Feynman motives of banana graphs

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

FEYNMANMOTIVESOFBANANAGRAPHS

PAOLOALUFFIANDMATILDEMARCOLLI

arXiv:0807.1690v2 [hep-th] 16 Jul 2008Abstract.Weconsiderthein nitefamilyofFeynmangraphsknownasthe“bananagraphs”andcomputeexplicitlytheclassesofthecorrespondinggraphhypersurfacesintheGrothendieckringofvarietiesaswellastheirChern–Schwartz–MacPhersonclasses,usingtheclassicalCremonatransformationandthedualgraph,andablowupformulaforcharacteristicclasses.Weoutlinetheinterestingsimilaritiesbetweentheseoperationsandwegiveformulaeforconesobtainedbysimpleoperationsongraphs.Weformulateapositivityconjectureforcharacteristicclassesofgraphhypersurfacesanddiscussbrie ythee ectofpassingtononcommutativespacetime.1.IntroductionSincetheextensivestudyof[15]revealedthesystematicappearanceofmultiplezetavaluesastheresultofFeynmandiagramcomputationsinperturbativequantum eldthe-ory,thequestionof ndingadirectrelationbetweenFeynmandiagramsandperiodsofmotiveshasbecomearich eldofinvestigation.TheformulationofFeynmanintegralsthatseemsmostsuitableforanalgebro-geometricapproachistheoneinvolvingSchwingerandFeynmanparameters,asinthatformtheintegralacquiresdirectlyaninterpretationasaperiodofanalgebraicvariety,namelythecomplementofahypersurfaceinaprojectivespaceconstructedoutofthecombinatorialinformationofagraph.Thesegraphhyper-surfacesandthecorrespondingperiodshavebeeninvestigatedinthealgebro-geometricperspectiveintherecentworkofBloch–Esnault–Kreimerandmorerecently,fromthepointofviewofHodgetheory,inand[26].Inparticular,thequestionofwhetheronlymotivesofmixedTatetypewouldariseinthequantum eldtheorycontextisstillunsolved.Despitethegeneralresultofwhichshowsthatthegraphhypersur-facesaregeneralenoughfromthemotivicpointofviewtogeneratetheGrothendieckringofvarieties,theparticularresultsof[15]andpointtothefactthat,eventhoughthevarietiesthemselvesareverygeneral,thepartofthecohomologythatsupportstheperiodofinteresttoquantum eldtheorymightstillbeofthemixedTateform.Onecomplicationinvolvedinthealgebro-geometriccomputationswithgraphhyper-

surfacesisthefactthatthesearetypicallysingular,withasingularlocusofsmallcodi-mension.Itbecomesthenaninterestingquestioninitselftoestimatehowsingularthegraphhypersurfacesare,acrosscertainfamiliesofFeynmangraphs(thehalfopenladdergraphs,thewheelswithspokes,thebananagraphsetc.).Sincethemaingoalistode-scribewhathappensatthemotiviclevel,onewantstohaveinvariantsthatdetecthowsingularthehypersurfaceisandthatarealsosomehowadaptedtoitsdecompositionintheGrothendieckringofmotives.Inthispaperweconcentrateonaparticularexampleandillustratesomegeneralmethodsforcomputingsuchinvariantsbasedonthetheoryofcharacteristicclassesofsingularvarieties.

Partofthepurposeofthepresentpaperistofamiliarizephysicistsworkinginpertur-bativequantum eldtheorywithsometechniquesofalgebraicgeometrythatareusefulintheanalysisofgraphhypersurfaces.Thus,wetryasmushaspossibletospellouteverythingindetailandrecallthenecessarybackground.

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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

2ALUFFIANDMARCOLLI

In§1,webeginbyrecallingthegeneralformoftheparametricFeynmanintegralsforascalar eldtheoryandtheconstructionoftheassociatedprojectivegraphhypersurface.WerecalltherelationbetweenthegraphhypersurfaceofaplanargraphandthatofthedualgraphviathestandardCremonatransformation.Wethenpresentthespeci cexampleofthein nitefamilyof“bananagraphs”.Weformulateapositivityconjectureforthecharacteristicclassesofgraphhypersurfaces.

Fortheconvenienceofthereader,werecallin§2somegeneralfactsandresults,bothabouttheGrothendieckringofvarietiesandmotives,andaboutthetheoryofcharacteristicclassesofsingularalgebraicvarieties.Weoutlinethesimilaritiesanddi erencesbetweentheseconstructions.

In§3wegivetheexplicitcomputationoftheclassesintheGrothendieckringofthehypersurfacesofthebananagraphs.Weconcludewithageneralremarkontherelationbetweentheclassofthehypersurfaceofaplanargraphandthatofadualgraph.

In§4weobtainanexplicitformulafortheChern–Schwartz–MacPhersonclassesofthehypersurfacesofthebananagraphs.We rstproveageneralpullbackformulafortheseclasses,whichisnecessaryinordertocomputethecontributiontotheCSMclassofthecomplementofthealgebraicsimplexinthegraphhypersurface.Theformulaisthenobtainedbyassemblingthecontributionoftheintersectionwiththealgebraicsimplexandofitscomplementviainclusion–exclusion,asinthecaseoftheclassesintheGrothendieckring.

Wegivethen,in§5,aformulafortheCSMclassesofconesonhypersurfacesandusethemtoobtainformulaeforgraphhypersurfacesobtainedfromknownonebysimpleoperationsonthegraphs,suchasdoublingorsplittinganedge,andattachingsingle-edgeloopsortreestovertices.

Finally,in§6,welookatthedeformationsofordinaryφ4theorytoanoncommutativespacetimegivenbyaMoyalspace.Welookattheribbongraphsthatcorrespondtotheoriginalbananagraphsinthisnoncommutativequantum eldtheory.Weexplaintherelationbetweenthegraphhypersurfacesofthenoncommutativetheoryandoftheoriginalcommutativeone.WeshowbyanexplicitcomputationofCSMclassesthatinnoncommutativeQFTthepositivityconjecturefailsfornon-planarribbongraphs.

Acknowledgment.The rstauthorispartiallysupportedbyNSAgrantH98230-07-1-0024.ThesecondauthorispartiallysupportedbyNSFgrantDMS-0651925.WethanktheMax–Planck–InstituteandFloridaStateUniversity,wherepartofthisworkwasdone.WealsothankAbhijnanRejforexchangesofnumericalcomputationsofCSMclassesofgraphhypersurfaces.

1.1.ParametricFeynmanintegrals.Webrie yrecallsomewellknownfacts(cf.§6-2-3of[23],§18of[9],and§6of[27])abouttheparametricformofFeynmanintegrals.

Givenascalar eldtheorywithLagrangianwritteninEuclideansignatureas

2

wheretheinteractionpartisapolynomialfunctionofφ,aone-particle-irreducible(1PI)FeynmangraphofthetheoryisaconnectedgraphΓwhichcannotbedisconnectedbyremovingasingleedge,andwiththefollowingproperties.AllverticesinV(Γ)havevalenceequaltothedegreeofoneofthemonomialsintheLagrangian.ThesetofedgesE(Γ)=Eint(Γ)∪Eext(Γ)consistsofinternaledgeshavingtwoendverticesandexternaloneshavingonlyonevertex.AFeynmangraphwithoutexternaledgesiscalledavacuumbubble.

Inperturbativequantum eldtheory,theFeynmanintegralsassociatedtotheloopnumberexpansionofthee ectiveactionforascalar eldtheoryarelabeledbythe1PI(1.1)L(φ)=1φ2+Lint(φ),

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

BANANAMOTIVES3

Feynmangraphsofthetheory,eachcontributingacorrespondingintegraloftheform(1.2)U(Γ,p)=Γ(n D /2)

ΨΓ(t)D/2VΓ(t,p)n D /2dt1···dtn.

wherethesumisoverallthespanningtreesTofΓ.ThefunctionVΓ(t,p)isarationalfunctionoftheform

PΓ(t,p)(1.4)VΓ(t,p)=Heren=#Eint(Γ)isthenumberofinternaledgesofthegraphΓ,D∈Nisthespacetimedimensioninwhichthescalar eldtheoryisconsidered,and =b1(Γ)isthenumberofloopsinthegraph,i.e.therankofH1(Γ,Z).ThefunctionΨΓisapolynomialofdegree =b1(Γ).ItisgivenbytheKirchho polynomial te,(1.3)ΨΓ(t)=T Γe/∈E(T)

(4π)(n 1)D/2 PΓ(p,t) n+D(n 1)/2ωnσn

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

4ALUFFIANDMARCOLLI

thisassumption.However,forthealgebro-geometricargumentsthatconstitutethemaincontentofthispaper,the1PIconditionisnotstrictlynecessary.

1.2.Feynmangraphs,varieties,andperiods.ThegraphpolynomialΨΓ(t)of(1.3)alsoadmitsadescriptionasdeterminant(1.11)ΨΓ(t)=detMΓ(t)

n i=1ofan × -matrixMΓ(t)associatedtothegraph([27],§3and[9],§18),oftheform(1.12)(MΓ)kr(t)=tiηikηir,

afterchoosinganorientationoftheedges.

NoticehowtheresultisindependentofthechoiceoftheorientationoftheedgesandofthechoiceofthebasisofH1(Γ,Z).Infact,achangeoforientationinagivenedgeresultsinachangeofsigntooneofthecolumnsofthematrixηki,whichiscompensatedbythechangeofsigninthecorrespondingrowofthematrixηir,sothatthedeterminantdetMΓ(t)isuna ected.Similarly,achangeinthechoiceofthebasisofH1(Γ,Z)hasthee ectofchangingMΓ(t)→AMΓ(t)A 1forsomeA∈GL( ,Z)andthedeterminantisagainunchanged.

ThegraphhypersurfaceXΓisbyde nitionthezerolocusoftheKirchho polynomial,(1.14)XΓ={t=(t1:...:tn)∈Pn 1|ΨΓ(t)=0}.

SinceΨΓishomogeneous,itde nesahypersurfaceinprojectivespace.

Thedomainofintegrationσnde nesacycleintherelativehomologyHn 1(Pn 1,Σn),whereΣnisthealgebraicsimplex(theunionofthecoordinatehyperplanes,see(1.16)below).TheFeynmanintegral(1.2),(1.9)thencanbeviewed([11],[10])astheevaluationofanalgebraiccohomologyclassinHn 1(Pn 1 XΓ,Σ Σ∩XΓ)onthecyclede nedbyσn.Inthissense,itcanbeviewedastheevaluationofaperiodofthealgebraicvarietygivenbythecomplementofthegraphhypersurface.Tounderstandthenatureofthisperiod,oneisfacedwithtwomainproblems.Oneiseliminatingdivergences(regularizationandrenormalizationofFeynmanintegrals),andtheotherisunderstandingwhatkindofmotivesareinvolvedinthepartofthehypersurfacecomplementPn 1 XΓthatisinvolvedintheevaluationoftheperiod,hencewhatkindoftranscendentalnumbersoneexpectsto ndintheevaluationofthecorrespondingFeynmanintegrals.Adetailedanalysisoftheseproblemswascarriedoutin[11].Theexamplesweconcentrateoninthispaperarenotespeciallyinterestingfromthemotivicpointofview,sincetheyareexpressibleintermsofpureTatemotives(cf.[10]),buttheyprovideuswithanin nitefamilyofgraphsforwhichallcomputationsarecompletelyexplicit.

1.3.DualgraphsandCremonatransformation.Inthecaseofplanargraphs,thereisaninterestingrelationbetweenthehypersurfaceofthegraphandtheoneofthedualgraph.Thiswillbeespeciallyusefulintheexplicitcalculationweperformbelowinthespecialcaseofthebananagraphs.Werecallithereinthegeneralcaseofarbitraryplanargraphs.wherethen× -matrixηikisde nedintermsoftheedgesei∈E(Γ)andachoiceofabasisforthe rsthomologygroup,lk∈H1(Γ,Z),withk=1,..., =b1(Γ),bysetting +1edgeei∈looplk,sameorientation (1.13)ηik= 1edgeei∈looplk,reverseorientation 0otherwise,

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

BANANAMOTIVES5

thoughweseeinLemma1.2belowthatitiswellde nedalsoonthegeneralpointofΣn,itslocusofindeterminaciesbeingonlythesingularitysubschemeofΣn.

LetG(C)denotetheclosureofthegraphofC.ThenG(C)isasubvarietyofPn 1×Pn 1withprojections

(1.17)G(C)????π2 π1 ?? ? CPn 1______Pn 1ThestandardCremonatransformationofPn 1isthemap 1(1.15)C:(t1:···:tn)→.tnThisisaprioride nedawayfromthealgebraicsimplexofcoordinateaxes n 1ti=0} Pn 1,(1.16)Σn={(t1:···:tn)∈P|i

ingcoordinates(s1:···:sn)forthetargetPn 1,thegraphG(C)hasequations

(1.18)t1s1=t2s2=···=tnsn.

Inparticular,thisdescribesG(C)asacompleteintersectionofn 1hypersurfacesinPn 1×Pn 1withequationstisi=tnsn,fori=1,...,n 1.

Proof.Theequations(1.18)clearlycutoutG(C)overtheopensetU Pnwhereallt-coordinatesarenonzero.Sinceeverycomponentofaschemede nedbyn 1equationshascodimension≤n 1,itsu cestoshowthatequations(1.18)de neasetofcodimension>n 1overthecomplementofU.Nowassumethatatleastoneofthet-coordinatesequal0.Withoutlossofgenerality,supposetn=0.Intersectingwiththelocusde nedby(1.18)determinesthesetwithequations

t1s1=···=tn 1sn 1=tn=0,

whichhascodimensionn>n 1,aspromised.

ItisnothardtoseethatthevarietyG(C)hassingularitiesincodimension3.Itisnonsingularforn=2,3,butsingularforn≥4.

TheopensetUasaboveisthecomplementofthedivisorΣnof(1.16).TheinverseimageofΣninG(C)canbedescribedeasily.Itconsistsofthepoints

((t1:···:tn),(s1:···:sn))

suchthat

{i|ti=0}∪{j|sj=0}={1,...,n}.

Thislocusconsistsof2N 2componentsofdimensionn 2:onecomponentforeachnonemptypropersubsetIof{1,...,n}.ThecomponentcorrespondingtoIisthesetofpointswithti=0fori∈Iandsj=0forj∈I.

Thesituationforn=3iswellrepresentedbythefamouspictureofFigure1.Thethreezero-dimensionalstrataofΣ3areblownupinG(C)asweclimbthediagramfromthelowerlefttothetop.Thepropertransformsoftheonedimensionalstrataareblowndownaswedescendtothelowerright.Thehorizontalrationalmapisanisomorphismbetweenthecomplementsofthetriangles.TheinverseimageofΣ3consistsof23 2=6components,asexpected.

Ofcoursethesituationiscompletelysymmetric:thealgebraicsimplex(1.16)maybe 1 1(Σn)=π2(Σn).embeddedinthetargetPnaswell(withequationisi=0).Onehasπ1

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

6ALUFFIAND

MARCOLLI

Figure1.TheCremonatransformationinthecasen=3.

LetSn Pn 1bethesubschemede nedbytheideal

(1.19)ISn=(t1···tn 1,t1···tn 2tn,...,t1t3···tn,t2···tn).

TheschemeSnisthesingularitysubschemeofthedivisorwithsimplenormalcrossingsΣnof(1.16),givenbytheunionofthecoordinatehyperplanes.WecanplaceSninboththesourceandtargetPn 1.Finally,letLbethehyperplanede nedbytheequation(1.20)L={(t1:···:tn)∈Pn 1|t1+···+tn=0}.

Wethencanmakethefollowingobservations.

Lemma1.2.LetC,G(C),Sn,andLbeasabove.

(1)SnisthesubschemeofindeterminaciesoftheCremonatransformationC.

(2)π1:G(C)→Pn 1istheblow-upalongSn.

(3)LintersectseverycomponentofSntransversely.

(4)ΣncutsoutadivisorwithsimplenormalcrossingsonL.

Proof.(1)Noticethatthede nition(1.15)oftheCremonatransformation,whichisaprioride nedonthecomplementofΣnstillmakessenseonthegeneralpointofΣn.Thus,theindeterminaciesofthemap(1.15)arecontainedinthesingularitylocusSnofΣnde nedby(1.19).ItconsistsinfactofallofSnsinceafter‘clearingdenominators’,thecomponentsofthemapde ningCgivenin(1.15)canberewrittenas:

(1.21)(t1:···:tn)→(t2···tn:t1t3···tn:···:t1···tn 1),

sothatoneseesthattheindeterminaciesarepreciselythosede nedbytheideal(1.19).

(2)Using(1.21),themapπ1:G(C)→Pnmaybeidenti edwiththeblow-upofPnalongthesubschemeSnde nedbytheidealISnof(1.19).Thegeneratorsofthisidealare

thepartialderivativesoftheequationofthealgebraicsimplex.Thus,SnisthesingularitysubschemeofΣn.Itconsistsoftheunionoftheclosureofthedimensionn 2strataofΣn.Again,notethatthesituationisentirelysymmetrical:wecanplaceSninthetargetPnaswell,andviewπ2astheblow-upalongSn.

(3)and(4)areimmediatefromthede nitions.

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

BANANAMOTIVES

7

dual graph

dual graph

Figure2.Dualgraphsofdi erentplanarembeddingsofthesamegraph.

GivenaconnectedplanargraphΓ,onede nesitsdualgraphΓ∨by xinganembeddingofΓinR2∪{∞}=S2andconstructinganewgraphinS2thathasavertexineachcomponentofS2 ΓandoneedgeconnectingtwosuchverticesforeachedgeofΓthatisinthecommonboundaryofthetworegionscontainingthevertices.Thus,#E(Γ∨)=#E(Γ)and#V(Γ∨)=b0(S2 Γ).Thedualgraphisingeneralnon-unique,sinceitdependsonthechoiceoftheembeddingofΓinS2,seee.g.Figure2.Werecallhereawellknownresult(seee.g.[10],Proposition8.3),whichwillbeveryusefulinthefollowing.

Lemma1.3.SupposegivenaplanargraphΓwith#E(Γ)=n,withdualgraphΓ∨.Thenthegraphpolynomialssatisfy 1 1te)ΨΓ∨(t (1.22)ΨΓ(t1,...,tn)=(1,...,tn),

e∈E(Γ)

hencethegraphhypersurfacesarerelatedbytheCremonatransformationCof(1.15),(1.23)C(XΓ∩(Pn 1 Σn))=XΓ∨∩(Pn 1 Σn).

Proof.Thisfollowsfromthecombinatorialidentity ΨΓ(t1,...,tn)=T Γe/∈E(T)te 1=(e∈E(Γ)te)T Γe∈E(T)t e 1=(e∈E(Γ)te)T′ Γ∨e/∈E(T′)te 1 1,...,tn).=(e∈E(Γ)te)ΨΓ∨(t1

Thethirdequalityusesthefactthat#E(Γ)=#E(Γ∨)and#V(Γ∨)=b0(S2 Γ),sothatdegΨΓ+degΨΓ∨=#E(Γ),andthefactthatthereisabijectionbetweencomplementsofspanningtreeTinΓandspanningtreesT′inΓ∨obtainedbyshrinkingtheedgesofTinΓandtakingthedualgraphoftheresultingconnectedgraph.

Writteninthecoordinates(s1:···:sn)ofthetargetPn 1oftheCremonatransfor-mation,theidentity(1.22)gives 1s ΨΓ(t1,...,tn)=(e)ΨΓ∨(s1,...,sn)

e∈E(Γ∨)

fromwhich(1.23)follows.

Wethenhavethefollowingsimplegeometricobservation,whichfollowsdirectlyfromLemma1.2andLemma1.3above.

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

8ALUFFIAND

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Γ3Γ4

Figure3.ExamplesofbananagraphsΓ5

1Corollary1.4.ThegraphhypersurfaceofthedualgraphisXΓ∨=π2(π1(XΓ)),withn 1πi:G(C)→P,fori=1,2,asin(1.17).TheCremonatransformationCrestrictstoa

(biregular)isomorphism

(1.24)C:XΓ Σn→XΓ∨ Σn.

1π2:π1(XΓ Σn)→XΓ∨ Σn.Themapπ2:G(C)→Pn 1of(1.17)restrictstoanisomorphism(1.25)

Noticethattheformula(1.22)canbeusedasasourceofexamplesofcombinatoriallyinequivalentgraphsthathavethesamegraphhypersurface.Infact,thegraphpolynomialΨΓ∨(s1,...,sn)isthesameindependentlyofthechoiceoftheembeddingoftheplanargraphΓintheplane,whilethedualgraphΓ∨dependsonthechoiceoftheembeddingofΓintheplane.Thus,di erentembeddingsthatgiverisetodi erentgraphsΓ∨pro-videexamplesofcombinatoriallyinequivalentgraphswiththesamegraphhypersurface.Thishasdirectconsequences,forexample,onthequestionofliftingtheConnes–KreimerHopfalgebraofgraphs[17]atthelevelofthegraphhypersurfacesortheirclassesintheGrothendieckringofmotives.Anexplicitexampleofcombinatoriallyinequivalentgraphswiththesamegraphhypersurface,obtainedasdualgraphsofdi erentplanarembeddingsofthesamegraph,isgiveninFigure2.

Weseeadirectapplicationofthisgeneralresultforplanargraphsin§3.1below,wherewederivearelationbetweentheclassesintheGrothendieckring.Ingeneral,thisrelationaloneistooweaktogiveexplicitformulae,buttheexampleweconcentrateoninthenextsectionshowsafamilyofgraphsforwhichacompletedescriptionofboththeclassintheGrothendieckringandtheCSMclassfollowsfromthespecialformthattheresultofCorollary1.4takes.

1.4.Anexample:thebananagraphs.Inthispaperweconcentrateonaparticularexample,forwhichwecancarryoutcompleteandexplicitcalculations.Weconsideranin nitefamilyofgraphscalledthe“bananagraphs”.Then-thtermΓninthisfamilyisavacuumbubbleFeynmangraphforascalar eldtheorywithaninteractiontermoftheformLint(φ)=φn.ThegraphΓnhastwoverticesandnparalleledgesbetweenthem,asinFigure3.

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

BANANAMOTIVES9

AdirectcomputationusingtheMacaulay2program[20]forcharacteristicclassesde-velopedin[4]shows,forthe rstthreeexamplesinthisseriesofgraphsdepictedinFigure3,thefollowinginvariants(see§2forprecisede nitions).

n

t1t2+t2t3+t1t3

t1t3t4+t2t3t4

c(XΓ)

Mil(XΓ)

χ(XΓ)5H3+3H2+3H 4H34t1t2t3t4+t1t2t3t5+5

HereHdenotesthehyperplaneclassandc(XΓ)istheChern–Schwartz–MacPhersonclassofthehypersurfacepushedforwardtotheambientprojectivespace.WealsoshowtheMilnorclass,whichmeasuresthediscrepancybetweentheChern–Schwartz–MacPhersonclassandtheFultonclass,thatis,betweenthecharacteristicclassofthesingularhyper-surfaceXΓ={ΨΓ=0}andtheclassofasmoothdeformation.WealsodisplaythevalueoftheEulercharacteristic,whichonecanreado theCSMclass.ThereadercanpausemomentarilytoconsidertheCSMclassesreportedinthethreeexamplesaboveandnoticethattheysuggestageneralformulaforthisfamilyofgraphs,wherethecoe cientofHkintheCSMclassforthen-thhypersurfaceXΓnisgivenbytheformula nn 1n 1 =ifkiseven k kk 1(1.26) nn 1 +ifkisoddkk

for1<k<n,andn 1fork=1.Thus,forexample,forn≥3theEulercharacteristicχ(XΓn)ofthen-thbananahypersurface tsthepattern

(1.27)χ(XΓn)=n+( 1)n.

ThisisindeedthecorrectformulafortheCSMclassthatwillbeprovedin§4below.Thesamplecasereportedherealreadyexhibitsaninterestingfeature,whichweencounteragaininthegeneralformulaof§4andwhichseemscon rmedbycomputationscarriedoutalgorithmicallyonothersamplegraphsfromdi erentfamiliesofFeynmangraphs,namelytheunexpectedpositivityofthecoe cientsoftheChern–Schwartz–MacPhersonclasses.NoticethatasimilarinstanceofpositivityoftheCSMclassesarisesinanothercaseofvarietieswithastrongcombinatorial avor,namelythecaseoftheSchubertvarietiesconsideredin[7].Atpresentwedonothaveaconceptualexplanationforthispositivityphenomenon,butwecanstatethefollowingtentativeguess,basedonthesparsenumericalandtheoreticalevidencegatheredsofar.

Conjecture1.5.Thecoe cientsofallthepowersHkintheCSMclassofanarbitrarygraphhypersurfaceXΓarenon-negative.

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

10ALUFFIANDMARCOLLI

ForthegeneralelementΓninthefamilyofthebananagraphs,thegraphhypersurfaceXΓninPn 1isde nedbythevanishingofthegraphpolynomial

).tn

Thisiseasilyseen,sinceinthiscasespanningtreesconsistofasingleedgeconnectingthetwovertices.Equivalently,onecanseethisintermsofthematrixMΓ(t).

Lemma1.6.Forthen-thbananagraphΓn,thematrixMΓn(t)isoftheform t1+t2 t200···0 t2 t2+t3 t300 0 tt+t t03344 (1.29)MΓn(t)= .00 tt+t0445 ...... ...

0000···tn 1+tn(1.28)ΨΓn=t1···tn(1

Proof.Infact,ifwechooseasabasisofthe rstcohomologyofthegraphΓntheobviousoneconsistingofthe =n 1loopsei∪ ei+1,withi=1,...,n 1,weobtainthatthen×(n 1)-matrixηikisoftheform 10000··· 11000··· .0 1100···ηik= 00 110··· 000 11··· Thus,thematrix(MΓ)rk(t)=itiηriηikhastheform(1.29).Itiseasytocheckthatthisindeedhasdeterminantgivenby(1.28).Infact,from(1.29)oneseesthatthedeterminantsatis es

detMΓn(t)=(tn 1+tn)detMΓn 1(t) t2n 1detMΓn 2(t).

Itthenfollowsbyinductionthatthedeterminantsatis estherecursiverelation

(1.30)detMΓn(t)=tndetMΓn 1(t)+t1···tn 1.

Infact,assumingtheaboveforn 1weobtain

2detMΓn(t)=tndetMΓn 1(t)+t2n 1detMΓn 2(t)+t1···tn 1 tn 1detMΓn 2(t).

ItisthenclearthatdetMΓn(t)=ΨΓn(t),withthelattergivenbytheformula(1.28),

sincethisalsoclearlysatis esthesamerecursion(1.30).

ThedualgraphΓ∨nisjustapolygonwithnverticesandwecanidentifythehypersurfacen 1XΓ∨inPwiththehyperplaneLde nedin(1.20).nWerephraseherethestatementofCorollary1.4inthespecialcaseofthebananagraphs,sinceitwillbeveryusefulinourexplicitcomputationsof§§3and4below.

1Lemma1.7.Then-thbananagraphhypersurfaceisXΓn=π2(π1(L)),withπi:G(C)→n 1P,fori=1,2,asin(1.17).TheCremonatransformationCrestrictstoa(biregular)

isomorphism

(1.31)

(1.32)C:L Σn→XΓn Σn. 1(L Σn)→XΓn Σn.π2:π1Themapπ2:G(C)→Pn 1of(1.17)restrictstoanisomorphism

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

BANANAMOTIVES

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,inthefollowingwecaneitherregardtheclasses[XΓ]ofthegraphhypersurfacesintheGrothendieckringofvarietiesK0(VK)or,underthehomomorphism(2.2),aselementsintheGrothendieckringofmotivesK0(MK).Wewillnolongermakethisdistinctionexplicitinthefollowing.

2.2.CSMclassesasameasureofsingularities.TheChernclassofanonsingularcompletevarietyVisthe‘totalhomologyChernclass’ofitstangentbundle.Wewritec(V):=c(TV)∩[V] toindicatetheresultofapplyingtheChernclassofthetangentbundleofVtothefundamentalclass[V] ofV.(Weusethenotation[V] ratherthanthemorecommon[V]inordertoavoidanyconfusionwiththeclassofVintheGrothendieckgroup.)

Theclassc(V)residesnaturallyintheChowgroupA V.Forthepurposeofthispaper,thereaderwillmissnothingbyreplacingA Vwithordinaryhomology.

TheChernclassofavarietyVisaclassofevidentgeometricsigni cance:forex-ample,thedegreeofitszero-dimensionalcomponentagreeswiththetopologicalEulercharacteristicofV.ThisfollowsessentiallyfromthePoincar´e-Hopftheorem: c(TV)∩[V] =χ(V).

ItisnaturaltoaskwhetherthereareanalogsoftheChernclassde nedforpossiblysingularvarieties,forwhichatangentbundleisnotnecessarilyavailable.

Somewhatsurprisingly,one ndsthatthereareseveralpossiblede nitions,each‘natu-ral’fordi erentreasons,andallagreeingwitheachotherinthenonsingularcase.IfXisacompleteintersectioninanonsingularvarietyV,itisreasonabletoconsidertheFultonclassc(TV)cvir(X):=

behavesastheclassofa‘virtualtangent

bundle’toX.Itsde nitioncaninfactbeextended(andinmorethanoneway)toarbitraryvarieties,see§4.2.6in[18].

Theclasscvir(X)isinasenseuna ectedbythesingularitiesofX:forahypersurfaceXinanonsingularvarietyV,itisdeterminedbytheclassofXasadivisorinV.

Amuchmorere nedinvariantistheChern-Schwartz-MacPherson(CSM)classofX,whichdependsmorecruciallyonthesingularitiesofX,andwhichwewilluseasameasureofthesingularitiesbycomparisonwithcvir(X).

Thenameoftheclassretainssomeofitshistory.Inthemid-60s,M.-H.Schwartz([29],[30])introducedaclassextendingtosingularvarietiesPoincar´e-Hopf-typeresults,bystudyingtangentframesemanatingradiallyfromthesingularities.IndependentlyofSchwartz’work,GrothendieckandDeligneconjecturedatheoryofcharacteristicclasses ttingatightfunctorialprescription,andintheearly70sR.MacPhersonconstructedaclasssatisfyingthisrequirement([25]).ItwaslaterprovedbyJ.-P.BrasseletandM.-H.Schwartz([14])thattheclassesagree.

InthispaperwedenotetheChern-Schwartz-MacPhersonclassofasingularvarietyXsimplybyc(X)(thenotationcSM(X)isfrequentlyusedintheliterature).c(NXV)

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

14ALUFFIANDMARCOLLI

Thepropertiessatis edbyCSMclassesmaybesummarizedasfollows.Firstofall,c(X)mustagreewithitsnamesakewhenXisacompletenonsingularvariety:thatis,c(X)=c(TX)∩[X] inthiscase.Secondly,associatewitheveryvarietyXanabeliangroupF(X)of‘constructiblefunctions’:elementsofF(X)are niteintegerlinearcombinationsoffunctions1Z(de nedby1Z(p)=1ifp∈Z,1Z(p)=0ifp∈Z),forsubvarietiesZofX.TheassignmentX→F(X)iscovariantlyfunctorial:foreverypropermapY→Zthereisapush-forwardf :F(Y)→F(X),de nedbytakingtopologicalEulercharacteristicof bers.Moreprecisely,forW Yaclosedsubvariety,onede nesf (1W)=χ(W∩f 1(p)),andextendsthisde nitiontoF(Y)bylinearity.

GrothendieckandDeligneconjecturedtheexistenceofauniquenaturaltransformationc fromthefunctorFtothehomologyfunctorsuchthatc (1X)=c(TX)∩[X] ifXisnonsingular.MacPhersonconstructedsuchatransformationin[25].TheCSMclassofXisthende nedtobec(X):=c (1X).Resolutionofsingularitiesincharacteristiczeroimpliesthatthetransformationisunique,andinfactdeterminesc(X)foranyX.

AsanillustrationofthefactthattheCSMclassassemblesinterestinginvariantsofavariety,applythepropertyjustreviewedtotheconstantmapf:X→{pt}.Inthiscase,thenaturalitypropertyreadsf c (1X)=c f (1X),thatis,

f c(X)=c (χ(X)1pt)

(usingthede nitionofpush-forwardofconstructiblefunction).Takingdegrees,thisshowsthat c(X)=χ(X),

preciselyasinthenonsingularcase:thedegreeoftheCSMclassofa(possibly)singularvarietyequalsitstopologicalEulercharacteristic.

Itfollowsthat,ifXisahypersurfacewithoneisolatedsingularity,thenthedegreeoftheclass

Mil(X):=c(X) cvir(X)

equals(uptoasign)theMilnornumberofthesingularity.

Forhypersurfaceswitharbitrarysingularities,asthegraphhypersurfacesweconsiderinthepresentpapertypicallyare,thedegreeoftheCSMclassequalsParusi´nski’sgener-alizationoftheMilnornumber,[28].TheclassMil(X)iscalled‘Milnorclass’,andhasbeenstudiedrathercarefullyforXacompleteintersection,[13].

Forahypersurface,theMilnorclasscarriesessentiallythesameinformationastheSegreclassofthesingularitysubschemeofX(see[5]).Inthissense,itisameasureofthesingularitiesofthehypersurface.Forexample,thelargestdimensionofanonzerotermintheMilnorclassequalsthedimensionofthesingularlocusofX.

Thegraphhypersurfacesinthispaperarehypersurfacesofprojectivespace,henceitisconvenienttoviewtheCSMclassandtheMilnorclassofXasclassesinprojectivespace.Thispushforwardisunderstoodinthetablein§1.4,andwillbeoftenunderstoodintheexplicitcomputationsof§4.

2.3.CSMclassesversusclassesintheGrothendieckring.CSMclassesarede nedin[25]byrelatingthemtoadi erentclass,called‘Chern-Matherclass’,bymeansofalocalinvariantofsingularitiesknownasthe‘localEulerobstruction’.Asnotedabove,oncetheexistenceoftheclasseshasbeenestablished,thentheircomputationmaybeperformedbysystematicuseofresolutionofsingularitiesandcomputationsofEulercharacteristicsof bers.

Thefollowingdirectconstructionstreamlinessuchcomputations,byavoidinganycom-putationoflocalinvariantsorofEulercharacteristics.Thisisobservedin[1]and[2],whereitisusedtoprovideanalternativeproofoftheGrothendieck-Deligneconjecture,andas

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

BANANAMOTIVES15

HerethebundleTWi( logEi)isthedualofthebundle 1Wi(logEi)ofdi erentialformsonWiwithlogarithmicpolesalongEi.Eachterm

(2.4)νi c(TWi( logEi))∩[Wi] thebasisofageneralizationofthefunctorialityofCSMclassestopossiblynon-propermorphisms.GivenavarietyX,letZibea nitecollectionoflocallyclosed,nonsingularsubvarietiessuchthatX= iZi.Foreachi,letνi:Wi→Zi ZipullsbacktoadivisorwithnormalcrossingsEionWi.Then νi c(TWi( logEi))∩[Wi] .c(X)=i

computesthecontributionc(1Zi)totheCSMclassofXduetothe(possibly)non-compact

subvarietyZi.

Wewillusethisformulationintermsofdualsofsheavesofformswithlogarithmicpolestoobtaintheresultsof§4below.

Byabuseofnotation,wedenotebyc(Z)∈A Vtheclasssode ned,foranylocallyclosedsubsetZofalargeambientvarietyV.Withthisnotioninhand,notethatifY Xare(closed)subvarietiesofV,then

c(X)=c(Y)+c(X Y),

wherepush-forwardsare,asusual,understood.Thisrelationisveryreminiscentofthebasicrelation(2.1)thatholdsintheGrothendieckgroupofvarieties(see§2.1).Atthesametime,CSMclassessatisfya‘productformula’analogoustothede nitionofproductintheGrothendieckring([24],[1]).

Moreover,CSMclassessatisfyan‘embeddedinclusion-exclusion’ly,ifX1andX2aresubvarietiesofavarietyV,then

c(X1∪X2)=c(X1)+c(X2) c(X1∩X2).

Thisisclearbothfromtheconstructionpresentedaboveandfromthebasicfunctorialityproperty.

Inshort,thereisanintriguingparallelbetweenoperationsintheGrothendieckgroupofvarietiesandmanipulationsofCSMclasses.Thisparallelcannotbetakentoofar,sincethe‘embedded’Chern-Schwartz-MacPhersontreatedhereisnotaninvariantofisomorphismclasses.

Example2.1.LetZ1andZ2be,respectively,alinearlyembeddedP1andanonsingularconicinP2.DenotingbyHthehyperplaneclassinP2,we nd

c(Z1)=(H+2H2)·[P2] andc(Z2)=(2H+2H2)·[P2]

whileofcourse[Z1]=[Z2]asclassesintheGrothendieckgroup.

Thus,inparticular,theCSMclassc(X)doesnotde neanadditiveinvariantinthesenseof§2.1anddoesnotfactorthroughtheGrothendieckgroup,astheexampleaboveshows.

IncertainsituationsitishoweverpossibletoestablishasharpcorrespondencebetweenCSMclassesandclassesintheGrothendieckgroup.Forthenextresult,weadopttheratherunorthodoxnotationH rfortheclass[Pr] ofalinearsubspaceofagivenprojectivespace.Thus,1standsfortheclassofapoint,[P0] ,andthenegativeexponentsareconsistentwiththefactthatifHdenotesthehyperplaneclassthenHr·[Pr] =[P0] .

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

16ALUFFIANDMARCOLLI

whereT=[Gm]istheclassofthemultiplicativegroup,see§

3.ThentheclassofXintheGrothendieckgroupofvarietiesequals [X]=aiTi,Proposition2.2.LetXbeasubsetofprojectivespaceobtainedbyunions,intersections,di erencesoflinearlyembeddedsubspaces.Withnotationasabove,assume c(X)=aiH i.

Thus,adoptingavariableT=H 1intheCSMenvironment,andT=TintheGrothendieckgroupenvironment,theclassescorrespondingtosubsetsasspeci edinthestatementwouldmatchprecisely.

Proof.TheformulaholdsforalinearlyembeddedX=Pr,since

c(P)=((1+H)rr+1 Hr+1)·[P] =((1+H)rr+1 Hr+1)·H r=(1+H 1)r+1 1

.T

SinceembeddedCSMclassesandclassesintheGrothendieckgroupbothsatisfyinclusion-exclusion,thisrelationextendtoallsetsobtainedbyordinaryset-theoreticoperationsperformedonlinearlyembeddedsubspaces,andthestatementfollows.

Proposition2.2applies,forexample,tothecaseofhyperplanearrangementsinPN:forahyperplanearrangement,theinformationcarriedbytheclassintheGrothendieckgroupofvarietiesispreciselythesameastheinformationcarriedbytheembeddedCSMclass.Theseclassesre ectinasubtlewaythecombinatoricsofthearrangement.

Inamoregeneralsetting,itisstillpossibletoenhancetheinformationcarriedbytheCSMclassinsuchawayastoestablishatightconnectionbetweenthetwoenvironments.Forexample,CSMclassescanbetreatedwithinaframeworkwithstrongsimilaritieswithmotivicintegration,[3].

Inanycase,oneshouldexpectthat,inmanyexamples,theworkneededtocomputeaCSMclassshouldalsoleadtoacomputationofaclassintheGrothendieckgroup.Thecomputationsin§3and§4inthispaperwillcon rmthisexpectationforthehypersurfacescorrespondingtobananagraphs.

3.Bananagraphsandtheirmotives

Inthissectionwegiveanexplicitformulafortheclasses[XΓn]ofthebananagraph

hypersurfacesXΓnintheGrothendieckring.Theprocedureweadopttocarryoutthe

computationisthefollowing.WeusetheCremonatransformationof(1.17).ConsiderthealgebraicsimplexΣnplacedinthePn 1ontheright-hand-sideofthediagram(1.17).ThecomplementofthisΣninthegraphhypersurfaceXΓnisisomorphictothecomplementof

thesameunionΣninthecorrespondinghyperplaneLinthePn 1ontheleft-hand-sideof(1.17),byLemma1.7above.Sothisprovidestheeasypartofthecomputation,andonethenhastogiveexplicitlytheclassesoftheintersectionsofthetwohypersurfaceswiththeunionofthecoordinatehyperplanes.The nalformulafortheclass[XΓn]hasa

simpleexpressionintermsoftheclassesoftoriTk,withT:=[A1] [A0]theclassofthemultiplicativegroupGm.ThenTn 1istheclassofthecomplementofΣninsidePn 1.

Inthefollowingwelet1denotetheclassofapoint[A0].WeusethestandardnotationLfortheclass[A1]ofthea neline(theLefschetzmotive).Wealsodenote,asabove,byΣntheunionofcoordinatehyperplanesinPn 1andbySnitssingularitylocus.

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

BANANAMOTIVES17

ThisexpressioncanbethoughtofastakingplaceinalocalizationoftheGrothendieckring,butinfactthisisnotreallynecessaryifwetakethesefractionsasjustshorthandfortheirunambiguousexpansions.

Weintroducethefollowingnotation.SupposegivenaclassCintheGrothendieckringwhichcanbewrittenintheform

(3.2)

(3.3)C=a0[P0]+a1[P1]+a2[P2]+···f(P)=a0+a1P+a2P2+···Tosuchaclassweassignapolynomial

Remark3.1.NoticethattheformalvariablePdoesnotde neanelementintheGrothendieckring,sinceoneseeseasilythatPiPj=Pi+j.Infact,thevariablesPisatisfyadi erentmultiplicationrule,whichwedenoteby andwhichisgivenby(3.4)Pi Pj=Pi+j+Pi+j 1+···+Pj Pi 1 ··· 1

andwhichrecoversinthiswaytheclass[Pi×Pj].ThisfollowsfromLemma3.2,by

converting

each

of

the

twofactorsintothecorrespondingexpressionsinT,multiplyingtheseasclassesintheGrothendieckring,andthenconvertingtheresultbackintermsofthevariablesPi.

Lemma3.2.LetCbeaclassintheGrothendieckringthatcanbewrittenintermsofclassesofprojectivespacesintheform(3.2).OnecanconvertitintoafunctionoftheclassToftheform

(3.5)C=(1+T)f(1+T) f(1)FirstnoticethefollowingsimpleidentityintheGrothendieckring.r 1 Lr+1rrL=(3.1)[P]=.Ti=0

P

andthenreplacingPbytheclass[P]intheexpansionof(3.6)asapolynomialintheformalvariableP.rr

Proof.Theresultisobtainedbysolvingforfin(3.5),whichyieldstheformula(3.6).

Nextwede neanoperationonclassesoftheformC=g(T),whichonecanthinkofas“takingahyperplanesection”.Noticethatliterallytakingahyperplanesectionisnotawellde nedoperationattheleveloftheGrothendieckring,butitdoesmakesenseonclassesthatareconstructedfromlinearlyembeddedsubspacesofaprojectivespace,asisthecaseweareconsidering.

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

18ALUFFIANDMARCOLLI

Lemma3.4.Thetransformation

(3.7)H:g(T)→g(T) g( 1)

T,wehave

g(T) g( 1)T 1 1

T=[Pr],

or0ifr=0,sothattheoperation(3.7)indeedcorrespondstotakingahyperplanesection.Theoperationislinearing,viewedasalinearcombinationofclassesofprojectivespaces,soitworksforarbitraryg.

Wethenhavethefollowingpreliminaryresult.

Lemma3.5.TheclassofΣr+1 PrintheGrothendieckringisoftheform

+1

(3.8)[Σ 1 Tr+1

r+1]=(1+T)r

1

T Tr +Tr 2 Tr 3+···±1.

Proof.TheclassofthecomplementofΣr+1inPristhetorusclassTr.Infact,thecomplementofΣr+1consistsofall(r+1)-tuples(1: :···: ),whereeach isanonzeroelementoftheground eld.ItthenfollowsdirectlythattheclassofΣr+1hastheform(3.8),usingtheexpression(3.1)fortheclass[Pr].OnethenappliesthetransformationHof(3.7)toobtain

[L∩Σ (1+T)r+1 1 Tr+1

r+1]= 1

1 /(T+1)=(1+T)r

T+1

fromwhich(3.9)follows. De nition3.6.ThetraceΣ′r 1ofthealgebraicsimplexΣr+1

Σr+1 P Pristhein-

tersectionofr+1withageneralhyperplane.Itisaunionofr+1hyperplanesinPr 1meetingwithnormalcrossings.

Forinstance,Σ′4consistsofthetransversalunionoffourlinesasinFigure5andby(3.9)itsclassis

[Σ′(1+T)3 1

4]=

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

BANANAMOTIVES

19

23Figure5.ThetraceΣ′4 PofthealgebraicsimplexΣ4 P

Proof.WeknowbyLemma1.7that

X

Γn Σn～=L ΣnviatheCremonatransformation,withL=Pn 2thehyperplane(1.20).ThishyperplaneintersectsΣntransversely,sothat(3.9)appliesandgives

[L Σn]=[L] [L∩Σn]=Tn 1 ( 1)n 1

T

Thisgivestheformula(3.11).

Wethenhavethefollowingresult..

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

20ALUFFIANDMARCOLLI

Theorem3.10.TheclassintheGrothendieckringofthegraphhypersurfaceXΓnofthebananagraphΓnisgivenby

(3.12)[XΓn]=(1+T)n 1T+1 nTn 2.

Proof.Wewritetheclass[XΓn]intheform

[XΓn]=[XΓn Σn]+[Sn].

UsingtheresultsofLemma3.9andProposition3.7wewritethisas

=Tn 1 ( 1)n 1

T,

fromwhich(3.12)follows.

Theformula(3.12)expressestheclass[XΓn]as

n 2,[XΓn]=[Σ′n] nT

i.e.astheclassoftheunionΣ′nofnhyperplanesmeetingwithnormalcrossings(asinDe nition3.6),correctedbyntimestheclassofann 2-dimensionaltorus.

Example3.11.Inthecasen=3ofFigure3,(3.12)showsthattheclassofthehyper-surfaceXΓ3 P2isequaltotheclassoftheunionoffourtransversallines,minusthree

timesa1-dimensionaltorus,i.e.thatwehave

[XΓ3]=4T+2 3T=T+2=[P1].

Thiscanalsobeseendirectlyfromthefactthattheequation

ΨΓ3=t1t2+t2t3+t1t3=0

de nesanonsingularconicintheplane.

Example3.12.Inthecasen=4ofFigure3,thehypersurfaceXΓ4isacubicsurfacein

P3withfoursingularpoints.TheclassintheGrothendieckringis

[XΓ4]=T2+5T+5.

IntermsoftheLefschetzmotiveL,theformula(3.12)readsequivalentlyas

(3.13)L

InthecontextofparametricFeynmanintegrals,itisthecomplementofthegraphhypersurfaceinPn 1thatsupportstheperiodcomputedbytheFeynmanintegral.Thus,ingeneral,oneisinterestedintheexplicitexpressionforthemotiveofthecomplement.Itsohappensthatintheparticularcaseofthebananagraphstheexpressionfortheclassofthehypersurfacecomplementisinfactsimplerthanthatofthehypersurfaceitself.Corollary3.13.TheclassofthehypersurfacecomplementPn 1 XΓnisgivenby

(3.14)[Pn 1 XΓn]=Tn ( 1)n[XΓn]=Ln 1 n(L 1)n 2.

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

BANANAMOTIVES21

Proof.TheEulercharacteristicisanadditiveinvariant,henceitdeterminesaringho-momorphismfromtheGrothendieckringofvarietiestotheintegers.Moreover,torihavezeroEulercharacteristic,sothatχ(Tr)=0forallr≥1.Thentheformula(3.14)fortheclassofthehypersurfacecomplementshowsthat

χ(Pn 1 XΓ 1

n)=χ(Tn)+(n 1)χ(Tn 2)+χ(Tn 3) ···±1=( 1)n 1.

Sinceχ(Pn 1)=nweobtain

χ(XΓ1

n)=χ(Pn ) χ(Pn 1 XΓn)=n+( 1)n

asin(1.27).

In§4below,wederivethesameEulercharacteristicformulainadi erentway,fromthecalculationoftheCSMclassofXΓn.

Remark3.15.Noticethat,ifweexpandin(3.12)the rsttermintheform[Pn 1]=Tn 1+nTn 2+...,weseethatthedominanttermin[XΓ2n]isTn .Thisisnotsurprising,

sinceforthebananagraphsthehypersurfacesXΓnarerational.

Remark3.16.ThepreviousremarkexplainstheappearanceofatermnTn 2intheexpression(3.14).Theremainingtermsareanalternatingsumoftori.Thistermcanbeviewedas

(3.15)Tn ( 1)n

T+1,

forg(T)=Tn.AccordingtoLemma3.4,thisistheclassofthehyperplanesectionofthecomplementofthealgebraicsimplexΣn+1inPn.However,howgeometricallyonecanassociateaPntoagraphhypersurfaceXΓn Pn 1isunclear,sothatasatisfactory

conceptualexplanationoftheoccurrenceof(3.15)in(3.14)isstillmissing.

Forcompletenesswealsogivetheexplicitformulaoftheclass(3.14)writtenintermsofclasses[Pr].

Corollary3.17.Intermsofclassesofprojectivespacestheclass[Pn 1 XΓn]isgiven

by

n 1

(3.16)[Pn 1 XΓ

k=0 n+1+2 nn 1 kPk

n]=( 1)[]+n 2

k=0 n 1k+1 ( 1)n 2 k

k[Pk].

Proof.Theformula(3.16)isobtainedeasilyusingthetransformationrulesofLemma3.3togofromexpressionsinTtoexpressionsin[Pr],sothat

(Tn ( 1)n)/(T+1)→ (P 1)(P 1)n ( 1)n

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