Note on the N=2 super Yang-Mills gauge theory in a noncommutative differential geometry

The N = 2 super-Yang-Mills gauge theory is reconstructed in a non-commutative differential geometry (NCG). Our NCG with one-form bases dx(mu) on the Minkowski space M-4 and chi on the discrete space Z(2) is a generalization of the ordinary differential geometry on the continuous manifold. Thus, the generalized gauge field is written as A(x, y) = A(mu)(x, y)dx(mu)+Phi(x, y)chi where y is the argument in Z(2). Phi(x, y) corresponds to the scalar and pseudo-scalar bosons in the N = 2 super Yang-Mills gauge theory whereas it corresponds to the Higgs field in the ordinary spontaneously broken gauge theory. Using the generalized field strength constructed from A(x, y) we can obtain the bosonic Lagrangian of the N = 2 super Yang-Mills gauge theory in the same way as Chamseddine first introduced the supersymmetric Lagrangian of the N = 2 and N = 4 super Yang-Mills gauge theories within the framework of Connes's NCG. The fermionic sector is introduced so as to satisfy the invariance of the total Lagrangian with respect to supersymmetry.