Gauge theory and Higgs mechanism based on differential geometry on discrete space M4 * ZN

Weinberg-Salam theory and $SU(5)$ grand unified theory are reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_{\mathop{}_{N}}$. Our starting point is the generalized gauge field expressed by $A(x,n)=\!\sum_{i}a^\dagger_{i}(x,n){\bf d}a_i(x,n), (n=1,2,\cdots N)$, where $a_i(x,n)$ is the square matrix valued function defined on $M_4\times Z_{\mathop{}_{N}}$ and ${\bf d}=d+\sum_{m=1}^{\mathop{}_{N}}d_{\chi_m}$ is generalized exterior derivative. We can construct the consistent algebra of $d_{\chi_m}$ which is exterior derivative with respect to $Z_{\mathop{}_{N}}$ and the spontaneous breakdown of gauge symmetry is coded in ${d_{\chi_m}}$. The unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry is realized. Not only Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the gauge transformation, are reproduced through the inner product between the differential forms. Three sheets ($Z_3)$ are necessary for Weinberg-Salam theory including strong interaction and $SU(5)$ Gut. Our formalism is applicable to more realistic model like $SO(10)$ unification model.