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 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.
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