Bosonic part of $4d$ $N=1$ supersymmetric gauge theory with general couplings: local existence

Abstract

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 the scalar potential is turned on. Then, for the sake of simplicity, we set all fermions to vanish at the level of equations of motion, 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 motion by taking Kähler potential to be bounded above by $U(n)$ symmetric Kähler potential and the first derivative of gauge couplings to be at most linear growth functions.