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

SinceC2/U(1)=R×S2,thisquotientspaceisnowisomorphictoanR×S2bundleoveraV2.Similarlyto[25],equation(3.6)describesrealconesonaS2bundleoverV2.Mathematically,itisnoteasytorevealthatthesequotientspaceshaveG2holonomygroup.However,onecanshowthisusingaphysicalargument.Indeed,V2,withh1,0=h2,0=0,preserves1/4ofinitialsuperchargesandinthepresenceofS2itshouldbe1/8.Inthisway,thesupersym-metrytellsusthattheholonomyof(3.6)istheG2Liegroup.Thus,M-theoryontheaboveseven-dimensionalmanifoldleadstoN=1theoryinfourdimensions.

3.2ExplicitmodelsfromV2geometries

Tobetterunderstandthestructureof(3.3-6),letusgiveillustratingmodels.InparticularwewillconsiderspecialmodelscorrespondingtoN=4sigmamodelwithconformalinvariance.Forthisreason,wewillrestrictourselvestoeight-dimensionaltoricHKmanifoldsX8withtheCalabi-Yaucondition(2.10)inN=4supersymmetricanalysis.Inthisway,thegeometryofX8dependsonthemannerwechoosetheU(1)rmatrixgaugechargeQaisatisfyingtheCalabi-Yaucondition.We rststudycomplextwo-dimensionalweightedprojectivespacesWP2,afterwhichwewillconsidertheHirzebruchsurfaces.Otherextendedmodelsarealsopresented.

3.2.1V2asweightedprojectivespaces

Forconstructingthesemodels,weconsideranU(1)gaugesymmetrywiththreehypermulti-pletsφiofcharges(Q1,Q2,Q3)suchthatQ1+Q2+Q3=0.OnewaytosolvethisconstraintequationistotakeQ1=m1,Q2= m1 m2andQ3=m2.ThisgivesWP2m1,m1+m2,ingexamples,letusseehowweobtainthisgeometry.Example1:(m1,m2)=(1,1).ThisexamplecorrespondstothreehypermultipletsφiwiththevectorchargeQi=(1, 2,1).Afterpermutingtheroleofφ12and

φ2= 2,2

ψ1+ 3

3ψ3+2ψ2=0(3.8)