Factorization Algebras in Quantum Field Theory

399 Pages·2017·3.136 MB·English

by  Kevin Costello, Owen Gwilliam

FactorizationAlgebrasinQuantumFieldTheory
Volume1
Factorizationalgebrasarelocal-to-globalobjectsthatplayaroleinclassicalandquan-
tum field theory that is similar to the role of sheaves in geometry: they conveniently
organizecomplicatedinformation.Theirlocalstructureencompassesexamplessuchas
associativeandvertexalgebras;intheseexamples,theirglobalstructureencompasses
Hochschildhomologyandconformalblocks.
Inthisfirstvolume,theauthorsdevelopthetheoryoffactorizationalgebrasindepth,
butwithafocusuponexamplesexhibitingtheiruseinfieldtheory,suchastherecov-
eryofavertexalgebrafromachiralconformalfieldtheoryandaquantumgroupfrom
Abelian Chern–Simons theory. Expositions of the relevant background in homologi-
calalgebra,sheaves,andfunctionalanalysisarealsoincluded,thusmakingthisbook
idealforresearchersandgraduatesworkingattheinterfacebetweenmathematicsand
physics.
KEVIN COSTELLO istheKrembilFoundationWilliamRowanHamiltonChairin
TheoreticalPhysicsatthePerimeterInstituteinWaterloo,Ontario.
OWEN GWILLIAMisapostdoctoralfellowattheMaxPlanckInstituteforMathe-
maticsinBonn.
NEW MATHEMATICAL MONOGRAPHS
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Be´laBolloba´s,WilliamFulton,FrancesKirwan,
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1. M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups
2. J.B.GarnettandD.E.MarshallHarmonicMeasure
3. P.CohnFreeIdealRingsandLocalizationinGeneralRings
4. E.BombieriandW.GublerHeightsinDiophantineGeometry
5. Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6. S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7. A.ShlapentokhHilbert’sTenthProblem
8. G.MichlerTheoryofFiniteSimpleGroupsI
9. A.BakerandG.Wu¨stholzLogarithmicFormsandDiophantineGeometry
10. P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11. B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
12. J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
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14. G.MichlerTheoryofFiniteSimpleGroupsII
15. R.SchertzComplexMultiplication
16. S.BlochLecturesonAlgebraicCycles(2ndEdition)
17. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups
18. T.DownarowiczEntropyinDynamicalSystems
19. C.SimpsonHomotopyTheoryofHigherCategories
20. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesI
21. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesII
22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory
23. J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry
24. E.RiehlCategoricalHomotopyTheory
25. B.A.MunsonandI.Volic´CubicalHomotopyTheory
26. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups(2ndEdition)
27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpacesonMetric
MeasureSpaces
28. Y.-G.OhSymplecticTopologyandFloerHomologyI
29. Y.-G.OhSymplecticTopologyandFloerHomologyII
30. A.BobrowskiConvergenceofOne-ParameterOperatorSemigroups
31. K.CostelloandO.GwilliamFactorizationAlgebrasinQuantumFieldTheoryI
32. J.-H.EvertseandK.Gyo˝ryDiscriminantEquationsinDiophantineNumberTheory
Factorization Algebras in
Quantum Field Theory
Volume 1
KEVIN COSTELLO
PerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario
OWEN GWILLIAM
MaxPlanckInstituteforMathematics,Bonn
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Contents
1 Introduction page 1
1.1 TheMotivatingExampleofQuantumMechanics 3
1.2 APreliminaryDefinitionofPrefactorizationAlgebras 8
1.3 PrefactorizationAlgebrasinQuantumFieldTheory 8
1.4 ComparisonswithOtherFormalizationsofQuantum
FieldTheory 11
1.5 OverviewofThisVolume 16
1.6 Acknowledgments 18
PARTI PREFACTORIZATIONALGEBRAS 21
2 FromGaussianMeasurestoFactorizationAlgebras 23
2.1 GaussianIntegralsinFiniteDimensions 25
2.2 DivergenceinInfiniteDimensions 27
2.3 ThePrefactorizationStructureonObservables 31
2.4 FromQuantumtoClassical 34
2.5 CorrelationFunctions 36
2.6 FurtherResultsonFreeFieldTheories 39
2.7 InteractingTheories 40
3 PrefactorizationAlgebrasandBasicExamples 44
3.1 PrefactorizationAlgebras 44
3.2 AssociativeAlgebrasfromPrefactorizationAlgebrasonR 51
3.3 ModulesasDefects 52
3.4 AConstructionoftheUniversalEnvelopingAlgebra 59
3.5 SomeFunctionalAnalysis 62
vii
viii Contents
3.6 TheFactorizationEnvelopeofaSheafofLieAlgebras 73
3.7 EquivariantPrefactorizationAlgebras 79
PARTII FIRSTEXAMPLESOFFIELDTHEORIESAND
THEIROBSERVABLES 87
4 FreeFieldTheories 89
4.1 TheDivergenceComplexofaMeasure 89
4.2 ThePrefactorizationAlgebraofaFreeField
Theory 93
4.3 QuantumMechanicsandtheWeylAlgebra 106
4.4 PushforwardandCanonicalQuantization 112
4.5 AbelianChern–SimonsTheory 115
4.6 AnotherTakeonQuantizingClassicalObservables 124
4.7 CorrelationFunctions 129
4.8 Translation-InvariantPrefactorizationAlgebras 131
4.9 StatesandVacuaforTranslationInvariantTheories 139
5 HolomorphicFieldTheoriesandVertexAlgebras 145
5.1 VertexAlgebrasandHolomorphicPrefactorization
AlgebrasonC 145
5.2 HolomorphicallyTranslation-InvariantPrefactorization
Algebras 149
5.3 AGeneralMethodforConstructingVertex
Algebras 157
5.4 Theβγ SystemandVertexAlgebras 171
5.5 Kac–MoodyAlgebrasandFactorizationEnvelopes 188
PARTIII FACTORIZATIONALGEBRAS 205
6 FactorizationAlgebras:DefinitionsandConstructions 207
6.1 FactorizationAlgebras 207
6.2 FactorizationAlgebrasinQuantumField
Theory 215
6.3 VariantDefinitionsofFactorizationAlgebras 216
6.4 LocallyConstantFactorizationAlgebras 220
6.5 FactorizationAlgebrasfromCosheaves 225
6.6 FactorizationAlgebrasfromLocalLieAlgebras 230
7 FormalAspectsofFactorizationAlgebras 232
7.1 PushingForwardFactorizationAlgebras 232
7.2 ExtensionfromaBasis 232
Contents ix
7.3 PullingBackAlonganOpenImmersion 240
7.4 DescentAlongaTorsor 241
8 FactorizationAlgebras:Examples 243
8.1 SomeExamplesofComputations 243
8.2 AbelianChern–SimonsTheoryandQuantumGroups 249
AppendixABackground 273
AppendixBFunctionalAnalysis 310
AppendixCHomologicalAlgebrainDifferentiableVectorSpaces 351
AppendixDTheAtiyah–BottLemma 374
References 377
Index 383