Nontrivial Solutions of Seiberg–Witten Equations on the Noncommutative 4-Dimensional Euclidean Space

Noncommutative Seiberg–Witten equations on the noncommutative Euclidean space $\mathbb R^4_\theta$ are studied that are obtained from the standard Seiberg–Witten equations on $\mathbb R^4$ by replacing the usual product with the deformed Moyal $\star$-product. Nontrivial solutions of these noncommutative Seiberg–Witten equations are constructed that do not reduce to solutions of the standard Seiberg–Witten equations on $\mathbb R^4$ for $\theta \to 0$. Such solutions of the noncommutative equations on $\mathbb R^4_\theta$ exist even when the corresponding commutative Seiberg–Witten equations on $\mathbb R^4$ do not have any nontrivial solutions.