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.
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Received: 19 March 1997
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Okumura, Y. Note on the \(N=2\) super Yang-Mills gauge theory in a noncommutative differential geometry. Eur. Phys. J. C 1, 735–738 (1998). https://doi.org/10.1007/s100520050119
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DOI: https://doi.org/10.1007/s100520050119