Feynman motives of banana graphs(4)

11

Figure4.Bananagraphswithexternaledges

InordertocomputetheFeynmanintegral(1.9),weviewthebananagraphsΓnnotasvacuumbubbles,butasendowedwithanumberofexternaledges,asinFigure4.Itdoesnotmatterhowmanyexternaledgesweattach.Thiswilldependonwhichscalar eldtheorythegraphbelongsto,buttheresultingintegralisuna ectedbythis,aslongaswehavenonzeroexternalmomenta owingthroughthegraph.

Lemma1.8.TheFeynmanintegral(1.9)forthebananagraphsΓnisoftheform(1.33)U(Γ,p)=Γ((1 D/2)(n 1)+1)C(p)2 1)(n 1) 1

2ωn 1)n,

Proof.Theresultisimmediatefrom(1.9),usingn= +1andthefactthattheonlycut-setforthebananagraphΓnconsistsoftheunionofalltheedges,sothat

PΓ(t,p)=C(p)t1···tn.

Forexample,inthecasewithn=2andD∈2N,D≥4,theintegral(uptoadivergentGammafactorΓ(2 D/2)4π D/2)reducestothecomputationoftheconvergentintegral

((DD/2 2(t(1 t))dt=.(D 3)![0,1]

Ingeneral,apartfrompolesoftheGammafunction,divergencesmayarisefromtheintersectionsofthedomainofintegrationσnwiththegraphhypersurfaceXΓn.

Lemma1.9.TheintersectionofthedomainofintegrationσnwiththegraphhypersurfaceXΓnhappensalongσn∩SninthealgebraicsimplexΣn.

Proof.ThepolynomialΨΓ(t)≥0fort∈Rn+andbytheexplicitform(1.28)ofthepolynomial,onecanseethatzeroswillonlyoccurwhenatleasttwoofthecoordinatesvanish,i.e.alongtheintersectionofσnwiththeschemeofsingularitiesSnofΣn(cf.Lemma3.8below).

OneproceduretodealwiththissourceofdivergencesistoworkonblowupsofPn 1alongthissingularlocus(cf.[11],[10]).In[26]anotherpossiblemethodofregularizationforintegralsoftheform(1.33)whichtakescareofthesingularitiesoftheintegralonσn(thepoleoftheGammafunctionneedstobeaddressedseparately)wasproposed,basedonreplacingtheintegralalongσnwithanintegralthatgoesaroundthesingularitiesalongthe bersofacirclebundle.Ingeneral,thistypeofregularizationproceduresrequiresadetailedknowledgeofthesingularitiesofthehypersurfaceXΓtobecarriedout,andthatisoneofthereasonsforintroducinginvariantsofsingularvarietiesinthestudyofgraphhypersurfaces. withthefunctionoftheexternalmomentagivenbyC(p)=(Pv)2,withvbeingeither oneofthetwoverticesofthegraphΓnandPv=e∈Eext(Γn),t(e)=vpe.

We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the c

12ALUFFIANDMARCOLLI

2.CharacteristicclassesandtheGrothendieckring

Inordertounderstandthenatureofthepartofthecohomologyofthegraphhy-persurfacecomplementthatsupportstheperiodcorrespondingtotheFeynmanintegral(ignoringdivergenceissuesmomentarily),onewouldliketodecomposePn 1 XΓintosimplerbuildingblocks.Asin§8of[11],thiscanbedonebylookingattheclass[XΓ]ofthegraphhypersurfaceintheGrothendieckringofmotives.Oneknowsbythegeneralre-sultofBelkale–Brosnam[8]thatthegraphhypersurfacesgeneratetheGrothendieckring,hencetheyarequitearbitrarilycomplexasmotives,butonestillneedstounderstandwhetherthepartofthedecompositionthatisrelevanttothecomputationoftheFeynmanintegralmightinfactbeofaveryspecialtype,e.g.amixedTatemotiveastheevidencesuggests.Thefamilyofgraphsweconsiderhereisverysimpleinthatrespect.Infact,onecanseeveryexplicitlythattheirclassesintheGrothendieckringarecombinationsofTatemotives(cf.theformula(3.13)below).OnecanseethisalsobylookingattheHodgestructure.Forthegraphhypersurfacesofthebananagraphsthisisdescribedin§8of[10].

Herewedescribetwowaysofanalyzingthegraphhypersuracesthroughanadditiveinvariant,oneasaboveusingtheclass[XΓ]intheGrothendieckring,andtheotherusingthepushforwardoftheChern–Schwartz–MacPhersonclassofXΓtotheChowgroup(orhomology)oftheambientprojectivespacePn 1.Whilethe rstdoesnotdependonanambientspace,thelatterissensitivetothespeci cembeddingofXΓintheprojectivespacePn 1,henceitmightconceivablycarryalittlemoreinformationthatisusefulinrelationtothecomputationoftheFeynmanintegralonPn 1 XΓ.Werecallherebelowafewbasicfactsaboutbothconstructions.Thereaderfamiliarwiththesegeneralitiescanskipdirectlytothenextsection.

2.1.TheGrothendieckring.LetVKdenotethecategoryofalgebraicvarietiesovera eldK.TheGrothendieckringK0(VK)istheabeliangroupgeneratedbyisomorphismclasses[X]ofvarieties,withtherelation

(2.1)[X]=[Y]+[X Y],

forY Xclosed.Itismadeintoaringbytheproduct[X×Y]=[X][Y].

Anadditiveinvariantisamapχ:VK→R,withvaluesinacommutativeringR,satisfyingχ(X)=χ(Y)ifX～=Yareisomorphic,χ(X)=χ(Y)+χ(X Y)forY Xclosed,andχ(X×Y)=χ(X)χ(Y).TheEulercharacteristicistheprototypeexampleofsuchaninvariant.AssigninganadditiveinvariantwithvaluesinRisequivalenttoassigningaringhomomorphismχ:K0(VK)→R.

LetMKbethepseudo-abeliancategoryof(Chow)motivesoverK.WewritetheobjectsofMKintheform(X,p,m),withXasmoothprojectivevarietyoverK,p=p2∈End(X)aprojector,andm∈ZaccountingforthetwistbypowersoftheTatemotiveQ(1).LetK0(MK)denotetheGrothendieckringofthecategoryMKofmotives.Theresultsof[19]showthat,forKofcharacteristiczero,thereexistsanadditiveinvariantχ:VK→K0(MK).ThisassignstoasmoothprojectivevarietyXtheclassχ(X)=

[(X,id,0)]∈K0(MK),whileforXageneralvarietyitassignsacomplexW(X)inthecategoryofcomplexesoverMK,whichishomotopyequivalenttoaboundedcomplexwhoseclassinK0(MK)de nesthevalueχ(X).Thisde nesaringhomomorphism(2.2)χ:K0(VK)→K0(MK).

IfLdenotestheclassL=[A1]∈K0(VK)thenitsimageinK0(MK)istheLefschetzmotiveL=Q( 1)=[(Spec(K),id, 1)].SincetheLefschetzmotiveisinvertibleinK0(MK),itsinversebeingtheTatemotiveQ(1),theringhomomorphism(2.2)inducesa

We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the c

BANANAMOTIVES13

ringhomomorphism

(2.3)χ:K0(VK)[L 1]→K0(MK).

Thus,inthefollowingwecaneitherregardthecla …… 此处隐藏：5975字，全部文档内容请下载后查看。喜欢就下载吧 ……