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)