  We develop BRST quantization of gauge theories with a soft gauge algebra on
spaces with asymptotic boundaries. The asymptotic boundary conditions are
imposed on background fields, while quantum fluctuations about these fields are
described in terms of quantum fields that vanish at the boundary. This leads us
to construct a suitable background field formalism that is generally applicable
to soft gauge algebras, and therefore to supergravity. We define a nilpotent
BRST charge that acts on both the background and the quantum fields, as well as
on the background and quantum ghosts. When the background is restricted to be
invariant under a residual isometry group, the background ghosts must be
restricted accordingly and play the role of the parameters of the background
isometries. Requiring in addition that the background ghosts will be BRST
invariant as well then converts the BRST algebra into an equivariant one. The
background fields and ghosts are then invariant under the equivariant
transformations while the quantum fields and ghosts transform under both the
equivariant and the background transformations. We demonstrate how this
formalism is suitable for carrying out localization calculations in a large
class of theories, including supergravity defined on asymptotic backgrounds
that admit supersymmetry.
