The A-twisted Higgs branch conjecture for boundary vertex algebras

Let $\mathcal{V}$ be the boundary vertex algebra of an $A$-twisted gauge theory with purely free fermion boundary degrees of freedom. Its associated variety is

$$X_\mathcal{V} \coloneqq \operatorname{Spec}\left(R_\mathcal{V}\right)_{\mathrm{red}},$$

where $R_\mathcal{V}=\mathcal{V}/C_2(\mathcal{V})$ is Zhu's $C_2$ algebra. Let $\mathcal{H}$ denote the Higgs branch of the corresponding gauge theory. $A$-twisted Higgs branch conjecture. The associated variety and the Higgs branch agree:

$$X_\mathcal{V}\cong\mathcal{H}.$$

This is the proposed three-dimensional analogue of the four-dimensional Higgs branch conjecture, which identifies the associated variety of the vertex algebra with the Higgs branch of vacua. In the stated class of theories, the conjecture predicts that boundary vertex algebras capture the Higgs-branch geometry; its general status is not established. An analogous statement for $B$-twisted vertex algebras and the Coulomb branch is also suggested in the source.