Symmetry breaking patterns of E8×E′8 on Z3 orbifold are completely analyzed for physically acceptable models by using a kind of Weyl transformation and modular invariance. It is shown that SU(3)×SU(2)×U(1)n cannot be obtained by using any kind of a shift vector and Wilson lines and that the minimal group including SU(3)×SU(2)×U(1) is SU(3)2×U(1)n which is realized by a shift vector and Wilson lines. All models accessible to realistic low energy theories are listed out and an example is described to get SU(3)×SU(2)×UY(1) by using Higgs mechanism with anomalous U(1) channel. The example is able to give massless quark, lepton and Higgs to suppress proton decay by Planck mass scale and to make left-handed neutrino massless.

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Two-dimensional random surfaces are studied numerically by the dynamical triangulation method. In order to generate various kinds of random surfaces, two higher derivative terms are added to the action. The phases of surfaces in the twodimensional parameter space are classified into three states: flat, crumpled surface, and branched polymer. In addition, there exists a special point (pure gravity) corresponding to the universal fractal surface. A new probe to detect branched polymers is proposed, which makes use of the minbu(minimum neck baby universe) analysis. This method can clearly distinguish the branched polymer phase from another according to the sizes and arrangements of baby universes. The size distribution of baby universes changes drastically at the transition point between the branched polymer and other kind of surface. The phases of surfaces coupled with multi-Ising spins are studied in a similar manner. †

Four-dimensional (4D) simplicial quantum gravity coupled to U(1) gauge fields has been studied using MonteCarlo simulations. A negative string susceptibility exponent is observed beyond the phase-transition point, even if the number of vector fields (NV ) is 1. We find a scaling relation of the boundary volume distributions in this new phase. This scaling relation suggests a fractal structure similar to that of 2D quantum gravity. Furthermore, evidence of a branched polymer-like structure is suggested far into the weak-coupling region, even for NV > 1. As a result, we propose new phase structures and discuss the possibility of taking the continuum limit in a certain region between the crumpled and branched polymer phases.