Bosonic Part of 4d N=1 Supersymmetric Gauge Theory with General Couplings: Local Existence

TL;DR

This paper proves the local existence of solutions for the bosonic sector of 4d N=1 supersymmetric gauge theories with general couplings, using Segal's theory under specific boundedness and growth conditions.

Contribution

It establishes the local existence of solutions for the bosonic equations of motion in 4d N=1 supersymmetric gauge theories with general couplings, extending previous results.

Findings

Proved local existence of solutions under bounded Kahler potential.

Demonstrated solutions exist with gauge couplings having at most linear growth.

Applied Segal's theory to supersymmetric gauge theories.

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 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.