M-theory on G_2 manifolds and the method of (p,q) brane webs

Theaboveidentitiesinthemethodof(p,q)websareveryexciting.First,thesameequationformshavebeenusedinthegeometricengineeringofsuperconformalmodelswitheightsu-percharges.Inthiscase,thequivermatrixisidenti edwithana neADECartanmatrixKandthegaugegroupisG= iSU(sin).ThepositiveintegerssiappearinginGaretheusualDynkinweights.Theyformaspecialpositivede niteintegervectors=(si)satisfyingKijsj=0,asrequiredbythevanishingofthebetafunction.Second,for =3correspond-ingtocomplextwo-dimensionalweightedprojectivespacesintypeIIBgeometry,thephysicscontentwithunitarygaugegroupsandchargedchiralmatterseemstobesimilartofour-dimensionalN=1modelsobtainedfromM-theoryonsingularG2manifoldsstudied rstin

[25],seealso[26,27].Thesemanifoldsareconstructedascirclequotientsofeight-dimensionaltorichyper-K¨ahler(HK)manifolds.Following[25],thetwistorspaceovertheweightedpro-jectivespaceWP2m,m,nhasaninterpretationintypeIIAsuperstringasanintersectionofthreegroupsofD6-braneswithmultiplicitiesm,m,nleadingtoSU(m)×SU(m)×SU(n)gaugesymmetry.Accordingtothisfeature,onemightaskthefollowingquestion.Isthereaconnectionbetweentheapproachof(p,q)websandM-theoryonG2manifolds1?However,thisconnectionmaynaturallyleadtotheneedofareformulationofthemethodof(p,q)webs.ThereasonforthisisthatthegaugesymmetryintheM-theorycompacti cationinvolvestheweightsoftheweightedprojectivespaceWP2.InthispaperweaddressthisquestionusingtoricgeometrydataofG2manifoldsasU(1)quotientsofeight-dimensionalHKmanifolds,andbyreconsideringthemethodof(p,q)webs.Thisstudymaycompletetheanalysisof[28]dealingwithdiscreteG2orbifoldsusingtheMcKaycorrespondence[29].

Ourprogramwillproceedintwosteps:

(i)WestudyG2manifoldsasU(1)quotientsofeight-dimensionaltoricHKmanifolds,X7=X8/U(1).ThemanifoldX8isobtainedusingrelevantconstraintequationsintermsoftwo-dimensionalN=4sigma-modelswithU(1)rgaugesymmetryandr+2hypermultiplets

[25,26,30].Weshowthattheresultingseven-dimensionalmanifolds,ingeneral,aregivenbyrealconesonS2bundlesovercomplextwo-dimensionaltoricvarieties

V2=Cr+2/C r.(1.5)

Explicitmodelsarepresentedintermsoftwo-dimensionalN=2sigmamodelrealizationsofV2.

(ii)WediscussthelinkbetweenthephysicscontentofM-theoryonsuchG2manifoldsandthemethodsof(p,q)webs.Inparticular,wereconsiderandreformulatethe(p,q)webequations