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

Abstract

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 ${\cal A}(x,y)=A_\mu(x,y)dx^\mu+\Phi(x,y)\chi$ 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 ${\cal A}(x,y)$ 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.