Solving N=2 supersymmetric Yang-Mills theory by reflection symmetry of quantum vacua

The recently rigorously proved nonperturbative relation $u=\ensuremath{\pi}i(\mathcal{F}\ensuremath{-}a{\ensuremath{\partial}}_{a}\mathcal{F}/2)$, underlying $N=2$ supersymmetry Yang-Mills theory with the gauge group SU(2), implies both the reflection symmetries $\overline{u(\ensuremath{\tau})}=u(\ensuremath{-}\overline{\ensuremath{\tau}})$ and $u(\ensuremath{\tau}+1)=\ensuremath{-}u(\ensuremath{\tau})$ which hold exactly. The relation also implies that $\ensuremath{\tau}$ is the inverse of the uniformizing coordinate $u$ of the moduli space of quantum vacua ${\mathcal{M}}_{\mathrm{SU}(2)},$ that is, $\ensuremath{\tau}:{\mathcal{M}}_{\mathrm{SU}(2)}\ensuremath{\rightarrow}H$, where $H$ is the upper half plane. In this context, the above quantum symmetries are the key points to determine ${\mathcal{M}}_{\mathrm{SU}(2)}.$ It turns out that the functions $a(u)$ and ${a}_{D}(u)$, which we derive from first principles, actually coincide with the solution proposed by Seiberg and Witten. We also consider some relevant generalizations.