The Vertex on a Strip

arXiv:hep-th/0410174v211Jul2006February7,2008SMS-0402SLAC-PUB-10804SU-ITP4/39hep-th/0410174TheVertexonaStripAmerIqbal♠,Amir-KianKashani-Poor♣♠SchoolofMathematicalSciencesGCUniversity,Lahore,54600,Pakistan.♣DepartmentofPhysicsandSLACStanfordUniversity,Stanford,CA94305/94309,U.S.A.♠DepartmentofMathematicsUniversityofWashington,Seattle,WA,98195,U.S.A.AbstractWedemonstratethatforabroadclassoflocalCalabi-YaugeometriesbuiltaroundastringofP1’s–thosewhosetoricdiagramsaregivenbytriangulationsofastrip–wecanderivesimplerules,basedonthetopologicalvertex,forobtainingexpressionsforthetopologicalstringpartitionfunctioninwhichthesumsoverYoungtableauxhavebeenperformed.Byallowingnon-trivialtableauxontheexternallegsofthecorrespondingwebdiagrams,thesestripscanbeusedasbuildingblocksformoregeneralgeometries.Asapplicationsofourresult,westudythebehavioroftopologicalstringamplitudesunderflops,aswellascheckNekrasov’sconjectureinitsmostgeneralform.1IntroductionInthelastfewyears,dramaticprogresshasbeenmadeintechniquesforcalculatingthepar-titionfunctionofthetopologicalstringontoric(hencenon-compact)Calabi-Yaumanifolds[1,2,3].Theculminationofthisefforthasbeentheformulationofthetopologicalvertex[4](see[5]forarecentmathematicaltreatment).Withit,asetofdiagrammaticrulescanbeformulatedwhichallowanexpressionforthetopologicalstringpartitionfunctiontobereadofffromthewebdiagramofthetoricmanifold.Whiletheexpressionsobtainedsucharealgorithmicallycomplete,theycontainunwieldysumsoverYoungtableaux,onesumforeachinternallineofthewebdiagram.Startingwith[6],methodsweredevelopedtoperformaportionofthesesums[7,8,9,10].Inthisnote,weshowhowtoperformallsumswhichariseinanarbitrarysmoothtriangulationofastriptoricdiagram,suchaswitharbitraryrepresentationsonallexternallegsbutthefirstandlast.TheultimategoalofthisprogramistoprovideatechniqueforefficientlyextractingtheGopakumar-Vafainvariantsfromtheexpressionsthetopologicalvertexyieldsforthetopo-logicalstringpartitionfunction.Wewilloutlinetheobstaclestothisgoalusingthemethodsofthispaperasweproceed.Asotherapplications,weofferananalysisofthebehaviorofthetopologicalamplitudeunderflopsofthetargetmanifold.WedemonstratethattheGopakumar-Vafainvariantsforalltoricgeometriesdecomposableintostripsareinvariantunderflops.WealsoshowthatourresultsprovidetheframeworktocheckNekrasov’sresults[11]inthemostgeneralcaseofproductU(N)gaugegroupswithanynumberofallowedhypermultiplets.Theorganizationofthispaperisasfollows.Insection2,weelucidatethegeometriesweareconsideringandpresentandinterprettherulesforobtainingthetopologicalstringpartitionfunctiononthem.Insection3,wederivetheseresults.Weincludeabriefreviewofthetopologicalvertexatthebeginningofthissection,andenditwithacomparisontothenatural4-vertexobtainedfromChern-Simonstheory.WediscussthebehaviorofGopakumar-Vafainvariantsunderflopsongeometriesdecomposableintostripsinsection4.1.Section4.2providesthebasicbuildingblockstostudyNekrasov’sconjecture.Weendwithconclusions.AnappendixgivesabriefintroductiontoSchurfunctions,andcollectstheidentitiesforSchurfunctionsusedthroughoutthepaper.12Theresults2.1GeometryofthestripRecallthatasimplewayofvisualizingthegeometriesgivenbytoricdiagramsistothinkofthemasTnfibrationsoverndimensionalbasemanifoldswithcorners(seeeg.section4.1of[19]).Locally,onecanintroducecomplexcoordinatesonthetoricmanifold.Thebasemanifoldisthenlocallygivenbytheabsolutevalueofthesecoordinates,theTnbythephases.Theboundaryofthebaseiswheresomeofthesecoordinatesvanish,entailingadegenerationofthecorrespondingnumberoffiberdirections.In3complexdimensions,the3realdimensionalbasehasa2dimensionalboundarywithedgesandcorner.Webdiagrams,easilyobtainedfromtoricdiagramsassketchedinfigure1,representtheprojectionoftheedgesandcornersofthebaseontotheplane.ThereisafullT3fiberedovereachpointabovetheplane,correspondingtotheinteriorofthebasemanifold.Onagenericpointontheplane,representingagenericpointontheboundary,onecycleofthefiberdegenerates,twodegenerateonthelinesofthewebdiagram,whichcorrespondtoedgesofthebase,andtheentirefiberdegeneratesattheverticesofthediagram,thecornersofthebase.Figure1:Relationbetweentoricandwebdiagram.Returningtofigure1,wenowseethestringofP1’s(inredandblue)emergingbyfollowingtheS1fibrationalongtheinternallinerunningthroughthewebdiagram.ItiscappedofftoP1’sbytheS1’sdegeneratingateachvertex.Thetwonon-compactdirectionsofthegeometrylocallycorrespondtothesumovertwolinebundlesovereachP1.Thetwolocalgeometriesthatariseonthestripare(Ø(−2)⊕Ø)→P1(inred)and(Ø(−1)⊕Ø(−1))→P1(inblue).WerefertotherespectiveP1’sas(−2,0)and(−1,−1)curvesinthefollowing.22.2RulesonthestripEachvertexhasonenon-trivialYoungtableauassociatedtoit,thetwoouterverticesinadditionhaveonelegcarryingthetrivialtableau.Allotherindicesoftheverticesaresummedover.Welabelthenon-trivialtableauxbyβi,withiindexingthevertex.TheinternallinescarryafactorQi=e−ti,wheretiistheK¨ahlerparameterofthecurvetheinternallinerepresents.•EachvertexcontributesafactorofWβi=sβi(qρ)(thenotationfortheargumentoftheSchurfunctionisexplainedinthenextsection).•Eachpairofvertices(notjustadja