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

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

GNPHE/03-04

hep-th/0303198

arXiv:hep-th/0303198v3 25 Mar 2004M-theoryonG2manifoldsandthemethodof(p,q)branewebsAdilBelhajNationalGroupingofHighEnergyPhysics,GNPHEandLab/UFRHighEnergyPhysics,DepartmentofPhysicsFacultyofSciences,Rabat,MoroccoDepartmentofMathematicsandStatistics,ConcordiaUniversityMontr´eal,Qu´ebec,CanadaH4B1R6February1,2008AbstractUsingareformulationofthemethodof(p,q)webs,westudythefour-dimensionalN=1quivertheoriesfromM-theoryonseven-dimensionalmanifoldswithG2holon-

omy.We rstconstructsuchmanifoldsasU(1)quotientsofeight-dimensionaltorichyper-K¨ahlermanifolds,usingN=4supersymmetricsigmamodels.Weshowthatthesegeometries,ingeneral,aregivenbyrealconesonS2bundlesovercomplextwo-dimensionaltoricvarieties,V2=Cr+2/C r.ThenwediscusstheconnectionbetweenthephysicscontentofM-theoryonsuchG2manifoldsandthemethodof(p,q)webs.Mo-tivatedbyaresultofAcharyaandWitten[hep-th/0109152],wereformulatethemethodof(p,q)websandreconsiderthederivationofthegaugetheoriesusingtoricgeometryMorivectorsofV2andbranechargeconstraints.ForWP2w1,w2,w3,we ndthatthe

gaugegroupisgivenbyG=U(w1n)×U(w2n)×U(w3n).Thisisrequiredbytheanomalycancellationcondition.

KEYWORDS:M-theory,G2manifolds,Toricgeometry.

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

1Introduction

Sincethediscoveryofsuperstringdualities,four-dimensionalsupersymmetricquantum eldtheories(QFT4)havebeenasubjectofgreatinterestinconnectionwithsuperstringcompact-i cationonCalabi-YaumanifoldsandD-branephysics[1,2,3,4].Forexample,embeddingN=2QFT4intypeIIAsuperstringcompacti edonCalabi-Yauthreefolds,withK3 bration,hasfoundaverynicegeometricdescriptionusingtheso-calledgeometricengineeringmethod

[5,6,7,8,9,10].Inthisprogram,thesemodels,whichgiveexactresultsforthemodulispaceofthetypeIIACoulombbranch,arerepresentedbyDynkinquiverdiagramsofLiealgebras

[7,8,9,10].

Quiterecently,aspecialinteresthasbeendevotedtofour-dimensionalgaugemodelspre-servingonlyfoursupercharges[11,12].These eldmodelsadmitaverynicedescriptionintheso-called(p,q)webs[13-24].ThismethodconcernsthestudyofN=1four-dimensionalquivertheoriesarisingontheworld-volumeofD3-branestransversetosingularCalabi-Yau

3threefolds,CYB.ThesubscriptherereferstotypeIIBstringgeometry.Themanifoldsarecomplexconesovercomplextwo-dimensionaltoricvarietiesV2,e.g.delPezzosurfaces.They

3givenbyellipticandC brationsaremirrormanifoldsoflocalCalabi-YauthreefoldsCYA

overthecomplexplane.Underlocalmirrorsymmetry,aD3-braneintypeIIBgeometrybecomesaD6-branewrappingaT3intypeIIAmirrorgeometry.Inthisway,theN=1four-dimensionalquivertheoriescanbeobtainedfromD6-braneswrapping3-cyclesSiinthemirrormanifold.Forinstance,aD6-braneonT3,whosehomologyclassis

[T]=3

i=1

3where{Si,i=1,..., }formabasisofH3(CYA,Z),givesafour-dimensionalN=1super-niSi,(1.1)

symmetricgaugetheorywithgaugegroup

G=

andquivermatrix

Iij=Si·Sj.

Ineqs.(1.1)and(1.2),thevectorniisspeci edbytheanomalycancellationcondition

i=1 i=1U(ni),(1.2)(1.3)Iijni=0.

(1.4)

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

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.

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

Instringtheory,thepowerofthetoricgeometryrepresentationisduetothefollowingpoints:

(1)Thetoricdataofthepolytope nhavesimilarfeaturestotheADEDynkindiagramsleadingtonon-abeliangaugesymmetriesintypeIIsuperstringcompacti cationsonCalabi-Yaumanifolds[7,8,9,10].(2)Thetoric xedloci,whichcorrespondtothevanishingcycles,havebeenknowntobeassociatedwithD-branecharges[32].Thelatterwillbeusedinsection4todiscussthephysicscontentofM-theoryonourproposedmanifoldsofG2holonomy,usingareformulationofthemethodof(p,q)websintypeIIsuperstringonCalabi-Yauthreefolds.Toillustratethemainideaoftoricgeometry,letusdescribethephilosophyofthissubjectthroughcertainusefulexamples.

(i)P1projectivespace.

Thisisthesimplestexampleintoricgeometrywhichturnsouttoplayacrucialroleinthebuildingblocksofhigher-dimensionaltoricvarietiesandinthestudyofthesmallresolutionofADEsingularitiesoflocalCalabi-Yaumanifolds.P1hasanU(1)toricaction

z→eiθz(2.1)

withtwo xedpointsv1andv2ontherealline.Thelatterpoints,whichcanbegenerallychosenasv1= 1andv2=1,describerespectivelynorthandsouthpolesoftherealtwosphereS2～P1.Thecorrespondingone-dimensionalpolytopeisjustthesegment[v1,v2]joiningthetwopointsv1andv2.Thus,P1canbeviewedasasegment[v1,v2]withacircleontop,wherethecirclevanishesattheendpointsv1andv2.

(ii)P2projectivespace.

P2isacomplextwo-dimensionaltoricvarietyde nedby

P=2C3\{(0,0,0)}

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

stableunderthethreeU(1)subgroupsofU(1)2;twosubgroupsarejustthetwoU(1)factors,whilethethirdsubgroupisthediagonalone.P2canbeviewedasatriangleovereachpointofwhichthereisanellipticcurveT2.Thistorusshrinkstoacircleateachsegment[vi,vj]anditshrinkstoapointateachvi.Theabovetoricrealizationcanbepushedfurtherfordescribingthesamephenomenoninvolvingcomplexn-dimensionaltoricvarietiesthataremorecomplicatedthanprojectivespaces.Thelatterspacescanbeexpressedinthefollowingform

V=nCn+r\U

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

whichmeansthatthesystem owsintheinfra-redtoanon-trivialsuperconformaltheory

[35,36].Underlocalmirrorsymmetry,thistoricCalabi-YausigmamodelmapstoLandau-Ginsburg(LG)models[37,38,39,40].Inthisway,themirrorversionoftheconstraintequation(2.9),givingtheLGsuperpotential,reads

iyi=0(2.11)

subjectto

iQayii=e ta,(2.12)

whereyiareLGdualchiral eldswhichcanberelated,upsome eldchanges,tosigmamodel elds,andwhereta’sarethecomplexi edFIparametersde ningnowthecomplexdeforma-tionsoftheLGCalabi-Yausuperpotentials.

Notethattheabovetwo-dimensionalN=2toricsigmamodelscanbeextendedtoN=4sypersymmetrymodelswithhypermultipletsleadingtotoricHKgeometries[41].Intherestofthispaper,wewillusetoricgeometryandHKanalysistostudyseven-dimensionalmanifoldswithG2holonomy.ThelatterareU(1)quotientsofeight-dimensionaltoricHKmanifoldsX8.3

3.1G2manifoldsasU(1)quotientsG2manifoldsandN=4D- atnessconditions

Itisknownthatinordertogetasemi-realisticfour-dimensionaltheoryfromM-theoryitisnecessarytoconsideracompacti cationonaseven-dimensionalmanifoldX7withG2holonomy

[42-51].Inthisway,theresultingmodelswithN=1supersymmetrydependonthegeometricpropertiesofX7.Forinstance,ifX7issmooth,thelow-energytheorycontains,inadditiontoN=1supergravity,onlyabeliangaugesymmetryandnochargedchiralfermions.Non-abelaingaugesymmetriescanbeobtainedbyconsideringlimitswhereX7developsADEorbifoldsingularitiesusingwrappedM2-branesonvanishing2-cycles[43].However,thepresenceofconicalsingularitiesleadstochargedchiralfermions.Following[25],aninterestinganalysisforbuildingsuchgeometriesistoconsiderquotientsofeight-dimensionaltoricHKmanifoldsX8byanU(1)circlesymmetry.TheU(1)grouphasbeenchosensuchthatitcommuteswiththeSU(2)symmetry,permutingthethreecomplexstructuresofHKgeometries.Apriori,therearemanywaystochoosetheU(1)groupaction.Twosituationshavebeengivenin

[25]butherewewillidentifytheU(1)groupwiththetoricgeometrycircleactionofcomplex

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

subvarietieswithinHKgeometries.Inparticular,wewillusetheHKanalysistopresentexplicitmodelswithG2holonomygroupleadingtointerestingN=1supersymmetricgaugetheoriesinfourdimensions.Todoso,weconsidertwo-dimensionalN=4supersymmetricgaugetheorieswithU(1)rgaugesymmetriesandr+2hypermultipletswithaQaimatrixcharge

[36,41].TheN=4D- atnessequationsofsuchmodelsaregenerallygivenby

r+2

i=1α¯¯α σα,Qai[φiφiβ+φiβφi]=ξa βa=1,...,r.(3.1)

Intheseequations,φαi’sdenoter+2component elddoubletsofhypermultiplets,ξaarerFI3-

αvectorcouplingsrotatedbySU(2)symmetry,and σβarethetraceless2×2Paulimatrices.In

thisconstruction,foreachU(1)factor,therearethreerealconstraintequationstransformingasaniso-tripletofSU(2)R-symmetry(SU(2)R)actingontheHKstructures.

UsingtheSU(2)Rtransformations

φα=εαβφβ,φα,ε12=ε21=1,(3.2)

andreplacingthePaulimatricesbytheirexpressions,theidentities(3.1)canbesplitasfollows

k

i=1

k

i=1

1φi=ξa iξ2a.11Qaiφi12223Qai(|φi| |φi|)=ξa(3.3)(3.5)

Dividingtheresultingspaceof(3.3-5)byU(1)rgaugetransformations,we ndpreciselyaneight-dimensionaltoricHKmanifoldX8.However,explicitsolutionsofthesegeometries

123dependonthevaluesoftheFIcouplings.Takingξa=ξa=0andξa>0,(3.3-5)describethe

cotangentbundleovercomplextwo-dimensionaltoricvarieties[30].Indeed,ifwesetallφ2i=0,wegetacomplextwo-dimensionaltoricvarietyVde nedby

Equations(3.4-5)meanthattheφ2i’s22+r i=1123Qai|φi|=ξa,(a=1,...,r).

de nethecotangentorthogonal berdirectionsover

2V2.Thismanifoldhasfourtoricgeometrycircleactions:U(1)2base×U(1)fiber.Twoofthem

2correspondtotheV2toricbasespacedenotedbyU(1)2base,whiletheremainingones,U(1)fiber,

actonthe berorthogonalcotangentdirections.Togetthecorrespondingseven-dimensionalmanifoldswithG2holonomy,wewillidentifytheU(1)groupsymmetryofthequotientusedin[25]withone nitecircletoricaction.IdentifyingthisU(1)symmetrywithoneU(1)fiber,one ndsthefollowingseven-dimensionalmanifold

X7=X8/U(1)fiber.

(3.6)

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)

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

TheseequationsdescribeacotangentbundleoverWP21,2,1.Indeed,takingψ1=ψ2=ψ3=0,eq.(3.7)reducesto| 1|2+| 3|2+2| 2|2=ξ3andde nesaWP21,2,1weightedprojectivespace,whereξ3isaK¨ahlerrealparametercontrollingitssize.Eqs.(3.7-9),forgenericvaluesofψi,canbeinterpretedtomeanthatψiparameterizetheorthogonal berdirectionsonWP21,2,1.Dividingbyone nitetoricgeometry bercircleaction,we ndarealconeonanS2bundleoverWP21,2,1withG2holonomy.

Example2:(m1,m2)=(1,2).Asanotherexample,weconsideravectorchargeasfol-lowsQi=(1, 3,2).Thisexampleisquitesimilartothe rstone,anditstreatmentwillbeparalleltothe rstone.Aftermakingsimilar eldchanges,thisexampledescribesWP21,3,2inthebasegeometryofaneight-dimensionalmanifold.AftertheU(1)quotient,thecorre-spondingseven-dimensionalmanifoldX7willbearealconeonS2bundleoverWP21,3,2.Wewillseelaterthatthisgeometryleadstoafour-dimensionalmodelwhichmightberelatedtothegranduni edsymmetry.

3.2.2V2asHirzebruchsurfacesFn

Fnarecomplextwo-dimensionaltoricsurfacesde nedbynon-trivial brationsofaP1overaP1.Thesemaybeviewedasthecompacti cationofcomplexlinebundlesoverP1byaddingapointtoeach beratin nity.Suchlinebundlesareclassi edbyanintegern,beingthe rstChernclassintegratedoverP1.Forsimplicity,wewillrestrictourselvestoF0withatrivial bration.AwaytowritedowntheF0N=4sigmamodelistostartwithoneP1andthenextendtheresulttoF0.Indeed,oneP1correspondstoanU(1)two-dimensionalN=4linearsigmamodelwithtwohypermultipletswithavectorcharge(1, 1).Makingasimilaranalysisofpreviousexamples,theD- atnessconditions(3.1)reduceto

(| 1|2+| 2|2) (|ψ1|2+|ψ2|2)=ξ3

1

ψ1ψ2=0 2=0.(3.10)(3.11)(3.12)

anddescribethecotangentbundleoveraP1,de nedby| 1|2+| 2|2=ξ3.ThemodelcorrespondingtoF0isobtainedbyconsideringanU(1)2two-dimensionalN=4linearsigmamodelwithfourhypermultipletswiththefollowingcharges

Qi=(1, 1,0,0),

(1)Qi=(0,0,1, 1).(2)(3.13)

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

Inthisway,N=4D- atnessconstraintequationsdescribethecotangentbundleoverF0.Afterdividingbyone nitetoricgeometrycircleaction,wegetarealconeonS2bundleoverF0.

3.3OthermodelsfromWP2

Here,westudysomeextendedmodelsusingmoregeneralN=4two-dimensionalgaugetheories.Inparticular,weconsidertwopossiblegeneralizationsforWP2.The rstmodeldescribestheblowingupofWP2atonepoint.IthasasimilarfeatureasF2geometry.ThesecondmodeldealswithmodelwithADECartanmatrixgaugechargesleadingtoADEintersectinggeometries.

3.3.1BlowingupofWP2atonepoint

Forsimplicity,weconsiderWP21,2,1asanexample.ThisspacehasaZ2orbifoldsingularitycorrespondingtonon-trivial xedpointsunderthehomogeneousidenti cation

(z1,z2,z3)≡(λz1,λ2z2,λz3).(3.14)

Takingλ= 1,WP21,2,1hasaZ2orbifoldsingularityat(z1,z2,z3)=(0,1,0).Thissingularitymaybeblownupbyintroducinganexceptionaldivisor.Intwo-dimensionalN=2sigmamodel,thiscanbedeformedbyintroducinganextrachiral eldX4andanU(1)gaugegroupfactor.Inthisway,thecorrespondingeight-dimensionalmanifoldscanbedescribedbyanU(1)2linearsigmamodelwithfourhypermultipletswiththefollowingcharges

Qi=(1, 2,1,0)(1)Qi=(0, 1,0,1).(2)(3.15)

ThismodelgivesthesameG2manifoldcorrespondingtotheF2Hirzebruchsurface.

3.3.2ADEintersectinggeometry

AnothergeneralizationistoconsidertheintersectingweightedprojectivespacesaccordingtoADEDynkindiagramsbyimitatingtheanalysisofN=2sigmamodel.Thisinvolvestwo-dimensionalN=4supersymmetricU(1)rgaugetheorywith(r+2)φαihypermultipletswithADECartanmatricesasmatrixgaugecharges.Forsimplicity,letusconsidertheArLie

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

aaaalgebrawherethematrixchargeisgivenbyQai= 2δi+δi 1+δi+1,a=1,...,r.Puttingthese

equationsintotheD- atnessequations(3.1),onegetsthefollowingsystemof3requations

21212222222(|φ1a 1|+|φa+1| 2|φa|) (|φa 1|+|φa+1| 2|φa|)=ξa(3.16)

(3.18)φ1a 1φ1a 1+φ2a+1φ2a+1 2φ1aφ1a=0.

AnexaminationoftheseequationsrevealsthatV2consistsofrintersectingWP21,2,1accordingtotheArDynkindiagram[30].Actually,thisgeometrygeneralizestheusualADEgeometrycorrespondingtotwo-cyclesofK3surfaces[7,8,9,10].Oneexpectstohaveasimilarfeatureinthecompacti cationofM-theoryonG2manifoldswithintersectingWP21,2,1’s.

ThepreviousanalysisisalsopossibleformodelswithdelPezzosurfacesasabasegeometryofG2manifolds.Notethat,thesesurfaceshavebeenusedinthebuildingofN=1supersym-metricgaugetheoriesinfourdimensionsusingtheso-called(p,q)webs.Thesegaugetheories

3ariseontheworld-volumeofD3-branestransversetolocalCalabi-YauthreefoldsCYBgiven

bycomplexconesoverdelPezzosurfaces[14,22].Inthispresentwork,wewillshowthatthisphysicsisrelatedtoofM-theoryonG2manifoldswithtwocomplexdimensiontoricmanifoldsinthebasegeometry.

4OnM-theoryonG2Manifoldsand(p,q)webs

Sofar,wehaveconstructedaspecialtypeofG2manifoldsasU(1)quotientsofeight-dimensionaltoricHKmanifolds.ThissectionwillbeconcernedwithM-theoryonsuchmanifolds.Wewilltryto ndasuperstringinterpretationofthisusingD-branephysics.Inparticular,wewilldiscusstheconnectionbetweenthephysicscontentofM-theoryonsuchG2manifoldsandthemethodof(p,q)webs,leadingtoN=1supersymmetricgaugetheoriesinfourdimensions.Theanalysiswewillbeusinghereisbasedonareconsiderationofthemethodof(p,q)websandreformulatingtheintersectionnumberstructuresintermsoftoricgeometrydataofV2varieties.

Beforeproceeding,letusrecallquotesomecrucialpointssupportingourdiscussion.Ononehand,accordingto[25],M-theoryonG2manifoldsasU(1)quotientofeight-dimensionalconicaltoricHKmanifolds,hasaninterpretationintermsofintersectingD6-branesintypeIIAsuperstringmodel.Inparticular,thegeometryofWP2describestheintersectionofthreesetsofD6-branes.ForexamplethegeometryWP2m,m,nwithm,mandnrelativelyprime

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.

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

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

(2)Forwi=1,werecoverthesimplemodelwithgaugegroupU(n)3andmattertriplicationineachbifundamental[20,21].

(3)Forn=1,thecorrespondinggaugetheoryisnowquitesimilartotheinterpretationofM-theoryonG2manifoldsgivenin[25].Inthisway,thegaugegroupreads

G=3

i=1U(wi).(4.8)

Intheinfra-redlimittheU(1)factorsdecoupleandoneisleftwiththegaugesymmetry

G=3

i=1SU(wi).(4.9)

Takinganappropriatechoiceofweights,werecoverthephysicalmodelgivenin[25].

(4)Thecorresponding eldmodelsisrepresentedbyatrianglequiverdiagram

¡¡¡¡¡¡¡¡¡¡¡¡

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

5.1U(n)2×U(2n)gaugetheory

Consider, rst,thegeometryofWP21,2,1inM-theorycompacti cations.In(p,q)webs,thisisequivalenttotakingthreestacksofbraneseach,wrappingthefollowing1cycles

C1=( 2,0),C2=(0, 1),C3=(2,4).(5.1)

Inthiscase,theintersectionnumbersreadas

I12=4

I31=8

I23=4.

ForoneD6-brane,thisexampleleadstoaN=1spectrumwithgaugegroupU(1)2×U(2)gaugegroupandbifundamentalmatter.ThismodelagreeswiththeresultofAcharyaandWittengivenin[25].WhilefornD6-branes,theabovechargecon gurationsgivesaN=1spectrumwithgaugegroupU(n)2×U(2n)gaugesymmetryandbifundamentalmatter.(5.2)

5.2U(n)×U(2n)×U(3n)gaugemodel

ThegeometryofWP21,3,2isveryexcitinginthisanalysisbecauseitmayleadtothesymmetryofthegranduni edtheory(GUT)2.Forthisexample,weconsiderthreestacksofnD6-braneseach,wrappingthefollowing1cycles

C1=(4,9),C2=( 1,0),C3=(0, 1).(5.3)

Inthiscase,theintersectionnumbersreadas

I12=18

I31=12

I23=6

ThisyieldsaN=1spectrumwithgaugegroupU(n)×U(2n)×U(3n)gaugegroupandbifundamentalmatter.Forn=1,onegetsU(1)×U(2)×U(3)asgaugesymmetry.

Concludingthissection,itisinterestingtomakeacommentregardingthenumbersappear-ingin(5.2)and(5.4),countingthenumberofN=1chiralmultipletsfiinthecorresponding(5.4)

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

gaugesystems.Thelatterhavearemarkablefeaturewhichhasaniceinterpretationusingtherecentderivationoflocalmirrorsymmetryintwo-dimensional eldtheorywithN=2supersymmetry[39].Indeed,intheabovetwoexamples,ficanbewrittenasfollows

f1=w2w3d

f2=w1w3d

f3=w1w2d

wheredisthedegreeofthefollowinghomogeneousLGCalabi-Yausuperpotentials

y2+x4+z4+etxyz=0

y2+x3+z6+etxyz=0(5.6)(5.5)

mirrortotypeIIBN=2sigmamodelontheanti-canonicallinebundlesoverWP21,2,1andWP21,2,3respectively.

6Conclusion

Inthispaper,wehavestudiedN=1supersymmetricgaugetheoriesembeddedinM-theoryonlocalseven-dimensionalmanifoldswithG2holonomygroup.WehaveengineeredtheN=1quivermodelsfromG2manifolds,asU(1)quotientsofeight-dimensionaltoricHKmanifolds.Thecorrespondingquivermodelshavebeenobtainedusingareformulationofthemethodof(p,q)webs.Ourmainresultsmaybesummarizedasfollows:

(i)Usingtwo-dimensionalN=4sigma-modelswithU(1)rgaugesymmetryandr+2hy-permultiplets,wehaveconstructedaspecialkindofG2manifolds.ThelatterareU(1)quo-tientsofeight-dimensionaltoric(HK)manifolds,X7=X8

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

Acknowledgments

A.BelhajwouldliketothankDepartmentofMathematicsandStatistics,ConcordiaUniver-sity,Montreal,foritskindhospitalityduringthepreparationofthiswork.HeisverygratefultoJ.McKayfortheinvitation,discussionsandencouragement.HewouldliketothankJ.Rasmussenforcommentsonthemanuscript,andH.Kisilevsky,E.H.SaidiandA.Sebbarfordiscussionsandscienti chelp.

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

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