Cutting and gluing with running couplings in N=2 QCD

We consider the order parameter $u=\left<{\rm Tr}\phi^2\right>$ as function of running coupling constant $\tau \in \mathbb{H}$ asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets. If domain for $\tau$ is restricted to an appropriate fundamental $\mathcal{F}_{N_f}$, $u$ one-to-one. demonstrate that these domains consist six or less images ${\rm SL}(2,\mathbb{Z})$ keyhole domain, identifications boundaries. For special choices masses, does not give rise branch points cuts, such a modular congruence subgroup $\Gamma$ $\Gamma\backslash\mathbb{H}$. generic however, cuts are present, subsets $\mathcal{F}_{N_f}$ being cut glued upon varying mass. study this mechanism various phenomena, decoupling hypermultiplets, merging local singularities, well non-local singularities which superconformal Argyres-Douglas theories.