Tinkertoys for Gaiotto Duality

TL;DR

This work provides a systematic Class S framework for 4D N=2 SCFTs arising from compactifying the 6D A_{N-1} (2,0) theory on a punctured Riemann surface, using a tinkertoy decomposition into fixtures (3-punctured spheres) connected by cylinders. It derives precise rules for regular and irregular punctures, Coulomb-branch dimensions, and gluing via gauge groups, enabling explicit classifications up to N=5 and construction of infinite families of interacting SCFTs with detailed global-symmetry and central-charge data. The authors verify the structure with 3D mirrors and extensive S-duality checks, and they introduce multiple new SCFT series (R_{0,N}, R_{1,N}, S_N, T_N, U_N, V_N, W_N, R_{2,N}) that extend the known T_N framework. The results illuminate a rich web of dualities between Lagrangian and non-Lagrangian sectors, provide a resource for constructing novel SCFTs, and set the stage for further exploration into higher D/E-series and connections to Liouville theory. These developments have potential implications for understanding nonperturbative dynamics and dualities in 4D N=2 theories, as well as for cross-checks with protected operator indices.

Abstract

We describe a procedure for classifying N=2 superconformal theories of the type introduced by Davide Gaiotto. Any curve, C, on which the 6D A_{N-1} SCFT is compactified, can be decomposed into 3-punctured spheres, connected by cylinders. We classify the spheres, and the cylinders that connect them. The classification is carried out explicitly, up through N=5, and for several families of SCFTs for arbitrary N. These lead to a wealth of new S-dualities between Lagrangian and non-Lagrangian N=2 SCFTs.