M-theory on G_2 manifolds and the method of (p,q) brane webs(17)

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

gaugesystems.Thelatterhavearemarkablefeaturewhichhasaniceinterpretationusingtherecentderivationoflocalmirrorsymmetryintwo-dimensional eldtheorywithN=2supersymmetry[39].Indeed,intheabovetwoexamples,ficanbewrittenasfollows

f1=w2w3d

f2=w1w3d

f3=w1w2d

wheredisthedegreeofthefollowinghomogeneousLGCalabi-Yausuperpotentials

y2+x4+z4+etxyz=0

y2+x3+z6+etxyz=0(5.6)(5.5)

mirrortotypeIIBN=2sigmamodelontheanti-canonicallinebundlesoverWP21,2,1andWP21,2,3respectively.

6Conclusion

Inthispaper,wehavestudiedN=1supersymmetricgaugetheoriesembeddedinM-theoryonlocalseven-dimensionalmanifoldswithG2holonomygroup.WehaveengineeredtheN=1quivermodelsfromG2manifolds,asU(1)quotientsofeight-dimensionaltoricHKmanifolds.Thecorrespondingquivermodelshavebeenobtainedusingareformulationofthemethodof(p,q)webs.Ourmainresultsmaybesummarizedasfollows:

(i)Usingtwo-dimensionalN=4sigma-modelswithU(1)rgaugesymmetryandr+2hy-permultiplets,wehaveconstructedaspecialkindofG2manifolds.ThelatterareU(1)quo-tientsofeight-dimensionaltoric(HK)manifolds,X7=X8