Note on the N=2 super Yang-Mills gauge theory
in a noncommutative differential geometry
Department of Natural Science, Chubu University, Kasugai, 487, Japan (e-mail: email@example.com)
Received: 19 March 1997
The N=2 super-Yang-Mills gauge theory is reconstructed in a non-commutative differential geometry (NCG). Our NCG with one-form bases on the Minkowski space M4 and on the discrete space Z2 is a generalization of the ordinary differential geometry on the continuous manifold. Thus, the generalized gauge field is written as where y is the argument in Z2. 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 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.