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