two gauge groups can already get unied at considerably lower energies

Unication scales and the heterotic picture

the expression for the rigid prep otential in the SU case equation

S

Substituting M e gives

SU

k

X

i M M

SU SU

F S ln c expkS

rig k

k

het

we see that from the heterotic p oint Since Re S is identied with e



g

Shet

of view we havean innite series of lo op p erturbative and nonp erturbative

quantum corrections

In heterotic string theory the string scale M is related to the constant

S

app earing in string p erturbation theory M It is not directly related

het

S

however to the constant app earing in typ e IIB p erturbation theory

IIB

V Re S M we used and for

X

S

the second equality So it is the dual heterotic string scale rather than the

het

typ e I IB string scale which app ears naturally in the low energy gauge physics

IIB

we are studying This can b e understo o d from the fact that typ e I IB p erturbation

theory breaks down here at energies higher than the mass of the lightest massive

gauge particles since these particles are nonp erturbative states in typ e I IB while

heterotic p erturbation theory is valid for energies ab ove the gauge particle masses

and all the way up to and b eyond the unication scales it can break down at

low energies however b ecause the gauge coupling b ecomes strong in the infrared

for rigid mo duli of order So the relevant p erturbative string theory for energies

near M is heterotic hence the relation with the heterotic

S

Now though these N theories are probably not directly relevant to phe

nomenology let us nevertheless consider some exp erimental data to get a more

concrete idea of the physical meaning of the ab ove results Measurements of the

running of couplings together with the exp erimentally quite plausible assump

tion of minimal SU sup ersymmetric unication suggest M

GU T

GeV and g M Let us assume M M and see what we

GU T SU GU T

YM

get From using the unied coupling value and GeV we

k

nd M GeV Now in order for M M to b e equal to M

S SU S GU T

within the error bars wemust have k Using the relation b etween

M and M obtained in the context of heterotic string compactications

S SU

eq one obtains k This is clearly to o big Some p os

sible solutions are explained in In the example wewillwork out in the next

section we will however nd k which is compatible with exp eriment

From again using the unied coupling value we get Re S Thus

generalizing in an obvious way to arbitrary YangMills theories p ossibly



This is the minimal GUT sup ersymmetric extension of the SU SU U standard

mo del In this mo del the gauge group is SU at energy scales MM GU T

Chapter Quantum YangMills gravity from I IB strings

with matter we nd for the scale of such a theory taking the exp erimentally

most favored k

b

e GeV

where b is the beta function co ecient of the theory b N N for N

f c

SU N with N avors Since k is dep endent on the details of the chosen Calabi

c f

Yau manifold considerations of this kind might put together with sp ectrum con

siderations constraints on candidates to nd realistic typ e I I CalabiYau compact

ications But since such compactications seem to havetoomuch sup ersymmetry

to make them realistic anyway and since a serious analysis of the phenomenology

would lead us to o far aeld and by insucient exp ertise of the author we will

leave this sub ject here

Dynamics of the dynamical dynamically generated scale

In the following we will consider CalabiYau manifold X which gives rise at low

energies to an arbitrary asymptotically free gauge theory with simple gauge group

and assume the CalabiYau manifold X has no other complex structure mo duli

than the rigid mo duli u and S Furthermore we will assume S as in Gen

i

eralizing in an evidentway to arbitrary gauge theories with b etafunction

co ecient b gives the following expression for the dynamically generated scale

expSjbj

p

k

Re S

Since S is a dynamical eld we see that the dynamically generated scale itself

is dynamical it can vary over spacetime Let us see what we can sayabout its

dynamics

We will assume S is always large if not the whole weak gravity approximation

breaks down Then the contribution from spacetime variation of the denominator

of to spacetime derivatives is suppressed by a factor Re S

wrt the contribution of the exp onential We will assume that this factor is

suciently small such that this contribution can b e neglected Then we can rewrite

as

M expSjbj

SU

with the string unication scale M as in assumed to b e a constant

SU

Plugging this in the expression for the Kahler p otential weget

jj K lnln jM j K u u

SU SW

Unication scales and the heterotic picture

where K jjuujj K u u and wehave dropp ed the constant ln jbj

SW SW

The corresp onding eective action for the graviton and the scalars is

Z

p

d

GR S d x

ln jM j

SU

Z

i j

jj K du du

i j SW

i

Re K du dK jdj

i SW SW

From this we see that generically the eld will couple to the gauge theory

elds but almost as weakly as gravity b ecause of the factor in frontofthe

kinetic term Therefore in circumstances where gravity eects are very small

for instance in accelerator exp eriments the eects of having a dynamic will

Note also that there is actually no coupling be very small as well ie d

between the gauge theory elds and when K is the classical Kahler p otential

SW

R

i

and if one uses the p erio d variables instead of the u variables

SW i

i

i j

as follows immediately from together with K where

SWclass

ij

is constant From the microscopic YangMills eld theory p ointofview the

ij

backreaction of the gauge theory elds on the gauge theory scale is thus entirely

a stringy quantum eect

In a certain sense one can also interpret the spacetime variation of the scale

as a spacetime variation of an eective Newton constant with constant scale To

see this write

G e hi expRe G

and substitute this in

Z

p

d x G R d d S

M

SU

expRe

ln

Z

i j

jhij K du du

i j SW

i

Re K du d K d d

i SW SW

So we see gets replaced by an eective expRe A review on the physical

consequences of a varying Newton constant can b e found in This interpreta

tion is p erhaps less natural here considering the coupling of to the scalar elds

including hyp ermultiplet scalars this rescaling induces Of course this situation

and the action is very similar to what one has with the usual dilaton scalar in

the role of In the light of the discussion on heterotic typ e I I duality in section

this is not surprising at all

 

somewhat stronger actually b ecause of presence of the extra factor ln jM j SU

Chapter Quantum YangMills gravity from I IB strings

Finally note that dynamical gauge theory scales and hence dynamical gauge

coupling constants could pro duce interesting and p erhaps measurable physical

eects In particular considering the time dep endendence of the matter densityin

the universe and the fact that the gauge theory scale couples to this according to

the picture wehave develop ed one would exp ect the gauge coupling constants

to have some time variation There are indeed some cosmological exp erimental

indications that this is the case One could also contemplate the p ossibility

of strong spatial variations of gauge theory scales near black holes This would

certainly pro duce rather sp ectacular eects since some particles might suddenly

b ecome unstable bya change of the scales leading to violent explosions gamma

bursts dark matter pro duction black hole jets and all that Or more realisti

cally p erhaps ne structure constant uctuations waves might b e measured by

equipment of similar sensitivity as gravitational wave detectors Considering the

fact that extremely high precision measurements of the ne structure constantare

p ossible such exp eriments mightbeworthwhile

An explicit example

In this section we will study in detail an explicit example to illustrate and

p erhaps clarify some of the topics wehave discussed more or less in general in

this chapter

Fibration structure

We take as example the CalabiYau manifold X The details of the

general construction of algebraic CalabiYau manifolds will not b e imp ortantfor

us We will dene this manifold simply as the hyp ersurface W with

B

W x x x x x

x x x x x x x x x x

s

and identications

x x x x x x x x x x

x x x x x

x x x x x

where C and expi It is also understo o d that all singular p oints

induced by the identications ie the xed p oints are blown up This intro duces

An explicit example

additional Kahler mo duli which are however not imp ortant for us as we will only

deal with the complex structure mo duli Note that these singularities lie all on

the lo cus x x x As it turns out we can for our purp oses simply ignore

this lo cus and always work in the patch x x x as we will do from now

on X is the mirror of X which is dened by W with

W x x x x x the general p olynomial invariant under with

with identications but without and Similarly W could

have b een dened as the general p olynomial invariant under with as

well as under and This indeed yields up to isomorphisms

For an intro duction to algebraic CalabiYau manifolds see p or See

also

The space X has h The complex structure mo duli space

is hence parametrized by co ecients of W We cho ose to work in the gauge

instead of the more usual B

S

This CalabiYau manifold as well as its mirror is a K bration with K

bre X resp X The change of variables needed to exhibit

this is

x x

x x

which is well dened b ecause of the identications This gives as

dening p olynomial for the K at xed

W B x x x x

x x x x x x

where

B

B

and the following identications remain

x x x x x x x x

x x x x

The K bre itself is an elliptic bration Actually it can be b ered in

twoways The most evident one which also works for the mirror manifold and

is suitable for discussing the large complex structure limit can be obtained in

complete analogy with the ab ove construction The bre is X X

Chapter Quantum YangMills gravity from I IB strings

and there are no identications left The other p ossibility do es not apply to the

mirror where it would lead to a genus bre but is more suitable for discussing

the rigid limit Both p ossibilities are exhibited by the following change of variables

x y

x y

x x y

x y y x y

again well dened b ecause of the identications The pro jective gauge symmetry

acts on these variables as

y xy y x y

There are no discrete identications left Fix the pro jective gauge symmetry by

putting y which can b e done b ecause wework in the patch y x x x

This gives

B

W y P x

where

P xx x

The rst bration is obtained by considering the plane to b e the base manifold

the second one by taking the x plane instead Indeed in both cases

denes a punctured torus for any xed generic base p oint From now on we

will consider only the second bration

The holomorphic form on the CalabiYau manifold is

d d

dx

i y

d

is the up to a constant factor unique holomorphic Note that

i y

form on the torus bre and dx the holomorphic form on

i

the K



The punctures can b e removed and the tori compactied by going to a pro jective description



of the torus bres This yields X as the torus bre for the rst bration and X for



 the second bration

An explicit example

Figure Branch p oints cuts and cycles of the torus bre

Construction of cycles and p erio ds

We will explicitly construct a basis of cycles and their corresp onding p erio ds

Toachieve this we exploit the bration structure of the mo delrst w e study the

cycles and periods of the torus bre then those of the K and nally those of

the CalabiYau manifold itself The explicit construction of the cycles allows us

to compute mono dromies and intersection forms in a straightforward way For the

torus and K bres we furthermore obtain closed expressions of the p erio ds in

terms of hyp ergeometric functions

Torus cycles and periods

Branch points If we consider to b e an equation for y as a function of

we get the torus as a sheeted cover of the plane The sheets coincide at the



B

branch p oints which come in pairs symmetric under and are lo cated at

p

P B P

and at

Chapter Quantum YangMills gravity from I IB strings

p p

Cycles In the following z denotes the p ositive square ro ot of z z

p

and i Dene to b e the cycle j j jB j passing counterclo ckwise

p p

i

p

through the p oint i B y P and the shortest cycle encircling the

pair of branch p oints with orientation such that This is shown

in gure

Singularities and vanishing cycles The torus degenerates when two branch

p oints coincide There are two p ossibilities

P B the branch p oints coincide and vanishes The lo cus

in the xplane where this o ccurs is a genus Riemann surface as can

b e seen by substituting the expressions for P xandB We denote this

surface by Since collapses to a p oint on this surface can b e lifted

trivially to the full CalabiYau By slight abuse of notation we denote the

lifted surface by as well Thus is the lo cus of elliptic bre singularities of

the CalabiYau We could also view the full D spacetime M CY as an

elliptic bration Then the lo cus of elliptic bre singularities gets promoted

to a dimensional manifold M

B one of the branch p oints coincides with the branchpoint

and vanishes The surface where this o ccurs consists of two copies

of the xplane at xed p ositions and is denoted by Again this surface

can be lifted to the full CalabiYau or promoted to a dimensional

submanifold of spacetime

p

Cuts There are jumps or cuts in the denition of at values of P B where

there are two homologically dierent cycles encircling the two branchpoints with

equal minimal length This o ccurs when the branchpoints are colinear

P

p

is imaginary typ e A cut Jumps in the denition of o ccur when ie when



B

P

p

the branchpoints lie on the circle j j jB j ie when lies on the real



B

interval typ e B cut There is yet another typ e of cut for b oth and

namely where our prescription for the orientation of and hence isambiguous

For xed B this is the case when P is real and negativetyp e C cut

p

The cut structure in the P B plane is shown in g The transfor

mation rules for continuous transp ort of torus cycles across the cuts yielding the

mono dromies are

A A

B B

C

An explicit example

Figure Singularities and cuts in the PB plane

p

B plane is not really singular as the mondromy Notice that the origin of the P

ab out this p oint is in fact trivial

Perio ds The following expressions for the p erio ds can be obtained by direct

P

p

integration Denoting k we nd



B

p

Z

p

B Kk

P

p

for close to and



B

p

Z

p

B k Kk

P

p

far from Here Ku F u is the complete elliptic integral for



B

p

of the rst kind These expressions can b e extended to other values of P B by

standard analytic continuation

Chapter Quantum YangMills gravity from I IB strings

Figure Elliptic bre singularities cuts and cycles of the K manifold

K cycles and p erio ds

As explained earlier the K bre at a xed generic value of and hence of

p

B is itself an elliptic bration with base parametrized by x Accordinglythe

relevant cycles of the K bre that is those which are in the transcendental

lattice can b e constructed as circle brations over certain paths in the xplane

where the circle is a cycle in the torus bre

Points with degenerating elliptic bre The xplane can be viewed as a

p

B plane considered in as is xed and P is of fold covering of the P

degree in x Therefore in the xplane there are copies of every ingredient

cuts singularities of g This is shown in g In particular the

g Riemann surface on which vanishes intersects the xplane having a

xed value of inpoints As splits in two branches corresp onding to the

p

solutions of P x B we can divide those six points accordingly in two

groups of three whichwe lab el by and Cho ose numb ering

such that the typ e B cuts connect i with i see gure

Cycles The idea is to construct K cycles as circle brations by transp orting

a certain torus cycle c along a path in the xplane The path can either b e a

closed lo op without mono dromy for c or an op en path terminating on the p oints

i with c vanishing at b oth endp oints The rst p ossibility will pro duce a torus

An explicit example

and the second a sphere This gives us the following cycles see g

s c at i and running b etween i and j This is a sphere

ij

s c at i and running b etween i and j again a sphere

ij

t c at i and a closed path encircling i i j j This is a

ij

torus

Note that it is not p ossible to construct a cycle by taking to run from i to

j b ecause such a path necessarily passes through an A typ e cut so we cannot

hav e as circle bre at b oth endp oints

Actually we havent given a precise description yet of the ab ove cycles in

general only for the sp ecic case of g To dene this set in the general

case we can pro ceed as follows rst we require the cycles to b e compatible with

the ab ove description including the homological relations and intersections Any

set of cycles obtained by continuation from the sp ecic set of g will satisfy

this Because continuation is in general not uniqely dened due to mono dromies

there are still many p ossibilities corresp onding to the choice of cuts Wewillnot

try to x the remaining ambiguity here in general but assume that in anycasea

prescription is adopted such that s vanishes whenever i approaches j

ij

At the level of homologywehave the following relations

t s s

ij

ij ij

s s s

This implies that of the cycles constructed ab ove only four are indep endent

Intersections Combining the the intersections of the paths in the base as

shown in g with the known intersections of the torus cycles taking into

account their transformation when passing a cut all cycle intersections can b e

calculated in a straightforward way

t s s s s s

s s s s s t

t t t s s s

ij kl

and all cyclic p ermutations hereof The rst two equations can b e summarized as

s s

il jk ik jl

ij

kl

Note that this is precisely minus the intersection of the cycles j i and

l k

Chapter Quantum YangMills gravity from I IB strings

Singularities and vanishing cycles The K degenerates when two or more

of the p oints coincide There are several p ossibilities corresp onding to

the marked p oints in g

B here and the tori t degenerate to

ij

lines recall that vanishes at B

p

B Keeping in mind that wehave dened B to have p ositive

real part there are two p ossibilities

p

if Re two of the zeros of P B coincide

and the corresp onding sphere s vanishes

ij

p

B coincide if Re two of the zeros of P

and the corresp onding sphere s vanishes

ij

In each case we call the vanishing sphere at this p oint v

a

B Again there are two cases if Re is p ositive negative there is

s Call this vanishing sphere v avanishing s

b

ij ij

Dene t t if v s and similarly for t The tori t and t degenerate

a ji a b a b

ij

at B The set v v t t forms a basis of the Picard lattice which has

a b a b

rank four here

W ehave constructed dierent bases dep ending on the signs of Re

and Re We therefore also exp ect the corresp onding intersection matrix to

dep end on these signs To calculate this matrix when say Re

and Re we consider the case where is very close to which

is the case shown in g Then it is easy to see by direct insp ection of

the ro ots of P x B that v and v are to b e identied with s and s if

a b

one cho oses suitable numb ering of the ro ots This identication together with

provides the complete intersection matrix An analogous pro cedure can

b e followed for the other cases

The results are

If Re and Re have the same sign

B C

B C

I

A

We recognise the SU Cartan matrix in the upp er blo ck and therefore call

this part of muduli space the SU sector

An explicit example

If the real parts of and have opp osite sign

B C

B C

I

A

Here we recognize the SU SU Cartan matrix accordingly we call

this part of mo duli space the SU SU sector

Notice that our division of the mo duli space in an SU and an SU SU

p

sector is dep endent on the sign convention for B except on the subspace

Therefore though the convention we have taken is quite natural

esp ecially when we are close to a rigid limit one can not exp ect the b oundary

between these sectors to haveanyphysical signicance However we shall see that

there exists a certain region inside the SU sector close to the SU rigid limit

where a D low energy observer indeed sees SU YangMills physics weakly

coupled to gravity and similarly for SU SU Outside these regions D

low energy physics might not lo ok at all like a particular nonab elian gauge theory

Mono dromies From the explicit construction of the cycles and the Picard

Lefschetz formula it is easy to calculate the mono dromies ab out the singular

p oints B B B and B The mono dromies are of

the form

v v

a a

B C B C

v v

b b

B C B C

M

A A

t t

a a

t t

b b

The resulting mono dromy matrices are

Ab out B

In the SU sector of mo duli space Re Re

B C

B C

M T

a

A



Aphysically signicant denition of eg the SU sector would b e the region of mo duli

space where BPS states exist which can b e identied as SU gauge b osons Unfortunatelyfor

a generic p oint of mo duli space the existence of these states is very dicult to check analytically if not imp ossible

Chapter Quantum YangMills gravity from I IB strings

In the SU SU sector of mo duli space Re Re

B C

B C

M T

a

A

Ab out B In the SU sector

B C

B C

M T

b

A

In the SU SU sector

B C

B C

M T

b

A

Ab out B In b oth sectors this is

B C

B C

M B

A

The matrices T T T and T have Jordan form diag hence an

a b

a

b

expansion of the p erio ds in a variable z around the corresp onding singularityhas

n n

terms of the form z and z B on the other hand has Jordan form

B C

B C

B

Jordan

A

n n

so p erio d expansions have terms z and z ln z

The matrices T and T generate the group S the Weyl group of SU

a b

while T and T generate S S the Weyl group of SU SU

a b

An explicit example

Perio ds An integral representation for the K p erio ds can b e given by making

use of the elliptic bration structure and Note in particular that

when i and j come close to each other wehave

p p



Z Z

j

p p

 

dx B B x x

j i



i i



s i

ij

p



B where the x are simply found by solving P x

i

Though not really necessary for the discussion of the rigid limit it is p ossible

using PicardFuchs techniques to nd a closed expression for a basis of noninte

gral p erio ds of the K namely

r s

r s

B C C B

A A

r s

r s

where

u B u F

u

u B u F

u

with

p

B B B B

and

p p

B B

r

p p

B B

s

For a derivation of this we refer to

The mono dromies of these solutions can b e calculated using the well known

continuation formulae of the hyp ergeometric functions By comparing mono dromies

one obtains the expression of the p erio ds of v v t and t in this basis up to

a b a b

an overall factor The overall factor can b e determined by comparing asymptotic

expansions in a limit where these can b e calculated easily on b oth sides eg in a

Chapter Quantum YangMills gravity from I IB strings

Figure K bre singularities cuts and cycles of the CY manifold

i

rigid limit see later The result for the SU sector is with e

i i

v

a

B C

i i

v

B C B C

b

 

B C

S S

A A

v v

p p p p

A

t

a

p p p p

t

b

CY cycles and p erio ds

Since the CalabiYau fold under consideration is a K bration one can con

struct the CY cycles as K cycle brations over paths in the base manifold the

plane Denote the path in the base by and the K cycle which is transp orted

along by c can b e op en with c vanishing at b oth endp oints or closed with

trivial mono dromyfor c The corresp onding CY p erio d is given by

Z

d

i

where denotes the p erio d of c in the K bre ab ove

A basis of cycles can b e constructed as follows see g

An explicit example

B

V c v and running between the two solutions of

v a

a

This is an S

B

V c v and running b etween the two solutions of

v b

b

Again an S

T c v and the unit circle This has top ology S S

v a

a

T same as T but with c v

v v b

a

b

B

V c t and running b etween the two solutions of

t a

a

Top ology S S

V same as V but with c v

t t b

a

b

T c t and the unit circle Top ology T

t a

a

T same as T but with c t

t t b

a

b

Formal connection with brane picture

We can also consider the CY cycles constructed ab ove as circle brations over

certain branes in the x space These branes can either be closed with

trivial mono dromy for the circle bre or op en and ending on the vebrane M

bre or M bre The top ology of the brane can b e either a disc

gives S CY cycle a cylinder S S cycle or a torus T cycle This

is of course very similar to the brane picture solutions of eld theories via

M theory There are several other similarities like the conditions for a

brane to b e sup ersymmetric and the top ology multiplet typ e corresp ondence but

we will not discuss them here One could wonder whether this connection could

hint to a generalisation of the usual derivation of eective eld theories without

gravity from branes in a trivial background to eective eld theories with some

gravitational corrections from branes in a nontrivial background We will not

pursue this question here however also b ecause it is not clear what the physical

basis would b e for such gravitational corrections from branes since branes do not

have a dynamical intrinsic metric

The SU rigid limit

Denition

The SU rigid limit is reached when three sheets of the Riemann surface co

incide or equivalently when the CY considered as an elliptic bration acquires

Chapter Quantum YangMills gravity from I IB strings

an I singularity according to the Ko daira classication ie a curveof A singu

larities This o ccurs when simultaneously B and However

as some p erio ds are ill dened in this limit wehave to sp ecify howwe approach

this p oint We set L and take

B b

and let while keeping b and nite It will b ecome clear that this

prescription is indeed essentially the one given in equation of the general

discussion Note that a rescaling b b yields the same

rigid limithence these v ariables give a pro jective description of the residual mo duli

space in the rigid limit

The ab ovechoice of dep endence keeps the branch p oints in the plane with

vanishing v or v at nite p ositions resp given by

a b

O

b

O

b

while the branchpoints with vanishing t and t given by

a b

b

are sent to innity This choice gives rise to light BPS states namely Dbranes

wrapp ed around the basis cycles V V T T which can be identied as

v v v v

a a

b b

the massive gauge b osons and dyons of the pure N SU YangMills theory

Furthermore we will show that in this limit lo cal sp ecial geometry indeed reduces

to SU rigid sp ecial geometry on the rigid mo duli space parametrized by b

and justifying the ab ovechoice of dep endence

Choice of p erio d basis

We start with the basis of CY cycles as dened in section Closetothe

rigid limit we clearly are well inside what we earlier called the SU region of

mo duli space Thus the intersection matrix of v v t t is given by eq

a b a b

An explicit example

From this it follows that the full CY intersection matrix is

B C

B C

B C

B C

B C

B C

Q

B C

B C

B C

B C

A

where the order of the basis cycles is the order in which they were dened As

the periods of v and v for nite values of b ecome small in the rigid limit we

a b

will showthis explicitly b elow we exp ect the rst four CY basis cycles to give

rise to the light Dbrane states corresp onding to gauge b osons and dyons To

isolate the contribution of those to the Kahler p otential wehave to redene the

last four basis elements such that the intersection matrix b ecomes blo ck diagonal

To accomplish this we put

V V V V T

t v v v

t a a a

b

a

V V T V V

v v v t

a

b b b t

b

T T T T

t v v

a a

b t

a

T T T T

t v v

a

b b t

b

The new noninteger basis has intersection matrix

C B

C B

C B

C B

C B

C B

Q

C B

C B

C B

C B

A

To deduce the form of the dep endence of the p erio ds it is sucient to calculate

I

their mono dromies under e The calculation of these mono dromies is a

bit tedious but can b e done from the construction of the cycles without knowing

any detailed expressions for the p erio ds The result is denoting the p erio ds with

Chapter Quantum YangMills gravity from I IB strings

the corresp onding caligraphic letters

V V

v v

a a

C C B B

V V

v v

b b

C C B B

A A

T T

v v

a a

T T

v v

b b

i

where e and

V V

t t

a a

B C B C B C

V V

t t

b b

B C B C B C

A A A

T T

t t

a a

T T

t t

b b

It is convenient to go to a basis of p erio ds for which this mono dromy matrix is in

Jordan form Therefore we replace again the last four basis elements by

T V V

t t

a

b

V T T

t t

a

b



T T T

t t

a

b



V V V

t t

a

b

The new basis indeed has the mono dromy in Jordan form

T T

B C B C B C

V V

B C B C B C

A A A

 

T T

 

V V

This implies the following b ehaviour

V V T T analytic

v v v v

a a

b b

T analytic

ln

T analytic V

i



T analytic

ln

 

T analytic V

i

Small p erio ds and the Seib ergWitten curve

In the region of mo duli space under consideration the paths in the xplane dening

the K cycles v and v stretchbetween sheets of the branch of the g

a b

An explicit example

p

Riemann surface given by the equation P x B Expressed in

the variables b and this equation b ecomes

s

b

x x

or rescaling x x and expanding the square ro ot for nite

b

x x O

which is the equation for the genus SU Seib ergWitten Riemann surface

and u Thus we see that in if weputb u

the SU rigid limit a genus branch of our general genus Riemann surface

degenerates and pro duces the Seib ergWitten surface with punctures at innity

where the rest of the genus surface is attached

For nite we furthermore have using

p

Z

O x x

i j



s

ij

So for a CY cycle s obtained by transp orting s along a path in the plane

ij ij

we nd

p

Z Z

SW

i

s

j i

ij

where as in

d

p

x

SW

and

lifted to sheet i of

i

Note that is always a closed cycle on the SW Riemann surface single

j i

lo op if is op en double lo op if closed Thus since is also precisely the

SW

Seib ergWitten meromorphic oneform we nd that the rst CY periods are

nothing but the Seib ergWitten p erio ds Note that these p erio ds are indeed small

as they are prop ortional to Comparison with furthermore yields

p

p



Finally using and the comment b elowit it is not dicult to show

that the intersection matrix of the small CalabiYau basis cycles is precisely

equal to minus the intersection matrix of the corresp onding four SW cycles This

Chapter Quantum YangMills gravity from I IB strings

is crucial to show the reduction of the lo cal Kahler p otential to the rigid Kahler

p otential see b elow

It is clear that higher order corrections gravityand other stringy eects

to eg the SW BPS mass formula can in principle b e calculated systematically in

this setup There are two sources of corrections the surface itself is corrected as

well as the meromorphic oneform

Large p erio ds

From we see that the regular part of the CY p erio ds should

b ecome indep endent of the rigid mo duli b in the limit since on one

hand the regular part of those p erio ds should b e continuous in B at their

limiting values while on the other hand B and themselves b ecome

indep endent of the rigid mo duli when This together with

implies in particular

T k O

ln

V k O k O

i

with k k constants indep endent of the rigid mo duli and k Alternatively

one could follow the argument outlined in the general discussion to establish the

indep endence of the rigid mo duli of these constants

Though not really necessary for the conclusions it is p ossible to calculate

k from the exact solutions of the K periods given earlier The result is k

p

B

withB and B as in i

B



Kahler p otential and reduction to rigid SG

The Kahler p otential is expressed in terms of the p erio ds as follows in the second

line the smalllarge p erio ds and corresp onding intersection matrices are denoted

as in the general treatment

Z

K ln i

CY

Z Z Z Z

ln iQ iq



p

  

jT j jT j jj K ln ImV T V T

ln jj

p

jk j Im k k jk j jj K O ln

An explicit example



jk j

b Imk k This is precisely of the form with L a

p

We can therefore pro ceed exactly as jk j and K as in with j j



in the general discussion whichwe will not rep eat here Note in particular that

we nd for the SU scale

r

N

L

a ln jLj b

Moreover since wehave the exact value of k we can also compute

r

p p

B

a

B

and hence the constant k M M determining the ratio of string and string

SU S

gauge unication scales dened in is

p

B

k

B

Chapter Quantum YangMills gravity from I IB strings

Chapter

Attractors at weak gravity

Having studied the brane representation of BPS states in weak gravity quantum

YangMills theory one could wonder how these states lo ok from the eective eld

theory p ointofview Are they also attractors as the N black holes describ ed

in chapter From the general discussion there one would exp ect they are But

are they black holes then Have they naked curvature singularities Or are they

smo oth solutions with only weak backreaction on the spacetime metric like clas

sical nonab elian YangMills monop oles Physicallyonewould exp ect the latter

Indeed as long as the elds are suciently slowly varying which recalling the

general discussion in chapter can always b e arranged by taking the charge N of

the state suciently large the eects of the massive nonab elian degrees of free

dom are taken into account to an arbitrarily go o d approximation in the quantum

ab elian eective action Furthermore the nonab elian gauge b osons are always

massiveinan N theory and the monop olesdyons decouple from the massless

elds in the infrared so we dont exp ect the eective action description to break

down in a fatal wa yanywhere if the charge is suciently large Thus we exp ect at

least curvature singularity free solutions of the eective action corresp onding to

semiclassical monop oles Only then we can really say gravity can b e consistently

decoupled from the eective theory something whichwe exp ect to b e the case for

an asymptotically free theory

These arguments are of course rather heuristic but we will see that in a

quite subtle way our exp ectations are indeed conrmed By careful analysis of

the attractor ow equations in the weak gravity limit we will b e able to show that

we obtain indeed monop ole solutions free of physical singularities with nite mass

entirely residing in the elds thereby solving some paradoxes and puzzles which

were encountered in earlier studies of this sub ject Not unexp ectedly

Chapter Attractors at weak gravity

these solutions are very similar to the conifold states we discussed in section

Note that the ab ovephysical arguments do not hold for N YangMills

this theory has p oints in mo duli space where the gauge b osons b ecome massless

and is not asymptotically free The two derivative low energy ab elian eective

action for an N YangMills theory is indeed classical and it is well known that

classical ab elian gauge theories do not have singularityfree monop ole solutions

The discrepancy in this resp ect with the full nonab elian theory can b e understo o d

from the fact that at the attractor p ointoftheows in mo duli space the gauge

b osons are massless and the coupling constant nite so that they can denitely

not b e neglected near the monop ole core

The outline of this chapter is as follows We rst derive the form of the

eective reduced action for spherical solutions in the weak gravity limit

which in principle should yield all static spherically symmetric solutions BPS or

not To sharp en our intuition wegive again an in terpretation in terms of a particle

moving in a certain p otential We then restrict to BPS solutions in other words

the attractor ow equations in the weak gravity limit We discuss prop erties of

the solutions we nd and comment and sp eculate on p ossible extensions of our

results

An extensive review on classical dyons coupled to gravity can b e found in

while asp ects of quantum dyons without gravityare studied eg in

and in particular by Chalmers Ro cek and von Unge in Some related work

can b e found in

Attractor ow equations in the weak gravity

limit

In the following we will make the same assumptions as in section In par

ticular we will assume the ratio of gauge and Planck scales to b e very small

We will also frequently make use of the notations and formulas intro duced there

and elsewhere in chapter

Spherically symmetric congurations at weak gravity

The weak gravity metric on mo duli space derived from is as can also b e

read o rom

g K

sw

ln jM j jj SU

Attractor ow equations in the weak gravity limit

g g

sw ij ij

g K

i sw

i

where g K is the Seib ergWitten metric on rigid mo duli space and

i sw

sw ij j

the string unication scale M asintro duced in section is given by M

S

SU

 

jk j jk j

with b the b eta function co ecient of the gauge theory

 

jbj lnjj

Re S

 



under consideration and k a mo del dep endent constant In the approximation we

are making the inverse metric is

g ln jM j jj

SU

ij ij

g g

sw

i ij

g ln jM j g K

SU j sw

sw

We consider one of the light cycles The relation b etween the sup ergravity

central charge Z and the Seib ergWitten central charge Z is given by

sw

p

p

Z G Z Z

N sw sw

With this information we can calculate for a given charge the p otential V as

dened in

V krZ k G ln jM j jZ j G jZ j

sw N SU sw N sw

sw

where

hrln jZ j rK i

sw sw sw

and we have denoted the scalar pro duct norm on the Seib ergWitten mo duli

ij

h and krf k space by h i kk that is hrf rhi Reg f

sw sw sw i

j

sw sw

hrf rf i It is straightforward to check that when the mo duli space

sw

geometry is classical that is when K is quadratic and the metric at From

sw

the microscopic quantum eld theory or heterotic p oint of view the second term

in the right hand side of can therefore be interpreted as a gravitational

ie G backreaction correction to the p otential which is caused byloopand

N

anishes in the ultraviolet free nonp erturbative quantum eld theory eects also v

vanishing electric coupling limits of mo duli space in SU case since

the Kahler potential b ecomes eectively quadratic there However it do es not

vanish it even diverges for the infrared free vanishing dual magnetic coupling

Recall the physical meaning of the p otential V is the energy density of the electromagnetic

elds including the graviphoton

Chapter Attractors at weak gravity

Figure jZ j plotted as a function of u in a neighb orho o d of u fora

sw

purely magnetic charge

limits in the SU case as the Kahler potential do es not b ecome

D

eectively quadratic there see for example On the other hand the full

second term in V diverges in the ultraviolet free limits while it is zero in the

infrared free limits if the charge under consideration is that of the particle with

vanishing mass in this limit in fact the complete p otential then vanishes as it

should of course for the energy density of the elds of a zero momentum massless

particle In general can be p ositive or negative dep ending on the relative

angle of the gradients of K and ln jZ j but at weak electric coupling it is always

p ositive In g jZ j is plotted as a function of u in a neighb orho o d of

sw

u for a purely magnetic charge Note also that the factor G ln jM j

N SU



with M the string scale and b the b etafunction in frontof is equal to



S



G M b

N

S

co ecient of the gauge theory Hence in a large N ie large b limit with xed

string and Planck scale this term will b e suppressed as well

We will come back to this term and give a geometrical interpretation b elow

Finally weplugthe expressions for the mo duli space metric in the reduced

eective action with h for general static spherically symmetric solu

tions and obtain

Z

jj

S T d fU c g

D

G ln jM j

N SU

Attractor ow equations in the weak gravity limit

Z

j

U i i

cc e V g K u d fg u u

i sw

sw ij

with V as in

From this action by completion of squares or simply from and

by substitution of one obtains the weak gravity approxi

mation to the attractor ow equations

p

U

U G e jZ j

N sw

p

U

G e ln jM j f u u

N SU

p

ij U i

g e jZ j u

sw

j

sw

where

ij

ln jZ j f u u g K jZ j

sw i sw sw

j

sw

U

As exp ected the variation of the scale and the metric red shift factor e vanishes

when G The factor f app earing in the rhs of is closely related to

N

in jZ j Ref Analogous to the analysis of earlier it can b e seen

sw

that f vanishes in any classical limit of mo duli space where the coupling of the

charge to the electromagnetic eld vanishes Decomp osing cohomologically

SW

i i

as c c some elementary manipulations givethe

SW i i

SW SW

following geometrical interpretation to f

Z

i

f e

Here is the phase of Z This expression makes it obvious that f vanishes in

the classical case since itself is a holomorphic form then From the

SW

microscopic nonab elian eld theory p oint of view the space variation of the gauge

theory scale is therefore entirely a quantum eect in the presence of gravity

that is string theory

Massless and Schwarzschild charged black holes

The weak gravity solution can b e used to study Stromingers massless black holes

If we take Z to b e already zero at spatial innity the attractor ow equa

tions gives simply at space with constant mo duli but still the energyless

electromagnetic eld of a p ointcharge We shouldnt exp ect more of course for

a massless particle at rest However we can let the rest mass approach zero and



This name is purely historical They are just particles not black holes

Chapter Attractors at weak gravity

simultaneously b o ost the solution along the xaxis to approach the sp eed of light

p p p

keeping the energy E jZ j v xed When v

the b o osted metric is given by

ds dt dx U dt dx dy dz

p

and for nonzero x t Denoting U U and Y Z

jxtj

we have jY j E and from the weak gravity attractor ow equations

const

dU

G jY j

N

d

djY j k

d ln jY j

with k a p ositive constant This implies jY j const E and U G E

N

So we nd a simple but nontrivial solution with Z everywhere the

electromagnetic eld strength of a p oint particle b o osted to the sp eed of light and

ashockwave metric due to the combined eect of expansion of the core region

in the rest frame and longitudinal Lorentz contraction while taking the limit

G E

N

dx dt ds dt dx dy dz

jx tj

This is the AichelburgSexl metric for a massless particle It is lo cally at and

can b e brought to standard form bychanging co ordinates as t t GE ln jt xj

x x GE ln jt xj where the upp er lower sign is to b e used for t x

t x

At conifold p oints of mo duli space Z the low energy eective eld

theory also allows charged Schwarzschild black holes with arbitrary mass with

mo duli xed at the conifold as can b e seen immediately from the action

Their extremal limit are the Strominger massless BPS states This freedom corre

sp onds to the freedom to add a bare mass to the massless monop oles at the u

p oint in Seib ergWitten mo duli space thus breaking in eld theory explicitly

sup ersymmetry to N From the p oint of view of the eective particle motion

this corresp onds to a tra jectory on the top of the V p otential with arbitrary

initial velo city in the U direction as was already indicated in g

Attractor ow equations in the weak gravity limit

Rigid limit

If we send G all the way to zero ie if we take the rigid limit the dynamics of

N

the gravitational eld and the scale decouple and we can consistently put U

const Then b ecomes simply

Z

j

i

S T d fg u u krZ k g

D sw

sw ij

sw

which could also have been obtained directly from the rigid eective action of

course By completing squares one nds immediately the attractor ow equations

in the rigid limit

Z

p p

i ij

S T jZ jjZ j d ku g jZ jk

D sw sw sw

j

sw

which is minimized when

p

ij i

g jZ j u

sw

j

sw

The other sign again leads to an unphysical innite energy solution Avery useful

prop erty of the rigid attractor ows is the fact that the phase of the central charge

is constant along the ow as can b e veried by a short calculation Essentially

this is thanks to the holomorphicityof Z in the rigid limit

The rigid limit of the formulas of section is also very useful In particular

equation b ecomes

p p

i i

Z Im e Z Im e

sw

i

with e the constant phase of Z

sw

The rigid limit of is of practical interest as well

ij

Img Z Z

i sw sw

j

sw

ij

Reg Z Z

i sw sw

j

sw

Solutions in the rigid limit

Since the O G corrections are small it is sucient in rst approximation to

N

We will do so by reconsidering the analysis in investigate the exactly rigid case

section Again the system is equivalent to a particle moving in a certain

U

p otential V in the exactly rigid case e V if we do consider the gravity

p erturbation Crucial again are thus the critical p oints of V In the following we

will drop the subscript SW to indicate rigid quantities since we will restrict to

the rigid case anywayfornow

Chapter Attractors at weak gravity

Critical p oints

Evidently critical p oints of Z are automatically minima of V krZ k with

critical value V Also since Z is analytic and since we already know from

cr

section that jZ j can only have minima Z must b e zero at its critical p oints

R

Alternatively one can see this as follows From it follows that

at a critical p ointofZ where is the cycle under consideration Since the Ho dge

pro duct is p ositive denite this implies that is a vanishing cycle which has

Z at nite p oints in mo duli space

In the pure SU YangMills case completely decoupled from gravity the

converse is also true in the sense that minima of V must b e minima of jZ j Indeed





jnm j

m

for a central c harge Z n m wehave V with

D D

Im Im

m

Minima of V are therefore maxima of Im Since is analytic and

nm

this maximum has to b e a p ole so V krZ k at this p oint that is it is a

critical p ointofZ as well and by the previous argument also a zero of Z It is

not clear to us if and how this generalizes to higher rank gauge groups

Anyway let us fo cus now on critical p oints of V which are also critical p oints

U

of Z Then since V the p otential e V is at in the U direction at the

cr

critical p oint as shown in g

Rigid BPS solutions

As argued ab ove the cycle must vanish at the critical p oint Therefore we

exp ect the situation close to a critical p ointtobevery similar to our discussion of

the conifold attractor in section Take the example of SU YangMills The

candidates for spherically symmetric BPS solutions are the charges which can have

vanishing mass here the monop ole and the elementary dyon Note that these

are the only candidates there will b e no spherically symmetric BPS solution for

example for a purely electrical charge a Wb oson This is a good thing since

the Wb osons are in a BPS vector multiplet and by the arguments of section

we do not exp ect a rotationally invariant b osonic solution for N BPSvector

multiplets We will come back to this later A quick glance at and

learns that close to u the geometry is indeed that of a conifold p oint The

exact p otential V for a magnetic charge in a neighb orho o d of u is given

in g Note that wehave indeed a maximum for V Though the top is a

sharp spike the details of the equations are such that we can nd a continuously



We assume m Thecase m is completely analogous



In the usual cut conventions ie all cuts as much as p ossible to the left in the u plane the

elementary dyon has charge the relative sign b etween electric and magnetic charges

dep ending on how the singularity is approached if from Im u

Attractor ow equations in the weak gravity limit

U

Figure The eective particle p otential e V near a critical p ointintheweak

gravity limit The p otential is at in the U direction equal to zero at the critical

p ointofV Tra jectory a is a typical weakly gravitating BP S solution Tra jectory

b is a charged Schwarschild black hole with vanishing electromagnetic energy in

a Z vacuum Tra jectory c corresp ond to a generic nonBPS black hole V

itself is drawn as a smo oth p otential for the sake of the picture but actuallyit

will have a sharp spikelike top in general

Figure Potential V for the eective particle in a neighb orho o d of u in the Seib ergWitten plane

Chapter Attractors at weak gravity

Figure Value of ju j as a function of the radial distance for N units of

magnetic charge with ur i The attractor point is reached at

r r N Though the radial derivative of the solution do es not lo ok

continuous on this picture it actually is but the drop to zero slop e happ ens over

avery small distance

dierentiable BPS attractor ow solution for the eective particle motion ending

at the top as wehave seen for the conifold case in section For ows not close

to the critical p oint numerical analysis is needed As an example the numerically

integrated ow for N units of magnetic charge with ur i is shown

in g The mass of such a monop ole is N the mass of the gauge

b oson at this p oint in mo duli space is The attractor p oint is reached at

the nite radius r r N This core radius increases when ur

moves towards the attractor p oint Inside the region r r the mo dulus u is

constant u Only the electromagnetic eld F is nonvanishing here since it is

U

as always given by here with e h However this electromagnetic

eld contains no energy as can readily b e seen by recalling the fact that V is



r

nothing but the electromagnetic eld energy density A plot of the emf energy

densityasa function of radial distance for our example is given in gs and

Note that this is half of the total energy density for BPS solutions as implied

eg by or directly from the attractor equations

An exact formula for the core radius of a charge N magnetic monop ole can

b e obtained from together with and the fact that Z for

Attractor ow equations in the weak gravity limit

Figure Electromagnetic energy density as a function of the radial co ordinate

for the ow of g This is half the total energy density The density drops to

zero inside the core rr The dotted line is the energy density the electromag

netic eld would have if the mo duli were kept constant This diverges of course at

r

Figure Energy distribution over space as in g Most of the energy is lo calized in a shell surrounding the core

Chapter Attractors at weak gravity

as can b e read of from

N j j

D

p

r

Im

D

When u is large ie weak electric coupling this reduces approximately to

N N

p p

r

jj

juj

which is exactly equal to the core radius of a classical nonab elian t Ho oftPolyakov

monop ole

Force elds and p otentials

Apart from the energy another measurable prop erty of an electromagnetic eld is

the force it exerts on a test particle For a test particle of unit magnetic charge in

the eld of a magnetic monop ole this is given by as F NV ur r

plotted for our numerical example in g The force drops to zero inside the

core in agreement with the fact that the dual electromagnetic coupling constant

is zero at u in fact it is not dicult to see that V is actually prop ortional to

the dual coupling

More generally also for more general groups from and it

follows that the electromagnetic force on a test particle with charge in the

eld of a particle at rest with charge is given by

e e

r r

F v

em

r r

where v is the velo city of the test particle Using this b ecomes

e e

r r

v F hrZ rZ i

em

r r

If the elds satisfy the attractor ow equations for the charge of the particle at

rest a short calculation using shows that the force can b e nicely derived

from a p otential

p

i

r Re e v Z F r

em

r

where is the constant phase of Z For colinear charges ie the

p

jZ j the sign dep ending on the relative sign of the charges p otential is equal to

Attractor ow equations in the weak gravity limit

Figure Static electromagnetic force F as felt by a test particle of unit magnetic

charge in function of the radial distance from the monop ole of g This is

half the total force felt by a static test particle with unit magnetic charge opp osite

to the charge of the monop ole at r The force drops to zero inside the core in

agreement with the fact that the dual electromagnetic coupling constant is zero

at u The dotted line shows the force if the mo duli would b e kept constant

minus for equal signs The attractor ow equations imply this p otential to b e

strictly monotonous till the core is reached and approximately Coulomb at large

r

On the other hand the force on the test particle due to scalar exchange

p

R R

is easily seen from giving in the rigid limit S Mds jZ jds

DB I

to b e

p

F r jZ j

sc

Therefore the total static p otential W for the test particle in the BPS eld of

is

p

i

W jZ jRe e Z

Denoting the variable phase of Z with we can rewrite this as

p

sin jZ j W

Note that W is always p ositive and that it will reach a minimum zero when

it is of course at zero when the charges are p ositively colinear This

result is interesting it shows that when the attractor ow of passes through

a point where the phases of Z and Z are equal there is a stable point for the

Chapter Attractors at weak gravity

Figure The static force p otential W for a p ositive electric test particle in the

eld of a p ositivecharge N magnetic monop ole with mo dulus at innity equal to

i The minimum of W is lo cated at the radius where the owintersects the

curve of marginal stability

test particle in the eld of Note that in a vacuum with equal phases of Z

and Z the heaviest of the two particles is only marginally stable wrt decayto

the other plus a particle with complementary charge of course For the SU

case this means that such minimum if it exists is always lo cated at the radius

where the owintersects the curve of marginal stability Anumerically integrated

example is given in g The app earance of stable p oints suggest the p ossibility

of b ound states We will come back to this later

Returning to the SU example we get for the static electromagnetic force

co ecients linear combinations of the following cycle pro ducts Im

ReIm Im From eg the approximate expressions

and it is straightforward to check that also the em force on a static elec

tric test particle will drop to zero at a monop ole core However the

magnetic force on a moving electric particle will stay nite since always

The em force b etween two Coulomb electric test particles on the other hand is

prop ortional to which diverges when approaching the core region a signal

of electric charge connementatu



Note that this is due to a nonzero thetaangle Re

Attractor ow equations in the weak gravity limit

Figure The potential V for an electric charge cycle V is multivalued

when one allows encircling singularities of mo duli space such that transforms

to another cycle n in this case There are no extrema at nite u

Electric BPS states and suicidal ows

Having found these nice solutions for magnetic monop oles and elementary dyons

one of course wonders what happ ens for the other BPS states which exist at weak

coupling the n dyons and the pure electric charges Let us consider for

example the electric particle corresp onding to the cycle The p otential V do es

not have a minimum at nite u for this charge This follows from the argument

that the cycle under consideration must vanish in a critical p oint of V plus the

fact that neither the cycle nor any of the cycles generated by mono dromyfrom

it vanish in a p oint of mo duli space For example in a neighb orho o d of u

the potential V for an electric charge is shown in g which also elucidates

probably how it can be that V has no extrema at nite p oints of the covering

space of mo duli space for this charge

There is a minimum of V at u in the rigid approximationit can indeed

be checked from that lim V but since implies that the

u

p

energy of a solution is always larger than jZ Z j the energy of a solution

owing to innity is innite so certainly not acceptable as a BPS state This do es

not exclude solutions owing to large values of u when gravity is turned on again

only that these solutions then will involve gravity in an essential way It can indeed

b e seen from that the minimum at innity will get displaced to nite mo duli

values when the Newton constant G is turned on The corresp onding solution N

Chapter Attractors at weak gravity

will be a black hole and will not be BPS It cannot be identied with the eld

theory vector multiplet

If we try to solve the attractor ow equations for we nd a false or

suicidal ow as in case in section terminating on a regular p ointofthe

line of marginal stability where Z We will discuss this phenomenon in more

detail for the n dyons

All in all we conclude for SU that thereare no electrical ly charged spher

ical ly symmetric BPS solutions at weak gravity Weshouldbehappy with this of

course since the electric BPS states are in a vector multiplet and as explained in

section we do not exp ect a spherically symmetric solution without fermionic

excitations in this case

We can understand the unevitability of this situation purely from asymptotic

freedom and the unstability of the electrically charged particles at strong coupling

Asymptotic freedom means the coupling grows when the mass of the electric par

ticle decreases On the other hand attractor ows tend to minimal mass hence

maximal coupling in this case Given the fact that the particle do es not exist

at strong coupling it is therefore imp ossible to havea legitimate attractor solu

tion Turning this argument around we conclude that the eld theory describing

BPS hyp ermultiplets at weak gravity must be infraredfree This is also what one

exp ects from the general N b eta function results

Of course there does exist a purely electrical BPS multiplet in the N

SU theory So how is it realized in the eective eld theory We believe it

should b e considered as a b ound state of an elementary dyon and a monop ole with

opp osite magnetic charges Evidence for this prop osal is the charge matching the

stability of the constituents at all mo duli values and the app earance of a p otential

as in g for the force on a test dyon in a monop ole eld and vice versa

This suggests one should lo ok for a centered BPS solution with charges

resp in the centers Wehave not b een able to nd such a solution however

Other p ossibilities worth lo oking at are solutions with fermions turned on of which

some should give a spin zero solution according to the N multiplet logic and

solution with charge distributed over a surface in stead of concentrated in p oints

The connection with pronged strings see b elow and the brane picture of

N theories could provide useful hints towards the solution of this problem

Below for the higher dyons we will present a quantum mechanical picture of

such b ound states which is quite attractive and makes contact with the pronged string picture

Attractor ow equations in the weak gravity limit

Figure Monop ole attractor ows to u in the Re u part of mo duli

space The fat line represents a cut

Higher dyons

The higher dyons with electric charge jnj and magnetic charge jmj

present a puzzle In general the same problems as for the purely electric charge

arisea spherically symmetric BPS solution do es not exist On the other hand

such BPS states can be obtained at weak coupling either from a monop ole for

even n or an elementary dyon for o dd nby p erforming a numb er of times the

i

mono dromy u e u as can b e seen from It seems quite unlikely

that the attractor owwould all of a sudden cease to exist at a certain p oint while

i

we are continuously varying the mo duli at innityasu e u R for a large

initial value of u Nevertheless this is exactly what happ ens

To see this consider the ows to a monop ole core while we are thus

varying the mo dulus at innity In the part of mo duli space with Re u

nothing can go wrong as can be seen from g But when we keep on

varying counterclo ckwise the mo dulus at innity whichwe assumed to b e large

we get into the Re u part where eventually when exactly dep ends on

the precise choice of cut p osition the cycle under consideration will b e assigned



Dep ending on where one puts the cuts in mo duli space there can b e a limited range of

mo duli at innity for whichsuch solutions still exist for charges obtained from the monop ole or

elementary dyon by a single mono dromy ab out u Such ob jects should rather b e considered

as elementary dyons or monop oles however since the corresp onding ows necessarily pass through

this cut again such that suciently close to the attractor p oint the charge is again a multiple

of the vanishing monop ole or dyon charge See also g

Chapter Attractors at weak gravity

Figure Monop ole attractor ows to u in the Re u part of mo duli

space The dotted lines indicate the false ows which one obtains when one tries

to pull the ows through the u singularity The fat straight line to the right

of the singularity represents a cut In the usual conventions there is also a cut to

the left but wehave not indicated this one her

charge So let us follow the ows further there in g Here something

does happ en Everything go es well till we reach the ow touching the u

singularity When we try to pull the ow through the singularity ie when we

rotate the mo dulus at innity a little bit further a catastrophe happ ens the ow

now passes through the cut at the right side of u such that its charge changes

identity wrt the neigh b oring owat its left hand side and in particular is no

longer a go o d monop ole charge The solution breaks downthe o w turns into a

false one again terminating on a regular p oint of the line of marginal stability

as in case in section Let us clarify this a bit

First consider g where the p otential V u u for the eective particle

describing the u dynamics in this case is plotted as a function of u The

indicated tra jectory corresp onds to the fattened ow in g When moving

the starting point of the particle more and more to the right corresp onding to

counterclo ckwise rotating the mo dulus at innity at a certain p oint the particle

will no longer b e able to reach the top of the monop ole p otential at least not in

a BPS way that is such that its total energy is zero This makes it intuitively

clear why the spherically symmetric BPS solution ceases to exist

One can also understand this at the level of the ows The attractor ows are

gradientows for the function jZ j that is they owdo wn the hill towards minimal

Attractor ow equations in the weak gravity limit

Figure The potential V u u for the eective particle describing the u

dynamics in the situation shown in g The indicated tra jectory corresp onds

to the fattened owing When moving the starting p oint of the particle

more and more to the right at a certain p oint it will no longer b e able to reach

the top of the monop ole p otential at least not in a BPS way that is such that its total energy is zero

Chapter Attractors at weak gravity

Figure jZ j as a function of u for the ows of g The indicated ow

corresp onds to the fattened ow in g Beyond a critical ow down the

hill no longer leads to the monop ole zero of Z

jZ j In g jZ j is plotted as a function of u for the case at hand Again the

indicated ow corresp onds to the fattened ow in g Rotating the mo dulus

at innity at a certain point namely at the ow touching u down the

hill no longer leads to the monop ole zero at u but to a certain regular zero

of Z on the line of marginal stability as shown in g a This is a false

owhowever in that it do es not corresp ond to a genuine solution of the equations

of motion Indeed as one can see in g b the eective particle p otential

is not extremal at this p ointit is just an ordinary regular p oin t and the particle

will simply continue its journey down the p otential hill there corresp onding to

an inverted BPS ow which as wehave seen several times alreadyleadstoan

unphysical singular innite energy solution

Now if the higher dyons are not realized as spherically symmetric BPS solu

tions what are they then And how can they b e continuously connected to the

spherically symmetric monop ole or the elementary dyon Again we b elieve they

are realized as b ound states of monop oles and elementary dyons In the case at

hand this would b e two elementary dyons of charge b ound to the original

charge monop ole

There are good reasons to b elieve in this First of all when considering

the dyons as test particles in the monop ole eld or vice versa we nd in the

situation at hand a stable equilibrium at nite distance as in g Furthermore

quantum mechanicallysuch a b ound state would indeed b e continuously connected

Attractor ow equations in the weak gravity limit

Figure Left a jZ j as a function of u for the continuation of the ows of

g b eyond the critical ow This connects smo othly to the lower sheet of

g The indicated ow corresp onds to one of the dotted lines in g

It is a false ow terminating at a regular zero of Z Right b the corresp onding

eective particle p otential Here we see clearly that the ow is indeed a false one

not corresp onding to a solution of the equations of motion The dotted lines show

how the true particle motion continues b eyond the false attractor p oint

Chapter Attractors at weak gravity

to the monop ole in the following sense Supp ose wehave a monop ole with ur

innitesimally close to the critical ow Now deform ur to a tra jectory in mo duli

space which lies just at the other side of the critical ow this do es not havetobe

a solution to the equations of motion and in particular will b e excited in energy

ab ove the BPS b ound Denote the radius at which u in the critical owby

r Now imagine that just b efore the move a virtual monop oleantimonop ole pair

c

was created to be destroyed again just after the move and that the monop ole

happ ened to be at r r when the critical tra jectory was crossed while the

c

antimonop ole was at r r Then the spacetime tra jectory of the monop ole

c

antimonop ole pair mapp ed to mo duli space via u rt encircles the p oint u

Consequently there is a mono dromy on the monop ole charge of which the net

result is that we are left with two almost massless elementary dyons when the

monop oleantimonop ole pair is destroyed again Now if the presence of the dyon

charge manages to relax the energy of the complete conguration back to its

BPS b ound this state b orn out of a quantum uctuation will actually survive

as a stable state whichwe can deform further maintaining the BPS prop ertyby

further rotating the mo duli at innity

Howwould such a b ound state lo ok like quantum mechanically The following

picture has some satisfying features though it is not clear how serious one can take

it Fix the overall translational zeromo de by putting the origin of our co ordinate

frame at any of the particle p ositions Say we put it at the monop ole p osition

We exp ect the wave function of the dyons to be concentrated at r r the

ms

radius where the u mo dulus is at the line of marginal stability since there lies

the minimum of the p otential for the dyon considered as a test particle in the

monop ole background Acharge distribution with a monop ole at r and

units of dyon charge smeared out over a spherical shell at r r will

ms

pro duce a BPS eld conguration which is a attractor ow for r r

ms

and a ow for r r See g The mass of the dyonic shell is

ms

p

jZ msj with Z ms the dyon central charge at the marginal stability

p oint ur The mass in the elds outside the shell is according to equal

ms

p

to jZ jjZ msj with Z the central charge at r of

and Z ms the same thing at the marginal stability p oint The mass of

p

the elds inside the shell is in obvious notation jZ msj Thanks to the fact

that the phases of Z ms and Z are equal at the marginal stability point

the total mass of the system assuming sup ersymmetry eliminates the zero p oint

p

energies adds up to jZ j that is the BPS b ound is saturated The



This should b e considered as giving a b oundary condition to calculate exp ectation values of

elds with the eective eld theory Also for the case of a single monop ole the p osition had to

b e xed b efore we could use the eective eld theory In the same spirit the mo duli at innity

had rst to b e xed as they have a at p otential



The owforrr also determines the value of r

ms ms

Attractor ow equations in the weak gravity limit

Figure Two p ossible representations of the BPS b ound state of twodyons

and a monop ole dep ending on howwe eliminate the translational zeromo de Left

monop ole at the center dyons in a shell Right dyons at the center monop ole

in a shell The attractor ows are indicated by arrows and lab eled bythecharges

they corresp ond to

dyonic shell can also b e represented as a charge ow inserted at r r

ms

in the f g ow starting from the marginal stability p oint This is how

the eld exp ectation values would lo ok around the dyons if wewould have started

by taking the dyons at the origin of our co ordinate frame in stead of the monop ole

in that case the monop ole would b e represented as a ow insertion This can all

b e summarized nicely in a pronged ow picture as in g

Analogouslyn dyons are represented as b ound states of n elementary

dyons and n monop oles with opp osite magnetic charge

Note that the construction implies that as the mo dulus at spatial innity

approaches the line of marginal stability the distance between the comp onents

of the b ound state will grow since marginal stabilityisreached closer and closer

to r till eventually when one reaches the MS line the comp onents are at

innite distance and the state decays in its comp onents This is a very pleasing

physical picture of the decay of these states which is indeed known to o ccur when

the vacuum crosses the curve of marginal stability in mo duli space Also the

ab ove construction of b ound states is clearly not p ossible inside the MS curve

again in agreementwithphysical exp ectations

There is a remarkable connection with a completely dierent picture of N

SU YangMills theory in string theory in typ e IIB string theory one can

repro duce the Seib ergWitten eective action as the low energy eective action of

a noncompact brane in the background of a and a brane placed

at nite distance from each other The geometry transverse to the branes

Chapter Attractors at weak gravity

Figure The pronged ow picture representing a charge bound

state of a monop ole and twodyons The state can b e considered as a ow

followed bya ow with a ow inserted at the transition p ointat

marginal stabilityoras followed by with a insertion The

picture is the same in b oth cases The total energy is just the sum of the energies

of the dierentows and saturates the BPS b ound The dotted line indicates the

false ow obtained by continuing the original ow MS indicates the line

of marginal stability

Multi monop ole dynamics

turns out to be precisely the Seib ergWitten geometry BPS states corresp ond

to or strings stretching b etween the branes and the corresp onding

branes the former giving rise to monop oles the latter to elementary dyons

The minimal energy stretched strings turn out to coincide with the attractor ows

in the transverse plane Precisely at the critical ow considered ab ove also the

single stretched string ceases to be the minimal energy solution Instead the

socalled three pronged strings further represent the BPS states Such

pronged strings lo ok exactly as the pronged ows in g which was of

course also inspired by the string picture The pure electric charge corresp onds to

a threepronged string as well in this picture Incidentally the sp ontaneously cut

o solutions we found for the monop ole and the dyon were recently repro duced

in the brane picture in There are some similarities with aswell

Charges not in the BPS sp ectrum

Quantum N SU YangMills theory is b elieved not to have other BPS states

apart from those having a multiple of the charges wehave considered thus far

Can we see this from the low energy eective eld theory

Clearly none of these charges will have a spherically symmetric BPS solution

For many of them a b ound state construction as eg for the n dyons is

furthermore not p ossible b ecause an interaction p otential with stable p oints at

nite distance as in g do es not arise However it seems that the ab ove

considerations do not exclude b ound states which are not actually in the BPS

sp ectrum though more careful study could p erhaps achieve this

A similar problem arises in the pronged string picture of BPS states There

this is solved by lifting string theory to Mtheory

It would be interesting to study nonBPS states of the eld theory in this

context These have attracted quite some interest recently The relation with

the nonBPS b ound states found in could also b e worth investigating

Multi monop ole dynamics

As explained in section the results for spherically symmetric solutions can

immediately b e extended to equal charge N center solutions simply by replacing

by the p otential

N

X

N r

i

i

Chapter Attractors at weak gravity

Figure The b oundary of the core of an Nmonop ole conguration is simply

P

N

given by the equip otential surface r r where r is the one

i

i

N

monop ole core radius

where r is the distance to the ith center An example of how such a conguration

i

can lo ok like is given in g

An interesting and imp ortant question is what can b e said ab out the low en

ergy dynamics of such N center congurations A b eautiful and very p owerful to ol

to tackle this problem is the mo duli space approximation This amounts to

separating collective co ordinates from uctuations integrating out the uctuations

and considering the limit of small time derivatives of the collective co ordinates

a

This pro duces an eective action for the collective co ordinates x whichwillbe

of the general form

Z

b a

dx dx

S dt G x

coll ab

dt dt

The collective co ordinates parametrize a space called the mo duli space not to

be confused with the mo duli spaces encountered thus far and G x can be

ab

considered to b e the metric on this space The action is then simply the

action for nonrelativistic geo desic motion on mo duli space

The mo duli space metric for ReisnerNordstrom black holes was obtained in

Some asp ects of the classical N SU YangMills monop ole mo duli

space were discussed in

For a classical SU BPS N monop ole conguration the collective co or

dinates can be identied with the monop ole p ositions together with an internal

Multi monop ole dynamics

angle for each monop ole asso ciated with the unbroken U gauge symmetry If

the angles can eectively b e considered as compact S dimensions of mo duli space

one would exp ect these mo des to freeze out when doing quantum mechanics and

considering the extreme low energy limit as there will b e a mass gap to the ex

citations of the angles The internal angle is in a certain sense still visible in the

low energy theory however its excitations corresp ond to the n dyons So if

the mass gap indeed exists we exp ect only the monop ole p ositions to remain as

collective co ordinates This is indeed what we seem to get

Our aim is thus to obtain the mo duli space metric on our collective co ordinate

N

space M R S in the U low energy Seib ergWitten theory Denote

N

the vector p otential as in by A For a static multicenter monop ole BPS

solution wehave from generalized to the multicenter case

A A

Dir

where A is the Dirac multimonop ole vector p otential satisfying

Dir

rA r

Dir

p

Here wehave put for convenience in

We exp ect from general considerations that in the gauge rA the

action is here

Z Z

dA du

dt d x S

coll

dt dt

Z Z

dA du

dt d xg

uu

dt dt

Z Z

du dA

uu

dt d xg g j j

uu u D

dt dt

Wehave no direct pro of for this however A short calculation involving the rigid

attractor ow equations transforms the latter expression to

p

Z Z

djZ j d dA

sw

S dt d x

coll

d dt dt

The quantitybetween square brackets is purely geometrical and can in principle

b e calculated with elementary metho ds for arbitrary monop ole p ositions and ve

lo cities The eect of the interactions with the integrated out massive charged

Chapter Attractors at weak gravity

djZ j

sw

Figure Weight function foratwo monop ole conguration

d

particles resides entirely in the weighting of the integral over space with the factor

djZ j

sw

which can b e interpreted as a mass density p er unit of The heighest

d

weightisthus given to a shell surrounding the core of the N monop ole solution

as illustrated for N in g

For a single particle at the origin moving at sp eed v say in the z direction one

dA d

v cos and v sin Hence the collective co ordinate nds

dt dt

action is simply

Z

p

dt S jZ j

coll sw

p

jZ j which is indeed what one would exp ect for a particle of mass

sw

Note that the central charge is evaluated at spatial innity not at the particle

worldline That is the mass app earing in front of the action is not the bare mass

but the true physical quantum corrected exact mass of the monop ole

Studying the dynamics of the rigid BPS solutions in mo duli space approxima

tion and for example comparing this with would b e very interesting

We will keep this for another o ccasion however

App endix A

Samenvatting

Niets is wat het lijkt Als er iets is dat het b esluit mag zijn van de twintigste

eeuwse natuurkunde is dat het wel De zo ekto cht naar de fundamentele wetten van

de natuur leidde tot een b eeld van de werkelijkheid dat steeds verder verwijderd

scheen van de dagdagelijkse realiteit maar tegelijk steeds mo oier en eleganter werd

Op dit ogenblik wordt de top van die evolutie gedomineerd do or een merkwaardig

rijke en unicerende theoretische constructie stringtheorie

Ho e b evreemdend ecient esthetische overwegingen o ok gebleken zijn bij het

ontrafelen van dit b eeld de werkelijkewaarde van om het even welkekandidaat

fundamentele theorie zal steeds b epaald blijven do or wat die theorie ons kan leren

over onze observeerbare wereld Een veel geho ord clicheover stringtheorie is dat

ze enkel relevant is vo or pro cessen waarbij quantumgravitatie een b elangrijke

rol sp eelt Aangezien dit pas het geval is bij een energieschaal die zichverhoudt tot

de energie b ereikbaar in de huidige deeltjesversnellers zoals de afstand aardezon

zichverhoudt tot de dikte van een blad papier lijkt de situatie hier vrij hop elo os

Dat clicheisevenwel fout om volgende redenen

Er b estaat een twintigtal elementaire natuurkonstanten zoals deeltjesmassas

en kopp elingsconstanten die alleen zullen kunnen vo orsp eld worden do or een

theorie die een consistente b eschrijving geeft van de natuurwetten tot op wil

lekeurig hoge energieschalen zoals wellicht stringtheorie

Het is niet no o dzakelijk zo dat de schaal waarop sp ecieke stringfysica zicht

baar wordt dezelfde is als die waarop de gravitatiekracht b elangrijk wordt

dit kan al bij veel lagere energie het geval zijn In princip e is het zelfs moge

lijk dat er stringfysica tevo orschijn zal komen bij de eerstvolgende generatie

acceleratorexp erimenten

App endix A Samenvatting

Stringtheorie blijkt als kader o ok bijzonder nuttig te zijn vo or de b eschrijving

en analyse van lage energie fysica die traditioneel do or quantumveldentheo

rie wordt gegeven De recente do orbraken in nietp erturbatieve stringtheorie

hebb en geleid tot nieuwe resultaten in sup ersymmetrische quantumvelden

theorie die met traditionele technieken zo go ed als onb ereikbaar zouden ge

weest zijn Het succes van stringtheorie hierin is vo oral te danken aan het

diep meetkundige unicerende b eeld dat ze levert vo or tal van veldentheo

retische ob jecten Deze opmerkelijke resultaten tonen aan dat stringtheorie

meer is dan gewoon gravitatie in een quantumkleedje de theorie leidt tot

een nieuw b egripskader van de natuurkunde in haar geheel

Deze thesis sluit aan bij het derde punt We b ekijken een sp ecieke klasse

van mo dellen namelijk typ e IIB stringtheorie gecompacticeerd op CalabiYau

manifolds en we b estuderen in dit k ader een aantal asp ecten van de lage energie

fysica die uit deze mo dellen vo ortvlo eit We spitsen ons vo ornamelijk to e op twee

onderwerp en de aeiding van exacte lage energie quantum eectieve acties van

nietab else N sup ersymmetrische YangMills theorieen inclusief kopp eling

aan gravitatie en de studie van lage energie eigenschapp en van BPS to estanden

BPS to estanden zijn massieve geladen deeltjesachtige to estanden elementair of

niet die minimale energie hebb en gegeven hun lading en daardo or absoluut stabiel

zijn het prototyp e vo orb eeld is de magnetische monop o ol in sp ontaan gebroken

ijktheorieen

De b elangrijkste nieuwe resultaten in deze thesis zijn in orderofappearance

de uitwerking tot een nuttige analysetechniek van de analogie tussen sferisch

symmetrische oplossingen van de vier dimensionele lage energie eectieve

actie en de b eweging van een nietrelativistisch deeltje in een b epaalde p o

tentiaal op de mo duliruimte scalar manifold van de theorie

de directe expliciete aeiding van Y angMills lage energie eectieveactiesin

het kader van sp eciale meetkunde volledig binnen I IB string theorie zonder

gebruik te maken van T of Sdualiteiten

de fysische interpretatie van de parameters in de algemene limietpro cedure

van lo cale naar rigide sp eciale meetkunde

de exacte kopp eling van de eectieve quantum YangMills theorie aan gra

vitatie en aan de ijktheorie schaal die blijkt dynamischteworden wanneer

de gravitatiekrachtverschillend is van nul

de b eschrijving van quantum YangMills BPS to estanden bij zwakke gravi

tatie met bijzonder aandachtvoor de overgang van elementaire naar ge b onden to estanden

de eerste stapp en in de analyse van lage energie multimonop o ol dynamica in

N quantum YangMills theorieen inclusief een mogelijke connectie met

een veralgemeende Nahm constructie via een Maldacenatyp e corresp on

dentie

Deze samenvatting volgt de struktuur van de thesis met de b edo eling dat

het mogelijk wordt om b epaalde delen afzonderlijk in meer detail te gaan lezen

in de thesis zelf We richten ons in deze samenvatting in eerste instantie op de

geinteresseerde nietexp ert We zullen om die reden ook relatief veel aandacht

b esteden aan de omkadering van de resultaten

Ho ofdstuk Inleiding

De inleiding is o ok in eerste instantie op de geinteresseerde nietexp ert gericht We

motiveren de intro ductie van stringtheorie als fundamentele theorie en b espreken

de vo ornaamste sterke punten en tekortkomingen in de huidige p erturbatieve

formulering ervan Vervolgens intro duceren en b espreken we de ho ekstenen van

de recente vo oruitgang in het inzicht in nietp erturbatieve asp ecten van de theorie

Dbranen en dualiteit Tenslotte wordt het gebruik van stringtheorie als kader voor

lage energie fysica to egelicht We b esluiten het ho ofdstuk met een samenvatting

van de resultaten in deze thesis In wat volgt nemen we aan dat de b egripp en

ingevo erd in deze inleiding gekend zijn do or de lezer

Ho ofdstuk Lage energie eectieveveldentheorie

In dit ho ofdtuk voeren we het concept van een lage energie eectieve actie in

zo wel in string als in veldentheorie Het onderscheid tussen de PI eective actie

en de Wilson eectieve actie wordt geschetst en we leggen uit wat de b etekenis

is van de Seib ergWitten eectieve actie in deze context Rigide en lo cale sp e

ciale meetkunde worden gedenieerd en hun centrale rol voor N eectieve

acties wordt to egelicht Ons werk in volgend leggen we uit ho e rigide

sp eciale meetkunde tevo orschijn komt als een b epaalde limiet van lo cale sp eciale

meetkunde We b esluiten het ho ofdstuk met een comp endium van Seib ergWitten

theorie en haar veralgemeningen Dit ho ofdstuk bevat vo ornamelijk welb ekend

materiaal maar we do en een inspanning om een aantal losse eindjes in de gebrui

kelijke Seib ergWitten review literatuur aan elkaar te knop en

Het is vo or de geinteresseerde nietexp ert die wil weten wat precies de fysische

inhoud is van de eectieveveldentheorieen die b eschouwd worden do orheen deze

thesis wellicht nuttig hier even uit te weiden en een aantal elementen uit deze

korte samenvatting in meer detail to e te lichten

App endix A Samenvatting

Bij voldo ende lage energieen heeft om het even welke fysische quantumthe

orie die aan een aantal basisprincip es voldo et een b eschrijving als een eectieve

veldentheorie waarbij de lichte elementaire deeltjes van de theorie de quanta

van de b eschouwde velden zijn en de quantum gecorrigeerde b ewegingsvergelij

kingen van die velden gegeven worden do or extremalisatie van een eectieve actie

Dit is een functionaal die gegeven wordt do or de integraal over de ruimtetijd van

een b epaalde niet no o dzakelijk lokale uitdrukking in de velden Bij voldo ende

lage energie kan men deze uitdrukking meestal go ed b enaderen do or termen met

ten ho ogste twee afgeleiden dit komt neer op een momentumexpansie tot tweede

orde De eecten van de hoge energie vrijheidsgraden zijn uitgentegreerd en

b epalen enkel de precieze vorm van de eectieve actie

De theorieen die we b estuderen in deze thesis hebb en N supersymmetrie

dat wil zeggen dat hun deeltjessp ectrum en dynamica invariant is onder een b e

paalde symmetriegro ep die buiten translaties en Lorentz transformaties o ok trans

formaties bevat die b osonische en fermionische velden op een b epaalde manier

met elkaar vermengen Bovendien b ep erken we ons in eerste instantie tot de b e

schrijving van de lage energie vrijheidsgraden die in een zogenaamde massaloze

vectormultiplet representatie van deze sup ersymmetriegro ep vallen al dan niet

gekopp eld aan gravitatie Een dergelijk vector multiplet omvat twee spin deel

tjes de scalar b eschreven do or een complex scalair veld een spin deeltje het

foton b eschreven do or een vectorp otentiaal en twee spin deeltjes Wanneer o ok

sup ergravitatie b eschouwd wordt hebb en we in het sp ectrum van de theorie o ok

een spin deeltje het graviton b eschreven do or de ruimtetijd metriek G een

spin deeltje het gravifoton b eschreven do or een vectorp otentiaal en twee spin

deeltjes

Men kan aantonen dat N sup ersymmetrie vereist dat het b osonische

stuk van de lage energie quantum eectieve actie van een aantal aan sup er

gravitatie gekopp elde neutrale vectormultipletten b ep erkt tot maximaal twee

afgeleiden noodzakelijk de volgende vorm heeft

S S S S

graviton scalar f oton

waarbij S de gebruikelijke EinsteinHilb ert gravitationele actie is

graviton

Z

p

S GR d x

g r av iton

G

N

Aangezien het hier over massaloze deeltjes gaat komen met een spin s deeltje twee to e

standen overeen met heliciteit s als s



Dit zijn o ok de relevante lage energie vrijheidsgraden voor spontaan gebroken niet ab else

N YangMills theorieen aangezien de nietneutrale vector multipletten massief worden do or

de sp ontane ijksymmetriebreking

R is de scalaire kromming G de determinant van de metriek G en G de

N

Newton constante en waarbij de actie vo or de scalars de volgende vorm heeft

Z

p

a b

S Gg z z z d x

scalar

ab

G

N

a

Hierbij is z a het complexe scalaire veld van het ade vectormultiplet

en g z z een hermitische p ositief deniete matrix die in het algemeen afhanke

ab

a

lijk is van de z endeinterpretatie van een metriek heeft op de scalaire manifold

M Sup ersymmetrie vereist dat deze metriek van een zeer bijzondere vorm is

namelijk een zogenaamde locale speciale Kahler metriek Voordetechnische de

nitie van sp eciale Kahler meetkunde verwijzen we naar de thesis Essentieel komt

het erop neer dat de metriek aeidbaar is van een Kahler p otentiaal die op haar

b eurt aeidbaar is v an een b epaalde meromorfe eventueel meerwaardige functie

op M die de prepotentiaal geno emd wordt

De actie S vo or de fotonen en het gravifoton heeft o ok een zeer sp ecieke

f oton

vorm eveneens aeidbaar van de prep otentiaal maar daarvoor verwijzen we naar

de thesis zelf Hetzelfde geldt vo or het fermionische stuk van de actie

Wanneer een theorie zonder gravitatie wordt b eschouwd krijgen weeenana

loge vorm voor de eectieve actie met analoge b ep erkingen maar nu met als

relevante meetkundige structuur rigide speciale meetkunde

Het is duidelijk dat sp eciale meetkunde zeer sterke b ep erkingen oplegt op de

mogelijke eectieve acties Sterk geno eg zo bleek uit het baanbrekend werk van

Seib erg en Witten om tot niettriviale exacte oplossingen te kunnen komen Uit

het b ovenstaande volgt dat het vinden van een exacte oplossing neerkomtophet

vinden van de relevante sp eciale Kahler manifold Seib erg en Witten ontdekten dat

de sp eciale Kahler manifold die de exacte oplossing van de lage energie eectiev e

actie van N SU YangMills theorie geeft niets anders is dan de complex

dimensionale moduliruimte die de complexe structuren op een genus Riemann

opp ervlak een torus parametrizeert met de sp eciale Kahler metriek gegeven in

termen van integralen over nietsamentrekbare lussen van een b epaalde meromorfe

vorm op dat Riemann opp ervlak

De Seib ergWitten oplossing gaf het startschot voor een massale zo ekto cht

naar het antwo ord op twee vo or de hand liggende vragen

Ho e kan de enigmatische verschijning van een meetkundige mo duliruimte in

veldentheorie b egrep en worden

Ho e kan dit veralgemeend worden naar andere ijktheorieen



De globale structuur van deze manifold hangt af van het concrete fysische mo del dat b e

schouwd wordt

App endix A Samenvatting

Het antwo ord op de eerste vraag en op die manier o ok een systematische aanpak

van de tweede vraag werd op een bijzonder mo oie manier gegeven do or string

theorie Ijktheorieen hebb en vaak een geometrische o orsprong in stringtheorie en

dit leverde de link tussen de exacte oplossingen van lage energie eectieve acties

enerzijds en meetkundige structuren anderzijds Het unicerende geometrische

kader van stringtheorie gaf bovendien ook een elegante en nuttige meetkundige

interpretatie aan tal van veldentheoretische ob jecten zoals bijvo orb eeld BPS to e

standen Deze laatste blijken in stringtheorie dikwijls vo orgesteld te kunnen wor

den do or Dbranen cf ho ofdstuk gewikkeld rond niettriviale cycles gesloten

niet samentrekbare lussen opp ervlakken volumes etc in de manifold die de zes

kompakte ruimtetijd dimensies vormt

In het overblijvende deel van deze thesis b estuderen we ho e de exacte oplossin

gen van lage energie eectieve acties precies tevo orschijn komen uit stringtheorie

en analyseren we een aantal implicaties v o or lage energiefysica inclusief gravita

tie vo or unicatie en vo or de dynamica van BPS to estanden

Ho ofdstuk CalabiYau compacticaties van typ e I I stringtheorie

In dit ho ofdstuk b estuderen we uitvo erig de compacticatie van typ e IIB string

theorie op een CalabiYau manifold X g A Dit geeft een N sup er

symmetrische theorie in vier dimensies Het b osonische massaloze sp ectrum in

vier dimensies en de overeenkomstige eectieve actie worden afgeleid De vec

tormultiplet sector hiervan heeft de vorm b epaald do or sp eciale meetkunde zoals

b esproken in ho ofdstuk De sp ecieke sp eciale Kahler manifold M is hier be

paald do or klassieke meetkunde het is de mo duliruimte van complexe structuren

van de CalabiYau manifold X De metriek is b epaald als volgt Op een Calabi

au manifold b estaat er een unieke holomorfe vorm die holomorf afhangt van Y

a a

de complexe structuur mo duli z van X de z vormen tevens de scalairen van de

D vectormultipletten De metriek op M is dan

g K

a

ab b

waarbij de Kahler p otentiaal K gegeven is do or

Z

K lni

X

De integraal in het rechterlid kan b erekend worden in termen van de integralen van

over een homologiebasis van niettriviale cycles de zogenaamde perioden



De reden waarom men zich hier b ep erkt tot BPS to estanden is dat die een deel van de sup er

symmetrie van het vacuum b ehouden net geno eg zo blijkt om nog een aantal exacte uitspraken

te kunnen do en zoals bijvo orb eeld een quantum exacte massaformule

Figuur A Schets van een CalabiYau compacticatie De complexe structuur

mo duli van de compacte CalabiYau manifold X kunnen varieren over de niet

compacte gekromde vier dimensionale ruimtetijd M Deze mo dulivelden vor

men de scalairen van de vectormultipletten in de vierdimensionale theorie

van X De andere elementen in de D lage energie eectieve actie worden ook

uitgedrukt in termen van CalabiYau meetkunde Een b elangrijke eigenschap van

typ e I IB CalabiYau compacticaties is dat b ovendien de vector multiplet sector

van de klassieke lage energie actie tot tweede orde in de afgeleiden exact is

quantumcorrecties zijn verb o den do or sup ersymmetrie

Vervolgens sectie spitsen we ons to e op BPS to estanden in N sup er

symmetrische theorieen in vier dimensies en de verschillende multipletten waarin

ze georganizeerd kunnen zijn Deeltjesachtige BPS to estanden in vier dimensies

kunnen uit typ e IIB stringtheorie bekomen worden do or Dbranen rond niet

triviale cycles te wikkelen Er zijn geen andere mogelijkheden omdat typ e I IB

stringtheorie alleen oneven BPS Dbranen heeft en er geen niettriviale of

cycles in een CalabiYau manifold b estaan De BPS vo orwaarde legt op dat de

inb edding van de Dbraan in X sp eciaal Lagrangiaans is hetgeen oa impliceert

dat het volume ervan minimaal is Ons werk in volgend leiden we de preciese

reductie af van de Dbraan actie tot een deeltjesactie in vier dimensies waaruit

onmiddellijk de massa van deze to estanden volgt Vo or een Dbraan gewikkeld

rond een cycle is dit

p

M jZ j

G N

App endix A Samenvatting

waarbij

Z

K

Z e

met K gedenieerd in Deze uit klassieke meetkunde b ekomen formule is me

teen o ok quantummechanisch exact omdat correcties verb o den zijn do or sup er

symmetrie Als to epassing b erekenen we de quantum exacte electromagnetische

kracht tussen twee b ewegende BPS testdeeltjes met willekeurige lading Tenslotte

b espreken we het probleem van de b epaling van het typ e sup ersymmetrie multi

plet waarto e een b epaalde gewikkelde Dbraan aanleiding geeft Dit hangt af van

inb edding en top ologie van de braan We stellen een eenvoudige shortcut voor

die een aantal gevallen kan worden to egepast

De rest van het ho ofdstuk sectie is in ho ofdzaak gewijd aan het eectieve

veldentheorie b eeld van de BPS to estanden dat is de BPS to estanden b eschouwd

als oplossingen van de lage energie eectieve actie Centraal hier staat het attractor

mechanismeontdekt in de scalairen van de vectormultipletten vlo eien steeds

naar vaste waarden wanneer men naar het centrum v an de BPS oplossing to e gaat

di de horizon wanneer dit een zwart gat is Die vaste waarden worden b epaald

do or de lading maar zijn onafhankelijk van de waarde van de scalairen op oneindig

We leiden in een invariant meetkundig formalisme de gereduceerde eectieve actie

af voor sferisch symmetrische oplossingen vertrekkend van een ansatz die wat

algemener is dan gebruikelijk in de literatuur en in het bijzonder werken we in

detail de analogie uit met een niet relativistisch deeltje b ewegend in een b epaalde

p otentiaal op de mo duliruimte

Meer in detail ziet dit eruit als volgt De ruimtetijd metriek is van de vorm

U r U r

ds e dt e fcr dr r d sin d g

met c een constante Vo or een BPS to estand afkomstig van een Dbraan gewik

keld rond een cycle is de gereduceerde actie vgl in thesis

Z

b

a U

S d fU g z z c G e V z g

red N

ab

G

N

waarbij de tijd gerelateerd is aan de radiele coordinaat r via r c sinh c een

punt de afgeleide naar vo orstelt c de energie van de b eweging is c voor

U

BPS oplossingen en de eectieve p otentiaal gegeven is do or G e V z met

N

ab

V z jZ j g jZ j jZ j

a

b

G is nog steeds de Newton constante en g de metriek op de complexe

N

ab

structuur mo duli ruimte We gebruiken deze equivalentie met een b ewegend deeltje

vo or een intuitieve analyse van de algemene oplossing inclusief nietBPS zwarte

U

Figuur A Schets van de p otentiaal e V waarin het eectieve deeltje b eweegt

U

uitgezet als functie van e en de mo duli z Tra ject a generieke b eginvo orwaar

den corresp ondeert met een onfysiche sup ergravitatieoplossing sterk singulier en

antigraviterend Tra ject b corresp ondeert met een nietBPS zwart gat Tra ject

c corresp ondeert met een BPS zwart gat

gaten g A We maken zo o ok manifest dat het attractor mechanisme niet

vero orzaakt wordt do or dissipatieve demping zoals dikwijls b eweerd wordt in de

literatuur maar integendeel juist als gevolg van een eindige energie voorwaarde

samen met de sterke instabiliteit van het systeem zoals gewo onlijk in solitonfysica

Het ruwe verband tussen deze instabiliteit en de entropie van zwarte gaten wordt

kort geschetst

BPS oplossingen hebb en c en voldo en aan de zogenaamde attractor ow

vergelijkingen

p

U

U G e jZ j

N

p

a U ab

z G e g jZ j

N

b

De mo duli convergeren voor steeds naar een welb epaalde waarde enkel

afhankelijk van de keuze van namelijk die waarde die jZ j minimaliseert

In sectie b ekijken we een de oplossingen van de attractor owvergelijkin

gen vo or een aantal mo dellen met sp eciale nadruk op de nietgenerieke gevallen

en we bekomen in het bijzonder een totnogto e onopgemerkt gebleven oplossing

overeenkomend met een Dbraan gewikkeld over een zogenaamde conifold

cycle vo orb eeld Dit is een niet triviale cycle die tot nul volume kan krim

App endix A Samenvatting

Figuur A Een schets van een CalabiYau manifold die in vier dimensies een niet

ab else N YangMills theorie zwak gekopp eld aan gravitatie oplevert Branen

gewikkeld rond een generieke cycle hebb en typisch Planckschaal massas Do or

jnregeling van de mo duli kunnen sommige cycles echter veel lichter gemaakt

worden dan de Planck massa en zo aanleiding geven tot de b eno digde zwak gra

viterende YangMills deeltjes

pen door variatie van de mo duli Deze oplossing zal een prominente rol sp elen in

ho ofdstuk We eindigen dit deel over attractors met een discussie in van het

multicenter geval en de presentatie in van een aantal krachtige technieken om

de attractor owvergelijkingen op te lossen daarbij een intrinsiek meetkundige

Kahler ijk invariante formulering van de metho des ontwikkeld in leverend

We sluiten dit ho ofdstuk af met een analyse van de geldigheidsdomeinen van

de Dbraan en veldentheorie b eelden van de BPS to estanden Weschetsen de b e

ro emde Maldacena corresp ondentie in deze kontekst Wanneer dezelfde redenering

to egepast wordt op BPS to estanden die geen zwarte gaten zijn schijnen we een

nieuw so ort corresp ondentie te vinden die sterk do et denken aan de welb ekende

wiskundige Nahm dualiteit tussen N monop olen en een b epaald N N hermitisch

matrix systeem We laten dit als een intrigerende op ening naar verder onderzo ek

Ho ofdstuk Quantum YangMills gravitatie uit I IB strings

In dit ho ofdstuk b estuderen we in detail ho e de lage energie eectieve actie van

N quantum YangMills theorie zwak gekopp eld aan gravitatie bekomen

wordt vanuit typ e I IB string theorie gecompacticeerd op een CalabiYau mani

fold X We baseren ons hier op welb ekende geometrical engineering technieken

De vo ornaamste nieuwe elementen in onze b ehandeling zijn de

expliciete kopp eling aan gravitatie en aan de dynamische dynamisch gegenereerde

schaal van de quantum YangMills theorie en het feit dat we alles rechtstreeks en

volledig binnen typ e I IB string theorie aeiden zonder ons te b ep erken tot lo cale

overwegingen of gebruik te maken van string S of Tdualiteiten Ook de reductie

van lo cale tot rigide sp eciale meetkunde wordt expliciet aangeto ond

De basisidee is als volgt g A Wanneer we een Dbraan wikkelen rond

een generieke cycle van X krijgen wetypisch een BPS deeltje met massa van de

orde van de Planck massa in de vier dimensionale theorie dus zeker geen go ede

kandidaat vo or een ijkdeeltje dat slechts zwak interageert met de gravitatiekracht

Do or jnregeling van de complexe structuur mo duli van X namelijk do or X op

een b epaalde manier bijnasingulier te maken kunnen sommige cycles echter veel

lichter gemaakt worden dan de Planck massa Als weditbovendien op een ma

nier do en die ervo or zorgt dat de overeenkomstige BPS deeltjes in vier dimensies

precies het massieve BPS sp ectrum opleveren van een of andere N YangMills

theorie met ijkgro ep G met in het bijzonder een aantal massieve vector multi

pletten die samen een irreducib ele representatie dragen van de Weyl gro ep van G

geinduceerd do or mono dromieen in de mo duliruimte dan kan aangenomen wor

den dat de dynamica van deze massieve BPS to estanden samen met de massaloze

vectormultipletten die we al hadden gegeven wordt do or die N YangMills

theorie met ijkgro ep G Anderzijds hadden wevanuit typ e I IB string theorie al de

exacte quantum lage energie eectieve actie tot tweede orde in de afgeleiden van

de massaloze vector multipletten gevonden cf ho ofdstuk Dus we bekomen

op deze manier bijna zonder inspanning de exacte oplossing van de lage energie

quantum eectieve actie van de b eschouwde YangMills theorie

Het mo eilijkste deel van de constructie is dus te b epalen welke CalabiYau

manifolds en welkelimietenvan de complexe structuurmo duli overeenkomen met

een b epaalde ijktheorie Hieraan is sect gewijd We b eargumenteren dat een

ijkgro ep met Lie algebra S waarbij S van typ e A D of E is b ekomen wordt bij

k k k

een CalabiYau die lokaal de structuur heeft van een licht gedeformeerde S typ e

singulariteit gebreerd over een cylinder Een S typ e singulariteit is ruwweg

een singulariteit bekomen do or in een twee complex dimensionale varieteit een

aantal niet triviale sferen die elkaar snijden volgens het Dynkin diagram van de

Lie algebra S zie g A vo or het vo orb eeld van een D of so singulariteit

tot punten te laten krimp en Dit geeft een to chwel zeer mo oie corresp ondentie

tussen de classicatie van singulariteiten en de classicatie van ijkgro ep en

In sect b estuderen wedezwakke gravitatielimiet waarbij we de cycles

corresp onderend met de ijktheoriedeeltjes laten degeneren tot volume massa

nul We b ewijzen de reductie van lo cale tot rigide sp eciale meetkunde in deze limiet

en bekomen de volledige exacte quantum eectieve actie inclusief de kopp eling

aan gravitatie en aan de dynamisch geworden dynamisch gegenereerde ijktheorie

App endix A Samenvatting

Figuur A De gedeformeerde D of so singulariteit heeft een basis van vier

tot punten krimp ende sferen die elkaar snijden zoals aangegeven do or het D

Dynkin diagram

schaal Dit repro duceert en veralgemeent de Seib ergWitten oplossing In het bij

zonder wordt de metriek op de rigide sector van de mo duliruimte gegeven do or de

sp eciale meetkunde van een b epaalde klasse Riemannopp ervlakken voorzien van

een zekere meromorfe vorm b eide eenduidig b epaald uit de lokale Calabi

sw

Yau meetkunde De lichte cycles op de CalbiYau manifold corresp onderen

met welb epaalde cycles op het Riemann opp ervlak

In sectie analyseren wedeverschillende unicatieschalen die tevo orschijn

komen en wemaken de connectie met het Sduale heterotische b eeld expliciet

Om wat inzicht te krijgen in de fysische inhoud van onze resultaten b eschouwen we

e b ekijken uiteraard niet wat exp erimentele data ho ewel de N theorieen die w

volledig overeenkomen met observaties Tenslotte b estuderen we de dynamica van

de dynamische dynamisch gegenereerde schaal We stellen vast dat deze ont

kopp elt van de ijktheorie wanneer de verhouding tussen ijkschaal en Planckschaal

nul is of wanneer de kopp elingsconstante van de ijktheorie nul is We concluderen

dus dat de bevlo eding van de schaal do or de YangMillsvrijheidsgraden vanuit

microscopischveldentheorie standpunt puur een quantum stringy eect is

De lage energie quantum eectieve actie die we vinden vo or het graviton de

i

YangMills massaloze scalars u en de schaal van de YangMills theorie is

Z

p

d

S GR d x

ln jM j

SU

Z

i j

K du du jj

j SW i

i

K du dK jdj Re

SW SW i

Hier is K de dimensieloze veralgemeende Seib ergWitten Kahler p otentiaal

SW

voor de b eschouwde YangMillstheorie gegeven in termen van de sp eciale meet

kunde van een welb epaalde klasse Riemann opp ervlakken M is de in go ede

SU

b enadering constante string unicatie schaal

jk j

M

SU

jbj lnjj

met G b de b etafunctie co ecientvan de b eschouwde ijktheorie en k

N

een CalabiYau mo del afhankelijke constante

In sectie werken we in uitvo erig detail een expliciet vo orb eeld uit vo or een

sp ecieke CalabiYau we onderzo eken een limiet van de X Calabi

Yau manifold die in vier dimensies pure SU YangMills theorie zwak gekopp eld

aan gravitatie oplevert Cycles p erio den en mono dromieen worden expliciet ge

construeerd De resultaten ondersteunen de algemene argumentatie Voor de

waarde van k in de formule voor M hierb oven geeft dit vo orb eeld de waarde

SU

p





k







Ho ofdstuk Attractoren bij zwakke gravitatie

Hier b estuderen we de BPS to estanden bij zwakke gravitatie in het eectievevel

dentheorie beeld Het gros van de resultaten in dit ho ofdstuk zijn nieuw We

herb eschouwen het equivalente eectieve deeltje bewegend in een p otentiaal op

de mo duliruimte om intuitief inzicht te krijgen in de sferisch symmetrische oplos

singen We isoleren en b espreken een interessante O G correctieterm in de

N

gereduceerde eectieve actie die vanuit het microscopische quantumveldentheorie

standpuntvolledig afkomstig is van de interactie tussen de dynamische ijktheorie

de quantum uctuaties van de ijktheorie vrijheidsgraden De zwakke schaal en

gravitatielimiet van de attractor owvergelijkingen wordt afgeleid en gebruikt om

een expliciete b eschrijving te geven van Stromingers massaloze zwarte gaten

bewegend aan de snelheid van het licht

Omdat de O G correcties klein zijn vo or de BPS to estanden in het sp ectrum

N

van de YangMills theorie en in het bijzonder niet van b elang voor de kwalitatieve

eigenschapp en van de oplossingen sp ecialiseren we verder tot de volledig rigide

limiet G met als prototyp e vo orb eeld SU YangMills theorie Gravitatie

N

en schaal ontkopp elen volledig zo dat U en constant gesteld kunnen worden De

attractor owvergelijkingen vereenvoudigen in dit SU geval tot

p

uu

jZ j u g

u sw sw

App endix A Samenvatting

Figuur A Energiedichtheid als functie van de radiale co ordinaat r vo or magne

tische lading N met ur inumerisch geintegreerd De dichtheid wordt

nul in een kern met straal r N De streep jeslijn to ontdeenergiedicht

heid die het electromagnetische veld zou hebb en indien de mo duli niet dynamisch

zouden zijn zoals in Maxwell electromagnetisme Dit divergeert uiteraard voor

r

uu

Hier is g de inverse metriek op de Seib ergWitten mo duli ruimte en Z de p eri

sw

sw

ode van de meromorfe Seib ergWitten vorm bij mo dulus uover de b eschouwde

R

Een interessante eigenschap van de rigide limiet is dat cycle Z

sw sw

de faze van Z constant is langs de o w

sw

Enkel voor magnetische monop olen en elementaire dyonen dit zijn precies

de BPS to estanden die massalo os kunnen worden vo or b epaalde waarden van de

mo duli vinden we een sferisch symmetrische oplossing Deze zijn van dezelfde

vorm als de eerder al b esproken conifold oplossingen sectie vo orb eeld

Ze zijn continu dierentieerbaar maar hebb en een eindige energieloze kern waarin

de mo dulus constant is gelijk aan zijn attractorwaarde g A De totale massa

is eindig en zit volledigindevelden rond de kern De straal van de kern en de hele

schaal van de oplossing eigenlijk is prop ortioneel met de lading N van de to estand

Voor geen enkele waarde van de lading vormen deze to estanden zwarte gaten

do or het attractor mechanisme geraakt de massa no oit geconcentreerd binnen haar

Schwarzschild straal

We onderzo eken een aantal eigenschapp en van deze oplossingen en vinden oa

dat de krachtp otentiaal vo or een testdeeltje afk omstig van een Dbraan gewikkeld

rond een cycle in het BPS veld van een deeltje afkomstig van een Dbraan t

Figuur A De kern van een kubische monop o ol conguratie

gewikkeld rond een cycle gegeven is do or

p

r

t

sin W r jZ r j

t

waarbij de constante faze is van Z en de varierende faze van Z in het

t t

BPS veld van Wanneer de attractor owvan dus do or een punt passeert

waarin de fazes van Z en Z identiek zijn vinden we een stabiele afstand voor

t

een testdeeltje in het veld van Dit suggereert sterk de mogelijkheid van

t

geb onden to estanden

Vervolgens b estuderen weindetailwat er gelukkig zoals we b eargumen

teren fout lo opt wanneer we prob eren een sferisch symmetrische BPS oplossing

te construeren voor puur electrisch geladen deeltjes het W b oson multiplet en

de hogere dyonen De puzzel stelt zich het scherpst bij de dyonen aangezien die

do or continue mono dromietransformaties met elkaar verb onden zijn We komen

zo tot een aantrekkelijk b eeld van een overgang naar een geb onden to estand van

monop olen en elementaire dyonen Het W b oson is in dit b eeld bijvo orb eeld een

geb onden to estand van monop o ol en elementair dyon met tegengestelde mag

netische lading Op deze manier wordt o ok kwantitatief contact gemaakt met

de pronged string representatie van BPS to estanden Een interessante vraag is

of dit mechanisme kan veralgemeend worden naar BPS zwarte gaten

We b esluiten dit ho ofdstuk in sect met de eerste stapp en in een analyse

van de lage energie dynamica van N monop olen g A Wegeven een algemene



dit is een punt waarop het zwaarste deeltje slechts marginaal stabiel is voor verval naar

het lichtste plus nog iets dus dit punt ligt no o dzakelijk op de zogenaamde lijn van marginale

stabiliteit in de mo duliruimte

App endix A Samenvatting

formule voor de N monop o ol mo duli ruimte metriek en evalueren dit voor N

hetgeen het verwachte resultaat oplevert Een verdergaande analyse van de

multimonop o ol metriek zou bijzonder interessant zijn zeker in het lichtvan een

mogelijke connectie met een veralgemeende Nahm constructie via de eerder

aangehaalde Maldacenatyp e corresp ondentie

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