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Associated variety of the A-model boundary VOA equals the Higgs branch (abelian case)

Establish that, for abelian gauge theories in the 3d A-twist with additional boundary free fermions and boundary vertex operator algebra V^A defined via relative BRST reduction, the associated variety X_{V^A} is isomorphic to the Higgs branch of the theory.

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Background

Motivated by dimensional reduction ideas and free field realizations, the authors restate a conjecture formulated in earlier work asserting that the associated variety of the A-model boundary VOA captures the Higgs branch. This extends 4d insights to the 3d A-model setting.

In this paper they verify the conjecture in the concrete SQED[n] setup by identifying VA with L_1(psl_{n|n}) and proving that its associated variety is the minimal nilpotent orbit closure. However, the general abelian case beyond this family remains conjectural as stated.

References

..., the following conjecture was formulated For an abelian gauge theory in the A-twist with free fermions as additional boundary degrees of freedom and boundary VOA $VA$, the associated variety $X_{VA}$ is isomorphic to the Higgs branch of the theory.

— $L_1(\mathfrak{psl}_{n|n})$ from BRST reductions, associated varieties and nilpotent orbits (2409.13028 - Ferrari et al., 19 Sep 2024) in Conjecture (label mainconjecture), Section 1 (Introduction)