Long Exact Sequence in the Symmetry Broken Phase: defect anomaly matching and Berry phase protection.

We investigate 't Hooft anomaly matching in the symmetry broken phase by studying the Jackwi-Rebbi gapless modes living on defects such as domain wall and vortices. We show that the anomaly of the effective lower dimensional system living on the core of the defect matches the bulk anomaly under the Smith homomorphism map, extending the results of Callan and Harvey. We extend this map to a long exact sequence and relate it to physical phenomona such as Berry phase protecting the diabolical point and higher dimensional version of bulk-boundary correspondence. Additionally, we give a crystalline-symmetric interpretation for this anomaly matching condition and the long exact sequence. The twisting of the symmetry on the defect is explained by the crystalline equivalence principle (CEP), and the anomaly matching is related to Lieb-Schultz-Mattis (LSM) theorems. Lastly, we explicitly apply these anomaly matching conditions to free fermion symmetry breaking and abelian Higgs models.