  Three-dimensional $\mathcal{N}=4$ supersymmetric field theories admit a
natural class of chiral half-BPS boundary conditions that preserve
$\mathcal{N}=(0,4)$ supersymmetry. While such boundary conditions are not
compatible with topological twists, deformations that define boundary
conditions for the topological theories were recently introduced by Costello
and Gaiotto. Not all $\mathcal{N}=(0,4)$ boundary conditions admit such
deformations. We revisit this construction, working directly in the setting of
the holomorphically twisted theory and viewing the topological twists as
further deformations. Properties of the construction are explained both purely
in the context of holomorphic field theory and also by engineering the
holomorphic theory on the worldvolume of a D-brane. Our brane engineering
approach combines the intersecting brane configurations of Hanany-Witten with
recent work of Costello and Li on twisted supergravity. The latter approach
allows to realize holomorphically and topologically twisted field theories
directly as worldvolume theories in deformed supergravity backgrounds, and we
make extensive use of this.
