$\text{Spin}^c$-structures and Seiberg–Witten equations

The Seiberg–Witten equations, found at the end of the $20$th century, are one of the main discoveries in the topology and geometry of four-dimensional Riemannian manifolds. They are defined in terms of a $\text{Spin}^c$–structure that exists on any four-dimensional Riemannian manifold. Like the Yang–Mills equations, the Seiberg–Witten equations are the limit case of a more general supersymmetric Yang–Mills equations. However, unlike the conformally invariant Yang–Mills equations, the Seiberg–Witten equations are not scale invariant. Therefore, in order to obtain “useful information” from them, one must introduce a scale parameter $\lambda$ and pass to the limit as $\lambda\to\infty$. This is precisely the adiabatic limit studied in this paper.