M-theory on G_2 manifolds and the method of (p,q) brane webs(13)

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

correspondstoapairoftwospheresofAmsingularitiesandasingleofAnsingularitiestwospheres.IntypeIIAsuperstringpicture,thisisequivalenttotheintersectionofthreesetsofD6-braneswithmultiplicitiesm,mandnleadingtoSU(m)×SU(m)×SU(n)gaugesymmetrywithchiralmultipletsinthe(m,m,¯1)+(1,m,n¯)+(m,¯1,n)bi-fundamentalrepresentations.ThisgaugesystemisrepresentedbyaquivertrianglewhichmaybeviewedasthetoricgeometrygraphofWP2.Ontheotherhand,thesamephysicscontentmodelscanappear

3intypeIIAsuperstringcompacti edonlocalelliptic brationCalabi-YauthreefoldsCYAin

thepresenceofD6-braneswrapping3-cyclesSiand llingthefour-dimensionalMinkowskispace-time[11,12].Intermsofgaugetheory,each3-cycleSiisassociatedtoasinglegaugegroupfactorandtheintersectionnumbers(1.4)countthenumberofN=1chiralmultipletswhichtransforminthebi-fundamentalrepresentation.Wewillseelaterthattheinformationofgaugesystemisencodedintheintersectionnumbers.Undermirrorsymmetry,thephysicsofD6-braneswrapping3-cyclesmapstoIIBD3-branestransversetolocalCalabi-Yauthreefolds

3CYA.ThelatterarecomplexconesoverdelPezzosurfacesormoregenerallycomplextwo-dimensionaltoricmanifoldsV2.Inthisway,theN=1four-dimensionalquivertheorycanbeobtainedfromtypeIIBgeometryusingthesocalled(p,q)branewebs.Indeed,thetoricskeletonsofthesevarietiesarede nedbythe(p,q)webchargesofD5-branes.Theycorrespondtothelociofpointsatwhichsome1-cyclesoftheellipticcurve brationofV2shrinkstozeroradius[32].Thephysicscontentofthesemodelscanbedeterminedexplicitlyfromthegeometryofthe(p,q)webs.Moredetailsonthismethod,see[14,15].Inparticular,ifthevanishing1-cyclesoftheelliptic brationCi≡(pi,qi),theintersectionnumbers,intypeIIAmirrorgeometry,readas

Iij=Ci·Cj=piqj pjqi

Thesetoftheranksofthegaugegroupsniisanullvectorofthismatrix,i.e

i(4.1)(piqj pjqi)ni=0.(4.2)

Untilthislevel,theconnectionbetweenthemethodof(p,q)branewebsandtheAcharya-Wittenmodel[25]isnotobvious.WeproposethatthisconnectionrequirestheintroductionoftheMorivectorchargeQaiinthedescriptionof(p,q)webs.Oursolutionwasinspiredbythefollowing:

(1)ThestudyofM-theoryonlocalgeometryofCalabi-Yauthreefoldshavingtoricrealizationintermsof(p,q)D5-branesoftypeIIBsuperstrings[32].Inthisway,theD-branechargesareassociatedwiththevanishingcyclesinthetoricrepresentation.