GAUGE-THEORY ON DISCRETE SPACE M4XZN AND SU(5) GUT IN NONCOMMUTATIVE GEOMETRY

The previous differential calculus on discrete space M4 x Z2 which is an underlying space-time in the non-commutative geometry for the standard model is reformulated by introducing two kinds of extra one-form basis. It is shown that the result is easily generalized to discrete space M4 X Z(N), where N signifies number of sheets or copies of M4. It is also shown that, for N = 3, SU(5) GUT with two energy scales of symmetry breaking fits naturally into the scheme, where the Higgs sector possesses the same structure as obtained by Chamseddine et al., although no spinor quantities appear in our formalism in the bosonic sector. One consequence from not using the Dirac operator at the outset is that coupling parameters in the Higgs potential no longer depend on fermion mass matrix in generation space, in particular, Higgs potentials survive even for one generation. The present formalism is proved to be equivalent to matrix formulation when N=2,3.