Feynman motives of banana graphs(14)

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

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新工程科技Feynman motives of banana graphs(14)全文阅读和word下载服务。