The Duality Generator

Argyres-Douglas theories are a family of superconformal field theories that arise at special points in the Coulomb branch of gauge theories. Recent work by Beem, Martone, Sacchi, Singh, and Stedman cataloged a network of dualities between type A Argyres-Douglas theories — different-looking theories that turn out to describe the same physics. The catalog is rich but the mechanism producing the dualities remained opaque.

arXiv:2603.15942 gives the mechanism: all of these dualities decompose into compositions of exactly two basic operations. The first is the Fourier transform of irregular connections on P¹. The second is a Möbius transformation exchanging zero and infinity.

Via the wild nonabelian Hodge correspondence, the data defining an Argyres-Douglas theory amount to singularity data for irregular connections of a specific form. The Fourier transform acts on these connections, and the stationary phase formula gives explicit expressions for how the singularity data transform. The Möbius transformation permutes the singular points. Every duality in the catalog is a finite sequence of these two moves.

The proof also clarifies the 3d mirror description: the quiver describing the 3d mirror corresponds to the unique nonabelian Hodge diagram with no negative edges among all diagrams in the orbit under basic operations. The physical duality selects, among all mathematically equivalent presentations, the one with the simplest combinatorial structure.

Two operations generate an entire duality network. The dualities are not coincidences discovered one at a time — they are the orbit of a group action, and the group has two generators.