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

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