  We study quantum corrections to hypersurfaces of dimension $d+1>2$ embedded
in generic higher-dimensional spacetimes. Manifest covariance is maintained
throughout the analysis and our methods are valid for arbitrary co-dimension
and arbitrary bulk metric. A variety of theories which are prominent in the
modern amplitude literature arise as special limits: the scalar sector of
Dirac-Born-Infeld theories and their multi-field variants, as well as generic
non-linear sigma models and extensions thereof. Our explicit one-loop results
unite the leading corrections of all such models under a single umbrella. In
contrast to naive computations which generate effective actions that appear to
violate the non-linear symmetries of their classical counterparts, our
efficient methods maintain manifest covariance at all stages and make the
symmetry properties of the quantum action clear. We provide an explicit
comparison between our compact construction and other approaches and
demonstrate the ultimate physical equivalence between the superficially
different results.
