CMSA Quantum Matter in Mathematics and Physics: A Long Exact Sequence in Symmetry Breaking

We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations that 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 the Smith homomorphism, which gives a family of maps between twisted bordism groups. Though many examples have been studied, we give the first completely general account of the Smith homomorphism. We lift it to a map of Thom
spectra and identify the cofiber, producing a long exact sequence of twisted bordism groups; the SBLES is the Anderson dual of that long exact sequence. Our work develops further the theory of higher Berry phase and its bulk-boundary correspondence and serves as a new computational tool for classifying symmetry protected topological phases.

--------
Subscribe to Harvard CMSA Quantum Matter and other seminar videos
(more to be uploaded):
https://www.youtube.com/playlist?list=PL0NRmB0fnLJQAnYwkpt9PN2PBKx4rvdupSubscribe to Harvard CMSA seminar mailing list:
https://forms.gle/1ewa7KeP6BxBuBeRA