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

TL;DR

This paper explores the symmetry algebra and semi-chiral ring structure of a twisted 4d supersymmetric $eta \gamma$ system derived from a $ =1$ supersymmetric $^4$ theory, revealing complex modules and challenging analogies with 2d models.

Contribution

It provides a detailed analysis of the semi-chiral ring as a module over the symmetry algebra in a twisted 4d supersymmetric theory, introducing new insights into its structure.

Findings

Identifies the semi-chiral ring as a module over the stress tensor algebra.

Reveals an intricate module structure that does not match 2d Ising Virasoro minimal model.

Shows the algebraic complexity of the twisted supersymmetric $^4$ theory.

Abstract

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.