Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics

493 Pages·2006·3.43 MB·English

by  Terry Gannon

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Moonshine Beyond the Monster
TheBridgeConnectingAlgebra,
ModularFormsandPhysics
Moonshineformsawayofexplainingthemysteriousconnectionbetweenthemonsterfinitegroup
andmodularfunctionsfromclassicalnumbertheory.Thetheoryhasevolvedtodescribetherela-
tionshipbetweenfinitegroups,modularformsandvertexoperatoralgebras.MoonshineBeyond
theMonster,thefirstbookofitskind,describesthegeneraltheoryofMoonshineanditsunderly-
ingconcepts,emphasisingtheinterconnectionsbetweenmodernmathematicsandmathematical
physics.
Writteninaclearandpedagogicalstyle,thisbookisidealforgraduatestudentsandresearchers
workinginareassuchasconformalfieldtheory,stringtheory,algebra,numbertheory,geometry
andfunctionalanalysis.Containingalmosttwohundredexercises,itisalsoasuitabletextbook
forgraduatecoursesonMoonshineandassupplementaryreadingforcoursesonconformalfield
theoryandstringtheory.
Terry Gannon iscurrentlyProfessorofMathematicsattheUniversityofAlberta.Overthe
years he has had extended research visits to the Erwin-Schrodinger Institute in Vienna, IHES
Bures,Max-PlanckInstituteinBonn,St.John’sCollege,Cambridge,theFezaGu¨rseyInstitutein
Istanbul,Universita¨tHamburgandtheUniversityofWales.Hebecameanassistantprofessorat
YorkUniversityin1996andthenmovedtotheUniversityofAlbertain1998.Hisresearchinterests
coverawiderangeofmathematics,inparticulartheinteractionsofalgebra,numbertheoryand
mathematicalphysics,withafocusonconformalfieldtheory.
cambridge monographs on
mathematical physics
Generaleditors:P.V.Landshoff,D.R.Nelson,S.Weinberg
S.J.Aarseth GravitationalN-BodySimulations
J.Ambjørn,B.DurhuusandT.Jonsson QuantumGeometry:AStatisticalFieldTheoryApproach
A.M.Anile RelativisticFluidsandMagneto-Fluids
J.A.deAzca´rrageandJ.M.Izquierdo LieGroups,LieAlgebras,CohomologyandSomeApplicationsinPhysics†
O.Babelon,D.BernardandM.Talon IntroductiontoClassicalIntegrableSystems
F.BastianelliandP.vanNieuwenhuizenPathIntegralsandAnomaliesinCurvedSpace
V.BelinkskiandE.Verdaguer GravitationalSolitons
J.Bernstein KineticTheoryintheExpandingUniverse
G.F.BertschandR.A.Broglia OscillationsinFiniteQuantumSystems
N.D.BirrellandP.C.W.Davies QuantumFieldsinCurvedSpace†
M.Burgess ClassicalCovariantFields
S.Carlip QuantumGravityin2+1 Dimensions
J.C.Collins Renormalization†
M.Creutz Quarks,GluonsandLattices†
P.D.D’Eath SupersymmetricQuantumCosmology
F.deFeliceandC.J.S.Clarke RelativityonCurvedManifolds†
B.S.DeWitt Supermanifolds,2ndedition†
P.G.O.Freund IntroductiontoSupersymmetry†
J.Fuchs AffineLieAlgebrasandQuantumGroups†
J.FuchsandC.Schweigert Symmetries,LieAlgebrasandRepresentations:AGraduateCourseforPhysicists†
Y.FujiiandK.Maeda TheScalar–TensorTheoryofGravitation
A.S.Galperin,E.A.Ivanov,V.I.OrievetskyandE.S.Sokatchev HarmonicSuperspace
R.GambiniandJ.Pullin Loops,Knots,GaugeTheoriesandQuantumGravity†
T.Gannon MoonshineBeyondtheMonster:TheBridgeConnectingAlgebra,ModularFormsandPhysics
M.Go¨ckelerandT.Schu¨cker DifferentialGeometry,GaugeTheoriesandGravity†
C.Go´mez,M.RuizAltabaandG.Sierra QuantumGroupsinTwo-DimensionalPhysics
M.B.Green,J.H.SchwarzandE.Witten SuperstringTheory,volume1:Introduction†
M.B.Green,J.H.SchwarzandE.Witten SuperstringTheory,volume2:LoopAmplitudes,Anomaliesand
Phenomenology†
V.N.Gribov TheTheoryofComplexAngularMomenta
S.W.HawkingandG.F.R.Ellis TheLarge-ScaleStructureofSpace-Time†
F.IachelloandA.Arima TheInteractingBosonModel
F.IachelloandP.vanIsacker TheInteractingBoson–FermionModel
C.ItzyksonandJ.-M.Drouffe StatisticalFieldTheory,volume1:FromBrownianMotiontoRenormalizationand
LatticeGaugeTheory†
C.ItzyksonandJ.-M.Drouffe StatisticalFieldTheory,volume2:StrongCoupling,MonteCarloMethods,
ConformalFieldTheory,andRandomSystems†
C.Johnson D-Branes
J.I.KapustaandC.Gale Finite-TemperatureFieldTheory,2ndedition
V.E.Korepin,A.G.IzerginandN.M.Boguliubov TheQuantumInverseScatteringMethodandCorrelation
Functions†
M.LeBellac ThermalFieldTheory†
Y.Makeenko MethodsofContemporaryGaugeTheory
N.MantonandP.Sutcliffe TopologicalSolitons
N.H.March LiquidMetals:ConceptsandTheory
I.M.MontvayandG.Mu¨nster QuantumFieldsonaLattice†
L.O’Raifeartaigh GroupStructureofGaugeTheories†
T.Ort´ın GravityandStrings
A.OzoriodeAlmeida HamiltonianSystems:ChaosandQuantization†
R.PenroseandW.Rindler SpinorsandSpace-Time,volume1:Two-SpinorCalculusandRelativisticFields†
R.PenroseandW.Rindler SpinorsandSpace-Time,volume2:SpinorandTwistorMethodsinSpace-Time
Geometry†
S.Pokorski GaugeFieldTheories,2ndedition
J.Polchinski StringTheory,volume1:AnIntroductiontotheBosonic,String†
J.Polchinski StringTheory,volume2:SuperstringTheoryandBeyond†
V.N.Popov FunctionalIntegralsandCollectiveExcitations†
R.J.Rivers PathIntegralMethodsinQuantumFieldTheory†
R.G.Roberts TheStructureoftheProton†
C.Rovelli QuantumGravity
W.C.Saslaw GravitationalPhysicsofStellarandGalacticSystems†
H.Stephani,D.Kramer,M.A.H.MacCallum,C.HoenselaersandE.Herlt ExactSolutionsofEinstein’sField
Equations,2ndedition
J.M.Stewart AdvancedGeneralRelativity†
A.VilenkinandE.P.S.Shellard CosmicStringsandOtherTopologicalDefects†
R.S.WardandR.O.WellsJr TwistorGeometryandFieldTheories†
J.R.WilsonandG.J.Mathews RelativisticNumericalHydrodynamics
†Issuedasapaperback
Moonshine Beyond the Monster
The Bridge Connecting Algebra,
Modular Forms and Physics
TERRY GANNON
UniversityofAlberta
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
TheEdinburghBuilding,Cambridgecb22ru,UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521835312
© Terry Gannon 2006
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2006
isbn-13 978-0-511-24514-5eBook (EBL)
isbn-10 0-511-24514-9 eBook (EBL)
isbn-13 978-0-521-83531-2hardback
isbn-10 0-521-83531-3 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
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Contents
Acknowledgements pagexiii
0 Introduction:glimpsesofthetheorybeneath
MonstrousMoonshine 1
0.1 Modularfunctions 1
0.2 TheMcKayequations 3
0.3 Twisted#0:theThompsontrick 4
0.4 MonstrousMoonshine 5
0.5 TheMoonshineof E andtheLeech 6
8
0.6 MoonshinebeyondtheMonster 8
0.7 PhysicsandMoonshine 9
0.8 Braided#0:themeaningofMoonshine 11
0.9 Thebook 11
1 Classicalalgebra 14
1.1 Discretegroupsandtheirrepresentations 14
1.1.1 Basicdefinitions 15
1.1.2 Finitesimplegroups 17
1.1.3 Representations 20
1.1.4 Braided#1:thebraidgroups 26
1.2 Elementarygeometry 29
1.2.1 Lattices 29
1.2.2 Manifolds 32
1.2.3 Loops 40
1.3 Elementaryfunctionalanalysis 44
1.3.1 Hilbertspaces 45
1.3.2 Factors 49
1.4 LiegroupsandLiealgebras 52
1.4.1 DefinitionandexamplesofLiealgebras 53
1.4.2 Theirmotivation:Liegroups 55
1.4.3 SimpleLiealgebras 59
1.5 RepresentationsofsimpleLiealgebras 65
1.5.1 Definitionsandexamples 65
1.5.2 ThestructureofsimpleLiealgebras 68
1.5.3 Weylcharacters 73
viii Contents
1.5.4 Twisted#1:automorphismsandcharacters 78
1.5.5 RepresentationsofLiegroups 82
1.6 Categorytheory 87
1.6.1 Generalphilosophy 87
1.6.2 Braidedmonoidalcategories 88
1.7 Elementaryalgebraicnumbertheory 95
1.7.1 Algebraicnumbers 95
1.7.2 Galois 98
1.7.3 Cyclotomicfields 101
2 Modularstuff 104
2.1 Theunderlyinggeometry 104
2.1.1 Thehyperbolicplane 104
2.1.2 Riemannsurfaces 110
2.1.3 Functionsanddifferentialforms 116
2.1.4 Moduli 119
2.2 Modularformsandfunctions 126
2.2.1 Definitionandmotivation 126
2.2.2 Thetaandeta 131
2.2.3 Poissonsummation 135
2.2.4 Hauptmoduls 138
2.3 Furtherdevelopments 140
2.3.1 Dirichletseries 140
2.3.2 Jacobiforms 142
2.3.3 Twisted#2:shiftsandtwists 144
2.3.4 Theremarkableheatkernel 147
2.3.5 Siegelforms 150
2.4 Representationsandmodularforms 154
2.4.1 Automorphicforms 154
2.4.2 Thetafunctionsasmatrixentries 159
2.4.3 Braided#2:fromthetrefoiltoDedekind 164
2.5 Meta-patternsinmathematics 168
2.5.1 Twenty-four 168
2.5.2 A–D–E 169
3 Goldandbrass:affinealgebrasandgeneralisations 176
3.1 Modularityfromthecircle 176
3.1.1 Centralextensions 176
3.1.2 TheVirasoroalgebra 180
3.2 Affinealgebrasandtheirrepresentations 187
3.2.1 Motivation 187
3.2.2 Constructionandstructure 189
3.2.3 Representations 192