FactorizationAlgebrasinQuantumFieldTheory Volume2 Factorization algebras are local-to-global objects that play a role in classical and quantumfieldtheorythatissimilartotheroleofsheavesingeometry:theyconveniently organizecomplicatedinformation.Theirlocalstructureencompassesexamplessuchas associativeandvertexalgebras;intheseexamples,theirglobalstructureencompasses Hochschildhomologyandconformalblocks. In this second volume, the authors show how factorization algebras arise from interacting field theories, both classical and quantum, and how they encode essential information such as operator product expansions, Noether currents, and anomalies. Along with a systematic reworking of the Batalin–Vilkovisky formalism via derived geometryandfactorizationalgebras,thisbookoffersconcreteexamplesfromphysics, rangingfromangularmomentumandVirasorosymmetriestoafive-dimensionalgauge theory. Kevin Costello is Krembil William Rowan Hamilton Chair in Theoretical PhysicsatthePerimeterInstituteforTheoreticalPhysics,Waterloo,Canada.Heisan honorarymemberoftheRoyalIrishAcademyandaFellowoftheRoyalSociety.Hehas wonseveralawards,includingtheBerwickPrizeoftheLondonMathematicalSociety (2017)andtheEisenbudPrizeoftheAmericanMathematicalSociety(2020). Owen Gwilliam is Assistant Professor in the Department of Mathematics and StatisticsattheUniversityofMassachusetts,Amherst. NEW MATHEMATICAL MONOGRAPHS EditorialBoard JeanBertoin,Be´laBolloba´s,WilliamFulton,BrynaKra,IekeMoerdijk, CherylPraeger,PeterSarnak,BarrySimon,BurtTotaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversity Press.Foracompleteserieslistingvisitwww.cambridge.org/mathematics. 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 13. M.GrandisDirectedAlgebraicTopology 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 33. G.FriedmanSingularIntersectionHomology 34. S.SchwedeGlobalHomotopyTheory 35. M.Dickmann,N.SchwartzandM.TresslSpectralSpaces 36. A.BaernsteinIISymmetrizationinAnalysis 37. A.Defant,D.Garc´ıa,M.MaestreandP.Sevilla-PerisDirichletSeriesandHolomorphic FunctionsinHighDimensions 38. N.Th.VaropoulosPotentialTheoryandGeometryonLieGroups 39. D.ArnalandB.CurreyRepresentationsofSolvableLieGroups 40. M.A.Hill,M.J.HopkinsandD.C.RavenelEquivariantStableHomotopyTheoryandthe KervaireInvariantProblem Factorization Algebras in Quantum Field Theory Volume 2 KEVIN COSTELLO PerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario OWEN GWILLIAM UniversityofMassachusetts,Amherst UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107163157 DOI:10.1017/9781316678664 ©KevinCostelloandOwenGwilliam2021 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2021 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:Costello,Kevin,1977-author.|Gwilliam,Owen,author. Title:Factorizationalgebrasinquantumfieldtheory/KevinCostello,PerimeterInstitute forTheoreticalPhysics,Waterloo,Ontario,OwenGwilliam,MaxPlanckInstituteforMathematics,Bonn. Othertitles:Newmathematicalmonographs;31. Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversityPress,2017-.| Series:Newmathematicalmonographs;31|Includesbibliographicalreferencesandindex. Contents:FromGaussianmeasurestofactorizationalgebras–Prefactorizationalgebrasand basicexamples–Freefieldtheories–Holomorphicfieldtheoriesandvertexalgebras– Factorizationalgebras:definitionsandconstructions–Formalaspectsof factorizationalgebras–Factorizationalgebras:examples. Identifiers:LCCN2016047832|ISBN9781107163102(hardback;v.1)|ISBN1107163102 (hardback;v.1)| ISBN9781107163157(hardback;v.2)|ISBN9781316678664(epub;v.2) Subjects:LCSH:Quantumfieldtheory–Mathematics.|Factorization(Mathematics)|Factors(Algebra)| Geometricquantization.|Noncommutativealgebras. Classification:LCCQC174.45.C682017|DDC530.14/30151272–dc23LCrecordavailable athttps://lccn.loc.gov/2016047832 ISBN–2VolumeSet978-1-009-00616-3Hardback ISBN–Volume1978-1-107-16310-2Hardback ISBN–Volume2978-1-107-16315-7Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. To Josie and Dara & To Laszlo and Hadrian Contents ContentsofVolume1 pagexi 1 IntroductionandOverview 1 1.1 TheFactorizationAlgebraofClassicalObservables 1 1.2 TheFactorizationAlgebraofQuantumObservables 2 1.3 ThePhysicalImportanceofFactorizationAlgebras 3 1.4 PoissonStructuresandDeformationQuantization 6 1.5 TheNoetherTheorem 8 1.6 BriefOrientingRemarkstowardtheLiterature 11 1.7 Acknowledgments 12 PARTI CLASSICALFIELDTHEORY 15 2 IntroductiontoClassicalFieldTheory 17 2.1 TheEuler–LagrangeEquations 17 2.2 Observables 18 2.3 TheSymplecticStructure 19 2.4 TheP Structure 19 0 3 EllipticModuliProblems 20 3.1 FormalModuliProblemsandLieAlgebras 21 3.2 Examples of Elliptic Moduli Problems Related toScalarFieldTheories 25 3.3 Examples of Elliptic Moduli Problems Related toGaugeTheories 28 3.4 CochainsofaLocalL∞Algebra 32 3.5 D-modulesandLocalL∞Algebras 34 vii viii Contents 4 TheClassicalBatalin–VilkoviskyFormalism 43 4.1 TheClassicalBVFormalisminFiniteDimensions 43 4.2 TheClassicalBVFormalisminInfiniteDimensions 45 4.3 TheDerivedCriticalLocusofanActionFunctional 48 4.4 ASuccinctDefinitionofaClassicalFieldTheory 54 4.5 ExamplesofScalarFieldTheoriesfromActionFunctionals 57 4.6 CotangentFieldTheories 58 5 TheObservablesofaClassicalFieldTheory 63 5.1 TheFactorizationAlgebraofClassicalObservables 63 5.2 TheGradedPoissonStructureonClassicalObservables 64 5.3 ThePoissonStructureforFreeFieldTheories 66 5.4 ThePoissonStructureforaGeneralClassicalFieldTheory 68 PARTII QUANTUMFIELDTHEORY 73 6 IntroductiontoQuantumFieldTheory 75 6.1 TheQuantumBVFormalisminFiniteDimensions 76 6.2 The“FreeScalarField”inFiniteDimensions 79 6.3 AnOperadicDescription 81 6.4 EquivariantBDQuantizationandVolumeForms 82 6.5 HowRenormalizationGroupFlowInterlockswiththeBV Formalism 83 6.6 OverviewoftheRestofThisPart 84 7 EffectiveFieldTheoriesandBatalin–VilkoviskyQuantization 86 7.1 LocalActionFunctionals 87 7.2 TheDefinitionofaQuantumFieldTheory 88 7.3 FamiliesofTheoriesoverNilpotentdgManifolds 99 7.4 TheSimplicialSetofTheories 105 7.5 TheTheoremonQuantization 109 8 TheObservablesofaQuantumFieldTheory 111 8.1 FreeFields 111 8.2 TheBDAlgebraofGlobalObservables 115 8.3 GlobalObservables 124 8.4 LocalObservables 126 8.5 LocalObservablesFormaPrefactorizationAlgebra 128 8.6 LocalObservablesFormaFactorizationAlgebra 132 8.7 TheMapfromTheoriestoFactorizationAlgebrasIsaMap ofPresheaves 138 Contents ix 9 FurtherAspectsofQuantumObservables 144 9.1 TranslationInvarianceforFieldTheoriesandObservables 144 9.2 HolomorphicallyTranslation-InvariantTheories andObservables 148 9.3 RenormalizabilityandFactorizationAlgebras 154 9.4 CotangentTheoriesandVolumeForms 168 9.5 CorrelationFunctions 177 10 OperatorProductExpansions,withExamples 179 10.1 PointObservables 179 10.2 TheOperatorProductExpansion 185 10.3 TheOPEtoFirstOrderin(cid:2) 188 10.4 TheOPEintheφ4Theory 197 10.5 TheOperatorProductforHolomorphicTheories 201 10.6 QuantumGroupsandHigher-DimensionalGaugeTheories 210 PARTIII AFACTORIZATIONENHANCEMENT OFTHENOETHERTHEOREM 225 11 IntroductiontotheNoetherTheorems 227 11.1 SymmetriesintheClassicalBVFormalism 228 11.2 KoszulDualityandSymmetriesviatheClassicalMaster Equation 233 11.3 SymmetriesintheQuantumBVFormalism 239 12 TheNoetherTheoreminClassicalFieldTheory 245 12.1 AnOverviewoftheMainTheorem 245 12.2 SymmetriesofaClassicalFieldTheory 246 12.3 The Factorization Algebra of Equivariant Classical Observables 259 12.4 TheClassicalNoetherTheorem 263 12.5 ConservedCurrents 268 12.6 ExamplesoftheClassicalNoetherTheorem 270 12.7 TheNoetherTheoremandtheOperatorProductExpansion 279 13 TheNoetherTheoreminQuantumFieldTheory 289 13.1 TheQuantumNoetherTheorem 289 13.2 ActionsofaLocalL∞AlgebraonaQuantumFieldTheory 294 13.3 ObstructionTheoryforQuantizingEquivariantTheories 299 13.4 The Factorization Algebra of an Equivariant QuantumFieldTheory 303 13.5 TheQuantumNoetherTheoremRedux 304