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Explore the fundamental concepts of generalized symmetries in quantum field theory through this comprehensive lecture delivered at the International Centre for Theoretical Sciences. Delve into the revolutionary developments in symmetry studies within quantum many-body systems, examining how new forms of symmetry—including higher-form, non-invertible, and subsystem symmetries—extend far beyond traditional particle and condensed matter physics paradigms. Learn how these generalized symmetries provide powerful new frameworks for understanding and classifying quantum phases of matter, including exotic states that fall outside conventional classification schemes. Discover the evolving role of anomalies, originally from high-energy physics, in condensed matter contexts and their crucial function in understanding microscopic symmetry realizations at low energies. Understand how these concepts protect and constrain exotic quantum phases such as symmetry-protected topological (SPT) states. Gain insights into how these developments are reshaping our understanding of quantum entanglement, conservation laws, hydrodynamics in interacting systems, and their emerging influence on mixed states and open quantum systems. This lecture forms part of a broader program covering topics including Chern-Simons theories, fractons, categorical approaches in condensed matter, defects and symmetries, emergent symmetries, Hilbert space fragmentation, hydrodynamics, and numerical methods for topological phases.
