String theory limits and dualities

182 Pages·2000·1.205 MB·English

by  Van der Schaar J.P.

String Theory Limits and Dualities
Voor Heiten Mem.
Omslagillustratie : bewerkte foto van de passage van boei nummer D24 tijdens
eenzeiltochtnaar hetWaddeneiland Texel,augustus 1999.
Rijksuniversiteit Groningen
STRING THEORY LIMITS AND
DUALITIES
PROEFSCHRIFT
terverkrijging van het doctoraatin de
Wiskunde en Natuurwetenschappen
aan de RijksuniversiteitGroningen
op gezag van de
RectorMagnificus, Dr. D.F.J.Bosscher,
in hetopenbaar teverdedigen op
vrijdag 30 juni 2000
om 14.15 uur
door
Jan Pieter van der Schaar
geboren op 28 december1972
teHeerenveen
Promotor: Prof. Dr. D. Atkinson
Referenten: Dr. E.A.Bergshoeff
Dr. M. de Roo
Beoordelingscommissie: Prof. Dr. P. DiVecchia
Prof. Dr. P.K. Townsend
Prof. Dr. E.P.Verlinde
ISBN 90-367-1254-8
Contents
1 Introduction 7
2 String Theory 13
2.1 Thebosonic string . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Closedstring spectrum . . . . . . . . . . . . . . . . . . . 18
2.1.2 Open string spectrum . . . . . . . . . . . . . . . . . . . . 24
2.1.3 String interactions . . . . . . . . . . . . . . . . . . . . . 26
2.1.4 T-dualityand theappearance of D–branes . . . . . . . . . 28
2.1.5 D–branes . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.6 Strings in generalbackground fields . . . . . . . . . . . . 35
2.2 Thesuperstring . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.1 TheGSO–projection . . . . . . . . . . . . . . . . . . . . 44
2.2.2 ClosedType IIA and Type IIB superstrings . . . . . . . . 45
2.2.3 TheHeteroticsuperstring . . . . . . . . . . . . . . . . . . 47
2.2.4 TheType I superstring . . . . . . . . . . . . . . . . . . . 48
2.2.5 Superstring T–dualityand D–branes . . . . . . . . . . . . 49
3 Superstring low energylimitsand dualities 51
3.1 Supersymmetry algebrasand BPSstates . . . . . . . . . . . . . . 51
3.1.1 TheD 11 supersymmetry algebra . . . . . . . . . . . . 52
3.1.2 Supersy=mmetry algebrasin tendimensions . . . . . . . . 57
3.2 Supergravities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 Supergravity in elevendimensions . . . . . . . . . . . . . 62
3.2.2 Supergravities inten dimensions . . . . . . . . . . . . . . 65
3.3 Dualities,M–theory and p–branes . . . . . . . . . . . . . . . . . 72
3.3.1 BPS p–brane solutions . . . . . . . . . . . . . . . . . . . 72
3.3.2 Dualitiesand effectiveworldvolume theories . . . . . . . 79
5
CONTENTS
4 Matrix theory 83
4.1 The Matrixmodel conjecture . . . . . . . . . . . . . . . . . . . . 83
4.1.1 D0–branes and DLCQ M–theory . . . . . . . . . . . . . . 84
4.2 BPS objectsin Matrix theory . . . . . . . . . . . . . . . . . . . . 90
4.2.1 BasicMatrix theory extended objects . . . . . . . . . . . 91
4.2.2 IntersectingBPS objectsin Matrix theory . . . . . . . . . 99
4.2.3 Matrix BPS statesand theM–theory algebra . . . . . . . 103
4.3 The statusof Matrix theory . . . . . . . . . . . . . . . . . . . . . 107
5 String solitons and the field theory limit 109
5.1 String soliton geometries . . . . . . . . . . . . . . . . . . . . . . 109
5.1.1 Near–horizon geometriesof p–branes . . . . . . . . . . . 110
5.1.2 Domain–walls and Anti–de Sitterspacetimes . . . . . . . 116
5.2 The fieldtheory limit . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 The generalsetup . . . . . . . . . . . . . . . . . . . . . . 126
5.2.2 The AdS/CFT examples . . . . . . . . . . . . . . . . . . 134
5.2.3 Non–trivial dilatonDp–branes in D 10 . . . . . . . . . 138
5.2.4 Non–trivial dilatondp–branes in D =6 . . . . . . . . . . 146
=
6 Emergingstructureand discussion 155
A String theory units and charge conventions 159
Bibliography 163
Samenvatting 173
Dankwoord 181
6
Chapter 1
Introduction
At this moment the physics of elementary particlesis well described by the Stan-
dardModel. Atthesmallestscaleswecanexperimentallyprobe( 100GeV),the
Standard Model predicts the outcomes of (scattering) experiments with incredi-
(cid:24)
ble accuracy. The Standard Model accommodates the (observed) constituents of
matter, the quarks and leptons, and the vector particles responsible for the medi-
ation of the strong and electroweak forces. The only ingredient of the Standard
ModelwhichstilllacksexperimentalsupportistheHiggsbosonparticle,whichis
thoughttoberesponsibleforthebreakingoftheelectroweakgaugesymmetryand
the masses of the different particles. One can therefore conclude that at this mo-
ment there is no direct experimental need to construct and investigate theoretical
modelsthatgo beyond the Standard Model1.
On the other hand there are many theoretical reasons to go beyond the Stan-
dard Model. The pillars of contemporary theoretical physics are quantum me-
chanics and general relativity. The Standard Model is a collection of quantum
(gauge) field theories which can be considered to be a merger of quantum me-
chanics and special relativity, describing physics at small scales and relativistic
energies. Generalrelativityhasprovenitsaccuracyonlargescalesdescribingvery
massiveobjects. Whenwekeepincreasingtheenergyscalesandatthesametime
keep decreasing our length scales, we expect new physics which is not described
byeithergeneralrelativityortheStandardModel. Generalrelativitybreaksdown
atshortdistancesandintheStandardModelorinquantumfieldtheoryweshould
incorporatetheeffectsof(quantum)gravitationalinteractions. Thetypicalenergy
1Wehavetoremarkherethatrecentexperimentshavemostprobablyexcludedthepossibility
thatallneutrinosaremassless,whichrequiresamodificationoftheStandardModel. Still,inthe
contextofthisthesiswewouldliketoconsiderthisaminormodification.
7
Chapter1.Introduction
scaleat which this happens is called the Planck mass ( 1019 GeV). This scale is
waybeyondtheexperimentalenergiescurrentlyaccessibleandonemightwonder
(cid:24)
whetheritwilleverbepossibletoattaintheseenergiesinacontrolledexperiment.
Thisdoesnotmean,however,thatthisproblemisofpurelyacademicinterest.
Immediately after the big bang ( 10 43 s), from which our observable universe
evolved,theenergieswereof theorde(cid:0)r of thePlanckscaleandthephysics during
(cid:24)
that(short)timedeterminedthefurtherdevelopmentoftheuniverse. Itistherefore
ofdirectinterestincosmologytosearchforatheorythatisabletounifyormerge
quantum mechanicsand generalrelativity.
Other arguments are of a more theoretical and/or aesthetic nature. The Stan-
dardModelcontainsahugeamountofparameterswhichhavetobedeterminedby
experiments,e.g. masses,couplingconstants,anglesandsoon. Onewouldexpect
(orperhapsonelikestoexpect)thatafundamentaltheoryofnaturewillnotallow
too many adjustableparameters. In thebest casescenario, wewould likeour the-
ory to be unique. Many people therefore like to think of the standard model asan
effective theory, only applicable at our current available energy scales [1]. This
pointofviewisbackedupbyconsideringtherunningofthecouplingconstantsof
thedifferentgaugetheoriesthatarepartoftheStandardModel. Whenplottingthe
coupling constants as a function of the energy scale, one finds that at a particular
high energy scale, referred to as the Grand Unified Theory (GUT) energy scale
( 1014 GeV), the three coupling constants all seem to meet in (approximately)
thesamepoint. ThissuggestsapossibleunifieddescriptionatandabovetheGUT
(cid:24)
energy scale.
TheGUTenergyscaleliesseveralordersofmagnitudebelowthePlanckscale,
so this unified theory would not involve quantum gravity. However, for the three
gaugetheorycouplingconstantstomeetatthesamepoint,thetheoreticalconcept
of supersymmetry improves on the approximate result without supersymmetry.
Supersymmetry isasymmetry thatconnectsbosons and fermions: startingwitha
bosonicparticleonecanperformasupersymmetrytransformationandendupwith
a fermionic particle. In our world supersymmetry, if it exists, must be a broken
symmetry,becausewehavenotdetectedanysupersymmetricpartnersoftheStan-
dardModelparticles. Globalsupersymmetricquantumfieldtheorieshaveslightly
different properties than their non–supersymmetric counterparts. Most impor-
tantly, in the perturbative expansion cancellations take place between bosons and
fermions, generically making supersymmetric quantum theories better behaved.
Localsupersymmetrictheories automaticallyinclude supergravity, the supersym-
metric version of general relativity. So Grand Unified Theory, supersymmetry
and supergravity are intimately linked, which again suggests a possible unified
8
descriptionof gauge field theoriesand quantum gravity atthePlanck scale.
From another point of view the basic fact that general relativity is a classical
field theory is unsatisfactory. At scales around the Planck length, quantum gravi-
tationaleffectsareboundtobecomeimportantandwewillneedaquantumtheory
of gravity. However, so far general relativity has resisted all standard methods of
quantization and is said to be non–renormalizable (for an overview see [2]). One
may think this is just a technical problem, but there are fundamental interpreta-
tional problems as well when trying to quantize general relativity because we are
trying to quantize space and time. Many theoretical physicists agree that in order
to deal with the problem of quantization of space and time a radically new ap-
proach is calledfor. This is also emphasized by the confusing properties of black
holes in general relativity, which in a semiclassical approach are not black at all
andemitblackbodyHawkingradiation. Studyingtheseobjectsingeneralrelativ-
ity, it turns out that one can formulate black hole laws that are strikingly similar
to the laws of thermodynamics. For example, one can assign a temperature and
anentropy toablackhole. Atthismomentoneofthekeyquestionsintheoretical
physics is to try to understand what the fundamental degrees of freedom are that
make up the entropy of the black hole and how these thermodynamic degrees of
freedom arisefrom (quantum) generalrelativity.
Atthismoment,stringtheoryistheonlytheoreticalconstructionthatcandeal
withquantumgravity2 albeitinaperturbative,backgrounddependentway. String
theory needs supersymmetry and extra spacetime dimensions to be set up consis-
tently (free of anomalies). In fact, there exist five different superstring theories
which are all living in ten spacetime dimensions and which are distinguished by
the number of supersymmetries and by the kind of strings (open and/or closed).
The construction of these five different anomaly free string theories is referred to
as the first string revolution (1984 1985). Through the method of compactifi-
cation one can try to make contact with our observed universe, containing four
(cid:0)
extended spacetime dimensions. String theory not only contains (quantum) grav-
ity,butgaugetheoriesalsoappearnaturally. Alltheseingredients,extraspacetime
dimensions, supersymmetry, quantum gravity and gauge theories, which are part
of any consistent string theory, make them interesting and promising candidates
for aunified theory [4].
Oneofthebiggestproblemsofstringtheoryisthefactthatonlyaperturbative,
background dependent formulation exists. This makes it very hard to gain any
information about non–perturbative string theory. To make contact with our ob-
2AsWeinbergsaid[3]: “Stringtheoryistheonlygameintown”.
9
Chapter1.Introduction
servable universe this is an important complication because it turns out that there
exist millions of ways to compactify to four dimensions. Perturbative string the-
orywillnotpredictwhichcompactificationisactuallygoingtobepreferredandit
is expected that non–perturbative information is needed in order to determine the
energetically favored compactification (vacuum). Another way of saying this is
that string theory, as it is formulated at this moment, is incapable of dynamically
generating itsown vacuum.
An immediate drawback of string theory as a candidate of a unified theory is
thefactthatthereexistfiveofthem. Ideally,onewouldlikeasingleuniquestruc-
ture. The developments in string theory over the last ten years seem to indicate
that this is in fact true. The reason why this was not recognized before again has
itsrootsinthefactthatstringtheoriesareonlydefinedperturbatively. Onlyinthe
last ten years was it discovered that all five string theories are related by duality
transformations. The concept of duality, a general term used to describe a rela-
tion between two physical theories or mathematical structures, has become very
important in string theory [5]. Through the concept of duality it has now been
recognized that all five string theory formulations describe different perturbative
cornersofasingleuniquetheoreticalstructure,whichhasbeennamedM–theory3
[6]. M–theory is supposed to live in eleven spacetime dimensions and its low en-
ergylimitisdescribedbyeleven–dimensionalsupergravity, whichisthemaximal
spacetime dimension for a supergravity theory with Minkowski signature. The
theoretical tools used in establishing these duality relations were supersymmetry
and the use of D–brane string solitons, which enabled one to study string theory
beyond the perturbativeregime.
Oneofthemaintopicsofresearchfollowingthesecondsuperstringrevolution
(1994 1996) was to find a formulation of M–theory. Following up on work
doneonaregularizedquantizationofsupermembranes,Matrixtheoryemergedas
(cid:0)
a possible non–perturbative candidate capturing the dynamics of discrete light–
cone quantized M–theory. Although thisformulation of M–theory certainlylacks
generalcovarianceandisnotbackgroundindependent,itwasthefirsttimeanon–
perturbativedescriptionof quantum gravity had beenput forward. The surprising
thing about Matrix theory is that it is a 0 1–dimensional quantum mechanics
model of N particles. It can be obtained +by considering a low energy limit of
string theory in the background of N D0–brane solitons. These non–trivial low
energylimitsofstringtheoryinthebackground ofD–branesolitonswerestudied
further and have led to all kinds of interesting relations between gauge theories
3TheMstandsforanythingyoulike,forexampleMother,MembranesorMystery.
10