Geometric response approach to the filling anomaly

The filling anomaly two-dimensional higher order topological insulators without chiral symmetry feature an excess charge at corners even in the absence of discrete zero modes. These systems exhibit a similar effect in the presence of rotational defects (disclinations), where additional bound charge is localized at disclination cores. In this work we provide a unifying field-theoretic description to these effects in the language of geometric response. By focusing on the low-energy theory of the gapped boundary, we derive the Wen-Zee response familiar from continuum topological phases, taking care to include the anomaly-canceling Gromov-Abanov-Jensen (GJA) boundary term, which gives rise to the filling anomaly. As a result we provide a concrete through-line between higher order topological insulators (HOTIs) and geometric effective action descriptions of topological phases.