M-theory on G_2 manifolds and the method of (p,q) brane webs

TheseequationsdescribeacotangentbundleoverWP21,2,1.Indeed,takingψ1=ψ2=ψ3=0,eq.(3.7)reducesto| 1|2+| 3|2+2| 2|2=ξ3andde nesaWP21,2,1weightedprojectivespace,whereξ3isaK¨ahlerrealparametercontrollingitssize.Eqs.(3.7-9),forgenericvaluesofψi,canbeinterpretedtomeanthatψiparameterizetheorthogonal berdirectionsonWP21,2,1.Dividingbyone nitetoricgeometry bercircleaction,we ndarealconeonanS2bundleoverWP21,2,1withG2holonomy.

Example2:(m1,m2)=(1,2).Asanotherexample,weconsideravectorchargeasfol-lowsQi=(1, 3,2).Thisexampleisquitesimilartothe rstone,anditstreatmentwillbeparalleltothe rstone.Aftermakingsimilar eldchanges,thisexampledescribesWP21,3,2inthebasegeometryofaneight-dimensionalmanifold.AftertheU(1)quotient,thecorre-spondingseven-dimensionalmanifoldX7willbearealconeonS2bundleoverWP21,3,2.Wewillseelaterthatthisgeometryleadstoafour-dimensionalmodelwhichmightberelatedtothegranduni edsymmetry.

3.2.2V2asHirzebruchsurfacesFn

Fnarecomplextwo-dimensionaltoricsurfacesde nedbynon-trivial brationsofaP1overaP1.Thesemaybeviewedasthecompacti cationofcomplexlinebundlesoverP1byaddingapointtoeach beratin nity.Suchlinebundlesareclassi edbyanintegern,beingthe rstChernclassintegratedoverP1.Forsimplicity,wewillrestrictourselvestoF0withatrivial bration.AwaytowritedowntheF0N=4sigmamodelistostartwithoneP1andthenextendtheresulttoF0.Indeed,oneP1correspondstoanU(1)two-dimensionalN=4linearsigmamodelwithtwohypermultipletswithavectorcharge(1, 1).Makingasimilaranalysisofpreviousexamples,theD- atnessconditions(3.1)reduceto

(| 1|2+| 2|2) (|ψ1|2+|ψ2|2)=ξ3

1

ψ1ψ2=0 2=0.(3.10)(3.11)(3.12)

anddescribethecotangentbundleoveraP1,de nedby| 1|2+| 2|2=ξ3.ThemodelcorrespondingtoF0isobtainedbyconsideringanU(1)2two-dimensionalN=4linearsigmamodelwithfourhypermultipletswiththefollowingcharges

Qi=(1, 1,0,0),

(1)Qi=(0,0,1, 1).(2)(3.13)