  Two-dimensional higher-order topological insulators can display a number of
exotic phenomena such as half-integer charges localized at corners or
disclination defects. In this paper, we analyze these phenomena, focusing on
the paradigmatic example of the quadrupole insulator with $C_4$ rotation
symmetry, and present a topological field theory description of the mixed
geometry-charge responses. Our theory provides a unified description of the
corner and disclination charges in terms of a physical geometry (which encodes
disclinations), and an effective geometry (which encodes corners). We extend
this analysis to interacting systems, and predict the response of fractional
quadrupole insulators, which exhibit charge $e/2(2k+1)$ bound to corners and
disclinations.
