  In two dimensions, magnetic higher-order topological insulators (HOTIs) are
characterized by excess boundary charge and a compensating bulk ``filling
anomaly.'' At the same time, without additional noncrystalline symmetries, the
boundaries of two-dimensional HOTIs are gapped and featureless at low energies,
while the bulk of the system is predicted to have a topological response to the
insertion of lattice (particularly disclination) defects. Until recently, a
precise connection between these effects has remained elusive. In this work, we
point the direction towards a unifying field-theoretic description for the bulk
and boundary response of magnetic HOTIs. By focusing on the low-energy
description of the gapped boundary of a two-dimensional magnetic HOTI with no
time-reversing symmetries, we show that the boundary charge and filling anomaly
arise from the gravitational ``Gromov-Jensen-Abanov'' (GJA) response action
first introduced in [Phys. Rev. Lett. 116, 126802 (2016)] in the context of the
quantum Hall effect. As in quantum Hall systems the GJA action cancels apparent
anomalies associated with bulk response to disclinations, allowing us to derive
a concrete connection between the bulk and boundary theories of HOTIs. We show
how our results elucidate the connection between higher order topology and
geometric response both in band insulators, and point towards a new route to
understanding interacting higher order topological phases beyond the simple
cases considered here.
