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

Inthisway,N=4D- atnessconstraintequationsdescribethecotangentbundleoverF0.Afterdividingbyone nitetoricgeometrycircleaction,wegetarealconeonS2bundleoverF0.

3.3OthermodelsfromWP2

Here,westudysomeextendedmodelsusingmoregeneralN=4two-dimensionalgaugetheories.Inparticular,weconsidertwopossiblegeneralizationsforWP2.The rstmodeldescribestheblowingupofWP2atonepoint.IthasasimilarfeatureasF2geometry.ThesecondmodeldealswithmodelwithADECartanmatrixgaugechargesleadingtoADEintersectinggeometries.

3.3.1BlowingupofWP2atonepoint

Forsimplicity,weconsiderWP21,2,1asanexample.ThisspacehasaZ2orbifoldsingularitycorrespondingtonon-trivial xedpointsunderthehomogeneousidenti cation

(z1,z2,z3)≡(λz1,λ2z2,λz3).(3.14)

Takingλ= 1,WP21,2,1hasaZ2orbifoldsingularityat(z1,z2,z3)=(0,1,0).Thissingularitymaybeblownupbyintroducinganexceptionaldivisor.Intwo-dimensionalN=2sigmamodel,thiscanbedeformedbyintroducinganextrachiral eldX4andanU(1)gaugegroupfactor.Inthisway,thecorrespondingeight-dimensionalmanifoldscanbedescribedbyanU(1)2linearsigmamodelwithfourhypermultipletswiththefollowingcharges

Qi=(1, 2,1,0)(1)Qi=(0, 1,0,1).(2)(3.15)

ThismodelgivesthesameG2manifoldcorrespondingtotheF2Hirzebruchsurface.

3.3.2ADEintersectinggeometry

AnothergeneralizationistoconsidertheintersectingweightedprojectivespacesaccordingtoADEDynkindiagramsbyimitatingtheanalysisofN=2sigmamodel.Thisinvolvestwo-dimensionalN=4supersymmetricU(1)rgaugetheorywith(r+2)φαihypermultipletswithADECartanmatricesasmatrixgaugecharges.Forsimplicity,letusconsidertheArLie