Symmetry and many-body topology in quantum matters

Symmetry and topology play crucial roles in characterizing and classifying quantum phases of matter. Many quantum matters can be characterized by how symmetry is implemented and its intricate interplay with the entanglement structure in many-body wavefunctions. Recently, remarkable progress has been made in generalizing the concept of symmetry and elucidating its connection to topological defects. This advancement has propelled the study of symmetry and topological defects to the forefront, reaching beyond condensed matter physics into mathematics, high-energy physics, and quantum information science. In this talk, I will introduce a general framework to classify and characterize topological phases with crystalline symmetry as networks of topological defects. This approach provides a simple physical picture for crystalline topological phases and enables an understanding of various characteristic boundary properties, including higher-order topology. I will discuss how this framework can help in the search for topological superconducting materials. Additionally, I will delve into recent advancements in our understanding of generalized symmetries, topological defects, and their connection to logical operations in topological quantum error correcting codes.