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)}