One constant to rule them all

A bstract We study the coupling matrix of 𝒩 = 2 SU( N ) gauge theories with 2 N fundamental hypermultiplets in the special vacuum, where a residual ℤ N symmetry restores nontrivial modular structure. Using symmetry and dimensional arguments, we construct its general form and identify ⌊ N /2⌋ coupling constants in their most natural basis. We show that in the massless theory these couplings transform independently under S -duality and that the bare coupling is a modular function of any of them. One coupling constant, however, plays a distinguished role, emerging in the asymptotic regime and in instanton recursion relation. In the massive case, this structure is deformed but the distinguished coupling retains its privileged role.