Appears in collection : 2024 IHES Summer School – Symmetries and Anomalies : a Modern Take

1. General "definition" of topological phases and of invertible phases.
2. Illustration of invertible phases with a 1+1d Majorana chain and connection to continuum field theory (massive Majorana fermion).
3. Introduction of bordism invariance and classification of invertible phases.
4. If time allows, invertible phases with Z classification (Chern Simons response) as illustrated by 2+1d p+ip superconductor.
5. Phases protected by symmetry. Symmetry on the lattice.
6. In cohomology SPT and Dijkgraaf-Witten theories. If time allows, a detailed analysis of the 2+1D Levin-Gu bosonic Z 2 SPT.