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

(2)TheresultofAcharya-WittenonM-theoryonG2manifolds,wherethesetofranksofgaugegroupscoincidewiththeweightvectoroftheWP2[25].

(3)ThelocalmirrorsymmetryapplicationintypeIIsuperstrings,wherethemirrorconstraintequationsinvolvethetoricgeometrydataoftheoriginalmanifolds[37,38,39,40].

Besidesthesepoints,acloseexaminationoftheformulationofthe(p,q)websreveals,how-ever,thatthematrixintersection(4.1)appearsintheordinaryandweightedprojectivespaces.Moreover,itdoesnotcarryanytransparenttoricgeometrydatadistinguishingthesegeome-tries.Takingintoaccountthisobservation,theconnectionweareafterleadsustoreformulatetheintersectionnumberstructuresbyintroducingthetoricgeometryMorivectorsQaiandasetofbranechargeconstraintequations.Tomakeconnectionwith[25],werestrictourselves

1=(w1,w2,w3).Givenasetofcharges(pi,qi),totheweightedprojectivespaceswhereQ

i=1,2,3,weproposetheintersectionnumberformula

Iij=wiwj(piqj piqj)

withthefollowingconstraintequations

222w1p1+w2p2+w3p3=0

222q2+w3q3=0.q1+w2w1(4.3)(4.4)

Now,thesetofranksofthegaugegroupsnishouldsatisfythefollowingconstraint

iIijni=0,(4.5)

asrequiredbytheanomalycancellationcondition[14,15].Usingequation(4.4),itiseasytoseethatthisconditioncanbesatis edintermsoftheweightsofWP2asfollows

ni=win,

andsothecorrespondinggaugesymmetryisgivenby

G=

i=1(4.6)U(win).(4.7)

Ourreformulationofthe(p,q)webshasthefollowingnicefeatures:

(1)Thisformulationisquitesimilartothegeometricengineeringoffour-dimensionalN=2superconformal eldtheorieswithgaugegroupG=

betafunction.

Dynkinlabelsbeinganullvectorofa neCartanmatricesasrequiredbythevanishingofthei=1 SU(sin)wherethesi’saretheusual