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

Yoshitaka Okumura

doi:doi:10.1007/s100520050119

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2018 Impact factor 4.843

Particles and Fields

Eur. Phys. J. C 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 $dx^\mu$ on the Minkowski space M₄ and $\chi$ 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 ${\cal A}(x,y)=A_\mu(x,y)dx^\mu+{\mit\Phi}(x,y)\chi$ where y is the argument in Z₂. ${\mit\Phi}(x,y)$ 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.