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Three-dimensional $\mathcal{N}=4$ supersymmetric field theories admit anatural class of chiral half-BPS boundary conditions that preserve$\mathcal{N}=(0,4)$ supersymmetry. While such boundary conditions are notcompatible with topological twists, deformations that define boundaryconditions for the topological theories were recently introduced by Costelloand Gaiotto. Not all $\mathcal{N}=(0,4)$ boundary conditions admit suchdeformations. We revisit this construction, working directly in the setting ofthe holomorphically twisted theory and viewing the topological twists asfurther deformations. Properties of the construction are explained both purelyin the context of holomorphic field theory and also by engineering theholomorphic theory on the worldvolume of a D-brane. Our brane engineeringapproach combines the intersecting brane configurations of Hanany-Witten withrecent work of Costello and Li on twisted supergravity. The latter approachallows to realize holomorphically and topologically twisted field theoriesdirectly as worldvolume theories in deformed supergravity backgrounds, and wemake extensive use of this.

Holomorphic boundary conditions for topological field theories via branes in twisted supergravity