Factorization algebras and free field theories

172 Pages·2012·1.027 MB·English

by  Owen Gwilliam

Factorization algebras and free field theories
Owen Gwilliam

Contents
Acknowledgements vii
Chapter1. Introduction 1
1.1. Anoverviewofthechapters 3
1.2. Notations 4
Chapter2. Motivationandalgebraictechniques 7
2.1. ClassicalBVformalism: thederivedcriticallocus 8
2.2. QuantumBVformalism: thetwisteddeRhamcomplex 10
2.3. Wick’slemmaandFeynmandiagrams,homologically 17
2.4. Acompendiumofessentialdefinitionsandconstructions 24
2.5. ThehomologicalperturbationlemmaintheBVformalism 27
2.6. GlobalobservablesandformalHodgetheory 34
Chapter3. BVformalismasadeterminantfunctor 39
3.1. Cotangentquantizationofk-vectorspaces 41
3.2. Recollections 44
3.3. Propertiesofcotangentquantizationoveranycommutativedgalgebra 48
3.4. Invertibilitysurvivesoverartiniandgalgebras 51
Chapter4. Factorizationalgebras 59
4.1. Definitions 60
R
4.2. Associativealgebrasasfactorizationalgebrason 63
4.3. Associativealgebrasandthebarcomplex 66
4.4. Thecategoryoffactorizationalgebras 73
4.5. Generalconstructionmethodsforfactorizationalgebras 75
4.6. Anovelconstructionoftheuniversalenvelopingalgebra 78
4.7. Extensionfromafactorizingbasis 81
4.8. PushforwardandPullback 83
Chapter5. Freefieldsandtheirobservables 87
5.1. Introduction 87
5.2. EllipticcomplexesandfreeBVtheories 90
5.3. Observablesasafactorizationalgebra 94
5.4. BVquantizationasaHeisenbergLiealgebraconstruction 98
5.5. BVquantizationasadeterminantfunctor 100
iii
5.6. Implicationsforinteractingtheories 101
5.7. TheorieswithaPoincare´ lemma 102
Chapter6. Freeholomorphicfieldtheoriesandvertexalgebras 105
6.1. The βγsystem 106
6.2. Thequantumobservablesofthe βγsystem 108
6.3. Recoveringavertexalgebra 122
6.4. VertexalgebrasfromLiealgebras 127
6.5. Definitionsandaconjecture 133
Chapter7. Anindextheorem 137
7.1. Amotivatingexample 139
7.2. Aprecisestatementofthetheorem 140
7.3. Settinguptheproblem 144
7.4. BackgroundaboutBVtheoriesandrenormalization 148
7.5. Theproof 155
7.6. Globalstatements 159
Bibliography 163
iv
ABSTRACT. WestudytheBatalin-Vilkovisky(BV)formalismforquantizationoffieldtheoriesinsev-
eralcontexts.First,weextracttheessentialhomologicalprocedureandstudyitfromtheperspective
of derived algebraic geometry. Our main result here is that the BV formalism provides a natural
determinantfunctorwecall“cotangentquantization,”sendingaperfect R-moduletoaninvertible
R-moduleandquasi-isomorphismstoquasi-isomorphisms,whereRisanartiniancommutativedif-
ferential graded algebra over a field of characteristic zero. Second, we introduce the formalism of
factorizationalgebras, alocal-to-globalobjectmuchlikeasheaf, anddescribeseveralperspectives
onhowtheBVformalismmakestheobservablesofafreequantumfieldtheoryintoafactorization
algebra.Westudyindetailthefreeβγsystem,aholomorphicfieldtheorylivingonanyRiemannsur-
face,andwerecovertheβγvertexalgebrafromthefactorizationalgebraofquantumobservables.We
alsoconstructthefactorizationalgebrasonaRiemannsurfacethatrecoverthevertexalgebrasarising
fromaffineKac-MoodyLiealgebras.Finally,westudyquantizationoffamiliesofellipticcomplexes.
Ourmainresulthereisanindextheoremrelatingtheassociatedfamilyoffactorizationalgebrasto
thedeterminantlineofthefamilyofellipticcomplexes.Attheheartofourworkistheformalismfor
perturbativequantumfieldtheorydevelopedbyCostello[Cos11]andfortheassociatedobservables
byCostello-Gwilliam[CG],andthisthesisprovidesanexpositionoftheideasandtechniquesinan
accessiblecontext.
v

Acknowledgements
Thisthesisappearsthankstothesupportandencouragementofmanypeople.
Throughallmyendeavors,mathematicalandotherwise,Iamsustainedbytheloveandtrust
ofmyfamily. TheGwilliamsofEvanstonhavemadethistownhomeformeandhostedgracefully
some of the most important events of my life. With profound warmth, the Hermanns have wel-
comedmeintotheirmidstandmademeapartoftheirfamily. MysisterNatalieoffersunwavering
encouragement and also challenges me always to do better; I admire the passion with which she
pursuesherownprojectsandtrytocultivatesuchpassionformyown. Inmyfather,Ifoundmy
intellectual role model and mentor: he taught me the joy of explaining an idea clearly, to ask for
what’s missing in my current point of view, and to have faith in my intuition but test its sugges-
tions thoroughly. He made me a scientist. Beyond the intellectual, I admire and seek to emulate
hisdevotiontothequirkyjokeandthethoughtfulgesture. Finally,Ithankmymother,whoismy
originalteacher,interlocutor,andcollaborator. Shetaughtmetowritethroughhoursofreflective
editing;shefoughttogivemeopportunitieswhereIcouldgrowintellectuallyandwhereIfound
mathematics;andsheshowedmethepowerofempathy. Throughherdeepengagementwithmy
life, I am lucky never to have felt unloved or unimportant. My parents’ unconditional support
andlovehaveallowedmetothriveandgrowintoanadultandmathematician.
AttheheartofitallismywifeSophie. Thankstoherenthusiasticpursuitofnewadventures,
we have traveled around the globe and gained the stories that define us, teaching me to open
myselfuptotheworld,evenwhenI’mafraid. Thankstoherpreternaturalattunementtoemotion,
shehastutoredmetowardgreaterself-insightandmademeabetter,morehonestperson. Thanks
to her capacity to find and share joy, our partnership grows daily more intimate and rewarding.
As we grow together into adulthood, I am continually astonished and bolstered by her steadfast
loveofme. Attheendoftheseeventfulfiveyears,Icansaythatmyproudestaccomplishmentis
marryingher.
Amongmathematicians,mygratitudebeginsfirstandforemostwithmyadvisorKevinCostello.
Beyond introducing me to beautiful ideas and teaching me techniques to make them precise, he
hasshownanexquisitesenseforhowtonurturemetowardsbecominganindependentresearcher,
by daydreaming with me about what might be true, by giving me the space to slowly grope to-
wardsrealizingthosedaydreams,andbyofferingthecrucialinsightswhenIneedguidance. His
vii
generousoffertocollaborateonthedevelopmentoffactorizationalgebrasgavemeajoltofconfi-
dence,andhisongoingsupportofmeduringthatprojecthasmaturedmeconsiderably. I’mgoing
tomissourregularcoffeeshopconversationstremendouslyinthenextfewyears.
The warm, supportive community of the Northwestern mathematics department has made
mytimethereapleasure. DavidNadlerhasprofoundlyshapedmyapproachtoandperspective
onmathematics,startingwithhisguidanceofthewonderfulexplorationsIundertookwithMike
and Ian that first year and continuing on in his demanding but rewarding seminar, and through
numerous conversations over the years. Eric Zaslow has provided many insights about the in-
terplay of mathematics and physics and much encouragement to understand both. John Francis
introduced me to factorization algebras, continues to teach me about them, and gave me crucial
adviceatimportantjuncturesingraduateschool. Iamgratefulformanyfunandeducationalcon-
versations with Erik Carlsson, Nick Rozenblyum, and David Treumann. My fellow travelers in
geometryandphysics—VivekDhand,SamGunningham,BorisHanin,IanLe,TakuoMatsuoka,
JoshShadlen, NicoloSibilla, TedStadnik, JesseWolfson—providedthediscussionandcompan-
ionship that enlivened this journey. My office mates in B20 have endured my mess and conver-
sational interruptions with good humor and made the office a pleasant haven for mathematics.
Yuan, thank you for being a patient and good-natured brother in arms against the challenges of
BVandobstructioncomputations. Hiro,ouradventuresinMathcampandabroadarethestuffof
privatelegend. Austin, yourforbearanceinthefaceofmyjibes, yourdeliciousbaking, andyour
helpwithanalysisaredeeplyappreciated. Mike,Ihavedependedforfiveyearsonaregulardose
ofyourmathematicalcamaraderie,absurdcommentary,andbeerexpertise.
OutsideNorthwestern,Ihavebenefitedfromthefriendshipandconversationofmanymath-
ematicians. At Notre Dame, Stephan Stolz bolstered my confidence that Kevin and I were up
to something worthwhile and forced me to articulate and clarify my thoughts about field the-
ory. Ryan Grady has been a fantastic friend, interlocutor, and collaborator; my understanding of
theseideaswouldbemuchpoorerwithoutourdiscussions. AtBerkeley,DanBerwick-Evansand
TheoJohnson-Freydhaveprovidedgoodconversationandcollaboration. GregGinotandDamien
Calaque have shared wonderful insights into factorization algebras and field theory, and I thank
them for our conversations and the opportunities to visit them in beautiful locations. Over the
past year, Maciej Szczesny’s enthusiasm and encouragement provided much-appreciated impe-
tus for developing the ideas in this thesis. I thank Fred Paugam for several wide-ranging and
stimulatingdiscussionsthisspring.
IwouldneverhaveknownthatIshouldbeamathematicianwithouttheencouragementand
friendship of several people before graduate school. Alina Marian revived my mathematical in-
terest and self-confidence in college, and her encouragement ever after has helped me a lot. The
WaverlyAlgebraSalon—KeithPardue,WillTraves,EmilVolcheck—andthefriendshipofRobert
Stingley, Kevin Player, and Bob Shea were crucial in crystallizing my mathematical taste and my
desiretogotograduateschool.
viii
CHAPTER 1
Introduction
An ongoing endeavor of mathematics is to provide a language adequate for expressing rig-
orously the ideas of physics, and this thesis is a product of that endeavor. Before discussing the
contentsofthisthesis,weexplainthegeneralcontextandsomemathematicalquestionsitraises.
Our starting point is the path integral approach to quantum field theory. In this formalism
a physical system consists of a bundle P → M over a smooth manifold, whose space of smooth
sectionsM := Γ(M,P)wecallthefields,equippedwithalocal1functionalS : M → Rcalledthe
action. AnobservableofthesystemisafunctionO : M → R,anditsexpectedvalueiscomputed
as
1 (cid:90)
(cid:104)O(cid:105) := O(φ)e−S(φ)/h¯ Dφ,
ZS φ∈M
wheree−S(φ)/h¯ DφisaputativemeasureonMandthepartitionfunction
(cid:90)
Z := e−S(φ)/h¯ Dφ
S
φ∈M
makes this measure into a probability measure.2 This perspective on field theory, as a kind of
probabilistic system, leads to beautiful insights into many areas of mathematics and physics, but
itisoftenmerelyaheuristicbecausemeasuretheoryoninfinite-dimensionalspacesrarelyhasthe
propertieswedesire.
Nonetheless,physicistshaveprovidedalgorithmsforcomputingexpectationvaluesofobserv-
ables, rooted in this perspective, that are wildly successful. It is a challenge for mathematicians
to find explanations and formalisms that justify mathematically these algorithms. In [Cos11],
Costello has developed a theory that provides a rigorous approach to the algorithms that consti-
tute perturbative quantum field theory (i.e., viewing h¯ as a formal parameter). In [CG], we have
studiedthemathematicalstructureoftheobservablesofsuchaperturbativequantumfieldtheory,
organized around the idea of a factorization algebra. The basic concept is simple. In a classical
fieldtheory,westudythesolutionstotheEuler-LagrangeequationsofS,whichpickoutthecriti-
calpointsofS. AstheEuler-Lagrangeequationsarepartialdifferentialequations,thereisasheaf
EL of solutions on the manifold M, so the functions O(EL) on these solutions form a cosheaf of
commutative algebras on M. For each open set U ⊂ M, the algebra O(EL(U)) consists of the
1“Local”meansthatSisgivenbyintegratingapointwisefunctionofthejetsofasectionagainstameasureonthe
manifoldM.
2WearediscussinghereEuclideanfieldtheories,sinceweweightSby−1ratherthani.
1
observablesfortheclassicaltheorywithsupportinU. Inaquantumfieldtheory,onecanstilltalk
aboutthesupportofobservables,buttheexpectedvalueofaproductofobservables(withdisjoint
support) includes quantum corrections, depending on h¯, to the classical expected value. Indeed,
these quantum corrections satisfy algebraic relations arising from the Feynman diagram expan-
sion used to compute them. For the precosheaf Obsq of quantum observables, these algebraic
relationsmodifythestructuremaps
Obsq(U)⊗Obsq(V) → Obsq(W),
where U and V are disjoint opens contained in the open W, by adding h¯-dependent terms to the
structuremapsofthecosheafofclassicalobservablesObscl = O(EL). Inparticular,theprecosheaf
Obsqisnolongeracosheafofcommutativealgebras. Instead,itisafactorizationalgebra,anotion
introducedbyBeilinsonandDrinfeld[BD04]intheirworkonconformalfieldtheory.
Perturbativequantumfieldtheoriesarerichandsubtleobjects,andtheconstructionsin[CG],
whileexplicit,canbeveryinvolvedbecausetheymixanalysis,homologicalalgebra,andcategory
theory in complicated ways. The central aim of this thesis is to study a special class of theories
where the constructions are much simpler. We focus on free field theories, in which the action
functional S isaquadraticfunctionofthefields. Thisrestrictionmightseemtolimitthepossibil-
ityofinterestingresults,buttheframeworkof[Cos11]allowsanyellipticcomplexonamanifold
to provide a free theory. Thus, there is a plethora of examples and the possibility that one might
obtainnewinsightsintogeometry,whereellipticcomplexesareubiquitous. Moreover,thefactor-
izationalgebrasarisingfromfreefieldtheoriesareasmallstepawayfromfamiliarconstructions
withellipticcomplexes,andthustheyaremoreamenabletohumanunderstanding.
AttheheartofCostello’sapproachtoquantumfieldtheoryistheBatalin-Vilkovisky(BV)for-
malism, which is a homological approach to defining the path integral. It forms the basic mech-
anism by which we obtain the quantum observables Obsq from the classical observables Obscl.
Unfortunately, it is notoriously difficult to learn and hard to motivate. Thus a secondary aim of
thisthesisistoprovideanintroductiontotheBVformalismwhereitsvirtuesareapparent. Again,
freetheoriesprovidesuchacontext. Infact,weshowhowthehomologicalalgebraoftheBVfor-
malismcanbedeployedoutsidefieldtheoryandapplyitto(well-behaved,i.e.,perfect)modules
overanycommutativedgalgebra.
Ourmainresultsinthisthesisarethefollowing.
(1) BV quantization defines a determinant functor from perfect R-modules to invertible R-
modules,for Ranartiniancommutativedgalgebra.
(2) One can recover rigorously a vertex algebra from an action functional. In particular, we
C
start with the free βγ system on and show that its factorization algebra of quantum
observablesrecoversthe βγvertexalgebra.
(3) We prove an index theorem arising from the study of quantization of free field theories
in families (i.e., families of elliptic complexes). In particular, the global observables on a
2