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)