Chiral Algebras for Trinion Theories
Madalena Lemos, Wolfger Peelaers
TL;DR
The paper advances the understanding of chiral algebras for 4D $\mathcal{N}=2$ SCFTs by focusing on the $T_n$ class. It derives a conjectured complete generator set for $\chi(T_n)$ based on the affine-module structure visible in the Schur limit of the superconformal index and provides an explicit $\chi(T_4)$ construction. Through a detailed Jacobi-identity bootstrap, it fixes the OPEs, identifies null relations, and shows how these yield Higgs-branch chiral ring relations (including new ones), while clarifying when the stress tensor can be composite. The work also discusses the relationship to critical-level AKM modules, the potential emergence of $W$-algebras, and possible extensions to larger $n$ via DRinfeld–Sokolov reductions and index consistency checks. Overall, the paper offers a concrete framework to engineer and test chiral algebras for class-$\mathcal{S}$ theories and highlights a path toward a full, index-consistent description for all $T_n$.
Abstract
It was recently understood that one can identify a chiral algebra in any four-dimensional N=2 superconformal theory. In this note, we conjecture the full set of generators of the chiral algebras associated with the T_n theories. The conjecture is motivated by making manifest the critical affine module structure in the graded partition function of the chiral algebras, which is computed by the Schur limit of the superconformal index for T_n theories. We also explicitly construct the chiral algebra arising from the T_4 theory. Its null relations give rise to new T_4 Higgs branch chiral ring relations.