**1**, 735-738

## Note on the *N*=2 super Yang-Mills gauge theory

in a noncommutative differential geometry

**Yoshitaka Okumura**

Department of Natural Science, Chubu University, Kasugai, 487, Japan (e-mail: okum@isc.chubu.ac.jp)

Received: 19 March 1997

**Abstract**

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 *M _{4}* and
on the discrete space

*Z*is a generalization of the ordinary differential geometry on the continuous manifold. Thus, the generalized gauge field is written as where

_{2}*y*is the argument in

*Z*. corresponds to the scalar and pseudo-scalar bosons in the

_{2}*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.

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