Large Lepton Mixing in A String Brane World

It is very interesting if the observed disparity between mixing angles for neutrinos and for quarks has a geometrical origin in an extra dimensional space. We construct such an example in the type IIB string theory, where wave functions of left-handed leptons spread in an effective two dimensional extra space while wave finctions of left-handed quarks are localized at separate fixed points. This leads to the anarchy hypothesis for the left-handed leptons which explains naturally the large mixing angles for neutrinos. We emphasize, however, that this is only a projection of the six dimensional extra space onto a specific two dimensions. We have, in fact, another projected two dimensional extra space where the wave functions of left-handed quarks spread there and those of left-handed leptons are localized at fixed points.