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

The recently rigorously proved nonperturbative relation u=πi(F−a∂aF/2), underlying N=2 supersymmetry Yang-Mills theory with the gauge group SU(2), implies both the reflection symmetries ¯u(τ)=u(−¯τ) and u(τ+1)=−u(τ) which hold exactly. The relation also implies that τ is the inverse of the uniformizing coordinate u of the moduli space of quantum vacua MSU(2), that is, τ:MSU(2)→H, where H is the upper half plane. In this context, the above quantum symmetries are the key points to determine MSU(2). It turns out that the functions a(u) and aD(u), which we derive from first principles, actually coincide with the solution proposed by Seiberg and Witten. We also consider some relevant generalizations.