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)