Cutting and gluing with running couplings in N=2 QCD

We consider the order parameter u = < Tr phi(2)> as a function of the running coupling constant a is an element of H of asymptotically free N = 2 QCD with gauge group SU(2) and N-f < 3 massive hypermultiplets. If the domain fora is restricted to an appropriate fundamental domain F-Nf , then the function u is one to one. We demonstrate that these domains consist of six or less images of an SL(2 , 7L) 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 Gamma of SL(2 , 7L) and the fundamental domain is Gamma\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 the decoupling of hypermultiplets, the merging of local singularities, as well as the merging of nonlocal singularities which give rise to superconformal Argyres-Douglas theories.

Automated summary

In this study, we investigate the behavior of the order parameter u = < Tr phi(2)> as a function of the running coupling constant a in asymptotically free N = 2 QCD with SU(2) gauge group and less than 3 massive hypermultiplets. By restricting the domain of a to a suitable fundamental domain, we show that u is one to one, and these domains consist of six or less images of an SL(2 , 7L) keyhole fundamental domain.