Twisted formalism for 3d \({\mathcal {N}}=4\) theories

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.