M-theory on G_2 manifolds and the method of (p,q) brane webs(6)

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

stableunderthethreeU(1)subgroupsofU(1)2;twosubgroupsarejustthetwoU(1)factors,whilethethirdsubgroupisthediagonalone.P2canbeviewedasatriangleovereachpointofwhichthereisanellipticcurveT2.Thistorusshrinkstoacircleateachsegment[vi,vj]anditshrinkstoapointateachvi.Theabovetoricrealizationcanbepushedfurtherfordescribingthesamephenomenoninvolvingcomplexn-dimensionaltoricvarietiesthataremorecomplicatedthanprojectivespaces.Thelatterspacescanbeexpressedinthefollowingform

V=nCn+r\U