26583 2309.16749 10.1007/JHEP07(2025)007 7 7 A long exact sequence in symmetry breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases 01 Mathematical Sciences 0101 Pure Mathematics Regular Article - Theoretical Physics 1 48 2025 7 1 2024 9 3 2025 4 7 2025 7 1 2025 7 1 The Author(s) 2025 Open Access. This article is distributed under the terms of the Creative Commons Attribution License ( CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 13130 2025 2025 7 7 Arun Debray adebray@purdue.edu Sanath K. Devalapurkar sdevalapurkar@math.harvard.edu Cameron Krulewski camkru@mit.edu Yu Leon Liu yuleonliu@math.harvard.edu Natalia Pacheco-Tallaj nataliap@mit.edu Ryan Thorngren ryan.thorngren@physics.ucla.edu https://ror.org/02dqehb95 grid.169077.e 0000 0004 1937 2197 Department of Mathematics Purdue University 150 N University Street West Lafayette IN 47907 USA https://ror.org/03vek6s52 grid.38142.3c 0000 0004 1936 754X Harvard University Department of Mathematics 1 Oxford Street Cambridge MA 02138 USA https://ror.org/042nb2s44 grid.116068.8 0000 0001 2341 2786 Massachusetts Institute of Technology, Department of Mathematics Simons Building, 77 Massachusetts Avenue Cambridge MA 02139 USA https://ror.org/046rm7j60 grid.19006.3e 0000 0000 9632 6718 Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy University of California 475 Portola Plaza Los Angeles CA 90095 USA Abstract We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an ’t Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the “symmetry breaking long exact sequence” (SBLES). The mathematical backbone of the SBLES is studied in a companion paper [1]. Our work further develops the theory of higher Berry phase and its bulk-boundary correspondence, and serves as a new computational tool for classifying symmetry protected topological phases. Keywords Anomalies in Field and String Theories Topological Field Theories ArXiv ePrint: 2309.16749