Local expression of fractional corner charges in obstructed atomic insulators and relationship with the fractional disclination charges

In obstructed atomic insulators, fractionally quantized charges appear at the corners of the crystals in the shapes of vertex-transitive polyhedra, and are given by the filling anomaly divided by the number of corners. Recent studies reveal that the filling anomaly for the cases with genus $0$ is universally given by the total charge at the Wyckoff position $1a$. In this study, we rewrite the formula in terms of the degree of sharpness of the corner, and show that the corner charge formula also holds for cases with arbitrary genus. We also extend our formula to vertex-transitive shell polyhedra, which are closed or open polyhedra without the bulk region, with all the vertices related by symmetry. Then, we show that the corner charges of such shell polyhedra are equal to the two-dimensional disclination charges of the corresponding disclinations. By identifying it with the disclination charge under the Wen-Zee action, we show that the coupling constant of the Wen-Zee action for a crystalline insulator is given by the total charge at the Wyckoff position at the disclination core.