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

In the study of Riemann surfaces the key role is played by the complex structure compatible with Riemannian metric and Cauchy–Riemann operator related to this structure. However, in the case of 4-dimensional Riemannian manifolds the subclass of the manifolds having the complex structures is comparatively narrow and it is hard to understand general properties of Riemannian 4-manifolds investigating only this subclass. So in the study of such manifolds two natural questions arise: the first one — what can replace the complex structure on 4-dimensional Riemannian manifolds, and the second one — which linear differential operator should play the role of $\bar\partial$-operator in the 4-dimensional case.
To answer the first question it is proposed to replace the complex structure by the ${Spin}^c$-structure existing on any 4-dimensional Riemannian manifold. To answer the second question we replace the $\bar\partial$-operator on the 4-dimensional Riemannian manifold by the Dirac operator associated with the given ${Spin}^c$-structure. Having the ${Spin}^c$-structure one can introduce the Seiberg–Witten action functional. The local minima of this functional satisfy the Seiberg–Witten equations being the main subject of our talk.
These equations, found at the end of XXth century, are one of the principal discoveries in topology and geometry of 4-dimensional Riemannian manifolds. As the Yang–Mills equations they are the limiting case of more general supersymmetric Yang–Mills equations. But in contrast with conformally invariant Yang–Mills equations the Seiberg–Witten equations are not invariant under the change of scale. So in order to derive a “useful information” from them it is necessary to introduce the scale parameter $\lambda$ and take the limit for $\lambda\to\infty$. This limit is called adiabatic and is another main subject of our talk.