Left-Right Symmetric Gauge Theory in Non-Commutative Geometry on M4 xZN

The left-right symmetric gauge model (LRSM) is reconstructed using the previously proposed formalism based on the non-commutative differential geometry extended on the discrete space M4 ×ZN. This formalism is so flexible and applicable that not only the standard model but also the SU(5) grand unified model have already been reformulated in this formalism, which presents many attractive points such as the unified picture of the gauge field and Higgs field as the generalized connection in noncommutative geometry. LRSM is still alive as a model with the intermediate symmetry of the spontaneously broken SO(10) grand unified theory (GUT). Six sheets are prepared for LRSM (N = 6), one is for SU(3)c color symmetry and the rest of five are for SU(2)L ×SU(2)R ×U(1) symmetry. We can achieve the reformulation of LRSM with the quite different configurations of Higgs particles from the ordinary one. Namely, the left-right symmetric gauge groups are broken owing to two (2, 1) and two (1, 2) doublet Higgs fields with hypercharge 1, one (2, 2*) Higgs field, and one (1, 3) Higgs field with hypercharge −2. The fermion sectors are nicely incorporated so that the seesaw mechanism works well to make the right-handed neutrino super heavy and the left-handed neutrino super light.