Table Of ContentOUPCORRECTEDPROOF – FINAL,16/2/2021,SPi
COMBINATORIAL PHYSICS
OUPCORRECTEDPROOF – FINAL,16/2/2021,SPi
OUPCORRECTEDPROOF – FINAL,16/2/2021,SPi
Combinatorial Physics
Combinatorics, quantum field theory, and quantum
gravity models
Adrian Tanasa
UniversityofBordeaux,France
1
OUPCORRECTEDPROOF – FINAL,16/2/2021,SPi
3
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PublishedintheUnitedStatesofAmericabyOxfordUniversityPress
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LibraryofCongressControlNumber:2021932400
ISBN 978–0–19–289549–3
DOI:10.1093/oso/9780192895493.001.0001
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OUPCORRECTEDPROOF – FINAL,16/2/2021,SPi
ToLuca,Brittany,andtoourfamily
OUPCORRECTEDPROOF – FINAL,16/2/2021,SPi
OUPCORRECTEDPROOF – FINAL,16/2/2021,SPi
Contents
1 Introduction 1
2 Graphs,ribbongraphs,andpolynomials 7
2.1 Graphtheory:TheTuttepolynomial 7
2.2 Ribbongraphs;theBollobás–Riordanpolynomial 12
2.3 Selectedfurtherreading 15
3 Quantumfieldtheory(QFT)—built-incombinatorics 17
3.1 DefinitionofthescalarΦ4 model 18
3.2 Perturbativeexpansion—Feynmangraphsandtheircombinatorialweights 20
3.3 Fouriertransform—themomentumspace 23
3.4 ParametricrepresentationofFeynmanintegrands 24
3.5 Thepropagatorandtheheatkernel 26
3.6 Aglimpseofperturbativerenormalization 27
3.6.1 Thepowercountingtheorem 29
3.6.2 Locality 30
3.6.3 Multi-scaleanalysis 32
3.6.4 ThesubtractionoperatorforageneralFeynmangraph 33
3.6.5 Dimensionalrenormalization 35
3.7 Dyson–Schwingerequation 36
3.8 Combinatorial(or0-dimensional)QFTandtheintermediatefieldmethod 36
3.8.1 Combinatorial(or0-dimensional)QFT 36
3.8.2 Theintermediatefieldmethod 37
3.9 Selectedfurtherreading 38
4 TreeweightsandrenormalizationinQFT 39
4.1 Preliminaryresults 41
4.2 Partitiontreeweights 43
4.3 Selectedfurtherreading 49
5 CombinatorialQFTandtheJacobianConjecture 50
5.1 TheJacobianConjectureascombinatorialQFTmodel
(theAbdesselam–Rivasseaumodel) 52
5.2 TheintermediatefieldmethodfortheAbdesselam–Rivasseaumodel 53
5.3 Selectedfurtherreading 55
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viii Contents
6 FermionicQFT,Grassmanncalculus,andcombinatorics 56
6.1 GrassmannalgebrasandGrassmanncalculus 57
6.1.1 TheGrassmannalgebra 57
6.1.2 Grassmanncalculus;PfaffiansasGrassmannintegrals 58
6.2 OnGrassmannGaussianmeasures 59
6.3 Lingström–Gessel–Viennot(LGV)formulaforgraphswithcycles 60
6.4 Stembridge’sformulasforgraphswithcycles 63
6.5 Ageneralization 66
6.6 TuttepolynomialandtheparametricrepresentationinQFT 67
6.7 Selectedfurtherreading 71
7 AnalyticcombinatoricsandQFT 72
7.1 TheMellintransformtechnique 72
7.2 Thesaddlepointmethod 74
7.3 Selectedfurtherreading 75
8 AlgebraiccombinatoricsandQFT 76
8.1 Algebraicreminder;CombinatorialHopfAlgebras(CHAs) 77
8.2 TheConnes–KreimerHopfalgebraofFeynmangraphs 79
8.3 TheB operator,HochschildcohomologyoftheConnes–Kreimeralgebra 83
+
8.4 Multi-scalerenormalization,CHAdescription 85
8.5 Selectedfurtherreading 94
9 QFTonthenon-commutativeMoyalspaceandcombinatorics 95
9.1 Mathematicalsetting:Renormalizability 96
9.2 TheMehlerkernelandtheGrosse–Wulkenhaarmodel 99
9.3 ParametricrepresentationofGrosse–Wulkenhaar-likemodels 100
9.4 TheMellintransformandtheGrosse–Wulkenhaarmodel 104
9.5 DimensionalrenormalizationfortheGrosse–Wulkenhaarmodel 107
9.6 Aheatkernel–basedrenormalizablemodel 108
9.7 ParametricrepresentationandtheBollobás–Riordanpolynomial 110
9.7.1 Parametricrepresentation 110
9.7.2 Relationbetweenthemulti-variateBollobás–Riordanandthe
polynomialsoftheparametricrepresentation 111
9.8 CombinatorialConnes–KreimerHopfalgebraandits
Hochschildcohomology 112
9.8.1 CombinatorialConnes–KreimerHopfalgebra 112
9.8.2 HochschildcohomologyandthecombinatorialDSE 117
9.9 Selectedfurtherreading 120
10 Quantumgravity,groupfieldtheory(GFT),andcombinatorics 121
10.1 Quantumgravity 121
10.2 Maincandidatesforatheoryofquantumgravity:Theholographicprinciple 122
10.3 GFTmodels:theBoulatovandthecolourablemodels 123
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Contents ix
10.4 Themulti-orientableGFTmodel 125
10.4.1 Tadpolesandgeneralizedtadpoles 127
10.4.2 Tadfaces 128
10.5 SaddlepointmethodforGFTFeynmanintegrals 129
10.6 AlgebraiccombinatoricsandtensorialGFT 133
10.6.1 TheBenGeloun–Rivasseau(BGR)model 133
10.6.2 Cones–KreimerHopfalgebraicdescriptionofthecombinatorics
oftherenormalizabilityoftheBGRmodel 143
10.6.3 HochschildcohomologyandthecombinatorialDSE
fortensorialGFT 153
10.7 Selectedfurtherreading 165
11 Fromrandommatricestorandomtensors 166
11.1 ThelargeN limit 169
11.2 Thedouble-scalinglimit 169
11.3 Frommatricestotensors 170
11.4 Tensorgraphpolynomials—ageneralizationoftheBollobás–Riordan
polynomial 174
11.5 Selectedfurtherreading 176
12 Randomtensormodels—theU(N)D-invariantmodel 178
12.1 DefinitionofthemodelanditsDSE 179
12.1.1 U(N)D-invariantbubbleinteractions 179
12.1.2 Bubbleobservables 182
12.1.3 TheDSEforthemodel 185
12.1.4 Navigatingthefollowingsectionsofthechapter 187
12.2 TheDSEbeyondthelargeN limit 188
12.2.1 TheLO 188
12.2.2 MomentsandCumulants 189
12.2.3 Gaussianandnon-Gaussiancontributions 192
12.2.4 TheDSEatNLO 198
12.2.5 Theorder1/ND inthequarticmodel 199
12.3 Thedouble-scalinglimit 202
12.3.1 Double-scalinglimitintheDSE 202
12.3.2 Fromthequarticmodeltoagenericmodel 206
12.4 Selectedfurtherreading 208
13 Randomtensormodels—themulti-orientable(MO)model 209
13.1 Definitionofthemodel 209
13.2 The1/N expansionandthelargeN limit 212
13.2.1 Feynmanamplitudes;the1/N expansion 212
13.2.2 ThelargeN limit—theLO(melonicgraphs) 214
13.2.3 ThelargeN limit—theNLO 215
13.2.4 LeadingandNLOseries 216
