Twisted Formalism for 3d $\mathcal{N}=4$ Theories

Abstract: We describe the topological $A$ and $B$ twists of 3d $\mathcal{N}=4$ theories of hypermultiplets gauged by $\mathcal{N}=4$ vector multiplets as certain deformations of the holomorphic-topological ($HT$) twist of those theories, utilizing the twisted superfields of Aganagic-Costello-Vafa-McNamara describing $HT$-twisted 3d $\mathcal{N}=2$ theories. We rederive many known results from this perspective, including state spaces on Riemann surfaces, deformations induced by flavor symmetries, the boundary VOAs of Costello-Gaiotto, and the category of line operators as proposed by Costello-Dimofte-Gaiotto-Hilburn-Yoo. Along the way, we show how the secondary product of local operators in the holomorphic-topological twist is related to the secondary product in the fully topological twist.