LAUSR

We consider the order parameter u = 〈Trφ2〉 as function of the running coupling constant τ ∈ H of asymptotically free N = 2 QCD with gauge group SU(2) and Nf ≤ 3 massive hypermultiplets. If the domain for τ is restricted to an appropriate fundamental domain FNf , the function u is one-to-one. We demonstrate that these domains consist of six or less images of an SL(2,Z) keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, u does not give rise to branch points and cuts, such that u is a modular function for a congruence subgroup Γ of SL(2,Z) and the fundamental domain is Γ\H. For generic masses, however, branch points and cuts are present, and subsets of FNf are being cut and glued upon varying the mass. We study this mechanism for various phenomena, such as decoupling of hypermultiplets, merging of local singularities, as well as merging of non-local singularities which give rise to superconformal Argyres-Douglas theories. ar X iv :2 10 7. 04 60 0v 1 [ he pth ] 9 J ul 2 02 1