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COMBINATORIALPHYSICS CombinatorialPhysics Combinatorics,quantumfieldtheory,andquantum
gravitymodels
AdrianTanasa
UniversityofBordeaux,France
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
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©AdrianTanasa2021
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Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
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LibraryofCongressControlNumber:2021932400 ISBN978–0–19–289549–3 DOI:10.1093/oso/9780192895493.001.0001
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ToLuca,Brittany,andtoourfamily
2.1Graphtheory:TheTuttepolynomial7
2.2Ribbongraphs;theBollobás–Riordanpolynomial12
2.3Selectedfurtherreading15
3Quantumfieldtheory(QFT)—built-incombinatorics 17
3.1Definitionofthescalar Φ4 model18
3.2Perturbativeexpansion—Feynmangraphsandtheircombinatorialweights20
3.3Fouriertransform—themomentumspace23
3.4ParametricrepresentationofFeynmanintegrands24
3.5Thepropagatorandtheheatkernel26
3.6Aglimpseofperturbativerenormalization27
3.6.1Thepowercountingtheorem29
3.6.2Locality30
3.6.3Multi-scaleanalysis32
3.6.4ThesubtractionoperatorforageneralFeynmangraph33
3.6.5Dimensionalrenormalization35
3.7Dyson–Schwingerequation36
3.8Combinatorial(or 0-dimensional)QFTandtheintermediatefieldmethod36
3.8.1Combinatorial(or 0-dimensional)QFT36
3.8.2Theintermediatefieldmethod37 3.9Selectedfurtherreading38
4.1Preliminaryresults41 4.2Partitiontreeweights43 4.3Selectedfurtherreading49
5.1TheJacobianConjectureascombinatorialQFTmodel (theAbdesselam–Rivasseaumodel)52
5.2TheintermediatefieldmethodfortheAbdesselam–Rivasseaumodel53
5.3Selectedfurtherreading55
6FermionicQFT,Grassmanncalculus,andcombinatorics 56
6.1GrassmannalgebrasandGrassmanncalculus57
6.1.1TheGrassmannalgebra57
6.1.2Grassmanncalculus;PfaffiansasGrassmannintegrals58
6.2OnGrassmannGaussianmeasures59
6.3Lingström–Gessel–Viennot(LGV)formulaforgraphswithcycles60
6.4Stembridge’sformulasforgraphswithcycles63
6.5Ageneralization66
6.6TuttepolynomialandtheparametricrepresentationinQFT67
6.7Selectedfurtherreading71
7AnalyticcombinatoricsandQFT
8.1Algebraicreminder;CombinatorialHopfAlgebras(CHAs)77
8.2TheConnes–KreimerHopfalgebraofFeynmangraphs79
8.3The B+ operator,HochschildcohomologyoftheConnes–Kreimeralgebra83
8.4Multi-scalerenormalization,CHAdescription85
8.5Selectedfurtherreading94
9QFTonthenon-commutativeMoyalspaceandcombinatorics 95
9.1Mathematicalsetting:Renormalizability96
9.2TheMehlerkernelandtheGrosse–Wulkenhaarmodel99
9.3ParametricrepresentationofGrosse–Wulkenhaar-likemodels100
9.4TheMellintransformandtheGrosse–Wulkenhaarmodel104
9.5DimensionalrenormalizationfortheGrosse–Wulkenhaarmodel107
9.6Aheatkernel–basedrenormalizablemodel108
9.7ParametricrepresentationandtheBollobás–Riordanpolynomial110
9.7.1Parametricrepresentation110
9.7.2Relationbetweenthemulti-variateBollobás–Riordanandthe polynomialsoftheparametricrepresentation111
9.8CombinatorialConnes–KreimerHopfalgebraandits Hochschildcohomology112
9.8.1CombinatorialConnes–KreimerHopfalgebra112
9.8.2HochschildcohomologyandthecombinatorialDSE117
9.9Selectedfurtherreading120
10Quantumgravity,groupfieldtheory(GFT),andcombinatorics 121
10.1Quantumgravity121
10.2Maincandidatesforatheoryofquantumgravity:Theholographicprinciple122
10.3GFTmodels:theBoulatovandthecolourablemodels123
10.4Themulti-orientableGFTmodel125
10.4.1Tadpolesandgeneralizedtadpoles127
10.4.2Tadfaces128
10.5SaddlepointmethodforGFTFeynmanintegrals129
10.6AlgebraiccombinatoricsandtensorialGFT133
10.6.1TheBenGeloun–Rivasseau(BGR)model133
10.6.2Cones–KreimerHopfalgebraicdescriptionofthecombinatorics oftherenormalizabilityoftheBGRmodel143
10.6.3HochschildcohomologyandthecombinatorialDSE fortensorialGFT153
10.7Selectedfurtherreading165
11Fromrandommatricestorandomtensors 166
11.1Thelarge N limit169
11.2Thedouble-scalinglimit169
11.3Frommatricestotensors170
11.4Tensorgraphpolynomials—ageneralizationoftheBollobás–Riordan polynomial174
11.5Selectedfurtherreading176
12Randomtensormodels—the U (N )D -invariantmodel 178
12.1DefinitionofthemodelanditsDSE179
12.1.1 U(N)D -invariantbubbleinteractions179
12.1.2Bubbleobservables182
12.1.3TheDSEforthemodel185
12.1.4Navigatingthefollowingsectionsofthechapter187
12.2TheDSEbeyondthelarge N limit188
12.2.1TheLO188
12.2.2MomentsandCumulants189
12.2.3Gaussianandnon-Gaussiancontributions192
12.2.4TheDSEatNLO198
12.2.5Theorder 1/N D inthequarticmodel199
12.3Thedouble-scalinglimit202
12.3.1Double-scalinglimitintheDSE202
12.3.2Fromthequarticmodeltoagenericmodel206
12.4Selectedfurtherreading208
13Randomtensormodels—themulti-orientable(MO)model 209
13.1Definitionofthemodel209
13.2The 1/N expansionandthelarge N limit212
13.2.1Feynmanamplitudes;the 1/N expansion212
13.2.2Thelarge N limit—theLO(melonicgraphs)214
13.2.3Thelarge N limit—theNLO215
13.2.4LeadingandNLOseries216
13.3Combinatorialanalysisofthegeneraltermofthelarge N expansion219
13.3.1Dipoles,chains,schemes,andallthat220
13.3.2Generatingfunctions,asymptoticenumeration, anddominantschemes226
13.4Thedouble-scalinglimit230
13.4.1Thetwo-pointfunction231
13.4.2Thefour-pointfunction232
13.4.3The 2r -pointfunction232
13.5Selectedfurtherreading233 14Randomtensormodels—the
14.1Generalmodelandlarge N expansion234
14.2Quarticmodel,large N expansion241
14.2.1LargeNexpansion:LO242
14.2.2NLO247
14.3Generalquarticmodel:Criticalbehaviour248
14.3.1Explicitcountingofmelonicgraphs248
14.3.2Diagrammaticequations,LOandNLO252
14.3.3Singularityanalysis253
14.3.4Criticalexponents256
14.4Selectedfurtherreading259
15TheSachdev–Ye–Kitaev(SYK)holographicmodel
15.1DefinitionoftheSYKmodel:ItsFeynmangraphs261
15.2Diagrammaticproofofthelarge N melonicdominance264
15.3ThecolouredSYKmodel271
15.3.1Definitionofthemodel,real,andcomplexversions271
15.3.2Diagrammaticsoftherealandcomplexmodel272
15.3.3MoreonthecolouredSYKFeynmangraphs282
15.3.4Non-Gaussiandisorderaverageinthecomplexmodel284
15.4Selectedfurtherreading290
16SYK-liketensormodels 291
16.1TheGurau–Wittenmodelanditsdiagrammatics292
16.1.1Two-pointfunctions:LO,NLO,andsoon293
16.1.2Four-pointfunction:LO,NLO,andsoon295
16.2The O (N )3 -invariantSYK-liketensormodel300
16.3TheMOSYK-liketensormodel303
16.4RelatingMOgraphsto O (N )3 -invariantgraphs304
16.5Diagrammatictechniquesfor O (N )3 -invariantgraphs306
16.5.1Two-edge-cuts306
16.5.2Dipoleremovals307
16.5.3Dipoleinsertions309
16.5.4Chainsofdipoles310
16.5.5Facelength312
16.5.6Thestrategy314
16.6Degree 1 graphsofthe O (N )3 -invariantSYK-liketensormodel316
16.6.12PI,dipole-freegraphofdegreeone316
16.6.2Thegraphsofdegree1319
16.7Degree 3/2 graphsofthe O (N )3 -invariantSYK-liketensormodel323
AExamplesoftreeweights
A.1Symmetricweights—completepartition331
A.2Onesingletonpartition—rootedgraph332
A.3Twosingletonpartition—multi-rootedgraph333
BRenormalizationoftheGrosse–Wulkenhaarmodel,one-loopexamples
CThe B + operatorinMoyalQFT,two-loopexamples
C.1One-loopanalysis338 C.2Two-loopanalysis338
DExplicitexamplesofGFTtensorFeynmanintegralcomputations
D.1Anon-colourable,MOtensorgraphintegral345
D.2Acolourable,multi-orientabletensorgraphintegral345
D.3Anon-colourable,non-multi-orientabletensorgraphintegral347
G.1Bijectionwithconstellations362
G.1.1Bijectioninthebipartitecase362
G.1.2Thenon-bipartitecase365
G.2Enumerationofcolouredgraphsoffixedorder366
G.2.1Exactenumeration366
G.2.2Singularityanalysis369
G.3TheconnectivityconditionandSYKgraphs371
G.3.1Preliminaryconditions371
G.3.2Thecase q> 3
G.3.3Thecase q =3
G.3.4Thenon-bipartitecase374 HProofofTheorem16.1.1
ISummaryofresultsonthediagrammaticsofthecolouredSYK modelandoftheGur ˘ au–Wittenmodel
Introduction Theinterplaybetweencombinatoricsandtheoreticalphysicsisarecenttrendwhich appearstousasparticularlynatural,sincetheunfoldingofnewideasinphysicsis oftentiedtothedevelopmentofcombinatorialmethods,and,conversely,problems incombinatoricshavebeensuccessfullytackledusingmethodsinspiredbytheoretical physics.Alotofproblemsinphysicsarethusrevealedtobeenumerative.Ontheother hand,problemsincombinatoricscanbesolvedinanelegantwayusingtheoretical physics-inspiredtechniques.Wecanthusspeaknowadaysofanemergingdomainof CombinatorialPhysics.
Theinterferencebetweenthesetwodisciplinesismoreoveraninterferenceofmultiple facets.Thus,itsmostknownmanifestation(bothtocombinatorialistsandtheoretical physicists)hassofarbeentheonebetweencombinatoricsandstatisticalphysics, orcombinatoricsandintegrablesystems,asstatisticalphysicsreliesonanaccurate counting ofthevariousstatesorconfigurationsofaphysicalsystem.
However,combinatoricsandtheoreticalphysicsinteractinvariousotherways.One oftheseinteractionsistheonebetweencombinatoricsandquantummechanics,because combinatorialtoolscanbeusedhereforabettermathematicalunderstandingofthe algebrasunderlyingquantummechanics.
Inthisbook,wemainlyfocusonyetanothertypeofthesemultipleinteractions betweencombinatoricsandtheoreticalphysics,theonebetweencombinatoricsand quantumfieldtheory(QFT).Weestimatethatcombinatoricsis builtinto themathematicalformulationofQFT.Thisstemsinitiallyfromthefactthatthemostpopular toolofQFTisperturbationtheoryinthecouplingconstantofthemodel,whichmeans thatoneconsidersFeynman graphs,withappropriatecombinatorialweights,inorderto encodethephysicalinformationoftherespectivesystem.Moreover,oneelegantwayof expressingtheFeynmanintegralsassociatedwiththesegraphsistousetheKirchhoff–Symanzikpolynomialsoftheparametricrepresentation,polynomialswhichcanbe proventoberelatedtosomemulti-variateversionofthecelebratedTuttepolynomial ofcombinatorics.Aparticularlyelegantwaytoprovethisistousethe Grassmann development ofthedeterminantsandPfaffiansinvolvedinthesecomputations.Letus emphasizeherethatthisGrassmanndevelopmentusesGrassmanncalculus,whichwere developedbyphysiciststoexpressfermionicQFT.Grassmanncalculusisfurtherusedin thisbooktogiveasimpleproofofthecelebratedLingström–Gessel–Viennot(forgraphs
Introduction withcycles)andtofurthergeneralizesomeidentities,initiallyprovedbyStembridge,in thesamecontextofgraphswithcycles.
Theso-called 0 dimensionalQFT(calledbysomeauthors,combinatorialQFT),or morepreciselytheuseoftheintermediatefieldmethodinthissetting,allowstoestablisha theoremconcerningpartialeliminationofvariablesinthecelebratedJacobianconjecture (whichconcernstheglobalinvertibilityofpolynomialsystems).
Moreover, analyticcombinatorial techniquesareusedonaregularbasisinQFT computations.Thus,thepropagatorofanyscalarmodelcanberepresentedusingthe heatkernel.TheMellintransformtechniquecanalsobeusedinordertorapidlyprove themeromorphyofFeynmanintegrands.Thesaddlepointmethodisfrequentlyusedto tamethedivergentbehaviouroftheseintegrals.
Lastbutnotleast,renormalizationinQFT(whichonecansayliesattheveryheartof QFT)hasahighlynon-trivialcombinatorialcore,andthishasbeenrecentlypresented ina combinatorialHopfalgebra form—theConnes–KreimerHopfalgebra.Relatedtothis, theHochschildcohomologyofthiscombinatorialHopfalgebracanbeusedtoexpress thecombinatoricsoftheDyson–Schwingerequation(DSE)asasimplepowerseriesin someappropriateinsertionoperatorofFeynmangraphs.
Allthesecombinatorialtechniques(analyticoralgebraic)generalizetomoreinvolved QFTmodels.Thus,non-commutativeQFT(thatis,QFTonanon-commutativespacetime)alsopossessesmostofthesecombinatorialproperties.First,thegraphsusedin QFTareupliftedtoribbongraphs(or combinatorialmaps).Furthermore,onecanstill usetheheatkernelforpropagatorsofthetheories,butinordertohaverenormalizable models,oneneedstouseamoreinvolvedspecialfunction,theMehlerkernelorsome non-trivialmodificationoftheheatkernel.Moreover,theMellintransformtechniquecan againbeused,asinthecaseofcommutativeQFT.Thecorrespondingnon-commutative Kirchhoff–Symanzikpolynomialsareproventobealimitofamulti-variateversionof theBollobás–Riordanpolynomial(whichisanaturalgeneralizationforribbongraphs oftheuniversalTuttepolynomial).Finally,algebraiccombinatorialtechniquescanalso beusedinnon-commutativeQFT.ThecorrespondingcombinatorialConnes–Kreimer HopfalgebraofribbonFeynmangraphscanbedefinedandrelatedtonon-commutative renormalization.Furthermore,theappropriateHochschildcohomologythendescribes thecombinatoricsoftheDSEofthesemodels.
Non-commutativeQFTcanalsobeseenasaspecialcaseofthecelebratedmatrix models.Followingthislineofreasoning,onecannaturallygeneralizerandommatrix modelstorandomtensormodels,
Thecombinatoricsoftensormodelsperseisextremelyinvolved.Onecannot just use thegenustocharacterizetheso-calledlarge N expansion, N beingthesizeofthematrix resp.ofthetensor.Itisworthemphasizingherethatthelarge N expansionis,froma combinatorialpointofview,acertainasymptoticexpansion(correspondingtothelimit N →∞).
However,inordertomakethecombinatoricssimpler,severalQFT-inspiredsimplificationsoftensormodelscanbeproposed.Thefirsttwosuchsimplicationswerethe colouredmodelandthemulti-orientablemodels.Forbothofthesemodels,onecan implementthelarge N expansionandthedouble-scalingmechanism,which,inthecase
3 ofmatrixmodels,areveryimportantmathematicalphysicstools.Severalothermodels (basedon U (N ) andthen O (N ) modelshavealsobeenstudied.
Feynmangraphsassociatedtotensormodels,throughthecelebratedQFTperturbativeexpansions,arecalledtensorgraphsandcanbeseenasanatural3Dgeneralizationof mapsorofribbongraphs.Thedominanttermofthelarge N expansionofthepreviously mentionedtensormodelsaretheso-calledmelonicgraphs,whichare,fromagraph theoreticalpointofview,aparticularcaseofseries-parallelgraphs.
Thelarge N expansioniscontrolledinthe2Dcase,thematrixmodelcase,bythe genusofthecorrespondingcombinatorialmaps.Indimensionhigherthantwo,there isnodirectanalogueofthegenus.Nevertheless,thetensorasymptoticexpansionin N iscontrolledbyaninteger,calledthedegree,whichisdefinedasthehalf-sumofthe non-orientablegenusofribbongraphscanonicallyembeddedinatensorgraph(called thejacketsoftherespectivetensorgraphs).Thedegreeisthusahalfinteger,naturally generalizingthe2Dnotionofgenusfortensormodels.
Inordertostudythegeneraltermofthelarge N expansionofvarioussuchtensor models,weextensivelyuse,inthisbook,variousgraphtheoreticaland enumerative combinatorics techniquestoperformtheirenumerationandweestablishwhicharethe dominantconfigurationsofagivendegree.
Itisworthemphasizingherethattensormodelshaverecentlybeenprovenby WittentoberelatedtothecelebratedholographicSachdev–Ye–Kitaev(SYK)quantum mechanicalmodel.Thiscomesfromthefactthat,intheso-calledlarge N expansion (N beinginthecaseoftheSYKmodelthetotalnumberoffermionsofthemodel), bothtypesofmodelsaredominatedbythemelonicgraphs.Thelarge N expansion forvariousSYK-liketensormodelsisthenstudiedusingagaingraphtheoreticaland enumerativecombinatoricstechniques.Thesetechniquesallowustoasymptotically enumerateFeynmangraphsofvariousSYK-liketensormodels.
Thebookisorganizedasfollows.InChapter 2,wepresentsomenotionsofgraph theorythatwillbeusefulintherestofthebook.Itisworthemphasizingthatgraph theoristsandtheoreticalphysicistsadopt,unfortunately,differentterminologies.We presentherebothterminologies,suchthatasortofdictionarybetweenthesetwo communitiescanbeestablished.Wethenextendthenotionofgraphtothatofmaps(or ofribbongraphs).Moreover,graphpolynomialsencodingthesestructures(theTutte polynomialforgraphsandtheBollobás–Riordanpolynomialforribbongraphs)are presented.
InChapter3,webrieflyexhibitthemathematicalformalismofQFT,which,as mentionedpreviously,hasanon-trivialcombinatorialbackbone.TheQFTsettingcan beunderstoodasaquantumdescriptionofparticlesandtheirinteractions,adescription whichisalsocompatiblewithEinstein’stheoryofspecialrelativity.Withintheframework ofelementaryparticlephysics(orhighenergyphysics),QFTledtotheStandard ModelofElementaryParticlePhysics,whichisthephysicaltheorytestedwiththebest accuracybycolliderexperiments.Moreover,theQFTformalismsuccessfullyapplies tostatisticalphysics,condensedmatterphysics,andsoon.Weshowinthischapter howFeynmangraphsappearthroughtheso-calledQFTperturbativeexpansion,how FeynmanintegralsareassociatedtoFeynmangraphs,andhowtheseintegralscanbe
expressedviathehelpofgraphpolynomials,theKirchhoff–Symanzikpolynomials. Finally,wegiveaglimpseofrenormalization,oftheDSE,andoftheuseofthesocalledintermediatefieldmethod.Thischaptermainlyfocusesontheso-called Φ4 QFT scalarmodel.
InChapter4,wedefinespecifictreeweightswhichappearnaturalwhenconsidering acertainapproachtonon-perturbativerenormalizationinQFT,namelyconstructiverenormalization.Severalexamplesofsuchtreeweightsareexplicitlygivenin AppendixA.
Chapter5dealswithacombinatorialQFTapproachtotheJacobianconjecture.The JacobianConjecturestatesthatanycomplex n-dimensionallocallyinvertiblepolynomial systemisgloballyinvertiblewithapolynomialinverse.In1982,Bassetal.proved animportantreductiontheoremstatingthattheconjectureistrueforanydegreeof thepolynomialsystemifitistrueindegreethree.Weshow,inthischapter,aresult concerningpartialeliminationofvariables,whichimpliesareductionofthegenericcase tothequadraticone.Thepricetopayistheintroductionofasupplementaryparameter, 0 ≤ n ≤ n parameter,whichrepresentsthedimensionofalinearsubspacewheresome particularconditionsonthesystemmusthold.Weexhibitaproof,inaQFTformulation, usingtheintermediatefieldmethodexposedinChapter3.
InChapter6,weuseGrassmanncalculus,usedinfermionicQFT,tofirstgivea reformulationoftheLingström–Gessel–Viennotlemmaproof.Wefurthershowthat thisproofgeneralizestographswithcycles.WethenusethesameGrassmanncalculus techniquestogivenewproofsofStembridge’sidentitiesrelatingappropriategraph Pfaffianstoasumovernon-intersectingpaths.Theresultspresentedheregofurther thantheonesofStembridge,becauseGrassmannalgebratechniquesnaturallyextend (withoutanycost!)tographswithcycles.Wethusobtain,insteadofsumsovernonintersectingpaths,sumsovernon-intersectingpathsandnon-intersectingcycles.In thefifthsectionofthechapter,wegiveageneralizationoftheseresults.Inthesixth sectionofthischapterweuseGrassmanncalculustoexhibittherelationshipbetweena multi-variateversionofTuttepolynomialandtheKirchhoff–Symanzikpolynomialsof theparametricrepresentationofFeynmanintegrals,polynomialsalreadyintroducedin Chapters1andresp.3.
InChapter7,wepresenthowseveralanalytictechniques,oftenusedincombinatorics, appearnaturallyinvariousQFTissues.Inthefirstsection,weshowhowonecanuse theMellintransformtechniquetore-expressFeynmanintegralsinausefulwayforthe mathematicalphysicist.Finally,webrieflypresenthowthesaddlepointapproximation techniquecanbealsousedinQFT.
InChapter8,afterabriefalgebraicreminder,weintroduceinthesecondsectionthe Connes–KreimerHopfalgebraofFeynmangraphsandweshowitsrelationshipwith thecombinatoricsofQFTperturbativerenormalization.Wethenstudythealgebra’s HochschildcohomologyinrelationwiththecombinatorialDSEinQFT.Inthefourth, sectionwepresentaHopfalgebraicdescriptionoftheso-calledmulti-scalerenormalization(themulti-scaleapproachtotheperturbativerenormalizationbeingthestarting pointfortheconstructiverenormalizationprogramme).
InChapter9,wepresentthe Phi4 QFTmodelonthenon-commutativeMoyal spaceandtheUV/IRmixingissue,whichpreventsitfrombeingrenormalizable.We thenpresenttheGrosse–Wulkenhaar Phi4 QFTmodelonthenon-commutativeMoyal space,whichchangestheusualpropagatorofthe Φ model(basedontheheatkernel formula)toaMehlerkernel-basedpropagator.ThisGrosse–Wulkenhaarmodelis perturbativelyrenormalizablebutitisnottranslation-invariant(translation-invariance beingausualpropertyofhigh-energyphysicsmodels).WethenshowhowtheMellin transformtechniquecanbeusedtoexpresstheFeynmanintegralsoftheGrosse–Wulkenhaarmodel.Inthelastpartofthechapter,wepresentanother Phi4 QFT modelonthenon-commutativeMoyalspace,whichishoweverbothrenormalizable andtranslation-invariant.Weshowtherelationbetweentheparametricrepresentation ofthismodelandtheBollobás–Riordanpolynomial.Finally,weshowhowtodefinea Connes–KreimerHopfalgebrafornon-commutativerenormalizationandhowtostudy itsHochschildcohomologyinrelationtothecombinatorialDSEoftheseQFTmodels.
Thelastpartofthebookisdedicatedtothestudyofcombinatorialaspectsof quantumgravitymodels.Thus,inChapter10,afterabriefintroductorysectionto quantumgravity,wementionthemaincandidatesforaquantumtheoryofgravity: stringtheory,loopquantumgravity,andgroupfieldtheory(GFT),causaldynamical triangulations,andmatrixmodels.ThenextsectionsintroducesomeGFTmodelssuch astheBoulatovmodel,thecolourable,andthemulti-orientablemodel.Thesaddlepoint methodforsomespecificGFTFeynmanintegralsispresentedinthefifthsection. Finally,somealgebraiccombinatoricsresultsarepresented:definitionofanappropriate Connes–KreimerHopfalgebradescribingthecombinatoricsoftherenormalizationof acertaintensorGFTmodel(theso-calledBenGeloun–Rivasseaumodel)andthe useofitsHochschildcohomologyforthestudyofthecombinatorialDSEofthis specificmodel.
InChapter11,afterabriefpresentationofrandommatricesasarandomsurfaceQFT approachto2Dquantumgravity,wefocusontwocrucialmathematicalphysicsresults: theimplementationofthelarge N limit(N beingherethesizeofthematrix)andofthe doublescalingmechanismformatrixmodels.Itisworthemphasizingthat,inthelarge N limit,itistheplanarsurfaceswhichdominate.Inthethirdsectionofthechapter,we introducetensormodels,seenasanaturalgeneralization,indimensionshigherthentwo, ofmatrixmodels.Thelastsectionofthechapterpresentsapotentialgeneralizationof theBollobás–Riordanpolynomialfortensorgraphs(whicharetheFeynmangraphsof theperturbativeexpansionofQFTtensormodels).
InChapter12,wefirstbrieflypresentthe U (N )D -invarianttensormodels(N being againthesizeofthetensor,and D beingthedimension).Thenextsectionisthen dedicatedtotheanalysisoftheDyson–Schwingerequations(DSEs)inthelargeN limit.Theseresultsareessentialtoimplementthedoublescalinglimitmechanismofthe DSEswhichisdoneinthethirdsection.Themainresultofthischapteristhedoublyscaledtwo-pointfunctionforamodelwithgenericmelonicinteractions.However,several assumptionsonthelarge N scalingofcumulantsaremadealongtheway.Theyare provedusingvariouscombinatorialmethods.
Chapter13isdedicatedtothepresentationofthemulti-orientabletensormodel.After definingthemodel,the 1/N expansionandthelarge N limitareexaminedinthesecond sectionofthechapter.Inthethirdsection,athoroughenumerativecombinatorialanalysis ofthegeneraltermofthe 1/N expansionispresented.Theimplementationofthedouble scalingmechanismisthenexhibitedinthefourthsection.
InChapter14,wedefineyetanotherclassoftensormodels,endowedwith O (N )3 invariance,Nbeingagainthesizeofthetensor.Thisallowstogenerate,via theusualQFTperturbativeexpansion,aclassofFeynmantensorgraphswhichis strictlylargerthantheclassofFeynmangraphsofboththemulti-orientablemodeland the U (N )3 -invariantmodelstreatedintheprevioustwochapters.Wefirstexhibitthe existenceofalargeNexpansionforsuchamodelwithgeneralinteractions(notnecessary quartic).Wethenfocusonthequarticmodelandweidentifytheleadingorder(LO) andnext-to-leading(NLO)Feynmangraphsofthelarge N expansion.Finally,weprove theexistenceofacriticalregimeandwecomputetheso-calledcriticalexponents.This isachievedthroughtheuseofvariousanalyticcombinatoricstechniques.
InChapter15,wefirstreviewtheSYKmodel,whichisaquantummechanical modelof N fermions.Themodelisaquenchedmodel,whichmeansthatthecoupling constantisarandomtensorwithGaussiandistribution.TheSYKmodelisdominated inthelargeNlimitbymelonicgraphs,inthesamewaythetensormodelspresented inthepreviousthreechaptersaredominatedbymelonicgraphs.Wethenpresenta purelygraphtheoreticalproofofthemelonicdominanceoftheSYKmodel.Asalready mentioned,itisthispropertywhichledE,wittentorelatetheSYKmodeltothecoloured tensormodel.Intherestofthechapterwedealwiththeso-calledcolouredSYKmodel, whichisaparticularcaseofthegeneralizationoftheSYKmodelintroducedbyD. GrossandV.Rosenhaus.WefirstanalyseindetailtheLOandNLOordervacuum,twoandfour-pointFeynmangraphsofthismodel.Wethenexhibitathoroughasymptotic combinatorialanalysisoftheFeynmangraphsatanarbitraryorderinthelarge N expansion.Weendthechapterbyananalysisoftheeffectofnon-Gaussiandistribution forthecouplingofthemodel.
InChapter16,weanalyseindetailthediagrammaticsofvariousSKY-liketensor models:theGurau–Wittenmodel(inthefirstsection),andthemulti-orientableand O (N )3 -invarianttensormodels,intherestofthechapter.Variousexplicitgraph theoreticaltechniquesareused.
2 Graphs,ribbongraphs, andpolynomials Inthischapter,wepresentsomenotionsofgraphtheorythatwillbeusefulinthe restofthisbook.Letusemphasizethatgraphtheoristsandquantumfieldtheorists adopt,unfortunately,differentterminologies.Wepresentbothhere,suchthatasortof dictionary betweenthesetwocommunitiesmaybeestablished.
Wethenextendthenotionofgraphstothatofmaps(orofribbongraphs).Moreover, graphpolynomialsencodingthesestructures(theTuttepolynomialforgraphsandthe Bollobás–Riordanpolynomialforribbongraphs)arepresented.
Inthischapter,wefollowtheoriginalarticle(ThomasKrajewskietal.2010)andthe reviewarticle(AdrianTanasa2012).
2.1Graphtheory:TheTuttepolynomial Forageneralintroductiontographtheory,theinterestedreadermayrefertoClaude Berge(1976).Letusnowdefineagraphinthefollowingway:
Definition2.1.1 Agraph Γ isdefinedasasetofvertices V andofedges E togetherwithan incidencerelationshipbetweenthem.
Noticethatweallowmulti-edgesandself-loops(seedefinition2.1.2 4),butstilluse theterm‘graph’(andnot‘pseudograph’).
Thenumberofverticesandedgesinagrapharealsonoted V and E forsimplicity, sinceourcontextpreventsconfusion.
OneneedstoemphasizethatinQFTasupplementarytypeofedgeexists, external edges.Theseedgesareonlyhookedtooneoftheverticesofthegraph,theotherend oftheedgebeing‘free’(seeFig.2.1foranexampleofsuchagraph,withfourexternal edges).Inelementaryparticlephysics,theseexternaledgesarerelatedtotheobservables insomeexperiments.
Graphs,ribbongraphs,andpolynomials
Figure2.1 A Φ4 graph,withfourinternaledgesandfourexternaledges
Letusnowgivethefollowingdefinition:
Definition2.1.2
1. Thenumberofedgesatavertexiscalledthe degree oftherespectivevertex(field theoristsrefertothisasthe coordinationnumber oftherespectivevertex).
2. Anedgewhoseremovalincreasesthenumberofconnectedcomponentsofthe respectivegraphiscalleda bridge (fieldtheoristsrefertothisasa 1-particle reducible edge).
3. Aconnectedsubsetofequalnumberofedgesandofverticeswhichcannotbe disconnectedbyremovinganyoftheedgesiscalleda cycle (fieldtheoristsrefer tothisasa loop).
4. Anedgewhichconnectsavertextoitselfiscalleda self-loop (fieldtheoristsreferto thisasa tadpole edge).
5. Anedgewhichisneitherabridgenoraself-loopiscalled regular
6. Anedgewhichisnotaself-loopiscalled semi-regular
7. Agraphwithnocyclesiscalleda forest.
8. Aconnectedforestiscalleda tree.
9. A two-tree isaspanningtreewithoutoneofitsedges.
10. The rank ofasubgraph A isdefinedas
where k (A) isthenumberofconnectedcomponentsofthesubgraph A.
11. The nullity (or cyclomaticnumber)ofasubgraph A isdefinedas
Remark2.1.3 Foraconnectedgraph,thenullitydefinedpreviouslyrepresentsthenumberof independentcircuits.
InQFT,oneoftenusestheterm(numberof)loopstodenote(thenumberof) independentloops.
Figure2.2 Anexampleofagraph(withsevenedgesandsixexternaledges).Wechoseaspanningtree andlabelleditsedgesby1,..., 4.Wethenchosetheedge3toberemoved.Theset{1,2,4}isatwo-tree; onehastwoconnectedcomponents(thefirstoneformedbytheedges1and2andthesecondoneformed byedge4).Theexternaledgesareattachedtooneofthesetwoconnectedcomponents
Remark2.1.4 Atwo-treegeneratestwoconnectedcomponentsontherespectivegraph.Letus alsonotethatatwo-treecanbedefinedasaspanningforestwithtwoconnectedcomponents(in thisway,norelationwithatreeisgiven).
LetusillustratethisinFig.2.2.Onecandefinetwonaturaloperationsforanarbitrary edge e ofsomegraph Γ:
1.thedeletion,whichleadstoagraphnoted Γ e;
2.thecontraction,whichleadstoagraphnoted Γ/e.Thisoperationidentifiesthe twovertices v1 and v2 attheendsof e intoanewvertex v12 ,attributingallthe edgesattachedto v1 and v2 to v12 ;finally,thecontractionoperationremoves e
Remark2.1.5 If e isaself-loop,then Γ/e isthesamegraphas Γ e.
Foranillustrationofthesetwooperations,onecanrefertoFig.2.3,wherethese operationsareiterateduntilonereaches terminalforms namelygraphsformedonly ofself-loopsandbridges.
LetusnowgiveafirstdefinitionoftheTuttepolynomial:
Definition2.1.6 If Γ isagraph,thenitsTuttepolynomial TΓ (x,y ) isdefinedas
AfundamentalpropertyoftheTuttepolynomialisadeletion/contractionproperty:
Graphs,ribbongraphs,andpolynomials
Figure2.3 Thedeletion/contractionofsomegraph.Oneisleftwithvariouspossibilities(herefive)of terminalforms(thatis,graphswithonlybridgesorself-loops)
Theorem2.1.7 If Γ isagraph,and e isaregularedge,then
ThispropertyoftheTuttepolynomialisoftenusedasitsdefinition,ifonecompletes itbygivingtheformoftheTuttepolynomialonterminalforms:
where m isthenumberofbridgesand n isthenumberofself-loops.
Multi-variate(orweighted)versionsoftheTuttepolynomialexistintheliterature. Thus,intheseminalpaper,AlanD.Sokal(2005)analysedindetailsuchamulti-variate polynomial.Themainideaisthefollowing:oneintroducesasetofvariables β1 ,...,βE , oneforeachedge,andavariable q ,insteadofthecoupleofvariables x and y oftheTutte polynomial.
Othermulti-variateversionscanalsobefoundintheliteraturebuttheyareessentially equivalenttotheSokalpolynomialafterappropriatechangesofvariables.Nevertheless, thisisnotthecaseforthepolynomialsdefinedinZaslavsky(1992)andBollobásand Riordan(1992)(seealsoEllis-MonaghanandTraldi(2006)forgeneralizations).
Letusgivethedefinitionofthefollowingmulti-variateversionoftheTutte polynomial:
Graphtheory:TheTuttepolynomial 11
Definition2.1.8 If Γ isagraph,thenitsmulti-variateTuttepolynomialisdefinedas
Similarly,onecanprovethatthemulti-variateTuttepolynomial (2.6) satisfiesthe deletion/contractionrelation,foranyedge e.Thedefinitionofthepolynomialonthe terminalforms(graphswith v isolatedvertices)is
Onecanprove(throughdirectinspection)therelationbetweentheTuttepolynomial (2.3) anditsmulti-variatecounterpart (2.6):
Itisthisversionofthemulti-variateTuttepolynomial (2.6) thatweuseinthisbookto provetherelationwiththeparametricrepresentationofFeynmanintegralsinQFT(see Chapter6.6).
PuttingasidetheTuttepolynomial,severalgraphpolynomialshavebeendefinedand extensivelystudiedintheliterature.Aswehaveseenpreviously,theTuttepolynomial isatwo-variablepolynomial.Ithasone-variablespecializations,suchasthechromatic polynomialortheflowpolynomial.
The chromaticpolynomial isagraphpolynomial PΓ (k ) (k ∈ N )whichcountsthe numberofdistinctwaystocolourthegraph Γ with k orfewercolours,colouringsbeing countedasdistincteveniftheydifferonlybypermutationofcolours.Foraconnected graph,thispolynomialisrelatedtotheTuttepolynomial (2.3) bytherelation
Inordertodefinetheflowpolynomial,weneedafiniteabeliangroup G.One canarbitrarilychooseanorientationforeachedgeofthegraph Γ,theresultbeing independentofthischoice(thesametypeofsituationappearswhencomputingFeynman integrals,seenextsection).A G flow on Γ isamapping
thatsatisfiescurrentconservationateachvertex.A G flowon Γ issaidtobe nowherezero if ψ (e) =0 forall e.Let FΓ (G) bethenumberofnowhere-zero G flowson Γ.One canprovethatthisnumberdependsonlyontheorder k ofthegroup G;itcanthusbe written FΓ (k )—itistherestrictiontonon-negativeintegersofapolynomialin k ,the flow polynomial
Graphs,ribbongraphs,andpolynomials
Onehas:
AcrucialpropertyoftheTuttepolynomialisitspropertyof universality.This propertystatesthatanyTutteinvariant(i.e.anygraphpolynomialsatisfyingdeletion/contractionpropertyandamultiplicativelawongraphdisjointreunionandonevertexjoint)isanevaluationoftheTuttepolynomial.Thispropertyisprovedinthe combinatoricsliteraturebycarefullyusinginductionargumentsontheedgesofthe graph.
LetusendthissectionbyemphasizingthattheTuttepolynomial(anditsmulti-variate versionthatwehavepresentedhere)extendsinanaturalmannertothemoreinvolved combinatorialnotionof matroids (seeagainSokal(2005)).
2.2Ribbongraphs;theBollobás–Riordanpolynomial Inthissection,weintroduceanaturalgeneralizationofthenotionofgraphs—ribbon graphsormaps.Forageneralintroductiontocombinatorialmaps,theinterestedreader mayrefertoGuillaumeChapuy’sPhDthesisChapuy(2009)ortoGillesSchaeffer’suse (2009).
Letusdefinesucharibbongraphinthefollowingway:
Definition2.2.1 A ribbongraph Γ isanorientablesurfacewithitsboundaryrepresented astheunionofcloseddisks,alsocalled vertices,andribbonsalsocalled edges,suchthat: thedisksandribbonsintersectindisjointlinesegments,eachsuchlinesegmentlyingonthe boundaryofpreciselyonediskandoneribbonandfinally,everyribboncontainingtwosuch linesegments.
Notethat,asinthecaseofgraphs,thisdefinitioncanbeextendedbyaddinganew typeofedge(notwithtwolinesegments,seethepreviousdefinition)suchthat external edges areallowed(seeprevioussubsection).Examplesofsuchgraphsaregivenin Figs.2.4and2.5.
Letusalsomentionthatribbongraphscanbedefinedasgraphsequippedwitha cyclicorderingoftheincidenceedgesateachvertexorasgraphsembeddedinsurfaces (thelatterwasactuallythemathematicalobjectonwhichB.BollobásandO.Riordan definedtheirgeneralizationoftheTuttepolynomialinBollobásandRiordan(2001)and BollobásandRiordan(2002),seefollowingsection).
Aninteresting connectionbetweentheTuttepolynomialofgraphsandcombinatorialmaps wasproveninBernardi(2008).HegaveacharacterizationoftheTuttepolynomialof graphswhichisdifferenttotheinitialonegivenbyTutte(whichrequiredsomechoice oflinearorderontheedgeset,inordertowritetheTuttepolynomialasafunctionof
Ribbongraphs;theBollobás–Riordanpolynomial
Figure2.4 Anexampleofaribbongraphwithonevertex,oneinternaledge,andtwoexternaledges
Figure2.5 Anexampleofaribbongraphwithtwovertices,threeinternaledges,andtwoexternaledges spanningtrees).O.BernardiprovedthattheTuttepolynomialcanalsobewrittenasthe generatingfunctionofspanningtreescountedwiththehelpofacyclicorderoftheedges aroundeachvertex.
Letusnowgivethefollowingdefinition:
Definition2.2.2 A face ofaribbongraphisaconnectedcomponentofitsboundaryasa surface.
Forexample,thegraphofFig.2.4hastwofaces,whiletheoneofFig.2.5hasasingle face.IfwegluedisksalongthefacesweobtainaclosedRiemannsurfacewhosegenusis alsocalledthe genus ofthegraph.
Definition2.2.3 Theribbongraphiscalled planar ifithasvanishinggenus.
Forexample,thegraphofFig.2.4isplanarwhiletheoneofFig.2.5isnon-planar(it hasgenus 1).
