The semi-chiral ring of supersymmetric $\phi^4$ theory as a representation

In this short note, we study the infinite-dimensional symmetry algebras which appear in holomorphic twists of 4d $\mathcal{N}=1$ supersymmetric quantum field theories. In particular, we investigate whether their representation theory helps us understand the semi-chiral ring of $\frac{1}{4}$-BPS operators. We focus on the supersymmetric analogue of $\phi^4$ theory. Upon twist, this becomes a 4d $\beta \gamma$ system deformed by a cubic superpotential. We compute the semi-chiral ring in this example and organize it into modules for the algebra generated by the stress tensor. We find an intricate module structure and falsify the hypothesis that this could be a 4d analogue of the 2d Ising Virasoro minimal model.