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

whichmeansthatthesystem owsintheinfra-redtoanon-trivialsuperconformaltheory

[35,36].Underlocalmirrorsymmetry,thistoricCalabi-YausigmamodelmapstoLandau-Ginsburg(LG)models[37,38,39,40].Inthisway,themirrorversionoftheconstraintequation(2.9),givingtheLGsuperpotential,reads

iyi=0(2.11)

subjectto

iQayii=e ta,(2.12)

whereyiareLGdualchiral eldswhichcanberelated,upsome eldchanges,tosigmamodel elds,andwhereta’sarethecomplexi edFIparametersde ningnowthecomplexdeforma-tionsoftheLGCalabi-Yausuperpotentials.

Notethattheabovetwo-dimensionalN=2toricsigmamodelscanbeextendedtoN=4sypersymmetrymodelswithhypermultipletsleadingtotoricHKgeometries[41].Intherestofthispaper,wewillusetoricgeometryandHKanalysistostudyseven-dimensionalmanifoldswithG2holonomy.ThelatterareU(1)quotientsofeight-dimensionaltoricHKmanifoldsX8.3

3.1G2manifoldsasU(1)quotientsG2manifoldsandN=4D- atnessconditions

Itisknownthatinordertogetasemi-realisticfour-dimensionaltheoryfromM-theoryitisnecessarytoconsideracompacti cationonaseven-dimensionalmanifoldX7withG2holonomy

[42-51].Inthisway,theresultingmodelswithN=1supersymmetrydependonthegeometricpropertiesofX7.Forinstance,ifX7issmooth,thelow-energytheorycontains,inadditiontoN=1supergravity,onlyabeliangaugesymmetryandnochargedchiralfermions.Non-abelaingaugesymmetriescanbeobtainedbyconsideringlimitswhereX7developsADEorbifoldsingularitiesusingwrappedM2-branesonvanishing2-cycles[43].However,thepresenceofconicalsingularitiesleadstochargedchiralfermions.Following[25],aninterestinganalysisforbuildingsuchgeometriesistoconsiderquotientsofeight-dimensionaltoricHKmanifoldsX8byanU(1)circlesymmetry.TheU(1)grouphasbeenchosensuchthatitcommuteswiththeSU(2)symmetry,permutingthethreecomplexstructuresofHKgeometries.Apriori,therearemanywaystochoosetheU(1)groupaction.Twosituationshavebeengivenin

[25]butherewewillidentifytheU(1)groupwiththetoricgeometrycircleactionofcomplex