# The exact solutions to two-dimensional Yang-Mills theory: $su(2)$ and $u_\star(1)$

## TL;DR

This paper demonstrates the existence of exact solutions to two-dimensional Yang-Mills theories, explores their stability, and investigates source-free solutions in noncommutative two-dimensional space-time, revealing similarities between nonabelian and $u_igstar(1)$ gauge theories.

## Contribution

It provides explicit exact solutions for 2D Yang-Mills theories with $su(2)$ and $u_igstar(1)$, and analyzes their stability and properties in noncommutative space-time.

## Key findings

- Exact solutions exist for 2D Yang-Mills theories.
- Stationary solutions exhibit instability depending on space-time dimensions.
- Similarity found between bound states in nonabelian and $u_igstar(1)$ gauge theories.

## Abstract

The purpose of this study is to show that the exact solutions to Yang-Mills theory can occur in two-dimensional space-time. We show that the instability of the stationary solutions for the nonabelian gauge field theories questioned by the dimensions of space-time. We show that instability of the stationary solution for the nonabelian gauge field theory questioned by the dimension of space-time. Another intention of this work is research of the source-free solutions to Maxwell's equations in a two-dimensional noncommutative temporal state space-time. We will show that there is comprehensively similarity between the existence of the bound states of nonabelian gauge theory and $u_\star(1)$.