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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