  In this paper, we prove the local existence of the bosonic part of N=1
supersymmetric gauge theory in four dimensions with general couplings. We start
with the Lagrangian of the vector and chiral multiplets with general couplings
and scalar potential turned on. Then, for the sake of simplicity, we set all
fermions vanish at the level of equations of motions, so we only have the
bosonic parts of the theory. We apply Segal's general theory to show the local
existence of solutions of equations of motions by taking Kahler potential to be
bounded above by U(n) symmetric Kahler potential and the first derivative of
gauge couplings to be at most linear growth functions.
