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From4D

ReducedSupersymmetric

Yang-MillsIntegralsto

BranchedPolymers

Dissertation

zurErlangungdesDoktorgrades

anderFakulta¨tfu¨rPhysik

derUniversit¨atBielefeld

vorgelegtvon

MarcWattenberg

October2004

’Therearereallyfourdimensions,threewhichwecallthe
threeplanesofSpace,andafourth,Time.Thereis,however,
atendencytodrawanunrealdistinctionbetweentheformer
threedimensionsandthelatter,becauseithappensthatour
consciousnessmovesintermittentlyinonedirectionalongthe
latterfromthebeginningtotheendofourlives.’
‘That,’saidaveryyoungman,makingspasmodiceortsto
relighthiscigaroverthelamp;‘that...veryclearindeed.’

.H

.G

Wells,

ehT

emiT

Machine,1895

Contents

1Introduction1
1.1Generalsurvey...........................1
1.2FromYang-MillsgaugetheorytoYang-Millsmatrixmodels..2
1.3Thesisplan.............................7

2FromtheIIbMatrixModeltoBranchedPolymers9
2.1TheIIbmatrixmodel........................9
2.2One-loopapproximation......................11
2.2.1Perturbativeexpansion...................12
2.2.2Longdistancedynamics..................20
2.2.3Shortdistances.......................24
2.3Polyakov-lineoperatorwithinthebranchedpolymermodel...25
2.3.1DerivationofthePolyakov-lineoperatorintheone-loop
approximation........................25
2.3.2Polyakovone-pointcorrelationfunction.........27

3DynamicalTrees37
3.1Treestructures...........................37
3.1.1Generalintroduction....................37
3.1.2Criticalbehaviouranduniversalityclasses........43
3.1.3Correlationfunctions....................46
3.1.4Fractalgeometry......................50
3.1.5Internaltwo-pointfunction.................51
3.1.6Externaltwo-pointfunction................54

i

3.2Numericalsimulations.......................64
3.2.1MonteCarloalgorithms..................64
3.2.2Simulationofthe4Dbranchedpolymermodel.....67
3.2.3Transformations.......................70
3.2.4Autocorrelations&Jackknifealgorithm..........71

4Simulationof4DBranchedPolymers75
4.1Gaussianbranchedpolymers....................76
4.2Power-tailbranchedpolymers...................80
4.2.1Shortdistancebehaviourregulatedbya=1.......81
4.2.2Shortdistancebehaviourregulatedbyc=1.......85
4.3Polyakov-lineoperatorinthebranchedpolymermodel.....90
4.3.1ScalingbehaviourofthePolyakovone-pointcorrelation
functionwithrespecttoaandc..............91
4.3.2Polyakovone-pointcorrelationfunctions.........97

5Summary&Outlook

ACalculationoftheLinkWeightsWijandUij

101

015

BHigherOrderCorrectionstothePolyakov-lineOperator109

CSU(2)PolyakovOne-PointCorrelationFunction113

DIntegrationoverMatricesandGhosts117
D.1Gaussianintegrals..........................117
D.2Integratingspinorcomponents...................120
D.3Integratingghosts..........................121

ii

Chapter1

Introduction

1.1Generalsurvey

Themaintaskofmoderntheoreticalphysicsistheunicationofallfunda-
mentalinteractions.Thereexistfourforcesthatdescribeanyphysicalphe-
nomenon.Threeofthem,theelectromagnetic,thestrongaswellastheweak
interaction,arealreadycombinedinthestandardmodelofelementaryparti-
clephysics.Unfortunatelythefourthforce,gravitation,cannotbeembedded
intothestandardmodelwithnowadaysknowledge.Thereforeitisthegreatest
challengetondatheorycomprisingallfourfundamentalinteractions.The
mostpromisingcandidateisthetheoryofsuperstrings.Amainproblemis
thatitscriticaldimensionturnsouttobeequaltoD=10,whichisfarfrom
ourobservedphysicaldimensionofD=4.
Onewayoutisthecompacticationoftheadditionalsixdimensionsinto
so-calledCalabi-Yaumanifolds,whichhaveanextensionoftheorderofthe
Plancklength,sosmallthatitisimpossibletoeverbedetectedexperimentally.
Anotherwayoutcouldbeanonperturbativeapproachtosuperstringtheory,
similartothehighlyproductivelatticegaugetheoryapproachtoQCD.Two
constructivedenitionsofthistypehavebeenproposed:theBFSSmodel[1]
andtheIIbmatrixmodel,alsoknownastheIKKTmodel[2].Bothmodels
aredenedasreducedsupersymmetricYang-Millsintegrals.Actuallyreduced
modelswererstintroducedbyEguchiandKawaiinconnectionwithare-

1

2

Chapter1:Introduction

ductioninthedynamicaldegreesoffreedominthelargeNlimitoflattice
gaugetheory[3].ItwasshownthatlargeNYang-Millstheorycanbeequiva-
lentlydescribedbyitsreducedmodel[3–5].Therefore,reducedsupersymmet-
ricYang-MillstheoriesaredirectlyrelatedtolargeNsupersymmetricQCD.
Ananalysisofthelow-energyeectivepropertiesoftheIIbmatrixmodel
hasledtoaconjecturethatthe4Dspacetimeisgenerateddynamicallyviaa
spontaneoussymmetrybreakingoftheLorentzinvariancefromD=10to4.
Thislow-energyeectivetheoryiscloselyrelatedtoageometricalmodelof
branchedpolymers.
Themainfocusofthisthesisisconcentratedonthebehaviouroftheun-
derlyingbranchedpolymermodelanditsrelationtothe4DIIbmatrixmodel.

1.2FromYang-MillsgaugetheorytoYang-
Millsmatrixmodels
ReducedYang-Millsmatrixmodelsarerelatedtoordinarygaugetheoriesby
dimensionalreduction.Itwasconjecturedthattheyreproducethefullun-
derlyingYang-MillsgaugetheoryinthelargeNlimit.Thisfeaturehasbeen
rstdiscoveredbyEguchiandKawaiwithinthelatticeformulationofgauge
theory[3],whointroducedamodelona1Dtoruswiththeaction
D††SEK=TrUUUU,(1.1)
X6==1
whereUareSU(N)linkvariables,asamaximallyreducedmodeloflattice
gaugetheory.Theysketchedtheproofoftheequivalenceofthismodelto
theordinarySU(N)latticegaugetheoryonainnitelatticeinthelargeN
limit,withN/