; TeX output 2001.11.28:1907X'? "V cmbx10QUANTUMCOHOMOLOGYANDGEOMETR Y0Ե2+- cmcsc10MarUTtinLA.Guest8Ӎ K`y cmr10In /theselecturesweshalldescribGesomeaspectsofquantumcohomologytheorythat arerelatedtointegrablesystems.sRoughlyspGeaking,GthefollowinginclusionsarealreadywellUUknown:g[integrableUUsystems{D !", cmsy10 F*robGeniusUUmanifoldsqy quantumUUcohomologyRecent/workofDubrovin([Du2])andGivental([Gi2])suggeststhatthesethreeob8jects are)infact$': cmti10verykcloselyrelated.c.W*eshallgiveanintroGductiontothisarea,2MbeginningwithF*robGeniusmanifoldsinx1,Qthenquantumcohomologyinx2.4Inx3weshallgiveanexampleofUUhowthetheoryofintegrablesystemscanbGeappliedtoquantumcohomology*. There4aremanyrecentarticlesinthisarea.qAsasampleofbGooks4andfundamentalreferences werecommend[Mc-Sa],8[Du1],[Gi1],[Co-Ka] and[Ma].STheauthor'ssurveyarticles/[Gu1],%[Gu2]mayalsobGeconsulted.USectionxR2isinfactashortenedversionofpartUUofr[Gu1].'Yލ|xE1LFrobeniusmanifolds Consideratriple(R^ 0ercmmi7nq~ b> cmmi10;g[;!),where,foranyt2R^nq~,gtV(;L«)isanondegeneratesymmetricbilinear[formonR^m,and!tisalinearmapR^m !)wEnd%(R^m)."W*eassumethefollowingcompatibilityUUcondition:g(F)tñg[٫(x;!(y)(zp))UUissymmetricinIx;y;zp:It2cisconvenient2ctointroGducethe\productnotation"![٫(x)(y)7=x>y<(more2caccurately*, !tV(x)(y[٫)=xtLJy,\but^weusuallyomitt).4 Condition(F)/impliesthatiscommutative(butitIJisnotnecessarilyassoGciative).W*edenotebyrtheLevi-CivitaconnectionassoGciatedtog٫(regardinggasanotnecessarilypGositivede niteRiemannianmetric).ThisisaconnectionUUinthetrivialbundleR^n^8R^m _!R^nq~.|{Ycmr81*XDe nition.(R^nq~;g[;!)isaF;r}'obeniusstructureifthefollowingconditionshold:(1)t=isasso}'ciative,forallt2R^n(2)d!"=0(3)ris at,i.e.haszer}'ocurvatureWhen֢n=m,thisde nitionsays,essentially*,that֢\R^n H isaFrobGeniusmanifold"inthe senseUUofr[Du1]. Thev\integrabilityconditions(1),(2),(3)(togetherwith(F)v#abGove)areverycomplicated;apracticalapproachistoassumesomeofthemandthenstudythesystemofp.d.e.which!nisequivqalenttother}'emainingconditions.A!9particularsolution|orrather,TtitsunderlyingYF*robGeniusstructure|willgenerallyappearinmorethanonesuchway*.MThus,F*robGeniusstructureshavethepotentialtorelatequitedi erentsystemsofp.d.e.,andseveralUUexamplesofthisphenomenoncanbGefoundin[Du1]. ThesituationrelevqanttoquantumcohomologytheoryiswheregtV(;/)q=(;),Ta\constant"metriconR^nq~,fromwhichitfollowsthatr=d(thestandard atconnection).F*romnowonweshallassumethatg璫isofthisform.ItiseasytoseethattheremainingintegrabilityUUconditions(1)and(2)areequivqalentto(1)UU!^8!"=0(2)UU(x8y[;zp)=xyzpSforUUsome\pGotentialfunction"SZ:R^n8!R.(Here,!thenotationzpSmeansthederivqativeofthefunctionSbytheconstantvector eldzp,UUi.e.qDZzSZ=dS(z).) TheseUUintegrabilityconditionscanbGerephrasedfurtherasfollows:PropQosition.(R^nq~;UP(W;);![٫)isaF;r}'obeniusstructureifandonlyiftheconnectiondp+!is atforall2R.Pr}'oof.pThe0connectiond߫+!is0 atifandonlyifd(![٫)+(!)^)!)=00forall,8!i.e.eifandUUonlyifd!"=!^8!=0. E msam10E ThisՇ\zerocurvqatureformulation"justi estheterminology\integrable"andprovidesalinkwiththetheoryofintegrablesystems.Infactsomeauthorscallanyp.d.e.whichadmitsUUazerocurvqatureformulationanintegrablep.d.e.2 UX T*oobtainareasonablyspGeci cp.d.e.indepGendentoft.UjTheF*robeniuscondition(xy[;zp)=(y;xzp)>followsfromthefactthatbGothUUsidesareequaltotheevqaluationofxy[zonM.$ xkY2LQuantumcohomology LetxM͓bGeasimplyconnected(andcompact,connected)Kahlermanifold,ofcomplexdimension8n.RW*eshallassumethattheintegralcohomologyofM Sisevendimensionalandtorsion-free,UUi.e.qthat(A1) UVH^h(M;Z)m=u cmex10Lލ8n%8i=0 WرH^2i (M;Z)T͍+3=Z^s+1f=.#܍ Theroleofthevectorspaceofx1willbGeplayedbyH^h(M)=H^(M;C)T͍+3= UNC^s+1f=.6_Thereis}afamilyofproGductoperationstӫonthisvectorspace,parametrizedbyt2H^2Lq(M)T͍+3= UNC^kcalled+thequantummpr}'oduct.cIn+thissectionweshallde net-intermsofcertainGr}'omov-WitteninvariantsUUhAjBqjCiD@. W*eUUneedsomenotationfromordinarycohomologytheory.qLet ~HiPDJ:Hi$J(M;Z)!H2ni>(M;Z)bGemqthePoincarGemqdualityisomorphism.ThismaybGede nedasthemapwhichsendsa cohomology0oclassxtothe\capproGduct"ofxwiththefundamentalclassofM.ezAsfaraspGossibleUUweshallusethenotation hTIa;b;c;:::q2mHh(M;Z) (UUH(M;C)UU)3Xforicohomologyclasses,o andwewritejaj;jbj;jcj;:::iݫforithedegreesofa;b;c;:::.yW*eishall writey(tA=PD8(a);BG=PD(b);C~4=PD(c);:::q2mH(M;Z) (UUH(M;C)UU)forUUthePoincarGeUUdualhomologyclasses,andjAj;jBqj;jCj;:::ULforUUtheirdegrees. LetzhUU;i:Hi$J(M;Z)8HiTL(M;Z)!ZzdenotetheKronecker(or\evqaluation")pairing;_weusethesamenotationfortheextendedpairingxhUU;i:Hh(M;Z)8H(M;Z)!Z(thus,uha;Bqi iszerowheneverjaj6=jBqj).YSince thereisnotorsion,ubGoththesepairingsarenondegenerate. The4roleofthebilinearformofxW1willbGeplayed4bythe(C-linearextensionofthe)\intersectionUUpairing",whichisde nedbyyM(UU;):Hh(M;Z)8H(M;Z)!Z; (a;b)=hab;Mi:Onθtherighthandside,MӫdenotesthefundamentalhomologyclassofthemanifoldM.ItVZisanelementofH2n m(M;Z),anditsPoincarGedualcohomologyclass|theidentityelementreofthecohomologyalgebra|willbGedenotedby12H^0Lq(M;Z)re(anexceptiontoourgnotationalconventiongforcohomologyclasses!).VThehomologyclassrepresentedbyasinglelpGointofMwillbedenotedbyZ}Y2=H0|s(M;Z),anditsPoincarGedualcohomologyclassUUwillbGedenotedinaccordancewithourconventionUUbyzp. Now,ktheg-wellknowndualitybGetweenthecupproGductandtheKroneckerpairingmaybGeUUexpressedbytheformulaN{hab;Mi=ha;Bqi=hb;Ai:F*romthenondegeneracyoftheKroneckerpairing(inthissituation),itfollowsthattheintersectionUUpairing(;)isnondegerate. ThecupproGductonH^h(M;Z)maybespeci edintermsofits\structureconstants".T*oUUdothis,wechoGosegeneratorsasfollows:IэHH(M;Z)=r6s M t+i=05ZAi!Q(CJHh(M;Z)=r6s M t+i=05Zai4$Xand&?wede ne\Kroneckerdual"cohomologyclassesa^cl0|s;:::;a^csly(i.e.thedualbasiswith respGectUUto(;))byaiTLa^c;Zjī=ij zp.qThenforanyi;jwehave%{ aiTLajī= HX jf UUjja O \cmmi5  j=jai*j+jajajgNci;j jac %ԍforUUsome1ɍi;jxݍ1 j;:::;^i;js 22Z.qThesestructureconstantsaregivenby鍒ȱ1ɍi;jvk 2=haiTLaj6ak됱;Mi:"ObserveUUthatif1ɍi;jvk 26=0thenthenumericalconditionF>·jaiTLj8+jaj6j+jak됷j=2nmustUUbGesatis ed.׍De nition.F;orc}'ohomologyclassesa;b;cwede nehAjBqjCi0C=hab;Ci=habc;Mi.Giving8allthestructureconstantsisthesamethingasgivingall\tripleproGducts"hAjBqjCi0|s. The.quantumproGductwillbedeterminedbyalargerfamilyoftripleproductsdenotedby\chAjBqjCiD@,&whereDvqariesinH2|s(M;Z).(ThesearetheGromov-Witteninvqariants ^[ٴDl3;0 (A> B C).)HW*e4shallde nethesenewtripleproGductslater;forthemomentweshalljustUUassumethat(A2) hAjBqjCiD 2^Zisde nedforanyA;B;C22^H(M;Z),ƔD32H2|s(M;Z).Moreover,hAjBqjCiD 2ZUUislinearandsymmetricalinA;B;C. TheUUquantumproGductisde nedasfollows:׍De nition.ha8t6b;Ci=P USD72HZcmr52 (M,;f$cmbx7Z)(;);t=)de nesaF;r}'obeniusstructure. ThereUUisamoGdi cationofthequantumproduct,whichisaproductoperationCam:Hh(M;C)8H(M;C)!H(M;C)8 wherebisthegroupalgebraC[H2|s(M;Z)].\F*ormally,anbelementofisa nitesum P ;i/iTLq[ٟ^Di ,1Fwhere(Bid2C,Di2H2|s(M;Z),and(BwherethesymbGolsq[ٟ^D aremultipliedinthe iobviousUUway*,i.e.qDZq[ٟ^Dбq[ٟ^E ^=q[ٟ^D7+Eͫ.Thede nitionis:De nition.a]b=P USD72H2 (M,;Z)<(a]b)D럱q[ٟ^D,wher}'e?(ab)D 6isde ne}'dbyh(ab)D@;Ci= @hAjBqjCiD forallC~42H(M;C).W*eshallalsorefertothisopGerationasthequantumproduct.Byourassumption(A3),theUUsumontherighthandsideofthede nitionofa8bUUis nite. ItUUisconvenientUUtode neagradingonthealgebraH^h(M;C)8 UUbyde ning͍ 6jaq[ٟDзj=jaj8+2hc1|sTcM;DGi:CW*elshallassumethatthequantumproGductspreservethisgrading,i.e.$/thatjagRbj=jajgR+jbj.ItPNfollowsthatj(a0b)D@ji`=jaj0+jbj2hc1|sTcM;DGi.bHence, ifPNhAjBqjCiD W6=i`0,thenthenumericalUUconditionjaj8+jbj2hc1|sTcM;DGi=jCjUUmustbGesatis ed,i.e.yjaj8+jbj+jcj=2n8+2hc1|sTcM;DGi:TheigeometricalmeaningofthisnumericalconditionwillbGecomeclearlater.W*ewriteQH^i$J(M;C)=fx2QH^h(M;C)UUjjxj=ig. OneRfurtherpieceofnotationwillbGeuseful.jInallofourexamplesweshallchoGosean-identi cationH2|s(M;Z)T͍/з+3/Ы=&Z^rf(forsomerv/Ы1).Havingmadethischoice,cwewriteD5=(s1|s;:::;srm),UUandq[ٟ^D c=q:[ٴs1l1 3۱:::*q^[ٴsr፴rʫ. Before-de ningtheGromov-WitteninvqariantshAjBqjCiD@,weshallbGeginbyreviewinganspGecialcase,u5namelythede nitionofthetripleproducthAjBqjCi0n =hab;Ci=habc;MiinUUordinarycohomology*. TheUUnaivede nitionis}hAjBqjCi0 =mjx䍑~A \x䍑S~8B \x䍑|~8C j;iwhereytherighthandsidemeansthenumbGerofpoints(countedwithmultiplicity)inthe ԍintersectionx䍑~-A P\x䍑~B [\x䍑N~C ],"wherex䍑~-A .;x䍑 ~B ;x䍑e~C are-suitablerepresentativesofthehomologyclassesA;Bq;C.In8 certainsituationsthisde nitionis\correct",pinthesensethatitgivesthe6:Xusual%tripleproGducthabc;Mi.a(Similarly*,/hab;Mi%maybGede nednaivelyasjx䍑~A YG\x䍑9~FB j.)aF*or example,j"inethecomplexalgebraiccategory*,thede nitioniscorrectwheneverthereexist &representativealgebraicsubvqarietiesx䍑|~A f;x䍑 ~B ;x䍑e~C n_whoseintersectionis nite(orempty).LThus,wheneverUUweareluckyenoughto ndsuchrepresentatives,wecancalculatehAjBqjCi0|s. TheqmostfamousexamplewherethismethoGdworksisthecaseM3=Grk됫(C^nq~)(SchubGertcalculus).yHereUallthegeneratorsofthehomologygroupsarerepresentablebyalgebraiccycles(SchubGertvqarieties),aandforanythreesuchgeneratorsa;b;csatisfyingtheconditionjaj8+jbj+jcj=dimnMlpthereUUexistrepresentativesUUwhoseintersectionis nite(orempty). W*ewishtode nehAjBqjCiD $}asacertainintersectionnumbGer. [F*romthepreviousdiscussion,UUitshouldhaveUUthepropGerties(a) D5=00=UX)hAjBqjCiD =habc;Mi(b) hAjBqjCiD 6=00=UX)jaj8+jbj+jcj=2n+2hc1|sTcM;DGi.(Actually*,`{(a)willbGeobviousfromthede nition,but(b)willhavethestatusofanassumption.)TheUUnaivede nitionismyhAjBqjCiD =jHol1ɍ;~ǴAW;pDQ\Hol1ɍ;~ǴBS;qD\Hol1ɍ;Ѯ~ǴC+;rDjqwhere,%Hol1ɍ;>Gd~<0DAB3;p<0DDK>=fholomorphicUUmapsSjfڧ:CPc1!MlpjUUf(p)2x䍑j~A nand!8[f]=DGgɜand@wherex䍑_~Aisarepresentative@ofthehomologyclassA.ThepGointsp;q[;r]are@three distinctsbasepGointsinCPc^1. Thenotation[f]denotesthehomotopyclassoff,zwhichisanUUelementof2|s(M)T͍+3= UNH2(M;Z).ԍ W*eassumethatthereexistx䍑Ls~A (;x䍑 ~B ;x䍑e~C 0VsuchthattheabGoveintersectionis nite(orempty);thishisessentiallythemeaningofassumption(A2)oftheprevioussection.Morefunda-mentally*,UUweshallmakethefollowingthreeassumptions:(A2a) Hol1ɍG MQ;pG D,isXa(smoGoth)complexmanifold,ofcomplexdimensionnT+hc1|sTcM;Di.ThisUUwillbGedenotedmorebrie ybyHoltDk.:O(A2b) Hol1ɍ;^@~G AJ;pG D)Gisacomplexsubmanifold(orsubvqariety)ofHol1ɍEMQ;pED!6u,_andthecomplexcoGdi-8ߍmensionUUofHol1ɍ;~tA;ptD"inHol1ɍtMQ;ptD$>isequaltothecomplexcoGdimensionofx䍑~A*inM.8ߍ(A2c) Ifx䍑~UUA V;x䍑 ~B ;x䍑e~C ɫintersectUUtransversely*,thensodoHol1ɍ;~tA;ptDV;Hol1ɍ;~ǴBS;qDh;Hol1ɍ;Ѯ~ǴC+;rDC.7JXW*eshallnotdiscusstheextenttowhich(A2)isequivqalentto(A2a),^(A2b),(A2c),nor the,extenttowhichtheseconditionsaretrue.(CertainlyallfourassumptionsholdforhomogeneousKahlermanifolds.)KHowever,{abriefcommentontheoriginofthenumerical expressionsQin(A2a)and(A2b)maybGehelpful.TFirst,thetangentspaceatfڧ2Hol1ɍ7MQ;p7D#| assumingߤthatHol1ɍ&ôMQ;p&D%ܫisamanifold|maybGeidenti edwiththespaceofholomorphicsectionsm9ofthebundlef^sTcM.rBytheRiemann-RoGchtheorem,s2thecomplexdimensionofthisvectorspaceisn5+hc1|sTcM;DGi,asstatedin(A2a).FRegarding(A2b),thiswouldfollowfromUUthecommutativeUUdiagram6P߭Hol1ɍ;=~&̴A*\;p&D )䍍>UM!)Hol1ɍpʹMQ;ppDX?X?Xy??yax䍒>~DA )䍍>UM!ࣱM9/inwhichtheverticalmapsaregivenbyfڧ7!f(p),whenevertheevqaluationmapHol1ɍMQ;pD#Ҍ!MlpisUUaloGcallytrivial brebundle(asitwillbewhenMlpishomogeneous,forexample). ItkisimpGortanttobearinmindthatthespaceHol>Dvis nite-dimensionalandalgebraic,so&thereisnoquestionofin nite-dimensionalanalysishere.Thetechnicalproblemsingivingarigorousde nitionarisefromthenoncompactnessofHolD,އandthefactthatthe"transversalityconditionmaynotbGesatis ed.-W*econcludewithaverybriefdiscus-sion+ofwhatisneeded.First,aAaverygeneralde nitionofhAjBqjCiD l(andhenceofthequantumJproGduct)ispossibleundertheassumptionthatthe(connectedsimplyconnectedKahler|oroOevenmerelysymplectic)manifoldMjis\pGositive"insomesense,uforexamplethat{hc1|sTcM;DGiV>0foreachhomotopyclassDs2V2|s(M)whichcontainsaholomorphicmapUCPc^1 !M.iA)F*anomanifold|thatis,'TamanifoldMpforwhichthecohomologyclassGc1|sTcM^īcanbGerepresentedbyaKahler2-form|isautomaticallypGositiveinthissense.LZIt1canbGeshownthatthequantumproGductiscommutativeandassoGciative,hwhenMӫispGositive.Sofar,however,thereisnoguaranteethattheGromov-WitteninvqarianthAjBqjCiD Jtcan }bGec}'omputed }bythenaiveformulagivenearlierinthissection.?F*orthis,one*needsanadditionalassumption,`'forexamplethatMAis\convex"*inthesensethatH^1Lq(CPc^1;f^sTcM)/y=0-forallholomorphicfC:/yCP^1 {!M.xConvexity-impliesinpartic-ularthatHol Dګisamanifold,andthatithasthe\expGected"dimensionn+hc1|sTcM;Di.HomogeneousUUKahlermanifoldsareconvex,UUforexample. Theseriesde ningthequantumproGductwillingeneralcontainin nitelymanypGowersofQq*(bGothpositiveandnegative).RHowever,ifthereexistD1|s;:::;Dr2D2(M)QsuchthatallholomorphicallyrepresentableclassesDareoftheformPލ f r% f i=1niTLDi,withnid0,thenitfollowsfromthepGositivityconditionthateachsuchseriescontainsonlya nitenumbGeroftermsmq[ٟ^D ^=q:[ٴs1l1 3۱:::*q^[ٴsr፴rʫ,randthatsiBڷ0ineachcase.AnysimplyconnectedhomogeneousKahlerUUmanifoldsatis esthiscondition,aswellaspGositivityandconvexity*.8 YŠXr0x3LAnapplicaUTtionofintegrablesystemstoquantumcohomology LetWK+ث=tR[q1|s;:::;qrm]bGetheringofpolynomialsinq1|s;:::;qrwithrealcoecients. DepGending0onthecontext,8we0regardqi;|eitherasaformalvqariableorasafunctiont7!e^tiwherei1t٫=(t1|s;:::;trm)2R^r.ZW*ei1introGducethenotation@i%= 3@ &fe ş@n9tioݫ=ٱqi  @&fe @n9qiCj,'andde ne D$Wto:bGetheringofdi erentialoperatorsgeneratedby@1|s;:::;@rJܫwithcoecientsinK.Let+M\=E}DG=(D1|s;:::;Du:b)beacyclicD-module(aleftmoduleover+D,!generatedbytheconstantHWdi erentialopGerator1),where(D1|s;:::;Du:b)meanstheleftidealgeneratedbydi erential}opGeratorsD1|s;:::;Du:b.F?W*eshallassumethatMisholonomic,ofranks+1.Thisimpliesinparticularthat, -asaK-moGdule,Misisomorphictothedirectsumofs|+1copiesUUofK. TheZD-moGduleMqūisanalgebraicversionofthesystemofpartialdi erentialequationsD1|sfi=UU=UDu:bf=0.r&Here,'fYbGelongstoagivenfunctionspaceFc,butMisofcourseindepGendentڑofF> (andthisisitsadvqantage). {T*osaythatMisholonomicistosaythatthesystemis\maximallyoverdetermined"; cmmi10Zcmr5Aacmr6ٓRcmr7|{Ycmr8K`y cmr10`