Gauge theory and the Higgs mechanism based on differential geometry in the discrete space M4xZN

Weinberg-Salam theory and SU(5) grand unified theory (GUT) are reconstructed using the generalized differential calculus extended on the discrete space M4xZN. Our starting point is the generalized gauge field expressed by A(x,n)=at sign mJi aidegree(x,n)scrdai(x,n), (n=1,2,...,N), where ai(x,n) is the square matrix valued function defined on M4xZN and scrd=d+at sign mJm=1Ndχm is a generalized exterior derivative. We can construct the consistent algebra of dχm which is an exterior derivative with respect to ZN and the spontaneous breakdown of gauge symmetry is coded in dχm. The unified picture of the gauge field and Higgs field as the generalized connection in noncommutative geometry is realized. Not only the Yang-Mills-Higgs Lagrangian but also the Dirac Lagrangian, invariant against the gauge transformation, is reproduced through the inner product between the differential forms. Three sheets (Z3) are necessary for Weinberg-Salam theory including strong interaction and the SU(5) GUT. Our formalism is applicable to a more realistic model such as the SO(10) unification model