One constant to rule them all

TL;DR

This paper analyzes the coupling matrix of certain $ ext{SU}(N)$ gauge theories, revealing a modular structure with a distinguished coupling constant that governs duality transformations and instanton relations.

Contribution

It constructs the general form of the coupling matrix in these theories, identifying multiple coupling constants and their transformation properties under S-duality.

Findings

Coupling constants transform independently under S-duality in the massless case.

A distinguished coupling emerges in the asymptotic regime and instanton recursion.

The structure persists with deformation in the massive case, maintaining the special role of one coupling.

Abstract

We study the coupling matrix of $\mathcal{N}=2$ $SU(N)$ gauge theories with $2N$ fundamental hypermultiplets in the special vacuum, where a residual $\mathbb{Z}_N$ symmetry restores nontrivial modular structure. Using symmetry and dimensional arguments, we construct its general form and identify $\lfloor N/2 \rfloor$ 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.