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

usingthetoricgeometryMorivectorsofV2andsetofbranechargeconstraintequations.FortheweightedprojectivespaceWP2w1,w2,w3,forexample,we ndthefollowinggaugegroup

G=U(w1n)×U(w2n)×U(w3n).(1.6)

Thisisrequiredbytheanomalycancellationcondition.Withanappropriatechoiceofweightvectors,werecovertheresultofAcharyaandWittengivenin[25].

Theplanofthispaperisasfollows.Insection2,webrie yreviewthemainlinesoftoricgeometrymethodfortreatingcomplexmanifolds.Thenwegivetheinterplaybetweenthetoricgeometryandtwo-dimensionalN=2supersymmetricgaugetheories.Insection3,westudyG2manifoldsasU(1)quotientsofeight-dimensionaltoricHKmanifoldsX8constructedfromD- atnessconditionsoftwo-dimensional eldtheorywithN=4supersymmetric.ThenweidentifytheU(1)symmetrygroupwiththetoricgeometrycircleactionsofX8topresentquotientsX7=X8/U(1)ofG2holonomy.ExplicitmodelsaregivenintermsofrealconesonanS2bundleovercomplextwo-dimensionaltoricvarietiesV2.Insection4,weengineerN=1quivermodelsfromG2manifolds.WediscussthelinkbetweenthephysicscontentofM-theoryonsuchG2manifoldsandthemethodof(p,q)webs.Wereconsiderandreformulatethe(p,q)equationsusingthetoricgeometryMorivectorsofV2andsetofbranechargeconstraintequations.Inparticular,fortheweightedprojectivespaceWP2w1,w2,w3,we nd

thatthegaugegroupisgivenby(1.6).Insection5,wegiveillustratingapplications.Insection6,wegiveourconclusion.

2Toricgeometry

Inthissection,wecollectafewfactsontoricgeometryofcomplexmanifolds.ThesefactsareneededlatertoconstructaspecialtypeofG2manifolds,asU(1)quotientsofeight-dimensionaltoricHKmanifolds.Roughlyspeaking,toricmanifoldsarecomplexn-dimensionalmanifoldswithTn brationovern-dimensionalbasespaceswithboundary[7,10,31,32,33,34].TheyexhibittoricactionsU(1)nallowingustoencodethegeometricpropertiesofthecomplexspacesintermsofsimplecombinatorialdataofpolytopes noftheRnspace.Inthiscorrespondence, xedpointsofthetoricactionsU(1)nareassociatedwiththeverticesofthepolytope n,theedgesare xedone-dimensionallinesofasubgroupU(1)n 1ofthetoricactionU(1)n,andsoon.Geometrically,thismeansthattheTn berscandegenerateovertheboundaryofthebase.Notethatinthecasewherethebasespaceiscompact,theresultingtoricmanifoldwillbecompactaswell.