  We study defects in symmetry breaking phases, such as domain walls, vortices,
and hedgehogs. In particular, we focus on the localized gapless excitations
which sometimes occur at the cores of these objects. These are topologically
protected by an 't Hooft anomaly. We classify different symmetry breaking
phases in terms of the anomalies of these defects, and relate them to the
anomaly of the broken symmetry by an anomaly-matching formula. We also derive
the obstruction to the existence of a symmetry breaking phase with a local
defect. We obtain these results using a long exact sequence of groups of
invertible field theories, which we call the "symmetry breaking long exact
sequence" (SBLES). The mathematical backbone of the SBLES is studied in a
companion paper. Our work further develops the theory of higher Berry phase and
its bulk-boundary correspondence, and serves as a new computational tool for
classifying symmetry protected topological phases.
