  This paper is concerned with the physics of parametrized gapped quantum
many-body systems, which can be viewed as a generalization of conventional
topological phases of matter. In such systems, rather than considering a single
Hamiltonian, one considers a family of Hamiltonians that depend continuously on
some parameters. After discussing the notion of phases of parametrized systems,
we formulate a bulk-boundary correspondence for an important bulk quantity, the
Kapustin-Spodyneiko higher Berry curvature, first in one spatial dimension and
then in arbitrary dimension. This clarifies the physical interpretation of the
higher Berry curvature, which in one spatial dimension is a flow of (ordinary)
Berry curvature. In d dimensions, the higher Berry curvature is a flow of
(d-1)-dimensional higher Berry curvature. Based on this, we discuss
one-dimensional systems that pump Chern number to/from spatial boundaries,
resulting in anomalous boundary modes featuring isolated Weyl points. In higher
dimensions, there are pumps of the analogous quantized invariants obtained by
integrating the higher Berry curvature. We also discuss the consequences for
parametrized systems of Kitaev's proposal that invertible phases are classified
by a generalized cohomology theory, and emphasize the role of the suspension
isomorphism in generating new examples of parametrized systems from known
invertible phases. Finally, we present a pair of general quantum pumping
constructions, based on physical pictures introduced by Kitaev, which take as
input a d-dimensional parametrized system, and produce new (d+1)-dimensional
parametrized systems. These constructions are useful for generating examples,
and we conjecture that one of the constructions realizes the suspension
isomorphism in a generalized cohomology theory of invertible phases.
