Chiral algebras for trinion theories

It was recently understood that one can identify a chiral algebra in any four-dimensional N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} superconformal theory. In this note, we conjecture the full set of generators of the chiral algebras associated with the Tn 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 Tn theories. We also explicitly construct the chiral algebra arising from the T4 theory. Its null relations give rise to new T4 Higgs branch chiral ring relations.