From 4D reduced supersymmetric Yang-Mills integrals to branched polymers [Elektronische Ressource] / vorgelegt von Marc Wattenberg

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From 4DReduced SupersymmetricYang-Mills Integrals toBranched PolymersDissertationzur Erlangung des Doktorgradesan der Fakult¨at fu¨r Physikder Universit¨at Bielefeldvorgelegt vonMarc WattenbergOctober 2004’There are really four dimensions, three which we call thethree planes of Space, and a fourth, Time. There is, however,a tendency to draw an unreal distinction between the formerthree dimensions and the latter, because it happens that ourconsciousness moves intermittently in one direction along thelatter from the beginning to the end of our lives.’‘That,’ said a very young man, making spasmodic efforts torelight his cigar over the lamp; ‘that... very clear indeed.’H. G. Wells, The Time Machine, 1895Contents1 Introduction 11.1 General survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 From Yang-Mills gauge theory to Yang-Mills matrix models . . 21.3 Thesis plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 From the IIb Matrix Model to Branched Polymers 92.1 The IIb matrix model. . . . . . . . . . . . . . . . . . . . . . . . 92.2 One-loop approximation . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Perturbative expansion . . . . . . . . . . . . . . . . . . . 122.2.2 Long distance dynamics . . . . . . . . . . . . . . . . . . 202.2.3 Short distances . . . . . . . . . . . . . . . . . . . . . . . 242.3 Polyakov-line operator within the branched polymer model . . . 252.3.
Publié le : jeudi 1 janvier 2004
Lecture(s) : 12
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Source : D-NB.INFO/973135549/34
Nombre de pages : 133
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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/

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