# The Continuity of the Gauge Fixing Condition $n\cdot\partial n\cdot A=0$ for $SU(2)$ Gauge Theory

## TL;DR

This paper proves that the gauge fixing condition $n ext{·}
abla n ext{·}A=0$ remains continuous for $SU(2)$ gauge theory on a compact manifold, assuming differentiable gauge potentials with continuous derivatives.

## Contribution

It establishes the continuity of the gauge fixing condition on a compact manifold under differentiability assumptions, contributing to the mathematical understanding of gauge fixing in $SU(2)$ theories.

## Key findings

- Gauge fixing condition is continuous on the manifold.
- Continuity holds if gauge potentials are differentiable with continuous derivatives.
- The proof applies to the compact manifold $R\otimes S^1\otimes S^1\otimes S^1$.

## Abstract

The continuity of the gauge fixing condition $n\cdot\partial n\cdot A=0$ for $SU(2)$ gauge theory on the manifold $R\bigotimes S^{1}\bigotimes S^{1}\bigotimes S^{1}$ is studied here, where $n^{\mu}$ stands for directional vector along $x_{i}$-axis($i=1,2,3$). It is proved that the gauge fixing condition is continuous given that gauge potentials are differentiable with continuous derivatives on the manifold $R\bigotimes S^{1}\bigotimes S^{1}\bigotimes S^{1}$ which is compact.