Feynman motives of banana graphs(2)

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 …… 此处隐藏：5512字，全部文档内容请下载后查看。喜欢就下载吧 ……