Abstract: A manifestation of S-duality or strong-weak coupling duality is the equivalent dynamics of a quantum field theory at distinct values of its coupling constant. A natural question for such a quantum field theory is the determination of a domain for the coupling constant parametrizing inequivalent quantum field theories. We address this question for asymptotically free N = 2 Yang-Mills theories with gauge group SU(2) and Nf ≤ 3 fundamental hypermultiplets. To this end, we consider the order parameter for the Coulomb branch, which is a function of the running coupling τ invariant under S-duality. If the domain for τ is restricted to an appropriate fundamental domain F, the function u is one-to-one. 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,Z) and the fundamental domain is \Gamma \in H. For generic masses, however, branch points and cuts are present, and subsets of F 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. Joint work with Johannes Aspman and Elias Furrer (https://arxiv.org/abs/2107.04600).
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